Biochimica et Biophysica Acta, 1031-3 (1990) 311-382 Elsevier
311
BBAREV 85371
Membrane electrostatics Gregor Cevc Medizinische Biophysik, Urologische Klinik und Poliklinik der Technischen UniversitSt Mi~nchen, Klinikum r.d.l., Miinchen (F.R.G.) (Received 16 October 1989) (Revised manuscript received 10 April 1990)
Contents I.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
311
II.
Electrostatic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-A. Gouy-Chapman model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-B. Modern electrostatic double layer theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
316 318 320
III.
Interracial structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III-A. Charge distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III-B. Interfacial curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
321 321 323
IV.
Interfacial polarity and the dielectric constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
324
V.
Hydration effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
325
VI.
Ion distribution and binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-A. Ion distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-B. Ion binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-C. Ion transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
330 330 332 338
VII. Thermodynamics of charged membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-A. Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-B. Electrostatic intermembrane pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-C. Electrostatic lateral pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-D. Transmembrane pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
340 340 342 344 344
VIII. Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-A. Direct measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-B. Electrostatic probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
345 345 349
IX.
Biological significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
354
X.
Summary
356
I. Introduction Electrostatic phenomena play a crucial role in many biological processes. For membranes they are signifi-
Correspondence: G. Cevc, Medizinische Biophysik, Urologische Klinik und Poliklinik der Technische Universitllt Mllnchen, Klinikum r.d.I., Ismaningerstr. 22, D-8000 Mtinchen 80, F.R.G.
cant, for example, because the electrostatic interactions can lead to an accumulation of charged species near the m e m b r a n e s u r f a c e w h i c h t h e n a c t s as a c a t a l y t i c site. The electrostatics effects, moreover, influence the conformation and function of many molecules and control in part the intracellular and intercellular recognition and transport. T h i s is p o s s i b l e b e c a u s e t h e b i o l o g i c a l m e m b r a n e s normally carry numerous ionized or polar groups. These
0304-4157/90/$03.50 © 1990 Elsevier Science Publishers B.V. (Biomedical Division)
312 g r o u p s are e i t h e r p a r t s of t h e lipid, g l y c o l i p i d , o r ( g l y c o ) p r o t e i n m e m b r a n e c o m p o n e n t s , o r else s t e m f r o m the c h a r g e d m o l e c u l e s w h i c h are b o u n d to o r a d s o r b e d o n t o the m e m b r a n e surface. ( M e m b r a n e i n t e r i o r is n o r m a l l y free of charge, o w i n g to the h i g h e n e r g y cost o f i o n - t r a n s f e r f r o m the s o l u t i o n i n t o the h y d r o c a r b o n r e g i o n w i t h a l o w d i e l e c t r i c c o n s t a n t of a r o u n d 2.5). B i o l o g i c a l m e m b r a n e s t h u s are c o v e r e d w i t h a l a y e r o f the ' s u r f a c e ' c h a r g e (Fig. 1). T h i s l a y e r m a y b e u p to 20 n m thick, c o m p a r a b l e to the d e p t h of the e x t r a c e l l u l a r m e m b r a n e g l y c o c a l i x region. M e m b r a n e s of cells a n d organelles, m o r e o v e r , s o m e t i m e s f o r m p r o t r u s i o n s , are c r e n a t e d o r folded. T h e m i c r o v i l l i of the p l a s m a m e m b r a n e , for e x a m p l e , t y p i c a l l y m e a s u r e 5 0 - 2 0 0 n m in d i a m e t e r a n d m a y e x t e n d as far as 2000 n m a w a y f r o m t h e b a s i c s u r f a c e p l a n e ; cillia h a v e c o m p a r a b l e d i m e n sions. B i o l o g i c a l m e m b r a n e s , t h e r e f o r e , s e l d o m - i f - e v e r - possess the h o m o g e n e i t y o r the r i g i d i t y of t h e classical
e l e c t r i f i e d surfaces. B u t artificial b i l a y e r m e m b r a n e s c a n be p r e p a r e d f r o m s y n t h e t i c lipids w h i c h are v e r y h o m o g e n e o u s a n d o n l y a little c u r v e d . N e t c h a r g e o n the b i o l o g i c a l m e m b r a n e s in the m a j o r i t y of cases is n e g a t i v e ( T a b l e I). T h i s is b e c a u s e m o s t of the m e m b r a n e p r o t e i n s , all c h a r g e d n a t i v e lipids, as well as m a n y i n t e r f a c i a l l y a d s o r b e d p o l y e l e c trolytes have isoelectric points below neutral. The overall c h a r g e d e n s i t y o n a t y p i c a l b i o l o g i c a l m e m b r a n e is n o t high, h o w e v e r ; it s e l d o m e x c e e d s 0.05 C • m -2. T h i s is d i f f e r e n t w i t h t h e m o d e l m e m b r a n e s w h i c h m a y c a r r y several c h a r g e d g r o u p s p e r m o l e c u l e a n d p r o v i d e s u r f a c e c h a r g e d e n s i t i e s in t h e r a n g e o f - 0.4 _< % _< 0.4 C • m - 2 S t r u c t u r a l c h a r g e s a s s o c i a t e d w i t h the m e m b r a n e s u r f a c e give rise to an e l e c t r o s t a t i c m e m b r a n e s u r f a c e p o t e n t i a l , ~k0 (Fig. 2). T h i s p o t e n t i a l p a r t i c i p a t e s in the p r o c e s s e s of m e m b r a n e i n t e r a c t i o n s , r e c o g n i t i o n , o r sol u t e b i n d i n g . I n a d d i t i o n to this, an e l e c t r o s t a t i c trans-
TABLE I Typical values of the main electrostatic and structural parameters for the charged biological and model membranes under physiological conditions
All relational signs refer to absolute magnitudes. Distinction of the separate membrane regions, such as the hydrophobic core or the polar-apolar interface, is a matter of convention and convenience; therefore, it may differ from source to source. Abbreviations used are: oet net surface charge density; op surface density of the local excess charge; ~kel,oelectrostatic surface potential; ~bh.o hydration potential at the membrane surface; A~pm.~t electrostatic transmembrane potential; E 0 interfacial electric field strength; AE,n transmembrane field gradient; c,, relative permittivity of the hydrocarbon membrane interior; c~,t, effective value of the relative permittivity in the interfacial region; R m conductance of the membrane core; Cm capacitance of the membrane core; Rim and C~ t conductance and capacitance of the inner part of the interracial region; Cm capacitance of the outer interracial zone; dm thickness of the membrane core; dp effective interracial width; d~ average charge displacement perpendicular to the membrane; A L molecular area per lipid molecule; A e typical area per protein in the membrane; A c area per charge; ~ / , effective decay length for the hydration phenomena. Parameter
oet
%
~Pel,o
~Ph,0
A ~Pm, el
EO
A Em
unit
C. m - 2
C. m - 2
mV
mV
mV
MV. cm - 1
MV. cm
Electrostatic Cell membrane Lipid bilayer
- (0.02-0.2) a - 0.4-0.4
> - 1 - (0.1-0.5)
- 15-30 < - 200
- (200-300) - (300-800)
- 70( < 200) b < + 300
1 25
6 4-8
~,,
~,~t
Parameter unit Electrostatic Cell membrane Lipid bilayer Parameter unit Structural Cell membrane Lipid bilayer
2.5-5 2.5
30-50 30
Cm
Rm
Ci,t
R int
Rout
mF.m -2
mS.m -2
mF.m 2
mS.m-2
mS.m-2
1-10 e 5d
0.5 d
200 d
800 d
24000 d
dm
dp
dc
AL
Ae
A~
~f
nm
nm
nm
nm2
nm 2
nm2
nm
3 3-5
10 0.6
1-5 0.3
0.8 0.4-0.9
>_10 -
>2 > 0.9
0.5-5 0.1-0.4
a The effective surface charge density of vacuoles from A. pseudoplatanus,
for example, is greater than 0.13 C.m -2 [483]; for horse-bean
microsomes it is approx. 0.03 C. m - 2 [486].
b Owing to the membrane asymmetry the electrostatic potential at the inner surface can be appreciably higher. For erythrocytes it is believed to be approx. - 6 0 mV [76]. For Ehrlich ascites cells it is around - 2 0 mV [441]; for synaptic vesicles inside neurons it is on the order of - 2 0 - 4 0 mV. ¢ Transmembrane potential in Amoeba proteus is - 6 0 - 7 0 mV [586]; in freshwater algae approx. - 1 2 0 - 1 7 0 mV and in sea algae - 8 0 - 1 8 0 mV [587]; similar electrostatic transmembrane potentials, on the order of -100-200 mV, are also observed for bacteria and across the inner membranes of chloroplasts and mitochondria. In eucaryotic cells, such as neuronal or tumour cells, the resting transmembrane (polarization) potential is normally between - 6 0 and - 7 5 mV [434]. The spatial profile of all these potential differences is always strongly sensitive to the nature of the outer membrane and ionic conditions in the system. d These values were deduced from data on black lipid membranes [103]. The conductance of the bulk physiological solution is around 10000 m S . m -2.
e For specialized, such as neural cells, the corresponding value can be orders of magnitude greater, approx. 10000 mF-m 2.
313
extracellular o
> *
e
~ ~--qo-~+® e~ 7~o~-'~'i~
e~
.'.-~.o
-i
in a variety of biological processes including transport, bioenergetics, excitation, etc. Charges on membrane surface or transmembrane ion concentration gradients create electric fields (Fig. 4). These fields act on, and depend upon, the distribution of ions near the membrane-solution interface, in a selfconsistent manner. They also regulate the adsorption or binding of charged species to the membrane surface. This is seen directly from one of the elementary laws of electrostatics, the Poisson equation, which states that the spatial variation of the electric field vector, this is, the field divergence, everywhere is proportional to the local net charge density. For planar membranes this can be written as d2~(x)/dx
intracellular
2 =- - d E ( x ) / d x
membrane potential, Arm , exists across many membranes and across all such semipermeable membranelike barriers which divide two aqueous compartments filled with different, differently concentrated, or asymmetric electrolytes [1-3] (Fig. 2). Such transmembrane potential is found in almost all living cells. Intracellular sodium concentration (9 mmol 1-1), for example, is typically much lower than in the extracellular space (120 mmol 1-1), whilst the concentration of K ÷ is higher (in the muscle cell interior, for example, 140 mmol 1-1) than outside the cell (2.5 mmol 1-1) owing to the activity of the Na ÷ K+-ATPase pumps. Chloride concentration is similar to sodium concentration. Consequently, in the resting cell, where the K+-selective channels contribute the largest proportion to the cation transport across plasma membrane, the electrostatic membrane potential is close to the equilibrium potential for the potassium ions, which is about - 70 mV (Fig. 3). Any cell- or organelle-membrane with a non-zero transmembrane potential is thus said to be polarized. * Electrostatic membrane polarization plays an essential role
* This polarization has nothing to do with the dielectric polarization, which occurs under the influence of the electric field at the membrane surface (cf. Section IV), or with the cell polarity, which is a manifestation of the cell asymmetry.
(1)
- (p~;.,,,(x) + pe;.;(X))/C%
o
Fig. 1. Schematic representation of a biological membrane from the structural and electrostatic point of view. The distribution of the structural and dissociated ionic charges (circles) is sketched as well as the most strongly bound water molecules (V). Structural membrane charges occupy a wide region perpendicular to the surface, especially on the ectoplasmatic side. Numerous water molecules are bound to the polar surface residues, most notably to the phosphate groups, some amino acids, and sugar moieties. See also Fig. 14 for comparison.
= - pet(X)/~0
where the integral charge density, Pet(X) has been assumed to consist of the contributions from the structural membrane charge, Pet,,,,( X ), and from the ions, pel, i ( x ) . c and % are the dielectric constant of the
-~N SOLUTION/ 0
A
OLUTION, ...
1
I
I
X Fig. 2. Ion distribution profile, Pet,i ( x ) (A), and electrostatic potential, 6 e i ( x ) = tk(x) (B), near a charged membrane with an applied transmembrane potential, A~m = V and surface potential % . Dashed line depicts the distribution of co-ions and full line of counterions. Owing to the juxtaposition of the structural and counterionic charges and because the concentration of the latter falls-off continuously with separation from the surface (see symbols with a plus sign in the upper picture) it is customary to speak of the surface associated ionic diffuse double layer. Far from the membrane the concentration of counterions and co-ions is the same as in the bulk (cb,tk). Dotted line in the lower graph delineates the range to which the electrostatic interactions typically play an important r61e; this range is proportional to Debye screening length, ~.
314
O
®
®
0
@
@
s
C'2,bMk CI ,bulk
SOLUTION 1
A _~z/~,,, 0
l~r
In ~'~'""
= T v q~"~g__
SOLUTION 2
/
A
""-
/ ~,,
B
3; Fig. 3. Schematic representation of the ion distribution, pet.i(x) (A), and electrostatic potential profiles, ~ke/(x) (B), near an asymmetrically screened, uncharged membrane. Transmembrane potential difference (A~km) is a consequence of the asymmetric electrolyte composition on both membrane sides only and is given, in the simplest approximation, by Nernst equation. Such electrolyte asymmetry gives rise to a small surface potential and to minute diffuse double layers.
aqueous subphase and the permittivity of free space, respectively. Long-range, coulombic, electrostatic fields * interactions cause gathering of counterions of the opposite sign to that of the membrane charges, typically cations, and dilution of ions of the same charge, coions, in the vicinity of the structural charges (Figs. 2 and 5). Shortrange interracial fields, which are largely but not exclusively of quantum-mechanical origin, to a large extent determine the tertiary molecular structure, perturb the solvent properties in the interfacial region, and also influence the ion distribution. All membranes, consequently, are embedded in diffuse double-layers in which the solvent structure, ion concentration, and in many cases also the distribution of the membrane associated charges, vary with the separation from the membranesolution interface.
* With the word 'coulombic' in this review I characterize all chargecharge interactions which occur across-and are screened by-a uniform, structureless solvent dielectricum. This term thus encompasses standard electrostatic forces of classical electrostatic theories. The dielectrieally unscreened intercharge forces are described as quantum-mechanic, unless they involve water molecules and their charges. The interactions of the latter subclass are here grouped into a special cathegory of 'hydration phenomena'. But strictly speaking they are both electrostatic and quantum-mechanical in origin.
Current understanding of the membrane electrostatics is still woefully deficient. Already discussing all controversies would fill-out a book. In this contribution I, therefore, deal mainly with various aspects of the membrane surface electrostatics. (Current comprehension of the transmembrane electrostatic potential is much better and consequently will not be reviewed here in any detail.) First of all, some standard electrostatic models for the description of charged membranes are introduced and the basic criteria for the decision about the best theoretical approximation for each given application are noted. Subsequently, a short treatise on the distribution functions and their application in the membrane research is given together with a brief survey of the fundamentals of the ion-membrane association and binding. In the last theoretical section I dwell on the problem of studying the thermodynamic aspects of membrane electrostatics and survey the inter- and trans-membrane interactions of electrostatic origin. In the second part of this review the principal experimental methods for studying the properties of charged membranes, their strengths and weaknesses are summarized. For the sake of brevity and transparency most of the illustrative examples are taken from the works on simple model membranes. Moreover, many pertinent remarks are banned into footnotes and nearly all theoretical results are given in appendices. Readers with only a
O
O
8
@
~
e
O
e e
@
~ O
g
o
• o
• +@ ~ , ~
+ o
@
+
0
+
®
tq
SOLUTION
~1
O'2
Cbulk
SOLUTION
Cbulk ~JO,1
%
~o,2
Fig. 4. Structural charge distribution, pet.,,,(x) (upper) and electrostatic potential, ¢et(X) (lower), close to a membrane with two different values of surface charge density, %ta, °et.2- The electrostatic surface potentials at each membrane surface are different (%.1, tk0.2) and proportional to the corresponding surface charge density. The corresponding profile of the electric field is represented by a dashed line; it makes a jump at the membrane surface owing to the dielectric discontinuity (c m ¢ ~)-
315
I
I
I
I
I
B
2.0
0.8
0.6
1.5
c(x)
c x,
teri°ns
0.4
÷÷ 1.0
r.-.7-"~-"
'-'T
-
-----
I
6- e- ~ ~
_~,..""~
~
coions
4t,, - / . / - - , ~ 1 ~
COlOnS
0.5 x [nm] Q2 0 -
I
i 2
I
i 4
i 6
X [nm] Fig. 5. Ion concentrationprofile for a symmetricalunivalent electrolytenear a charged surface in the form of a diffuse double layer. (A) oct = 0.062 C. m-2 and 0.1 M salt; (B) oel = 0.125 C. m-2 and 1.0 M salt. Results for counteri0ns and co-ions derived from Monte-Carlohardsphere/hard-wall model simulations are shown as crosses and dots, respectively,and those derived from the Gouy-Chapmanmodel are representedby the full lines. (Modified from Ref. 60).
general interest in the membrane electrostatics correspondingly should follow the main text solely and especially the short summaries at the end of each section; but those keen on learning more about the theoretical backgrounds (or those attracted to the more specialized aspects of the membrane surface electrostatics) should look into equations or consult original articles and reference works. A very clearly written introductory theoretical discussion of various aspects of the surface electrostatics, with an emphasis on the thermodynamic questions, is offered in the early, lucid treatise by Verwey and Overbeek [4]. Also old, but still relevant, articles by Levine and Bell [5,6] provide a fairly detailed, analytical survey of the classical electrostatic models. A relatively general and very informative discussion of the modern aspects of the interfacial electrochemistry is contained in the state of the art review by Carnie and Torrie [7] with a comprehensive section on computer simulations, or in the somewhat more specialized, but equally brilliant papers by Henderson [8,9], Blum [10], or Schmickler and Henderson [11]. Brief but well balanced and worthwhile reading are furthermore the review of computer results by WennerstrSm and J~Snnson [12] and the theoretical discourse on exactly solvable plasma models
by Alastuey [13]. (The latter only at first sight are irrelevant for the membrane research: with due reservations they shed light on the dynamic aspects of the membrane surface electrostatics which are not accessible to other analytical models.) Certain theoretical aspects of the membrane biophysics are covered by the volume by Lakshminarayanaiah [14]; membrane transport has been dealt with in extenso by Stein [15] and in a review article on channels by Hall [16]. The role of membrane electrostatics in membrane photobiology has been assessed by Tien [17]. A thorough survey of the spectroscopic methods for studying membrane electrostatics can be found in the volumes edited by Loew [18]. A discussion of the electrostatic properties of lipid membranes, which also tackles the problem of the solvent effects, can be found in the review article by Gruen, Mar~elja, and Parsegian, [19] and in the recent book on 'Phospholipid Bilayers' by Cevc and Marsh [20]. Some of the biological aspects of the membrane electrostatics are covered by the scholarly written papers by Weiss and Harlos [21], Dolowy [22], Warshell and Russel [23] and McLaughlin with colleagues [24,25,26, 27]. A review which covers some other aspects of the interfacial membrane research stems from the feather of Dimitrov [28].
316
Owing to their structural and adsorbed charges a n d / o r because of the perturbation which they induce in the electrolyte solution all membranes are embedded in diffuse double layers of ions. In such double layers counterions by far outnumber coions, indicative of an electrostatic 'surface' potential. In addition to this, transmembrane concentration gradients create a potential jump across the membrane. The electrostatic surface potential is important for the processes occurring near or at the membrane," the transmembrane potential by and large plays a r3le in the transmembrane transport.
•
0
o
@
® O
e
®
• e
• 0
®
e
GOUY--CHAPMAN
crd
=
®
®
'
DONNAN EQUILIBRIUM
p~t,mS(interface) s' t
]
p = constaT~t
S
....
4~ m u
II. Electrostatic models
Recently, electrostatic surface theories have experienced a renaissance of interest and success. This is largely because rigorous results, which initially were derived for the bulk solutions, have been generalized and adapted for the interfacial studies; but also because of the advent of computer simulations. For membranes this development has not yet fully set in. One reason for this is that the membrane surfaces are so complex that none of the available electrostatic theories a priori can be expected to be suitable and reliable for their general description. Each common electrostatic model has its own, frequently overestimated, range of validity. Attempts to theoretically analyze the experimental data for charged membranes therefore must start with a scrutiny of the main system parameters and a definition of the relevant variables and quantities of interest. Then, and only then, a decision can be made about the appropriate theoretical framework-or a combination thereof-to be used. A convenient, but not obligatory, starting point for the development of the electrostatic surface models is the Poisson equation, Eqn. 1. In combination with a suitable closure relation this permits an evaluation of the ion distribution or of the electrostatic potential profiles near the membrane. But in order to make the solutions for membranes reasonably tractable numerous simplifications and restrictions must be introduced. Two limiting simplified situations are habitually chosen. In the first case, the electrolyte solution is reduced in thought to a uniform 'medium' and the interracial region, in which all structural membrane charges are located, is assumed to be completely permeable to ions. Ions, consequently, are taken everywhere to have the same concentration as in the bulk (Fig. 6); this concentration thus appears in Poisson equation only as a constant. Screening of the membrane charges by ions from the solution in such case is completely disregarded (Oeu(X)=peU) and the entire theoretical interest is focused on the intrinsic electrostatic characteristics of the interphase between the membrane core and the bulk electrolyte (peU,,(X)). The Donnan equilibrium approximation results from this [29].
%
5/
I
x Fig. 6. Schematic representation of the structural surface charge distribution, Oet.m(X) (upper), and the membrane potential profiles, 4%t(x) (lower) in the approximation of Gouy-Chapman (left) and Donnan equilibrium (fight). In the GC-model all structural membrane charge is assumed to be accumulated in the surface plane which is screened by non-uniformly distributed ions from the solution. In the Donnan equilibrium approach the membrane charge is taken to occupy a zone of finite width which is freely accessible to the solution ions. The corresponding counterion concentration profiles are shown by dashed lines.
An alternative is to squeeze all structural information about the membrane-solution interface into the electrostatic boundary condition (pet,,,,(X)= ael6(X = 0), and devote all the theoretical interest to the evaluation of the spatial ion distribution (pet.i(X)) and its effects on the electrostatic membrane properties. Such a step leads to classical diffuse double layer approximations. The assumption that the ion distribution close to the membrane surface is governed solely by the coulombic surface-ion interactions in combination with the previously mentioned approximations results in Gouy-Chapman electrostatic theory [30,31]. An essential implication of this model is that the ion concentration near the membrane surface differs from the bulk over a zone of finite thickness, the so-called ionic diffuse double-layer (cf. Fig. 5). The width of this layer is approximately proportional to a characteristic, concentration dependent (Debye) screening length, ~t (Fig. 7), which under physiologic conditions (ion strength I = 0.154,otemperature T = 37°C) is on the order of 1 nm (10 A). Moreover, the net (excess) ion-charge within the double layer is deduced to be identical in size, but opposite in sign, to the total charge on the membrane, because of the overall neutrality of the system. Surface value of the electrostatic membrane potential from Gouy-Chapman theory is predicted to increase
317 30
causes charge separation and leads to a formation of the membrane-associated ionic double layers. In addition to this, most membrane surfaces exhibit a non-coulombic electrostatic 'dipole potential'. This can reach a magnitude of several hundred millivolts for zwitterionic amphiphiles. It arises, in part, from the molecular dipoles on the membrane constituents. Lipid headgroups and the carbonyl groups of the hydrocarbon chains are the most notable sources of these dipoles. But non-coulombic membrane potential to a nearly equal part also stems from the interfaciaUy bound water layers (Cevc and Gaub, unpublished data). The total electrostatic membrane potential therefore inevitably depends on a variety of chemical and physico-chemical membrane properties and is not always simply proportional to the net membrane charge. If the electrolyte compositions at the two membrane sides are different an electrostatic diffusion (trans)membrane potential, moreover, emerges. Such transmembrane Nernst potential, in the first approximation, is proportional to the logarithm of the ratio of the ion concentrations on the 'in-' and 'outside' of the membrane, A~k~. . . . t - Ad/m = ( k T / Z e ) ln(cin/Cout) *. In the absence of screening effects the Nernst potential can be envisaged as a simple potential jump from an inside value ~ki., which is independent of the distance from the membrane surface, to a constant value, ~out = tl% +
E t--
25
z:z
Jl= -,i.I~
'-
1:1
20
___~
1:2 2:2
"E:
15 2:3
f..)
:3
10 5
0
........
0.0001
'
........
0.001
I
.
.
.
0.01 c/M
.
.
.
.
.
.
.
0.1
.
.
,
1
0
Fig. 7. Debye screening length, which is an approximate measure of the diffuse double layer thickness, as a function of the bulk electrolyte concentration, c, and ion valency, Z. With increasing ion charge and n u m b e r of dissolved charges the electrostatic effects are better screened and their reach next to the m e m b r a n e is more and more short-ranged.
steadily, albeit non-linearly, with the density of the net surface charge (see Section II). But non-coulombic surface-ion interactions can give rise to an electrostatic membrane potential on their own, even in the absence of the net membrane charge, at least for real electrolytes. For example, different ion-surface affinities introduce a non-uniform ion distribution near the membrane; electrolyte asymmetry, this is, the asymmetry of colon and counterion valencies (Z+~= Z ) or size, also
* This can be seen, m o s t directly, from Eqn. 7. More general expression for the evaluation of N e r n s t potential also allows for the permeability a n d / o r mobility differences between the involved ion species.
T A B L E II
Effects of various phenomena on the electrostatic membrane properties * • The sign gives direction of effect; the n u m b e r of signs its magnitude. As a rule, the range of various effects increases as hydration < electrostatics < surface fluctuations. Phenomenon
Surface potential
Interfacial el. field
Surface polarity
Interfacial thickness
Interfacial concentration
D o u b l e layer thickness
(~o)
(Eo)
(~,,)
(dp)
(c(x = 0))
(X)
Charge density Charge mobility Discreteness of charge
+ + + (0) lateral variations
+ + + (0) lateral variations
+ + + 0 (+)
(+ ) (0) 0 0
+ + (+ ) laterally modulated
Hydration Interfacial polarity
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + +
Surface curvature Surface fluctuations Interracial width
(- ) (- )
(- ) (- ) (- - - )
+ 0 .
(+ ) 0 . .
Image effects Ionic correlations Salt concentration I o n valency
(- ) (- )
(+ ) + +
(0) 0 - (- -)
.
.
(0) 0 + -
Range
Ionic profile
0 0 0 0
(+ ) (- ) complex (0)
steeper steeper complex
+ +,- - -, + +
(+) (+ )
short, + short, +
bimodal bimodal
0
+ +
(very long) very long short
(steeper) 0 bimodal, shallow
(- ) modulated + + +
- -
short short + _
oscillatory oscillatory oscillatory, steepeer (binding), steeper
.
_ _
(0)
318 A~k,,, at the outside (see Figs. 2 and 3). But in reality, salt effects additionally cause the electrostatic membrane potential and the ionic distribution profiles to vary throughout the interracial region. The distribution of ions and membrane associated charges is, therefore, always continuous everywhere in the system. The Nernst potential is one of the driving forces for the passive membrane transport. But it is rather unimportant for the membrane surface electrostatics. Consequently, it is not discussed in this work except in section V-B which deals with the ion transport. Thus, if the Nernst potential is considered important for a given application, the appropriate transmembrane potential difference should be added to, or subtracted from, the corresponding electrostatic surface potential value. In any realistic, general membrane model a number of structural and physico-chemical aspects of the surface electrostatics should be considered. Furthermore, the complexity of the membrane-solution interface should be paid attention to. For example, the effects of spatial redistribution of the structural membrane and ionic charges, the direct and indirect interactions between such charges, their dependence on the membrane structure and vice versa, as well as the membrane hydration should all be thought of and, frequently, accounted for (see also Table II). As yet, no model is capable to cope will all these requirements at once. Nevertheless, in order to obtain tools for quantitatively analyzing the electrochemical membrane properties, special solutions are normally used. To meet a decision about the optimal theoretical model for a given investigation the following general criteria are helpful: (A) Smooth or rigid surfaces and interfaces which are narrow on the length scale of Debye screening length as well as ion polyvalency favour ion-layering and higherorder ion-ion correlation effects. Advanced, up-to-date electrochemical theories (such as modified PoissonBoltzmann or Hypernetted-Chain-Approximation) with allowance for the fine interfacial structure in principle should be suitable for the analysis of membrane data at this level. Comparison with Monte-Carlo and molecular dynamics results is also valuable. Highly ordered model membranes consisting of lipids with small head-groups in a crystalline or gel phase, small, flat and relatively rigid domains on the biological membranes, or interfaces between the multivalent macromolecular membrane components and the electrolyte solution (in the strong coupling regime) fall in this category. (B) Moderately curved or undulated (again on the length scale of Debye screening length) or relatively mobile surfaces with a narrow membrane-solution interface as well as membranes with a more or less uniform distribution of the structural surface charges represent nearly 'ideal electrified bio-surfaces'. As such, they are closest to the fulfillment of the requirements for the
validity of the Gouy-Chapman approximation. Standard electrostatic results, consequently, are normally applicable for such systems. If the water and ion binding effects are properly accounted for, it may then be inferred-but remains to be proven suitably-the generalized Poisson-Boltzmann equation can always yield reasonably accurate results for the systems of this type. Model lipid membranes in the fluid phase and other simple membranes with a smooth, charged surface provide examples for this. (C) Very irregular membranes with an interfacial region which is thick compared to the Debye screening length, finally, necessitate the allowance to be made for the spatial distribution of the ionic as well as of the structural membrane charges. The distribution of all polar and charged membrane residues perpendicular to the membrane surface must then be investigated and analyzed together with the electrolyte penetration and charge regulation in the interfacial region. Failure to do so may result in gross misinterpretation of the experimental data. Most cell membranes and other complex biological interfaces belong to this class. H-A. Gouy-Chapman model
The Gouy-Chapman (GC) approximation [30,31] is obtained directly from Eqn. 1 under the proviso that all structural membrane charges are confined to an infinitely narrow plane (cf. Fig. 4) and that the ion distribution is governed solely by Coulombic forces (see Appendix A). Owing to its straightforwardness the GC-model is extremely popular, especially for the routine membrane research. It predicts the electrostatic potential at the membrane surface to be: (2k T/ Ze ) sinh- l ( Zeo¢fl~ /2~c ok T ) ~o = (o¢/~/~o: % < 2kT/Ze -- 25 mV
(2)
for symmetrical electrolytes with counter-ions and coions of similar valency ( Z + = Z _ - Z). Parameters c, c0, and k are the dielectric constant of water, the permittivity of free space, and the Boltzmann constant, respectively. ~ is the Debye screening length (cf. Appendix A). The second form of Eqn. 2, which is an expanded and truncated version of the first one, suggests the interracial potential value to increase linearly with the charge density; for membranes with a higher charge density this increase is less, however, owing to the saturation of the interfacial double layer region with counterions (see Fig. 8 and also Fig. 5). Within the framework of the linear Gouy-Chapman model the variation of the electrostatic potential with distance from the membrane surface is given by the
319 02
I
I
20
.~'~
I
I
I
10
s
~,, >
I
t,
0.01 M
.--
I
~el = 0.22
C m -2
0.1 ;I
0
.. . - - - f
0.1 acl
0.2
.
0.3 I
(C m -2)
0
Fig. 8. Electrostatic surface potential, % , as a function of the membrane surface charge density, oet, for different values of the bulk electrolyte concentration, c. Solutions to Poisson-Boltzmarm equation (dashed), modified Poisson-Boitzrnann equation (full line), and an approximation to the latter (cf. Eqns. 27 and 28, dashed-and-dotted) are shown. Computer simulation data are represented by symbols. Results from linear Gouy-Chapman theory, deduced by solving linear Poisson-Boltzmann equation, are given by dotted lines and nearly always significantly overestimate the surface potential value (modified from Ref. 9).
hyperbolic sinus function, for constant surface charge density, and by the hyperbolic cosinus function, for constant surface potential. For membrane-membrane separations much greater than the Debye length, however, the spatial decay of the electrostatic membrane potential is always well described by a single exponential function ~ ( x ) = % exp(0x/X ) = (OetX/~%) e x p ( -
x/X)
(3)
Usually, and surely for most of the biologically relevant systems, the double layer properties are dominated by the counterions (cf. Fig. 5). This means that the electrostatic potential as well as the ionic profiles near a negatively charged membrane in the case of 1:1 and 1 : 2 (Z+ : Z_) electrolytes are rather similar; for positively charged surfaces the same holds for 1 : 1 and 2 : 1 salt solutions. When suitable analytic results for the membrane under investigation are not available, the solutions for the corresponding symmetrical electrolytes can then be used with the valency pertaining to the counterion; but more sophisticated approximations are also available [32]. The Gouy-Chapman approximation is imprecise from the membrane point of view in that it neglects the interfacial structure as well as the hydration effects. From the point of view of the electrolyte it also has shortcomings in that it treats the short range interactions between the charged surface and ions inconsistently [33]. No account is taken of the nonelectro-
4
8
12
× (ore) Fig. 9. Ion concentration profile, c~(x), near a charged surface in contact with a one molar monovalent salt solution as implied by various electrostatic models. In agreement with computer data (symbols) oscillations are predicted by the modified Poisson-Boltzmann equation (full line) which can not be accounted for by Gouy-Chapman approximation (dashed) (From Ref. 9). The latter model also overestimates the double layer thickness.
static, such as ion size effects, or soft, e.g., water-mediated, short-range membrane-ion interactions. Additional complications arise for elongated counterions [34,35]. * Gouy-Chapman approximation, moreover, neglects all interionic correlation effects: ionic distribution in the diffuse layer is assumed to be dictated solely by the coulombic direct forces between the ions and the interface. This is the reason for the monotonous change of the ion concentration profile with separation (Fig. 9). Consequently, Gouy-Chapman theory is strictly valid in the limit of infinitely dilute bulk solution bathing an infinitely narrow membrane-solution interface in a structureless solvent. Obviously, such a situation is not encountered for membranes. But fortunately this is not always an obstacle, since different corrections to GouyChapman results have different signs. The interfacial structure and higher order electrostatic effects, for example, lower the calculated electrostatic membrane potential. But the correction for the interracial hydration goes in the reverse direction (see further discussion). In fortunate cases-many lipid bilayers apparently belong to these-Gouy-Chapman theory may thus be nearly correct, despite the fact that it dramatically oversimplifies the reality. This mysterious mercy should be
* From the suitably modified Gouy-Chapman theory it is concluded that large, elongated divalent cations have a smaller effect on the surface potential than the corresponding point ions; the magnitude of this discrepancy decreasing with salt concentration. Large divalent cations also give rise to a negative contribution to the zeta potential.
320 accepted gratefully. But simultaneously the veritable complexity of the situation should be borne in mind, especially when conclusions are to be drawn from the theoretical data-analyses at the molecular level. Only then a clear distinction will be possible between the intrinsic and quantifiable experimental or theoretical errors on the one and the (in)adequacy of the model or working hypothesis on the other hand. II-B. Modern electrostatic double layer theories In contrast to Gouy-Chapman approximation all modern electrostatic theories incorporate the interionic correlations and other higher order effects into basic equations in a natural way. * This is possible because most of these theories do not rely on the simple Poisson-Boltzmann equation but rather on some expression of proven, rigorous validity for the bulk electrolyte solution. The most popular modern theories of electrified surfaces stem from the cluster expansion, the Bogolyubov-Born-Yvonne-Green set of equations [38] or from the Ornstein-Zernike equation (for reviews, see, for example, [8,9,7]. A combination of these with a suitable closure relation, the superposition closure of Kirkwood, the Loeb's closure, or the hypernetted-chain closure, affords a self-consistent and accurate electrostatic description of simple electrified surfaces. Some problems arise because of the interracial discontinuity which requires the exact results valid for a homogeneous bulk system to be adopted for an anisotropic case. But these can be circumvented by mapping the inhomogeneous three-dimensional system into a homogeneous two-dimensional one [36,37]. The hypernetted-chain/hypernetted-chain method [39,40,8,9], the hypernetted-chain/mean-sphere approximation [41,42,8,9,11], and the modified Poisson-Boltz-
* The simplest, but theoretically already formidably complicated, modern statistical mechanical model of the diffused part of the electrical double layer is that of a system of charged hard spheres (or even just points) immersed in a dielectric continuum next to an impermeable, planar charged wall. This is termed the primitive model double layer and has been studied extensively with various approaches over the last decades. One outcome of this is the conclusion that the solution of the non-linearized Poisson-Boltzmann equation for the point-ion version of such a model, which is equivalent to Gouy-Chapman approximation, gives a nearly quantitative description of surface electrostatic potential over a surprisingly large domain for 1 : 1 aqueous electrolytes. The accuracy of such model for other electrolytes is less and in general the ion and charge distribution profiles are poorly reproduced (cf. Fig.
9). Accordingly, the discrepanciesbetween the measured and calculated electrostatic membrane surface potential must be due to the indirect ion-surface interactions, to the effects of interfacial structure and polarity, to the consequences of solvent structure, or to any combination of these.
mann approximation [43,44-51] clearly stand out in their ability to make quantitative predictions for the structure and the thermodynamic properties of simple surfaces and their associated model double layers. Of these theories the modified Poisson-Boltzmann approximation is, in principle, optimally suited for the applications with membranes, mainly because of its formal familiarity with Gouy-Chapman model (see Appendix B). But in reality it is seldom going to be used as it stands. Partly owing to its complexity and partly due to the fact that in the present form it can not cope with the interfaces of finite width. The modified Poisson-Boltzmann approximation shows, among other things, that inclusion of higher order corrections has consequences which phenomenologically resemble the effects of the spatially variable Debye screening length (cf. Eqn. 26). * * For example, higher order electrostatic effects give rise to structural oscillations in the diffuse double layer. (cf. Figs. 6 and 9). These oscillations are small for dilute monovalent salt solutions. But for 1 : 1 electrolytes with a concentration in excess of 0.1 moi.1-1 near a charged planar surface the oscillation amplitudes may become substantial. Monte-Carlo simulations for lamellar systems suggest that under such conditions the ion-ion interactions are more important than the ion-surface interactions [52,53]. This explains why the electrostatic surface potential in the latter case is smaller than one would expect on the basis of Gouy-Chapman theory (Fig. 8). Higher order effects become progressively more important with increasing ion valency. For the divalent salt solutions the deviation from the results of standard electrostatic models, therefore, is significant already in the millimolar concentration range, at least if the charged surface is sufficiently uniform. (Fluorescence measurements with synchrotron light for manganese binding to a stearate monolayer show that, indeed, such oscillations form in a condensed but not in an expanded phase [54].) For a narrow, planar surface with a charge density OeZ= 0.0444 C" m 2 (one charge per 3.6 nm 2 or 12.5% charged lipids in a bilayer) in a 0.005 M solution of CaC12, for example, Gouy-Chapman approximation predicts the electrostatic surface potential value to be 30% too high (Fig. 8). Furthermore, the extension of the electrostatic potential and the thickness of the diffuse
The electrostatic potential felt by individual ions is never strictly identical to the local value of the mean membrane electrostatic potential (see Appendix B). Contributions from various cross-correlations impose fine structure to the interracial region and cause a deviation from the monotonous profile: layers of charge of alternating sign lead to oscillations in the charge and ion distribution profiles near e hard surface. The potential profile near a uniform surface therefore also oscillates and causes the diffuse layer potential and ionic adsorption isotherms to vary unmonotonously with the surface charge density.
321 double layer by Gouy-Chapman theory are overestimated (Fig. 9). Standard electrostatic models, consequently, are expected to predict excessive electrostatic interfacial repulsion for simple charged planar surfaces. * Experimental evidence for such effects exists for phosphatidylinositol bilayers interacting with polyvalent ions [57]. Ion correlation effects may be relevant for membrane systems. The more regular the membrane surface, the higher the surface charge density or the ion valency, and the more concentrated the bulk solution, the more likely this will be the case. But interfacial 'softness' or smallnes of the interracial dielectric constant can largely compensate for this. For a soft wall in the weak coupling limit, i.e., for a membrane in which the electrostatic interion potential is small compared to the kinetic energy of ions, the solution to the Poisson-Boltzmann equation may even become an exact result [13]. * * Highly mobile and elastic membranes or 'fluffy membrane surfaces' differ-from the electrostatic point of view-from rigid membranes and from the model membranes in the gel phase. Theoretical description of these two types of membranes consequently should be different. For the latter class of membranes the interionic correlations can play a rrle. Suitably modified mean field theories which account for the correlations, such as the modified Poisson-Boltzmann equation approach, therefore should be considered for the electrostatic analysis of rigid uniform membranes. But even when dealing with 'soft' membranes the higher order effects should be kept in mind. If for no other reason, in order to check whether or not the resulting effects will influence the electrostatic membrane properties on a small-scale. In the cases of (macro)molecular adsorption to the membrane, ion binding, or transmembrane mass-flow, this could be the case owing to the small size of the involved membrane areas.
No universal theory of the membrane electrostatics exists to date. Acceptable working models for different specific situations can be found, however, under appropriate simplifying assumptions. Gouy-Chapman approximation provides a reasonable description of the electrostatic properties for model membranes with an infinitely
* Correlation-dependent double layer narrowing, implied by all modern electrostatic theories, is confirmed by numerical MonteCarlo 'experiments' for charged surfaces in contact with an electrolyte solution. At high concentrationsand surface charge densities the counterions are reproducibly found to be tightly packed near the interfacein a layered structure; here, the potential drop is also appreciable [55,56]. ** The reason for this is that the three body, higher order effects, which underlie correlations and oscillations phenomena, demand hard,, regular surfaces to fully evolve. Indirect evidence for this comes from the studies of a one component plasma [58,13,59]and from the correspondingMonte-Carlo simulations [60,52,61,62].
narrow interface which does not perturb the water structure, provided that the higher order effects, arising predominantly from the direct ion-ion interactions, are negligible. This need not always be the case for rigid and~or highly charged membranes. Especially for the concentrated electrolytes the corresponding correction terms, based on up-to-date double-layer models, should be allowed for. The modified Poisson-Boltzmann equation approach is one example of this. Phenomenologically, this corresponds to using a spatially variable ion screening length within the framework of classical electrostatic membrane models. III. Interracial structure
Perhaps the weakest point of simple electrostatic membrane models is the neglect of the interracial structure. In Gouy-Chapman theory, for example, the interphase between the membrane and the solution is assumed to be a discontinuous interface which encompasses all structural membrane charges; the dielectric constant at this location is also assumed to change abruptly. In reality, the membrane charges are neither smeared-out uniformly over the surface plane nor is the transition between the bulk solvent and the membrane interior discontinuous. The membrane charges are always discrete and distributed throughout a mesh-like, water-containing region with parlous appendages ranging from glycolipid and protein molecules, with dimensions on the order of a few nanometers, to cilia and microvilli, on the order of micrometers. The solvent properties and ion concentration in this region may differ largely from the bulk. Any realistic analysis of the membrane electrostatics should pay attention to this.
III-A. Charge distribution Lateral discreteness of the membrane surface charge. To allow for the consequences of the discrete charge distribution on the membrane surface the continuous structural charge distribution function is replaced by the sum over all discrete membrane charges [63]. The latter, for the sake of simphcity, are normally postulated to be located on a regular lattice. From the appropriate solution to the Poisson-Boltzmann equation the electrostatic membrane potential is found to be identical to the lattice sum over the screened Coulomb potentials of individual charges [64]. As such, the Gouy-Chapman theory then implies the local potential maxima to exist at the charge-lattice sites (Fig. 10). The existence of well-defined potential peaks at the surface of real membranes is quite imProbable , however. First, experiments with the model membranes fail to show any firm indication of such discreteness effects [65-68]. Secondly, numerical simulations suggest that there may be little, if any, real difference between the
322
Fig. 10. Theoretical prediction for the spatial profile of the electrostatic membrane potential above a surface covered with a square-lattice of the structural electric charges. Discrete potential (in the original article ion-concentration) peaks are implied to exist at the lattice sites on which the structural surface charges are located. These, however, have never been observed experimentally. (From Ref. 63).
cases of discrete and continuous surface charge distribution [7,69,70]. * Also the lateral diffusion in planar double layers is virtually the same for a uniformly charged surface and for an interface with discrete charges [71]. In practical applications to membranes, therefore, the lateral surface charge distribution may be neglected, at least in the case of monovalent ions. Charge distribution perpendicular to the membrane. The distribution of the structural membrane charge perpendicular to the membrane in numerous of electrostatic studies is crucial, however. The rule of thumb is that it is necessary to consider such effects as soon as the thickness of the interracial charged layer is comparable to or greater than the Debye screening length. From the point of view of the membrane hydration the interfacial structure is always important (see Appendix D). The first reason for the charge smearing perpendicular to the membrane surface is the finite size of the charged membrane constituents. Structural charges often are distributed along extended, membrane-bound molecules, some of which may protrude up to 10 nm away from the average interface, for purely geometrical reasons. The second cause of the charge distribution per-
* This, on the one hand, may be due to the high mobility of the membrane components which partly alleviates the consequences of the charge singularity. On the other hand, the lack of the discreteness of charge effect may also be indicative of the relatively slow decay of the lateral pair correlation functions, which describe ionic distributions in the surface plane; such functions fall off as the inverse cube of the distance [78-80,59].
pendicular to the membrane are the thermal out-of-plane fluctuations of the individual membrane constituents or of the whole membrane domains. In consequence to this, the transverse smearing of the structural membrane charge can be appreciable even for the molecules with a relatively tiny head-group. For fatty acid monolayers the dynamic interfacial roughness has been measured to exceed 0.1 nm [72]; for phospholipids the width of the interphase with embedded charged residues may be close to a nanometer, owing to the greater headgroups [73] and to the headgroup motion. For even more extended molecules the effect is likely to be still more pronounced. Dealing with the interfacial charge distribution is an intrinsic feature of the Donnan equilibrium approach [29]. But such approach typically provides a less realistic description of the membrane surface than the corresponding generalizations of the Gouy-Chapman model [74-77]. The latter are briefly discussed in Appendix C and also account for the ion distribution effects negiected in classical Donnan approximation. Increasing the width of the membrane-solution interphase normally lowers the electrostatic potential at the membrane surface. This is mainly owing to the better screening of the structural surface charges by counterions because the latter are able to penetrate into the region occupied by the membrane charges. For the same reason, the electrostatic potential at the membrane surface is more sensitive to the concentration variations than standard Gouy-Chapman theory would imply. One further, but frequently disregarded, sequel of the transverse smearing of the structural membrane charge is that the ionic double-layer region extends further out into the electrolyte solution and is shallower near the very surface. This means that the interfacial electric field is diminished by the charge distribution perpendicular to the membrane. The dielectric saturation and attraction for ions at the membrane surface is thus often smaller than anticipated. For one particular system, in which the structural surface charge density is assumed to fall-off exponentially with the separation from the membrane-solution interface (Oet.m= (ocJd~)exp(-x/dc), the lowering of the membrane surface potential by the charge distribution perpendicular to the membrane has been calculated [75] to be: 60(dc) = q,o(dc = 0 ) / ( 1 +
dc/X )
(4)
Here, d c corresponds to the average transverse displacement of the structural surface charge and is proportional to the interracial thickness. This, admittedly oversimplified, relation shows that when the average charge displacement perpendicular to the membrane surface changes from 0.2 nm to 1 nm, in the case of physiological salt solution (0.15 M), the electrostatic potential
323 20 Oel = 0 . 0 1 5
16
Non-Coulombic phenomena exert their own effects on the ion penetration in the interfacial region [20,84,85] and also on other 'electrostatic' membrane properties. It is perhaps indicative of this that protein desialiation on the surface of the brush-border membranes increases membrane fluidity [86] despite the fact that the net membrane charge becomes less after the sialic acid residues have been stripped-off.
C m -2
c = o.15 M
>
E
0 12
t"
8
nrn
D.
III-B. lnterfacial curvature
0
i
0
1
i
2 x/nrn
i
3
4
Fig. 11. The electrostatic potential profile near a surface with transversely smeared-out structural charge in contact with a 0.15 molar monovalent salt solution. The density of the structural surface charge is assumed to decay exponentially(#el.,,(x) = (vet~de) exp(- x/dc) ) for three different values of d c in order to mimick the membranes with a thick interface or a 'fuzzy-coat' surface. The total surface charge density is eet = 0.015 C.m -2. The potential profile becomes progressivelymore shallow if the interracial thickness and the average charge displacement, de, increase and become substantially greater than Debye screeninglength, h = 0.9 nm.
at the membrane surface should decrease by 20 to 50% (Fig. 11). For real membranes the charge regulation and charge mobility interfere with the direct interfacial structure effects [81]. If a constant amount of charge is allowed to distribute throughout a finite region adjacent to the membrane surface in relatively diluted ionic solutions the electrostatic membrane potential becomes nearly independent of the separation from the surface. The ion distribution profile close to the membrane under such circumstances is also relatively uniform. But with increasing salt concentration this charge tends to pile-up at the outer border of the interracial region. The electrostatic potential then increases with the separation from the membrane and at the utmost end of the zone occupied by the structural membrane charge becomes the highest. Further out in the electrolyte solution, where no membrane charges are located, the electrostatic potential profile finally is such as expected for a simple, uniformly charged surface [83,811. Direct experimental evidence indicative of the importance of the interracial thickness for the electrostatic membrane properties comes from zeta potential measurements with red blood cells [76,3561 as well as from the measurements with ganglioside-contalning model membranes [821.
When the curvature radius is comparable to or smaller than the double-layer thickness [87,88] or smaller than the interfacial width the interracial curvature starts to influence the electrostatic membrane properties; typically it causes the electrostatic potential to be less than for a comparably charged planar membrane. But neither of these two conditions is likely to be satisfied for the membrane as a whole; only locally the membrane curvature may be sufficiently high for one of the previous requirement to be fulfilled. Notwithstanding this, the electrostatic properties, at least of small membrane areas [89,91], can be affected by the local curvature. This is especially true for the membranes suspended in dilute electrolytes for which Debye length, and thus the double layer thickness, is big. For such systems it may be important to discern whether the membrane curvature or some other, already mentioned, non-trivial effect is the origin of the theoretical deviations. To gauge this, any electrostatic model for a planar charged membrane can be set to work, because deviations between simple, such as Gouy-Chapman approximations and more sophisticated electrostatic models for a spherical geometry and for a charged plane are the same. Spheres. Membranes of small liposomes, of pinocytic vesicles, of mitochondria, and, perhaps, of thylacoids [89] in dilute electrolyte solutions belong to the few membraneous systems in which the curvature effects may be significant. From the theoretical point of view, these systems can be described with various forms of the appropriate approximate solutions to the nonlinear Poisson-Boltzmann model [92,88]. For the spherical symmetry an explicit solution has as yet not been found. Only the problem of electrostatic repulsion between two spherical particles has been solved accurately [90]. Cylinders. On the length scale of Debye screening length, which is the appropriate electrostatic yardstick, most of the biological surfaces including membranes are not even near spherical; normally, they are either fairly irregular or cylindric in shape. Quasi-cylindrical symmetry, for example, is characteristic for all membrane protrusions, such as microvilli. Moreover, many membrane-associated macromolecules or their aggregates, such as parts of the cytoskeleton are, on the average,
324 prolate. * Fortunately, the r61e of cylinder curvature is likely to be insignificant unless the curvature radius is less than approx. 8 nm. Planar model, consequently, remains a valuable approximation also for studying the cylindrical m e m b r a n e structures, irrespective of the lateral distribution of the structural charge [94,93]. This has been confirmed [96] by using hypernetted-chain approximation.
surface electrostatics. In contrast to this, the consequences of the charge distribution perpendicular to the membrane are frequently quite pronounced and far reaching. As a rule, the finite width of the membrane-solution interface causes the surface values of the electrostatic membrane potential and of the electric field to decrease and the double-layer thickness to increase. This is reminiscent of the smearing of the surface charge perpendicular to the membrane. This causes the interfacial structure often to dominate the electrostatic properties of the charged membranes. Membrane models which do not allow for the charge distribution throughout the interfacial region are prone to overestimate the electrostatic membrane effects.
Surface curvature and discreteness of charge effects normally do not play a significant role in the membrane * Solutions to linear [93] and nonlinear [97,95,98] Poisson-Boltzmann equation for a cylindrical particle or a supermacroion can prove useful for describing the electrostatic properties of such systems; an alternative is to use the corresponding perturbative treatments [92]. They show, as in the case of planar membranes, that Gouy-Chapman approximation tends to exaggerate the double-layer thickness of smooth cylindrical bodies. Gouy-Chapman theory therefore predicts too high values for the electrostatic surface potential of such charged cylinders. Maybe this is the reason why the measured mobilities of cylindrical particles in the electric field better agree with the predictions of hypernetted-chain/mean-sphere approximation than with the results of Gouy-Chapman approximation [96]. Caution is required in evaluating such observations, however, since the attempts to model zeta-potential data for charged bodies are not at all straightforward. The zeta potential of a cylindrical body, for example, is a non-monotonic function of various parameters, such as charge density. For 2:2 salts, the zeta potential has a maximum as a function of Oet.Similar maxima have been predicted for the spherical particles also [99].
IV. I n t e r f a c i a l polarity a n d t h e d i e l e c t r i c c o n s t a n t
M e m b r a n e polarity (hydrophilicity) is extremely important. It is the primary reason for the water binding to a m e m b r a n e surface; but it also affects, directly or via the hydration effects, the chemical reactivity and the electrical conductivity at the m e m b r a n e surface, and participates in the regulation of the lipid-ion association, ion binding [101,151] and transport. Moreover, the efficiency of electrogenic phases has been claimed to depend first of all u p o n the value of the dielectric constant of the respective m e m b r a n e regions and only in the second place u p o n the distance between the redox groups involved [102].
x (rim) 0
0.5
80
1.5
1
2
2.5
5
I
B
I
80 A
"
iiiiiiiiil
"
i n t e r p h ! ~
70
60
60
v
•
•
@
@
@
to
50 40
x
30 20 apolar 10 0 0
j
lr ..... poa
i
i
0.5
1
:::::::::: :':.:.:.:.
L
1.5
x/~
CsCI
2O 0
I
I
I
1
2
3
c [ m o l . l i t e r -1] CH2
CO-O-- CH
O
H 2 C - O - - P - OR I
oo
Fig. 12. Interfacial polarity profile, expressed in terms of an effective dielectric constant, c(x), as a function of the interfacial membrane width, dp (A) or bulk electrolyte concentration, c (B). (A) Curves were calculated from a nonlocal electrostatic model (cf. Eqn. 40) using 78 and 2.5 for the static and high-frequency dielectric constant, respectively. Upper scale refers to the case of the system with a water structure correlation length = 0.15 nm. (B) Effective dielectric constant in the plane of ion adsorption adjacent to a charged phosphatidylglycerol membrane. Data were deduced from thermodynamic results by means of indirect calibration procedure [110].)
325 Membrane interior, on the one hand, is apolar with a static dielectric constant between 2 and 5, depending on the degree of water penetration into the membrane. The aqueous subphase, on the other hand, is polar and structured, characterized by a bulk dielectric constant near 80. The transition between these two limiting values takes place in the interphase region and inevitably is continuous. Consequently, near the membrane surface the effective local dielectric constant must be realized to vary smoothly, albeit non-homogeneously, between the values of 2 and 80 [103]-as a function of temperature, as well as dependent on the membrane, solvent, and electrolyte properties. Indeed, the potentiometric [104,105], spectroscopic [106-108] and kinetic studies [109] with various probe molecules suggest that the effective dielectric constant near the surface of artificial bilayer membranes is 10-70, depending on the precise location, probe, composition of the aqueous subphase, and membrane type. More polar and/or charged membranes typically are characterized by a relatively low interfacial dielectric constant value. Likewise, molecules with a big polar head decrease the relative permittivity of the interface. Increasing the bulk salt concentration for the uncharged phosphatidylcholine bilayers has been claimed to decrease the interracial dielectric constant [108] below the initial value of approx. 70 [110]. A reversed trend has been reported for the charged phosphatidylglycerol membranes [110]. The latter, under physiological conditions, are characterized by interfacial constant of around 30 (Figure 12). This discrepancy, in part, is a consequence of the different locations of the probe ions in the two cases; but screening of the surface charge and its consequences also are likely to be important. By all means, the published estimates of the dielectric constant for the membrane vicinity should be used with utmost prudence. Firstly, owing to the uncertainty of the underlying calibration procedures; and secondly due to the fact that the polarity near the membrane surface varies so rapidly with distance from the membrane (Fig. 12) that even minute molecular displacement or small configurational changes in the headgroup region are liable to drastically change the estimated dielectric constant value. Conformation changes, for example, can be inferred to explain different values for the interracial dielectric constant obtained with various techniques. Reduced value of the interracial dielectric constant has been modelled by theoretically treating the ensemble of rotating lipid polar headgroups as a collection of interacting dipoles embedded in a dielectric with an exponential profile variation [111]. Such approximation conveys only part of the truth, however. It namely neglects the interdependence of the interracial and solution structures, as well as the direct influence of the latter on the local dielectric response. This missing part
of the information may come from the studies on the molecular origin and (electrostatic) consequences of the membrane-water association. Non-local dielectric response and dielectric inhomogeneity (see Sections V and VI) play a rrle in this. The former has to do with the inability of the solvent molecules to respond to the rapid spatial field variations for small wave-vector (k) values (see also Section V); the latter is simply a consequence of the non-uniform solvent distribution across the membrane and its adjacent solution (see also Sections V and VI). Relative permittivity and polarity in the interfacial region both vary smoothly between the values characteristic for the membrane interior and the bulk solution. The effective interracial dielectric constant can, therefore, be anywhere between 2 < c id,:,. however [172-174]. D i v a l e n t and p o l y v a l e n t c(~,.,,.n.~ b i n d to the n e u t r a l p h o s p h o h p i d m e m b r a n e s by ih~'~;racting with the m e m b r a n e q u a n t u m - m e c h a n i c a l l y , m .~ stereospecific m a n n e r . C h a n g e s in the fluorescence p r o p e r t i e s of the trivalent transition elements, such as t e r b i u m , i n d i c a t e this [359]. In general, the stoichiometry of ion b i n d i n g to the m e m b r a n e is p r o p o r t i o n a l to the valency of the i o , b o u n d . F o r example, in organic
TABLE IV Representative ion-binding constants (litre tool s) for model lipid membranes Abbreviations and selected References: PC phosphatidylcholine (unpubfished data and [588,176,177,232,592,589,173,174,591,310,172,590,225,205, 175]); PE phosphatidylethanolamine (unpublished data and [593]); PS phosphatidylserine [594,595,596,353,597,167,185,598,600,599,225,601,603. 602,607,193,606]; PI phosphatidylinositol [608,609,595]; PA phosphatidic acid [610,595]; PG phosphatidylglycerol [unpublished data and 595,187, 465,611]; CL cardiolipin [593,220,612]. Less certain values are given in brackets. Only some of the references quote binding constants together with the primary data. Gouy-Chapman theory has been used to calculate the binding constants throughout. 1:1 PC PGPSPI PA-
Li + (0.5) 0.8 0.9
Na +
K+
Rb +
Cs +
CI-
0 0.15 0.15 0.1 0.2 Sr 2+
0 0.1 0.08 0.05 0.1 Ba2+
0 0.05 0.05 0.05 0.1 Ni 2+
Co2+
Mn2+
Cd: ~
6.7 23 49.5
M;
1:2
1.1 Mg 2+
0 0.65 0.7 0.6 0.8 Ca 2+
0.9
PC PGPS1 :3
1.85 12 15 La3+
2.5 17 26.5 Pr 3+
0.7 10 26.5 Nd 3+
0.5 11 38.5 Eu 3+
1.7 15 80 Tb3+
1.7 10.5 56 AI3+
PC
(250)
(80)
120
280
(250)
(300)
Br 2.0
C10480
335 solution at the saturation limit the binding stoichiometry for palmitoyl-oleoyl-phosphatidylcholine is one Ca 2÷ per two phosphocholine headgroups [175]; for La 3÷ the corresponding ratio seems to be 1 : 3 [176,177]. But the decision about the precise stoichiometry of binding is not always unambiguous. Often it is a matter of convention. Different combinations of the stoichiometric ratio and of the binding constant may describe the same set of data comparably well (cf. Fig. 21). Moreover, neglecting the interfacial structure, this is, disregarding the finite interfacial thickness and the membrane defects can lead to a serious overestimation not only of the membrane surface potential but also of the amount of counterions bound [178]. * In fact, one recent analysis has shown [174] that the density of the binding sites on phosphatidylcholine bilayers for the divalent and polyvalent ions may be far lower than the assumed stoichiometry would suggest [179]. Perhaps it is also dependent on the number of the surface defects or domain boundaries. When several types of divalent ions simultaneously interact with a membrane their binding may either be enhanced or else suppressed by the mutual occupation, depending on their relative concentrations [180]. Divalent transition elements always bind better than alkali earth elements [182]. Polyvalent ions bind even more efficiently, the higher their charge the more so [183,184]. One would expect this from the laws of simple electrostatics. But simple electrostatics fails to explain the tendency for the binding constant of phosphatidylserine to increase with the atomic number of the transition element bound [167,185]. I trust that this is largely a consequence of the quantum mechanical effects and of the high ionic polarizability. It is, therefore, not surprising that the augmentation of the ion binding to phosphatidylglycerol [187] is less than for phosphatidylserine, despite the fact that these two lipids are similarly charged. The former has namely a less tight, better exposed binding site for which the stereochemical, steric effects must be less important. Several lines of evidence corroborate the conclusion that the selectivity of ion binding to phospholipids, in general, is of quantum mechanical origin rather than being a consequence of simple surface electrostatics. Experimental as well as theoretical studies [188,158,189] show that the selectivity of cation binding originates from the ion interactions with the phosphate group (Fig. 22), whether this headgroup is charged or not, But the strength of such binding depends on the net mem-
* It should also not be overlookedthat large oligo-valent and polyvalent ions may loose part of their screening capability because of purely geometrical reasons [34]; they may, moreover, exhibit nontrivial hydration effects. Counterion depletion from the solution, finally, may be appreciable for very dilute solutions in the case of extremely strong binding [181].
t
Fig. 22. The intrinsically preferred structural form of the complex between two serine phosphate and one Ca2+ molecules(left) and the complex form in extended configuration. This picture was deduced from ab initio quantum mechanical study of 11 pre-selectedconformations of serine phosphate. It is indicative of an intermolecular chelation with 6 or 4-fold coordination of calcium binding, depending on the relative arrangement of serine phosphate groups. Phosphate atoms are shown black, oxygens shadowed, and calcium ions as concentriccircles.(From Ref. 189.) brane charge. Calcium ion binds to dipalmitoylphosphatidylcholine within 0.1-0.2 nm of the phosphate moiety, as witnessed by the neutron diffraction experiments [190]. Charged phosphate and, less so, carboxylic groups on the phospholipid molecules, consequently, belong to the main membrane binding sites for cations [191-193]. The same is true for model and biological membranes because monovalent cations do not bind significantly to gangliosides [67], in contrast to divalent ions [194]. This is indicative of the poor binding capacity of the sialic acid residues for the cations. Anions normally bind to the membrane-associated charged amines. Non-coulombic effects observed in ion-binding studies may have other sources in addition to direct quantum-mechanical interactions. Anything that either influences the ion distribution near the membrane [114,110,195], modifies the m e m b r a n e structure [189,196,197], affects the surface hydration [198201,110,197], or creates the membrane defects [174] can also interfere with the membrane-ion association. Even noncharged species, such as highly concentrated sucrose and glucose in the membrane-bathing medium, can enhance ion binding to the membranes [202]. Uncharged anesthetics exhibit similar effects, as has been documented for the Ca 2÷ binding to the biological membranes [203,195]. One possible and probable explanation for this is that the noncharged but polar substances change the structure a n d / o r hydration of
336 the interfacial region. An indication for this is that the selectivity of binding is reversed upon the dilution of the negatively charged diacylglycerophosphoglycerol in the model lipid membranes by the uncharged glycolipid dimanosyldiacylglycerol [204]. Ion binding evokes structural changes both at the level of single molecules and of the whole membranes. Lipid headgroups with bound ions, for example, may turn out of the surface plane if the complex is sufficiently hydrophilic and charged, as witnessed by phosphorus nuclear magnetic resonance [172,205,206] and X-ray diffraction [207]. Surface charges increase the propensity for the molecular out-of-plane fluctuations [120] and (partial) elimination of the net charge leads to a reduced lipid headgroup mobility [208]. Headgroup deprotonation and ionization may lead to a hydrocarbon-chains tilt [211]; but also protonation of the headgroups which participate in hydrogen-bonding at very low pH may cause the lipids to form partly dehydrated bilayers with tilted chains [209]. Complete headgroup protonation decreases and maximal deprotonation, as a rule, increases the membrane hydration [210,211,123]. In final consequence, ions can force strongly hydrated lipids membranes in the ordered phase to form structures with interdigitated chains ([212,213], Kirchner and Cevc, unpublished). The major reason for such changes is that the membrane components or their parts are electrostatically and, even more so, hydrationally repelled from the interface. Salts always affect membrane polymorphism. Co-existing lamellar bilayer structures with different membrane-membrane separations are induced by Ca 2+ ions. This holds for pure [214,179,215] as well as for mixed [216], such as dipalmitoyl phosphatidylcholine/ cholesterol [217], membranes. In the latter, mixed systems the effects are somewhat smaller [218]. Ion binding, furthermore, affects the membrane phase transition temperatures, most notably - but not most strongly the chain melting transition temperature of the lipid bilayer (see section VII-A for a discussion of this effect). When the resulting process is associated with membrane dehydration the final phase can be of non-bilayer type [219,220]. Salt-induced phase transition temperature shift is in most cases [123] towards higher temperatures. For a one-molar monovalent salt solution it is on the order of 1 K or less, in the case of zwitterionic lipids, such as dimyristoylphosphatidylcholine *. Comparably concentrated divalent ions cause upward shifts of up to 7 degrees; the change of the phase transition temperature induced by a typical trivalent ion, such as La 3÷, is
* An exception is the lithium ion which produces similarly great shifts already in decimolar concentration range, owing to the strong phosphate-ion association.
similar [177]. But in diluted electrolytes the shift magnitude increases with the ion valency in the sequence: La 3+ >> Ca 2+ > Cd 2+ > Co 2+ >> Cs +-> Rb ~ > K * Na +, etc., below the saturation limit. Such dependence can not be described in the terms of mere electrostatics without invoking ion binding. Such binding, indeed, is very probable for the divalent and multivalent ions, but not inevitably necessarily for the monovalent ones; for the latter, other corrections discussed in Section II may also be important. The Poisson-Boltzmann equation suggests that the concentration dependence of the phase transition shift should increase approximately linearly with the square root of the bulk concentration. When this is not the case non-trivial ion-membrane interactions are probably a part of the game. Due to the gathering of counterions in the vicinity of the charged membranes the net membrane surface charge reduces the concentration required for an appreciable salt-induced shift of the bilayer transition temperature. The thermodynamic effects of the ion binding are therefore better expressed for the charged lipids. partly owing to the energetic contribution from the surface electrostatics and partly due to the changes in the membrane hydration. It would seem that the magnitude of the ordinary electrostatic phase transition shift is normally quite small, however, only a few degrees for the membranes composed from the biological lipids. By all means it is smaller than the corresponding hydration induced shift [20,123]. The reason for this is that the electrostatic membrane free energy is typically much smaller than the free energy of membrane hydration (cf. Fig. 24). Knowing this it is easy to comprehend the origin of the chain-melting transition temperature increase observed upon ion binding to the neutral phospholipids [177,221-223]. Such increase may exceed 50°C, if the interfacially bound ions have displaced most of the bound water from the interface. Essentially, anhydrous lipid-ion complexes in excess solution are no exception [224,143]. One example of this are metal-ion complexes of diacylphosphatidylserine bilayers [225]. These form highly ordered, essentially water-free bilayers with extremely high transition temperatures in the range between 151-155°C. Another example are (nearly) completely protonated lipid membranes and membranes with bound lithium which are also often nearly anhydrous [20,219]. As a rule, the highest chain-melting phase transition temperatures for diacylphospholipid membranes with bound monovalent ions or protons do not much exceed 100°C due to the lack of the strong intermolecular ionic coupling. Ions also induce lateral phase separation of phospholipids [226], acidic at least [227], gangliosides [228], and, perhaps, proteins in the membranes. Isothermal phase transitions upon ion binding also may occur [604].
337 1.2
B
DPPC C 0.5 m
0.8
A
o/°\
O.t,
ApH
v
/
0.3 0.2
=,. ..t)
0.4 k. ® rl
0
O
/
0.1
0.6
Tm 0.2
O o~
!
0
!
__L
6
2
° J"T"~ ' S
pH
0 20
i
i
i
i
i
25
50
55
40
45
50
T (*C)
Fig. 23. Effect of changing membrane properties on the ion binding to and transport through the lipid bilayers. (A) In the suspension of phosphatidic acid methyl-ester vesicles in 0.2 M NaCl-with a pK around 3.9-proton pulse is observed at the chain-melting phase transition (From Ref. 230.) (B) Similarly, a permeability maximum is caused by the hydrocarbon chain-melting in phosphatidylcholine bilayers which may be absent for other phospholipids.
Ion-induced phase transition shifts may go in either direction. When the membrane-ion complexes bind water more strongly than the membrane surface without bound ions the ion-induced shift of the bilayer main transition temperature is downwards, even for the charged membranes. This is the case with phosphatidylcholine in the presence of anions or with phosphatidylserine with bound organic counterions. The chain-melting phase transition temperature for such systems therefore decreases with the increasing bulk electrolyte concentration. Ion-lipid association is apt to change at all membrane phase transitions which change significantly the properties of the interfacial region. For zwitterionic lipids, such as phosphatidylcholine [229], and for certain anionic phospholipids, such as methyl ester of phosphatidic acid [230,167], calcium binding is stronger in the ordered gel phase; the affinity of phosphatidylserine bilayers for magnesium decreases with the degree of chain unsaturation and fluidity [170]. During the chain-melting phase transition, some of the bound protons [231] and nearly one half of the bound calcium ions consequently are expelled into the bulk solution [230] (Fig. 23A). The same is true for the interaction between Tl+-ion and dipalmitoylphosphatidylcholine, as witnessed by N M R [232]. Under proper conditions, membranes with ionizable groups may therefore function as a reservoir for protons and other ions: changes in the membrane structure can terminate with an ion pulse. Such peculiar behaviour is not observed for all membranes, however. The carboxylic group of phosphatidylserine, for example, which is located deep in the interfa-
cial region, is not fully accessible to divalent ions, such as calcium, unless membrane is in the fluid state. Binding pockets may also form with other headgroups if these interact strongly. Sensitivity of sodium binding to D- or L-phosphatidylglycerol head-group on the headgroup isomery corroborates this conclusion [233]. Permeability of membranes with bound ions differs from the leakiness of the corresponding virgin membranes in dilute salt solution. This is important because many transport studies are conducted in highly concentrated electrolytes or in solutions containing polyvalent ions. The latter may cause severe membrane perturbations and may even release the content of the membrane vesicles, probably by inducing membrane dehydration and lateral phase separation [183]. Ion-induced leakage is best established for phosphatidyserinecontaining lipid vesicles. It has also been observed with other anionic phospholipids, however. When it occurs during mutual membrane approach and partial surface dehydration it frequently terminates in a salt-induced fusion between the artificial lipid vesicles [234]. It is perhaps useful to remind that dimethonium, an organic cation with two charged quaternary nitrogens separated by approx. 0.3 nm, has been reported not to bind to the lipid membranes and yet to screen the electrostatic membrane potential [235]. This ion may therefore be used for studying certain aspects of the divalent-ion-membrane interactions. Dioalent and polyoalent ions normally bind to the charged membranes. Monooalent ions can but need not always do so. Deoiations from Gouy-Chapman theory, correspondingly, are sometimes indicatioe of the ion binding but may also be a consequence of other non-trioial
338
electrostatic, structural, or solvation effects. Association between membranes and ions always increases with increasing local ion concentration and ion valency. High ion polarizability also is often advantageous. The effective ion concentration near a membrane surface differs from that in the bulk, owing to the redistribution of the ions in the electrostatic, hydration, and steric-repulsion fields of the membrane. Consequently, increasing the net charge density on the membrane and decreasing the membrane hydration, which typically supports (counter)ion gathering in the interfacial region, enhances the ion binding to the membrane surface. Steric constraints are detrimental for the binding of monovalent ions," but intermolecular bridging or chelation of the membrane components in the case of the divalent and multivalent ions, as a rule, are only possible when the binding sites are mutually close. Membrane properties are strongly sensitive to the interfacial ion concentration and vice versa. Fine membrane structure often changes upon the ion binding. This is primarily owing to the changes in the membrane hydration and to the formation of the intermolecular bridges. The interfacial hydration typically increases and the membrane packing density decreases with the increasing polarity of the ion-membrane complexes. Membranes which bind small, especially polyvalent, inorganic ions at their main hydration sites become less hydrophilic and more rigid," sometimes they even form nearly anhydrous aggregates in excess solution. But larger bound ions, in particular the organic ones, often promote membrane solvation. Charged membranes expel small, interfacially bound ions during the phase transition. Ion pulses may be generated in this way. Organic ions, however, have a stronger affinity for the fluid than for the condensed bilayer phase. VI-C. Ion transport Transbilayer concentration gradient, which represents a sort of the chemical potential, or electrostatic transmembrane potentials, drive a flow of ions or other solute molecules across the membrane. This flow is either direct and diffusive, through the membrane core, or pores; otherwise, it involves ionophores or carrier molecules. Ionophore or carrier mediated flow is controlled by the properties of the ionophore or carrier itself; the interracial molecular concentration, which depends on the membrane surface electrostatics also plays some r61e. Such flow is normally modelled as a multi-step kinetic cascade, of which the first step, the interfacial ion-carrier association, is mostly the decisive one. In the simplest picture of the membrane permeability barrier the bilayer is envisaged to consist of the resistance of the hydrophobic membrane interior and of two interracial barriers (see Appendix F). For small ions the
former depends mainly on the membrane partition coefficient and is determined mainly by the large electrostatic (Born free) energy cost of charge transfer between the membrane interior and the solution [236], unless the ion permeation occurs via pores. The pore resistance to the ion permeation, as a rule, is much lower than the resistance of the bulk membrane core. as has been shown in the pioneering work by Parsegian [237]. This is true no matter whether or not such subtle electrostatic effects are important [238]. Ions thus normally trespass the membrane interior via channels, pores, or packing defects. Notwithstanding this, the overall membrane electrostatic may be important for the diffusive mass transport since, at least for the small solutes, the surface barrier frequently is substantially higher than the remaining transmembrane resistance [239]. Passive membrane conductance for ions controlled by the interracial permeability coefficient depends directly on the interracial barrier height. The latter can be evaluated by means of the Eyring absolute rate theory to be: P, nt = dpki, = ( d p R T / N ~ h ) e x p ( G * ( x ) / R T ) , where h is Planck's constant, and the total activation free energy is G*(x). In the first approximation, this activation energy can be set equal to the potential of mean force for the ion approaching a membrane (see Appendix E). Membrane permeability for ions increases with the ion size and decreases with the interfacial hydration, at least when the interfacial region is the rate-limiting zone and no specific ion-lipid association takes place. Direct ion binding to the membrane may compensate for this and the accumulation of ions in the interfacial region, brought about by the membrane electrostatics, as well as the membrane packing inhomogeneities are always advantageous for the ion transport [240]. Unstirred water layers also present a barrier for the transport of uncharged species across the membrane [241]. Ions which cross the membrane in the form of molecular complexes are less sensitive to the interfacial electrostatic permeability barrier than bare ions [242]. Perhaps this is the reason why halide ions permeate through the lipid bilayers relatively rapidly [243,244]. The other possible reason is that halides bind to the lipid heads. By all means, the halide ions, and possibly some other ions as well, cross the membrane in a bi-modal manner. The fast component, with a permeability of the order of 10-8_10 6 cm 2 s-1, is a function of the inter-ion or ion-lipid complex formation. It thus relies directly on the carrier activity of the lipid molecules. The slow component, for which permeability values on the order of 10 -H cm 2 s -~ are measured, is probably due to a single ion diffusion. The exchange transfer of H + / O H across the lipid membranes is also biphasic, but even faster than for the halide ions. The fast component with a permeability in the order of 1 0 - 6 "( P H , < 10 - 3 c m 2 s 1 has been pro-
339 posed [245,246,247], but not unambiguously proven [248] to occur via a network of pre-associated water molecules in the bilayer. The slower H÷/OH--flow depends on the counterion diffusion, however, and resembles that of other ions [246,249,250]. Ion transport across the membrane normally increases with the increasing temperature and sometimes exhibits a maximum near the chain melting phase transition [251,252] (see Fig. 23B). Critical fluctuations in the lateral membrane compressibility close to the bilayer phase transition [253,254], as well as the mismatch in molecular packing between coexisting gel and fluid regions at the phase transition [255] have been claimed to be responsible for such maxima. But other factors can be important as well. For example, stress in the lipid bilayer may open-up pores in the membrane interior [256]; the tendency of the lipid molecules to form highly curved surfaces, which is stronger for the more strongly charged and hydrophilic molecules [257,258], may also be beneficial [259]. * By all means, single ion channels have been reported to arise in pure phosphatidylcholine bilayers in the critical temperature region [260]. Moreover, the flip-flop rate upon binding of some divalent ions [261] or upon lipid chain fluidization in the presence of Na-ions [263] has also been shown to increase. Both latter effects could contribute to the ion permeability maxima in the case of lipidic ion-carriers [264]. In complex artificial lipid bilayer the permeability pattern is dominated by the short-chain lipids, especially by phosphatidylcholines [252], or by the charged lipids. It is unlikely, however, that the ion transport through the biological membranes goes directly through the hydrophobic bilayer core. It is more probable that it follows the paths along which the energy barrier for the permeation is lowered by the formation of the ion-lipid complexes and subsequent transmembrane movement of such complexes and/or is reduced by the lateral membrane density fluctuations. It may be, therefore, biologically relevant that the latter two phenomena are facilitated by the proton binding to the membrane and by the indirect surface charge effects [265]. The latter may also influence the ATP-dependent ion transport across membranes [266]. Charge transfer along the membrane surface has been rarely considered. But it is well worth concerning. In particular, proton transport is interesting from the
* The lack of an ion permeability maximum for phosphatidylethanolamines in the transition region, for example, might be caused by the inabilityof this lipid to form such curved areas. But the absence of a Rb+-permeation maximumat the transition of unilamellar phosphatidylcholinevesicles remains enigmatic [262] unless it is caused by the low capacity of this ion to bind to the lipids [230].
point of view of the membrane energetics and surface catalysis. Indeed, protons in the interracial region have first been theoretically hypothesized [267] and then claimed on the basis of measurements [268] to participate in the lateral membrane conductivity. There is some controversy about the interpretation of certain experimental data [269,270]. Anyhow, in order for such conductivity to be high, interfacial proton networks should exist. One possibility is that these involve the strongly bound, and thus well oriented and coordinated water molecules [271]. Such is the case with phosphatidylcholine membranes which, consequently, exhibit a very strong hydration dependence of the electric current along the membrane, reminiscent of Eqn. 40. Another, as yet tentative, possibility is that 'proton-wires' involve the membrane components which are mutually hydrogen bonded. Phosphatidylethanolamines and phosphatidylserine molecules are suitable candidates for the participation in the latter process. At the applicative, technological end of the electrostatic membrane research the development of solid-supported membranes attracts much attention. Bilayercoated semiconductors, through which charge transfer is induced, have been devised and proposed to act as a molecular energy device [272]. Other solid-supported membranes were used as bio-sensors [273-276]. Future applications could rely on the interfacial modulation of the enzyme activity, for example, or make use of the membrane transport system, the voltage dependent channels, etc. [277]. Many of the synthetic ionophores are also physiologically or therapeutically important. They act as antibiotics, presumably because they dissipate the transmembrane potential and concentration gradients. Calcium-transport behaviour of the channel-forming peptide alamethicin, for example, has been shown to be potential-dependent, in contrast to calcium transport by the ionophore A23187. The latter in one study has been found to be unaffected by the potential [278]; but in another report the depolarization of plasma membrane potential has been claimed to have a potent inhibitory effect also on A23187 and ionomycin mediated ion flux [279]. The reason why the membrane electrostatics is important for all such applications is that the activity of many membrane associated (macro)molecules is governed by the characteristics of the membrane-solution interface [281,282]. The conductance of many ion channels is determined by the electrolyte type and concentration at the two ends of the channel opening-up in the interfacial region. The membrane surface potential thus increases or decreases the interracial ion density, depending on its relative sign, and affects the ion flux through the channel. Gramicidin pores provide one [283], but not the only [284] example for this. The influence of the electrostatic effects on the conductance of gramicidin is greater for neutral channels embedded
340 in negatively charged bilayers than for related anionic channels in neutral lipid bilayers. For sufficiently low ion density or high transmembrane potential, the ion flux through all channels may be more sensitive to the electro-diffusion of ions through the electrolyte, in the close vicinity of the channel mouth, than to the intrinsic channel properties [285,286]. This does not hold for voltage-gated channels, of course. Transmembrane potentials, moreover, are prone to increase the rate of the membrane pore formation by enhancing the thermal membrane fluctuations [287] which then act as pore nuclei. Suppression of such fluctuations upon membrane rigidification therefore should lower the channel conductivity. But it also may be a consequence of the increased lateral pressure of the chains, which lower the probability for the internal opening-closing motions in proteins [288,289,291]. Channel conductivity may be thus indirectly, via a change in the membrane thickness [292], as well as directly regulated by the membrane surface electrostatics and polarity, and vice versa. Potential-induced changes of the channel conformation or orientation in the membrane enhance these effects further. The gap between simple electrostatic models of the ion permeation through the membrane channels [293-295] and more realistic descriptions of the transmembrane ion permeation process which include the structural details remains to be closed. Dramatic differences in the membrane permeability for organic anions and cations [296-299] have repeatedly been claimed to arise from the dipolar membrane potentials. It would seem, however, that the contribution from the interfacially bound water is at least comparably relevant. Two arguments speak in favour of such hypothesis. Firstly, the fact that phosphatidylethanolamines and phosphatidylcholines which have a similar dipolar headgroup moment [613,158,114,251,252] have entirely different permeation characteristics [251,252]. And secondly, the observation that the rate of transfer of (poly)ions across the bilayers of dialkyl (ether-bound) and diacyl (esterified) phospholipids are similar [300] despite the fact that the former lipids carry two extra, highly polarizable, carbonyl groups with a large dipole moment. When interpreting experimental data obtained with complex membranes one should be careful to consider different possible explanations. Valinomycin, for example, one of the most popular ionophores, also binds to the membrane proteins and thus may influence system properties in a non-trivial manner [290]. Gradients near or across the membrane cause a net flow of material along or through the membrane. The ion flow is governed by the electric potential difference and, in many cases, by the interfacial permeability barriers. Steric effects and hydration of the membrane surface contribute to the latter. Membrane permeability for an ion which has
not been dehydrated in the interfacial region is governed by the high energy cost of the ion-charge transfer from the highly polar bulk in the apolar membrane interior. Membrane pores or the formation of ion-carrier complexes can reduce this energy requirement dramatically. Consequently, such phases have a lions share in the ion transport across the biological membranes. The membrane surface electrostatics plays a role in transmembrane transport by affecting the interfacial ion concentration and by increasing the probability for the pore and ion-carrier complex formation.
VII. Thermodynamics of charged membranes Membrane and its surrounding aqueous subphase form a single thermodynamic entity. Regretfully, the basic thermodynamic quantities of such system are not directly amenable to experiments. Nevertheless, it is worthwhile to answer the question of their evaluation and relative magnitudes, since many properties of the charged membranes are affected by the surface thermodynamics (see also Appendix G). Lateral pressure of the membrane is believed to participate in the regulation of the membrane polymorphism; this pressure is nothing else but the surface density of the electrostatic membrane free energy. Similarly, the intermembrane pressure, which controls the interactions between the adjacent membranes, is, formally speaking, given by the variation of the membrane free energy with interfacial separation. VII-A. Free energy As long as the interfacial hydration is neglected the free energy of a charged surface [4,301,302] is identical to the electrical work done in charging up the surface (cf. Eqn. 53). * (In Appendix G it is briefly discussed how the electrostatic coulombic part of this work can be calculated.) In the linear regime of Gouy-Chapman approximation the dominant contribution to the electrostatic free energy comes from the internal energy, which is proportional to volume integral of the square of the electric field; in the nonlinear, high potential case the entropy term dominates [70]. As has been noted already, the surface charge density of the biological membranes is typically in the order of 0.025 C. m -2 or less. This means that the coulombic part of the electrostatic free energy for such membranes in a 0.1 molar, symmetrical, univalent solution is unlikely to be greater than 1 kJ mo1-1 (cf. Eqn. 54); this is below thermal energy and thus seldom significant. But the surface charge density of the artificial lipid bilayer
* F o r f o o t n o t e see n e x t p a g e .
341 membranes may exceed 0.25 C . m -2 ( = e / 0 . 7 nm2). The electrostatic free energy of such systems, correspondingly, could in principle be high, < 10 kJ mol- 1 (cf. Eqn. 56 and Fig. 24). I believe, however, that in reality the coulombic free energy of most membranes is only small, below 2 kJ mo1-1, owing to the interracial structure effects. Standard electrostatic membrane theory neither accounts for the interracial structure nor, and more importantly, does it allow for the membrane hydration effects. Especially the latter handicap may have far reaching consequences owing to the high value of the free energy of the membrane hydration. This is best seen if the integral free energy of a charged and hydrated membrane surface is written as a sum of the individual coulombic and hydration contributions (cf. Eqn. 57 in Appendix G) in analogy with the well-known result of classical electrostatics for coulombic free energy (Eqn. 53) [141,20]. For a given net surface charge density the membrane hydration causes the electrostatic part of the free energy to increase relative to the corresponding Gouy-Chapman result, the latter pertaining to a 'nonhydrated membrane'. This is because it includes the positive, 'coulombic' part of the total free energy of hydration, reminiscent of the increase of the electrostatic surface potential by the surface hydration. Correspondingly, it is the negative, 'chemical', contribution which minimizes the total free energy of the membrane-electrolyte system and guarantees the stability of hydrated membranes. This 'chemical' contribution, which is supplied
* If the electrostatic surface potential and not the net charge density at the membrane surface is fixed, which is rather improbable for the biological systems, a 'chemical part' of the double layer energy must also be included, which diminishes the total value somewhat. The reason for the difference between the cases of constant surface charge density and constant surface potential is the following [4]. If in the final state of equilibrium, an ionic excess of one species has been created in the double layer, there is obviously a chemical preference of these ions for the surface. During the transfer of ions from the bulk to the interfacial region there is, therefore, a gain in free energy, corresponding to the chemical free energy difference between the bulk and the bilayer surface. In the final state this free energy difference exactly compensates the electric potential difference due to the double layer and hence is equal to other part of the free energy change is associated with the build-up of the double layer. During the (hypothetical) charging process, the bilayer surface potential gradually increases from zero to the final potential value, ~k0, and the surface charge density correspondingly rises from zero to oct. The potential opposes ion transport from the bulk solution, where the electrical potential is zero, to the bilayer surface and hence the energy consumed in the whole charging process is that given by Eqn. 53. This may be called the electrical part of the free energy of the double layer in contrast to the chemical part, -
NAAcoetd/o.The
N.tA~oet~o.
40 c=0.1 M
.
s s
, ,,, "
o
E
~
-40 ( = 0.15 nm
0
0.1
%%% % %
0.2
"~.
0.3
Fig. 24. Electrostatic free energy (dashed curve) and hydration free energy (full curve) of a membrane as a function of the net surface charge density, or of the surface density of the local excess charge, ap. Curves were calculated for dilute electrolyte solution (c =10 -4 mol litre-1 ), where charge screening is essentially negligible, by means of Gouy-Chapman approximation and nonlocal electrostatic model, respectively.
act,
by the energy gain from the direct water binding, often is by more than one order of magnitude greater than the coulombic part. For phosphatidylcholines, for example, the absolute magnitudes of both hydration free energy components are > 10 and < 1 kJ tool -1, respectively. The nonlinear version of Eqn. 10 [122] and experiments [122,123] show this in detail. A feeling for the system characteristics is provided, however, already by the approximate linear expression for the total free energy of a hydrated charged membrane:
Gtot=NAAc{+ ~a2h[ 1 +
~(~-1)]
-t- c02` -~, (1- ~-~))
(10)
where the positive and negative signs refer to the cases of fixed surface charge density and potential, respectively. More general result is given in Appendix G. Eqn. 10 is valid for dilute electrolyte solutions within the framework of nonlocal electrostatics in the linear approximation (see Eqns. 37, 38 and 57). The underlaying approach is thus not exact and may result in 50% misjudgements. But it has the advantage that it permits estimates to be made about the effects of the ion screening on the total membrane free energy with the hydration being included.
342 VII-B. Electrostatic intermembrane pressure
Electrostatic intermembrane forces can be repulsive or attractive, depending on the interfacial separation, on the surface hydration, or on the sign and distribution of the surface charges. Electrostatic repulsion occurs between adjacent membranes because there is an excess of counterions in the region of overlapping double layers (Fig. 25). These ions are not at equilibrium. Therefore, a local excess osmotic pressure is set up between the membranes which drives solvent molecules between the interacting surfaces and tends to push the membranes apart. Standard coulombic repulsion between the charged surfaces is thus driven by the entropy of counterion distribution. The electrostatic repulsive force decays approximately exponentially on the length scale of DebyeHtickel length (see Appendix H). At separations between 2 and 5 nm this force for the model membranes is the prevailing repulsion (cf. Table III and [304]). Membranes with a thick, (partly) ionized surface-coat [303] or flexible membranes [305-307] may, however, exhibit a strong electrostatic repulsion at even greater separations, because the range of the electrostatic force is extended by the effects of the interracial structure and
I
hydration
~ 6
Ac [nm2]
k
0.7
~II.KX~I el. repuls,on
~,.0 A 7.0 o
5 I --__1__// , L|
0
t
4.0
,
I
8.0 dw[nrn]
I
I
12.0
I
I
16.0
Fig. 25. Electrostatic repulsive pressure between the lipid bilayer membranes of increasing surface charge density and decreasing area per charge (oct = e/A¢). Data stem from the osmotic pressure measurements of Rand and Parsegian (From Ref. 179).
fluctuations. In any case, suspensions of charged membranes attain a gel-like appearance, owing to the electrostatic coupling between the membranes, when the sah concentration is sufficiently low and when the concentration of the membrane vesicles is high enough. This effect, and the electrostatic repulsion in general, are more pronounced for the fluid membranes because of the dynamic, fluctuation-mediated force-enhancement. On the one hand, cations bound to the negatively charged membranes-predominantly to the lipid and to the glycolipid molecules-may diminish or nearly alleviate the consequences of the net membrane charge. Charged lipid vesicles therefore often aggregate and fuse after the addition of salts (for a review see Ref. 234). Increasing ion valency and surface charge density enhance such phenomena [383]. In complex electrolyte solutions the effects of the ion binding, especially for the polyvalent electrolytes, as well as the consequences of the ion-screening are simultaneously operative [308,309]. On the other hand, the ion binding to initially neutral membranes imposes a net electrostatic charge on the surface [179]. Consequently, even such membranes which are initialy per se uncharged may experience an electrostatic repulsion. For example, bound calcium or bound halide ions bring a net charge on the zwitterionic, such as phosphatidylcholine, membranes [310,311,175]. This creates a positive electrostatic membrane potential of 30-65 mV [179] in the millimolar or up to decimilar concentration range of this ion. Calcium and other bound ions thus cause appreciable electrostatic intermembrane repulsion in the appropriate concentration range. Only in relatively highly concentrated solutions of polyvalent ions such repulsion is screenedout or even replaced by a net, ion-dependent intermembrane attraction. Electrostatic attraction between membranes is a manifestation of van der Waals force [312] and of the higher-order electrostatic intermembrane correlations [20,313-319]. The latter reduce the repulsive entropic term and, moreover, give rise to an extra attractive energetic contribution. Membrane hydration decreases the former, close to the membrane by as much as 80% [320]; but it increases, which is more important, the latter contribution by more than one order of magnitude [321]. Electrostatic attraction arises, for example, between the membranes which attract dissolved ions as soon as the ionic distributions near the interacting surfaces are concerted. The resulting force depends sensitively on the interionic correlation function [313,315] but also on the dielectric properties of the interfacial region. The force magnitude, on the one hand, increases with the increasing density of the charges in the interfacial region and with the width of this region [69] (Fig. 26).
343 ,
,
7 -~~x~x ~ .~ ,,q-
Z
,
!
Ac=0/'*nm2 ~..
E 6-
"
-
"
-
4
0
t
0.5
I
I
1:0 1.5 dw [ nm]
2.0
Fig. 26. (Electrostatic) Attraction between two charged membranes caused by correlated distributions of interfacially compensated structural (full curve) or the ionic charges close to the membrane surface (dashed-and-dotted) in a 1.0 M monovalent electrolyte. The nominal surface charge densities are 0.4 and 0.23 C.m -2, respectively. The dashed lines refer to standard van der Waals force. (Modified from Refs. 61 and 316.)
Increasing salt concentration * and ion valency also enhances the intermembrane electrostatic attraction. Intermembrane attraction from the interion-correlations, moreover, is greater for the lower values of the dielectric constant. Such attraction, therefore, is relatively more important for the closdy apposed membranes since the dielectric constant in the interjacent region is then lower. This enhances the ion-correlation effects. Ultimately, the attractive component of the intermembrane force can be sufficiently strong to result in a net attraction even for equally charged membranes (cf. Fig. 26). (In other words: one can not squeeze out the solution if the two surfaces are not present simultaneously owing to the fact that the correlation effects cause the charge between membranes to be less than the charge on each separated membrane surface.) Numerical simulations [41,61,324-326] and analytical theories [69,327] prove this. A related 'electrostatic' attraction may arise between uncharged membranes immersed in water [313,330,20, 318,321]. A prerequisite for this is, that the distributions of the local atomic charges and/or of the dipoles on the interacting surfaces are correlated. I speculate that one of the reasons for the stronger attraction between the
* The reason for this is the following. Interion correlations near the membrane-solution interface cause the counterions to concentrate more towards the membrane, thus reducing the electrostatic repulsion between their associated electric double layers. Furthermore, correlated fluctuations in the ion clouds at the surfaces lead to an attraction of true van der Waals type. This type of interbilayer attraction is favoured in the 'strong-coupling' limit, i.e., with high surface charge density, low intedamellar dielectric permittivity, low temperature and with polyvalent ions [61].
artificial lipid membranes in the gel or crystalline phase is a correlation-dependent force of such an attractive type; excessive 'van der Waals' attraction between phosphatidylethanolamine bilayers might also have similar roots, as concluded from the fact that phosphatidylethanolamine headgroups have an intrinsic tendency for the formation of ribbon-like, interfacially correlated headgroup patterns [116]. Electrostatic attraction stemming from the intercharge correlations can exceed normal van der Waals attraction by more than one order of magnitude, at least when the interacting membranes are uniform and sufficiently close [61,316,321,324,322]. This attraction should be especially strong when the system contains multivalent or polyvalent ions. The correlations dependent force can thus be postulated to be one of the potential origins for the diminished lipid swelling in the solutions of divalent electrolytes, as compared to the monovalent electrolytes [217,179,70]. But unfortunately to date no simple analytic model of the correlations-dependent attraction has been devised. * For practical applications the attractive intermembrane electrostatic force, consequently, must be evaluated numerically from the detailed electrostatic surface models [69,328,321,331] or by using phenomenological expansion series [113]. Depending on whether the charges, which are the cause of anomalous attraction, are membrane-associated or ionic the correlationdependent force is expected to have a range of either less than 1.0 nm, or in excess of 1.5 nm, respectively [61,69]. At short separations from the surface such force is likely to be always appreciable. Consequently, the attractive force contribution should not be forgotten in the data analysis, especially when the results pertain to the specimen containing divalent and polyvalent ions. Electrostatic attraction between charged membranes has been measured directly for phosphatidylglycerol membranes [329]. Perhaps it is also the origin of the experimentally observed long range ' hydrophobic' interactions [335,334]. In sufficiently strong external electric fields the lipid vesicles, the membrane particles, and cells tend to aggregate and form long chains along the field direction (see, for example, [336,337]). This is probably caused by the induction of dipole moments on the (polyelectrolyte-like) membrane surface rather than being a consequence of the non-trivial intermembrane attraction. Such pearl-chain formation, consequently, is more pronounced in less concentrated electrolyte solutions. Field-induced membrane aggregation and fusion can be
* An exception is the 'counterions-only' model [332,333] in which the importance of correlations, and thus of the corresponding attraction between the charged surfaces, has been shown to be measured is by a dimensionless parameter oet(Ze)5/[dw(kTc%)3].
344 used for hybridoma production [338]; they are more efficient for the less charged cells because such cells repel each other relatively weakly. For the same reason electro-fusion efficiency varies with the cell cycle, the stage of cell differentiation, metabolic cell activity, etc. VII-C. Electrostatic lateral pressure
Surface density of the membrane free energy, or, strictly speaking, the derivative of this energy with regard to the surface (or molecular) area, gives an estimate of the lateral pressure within the membrane plane [301]. Within the framework of Gouy-Chapman theory this pressure is invariably found to be identical to twice the negative surface density of the internal energy, with an upper limit Ilet = -F~el, 0 in the high potential case [301]. The actual value of such pressure is probably < 18 mN • m -2, and thus less than the corresponding lateral pressure of hydration, 40-60 mN m-2 [143]. The main reason for this is the relatively slow decay of the electrostatic interactions compared to hydration. This causes the coulombic ('macroscopic electrostatic') interactions to prevail in the intermembrane interactions at large separations ( > 3 nm) but makes hydration ('quantum-mechanical electrostatics') the dominant factor of the short-range intermolecular forces. This explains why the lateral packing density and the molecular area of charged and uncharged lipids in the fluid phase are approx, the same, 0.75 < A,, < 0.85 nm 2 [339], as long as lipids are comparably hydrated. The curvature dependence of the electrostatic interaction, which has been proposed to determine the curvature elasticity of the fluid lipid bilayers [340], for the same reason is likely to be predominantly a function of the membrane surface structure and hydration [341]. VII-D. Transmembrane pressure
Transmembrane electric fields impose an electrostatic pressure across the hydrophobic membrane interior. High field strengths in excess of (1-3). 106 V. cm-1, which corresponds to the transmembrane potential jumps of 0.2-1 V, may dramatically diminish the membrane thickness and resistance. A high frequency field or a field pulse of sufficient strength may induce electric membrane breakdown within 10 -8 s. This rapid process is probably a manifestation of the field-induced changes in the lipid conformation and is less probable for the membranes stabilized with sterols [342,343]. Moreover, it reflects a concomitant local membrane expansion and defect formation. But the initial, reversible breakdown often leads to a mechanical breakdown during which a membrane ruptures. (In the case of cholesterol-free lipid bilayers the latter breakdown occurs within a period of 10-3 s or
longer [344,345].) The irreversible breakdown is an outcome of the membrane electrostriction and is theoretically discussed in Appendix I. Another possible source of the transmembrane pressure gradients is the membrane asymmetry. This is created and supported either by the asymmetric concentration or interactions between the transbilayer components and cell constituents or else, in the artificial systems, arises from the non-ideal mixing in the membrane. Membranes of certain small sonicated vesicles are asymmetric because of the latter reason [346]. The asymmetry of such unilamellar vesicles (r,, _< 30 nm) made of uncharged phospholipids phosphatidylcholine and phosphatidylethanolamine consequently decreases with increasing ion concentration [347]. Electrostatically triggered lateral phase separation or compartmentalization thus could be of relevance for the membrane structure and function, but this as yet has to be demonstrated experimentally. It has been hypothesized, furthermore, that the membrane asymmetry and transmembrane electrostatic potential participate in the processes of vesicle shedding, pinocytosis, and ameboid locomotion [348]. Asymmetric model membranes can be prepared [349,350] with the aid of nonspecific phospholipid exchange proteins [351] which catalize the lipid exchange. Excessive vesicle fragmentation is useful when lipids are sufficiently different. Asymmetric membranes are needed, for example, for the studies of the electrostatic transmembrane coupling and protein insertion. In the investigations involving proteins care must be taken to prepare low-defect vesicles in order to minimize the non-specific macromolecular adsorption. In thermodynamic terms each membrane and its bathing electrolyte are an inseparable system. Changing the properties of any of the system subcomponents therefore inevitably influences the characteristics of all the others. Especially the modifications in the region of the interfacially bound water are thermodynamically significant, owing to the fact that the hydration free energy is typically nearly one order of magnitude greater than the corresponding coulombic contribution, being 1 and < 10 kJ tool- 1, respectively. The intermembrane pressure, which is identical to the spatial derivative of the membrane-free energy, or the lateral membrane pressure, which corresponds to the free energy derivative with regard to the surface area, are both sensitive to this. In the membrane vicinity the repulsion and the lateral pressure of hydration are normally several orders of magnitude stronger than the corresponding coulombic contribution. But at separations greater than 2 - 3 nm the coulombic repulsion prevails over the hydration force, because the range of the former is normally larger than the reach of the former. When the electrostatic intermernbrane interaction is attractive owing to the charge-charge correlation effects
345 the resulting pressure in any case is stronger than standard van der Waals force, up to a distance of several nanometers.
VIII. Experimental studies Classical techniques for investigating the electrified surfaces by and large are based on the charge and potential measurements. Consequently, they mainly reveal the thermodynamic information about the solvent and the membrane surface. It is only with newer experimental techniques and models that the structure of the charged interface can be studied directly, often with a remarkable spatial and temporal resolution. Charged membranes offer an exceptionally promising and relevant experimental system not only for the biologically interested researcher but also for the electrochemists. This is chiefly owing to the fact that membranes typically provide a well defined, reproducible and easily modifiable surface which can be tailored to mimick the selected features of the living systems. Techniques for probing the electrostatic membrane properties are either direct or indirect, depending on whether or not they rely on the use of probes. In principle, direct methods are superior because they are non-perturbing. But in practice, indirect techniques frequently offer a much higher time-resolution and, sometimes, are far more sensitive. Often they are also more convenient and less costly to perform. VIII-A. Direct measurements
Direct studies of the electrostatic membrane properties by and large are based on the determination of the spatial variation of the ion concentration, its direct and indirect consequences, or on the ion transport studies (see further discussion). Alternatively, the mobility of the membrane vesicles (vesicle electrophoresis, zeta potential measurements) or the electrolyte flow past the membrane (electro-osmosis) in an electric field can be studied. For more direct, but classically low-resolution, experiments electrodes can be used directly. Ion distribution and binding studies. Visible or easily detectable ions offer the unique advantage of simultaneously acting as sensors of the electrostatic membrane properties as well as a natural system component. They thus provide direct insight into the electrostatically relevant structure of the membrane-solution interface without a need for foreign probes. Nuclear [352] and electron magnetic resonance [353,355], radioactive decay [357], and ion fluorescence upon optical [358,359], or X-ray excitation [54] can be used to make ions in the interfacial and in the bulk regions discernible, especially under the conditions of total reflection [360]. Replacement of the 'invisible' ions in the double layer by the
visible ones offers good experimental opportunities when the charged species under investigation can not be measured directly. * For the data analysis and the deduction of the electrostatic quantities of interest expression Wet(X)= Ziehbel(X ) o r related results can be used. The rationale for this is discussed in Section V.A and Appendix E. For example, if the signal pertaining to the probes at the interface and the signal stemming from the bulk are set in relation, the total potential of mean-force is found from the latter by using: Wtot( X = Xbind ) = R T
l n ( Cbulk / C ( X = Xbind ) )
(11)
Usually, in the studies of membrane electrostatics the coulombic membrane potential is segregated from the total membrane potential of mean-force. The assumption is then commonly made that such electrostatic potential is proportional to the logarithm of the relative probe concentration at the membrane surface Wel(X = Xbind ) ~
ZeJ/(x =
Xbirtd)
= ( R T / Z i ) In(Signalbulk/Signalmemb. . . . )
(12)
as is done in Gouy-Chapman approximation. Frequently this is a rather unrealistic oversimplification. Whilst it is clear that the coulombic electrostatic term is the leading term for each charged membrane other contributions should not be left out a priori. More general results, such as Eqn. 6, therefore offer a more reliable starting point for the electrostatic data analysis. The decision which terms should be included in the data analysis is neither an easy nor an unimportant one. As in any multi-parameter situation from a theoretical model it is relatively easy to find a solution. But it is far more difficult to prove that the obtained answer is the only, or at least the most plausible one. Wrong or too restrictive choice of the terms in Eqn. 6 may be fatal in this respect. In order to decide between various possibilities one must combine, therefore, the theoretical scrutiny with the time consuming, systematic experimental studies; various ions and closely spaced concentration intervals
* For example, binding of the trivalent, fluorescention Tb3+ to the plasma membrane can be used to study Ca2+ receptor sites on biological membranes [358]; electron paramagnetic resonance of manganese may be used to follow the substitution of this ion near the membrane surface by other divalent and monovalentions [353]. Deuteriated ions can be used for neutron experiments[354].
346 - - - l ---7 PA, MPA
F
7~
I
I
PS" PE" P G . P C ~'~ PS"
60
...... ~
50
""
~
~,
"'
£
k
I--
3 0,h \ \
',
pE ÷- " ,
X
',
\
PS PG
.PE ,PC*
-
',.
,,.
"\
~
I
,
~',,,
-,MPA
PA-
1.5 mol litre-~, where simple coulombic interactions are nearly completely screened out), for example, can largely compensate for the noncoulombic contribution to the total interaction potential. Approximate relation ~(x=xh.,d.,.)--(~)[ln(Signalbulk/Sig
z
hi ',.) ',/1 U.I n,"
1O0
J o/
/
J o/
o
tL
(15)
PS
=o i,i L)
5M]
therefore provides a far better basis for studying the electrostatic membrane properties than Eqn. 12.
1000
v
hal . . . . b. . . . ) l ,
--ln(Signalbulk/Signalmemb.... )l , ?
PC
I 0000
(14)
w . . ( x = xh, . d ) = R T In (Signal hulk/Signal ..... b..... )
/
10
z
1 0.001
0.01
0.1 VESICLE
0,001 CONCENTRATION
0.01
01
(mg/ml)
Fig. 30. Variation of the net fluorescence of a fluorescent probe (TNS = toluidino-naphthalene-sulphonate), as a measure of the membrane potential, for phosphatidylcholine (PC) and phosphatidylserine (PS) liposomes as a function of the vesicle concentration. TNS concentrations are 0.11, 0.22, 0.43, 0.82, 1.6, 3.5 and 7.0 #M. (From Ref. 595).
351 Calibration of the electrostatic membrane data obtained with charged probes is seldom possible without difficulties [429-435]. First of all, many probe characteristics, such as the quantum yield, strongly depend on the local properties of the binding site. The effective local polarity (dielectric constant) [436,437], or local (membrane) fluidity [437,438] play a rSle as well. The vicinity of other polarizable groups, on the proteins, say, is furthermore important. The calibration curve of cyanine and safranine dyes [439] or of diO-C6-3 [441], for example, is nonlinear at higher potential, mainly owing to the saturation effect or to the increased membrane permeability to protons [439]. Experimental conclusions are sensitive to the probe concentration. This concentration, consequently, can not be varied widely without side effects. Moreover, only at less than 5% molar concentration the commonly used optical potential probes dissolve ideally in the model lipid bilayers. At higher concentrations probe aggregation and m u t u a l i n t e r a c t i o n - b r o a d e n i n g or fluorescence-quenching become a problem. Lipophilic membrane probes frequently disturb the membrane quite strongly. They cause, among other things, marked changes in the lipid headgroup configuration and sometimes induce lipid vesicle fusion [432,433]. When interpreting the changes in the probe properties care must be exercised not to explain erroneously the consequences of the fluorescent dye destruction after (partial) lipid peroxidation, in terms of the potential changes [439]. In microscopic optical experiments, the analysis of the measured fluorescence intensities must be performed only after correction for the light collected from the outside plane of focus as well as for the non-potentiometric binding of the dye [434]. In biological systems, additional pitfalls open up. In the worst case, the potential sensing probes, carbocyanines in particular, monitor the potential of mitochondrial and not of plasma membranes [435]. More difficulties could be quoted. Optical Probes. Optically detectable probes include the safranine dyes [442], the fluorescent derivatives of anionic oxanol [443,A.~], the positively charged carbocyanines [435,445], the merocyanine containing fluorophores (see especially reviews [424] and [421]) and the more recently developed zwitterionic or positively charged styryl probes [446,447]. The mechanism and the rate of response, accumulation in cells, toxicity, potential sensitivity, etc, of all these probes vary widely. Nevertheless, such dyes have proved capable of functioning as optical indicators of charge separation on the biological (such as mitochondrial and related respiring preparations) erythrocytes, excitable squid tissues, heart tissue, cerebral cortex, as well as model membranes (see, for example, Ref. 433 for a brief survey). Certain fluorescent potential probes redistribute depending on the electrostatic membrane properties and also exhibit elec-
trochromic effect. Normally, however, just one of the two mechanism is exploited for the actual potential determination. The main strength of the fluorescent membrane probes are the kinetic potential studies. Their performance for the absolute membrane potential determination is much worse and requires cumbersome calibration protocols [423]. Carbocyanines and merocyanines, as well as the less toxic oxanols, offer a relatively low time resolution in the order of up to seconds. Their sensitivity to the membrane electric field is high as they mainly respond to the latter by a redistribution across the membrane. Cyanine dyes orient parallel to the hydrocarbon lipid chains as monomers and lay parallel to the surface if dimerized. Merocyanines exhibit a somewhat faster time response and have non-fluorescent dimers. This must be borne in mind, in particular, when measurements are performed with dyes at high concentrations. Oxanol dyes tend to aggregate less than the cyanine potential probes. They repartition in the membrane field and, moreover, respond to the latter by an intramolecular electronic redistribution which causes an electrochromic wavelength shift. The styryl probes apparently function by charge-shift mechanism. The aminostyryl dyes normally have their transition moments perpendicular to the membrane surface and thus are complementary to the class of long-chain cyanine dyes [448]. The response of styryl dyes to the changing membrane potential is fast, on the millisecond time scale. This makes such probes suitable for the imaging of rapid electrical phenomena in cells and on the membranes. Styryl dyes usually are non-fluorescent unless bound to membranes but have the disadvantage that their fluorescence changes upon field variation is relatively little, usually less than 5-10%. If suitable fluorescent probes are chosen, in principle at least, one can study both the overall membrane electrostatic properties as well as the space- and timedistribution of the electric activity on the membrane surface [449]. Moreover, the use of potential sensitive probes to estimate positive and negative electric transmembrane currents, may provide a methodology for monitoring changes in the membrane 'dipole potential' in vesicle systems [450]. But the analytic tools for extracting the electrostatically relevant information from measured data remain to be developed. A fluorescently labelled derivative of phosphoniumion has been reported to monitor the electrostatic potential deep in the interfacial region where no screening occurs [451]. It has been suggested that second harmonics signals could be obtained from living cells stained with fluorescent electrostatic potential probes [452]. Electrochromic effect. With the potential sensitive probes it is important to know where the fluorophores
352 are located. They all sense long-range electric fields, in which they redistribute. But the interfacial, intramembrane, or other local fields or gradients may dramatically change the probe properties and signal. The location of charge-shift electrochromic extrinsic probes in a bilayer membrane is particularly critical, since these indicators can respond only to that portion of the electrostatic potential difference which falls across the molecular dimension of the probe. In an electrochromic mechanism, molecules undergo an electronic redistribution upon excitation, and if the direction of the charge shift is parallel to the membrane field the energy of the electronic transition is altered. Because of the high anisotropy of many interfacial properties, such as the effective relative permittivity (dielectric constant, micropolarity), this may cause severe experimental problems. But it also ensures that the reaction is fast enough to follow essentially all physiologically relevant processes, the only movement upon excitation being that of an electron. * Electrochromic effect causes a shift of the fluorophore emisSion maximum. This shift, on the one hand, depends on all system characteristics which influence the polarizability of the valence electrons; variations in the solvent structure or changing electric fields of the membrane belong to these. On the other hand, it is decisive for the probe response whether or not the probe molecules participate in the intermolecular interactions. Changes in the protonation state of the probe, dimerization, or the formation of ion pairs are of paramount importance. Despite the complexity of phenomena underlying electrochromism, electrochromic probes are a valuable tool for the electrostatic membrane studies. With suitable calibration procedures they can provide simultaneous, albeit only semi-quantitative estimates of the effective dielectric constant at the binding site as well as of the electrostatic potential near the bound fluorophore [104,453-457]. Paramagnetic probes. An attractive technique for directly studying the electrostatic properties of strongly light absorbing, light scattering, and light sensitive specimen, including photo-active membranes, is to use electron spin resonance and paramagnetic ionic probes. Suitable natural paramagnetic probes are, for example, the common divalent ions such as manganese, cobalt, nickel, and copper. Synthetic probes have been devised, moreover, with attached paramagnetic spin-labels [459,418,460,297,461]. The advantage of the latter is that they can be tailored for a given application. Their disadvantage is the bulkiness of the paramagnetic group. * In practice, secondary, non-electrochromic,motion-dependent response mechanism may obscure the situation. In the best case, therefore, the sensitivityof such a method can reach 10%/100 mV I458].
20 13
T > E
). 0 ~0
20
40
60
Arginine addition (rnM)
80
20
b
> fo
O~
-~
. . . . . . . . . . . . . . . . . . . . .
I
I
I
I
40 60 80 NH4OAc oddition (raM) Fig. 31. Change of the membrane surface potential as a function of the bulk arginine concentration measured by the relaxationkinetics of a spin-label. Circles: outer surface potential changes; triangles: inner surface. All data were estimated for the phosphatidylcholinevesicles containing 8 mol% of phosphatidic acid. (From Ref. 461). 20
In typical electron spin resonance studies of the electrostatic membrane properties the distribution of paramagnetic [353] or spin-labelled ions [460], or of charged spin-labelled amphiphiles [459,418,375,451,65, 461,450,26] between the aqueous and the membrane subphase is followed. (Fig. 31). As long as the probe concentration is sufficiently low, so as not to change appreciably the membrane properties, the electrostatic membrane properties are determinable from Eqns. 14 or 15. Paramagnetic amphiphilic probes have been used to confirm that the discreteness of structural surface charge o n lipid bilayers does not significantly affect the electrostatic membrane potential [65]. Relaxation data recorded following the rapid mixing of a cationic spin probe with tetraphenylborate containing vesicles afforded simultaneous information about the electrostatic potentials at the outer and inner vesicles surfaces [461]. Nuclear magnetic resonance. This technique shares many of the nice features with the electron spin resonance method. Its additional advantage is that it does not depend on the use of perturbing labels; it may suffer, however, from the low natural abundance of the
353 detectable atomic species. With nuclear magnetic resonance, therefore, especially the model membranes which can easily be enriched with the latter, are frequently investigated. By and large the deuterium [466] or the narrow 31p signals from the sonicated vesicles and its variation with the concentration of shift reagents, such as paramagnetic ions, are measured [176,172,426,192]. The conclusions about the (electrostatic) interactions between the charged phospholipid headgroups and the ions are obtained after the data analysis, as in the case of other ionic labels, in terms of simplified relations such as Eqn. 14. Phosphorus resonance can be used also for the determination of the transmembrane Nernst diffusion potential [462]. From the 2H-nuclear magnetic resonance spectra of the molecules, which are specifically deuterated in the polar region, the molecular orientation can be apprehended. From this information conclusions can be drawn about the interracial membrane properties and, to a limited extent, about the membrane electrostatics. Examples with simple lipid [463,464,175,465, 466,419] or more complex [468,469] membranes are available. With nuclear magnetic resonance the main site of the phospholipid-ion association, the phosphate group, and its dependence on the electrolyte composition can be directly visualized. But other nuclei, such as 13C or 170 also are attractive and suitable for the electrostatic membrane NMR-studies, provided that a clear assignment of one part of the signal to the binding site or to the interfacial region is feasible. The latter goal is most readily achieved by measuring the specific resonances of various single ion nuclei, such as lithium [265,470], sodium or potassium [471,470,472], nitrogen [473], or others [474,475,232]. The main difficulty arises from masking of the bound-ion resonance peak by the dissolved ions; but this is a general trouble. Multi-nuclear resonance [476] under optimal conditions can thus unveil the distances between the membrane-bound monovalent and divalent cations. Owing to the sensitivity of the spin-lattice relaxation rate of the former on the distance between this ion and a divalent paramagnetic ion, such as Mn2÷, the relative separation between the binding sites is reflected in the nuclear magnetic resonance signals [352]. Seldom used, but potentially very powerful, for the studies of the membrane electrostatics is the nuclear Overhauser effect spectroscopy [477,478,479]. A recent such study with egg lecithin has revealed, for example, that hydrophobic ions (such as tetraphenylborate and tetraphenylphosphonium) at low concentrations partition preferentially in the depth of the hydrocarbon membrane core; conversely, at higher ion/lipid ratios specific interaction of tetrapehenylborate with the choline headgroup and ion insertion in the interfacial region occurs [4801.
Probe transport. Ion transport across a membrane depends on the transmembrane ion concentration and electric field gradients (cf. Eqn. 52). But it is also sensitive to the interfacial membrane potential which affects the interracial ion concentration and the on-off rate for the ion permeation (cf. Eqn. 51). Electrostatic membrane properties, therefore, can be studied by measuring the transmembrane flow of charged probes and by analyzing this in terms of Eqn. 52 or some other suitably generalized result. In membrane transport studies hydrophobic or amphiphilic ions are usually given preference over the hydrophilic ones, owing to their higher permeability. Such high permeability is mainly a result of the large effective ionic radius of the hydrophobic ions -and either is possible because the charge is surrounded by hydrophobic groups or because this charge is delocalized. This causes such ions to experience an electrostatic membrane barrier which is much lower than that felt by the smaller ions (cf. Eqn. 48). The difficulty with the hydrophobic ions is, however, that for the same reasons they are more susceptible to the non-trivial electrostatic or non-electrostatic effects. Ion polarization, direct binding, and other non-coulombic effects may lead to an undesired association, predominantly in the interracial region, and make quantitative data analysis difficult or impossible. For example, the suitability of the frequently used tetraphenyl phosphonium or triphenylmethylphosphonium ions as probes of transmembrane potential has been questioned [481] because of the binding of this ion to the membrane surface. Such binding mimics and masks the expected electrostatic phenomena. Results obtained with a series of organic ionic potential probes were reported to be in conflict with the corresponding data acquired with the charged fluorophores, the transmembrane potential values for the latter being in a rule higher [482]. With the organic cations as probes for the transmembrane potential of tonoplasts (vacuoles from higher plant cells) the potential value was found to be negative [483] whereas from the microelectrode studies it was concluded to be positive [484,485]. Most probably, these discrepancies are due to the binding of the lipophilic probes at the membrane-solution interface [486,487] and the resulting change of the local and overall electrostatic membrane characteristics. The teohniques used for monitoring the amount of transferred ions are similar to those used for the direct determination of ion concentration in general. Ion selective electrodes [274] or fluorescence measurements [488,489,490] are most popular. Radioisotopes. Owing to the low spatial resolution of the radiological techniques, radioactive ionic isotopes are of limited value for the investigations of the membrane electrostatics. Corresponding studies, to date, mainly deal with the binding or transport of simple
354 ions. They are either performed with the surface films and fl-sensitive detectors or with different versions of the dialysis technique with appropriate detection [357]. Other techniques. Uncommon, but potentially useful, techniques for studying the membrane surface electrostatics are (Fourier transform) infrared [491] and (surface enhanced) Raman spectroscopy [492-494]. Detection of the plasmon surface polaritons [495] is appealing as well. With any of these techniques the conclusions about the membrane electrostatic, in principle, can be deduced from the informations about the intramolecular conformational changes. Related but less direct are the second harmonic generation [496] and photo-acoustic spectroscopy [497]. Quartz microbalance technique [498] can directly monitor the mass of the interfacially adsorbed material. Lateral interfacial electrical conductivity is a sign, primarily, of the interfacial hydrogen bonding networks [271,499,268]. X-ray and neutron diffractometry [500,354] or spectroscopy, either in the normal mode [501], under the conditions of total external reflection (also in the combination with fluorescence detection [54]), or in the anomalous dispersion mode [502] can all potentially provide direct information about the ion distribution functions in the system. I trust that the scanning 'tunnel' microscope based on the ion conduction a n d / o r molecular force microscope in future will prove useful for monitoring the ion distribution near and on the surface of charged membranes. Last but not least, it should be realized that studies of the membrane electrostatics can not be conducted on their own. Different membrane properties are too intimately connected to allow separation of one aspect from the others. Detailed investigations of the membrane electrostatics should in my view be performed with keeping both the surface ionization and hydration in mind. The effects of membrane structure and polymorphism should also not be forgotten. Each of the techniques mentioned in this section, therefore, must be taken as a part of the experimental stream which leads us towards the ultimate membrane comprehension. A variety of experimental techniques can be used to probe the electrostatic membrane characteristics. Many are based on titrating the properties of intrinsic membrane components or of the whole membranes. Determinations of the molecular conformation, phase transition temperature, intermembrane force, mobility in external field, are a few examples. Experiments which rely on the use of (fluorescent, electron, nuclear, etc.) probes in the majority of cases are devised so as to monitor the adsorption or binding of these, suitably labelled, ionic substances to the membrane surface. Studying the probe transfer through the membrane can also be enlightening. In any case, the conclusions drawn from such studies are most relevant if the coulombic, as well as the non-trivial electrostatic effects are accounted for in the data analysis. Often, the accuracy can be increased by analyzing the differences
obtained after subtracting complementary sets of data measured under the conditions, which favor or disfavor certain effects, rather than by considering the original data. New methods for studying membrane electrostatics, such as surface-enhanced spectroscopy, microbalance methods, anomalous neutron and X-ray dispersion, etc. are being introduced These provide unprecedented spatial and occasionally temporal, resolution; some of them, moreover, can be tailored for optimal suitability for the scientific question assessed
IX. Biological significance Let me begin with a few 'macroscopic' examples, which are relatively remote to the topics discussed in this contribution, but demonstrate nicely the scope and wealth of the electrostatic phenomena in the biological systems. Electrostatic membrane potential changes in response to various odorants [503,504] and tasty substances [505]. Electric fields of the order of 4 V • cm 1 evolve in wounds [506] and electric field gradients along the cell may be expected to occur in moving or interacting cells, or cells in the external concentration or field gradients. Such electric fields may orient membranes or their parts [507] and induce the redistribution of charged cell surface components [508]. In consequence, membrane potential may change over minutes through to hours [509]. A spatially dependent change in the plasma membrane potential under the influence of such a field has been demonstrated for cells in an external, uniform electric field [510,387,434] and has been shown always to tend to oppose the change , l the field-induced transmembrane potential. Model calculations indicate that sodium pumps may produce a longitudinal electric field in the lateral intercellular spaces of renal proximal tubules, which should cause electromotive fluid flow in these tubules [511]. Longitudinal electric fields along neurons, moreover, can cause a flow of the charged ion channels along the membrane surface and might lead to the formation of distinct patterns on the cell-surface [512]. Net flow of charged intrinsic membrane proteins on other cells [509] as well as protein crowding on the plasma membrane [513] were already seen to occur in the external electric fields. Cells rotate in an outside revolving electric field (see, for example, ref. [389,514-516]) because of the different conductivities of the inner (e.g., cytoplasmatic) and outer (e.g., extracellular) compartments. Owing to the dependence of these conductivities on the ion concentration, membrane electrostatics can participate in this phenomenon indirectly. A direct contribution of the membrane potential to the interfacial instability at cell membranes has been
355 postulated to exist [517]. This, together with an increased capacity for the membrane aggregation, provides one possible explanation for the biological activity of certain ionic agents. For example, cations which bind hydrophobically to the lipid bilayers and thus increase the membrane charge necessitate a higher concentration of calcium for the induction of exocytosis of plasma membranes from sea urchin egg [518]. Surface potential or surface charge density also control the movement of synaptic vesicles inside the nerve cells, the vesicle-vesicle, as well as the vesicle-presinaptic membrane interactions [519]. The mechanism of regulation in all these cases probably is based on the variation of the interracial counterion and proton concentrations [520]. The electrostatic charge distribution is prone to play some rrle in proteins [521]. In a theoretical description of this problem it is important to allow for the low dielectric constant inside proteins, electrolyte screening effects and irregular protein surface topography [523,522]. But it is also essential to realize, that in certain cases, when specific effects are involved, such as ion transfer, exchange of a single amino acid residue can have dramatic consequences [524,525] and even may abolish the biological function. Under the influence of a trans-positive membrane potential many integral membrane proteins (such as glycophorin, the negatively charged asialoglycoprotein receptor, other cell coat proteins, immunoglobulins, and polypeptides, etc.) [526] appear to change their disposition with respect to the membrane. This implies that the electrostatic force can influence the location and perhaps the orientation of the integral membrane proteins. Thus, asymmetry in surface charge, which arises from the ionic or bilayer asymmetry, can be inferred to enhance the bias for correct protein orientation, at least with respect to the plasma membrane [527,528]. Moreover, the transmembrane electric fields may affect the protein conformation and folding in the membrane. For example, action of toxins has been reported to be potential-dependent [529]; colicin A, one of the E1 group types of bacterial toxins, can adopt two different conformation states within lipid bilayers depending on the bulk pH [530]; membrane-bound water soluble polypeptides, such as insulin, can trigger membrane fusion after a pH-drop. Many more examples could be given. Most of them have in common that an external electric field, irrespective of its.direction, decreases the a-helical fraction and increases the random fraction of the protein. This may be due to the unfolding of edges of helices as they submerge into the polar environment under the influence of the field [508]. (One practical aspect of the coupling between the electric field and membrane proteins is that protein-rich membrane fragments can be oriented in the external electric field [531].) Both charged peptide cationic groups and charged phospholipids, most notably phosphatidylglycerol, are
involved in the process of insertion and transport of the newly synthesized proteins across the inner bacterial membrane [532-534]. Binding of certain (cytoskeletal) proteins to the membrane surface seems to involve the charge-charge interactions as well [535]. Perhaps this observation has similar roots as the finding that upon transferring the macrophage-tumour cells from suspension onto a substratum the membrane potential increases from - 1 5 to - 7 0 mV over a period of 6-8 h [434]. Electrochemical phenomena, such as the variations in the cytoplasmic ion concentrations, sometimes are involved in the final steps of the cascade of activation of cells. As in the case of neurons, for example, the intracellular pH influences the resting membrane potential of isolated rat hepatocytes [536]. Intracellular alkalinization causes hyperpolarization and decreases the membrane resistance. Intracellular acidification causes just the opposite. Both these effects are probably a consequence of the change in the K+-channel conformation which is likely to depend on the variations of the overall electrostatic membrane properties. Physiological significance of the membrane electrostatics then can explain why the negative surface charge is a conserved feature of various Na-channel subtypes in the dog [537]; it also governs the binding of tetrodotoxin to sodium channels. Electric charges have been shown to participate in the binding of charged, predominantly anionic drugs [538] to a membrane. Certain second messengers, most notably calcium ions and phosphoinosites, are also charged. Electrostatic interactions, therefore, may play a rrle in their biological activities. The extracellular calcium concentration on excitable membranes, for example, is stabilized by such interactions [27]. Replacement of a specific arginine by a lysine, which reduces the local positive charge density on a molecule, increases the affinity constant for serine phosphorylation by phosphokinase C 20-fold. Addition of basic residues on either the N-terminus or on the COOH-terminal side of the phosphorylation site of the glycogen synthase peptide improves the kinetics of peptide phosphorylation [524,525]. Such effects are mimicked also by albumin, probably owing to its anionic character [539,540]. Anionic lipids are also beneficial in this respect [541]. A small net positive charge plays a role also as the targeting marker during the transport of proteins into thylakoids. This is recognized by the negative external membrane surface where the pool of proteins is located. Similar, (related?) processes probably exist for the membranes of the endoplasmatic reticulum [542] and for the export of bacterial proteins into the periplasmatic space and into the outer membrane [532,533]. Enzyme activity is affected by the interracial properties (such as diffusion resistance, local ion, or proton
356 concentration, etc.) for the membrane-bound [543,241, 544,545,281,102,546] as well as for the water-soluble proteins [547]. Electron transfer through membranes plays a central rSle in the energy transduction and utilization, in the cristae membranes of the mitochondria or in the thylakoid membranes of the chloroplasts, for example. It has been hypothesized that proton movement along the surface of such membranes, this is, in the region where the H + concentration is controlled by the surface electrostatic potential, is required for the coupling between the energy source and the ATP-synthase system [548]. By all means the interfacial charge is significant for the energy transduction [549,550,551] as well as for the protein binding [552]. Moreover, hormones, at least those which bind to the phospholipid (black) membranes, such as thyroxine and triiodothyronine [553], can strongly affect the transport of the hydrophilic ions across a bilayer. Electrostatic membrane potential and interracial polarity effects are influential in all these cases. It remains to be seen how successful this knowledge can be translated into the bio-technological and technological applications. Progress in the field of bio-sensors, for example, is encouraging [277]. In my view the membrane hydration is relevant for many, if not most, processes occurring near or at the membrane/solution interface. These include, among other things, ion binding and transport, (bio)chemical process at the interface, control of the membrane structure and phase behaviour by the electrolyte solution, as well as certain recognition phenomena at the membrane surface. Further from the membrane, however, coulombic phenomena prevail. As a final caveat I wish to remind that certain uncharged lipids, such as phosphatidylethanolamine, with special properties (lower hydrophilicity, or a tendency for the nonlamellar phase formation) may exhibit biological effects which commonly are attributed to the effect of the membrane surface charge, on the phosphatidylserine or phosphatidic acid molecules, for example. The reason for this is that such special properties are the same for all these lipids (see, for example, Ref. 554). Specific effects observed with the latter two lipids therefore may, but need not be of electrostatic origin.
X. Summaff In conclusion, charged membrane together with their adjacent electrolyte solution form a thermodynamic and physico-chemical entity. Their surfaces represent an exceptionally complicated interfacial system owing to intrinsic membrane complexity, as well as to the polarity and often large thickness of the interfacial region. Despite this, charged membranes can be described reasonably accurately within the framework of available theoretical models, provided that the latter are chosen
on the basis of suitable criteria, which are briefly discussed in Section A. Interion correlations are likely to be important for the regular a n d / o r rigid, thin membrane-solution interfaces. Lateral distribution of the structural membrane charge is seldom and charge distribution perpendicular to the membranes is nearly always electrostatically important. So is the interracial hydration, which to a large extent determines the properties of the innermost part of the interfacial region, with a thickness of 2-3 nm. Fine structure of the ion double-layer and the interfacial smearing of the structural membrane charge decrease whilst the surface hydration increases the calculated value of the electrostatic membrane potential relative to the result of common Gouy-Chapman approximation. In some cases these effects partly cancel-out; simple electrostatic models are then fairly accurate. Notwithstanding this, it is at present difficult to draw detailed molecular conclusions from a large part of the published data, mainly owing to the lack of really stringent controls or calibrations. Ion binding to the membrane surface is a complicated process which involves charge-charge as well as charge-solvent interactions. Its efficiency normally increases with the ion valency and with the membrane charge density, but it is also strongly dependent on the physico-chemical and thermodynamic state of the membrane. Except in the case of the stereospecific ion binding to a membrane, the relatively easily accessible phosphate and carboxylic groups on lipids and integral membrane proteins are the main cation binding sites. Anions bind preferentially to the amine groups, even on zwitterionic molecules. Membrane structure is apt to change upon ion binding but not always in the same direction: membranes with bound ions can either expand or become more condensed, depending on the final hydrophilicity (polarity) of the membrane surface. The more polar membranes, as a rule, are less tightly packed and more fluid. Diffusive ion flow across a membrane depends on the transmembrane potential and concentration gradients, but also on the coulombic and hydration potentials at the membrane surface. The hydration surface potential governs to a large extent the height of the interfacial permeability barrier and is often decisive for the overall rate of the ion permeation. But coulombic potential is the chief determinant of the number of ions approaching the membrane and thus affects the total transmembrane ion flow. In the presence of ion carriers or channels the diffusive transmembrane current normally makes out only a small contribution to the total flow. Numerous experimental methods can be used to study the membrane electrostatics but the evaluation of the specific electrostatic membrane parameters from the experimental data is never an easy, and frequently an
357 ambiguous task. In studies with charged probes and biological membranes, simple Gouy-Chapman theory is often a poor tool for the analysis of the experimental data. Suitable generalizations and control experiments are of help here, but for the model lipid membranes such theory works surprisingly well. New experimental techniques for studying the membrane electrostatics are emerging. These promise a significant increase in the depth and scope of our understanding of the electrostatic membrane properties. In this review I have attempted briefly to survey some of them. While doing so I have tried to point to the current consensi and conundrums in the field of electrostatic membrane research, especially with regard to the interracial structure and hydration. Some statements made in this review reflect my personal opinions which are not always the views of the majority. I hope that even in such cases they are a mirror of truth. By all means, I have made an effort to tear down the barrier separating the hitherto largely unconnected but closely related fields of classical electrochemistry and the studies with charged membranes. Both have grown rapidly and independently over the last years and would gain from mutual fertilization. Membrane researchers, on the one hand, should benefit from the new concepts, alternative experimental methods, and highly accurate analytic tools of the modern electrochemistry. And electrochemists, on the other hand, should get a chance of extending their area of interest to the challenging new realms of biology. It will be hoped that this will provide better insight into the molecular origin, physiological significance and practical applications of the electrostatic effects in the biological, biochemical, and biotechnological systems.
Appendix A Poisson-Boltzmann
approximation
= - Pel,i(X)//~CO
=-- -- ( P e t , i / / C O O )
peA x )
=
NAeY'.Ziq exp[ Zieq4 x ) / k T l i
=- F~_Z,c i e x p [ Z ~ F + ( x ) / R r ]
I
~(x)
(16)
with the boundary condition: d ~ ( i n t e r f a c e ) / d x = - o e t / c c , , . Parameters c, %, and co are the dielectric constants of the aqueous and membrane subphase and the permittivity of free space, respectively. Another key assumption of Gouy-Chapman approximation [30,31] is that the charge distribution near
(17)
i
e is the elementary electronic charge, Z~ is the ion valency, k is the Boltzmann constant, NA is the Avogadro number; F = e N A and R = k N A are Faraday and gas constant, respectively; T is the absolute temperature. In electrochemistry and in all electrostatic theories it is customary to introduce the Debye-Hiickel screening length by the general definition:
(
1
X = ~%kT/IO3NAe 2 Z c i
(18)
i
where the summation goes over all ionic species and the factor 103 appears because the bulk ion concentration is normally given in moles per litre. For a symmetrical electrolyte the Debye length X can be conveniently written as: .....
= ( c%k T/2O NAc ) ½/ l OZe
which after insertion of numerical values for T = 298 K (25°C) yields: h=0.304Z-lc 12nm Combination of Eqns. 16 and 17 affords PoissonBoltzmann equation for a planar charged surface. Specifically for the case of symmetrical ionic solution, this is for an A z+, B z- electrolyte, one obtains: d2~k ( x ) / d x 2 = ( 2"103NAZec / , % ) sinh[ Ze~ ( x ) / k T]
Traditionally, but not inevitably, the most popular electrostatic surface model, the Gouy-Chapman diffuse double layer theory, is derived from Poisson Eqn. 1 by neglecting the spatial distribution of the membrane associated charge. This is tantamount to writing: Pel,m(X) =--O~lS(interface) for the surface and P e l ( X ) = Pet,~(X) everywhere else. For a planar surface one then has: d2~b(x)/dx2
the membrane is governed solely by the electrostatic, Coulombic interactions according to the Boltzmann law:
(19)
Similar equation can be written for the membrane electric field. In terms of Debye length, the Poisson-Boltzmann equation for a symmetrical electrolyte reads: ~2d2~ ( x )//dx2 = ( k T / Z e ) sinh[ ZeqJ( x ) / k T ]
(20)
In principle, either the electrostatic surface potential, ~P0, or the surface density of the structural membrane charge, Oet= e / A o can be kept constant, say, by some metabolic process. In the majority of the biologically relevant cases the latter possibility is the more probable one. Therefore, the usual boundary conditions with which the solution to Eqn. 19 should be sought is oel = constant. For little charged surfaces, the relation between the electrostatic membrane potential and the surface charge
358 density from the Poisson-Boltzmann equation is found (by expanding Eqn. 2 in terms of the surface potential) to be:
roughness and the interfaoal hydration effects tend to diminish the significance of the correlation effects so that in many cases these, in fact, need not be included into the electrostatic membrane description.
% = %lh/cc0 > 2 R T / Z F --- - 5 1 . 4 Z -1 ln(0.36A~c~) mV, T= 25°C
(22)
In the numerical result the area per charge, A~, is in nm 2 and the concentration in moles litre -~. The result for spatial profile of the electrostatic potential near an isolated membrane in the linear approximation is given by Eqn. 3. The corresponding approximate nonlinear result is ~ ( x ) = (4kT/Ze) tanh l[exp( - x / X ) tanh(Ze%/4kT)]
(23)
The discrepancy between the latter approximation and the exact solution to the Poisson-Boltzmann equation is small [4], except in the region close to the membrane where the validity of Gouy-Chapman theory because of principal reasons anyhow breaks down. Much greater errors can be expected to result from the intrinsic over-simplifications of Gouy-Chapman model. The most obvious, but not the most severe one is the neglect of ion size. This can be improved in an ad hoc manner by postulating the existence of a distance of closest approach beyond which ions can not penetrate (the Stem layer). Calculations based on Eqns. 2 and 23, which neglect the steric, hard-core repulsion, therefore underestimate the double layer thickness. But cross correlations between the steric and electrostatic interactions give rise to an extra attractive potential for counterions and to an excessive repulsive potential for coions. Owing to the predominance of the former in the case of infinitely narrow interfaces, the net outcome of the ion size-contribution is an attractive correction to the potential of mean force. On the whole, the size effects thus cause the ionic double layer to be thinner and the electrostatic surface potential to be lower than predicted by Gouy-Chapman theory (see Fig. 9). Interionic correlations thus are more consequential than the ion size effects, at least for the simple surfaces with a narrow interface (see also Appendix B). GouyChapman theory which does not account for any correlations, therefore, can be qualitatively wrong for the description of charged membranes with a very regular surface especially when multivalent ions are present. Fortunately, however, for most membranes the surface
The modified Poisson-Boltzmann (MPB) approach represents the natural extension of standard electrostatic Gouy-Chapman approximation towards modern electrostatic theories [43-47,555,48-51]. It includes into the Poisson equation, via a fluctuation or self-atmosphere potential (~) but still within the framework of a mean field approximation, the most relevant higher order ion-surface and ion-ion effects, d 2 ~ ( x ) / d x 2 = -(Pet,,/C¢o) 1,¢~)+,~) = - ( NAe/C~o)~.~Z,2c, exp i
× [wi.m, ra ,.ore(x)/kT- Z , e + ( x ) / k T +(1/kT)
× lira fZ'el* r ~ 0"0
[
47rc~or
*--
~
bulk (24)
where r is the distance from the centre of the ion i and Wi.h~rd_core(X) represents the steric exclusion term. The fluctuation potential thus accounts for the difference between the actual mean-force potential (wi(x)) and Gouy-Chapman potential at any separation from the surface: ~ ~ ~j(x) - wi(x) - +et(X) (see Eqn. 45). In the above, typical form it is devised so as to allow for the higher order, such as volume exclusion, image force, and ion self-atmosphere (correlation), effects. * In the point-ion limit ion radius, and thus the hardcore term, is zero. MPB-equation then reduces to: d2q.,(x)/dx 2 = - ( N ~ e / c , o ) E Z ~ c ~e x p [ - Z~eq/(x)/kT i +(Z~e/8rrc¢o)× ( h - 1 - h ( x ) - l + [ ( c - ~ m ) /2x(~ + c,,,)] e x p ( - 2 x / ~ ( x ) ) ) k T ]
(25)
This shows that apart from the neglected ion-size improvement, which is also lacking in Eqn. 25, there are two essential-corrections to Gouy-Chapman point-ion theory. First, the local variation of the screening length,
* This is tantamount to replacing the ion profile, which corresponds to a singlet correlation function, by the pair correlation function in the calculation of the potential.
359 X(x); and secondly, the classical screened self-image term, which is proportional to the reflection coefficient (~ c,,)/(c + c=), where ~,, is the dielectric constant of the membrane core. For a planar surface and in the absence of images, the cavity term, which is a kind of cross-correlation between hard-core and electrostatic interactions [7], takes the simple form: -
~l'cavity = 0.5[ tp(x + ri) + ~p(x - ri) ]
This implies that the potential of mean force is no longer a local function of the mean electrostatic potential once the ion-size effects are included. An immediate consequence of this is that oscillations in the double layer may arise. The linearized modified Poisson-Boltzmann equation can be expanded in terms of higher order potential derivatives for all electrolytes and truncated [43,556,49,557]. This is equivalent to an alternative Poisson-Boltzmann equation of the approximate form: )~(x)2d2~(x)//dx
(26)
2 = ~(x)
with a solution, for an isolated surface: ~ ( x ) = (oelX(x)/c%) exp( - x / X ( x ) )
(27)
The effective, spatially variable screening length X(x) is given, in the first approximation [5], by: i
X(x)=(,%kT/lO3NAe2~_,Z2c,(x)) ~ i 1
- X[cosh(ZN~P(x,X(x))/kT)] -~
(28)
This reveals, after comparison with Eqn. 18, that the efficiency with which a charge in the interfacial region is screened is determined by the local, rather than by the bulk salt concentration. * Complete solution to the linearized modified Poisson-Boltzmann equation for a symmetrical electrolyte or for a mixture of such electrolytes has the form [555]: ~p(x) = ~ ~0.n e x p ( - x / X c ) n~l
where Xc are the roots, with positive real parts, of the transcendental equation: ( ri/Xc) cosh(ri/Xc) + ( ri/X ) sirth(ri/Xc) = (ri//)~c)3( 1 + ri//~.)//(ri//)~) 2
(29)
* When Eqn. 28 is used in order to evaluate the effective screening length, the agreement with more accurate models is the closest in the third iteration in terms of the corrected potential, the more approximate potential value being always too low. The discrepancy increases with decreasing salt concentration (cf. Fig. 11).
and r~ is the (hydrated) ion radius. Full expressions for nonlinear modified Poisson-Boltzmann equation and a thorough discussion of various aspects and implications of this approximation can be found in [43,45,555]. A brief discussion is given in Ref. 50. Effects of variable dielectric screening have been discussed by several russian scientists [558,148,138,139]. The first term in Eqn. 29 becomes identical to Gouy-Chapman result when the Debye screening length is far greater than ion radius. This leading term decays spatially on the length scale of XC> X until, for Debye lengths somewhat smaller than ion radius, all roots of Eqn. 29 are complex conjugates and MPB-theory implies a damped oscillatory behaviour (cf. Fig. 9). In comparison with MPB-result Gouy-Chapman theory predicts slightly too thick double layers. For 1 : 1 electrolyte solutions in contact with a planar charged surface the predictions of MPB model agree nearly quantitatively with the results of computer simulations for such a model system. This accuracy is greatly reduced for solutions of divalent ions, however, or for systems characterized by a low (interracial) dielectric constant. * * For a highly charged surface MPB approach suggests that nearly all of the counterion charges in the surface associated layer will condense into a very thin zone within the range of a few ion diameters from the wall. This should happen essentially independent of the excluded volume effects. The outcome is something like a dipole layer which contains anomalously high electrolyte concentrations; even for the colons this is approximately twice as high as in the bulk. Similar conclusions pertain to the effects of image charges. These arise because of the interfacial mismatch in dielectric properties ('constants') between the assumedly homogeneous solvent and the region behind the surface. A charge near such dielectric border can be taken to induce (via polarization) an image charge located at the equivalent distance 'behind' the interface. This induces its own image, etc, all of which are dependent on the precise interfacial dielectric profile. For membranes the image effects are likely to be less important than for solid electrified surfaces, owing to the continuum nature of the membrane-solution interface. Absolute magnitude of the image effects, however, is independent of the surface charge density and images are relatively more important for the little charged surfaces [149]. Electrostatic membrane potential is never strongly affected by the image interactions [48]. This is mainly because image force is independent of the sign of the
* * For membranes with a relatively hydrophilic surface, especially the latter, restriction is quite severe owing to the fact that solvent layers near the membrane-solution interface essentially behave as a system of low permittivity, c < 30.
360 charge on the ion and thus affects all ions in much the same way. For a realistic choice of the surface potential (< 100 mV) the image contribution to the electrostatic potential is less than 20%, leading in every case to a decreased surface potential, even for repulsive images and in the absence of specific adsorption [47]. But double-layer structure-and intermembrane interactions-may respond quite sensitively to image effects [48]. In the extreme case, when the ion concentration near a regular membrane surface is sufficiently low, all counterions near the interface are on the average separated far enough to experience a local surface charge determined by the self-image component. This can lead to local maxima or minima in the counterion and coion density profiles, respectively (cf. Fig. 5), and may cause the intermembrane repulsion to turn into attraction. Effects as dramatic as this, however, are expected for the closest membrane vicinity merely, which explains why image effects normally contribute so little to the value of the electrostatic membrane potential. For symmetrical 2 : 2 electrolytes the fractional change in the diffuse layer potential caused by image forces differs from the 1:1 case both by being much larger in the former system and by decreasing once the surface charge is increased beyond a certain point. For asymmetric, such as 2 : 1, electrolytes, moreover, a small charge separation is established even for uncharged surfaces owing to the stronger action of image forces on divalent ions relative to the monovalent ones. *
Appendix C Generalized Poisson-Boltzmann approximation When the structural surface charges are not confined all into one plane the spatial distribution of the membrane charge (Oel.,,(x)) as well as the distribution of ionic charges (Oel, i(x)) must be considered explicitly in the Poisson equation: d2~,( x ) / d x 2 = - ( Oel,i( X ) + pel.m( X ) ) /CCO
(30)
Under the assumption that the membrane charge distribution is unaffected by the electrolyte and if the
* It is noteworthy that potential of zero charge can arise also because of the size asymmetry. By solving the Poisson-Boltzmann equation for the case of coions and counterions of different size Valleau and Torrie have shown this [560,561]. They have, moreover, found that such ion-size effect gives rise to an asymmetrical charge-potential relationship. Identical positive and negative surface charge densities thus result in different electrostatic surface potentials. The latter conclusion is also in agreement with detailed Monte-Carlo simulations [562]. Corresponding analytic expressions, within the mean spherical approximation, for an arbitrary asymmetric electrolyte containing an unrestrained number of components with any diameter and charge, have been published by Khater and co-workers [563].
salt distribution is taken tO obey Boltzmann law, generalized Poisson-Boltzmann equation is obtained. For an isolated membrane, the solution to this equation is given [105] by the integral:
t~(X)=(~k/~o)(f0
Pel,m(X )exp(-x'/~)dx"
- foxPel'm(x , ) sinh[(x - x' ) / M d x ' )
(31)
Fortunately, a detailed knowledge of the membranecharge distribution is not really required. Different distribution profiles give approximately similar results, at least for the surface potential, provided that the average charge displacement perpendicular to the membrane, d C, remains the same [105]. For a hypothetical membrane with an exponentially decaying density of the structural charge density (Oel, m(X) = (ael/d C e x p ( - x / d c ) ), for example, one obtains in the linear approximation: ~k(x) = (o~)~/,Co) { (d,2/)~ 2 - 1) -~[(d,,/)~ ) exp( - x / d , . ) - e x p ( - x/)~)] }
(32)
This result contains explicit information about the spatial profile of the structural surface charge distribution and shows that the electrostatic potential profile near the membrane surface is governed by the membranecharge as well as by the ion-charge distribution. Far from the membrane the diffuse double layer has the form predicted by the ordinary Poisson-Boltzmann equation if the Debye length is greater than the interfacial thickness. In the opposite case the spatial variation of the electrostatic membrane properties are determined by the structural charge distribution. This limit corresponds to a special case of Donnan approximation. For a hypothetical membrane with a constant surface charge density distributed throughout a layer of thickness 2dp, which corresponds to a 'screened Donnan model', the electrostatic potential profile is found by straightforward integration of Eqn. 31 to be: ~ ( x ) = ( oei)~ / ~ o ) ( )~/ 2 d p ) e x p ( - x / )~ ) sinh(2dp/?~)
outside the charged region and again in the linear approximation.
Appendix D Nonlocal electrostatic model In classical electrostatics, valid for homogeneous dielectrics and structureless solvents, an external field of some form creates a polarization profile of similar shape. In dielectrically non-homogeneous media, which include
361 all structureable fluids and, in particular, aqueous solutions, the same applied field produces a polarization which is squewed, due to the correlations and coupling between the solvent molecules. The response to an applied field in such case is nonlocal. Its range is proportional to the space in which the water is correlated. (If these fluctuations are interpreted as a generalized form of 'polarization fluctuations' this corresponds formally to replacing the standard dielectric constant by the nonlocal dielectric response function, c(r,r'); the latter is especially important for the short wave-vectors k.) A very narrow field represents a local perturbation. This arises, for example, from individual polar residues on the membrane surface. The membrane surface field, therefore, must be assumed to create a response in water which is not limited to its site of origin, but rather extends over a finite distance of the order of the solvent correlation length ~. To account for the coulombic as well as for the hydration membrane properties from the electrostatic point of view a nonlocal electrostatic model proves useful. Within its framework two types of the 'membrane potential' are introduced in order to fully describe the membrane surface: the usual coulombic electrostatic potential, +et----~k, and the hydration potential, +h (or, alternatively to the latter, the polarization potential ~pp = --coo~ph/C ). Standard meaning of the Coulombic electrostatic potential has been discussed in previous sections and will be extended in the following. The hydration potential of the membrane originates primarily from the short-range fields associated with the polar membrane residues, this is, from the atomic charges on these residues and from their capacity to interact with the solution. Solvent 'polarization' and the corresponding membrane polarization potential therefore contain contributions from the electrical polarization as well as, and often more importantly, from the structural polarization described here in electrical terms. To model the membrane surface in nonlocal electrostatic approximation, consequently, either the surface values of the coulombic and hydration potentials, tk0 and tkh0, or else, the corresponding charge densities oel a n d op must be known or calculated (see further discussion). For many thermodynamic applications knowledge of the molecular area is also required. The distinctive water properties within the same model are described phenomenologicaUy by a static and a high frequency dielectric constant, c and 2 < coo < 6, respectively, and by some typical structural correlation length, ~. For water the latter is probably on the order of 0.1 nm or less, the effective value being sensitive to the electrolyte properties as well as to the interfacial properties and structure [113]. In the nonlinear approximation an effective 'charge of the water molecules', e~ -3.5.10 -20 C, which reflects the effects of the inter-
atomic charge-transfer in the hydrogen bond formation, is also needed [122]. Instead of a single Poisson-Boltzmann Eqn. (cf. 420) for the electrostatic potential, in the nonlocal electrostatic approximation two coupled equations x 2 [d2,~/dx 2 - (,/,oo - 1 ) d % , / d x ~ ] = ( c k T / , ~ h 2) s i n h ( Z e f / k T )
(33)
and I;2d2~p/dx 2 ~ ( , ~ k T / e w ) sinh([ew( ~ + t~p+ ¢/ho)]/kT }
(34)
for the electrostatic and for the 'polarization' or the hydration membrane potentials must therefore be solved simultaneously. The surface hydration potential is a measure of the membrane hydrophilicity (see further discussion). The main source of the membrane hydration potential are the atomic charges in the interracial region which give rise to the potential ~kh0. This is due to the fact that the chief mechanism of the membrane-water association is direct binding of the water molecules to the polar membrane residues and not the ordinary electrostatic water polarization. The sites and the strength of the water binding, consequently, are determined largely by the quantum-mechanical fields originating from the surface polar residues and much less so by the macroscopic fields from the net surface charges or from the interracial dipoles and higher multipoles. Water binding to a membrane is thus more chemical than electrical in nature. The use of the word polarization in this context must be interpreted correspondingly. This can be understood as follows. Many membrane components contain multiple charges of various signs or at least a few polarizable groups. Charged regions on these membrane components are typically separated by intramolecular barriers which prevent the charge exchange and charge neutralization within a single molecule. * Each membrane thus can be characterized by a surface density of the local excess charge, %, which provides a measure of the overall membrane capability for the direct binding of water. In the simplest approximation [371], the value of the phenomenological parameter op can be found by summing-up the corresponding individual charge density contributions from the appropriate atoms on the membrane surface (see Fig. 15). But the very first step is to weigh these contributions in order to account for the different
* Owing to the atomic origin of such charges the precise molecular configuration or organization(monomer or lattice) influencesonly little the intramoleculardistributionof excess charge [114].
362 accessibility, ct~, of the corresponding atoms or groups to water. Effective individual local charge density thus can be estimated from the relation: Op.~ = e r / A r. From this the total membrane local excess charge density is finally found by averaging [122] over the entire membrane surface:
potentials are found to be: % = { o.X[1 + ( ~ / c ~ ) ( ~ / X ) ]
+ oA(1 - ~ / ( ~ ) } Xl~%(X + 4)1 (37)
and -1 Op _- EOtrOp,r. EArAsurface
(35)
+~0 -- - [ o . ( , = X 2 - ~ )
Obviously, only such polar groups should be considered in this sum which are not compensated directly at the membrane surface; that is, only the polar residues which are not involved in the direct hydrogen bonds with other membrane constituents and, consequently, are free to bind water. If the availability of a given group or molecule for water decreases, in the present model, a lower value of the surface local excess charge density results from this as then: olr ~ 0 . * For mixed systems the corresponding expression is: Op.mix =
Y~l,pia ( Op,lipidA"pid ) ElipidAlipid
(36)
The dipolar and multipolar effects can be accounted for accordingly by introducing the appropriate surface densities ~e' ~'e' etc. Normally these contributions are relatively small, however. The main conclusion of nonlocal electrostatic approximation is that near a membrane surface both the ion as well as the hydration double layers should co-exist, owing to the simultaneous existence of the electrostatic and hydration interfacial fields [110,141,122,566, 20]. From such, admittedly oversimplified, approximation the surface values of these corresponding surface
* Protonation of any charged polar group on the m e m b r a n e surface, therefore, not only affects the ionization state of such a group but moreover-and sometimes more importantly-also decreases the capability of this group to bind water. When polar parts of the membrane constituents interact in the absence of water, the effective basic 'electroneutrality unit' of each molecular layer consists of the nearest-neighbour charged groups. For lipids, these are typically the phosphate and the a m m o n i u m groups. But the same groups also can form intermolecular hydrogen bonds and may partly exchange charges via the charge-transfer process along these bonds. This is the reason why in any realistic model of membrane hydration the intramolecular as well as the intermolecular charge neutralization must be considered. Moreover, even in the cases where no direct water-lipid hydrogen bonds exist, charge transfer between the lipid and water molecules takes place [564,565]. For the hydrated lipid bilayers-and surely also for other membrane types-local excess charges consequently are subject to the screening and charge-neutralization by the atomic local excess charges localized on the water molecules. This is mainly a consequence of the large strength with which the water molecules can participate in the hydrogen bonds and is also a consequence of the great separation between the positive and negative residues on the lipid headgroups.
+ o~(~X + ~)~][-o
, ~ ( x + ~)l
~(~/~) (38)
in the linear approximation. Implicit in Eqns. 37 and 38 is the assumption that ion screening length X is much greater than water structure length 4. Only if a given membrane does not interact with water (op = 0) and if the solvent structure is neglected (~ ~ 0) Eqns. 37 and 38 reduce to expression 22. Gouy-Chapman result is then recovered. In every other case the electrostatic potential at the hydrated membrane surface from the nonlocal model is implied to be greater than it would be near a comparable non-hydrated membrane (cf. Eqn. 5) being a 'superposition' of coulombic and hydration-dependent potential terms. This is also reflected in the expression for the electrostatic potential profile near an isolated, charged but nonpolar membrane surface (oe/~ O, Op = 0). In nonlocal electrostatic approximation [20] for the very dilute electrolyte solutions one has: + ( x , % = O) = (oe/?~/c%)[exp( - x / X ) + ( ~ / X ) ( c / c ~ - 1 ) exp( - x / ~ ) +Coulombic(X) 4- +hydration(X)
(39)
Recasting this in standard Gouy-Chapman form shows that the consequences of the membrane hydration resemble phenomenologically the effects of a low and variable interfacial dielectric constant: +(x) = oezX/c(x)%
This spatially variable dielectric response function in the expression for the electrostatic membrane potential of an infinitely narrow interface is approximately [148,20] given by: ~ ( x ) - c[1 + ( ~ / c ~ - 1) exp( - x / ~ ) ]
i ?~ >>
(40)
This suggests that the spatial profiles of the electrostatic and hydration membrane potentials are likely to depend on at least two characteristic decay lengths. The first pertains to the long ranged, primarily ionic, double layer. It is a combination of standard Debye screening length and of water order correlation length; from nonlocal electrostatic model its value is estimated [141] to be approximately: )~,,/-- )~[1 + (~/X)2(c - ~ ) / c ~ ] 1/2.
363 The second characteristic length approximately measures the thickness of the 'hydration double layer' near an infinitely narrow interface. In analogy with previous result its range is given approximately by: ~ei= ~[1 (~/)k)2(£ - - Eoc)/Coo]1/2. But this expression is model-dependent and thus of limited practical value. In reality, the spatial variation of the interfacial hydration potential, and consequently of all other hydration-dependent thermodynamic quantities, is apt to depend on the interfacial and not just on the solvent structure [113]. This should be true for the entire region accessible to or partly filled with the polar surface residues. To clarify this point, let me consider a simple uncharged membrane surface with an exponential distribution of the polar residues: pp(X) = (op/dp) e x p ( - x / d e). The spatial decay of the hydration variables, of the hydration potential, say, for such a model surface immersed in pure water is derived from a suitably generalized version of Poisson-Boltzmann equation [152] for hydration: Ch(X,~) = ~
( - - 0 . 5 f ~ ( X ') exp[(-- x ' ) / ~ l ~ '
+ ~Op(x') sinh[(x- x')/~ldx')
(41)
to be:
d~
~h( X'~'dP) = ¢~c0 dp-- ~ - exp( - x/~) + ~ e x p (
- x/dp)
]
(42) -- (ap~//,~Co)(~/dp) exp( - x/dp), dp >>~
(43)
and is bi-modal. Eqn. 41 can be derived in analogy to similar treatment of the double layer near a continuously charged surface (cf. Eqn. 31) [152]. The range of the first component is controlled by the solvent structure correlation length. The second component has the form of the interracial polarity profile. In the special, present case it falls-off exponentially on the length scale of the interfacial thickness (cf. Eqn. 43). Experimental aspects of the membrane hydration were recently reviewed in this journal by Rand and Parsegian [376] and previously by Rand [339].
Appendix E 1on distribution functions and membrane potential
mean-force
In the distribution function approach membrane vesicles are treated as one class of particles in the solution. Moreover, because the radius of these vesicles
is SO much greater than that of the other solutes each vesicle membrane is approximated with a planar bilayer situated at x = 0, the interactions with all other membranes being neglected. Concentration profiles. Each distribution function, gmi(X), normally corresponds to a density function. When multiplied by the appropriate bulk concentration values, c~, it thus gives the number density for any dissolved species i at a separation x from the membrane particle m. To determine the concentration profile of the dissolved molecules relatively to the membrane surface the radial distribution function for all dissolved species must be known. From the elementary theory of liquids [157] one learns that this function is related to the potential of mean force: gmi(X ) = exp[- w,m(x )/kT ]
(44)
where the mean-force potential, Wmi(X), is the integral potential associated with the force between a particle i and the membrane particle m, with the other particles canonically averaged over all positions. Eqn. 44 is a general result. It reduces only under certain restrictive conditions, discussed in this Appendix, to the common Boltzmann-factor approximation of Gouy and Chapman. To calculate the ion distribution profile from Eqn. 44 this result must be combined with Eqn. 6. Contributions to the mean-force potential are frequently assumed to be virtually independent. This leads to a superposition approximation: Wi( X ) = Wel,i( X ) q- d~( X ) -~- Whard . . . . . . i ( x ) -1- Whydrat,i( X ) + Wother,i(X)
(45)
where the index m for the membrane is omitted for simplicity. The choice and separation of individual contributions to the mean-force potential is largely a matter of convenience and of the characteristics of the membrane system under study. In Eqn. 45 the solute-surface mean-force potential has been chosen to consist of a coulombic term, wel,i(x), a fluctuation potential, ~(x), a term that depends solely on the steric, volume-exclusion, hard-core interactions, W h a r d _ c o r e , i ( X ) , and a hydration term, Whydrat,i(X ). In the membrane studies the first, the third and the fourth term are especially important; the last contribution, which accounts for all residual surface-solute interactions not otherwise included, Wother.i(X), is normally quite small. In all electrostatic models the coulombic contribution to the mean force potential is set equal to the ordinary electrostatic membrane potential: wet.i(x ) = Zie~bel( x )
(46)
364 where Z~ is the valency of the ith molecular species. From this result, together with Eqn. 6 and the PoissonBoltzmann equation, Gouy-Chapman electrostatic model is directly obtained. Additional allowance for the hard-core and fluctuation terms, which is described in some detail in Appendix B, yields the so-called 'modified Poisson-Boltzmann' approximation. The hard-core repulsion term, which is very important for the solid interfaces and in the immediate membrane vicinity, where it may give rise to notable volume effects, is probably relatively unimportant for the membranes, at least as far as the surface electrostatics is concerned [70]. On the one hand, this is owing to the mesh-like structure of the membrane-solution interface. On the other hand, the ion-packing constraints at the membrane surface are reduced by the interfacial mobility and the surface fluctuation effects. Steric terms in the membrane research therefore frequently can be omitted unless extended polar groups or the innermost interracial region are considered. Membrane hydration effects can be allowed for, in the simplest approximation, by assuming ad hoc that the dielectric properties of the aqueous sub-phase vary with the separation from the membrane surface. A connection between the mean-force potential and the dielectric 'constant' profile c(x) then can be made by identifying the potential Whydrat(X) with the Born [567] free energy of transfer of the solute charge between two sites with dielectric constants c and c(x) wB.... ,(x) = [Gh.i(~o)/NA][C/(c - 1)][1/c - 1/c(x)]
(47)
where Gh, i is the free energy of hydration for an isolated charge on particle i. This energy is tabulated and also can be calculated from suitable models [20,567]. The weakness of such ad hoc phenomenological model of the membrane hydration is that it gives no realistic description of the perturbed interfacial water structure and, in particular, provides no means for relating this with the membrane surface properties. Only if the large contributions from the interfacial ion and surface polarity effects are neglected, the effective dielectric profile caused by simple electric polarization can be calculated, for example, from the Booth [568] or some other [569] theory. Such, obviously unrealistic, assumption which allows only for the dielectric saturation effect leads to the conclusion that the interfacial counterion concentration are smaller by up to one order of magnitude compared to standard double layer theory results. The origin of this is the dielectric saturation of the interfacial region, this is the nonlinear electrostatic response. Such discrepancy should increase with decreasing ion size and become unimportant at separations larger than approximately twice the ion size [1501. A much better, but still only partly adequate, de-
scription of the membrane hydration is obtained if the hydration term in the mean-force potential is evaluated from the nonlocal electrostatic model. In brief, this is done by identifying the hydrational part of the meanforce interaction potential with the free energy of the solute-surface interaction. The latter is expressed in terms of the non-coulombic part of the electrostatic membrane potential, q~et(Oet,Op,~) and of Born hydration free energy, Ghj [151], to yield: Wh~,drat. i =
2(4~rco~/Zie )Gh,,/~t( oel,Op,~ )
(48)
in the absence of dielectric saturation. The potential value +el (Oct,Op, ~) corresponds to the short-range component of the electrostatic membrane potential, which is discussed in Appendix D. It can be evaluated, in the linear approximation and for a perfectly smooth membrane surface, from Eqn. 37, by retaining only the ~-dependent terms. Otherwise, complete solutions to Eqns. 33 and 34 must be used (see, for example, Refs. 141,20). A more general expression for the ion distribution profile which contains all relevant contributions to mean-force potential of the membrane is given as Eqn. 8 in the text.
Appendix F Permeability barrier and transmembrane currents
The direct, diffusive flow of charged particles can be calculated from the phenomenological relation: j =~~pi[ Ac, + (Zi~F/RT)A~,,, ]
(49)
i
where Ac, = c i ( x = O ) - c i ( x = - d i n ) is the transmembrane concentration gradient for the electrolyte component i, gi is the corresponding average ion concentration, and P~ the associated element of the permeability matrix. From the partial mass-flow the electric current across the membrane is calculated by summing-up all individual ion-flow components: I = Y'.~Z, Fji. In the steady state, the concentration driven diffusive flow is constant throughout the membrane. It is also proportional to the difference between the probability for the molecular species under investigation to be located at the two internal membrane interfaces. It is thus a function of the corresponding partition coefficients: j, = [(ci/par,)~_o--(ci/par,)x=
dm]Pm,i
(50)
Comparison with Eqn. 49 shows that the square bracket in previous result represents an effective concentration gradient between the internal sides of the two mere-
365 brane surfaces [239]. The inverse of the bilayer diffusion permeability: PZ.,] = - ~ D ( x )
-1 exp([Go,i(x) _ ZiF~ ( x ) l / R T }
Straightforward integration of Eqn. 53 by using the linear Gouy-Chapman result for the electrostatic potential, Eqn. 3, leads to:
(51)
G d --*oo × [ t a n h - l ( d w / 2 ) ~ ) - 2 ' % = c ° n s t a n t Get(dw) = et( ~ ) ttanh(dw/2)~), oel = constant
represents a sum of resistance elements d x / p a r ( x ) D ( x ) and gives the total resistance of the membrane hydrocarbon core (or pore) [239]. D ( x ) is the diffusion constant and G0.~(x) is the free energy profile for the solute i. The interracial partition coefficient can also be expressed in terms of the inward and outward interfacial rate constants, k~, and kou t [239]. The transmembrane permeability then becomes:
where dp is the interfacial thickness. From this, the height of membrane permeability barrier is seen to be determined either by the interfacial inward and outward rate constants, or else by the resistance of the hydrocarbon membrane interior, depending on the relative magnitudes of individual components. For ions crossing the lipid bilayers the former contribution, in the majority of cases, is the principal one. The electric current across the membrane from Eqns. 50 and 51 and from the basic ion-current definition is deduced to be: 1 =y'FZi(((i
cikout,i/ki..i ) Ix= -d~ exp(-- FZiA~bm/RT )
+ (ciko~,.i/ki,.i) l x=o } P,,,i)
(52)
But this relation may change, even qualitatively, if the surface or transmembrane potential values depend directly on the substratum concentration [570]. In such event the effects of ion binding on the (trans)membrane potential must be considered explicitly.
Appendix G Electrostatic free energy of the membrane
General result for the evaluation of the electrostatic membrane free energy is Get.° = NAAc fo°'~ ( o )do
(53)
and gives the electrical free energy part. The corresponding 'chemical part' is: -NAAcoe? k. It should be added to the electrical contribution in the biologically improbable case that the potential rather than the charge density at the membrane surface is fixed; this yields Gel,4, = Gel,o - NAAcoet~b.
(54)
with the maximal value 6,,( d, --, ~ ) = - ( NAA~o~X/ , o ) = O.O066(AeZcl/2 ) -1 j mol_ 1
311
BBAREV 85371
Membrane electrostatics Gregor Cevc Medizinische Biophysik, Urologische Klinik und Poliklinik der Technischen UniversitSt Mi~nchen, Klinikum r.d.l., Miinchen (F.R.G.) (Received 16 October 1989) (Revised manuscript received 10 April 1990)
Contents I.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
311
II.
Electrostatic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-A. Gouy-Chapman model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-B. Modern electrostatic double layer theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
316 318 320
III.
Interracial structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III-A. Charge distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III-B. Interfacial curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
321 321 323
IV.
Interfacial polarity and the dielectric constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
324
V.
Hydration effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
325
VI.
Ion distribution and binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-A. Ion distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-B. Ion binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI-C. Ion transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
330 330 332 338
VII. Thermodynamics of charged membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-A. Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-B. Electrostatic intermembrane pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-C. Electrostatic lateral pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII-D. Transmembrane pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
340 340 342 344 344
VIII. Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-A. Direct measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII-B. Electrostatic probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
345 345 349
IX.
Biological significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
354
X.
Summary
356
I. Introduction Electrostatic phenomena play a crucial role in many biological processes. For membranes they are signifi-
Correspondence: G. Cevc, Medizinische Biophysik, Urologische Klinik und Poliklinik der Technische Universitllt Mllnchen, Klinikum r.d.I., Ismaningerstr. 22, D-8000 Mtinchen 80, F.R.G.
cant, for example, because the electrostatic interactions can lead to an accumulation of charged species near the m e m b r a n e s u r f a c e w h i c h t h e n a c t s as a c a t a l y t i c site. The electrostatics effects, moreover, influence the conformation and function of many molecules and control in part the intracellular and intercellular recognition and transport. T h i s is p o s s i b l e b e c a u s e t h e b i o l o g i c a l m e m b r a n e s normally carry numerous ionized or polar groups. These
0304-4157/90/$03.50 © 1990 Elsevier Science Publishers B.V. (Biomedical Division)
312 g r o u p s are e i t h e r p a r t s of t h e lipid, g l y c o l i p i d , o r ( g l y c o ) p r o t e i n m e m b r a n e c o m p o n e n t s , o r else s t e m f r o m the c h a r g e d m o l e c u l e s w h i c h are b o u n d to o r a d s o r b e d o n t o the m e m b r a n e surface. ( M e m b r a n e i n t e r i o r is n o r m a l l y free of charge, o w i n g to the h i g h e n e r g y cost o f i o n - t r a n s f e r f r o m the s o l u t i o n i n t o the h y d r o c a r b o n r e g i o n w i t h a l o w d i e l e c t r i c c o n s t a n t of a r o u n d 2.5). B i o l o g i c a l m e m b r a n e s t h u s are c o v e r e d w i t h a l a y e r o f the ' s u r f a c e ' c h a r g e (Fig. 1). T h i s l a y e r m a y b e u p to 20 n m thick, c o m p a r a b l e to the d e p t h of the e x t r a c e l l u l a r m e m b r a n e g l y c o c a l i x region. M e m b r a n e s of cells a n d organelles, m o r e o v e r , s o m e t i m e s f o r m p r o t r u s i o n s , are c r e n a t e d o r folded. T h e m i c r o v i l l i of the p l a s m a m e m b r a n e , for e x a m p l e , t y p i c a l l y m e a s u r e 5 0 - 2 0 0 n m in d i a m e t e r a n d m a y e x t e n d as far as 2000 n m a w a y f r o m t h e b a s i c s u r f a c e p l a n e ; cillia h a v e c o m p a r a b l e d i m e n sions. B i o l o g i c a l m e m b r a n e s , t h e r e f o r e , s e l d o m - i f - e v e r - possess the h o m o g e n e i t y o r the r i g i d i t y of t h e classical
e l e c t r i f i e d surfaces. B u t artificial b i l a y e r m e m b r a n e s c a n be p r e p a r e d f r o m s y n t h e t i c lipids w h i c h are v e r y h o m o g e n e o u s a n d o n l y a little c u r v e d . N e t c h a r g e o n the b i o l o g i c a l m e m b r a n e s in the m a j o r i t y of cases is n e g a t i v e ( T a b l e I). T h i s is b e c a u s e m o s t of the m e m b r a n e p r o t e i n s , all c h a r g e d n a t i v e lipids, as well as m a n y i n t e r f a c i a l l y a d s o r b e d p o l y e l e c trolytes have isoelectric points below neutral. The overall c h a r g e d e n s i t y o n a t y p i c a l b i o l o g i c a l m e m b r a n e is n o t high, h o w e v e r ; it s e l d o m e x c e e d s 0.05 C • m -2. T h i s is d i f f e r e n t w i t h t h e m o d e l m e m b r a n e s w h i c h m a y c a r r y several c h a r g e d g r o u p s p e r m o l e c u l e a n d p r o v i d e s u r f a c e c h a r g e d e n s i t i e s in t h e r a n g e o f - 0.4 _< % _< 0.4 C • m - 2 S t r u c t u r a l c h a r g e s a s s o c i a t e d w i t h the m e m b r a n e s u r f a c e give rise to an e l e c t r o s t a t i c m e m b r a n e s u r f a c e p o t e n t i a l , ~k0 (Fig. 2). T h i s p o t e n t i a l p a r t i c i p a t e s in the p r o c e s s e s of m e m b r a n e i n t e r a c t i o n s , r e c o g n i t i o n , o r sol u t e b i n d i n g . I n a d d i t i o n to this, an e l e c t r o s t a t i c trans-
TABLE I Typical values of the main electrostatic and structural parameters for the charged biological and model membranes under physiological conditions
All relational signs refer to absolute magnitudes. Distinction of the separate membrane regions, such as the hydrophobic core or the polar-apolar interface, is a matter of convention and convenience; therefore, it may differ from source to source. Abbreviations used are: oet net surface charge density; op surface density of the local excess charge; ~kel,oelectrostatic surface potential; ~bh.o hydration potential at the membrane surface; A~pm.~t electrostatic transmembrane potential; E 0 interfacial electric field strength; AE,n transmembrane field gradient; c,, relative permittivity of the hydrocarbon membrane interior; c~,t, effective value of the relative permittivity in the interfacial region; R m conductance of the membrane core; Cm capacitance of the membrane core; Rim and C~ t conductance and capacitance of the inner part of the interracial region; Cm capacitance of the outer interracial zone; dm thickness of the membrane core; dp effective interracial width; d~ average charge displacement perpendicular to the membrane; A L molecular area per lipid molecule; A e typical area per protein in the membrane; A c area per charge; ~ / , effective decay length for the hydration phenomena. Parameter
oet
%
~Pel,o
~Ph,0
A ~Pm, el
EO
A Em
unit
C. m - 2
C. m - 2
mV
mV
mV
MV. cm - 1
MV. cm
Electrostatic Cell membrane Lipid bilayer
- (0.02-0.2) a - 0.4-0.4
> - 1 - (0.1-0.5)
- 15-30 < - 200
- (200-300) - (300-800)
- 70( < 200) b < + 300
1 25
6 4-8
~,,
~,~t
Parameter unit Electrostatic Cell membrane Lipid bilayer Parameter unit Structural Cell membrane Lipid bilayer
2.5-5 2.5
30-50 30
Cm
Rm
Ci,t
R int
Rout
mF.m -2
mS.m -2
mF.m 2
mS.m-2
mS.m-2
1-10 e 5d
0.5 d
200 d
800 d
24000 d
dm
dp
dc
AL
Ae
A~
~f
nm
nm
nm
nm2
nm 2
nm2
nm
3 3-5
10 0.6
1-5 0.3
0.8 0.4-0.9
>_10 -
>2 > 0.9
0.5-5 0.1-0.4
a The effective surface charge density of vacuoles from A. pseudoplatanus,
for example, is greater than 0.13 C.m -2 [483]; for horse-bean
microsomes it is approx. 0.03 C. m - 2 [486].
b Owing to the membrane asymmetry the electrostatic potential at the inner surface can be appreciably higher. For erythrocytes it is believed to be approx. - 6 0 mV [76]. For Ehrlich ascites cells it is around - 2 0 mV [441]; for synaptic vesicles inside neurons it is on the order of - 2 0 - 4 0 mV. ¢ Transmembrane potential in Amoeba proteus is - 6 0 - 7 0 mV [586]; in freshwater algae approx. - 1 2 0 - 1 7 0 mV and in sea algae - 8 0 - 1 8 0 mV [587]; similar electrostatic transmembrane potentials, on the order of -100-200 mV, are also observed for bacteria and across the inner membranes of chloroplasts and mitochondria. In eucaryotic cells, such as neuronal or tumour cells, the resting transmembrane (polarization) potential is normally between - 6 0 and - 7 5 mV [434]. The spatial profile of all these potential differences is always strongly sensitive to the nature of the outer membrane and ionic conditions in the system. d These values were deduced from data on black lipid membranes [103]. The conductance of the bulk physiological solution is around 10000 m S . m -2.
e For specialized, such as neural cells, the corresponding value can be orders of magnitude greater, approx. 10000 mF-m 2.
313
extracellular o
> *
e
~ ~--qo-~+® e~ 7~o~-'~'i~
e~
.'.-~.o
-i
in a variety of biological processes including transport, bioenergetics, excitation, etc. Charges on membrane surface or transmembrane ion concentration gradients create electric fields (Fig. 4). These fields act on, and depend upon, the distribution of ions near the membrane-solution interface, in a selfconsistent manner. They also regulate the adsorption or binding of charged species to the membrane surface. This is seen directly from one of the elementary laws of electrostatics, the Poisson equation, which states that the spatial variation of the electric field vector, this is, the field divergence, everywhere is proportional to the local net charge density. For planar membranes this can be written as d2~(x)/dx
intracellular
2 =- - d E ( x ) / d x
membrane potential, Arm , exists across many membranes and across all such semipermeable membranelike barriers which divide two aqueous compartments filled with different, differently concentrated, or asymmetric electrolytes [1-3] (Fig. 2). Such transmembrane potential is found in almost all living cells. Intracellular sodium concentration (9 mmol 1-1), for example, is typically much lower than in the extracellular space (120 mmol 1-1), whilst the concentration of K ÷ is higher (in the muscle cell interior, for example, 140 mmol 1-1) than outside the cell (2.5 mmol 1-1) owing to the activity of the Na ÷ K+-ATPase pumps. Chloride concentration is similar to sodium concentration. Consequently, in the resting cell, where the K+-selective channels contribute the largest proportion to the cation transport across plasma membrane, the electrostatic membrane potential is close to the equilibrium potential for the potassium ions, which is about - 70 mV (Fig. 3). Any cell- or organelle-membrane with a non-zero transmembrane potential is thus said to be polarized. * Electrostatic membrane polarization plays an essential role
* This polarization has nothing to do with the dielectric polarization, which occurs under the influence of the electric field at the membrane surface (cf. Section IV), or with the cell polarity, which is a manifestation of the cell asymmetry.
(1)
- (p~;.,,,(x) + pe;.;(X))/C%
o
Fig. 1. Schematic representation of a biological membrane from the structural and electrostatic point of view. The distribution of the structural and dissociated ionic charges (circles) is sketched as well as the most strongly bound water molecules (V). Structural membrane charges occupy a wide region perpendicular to the surface, especially on the ectoplasmatic side. Numerous water molecules are bound to the polar surface residues, most notably to the phosphate groups, some amino acids, and sugar moieties. See also Fig. 14 for comparison.
= - pet(X)/~0
where the integral charge density, Pet(X) has been assumed to consist of the contributions from the structural membrane charge, Pet,,,,( X ), and from the ions, pel, i ( x ) . c and % are the dielectric constant of the
-~N SOLUTION/ 0
A
OLUTION, ...
1
I
I
X Fig. 2. Ion distribution profile, Pet,i ( x ) (A), and electrostatic potential, 6 e i ( x ) = tk(x) (B), near a charged membrane with an applied transmembrane potential, A~m = V and surface potential % . Dashed line depicts the distribution of co-ions and full line of counterions. Owing to the juxtaposition of the structural and counterionic charges and because the concentration of the latter falls-off continuously with separation from the surface (see symbols with a plus sign in the upper picture) it is customary to speak of the surface associated ionic diffuse double layer. Far from the membrane the concentration of counterions and co-ions is the same as in the bulk (cb,tk). Dotted line in the lower graph delineates the range to which the electrostatic interactions typically play an important r61e; this range is proportional to Debye screening length, ~.
314
O
®
®
0
@
@
s
C'2,bMk CI ,bulk
SOLUTION 1
A _~z/~,,, 0
l~r
In ~'~'""
= T v q~"~g__
SOLUTION 2
/
A
""-
/ ~,,
B
3; Fig. 3. Schematic representation of the ion distribution, pet.i(x) (A), and electrostatic potential profiles, ~ke/(x) (B), near an asymmetrically screened, uncharged membrane. Transmembrane potential difference (A~km) is a consequence of the asymmetric electrolyte composition on both membrane sides only and is given, in the simplest approximation, by Nernst equation. Such electrolyte asymmetry gives rise to a small surface potential and to minute diffuse double layers.
aqueous subphase and the permittivity of free space, respectively. Long-range, coulombic, electrostatic fields * interactions cause gathering of counterions of the opposite sign to that of the membrane charges, typically cations, and dilution of ions of the same charge, coions, in the vicinity of the structural charges (Figs. 2 and 5). Shortrange interracial fields, which are largely but not exclusively of quantum-mechanical origin, to a large extent determine the tertiary molecular structure, perturb the solvent properties in the interfacial region, and also influence the ion distribution. All membranes, consequently, are embedded in diffuse double-layers in which the solvent structure, ion concentration, and in many cases also the distribution of the membrane associated charges, vary with the separation from the membranesolution interface.
* With the word 'coulombic' in this review I characterize all chargecharge interactions which occur across-and are screened by-a uniform, structureless solvent dielectricum. This term thus encompasses standard electrostatic forces of classical electrostatic theories. The dielectrieally unscreened intercharge forces are described as quantum-mechanic, unless they involve water molecules and their charges. The interactions of the latter subclass are here grouped into a special cathegory of 'hydration phenomena'. But strictly speaking they are both electrostatic and quantum-mechanical in origin.
Current understanding of the membrane electrostatics is still woefully deficient. Already discussing all controversies would fill-out a book. In this contribution I, therefore, deal mainly with various aspects of the membrane surface electrostatics. (Current comprehension of the transmembrane electrostatic potential is much better and consequently will not be reviewed here in any detail.) First of all, some standard electrostatic models for the description of charged membranes are introduced and the basic criteria for the decision about the best theoretical approximation for each given application are noted. Subsequently, a short treatise on the distribution functions and their application in the membrane research is given together with a brief survey of the fundamentals of the ion-membrane association and binding. In the last theoretical section I dwell on the problem of studying the thermodynamic aspects of membrane electrostatics and survey the inter- and trans-membrane interactions of electrostatic origin. In the second part of this review the principal experimental methods for studying the properties of charged membranes, their strengths and weaknesses are summarized. For the sake of brevity and transparency most of the illustrative examples are taken from the works on simple model membranes. Moreover, many pertinent remarks are banned into footnotes and nearly all theoretical results are given in appendices. Readers with only a
O
O
8
@
~
e
O
e e
@
~ O
g
o
• o
• +@ ~ , ~
+ o
@
+
0
+
®
tq
SOLUTION
~1
O'2
Cbulk
SOLUTION
Cbulk ~JO,1
%
~o,2
Fig. 4. Structural charge distribution, pet.,,,(x) (upper) and electrostatic potential, ¢et(X) (lower), close to a membrane with two different values of surface charge density, %ta, °et.2- The electrostatic surface potentials at each membrane surface are different (%.1, tk0.2) and proportional to the corresponding surface charge density. The corresponding profile of the electric field is represented by a dashed line; it makes a jump at the membrane surface owing to the dielectric discontinuity (c m ¢ ~)-
315
I
I
I
I
I
B
2.0
0.8
0.6
1.5
c(x)
c x,
teri°ns
0.4
÷÷ 1.0
r.-.7-"~-"
'-'T
-
-----
I
6- e- ~ ~
_~,..""~
~
coions
4t,, - / . / - - , ~ 1 ~
COlOnS
0.5 x [nm] Q2 0 -
I
i 2
I
i 4
i 6
X [nm] Fig. 5. Ion concentrationprofile for a symmetricalunivalent electrolytenear a charged surface in the form of a diffuse double layer. (A) oct = 0.062 C. m-2 and 0.1 M salt; (B) oel = 0.125 C. m-2 and 1.0 M salt. Results for counteri0ns and co-ions derived from Monte-Carlohardsphere/hard-wall model simulations are shown as crosses and dots, respectively,and those derived from the Gouy-Chapmanmodel are representedby the full lines. (Modified from Ref. 60).
general interest in the membrane electrostatics correspondingly should follow the main text solely and especially the short summaries at the end of each section; but those keen on learning more about the theoretical backgrounds (or those attracted to the more specialized aspects of the membrane surface electrostatics) should look into equations or consult original articles and reference works. A very clearly written introductory theoretical discussion of various aspects of the surface electrostatics, with an emphasis on the thermodynamic questions, is offered in the early, lucid treatise by Verwey and Overbeek [4]. Also old, but still relevant, articles by Levine and Bell [5,6] provide a fairly detailed, analytical survey of the classical electrostatic models. A relatively general and very informative discussion of the modern aspects of the interfacial electrochemistry is contained in the state of the art review by Carnie and Torrie [7] with a comprehensive section on computer simulations, or in the somewhat more specialized, but equally brilliant papers by Henderson [8,9], Blum [10], or Schmickler and Henderson [11]. Brief but well balanced and worthwhile reading are furthermore the review of computer results by WennerstrSm and J~Snnson [12] and the theoretical discourse on exactly solvable plasma models
by Alastuey [13]. (The latter only at first sight are irrelevant for the membrane research: with due reservations they shed light on the dynamic aspects of the membrane surface electrostatics which are not accessible to other analytical models.) Certain theoretical aspects of the membrane biophysics are covered by the volume by Lakshminarayanaiah [14]; membrane transport has been dealt with in extenso by Stein [15] and in a review article on channels by Hall [16]. The role of membrane electrostatics in membrane photobiology has been assessed by Tien [17]. A thorough survey of the spectroscopic methods for studying membrane electrostatics can be found in the volumes edited by Loew [18]. A discussion of the electrostatic properties of lipid membranes, which also tackles the problem of the solvent effects, can be found in the review article by Gruen, Mar~elja, and Parsegian, [19] and in the recent book on 'Phospholipid Bilayers' by Cevc and Marsh [20]. Some of the biological aspects of the membrane electrostatics are covered by the scholarly written papers by Weiss and Harlos [21], Dolowy [22], Warshell and Russel [23] and McLaughlin with colleagues [24,25,26, 27]. A review which covers some other aspects of the interfacial membrane research stems from the feather of Dimitrov [28].
316
Owing to their structural and adsorbed charges a n d / o r because of the perturbation which they induce in the electrolyte solution all membranes are embedded in diffuse double layers of ions. In such double layers counterions by far outnumber coions, indicative of an electrostatic 'surface' potential. In addition to this, transmembrane concentration gradients create a potential jump across the membrane. The electrostatic surface potential is important for the processes occurring near or at the membrane," the transmembrane potential by and large plays a r3le in the transmembrane transport.
•
0
o
@
® O
e
®
• e
• 0
®
e
GOUY--CHAPMAN
crd
=
®
®
'
DONNAN EQUILIBRIUM
p~t,mS(interface) s' t
]
p = constaT~t
S
....
4~ m u
II. Electrostatic models
Recently, electrostatic surface theories have experienced a renaissance of interest and success. This is largely because rigorous results, which initially were derived for the bulk solutions, have been generalized and adapted for the interfacial studies; but also because of the advent of computer simulations. For membranes this development has not yet fully set in. One reason for this is that the membrane surfaces are so complex that none of the available electrostatic theories a priori can be expected to be suitable and reliable for their general description. Each common electrostatic model has its own, frequently overestimated, range of validity. Attempts to theoretically analyze the experimental data for charged membranes therefore must start with a scrutiny of the main system parameters and a definition of the relevant variables and quantities of interest. Then, and only then, a decision can be made about the appropriate theoretical framework-or a combination thereof-to be used. A convenient, but not obligatory, starting point for the development of the electrostatic surface models is the Poisson equation, Eqn. 1. In combination with a suitable closure relation this permits an evaluation of the ion distribution or of the electrostatic potential profiles near the membrane. But in order to make the solutions for membranes reasonably tractable numerous simplifications and restrictions must be introduced. Two limiting simplified situations are habitually chosen. In the first case, the electrolyte solution is reduced in thought to a uniform 'medium' and the interracial region, in which all structural membrane charges are located, is assumed to be completely permeable to ions. Ions, consequently, are taken everywhere to have the same concentration as in the bulk (Fig. 6); this concentration thus appears in Poisson equation only as a constant. Screening of the membrane charges by ions from the solution in such case is completely disregarded (Oeu(X)=peU) and the entire theoretical interest is focused on the intrinsic electrostatic characteristics of the interphase between the membrane core and the bulk electrolyte (peU,,(X)). The Donnan equilibrium approximation results from this [29].
%
5/
I
x Fig. 6. Schematic representation of the structural surface charge distribution, Oet.m(X) (upper), and the membrane potential profiles, 4%t(x) (lower) in the approximation of Gouy-Chapman (left) and Donnan equilibrium (fight). In the GC-model all structural membrane charge is assumed to be accumulated in the surface plane which is screened by non-uniformly distributed ions from the solution. In the Donnan equilibrium approach the membrane charge is taken to occupy a zone of finite width which is freely accessible to the solution ions. The corresponding counterion concentration profiles are shown by dashed lines.
An alternative is to squeeze all structural information about the membrane-solution interface into the electrostatic boundary condition (pet,,,,(X)= ael6(X = 0), and devote all the theoretical interest to the evaluation of the spatial ion distribution (pet.i(X)) and its effects on the electrostatic membrane properties. Such a step leads to classical diffuse double layer approximations. The assumption that the ion distribution close to the membrane surface is governed solely by the coulombic surface-ion interactions in combination with the previously mentioned approximations results in Gouy-Chapman electrostatic theory [30,31]. An essential implication of this model is that the ion concentration near the membrane surface differs from the bulk over a zone of finite thickness, the so-called ionic diffuse double-layer (cf. Fig. 5). The width of this layer is approximately proportional to a characteristic, concentration dependent (Debye) screening length, ~t (Fig. 7), which under physiologic conditions (ion strength I = 0.154,otemperature T = 37°C) is on the order of 1 nm (10 A). Moreover, the net (excess) ion-charge within the double layer is deduced to be identical in size, but opposite in sign, to the total charge on the membrane, because of the overall neutrality of the system. Surface value of the electrostatic membrane potential from Gouy-Chapman theory is predicted to increase
317 30
causes charge separation and leads to a formation of the membrane-associated ionic double layers. In addition to this, most membrane surfaces exhibit a non-coulombic electrostatic 'dipole potential'. This can reach a magnitude of several hundred millivolts for zwitterionic amphiphiles. It arises, in part, from the molecular dipoles on the membrane constituents. Lipid headgroups and the carbonyl groups of the hydrocarbon chains are the most notable sources of these dipoles. But non-coulombic membrane potential to a nearly equal part also stems from the interfaciaUy bound water layers (Cevc and Gaub, unpublished data). The total electrostatic membrane potential therefore inevitably depends on a variety of chemical and physico-chemical membrane properties and is not always simply proportional to the net membrane charge. If the electrolyte compositions at the two membrane sides are different an electrostatic diffusion (trans)membrane potential, moreover, emerges. Such transmembrane Nernst potential, in the first approximation, is proportional to the logarithm of the ratio of the ion concentrations on the 'in-' and 'outside' of the membrane, A~k~. . . . t - Ad/m = ( k T / Z e ) ln(cin/Cout) *. In the absence of screening effects the Nernst potential can be envisaged as a simple potential jump from an inside value ~ki., which is independent of the distance from the membrane surface, to a constant value, ~out = tl% +
E t--
25
z:z
Jl= -,i.I~
'-
1:1
20
___~
1:2 2:2
"E:
15 2:3
f..)
:3
10 5
0
........
0.0001
'
........
0.001
I
.
.
.
0.01 c/M
.
.
.
.
.
.
.
0.1
.
.
,
1
0
Fig. 7. Debye screening length, which is an approximate measure of the diffuse double layer thickness, as a function of the bulk electrolyte concentration, c, and ion valency, Z. With increasing ion charge and n u m b e r of dissolved charges the electrostatic effects are better screened and their reach next to the m e m b r a n e is more and more short-ranged.
steadily, albeit non-linearly, with the density of the net surface charge (see Section II). But non-coulombic surface-ion interactions can give rise to an electrostatic membrane potential on their own, even in the absence of the net membrane charge, at least for real electrolytes. For example, different ion-surface affinities introduce a non-uniform ion distribution near the membrane; electrolyte asymmetry, this is, the asymmetry of colon and counterion valencies (Z+~= Z ) or size, also
* This can be seen, m o s t directly, from Eqn. 7. More general expression for the evaluation of N e r n s t potential also allows for the permeability a n d / o r mobility differences between the involved ion species.
T A B L E II
Effects of various phenomena on the electrostatic membrane properties * • The sign gives direction of effect; the n u m b e r of signs its magnitude. As a rule, the range of various effects increases as hydration < electrostatics < surface fluctuations. Phenomenon
Surface potential
Interfacial el. field
Surface polarity
Interfacial thickness
Interfacial concentration
D o u b l e layer thickness
(~o)
(Eo)
(~,,)
(dp)
(c(x = 0))
(X)
Charge density Charge mobility Discreteness of charge
+ + + (0) lateral variations
+ + + (0) lateral variations
+ + + 0 (+)
(+ ) (0) 0 0
+ + (+ ) laterally modulated
Hydration Interfacial polarity
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + +
Surface curvature Surface fluctuations Interracial width
(- ) (- )
(- ) (- ) (- - - )
+ 0 .
(+ ) 0 . .
Image effects Ionic correlations Salt concentration I o n valency
(- ) (- )
(+ ) + +
(0) 0 - (- -)
.
.
(0) 0 + -
Range
Ionic profile
0 0 0 0
(+ ) (- ) complex (0)
steeper steeper complex
+ +,- - -, + +
(+) (+ )
short, + short, +
bimodal bimodal
0
+ +
(very long) very long short
(steeper) 0 bimodal, shallow
(- ) modulated + + +
- -
short short + _
oscillatory oscillatory oscillatory, steepeer (binding), steeper
.
_ _
(0)
318 A~k,,, at the outside (see Figs. 2 and 3). But in reality, salt effects additionally cause the electrostatic membrane potential and the ionic distribution profiles to vary throughout the interracial region. The distribution of ions and membrane associated charges is, therefore, always continuous everywhere in the system. The Nernst potential is one of the driving forces for the passive membrane transport. But it is rather unimportant for the membrane surface electrostatics. Consequently, it is not discussed in this work except in section V-B which deals with the ion transport. Thus, if the Nernst potential is considered important for a given application, the appropriate transmembrane potential difference should be added to, or subtracted from, the corresponding electrostatic surface potential value. In any realistic, general membrane model a number of structural and physico-chemical aspects of the surface electrostatics should be considered. Furthermore, the complexity of the membrane-solution interface should be paid attention to. For example, the effects of spatial redistribution of the structural membrane and ionic charges, the direct and indirect interactions between such charges, their dependence on the membrane structure and vice versa, as well as the membrane hydration should all be thought of and, frequently, accounted for (see also Table II). As yet, no model is capable to cope will all these requirements at once. Nevertheless, in order to obtain tools for quantitatively analyzing the electrochemical membrane properties, special solutions are normally used. To meet a decision about the optimal theoretical model for a given investigation the following general criteria are helpful: (A) Smooth or rigid surfaces and interfaces which are narrow on the length scale of Debye screening length as well as ion polyvalency favour ion-layering and higherorder ion-ion correlation effects. Advanced, up-to-date electrochemical theories (such as modified PoissonBoltzmann or Hypernetted-Chain-Approximation) with allowance for the fine interfacial structure in principle should be suitable for the analysis of membrane data at this level. Comparison with Monte-Carlo and molecular dynamics results is also valuable. Highly ordered model membranes consisting of lipids with small head-groups in a crystalline or gel phase, small, flat and relatively rigid domains on the biological membranes, or interfaces between the multivalent macromolecular membrane components and the electrolyte solution (in the strong coupling regime) fall in this category. (B) Moderately curved or undulated (again on the length scale of Debye screening length) or relatively mobile surfaces with a narrow membrane-solution interface as well as membranes with a more or less uniform distribution of the structural surface charges represent nearly 'ideal electrified bio-surfaces'. As such, they are closest to the fulfillment of the requirements for the
validity of the Gouy-Chapman approximation. Standard electrostatic results, consequently, are normally applicable for such systems. If the water and ion binding effects are properly accounted for, it may then be inferred-but remains to be proven suitably-the generalized Poisson-Boltzmann equation can always yield reasonably accurate results for the systems of this type. Model lipid membranes in the fluid phase and other simple membranes with a smooth, charged surface provide examples for this. (C) Very irregular membranes with an interfacial region which is thick compared to the Debye screening length, finally, necessitate the allowance to be made for the spatial distribution of the ionic as well as of the structural membrane charges. The distribution of all polar and charged membrane residues perpendicular to the membrane surface must then be investigated and analyzed together with the electrolyte penetration and charge regulation in the interfacial region. Failure to do so may result in gross misinterpretation of the experimental data. Most cell membranes and other complex biological interfaces belong to this class. H-A. Gouy-Chapman model
The Gouy-Chapman (GC) approximation [30,31] is obtained directly from Eqn. 1 under the proviso that all structural membrane charges are confined to an infinitely narrow plane (cf. Fig. 4) and that the ion distribution is governed solely by Coulombic forces (see Appendix A). Owing to its straightforwardness the GC-model is extremely popular, especially for the routine membrane research. It predicts the electrostatic potential at the membrane surface to be: (2k T/ Ze ) sinh- l ( Zeo¢fl~ /2~c ok T ) ~o = (o¢/~/~o: % < 2kT/Ze -- 25 mV
(2)
for symmetrical electrolytes with counter-ions and coions of similar valency ( Z + = Z _ - Z). Parameters c, c0, and k are the dielectric constant of water, the permittivity of free space, and the Boltzmann constant, respectively. ~ is the Debye screening length (cf. Appendix A). The second form of Eqn. 2, which is an expanded and truncated version of the first one, suggests the interracial potential value to increase linearly with the charge density; for membranes with a higher charge density this increase is less, however, owing to the saturation of the interfacial double layer region with counterions (see Fig. 8 and also Fig. 5). Within the framework of the linear Gouy-Chapman model the variation of the electrostatic potential with distance from the membrane surface is given by the
319 02
I
I
20
.~'~
I
I
I
10
s
~,, >
I
t,
0.01 M
.--
I
~el = 0.22
C m -2
0.1 ;I
0
.. . - - - f
0.1 acl
0.2
.
0.3 I
(C m -2)
0
Fig. 8. Electrostatic surface potential, % , as a function of the membrane surface charge density, oet, for different values of the bulk electrolyte concentration, c. Solutions to Poisson-Boltzmarm equation (dashed), modified Poisson-Boitzrnann equation (full line), and an approximation to the latter (cf. Eqns. 27 and 28, dashed-and-dotted) are shown. Computer simulation data are represented by symbols. Results from linear Gouy-Chapman theory, deduced by solving linear Poisson-Boltzmann equation, are given by dotted lines and nearly always significantly overestimate the surface potential value (modified from Ref. 9).
hyperbolic sinus function, for constant surface charge density, and by the hyperbolic cosinus function, for constant surface potential. For membrane-membrane separations much greater than the Debye length, however, the spatial decay of the electrostatic membrane potential is always well described by a single exponential function ~ ( x ) = % exp(0x/X ) = (OetX/~%) e x p ( -
x/X)
(3)
Usually, and surely for most of the biologically relevant systems, the double layer properties are dominated by the counterions (cf. Fig. 5). This means that the electrostatic potential as well as the ionic profiles near a negatively charged membrane in the case of 1:1 and 1 : 2 (Z+ : Z_) electrolytes are rather similar; for positively charged surfaces the same holds for 1 : 1 and 2 : 1 salt solutions. When suitable analytic results for the membrane under investigation are not available, the solutions for the corresponding symmetrical electrolytes can then be used with the valency pertaining to the counterion; but more sophisticated approximations are also available [32]. The Gouy-Chapman approximation is imprecise from the membrane point of view in that it neglects the interfacial structure as well as the hydration effects. From the point of view of the electrolyte it also has shortcomings in that it treats the short range interactions between the charged surface and ions inconsistently [33]. No account is taken of the nonelectro-
4
8
12
× (ore) Fig. 9. Ion concentration profile, c~(x), near a charged surface in contact with a one molar monovalent salt solution as implied by various electrostatic models. In agreement with computer data (symbols) oscillations are predicted by the modified Poisson-Boltzmann equation (full line) which can not be accounted for by Gouy-Chapman approximation (dashed) (From Ref. 9). The latter model also overestimates the double layer thickness.
static, such as ion size effects, or soft, e.g., water-mediated, short-range membrane-ion interactions. Additional complications arise for elongated counterions [34,35]. * Gouy-Chapman approximation, moreover, neglects all interionic correlation effects: ionic distribution in the diffuse layer is assumed to be dictated solely by the coulombic direct forces between the ions and the interface. This is the reason for the monotonous change of the ion concentration profile with separation (Fig. 9). Consequently, Gouy-Chapman theory is strictly valid in the limit of infinitely dilute bulk solution bathing an infinitely narrow membrane-solution interface in a structureless solvent. Obviously, such a situation is not encountered for membranes. But fortunately this is not always an obstacle, since different corrections to GouyChapman results have different signs. The interfacial structure and higher order electrostatic effects, for example, lower the calculated electrostatic membrane potential. But the correction for the interracial hydration goes in the reverse direction (see further discussion). In fortunate cases-many lipid bilayers apparently belong to these-Gouy-Chapman theory may thus be nearly correct, despite the fact that it dramatically oversimplifies the reality. This mysterious mercy should be
* From the suitably modified Gouy-Chapman theory it is concluded that large, elongated divalent cations have a smaller effect on the surface potential than the corresponding point ions; the magnitude of this discrepancy decreasing with salt concentration. Large divalent cations also give rise to a negative contribution to the zeta potential.
320 accepted gratefully. But simultaneously the veritable complexity of the situation should be borne in mind, especially when conclusions are to be drawn from the theoretical data-analyses at the molecular level. Only then a clear distinction will be possible between the intrinsic and quantifiable experimental or theoretical errors on the one and the (in)adequacy of the model or working hypothesis on the other hand. II-B. Modern electrostatic double layer theories In contrast to Gouy-Chapman approximation all modern electrostatic theories incorporate the interionic correlations and other higher order effects into basic equations in a natural way. * This is possible because most of these theories do not rely on the simple Poisson-Boltzmann equation but rather on some expression of proven, rigorous validity for the bulk electrolyte solution. The most popular modern theories of electrified surfaces stem from the cluster expansion, the Bogolyubov-Born-Yvonne-Green set of equations [38] or from the Ornstein-Zernike equation (for reviews, see, for example, [8,9,7]. A combination of these with a suitable closure relation, the superposition closure of Kirkwood, the Loeb's closure, or the hypernetted-chain closure, affords a self-consistent and accurate electrostatic description of simple electrified surfaces. Some problems arise because of the interracial discontinuity which requires the exact results valid for a homogeneous bulk system to be adopted for an anisotropic case. But these can be circumvented by mapping the inhomogeneous three-dimensional system into a homogeneous two-dimensional one [36,37]. The hypernetted-chain/hypernetted-chain method [39,40,8,9], the hypernetted-chain/mean-sphere approximation [41,42,8,9,11], and the modified Poisson-Boltz-
* The simplest, but theoretically already formidably complicated, modern statistical mechanical model of the diffused part of the electrical double layer is that of a system of charged hard spheres (or even just points) immersed in a dielectric continuum next to an impermeable, planar charged wall. This is termed the primitive model double layer and has been studied extensively with various approaches over the last decades. One outcome of this is the conclusion that the solution of the non-linearized Poisson-Boltzmann equation for the point-ion version of such a model, which is equivalent to Gouy-Chapman approximation, gives a nearly quantitative description of surface electrostatic potential over a surprisingly large domain for 1 : 1 aqueous electrolytes. The accuracy of such model for other electrolytes is less and in general the ion and charge distribution profiles are poorly reproduced (cf. Fig.
9). Accordingly, the discrepanciesbetween the measured and calculated electrostatic membrane surface potential must be due to the indirect ion-surface interactions, to the effects of interfacial structure and polarity, to the consequences of solvent structure, or to any combination of these.
mann approximation [43,44-51] clearly stand out in their ability to make quantitative predictions for the structure and the thermodynamic properties of simple surfaces and their associated model double layers. Of these theories the modified Poisson-Boltzmann approximation is, in principle, optimally suited for the applications with membranes, mainly because of its formal familiarity with Gouy-Chapman model (see Appendix B). But in reality it is seldom going to be used as it stands. Partly owing to its complexity and partly due to the fact that in the present form it can not cope with the interfaces of finite width. The modified Poisson-Boltzmann approximation shows, among other things, that inclusion of higher order corrections has consequences which phenomenologically resemble the effects of the spatially variable Debye screening length (cf. Eqn. 26). * * For example, higher order electrostatic effects give rise to structural oscillations in the diffuse double layer. (cf. Figs. 6 and 9). These oscillations are small for dilute monovalent salt solutions. But for 1 : 1 electrolytes with a concentration in excess of 0.1 moi.1-1 near a charged planar surface the oscillation amplitudes may become substantial. Monte-Carlo simulations for lamellar systems suggest that under such conditions the ion-ion interactions are more important than the ion-surface interactions [52,53]. This explains why the electrostatic surface potential in the latter case is smaller than one would expect on the basis of Gouy-Chapman theory (Fig. 8). Higher order effects become progressively more important with increasing ion valency. For the divalent salt solutions the deviation from the results of standard electrostatic models, therefore, is significant already in the millimolar concentration range, at least if the charged surface is sufficiently uniform. (Fluorescence measurements with synchrotron light for manganese binding to a stearate monolayer show that, indeed, such oscillations form in a condensed but not in an expanded phase [54].) For a narrow, planar surface with a charge density OeZ= 0.0444 C" m 2 (one charge per 3.6 nm 2 or 12.5% charged lipids in a bilayer) in a 0.005 M solution of CaC12, for example, Gouy-Chapman approximation predicts the electrostatic surface potential value to be 30% too high (Fig. 8). Furthermore, the extension of the electrostatic potential and the thickness of the diffuse
The electrostatic potential felt by individual ions is never strictly identical to the local value of the mean membrane electrostatic potential (see Appendix B). Contributions from various cross-correlations impose fine structure to the interracial region and cause a deviation from the monotonous profile: layers of charge of alternating sign lead to oscillations in the charge and ion distribution profiles near e hard surface. The potential profile near a uniform surface therefore also oscillates and causes the diffuse layer potential and ionic adsorption isotherms to vary unmonotonously with the surface charge density.
321 double layer by Gouy-Chapman theory are overestimated (Fig. 9). Standard electrostatic models, consequently, are expected to predict excessive electrostatic interfacial repulsion for simple charged planar surfaces. * Experimental evidence for such effects exists for phosphatidylinositol bilayers interacting with polyvalent ions [57]. Ion correlation effects may be relevant for membrane systems. The more regular the membrane surface, the higher the surface charge density or the ion valency, and the more concentrated the bulk solution, the more likely this will be the case. But interfacial 'softness' or smallnes of the interracial dielectric constant can largely compensate for this. For a soft wall in the weak coupling limit, i.e., for a membrane in which the electrostatic interion potential is small compared to the kinetic energy of ions, the solution to the Poisson-Boltzmann equation may even become an exact result [13]. * * Highly mobile and elastic membranes or 'fluffy membrane surfaces' differ-from the electrostatic point of view-from rigid membranes and from the model membranes in the gel phase. Theoretical description of these two types of membranes consequently should be different. For the latter class of membranes the interionic correlations can play a rrle. Suitably modified mean field theories which account for the correlations, such as the modified Poisson-Boltzmann equation approach, therefore should be considered for the electrostatic analysis of rigid uniform membranes. But even when dealing with 'soft' membranes the higher order effects should be kept in mind. If for no other reason, in order to check whether or not the resulting effects will influence the electrostatic membrane properties on a small-scale. In the cases of (macro)molecular adsorption to the membrane, ion binding, or transmembrane mass-flow, this could be the case owing to the small size of the involved membrane areas.
No universal theory of the membrane electrostatics exists to date. Acceptable working models for different specific situations can be found, however, under appropriate simplifying assumptions. Gouy-Chapman approximation provides a reasonable description of the electrostatic properties for model membranes with an infinitely
* Correlation-dependent double layer narrowing, implied by all modern electrostatic theories, is confirmed by numerical MonteCarlo 'experiments' for charged surfaces in contact with an electrolyte solution. At high concentrationsand surface charge densities the counterions are reproducibly found to be tightly packed near the interfacein a layered structure; here, the potential drop is also appreciable [55,56]. ** The reason for this is that the three body, higher order effects, which underlie correlations and oscillations phenomena, demand hard,, regular surfaces to fully evolve. Indirect evidence for this comes from the studies of a one component plasma [58,13,59]and from the correspondingMonte-Carlo simulations [60,52,61,62].
narrow interface which does not perturb the water structure, provided that the higher order effects, arising predominantly from the direct ion-ion interactions, are negligible. This need not always be the case for rigid and~or highly charged membranes. Especially for the concentrated electrolytes the corresponding correction terms, based on up-to-date double-layer models, should be allowed for. The modified Poisson-Boltzmann equation approach is one example of this. Phenomenologically, this corresponds to using a spatially variable ion screening length within the framework of classical electrostatic membrane models. III. Interracial structure
Perhaps the weakest point of simple electrostatic membrane models is the neglect of the interracial structure. In Gouy-Chapman theory, for example, the interphase between the membrane and the solution is assumed to be a discontinuous interface which encompasses all structural membrane charges; the dielectric constant at this location is also assumed to change abruptly. In reality, the membrane charges are neither smeared-out uniformly over the surface plane nor is the transition between the bulk solvent and the membrane interior discontinuous. The membrane charges are always discrete and distributed throughout a mesh-like, water-containing region with parlous appendages ranging from glycolipid and protein molecules, with dimensions on the order of a few nanometers, to cilia and microvilli, on the order of micrometers. The solvent properties and ion concentration in this region may differ largely from the bulk. Any realistic analysis of the membrane electrostatics should pay attention to this.
III-A. Charge distribution Lateral discreteness of the membrane surface charge. To allow for the consequences of the discrete charge distribution on the membrane surface the continuous structural charge distribution function is replaced by the sum over all discrete membrane charges [63]. The latter, for the sake of simphcity, are normally postulated to be located on a regular lattice. From the appropriate solution to the Poisson-Boltzmann equation the electrostatic membrane potential is found to be identical to the lattice sum over the screened Coulomb potentials of individual charges [64]. As such, the Gouy-Chapman theory then implies the local potential maxima to exist at the charge-lattice sites (Fig. 10). The existence of well-defined potential peaks at the surface of real membranes is quite imProbable , however. First, experiments with the model membranes fail to show any firm indication of such discreteness effects [65-68]. Secondly, numerical simulations suggest that there may be little, if any, real difference between the
322
Fig. 10. Theoretical prediction for the spatial profile of the electrostatic membrane potential above a surface covered with a square-lattice of the structural electric charges. Discrete potential (in the original article ion-concentration) peaks are implied to exist at the lattice sites on which the structural surface charges are located. These, however, have never been observed experimentally. (From Ref. 63).
cases of discrete and continuous surface charge distribution [7,69,70]. * Also the lateral diffusion in planar double layers is virtually the same for a uniformly charged surface and for an interface with discrete charges [71]. In practical applications to membranes, therefore, the lateral surface charge distribution may be neglected, at least in the case of monovalent ions. Charge distribution perpendicular to the membrane. The distribution of the structural membrane charge perpendicular to the membrane in numerous of electrostatic studies is crucial, however. The rule of thumb is that it is necessary to consider such effects as soon as the thickness of the interracial charged layer is comparable to or greater than the Debye screening length. From the point of view of the membrane hydration the interfacial structure is always important (see Appendix D). The first reason for the charge smearing perpendicular to the membrane surface is the finite size of the charged membrane constituents. Structural charges often are distributed along extended, membrane-bound molecules, some of which may protrude up to 10 nm away from the average interface, for purely geometrical reasons. The second cause of the charge distribution per-
* This, on the one hand, may be due to the high mobility of the membrane components which partly alleviates the consequences of the charge singularity. On the other hand, the lack of the discreteness of charge effect may also be indicative of the relatively slow decay of the lateral pair correlation functions, which describe ionic distributions in the surface plane; such functions fall off as the inverse cube of the distance [78-80,59].
pendicular to the membrane are the thermal out-of-plane fluctuations of the individual membrane constituents or of the whole membrane domains. In consequence to this, the transverse smearing of the structural membrane charge can be appreciable even for the molecules with a relatively tiny head-group. For fatty acid monolayers the dynamic interfacial roughness has been measured to exceed 0.1 nm [72]; for phospholipids the width of the interphase with embedded charged residues may be close to a nanometer, owing to the greater headgroups [73] and to the headgroup motion. For even more extended molecules the effect is likely to be still more pronounced. Dealing with the interfacial charge distribution is an intrinsic feature of the Donnan equilibrium approach [29]. But such approach typically provides a less realistic description of the membrane surface than the corresponding generalizations of the Gouy-Chapman model [74-77]. The latter are briefly discussed in Appendix C and also account for the ion distribution effects negiected in classical Donnan approximation. Increasing the width of the membrane-solution interphase normally lowers the electrostatic potential at the membrane surface. This is mainly owing to the better screening of the structural surface charges by counterions because the latter are able to penetrate into the region occupied by the membrane charges. For the same reason, the electrostatic potential at the membrane surface is more sensitive to the concentration variations than standard Gouy-Chapman theory would imply. One further, but frequently disregarded, sequel of the transverse smearing of the structural membrane charge is that the ionic double-layer region extends further out into the electrolyte solution and is shallower near the very surface. This means that the interfacial electric field is diminished by the charge distribution perpendicular to the membrane. The dielectric saturation and attraction for ions at the membrane surface is thus often smaller than anticipated. For one particular system, in which the structural surface charge density is assumed to fall-off exponentially with the separation from the membrane-solution interface (Oet.m= (ocJd~)exp(-x/dc), the lowering of the membrane surface potential by the charge distribution perpendicular to the membrane has been calculated [75] to be: 60(dc) = q,o(dc = 0 ) / ( 1 +
dc/X )
(4)
Here, d c corresponds to the average transverse displacement of the structural surface charge and is proportional to the interracial thickness. This, admittedly oversimplified, relation shows that when the average charge displacement perpendicular to the membrane surface changes from 0.2 nm to 1 nm, in the case of physiological salt solution (0.15 M), the electrostatic potential
323 20 Oel = 0 . 0 1 5
16
Non-Coulombic phenomena exert their own effects on the ion penetration in the interfacial region [20,84,85] and also on other 'electrostatic' membrane properties. It is perhaps indicative of this that protein desialiation on the surface of the brush-border membranes increases membrane fluidity [86] despite the fact that the net membrane charge becomes less after the sialic acid residues have been stripped-off.
C m -2
c = o.15 M
>
E
0 12
t"
8
nrn
D.
III-B. lnterfacial curvature
0
i
0
1
i
2 x/nrn
i
3
4
Fig. 11. The electrostatic potential profile near a surface with transversely smeared-out structural charge in contact with a 0.15 molar monovalent salt solution. The density of the structural surface charge is assumed to decay exponentially(#el.,,(x) = (vet~de) exp(- x/dc) ) for three different values of d c in order to mimick the membranes with a thick interface or a 'fuzzy-coat' surface. The total surface charge density is eet = 0.015 C.m -2. The potential profile becomes progressivelymore shallow if the interracial thickness and the average charge displacement, de, increase and become substantially greater than Debye screeninglength, h = 0.9 nm.
at the membrane surface should decrease by 20 to 50% (Fig. 11). For real membranes the charge regulation and charge mobility interfere with the direct interfacial structure effects [81]. If a constant amount of charge is allowed to distribute throughout a finite region adjacent to the membrane surface in relatively diluted ionic solutions the electrostatic membrane potential becomes nearly independent of the separation from the surface. The ion distribution profile close to the membrane under such circumstances is also relatively uniform. But with increasing salt concentration this charge tends to pile-up at the outer border of the interracial region. The electrostatic potential then increases with the separation from the membrane and at the utmost end of the zone occupied by the structural membrane charge becomes the highest. Further out in the electrolyte solution, where no membrane charges are located, the electrostatic potential profile finally is such as expected for a simple, uniformly charged surface [83,811. Direct experimental evidence indicative of the importance of the interracial thickness for the electrostatic membrane properties comes from zeta potential measurements with red blood cells [76,3561 as well as from the measurements with ganglioside-contalning model membranes [821.
When the curvature radius is comparable to or smaller than the double-layer thickness [87,88] or smaller than the interfacial width the interracial curvature starts to influence the electrostatic membrane properties; typically it causes the electrostatic potential to be less than for a comparably charged planar membrane. But neither of these two conditions is likely to be satisfied for the membrane as a whole; only locally the membrane curvature may be sufficiently high for one of the previous requirement to be fulfilled. Notwithstanding this, the electrostatic properties, at least of small membrane areas [89,91], can be affected by the local curvature. This is especially true for the membranes suspended in dilute electrolytes for which Debye length, and thus the double layer thickness, is big. For such systems it may be important to discern whether the membrane curvature or some other, already mentioned, non-trivial effect is the origin of the theoretical deviations. To gauge this, any electrostatic model for a planar charged membrane can be set to work, because deviations between simple, such as Gouy-Chapman approximations and more sophisticated electrostatic models for a spherical geometry and for a charged plane are the same. Spheres. Membranes of small liposomes, of pinocytic vesicles, of mitochondria, and, perhaps, of thylacoids [89] in dilute electrolyte solutions belong to the few membraneous systems in which the curvature effects may be significant. From the theoretical point of view, these systems can be described with various forms of the appropriate approximate solutions to the nonlinear Poisson-Boltzmann model [92,88]. For the spherical symmetry an explicit solution has as yet not been found. Only the problem of electrostatic repulsion between two spherical particles has been solved accurately [90]. Cylinders. On the length scale of Debye screening length, which is the appropriate electrostatic yardstick, most of the biological surfaces including membranes are not even near spherical; normally, they are either fairly irregular or cylindric in shape. Quasi-cylindrical symmetry, for example, is characteristic for all membrane protrusions, such as microvilli. Moreover, many membrane-associated macromolecules or their aggregates, such as parts of the cytoskeleton are, on the average,
324 prolate. * Fortunately, the r61e of cylinder curvature is likely to be insignificant unless the curvature radius is less than approx. 8 nm. Planar model, consequently, remains a valuable approximation also for studying the cylindrical m e m b r a n e structures, irrespective of the lateral distribution of the structural charge [94,93]. This has been confirmed [96] by using hypernetted-chain approximation.
surface electrostatics. In contrast to this, the consequences of the charge distribution perpendicular to the membrane are frequently quite pronounced and far reaching. As a rule, the finite width of the membrane-solution interface causes the surface values of the electrostatic membrane potential and of the electric field to decrease and the double-layer thickness to increase. This is reminiscent of the smearing of the surface charge perpendicular to the membrane. This causes the interfacial structure often to dominate the electrostatic properties of the charged membranes. Membrane models which do not allow for the charge distribution throughout the interfacial region are prone to overestimate the electrostatic membrane effects.
Surface curvature and discreteness of charge effects normally do not play a significant role in the membrane * Solutions to linear [93] and nonlinear [97,95,98] Poisson-Boltzmann equation for a cylindrical particle or a supermacroion can prove useful for describing the electrostatic properties of such systems; an alternative is to use the corresponding perturbative treatments [92]. They show, as in the case of planar membranes, that Gouy-Chapman approximation tends to exaggerate the double-layer thickness of smooth cylindrical bodies. Gouy-Chapman theory therefore predicts too high values for the electrostatic surface potential of such charged cylinders. Maybe this is the reason why the measured mobilities of cylindrical particles in the electric field better agree with the predictions of hypernetted-chain/mean-sphere approximation than with the results of Gouy-Chapman approximation [96]. Caution is required in evaluating such observations, however, since the attempts to model zeta-potential data for charged bodies are not at all straightforward. The zeta potential of a cylindrical body, for example, is a non-monotonic function of various parameters, such as charge density. For 2:2 salts, the zeta potential has a maximum as a function of Oet.Similar maxima have been predicted for the spherical particles also [99].
IV. I n t e r f a c i a l polarity a n d t h e d i e l e c t r i c c o n s t a n t
M e m b r a n e polarity (hydrophilicity) is extremely important. It is the primary reason for the water binding to a m e m b r a n e surface; but it also affects, directly or via the hydration effects, the chemical reactivity and the electrical conductivity at the m e m b r a n e surface, and participates in the regulation of the lipid-ion association, ion binding [101,151] and transport. Moreover, the efficiency of electrogenic phases has been claimed to depend first of all u p o n the value of the dielectric constant of the respective m e m b r a n e regions and only in the second place u p o n the distance between the redox groups involved [102].
x (rim) 0
0.5
80
1.5
1
2
2.5
5
I
B
I
80 A
"
iiiiiiiiil
"
i n t e r p h ! ~
70
60
60
v
•
•
@
@
@
to
50 40
x
30 20 apolar 10 0 0
j
lr ..... poa
i
i
0.5
1
:::::::::: :':.:.:.:.
L
1.5
x/~
CsCI
2O 0
I
I
I
1
2
3
c [ m o l . l i t e r -1] CH2
CO-O-- CH
O
H 2 C - O - - P - OR I
oo
Fig. 12. Interfacial polarity profile, expressed in terms of an effective dielectric constant, c(x), as a function of the interfacial membrane width, dp (A) or bulk electrolyte concentration, c (B). (A) Curves were calculated from a nonlocal electrostatic model (cf. Eqn. 40) using 78 and 2.5 for the static and high-frequency dielectric constant, respectively. Upper scale refers to the case of the system with a water structure correlation length = 0.15 nm. (B) Effective dielectric constant in the plane of ion adsorption adjacent to a charged phosphatidylglycerol membrane. Data were deduced from thermodynamic results by means of indirect calibration procedure [110].)
325 Membrane interior, on the one hand, is apolar with a static dielectric constant between 2 and 5, depending on the degree of water penetration into the membrane. The aqueous subphase, on the other hand, is polar and structured, characterized by a bulk dielectric constant near 80. The transition between these two limiting values takes place in the interphase region and inevitably is continuous. Consequently, near the membrane surface the effective local dielectric constant must be realized to vary smoothly, albeit non-homogeneously, between the values of 2 and 80 [103]-as a function of temperature, as well as dependent on the membrane, solvent, and electrolyte properties. Indeed, the potentiometric [104,105], spectroscopic [106-108] and kinetic studies [109] with various probe molecules suggest that the effective dielectric constant near the surface of artificial bilayer membranes is 10-70, depending on the precise location, probe, composition of the aqueous subphase, and membrane type. More polar and/or charged membranes typically are characterized by a relatively low interfacial dielectric constant value. Likewise, molecules with a big polar head decrease the relative permittivity of the interface. Increasing the bulk salt concentration for the uncharged phosphatidylcholine bilayers has been claimed to decrease the interracial dielectric constant [108] below the initial value of approx. 70 [110]. A reversed trend has been reported for the charged phosphatidylglycerol membranes [110]. The latter, under physiological conditions, are characterized by interfacial constant of around 30 (Figure 12). This discrepancy, in part, is a consequence of the different locations of the probe ions in the two cases; but screening of the surface charge and its consequences also are likely to be important. By all means, the published estimates of the dielectric constant for the membrane vicinity should be used with utmost prudence. Firstly, owing to the uncertainty of the underlying calibration procedures; and secondly due to the fact that the polarity near the membrane surface varies so rapidly with distance from the membrane (Fig. 12) that even minute molecular displacement or small configurational changes in the headgroup region are liable to drastically change the estimated dielectric constant value. Conformation changes, for example, can be inferred to explain different values for the interracial dielectric constant obtained with various techniques. Reduced value of the interracial dielectric constant has been modelled by theoretically treating the ensemble of rotating lipid polar headgroups as a collection of interacting dipoles embedded in a dielectric with an exponential profile variation [111]. Such approximation conveys only part of the truth, however. It namely neglects the interdependence of the interracial and solution structures, as well as the direct influence of the latter on the local dielectric response. This missing part
of the information may come from the studies on the molecular origin and (electrostatic) consequences of the membrane-water association. Non-local dielectric response and dielectric inhomogeneity (see Sections V and VI) play a rrle in this. The former has to do with the inability of the solvent molecules to respond to the rapid spatial field variations for small wave-vector (k) values (see also Section V); the latter is simply a consequence of the non-uniform solvent distribution across the membrane and its adjacent solution (see also Sections V and VI). Relative permittivity and polarity in the interfacial region both vary smoothly between the values characteristic for the membrane interior and the bulk solution. The effective interracial dielectric constant can, therefore, be anywhere between 2 < c id,:,. however [172-174]. D i v a l e n t and p o l y v a l e n t c(~,.,,.n.~ b i n d to the n e u t r a l p h o s p h o h p i d m e m b r a n e s by ih~'~;racting with the m e m b r a n e q u a n t u m - m e c h a n i c a l l y , m .~ stereospecific m a n n e r . C h a n g e s in the fluorescence p r o p e r t i e s of the trivalent transition elements, such as t e r b i u m , i n d i c a t e this [359]. In general, the stoichiometry of ion b i n d i n g to the m e m b r a n e is p r o p o r t i o n a l to the valency of the i o , b o u n d . F o r example, in organic
TABLE IV Representative ion-binding constants (litre tool s) for model lipid membranes Abbreviations and selected References: PC phosphatidylcholine (unpubfished data and [588,176,177,232,592,589,173,174,591,310,172,590,225,205, 175]); PE phosphatidylethanolamine (unpublished data and [593]); PS phosphatidylserine [594,595,596,353,597,167,185,598,600,599,225,601,603. 602,607,193,606]; PI phosphatidylinositol [608,609,595]; PA phosphatidic acid [610,595]; PG phosphatidylglycerol [unpublished data and 595,187, 465,611]; CL cardiolipin [593,220,612]. Less certain values are given in brackets. Only some of the references quote binding constants together with the primary data. Gouy-Chapman theory has been used to calculate the binding constants throughout. 1:1 PC PGPSPI PA-
Li + (0.5) 0.8 0.9
Na +
K+
Rb +
Cs +
CI-
0 0.15 0.15 0.1 0.2 Sr 2+
0 0.1 0.08 0.05 0.1 Ba2+
0 0.05 0.05 0.05 0.1 Ni 2+
Co2+
Mn2+
Cd: ~
6.7 23 49.5
M;
1:2
1.1 Mg 2+
0 0.65 0.7 0.6 0.8 Ca 2+
0.9
PC PGPS1 :3
1.85 12 15 La3+
2.5 17 26.5 Pr 3+
0.7 10 26.5 Nd 3+
0.5 11 38.5 Eu 3+
1.7 15 80 Tb3+
1.7 10.5 56 AI3+
PC
(250)
(80)
120
280
(250)
(300)
Br 2.0
C10480
335 solution at the saturation limit the binding stoichiometry for palmitoyl-oleoyl-phosphatidylcholine is one Ca 2÷ per two phosphocholine headgroups [175]; for La 3÷ the corresponding ratio seems to be 1 : 3 [176,177]. But the decision about the precise stoichiometry of binding is not always unambiguous. Often it is a matter of convention. Different combinations of the stoichiometric ratio and of the binding constant may describe the same set of data comparably well (cf. Fig. 21). Moreover, neglecting the interfacial structure, this is, disregarding the finite interfacial thickness and the membrane defects can lead to a serious overestimation not only of the membrane surface potential but also of the amount of counterions bound [178]. * In fact, one recent analysis has shown [174] that the density of the binding sites on phosphatidylcholine bilayers for the divalent and polyvalent ions may be far lower than the assumed stoichiometry would suggest [179]. Perhaps it is also dependent on the number of the surface defects or domain boundaries. When several types of divalent ions simultaneously interact with a membrane their binding may either be enhanced or else suppressed by the mutual occupation, depending on their relative concentrations [180]. Divalent transition elements always bind better than alkali earth elements [182]. Polyvalent ions bind even more efficiently, the higher their charge the more so [183,184]. One would expect this from the laws of simple electrostatics. But simple electrostatics fails to explain the tendency for the binding constant of phosphatidylserine to increase with the atomic number of the transition element bound [167,185]. I trust that this is largely a consequence of the quantum mechanical effects and of the high ionic polarizability. It is, therefore, not surprising that the augmentation of the ion binding to phosphatidylglycerol [187] is less than for phosphatidylserine, despite the fact that these two lipids are similarly charged. The former has namely a less tight, better exposed binding site for which the stereochemical, steric effects must be less important. Several lines of evidence corroborate the conclusion that the selectivity of ion binding to phospholipids, in general, is of quantum mechanical origin rather than being a consequence of simple surface electrostatics. Experimental as well as theoretical studies [188,158,189] show that the selectivity of cation binding originates from the ion interactions with the phosphate group (Fig. 22), whether this headgroup is charged or not, But the strength of such binding depends on the net mem-
* It should also not be overlookedthat large oligo-valent and polyvalent ions may loose part of their screening capability because of purely geometrical reasons [34]; they may, moreover, exhibit nontrivial hydration effects. Counterion depletion from the solution, finally, may be appreciable for very dilute solutions in the case of extremely strong binding [181].
t
Fig. 22. The intrinsically preferred structural form of the complex between two serine phosphate and one Ca2+ molecules(left) and the complex form in extended configuration. This picture was deduced from ab initio quantum mechanical study of 11 pre-selectedconformations of serine phosphate. It is indicative of an intermolecular chelation with 6 or 4-fold coordination of calcium binding, depending on the relative arrangement of serine phosphate groups. Phosphate atoms are shown black, oxygens shadowed, and calcium ions as concentriccircles.(From Ref. 189.) brane charge. Calcium ion binds to dipalmitoylphosphatidylcholine within 0.1-0.2 nm of the phosphate moiety, as witnessed by the neutron diffraction experiments [190]. Charged phosphate and, less so, carboxylic groups on the phospholipid molecules, consequently, belong to the main membrane binding sites for cations [191-193]. The same is true for model and biological membranes because monovalent cations do not bind significantly to gangliosides [67], in contrast to divalent ions [194]. This is indicative of the poor binding capacity of the sialic acid residues for the cations. Anions normally bind to the membrane-associated charged amines. Non-coulombic effects observed in ion-binding studies may have other sources in addition to direct quantum-mechanical interactions. Anything that either influences the ion distribution near the membrane [114,110,195], modifies the m e m b r a n e structure [189,196,197], affects the surface hydration [198201,110,197], or creates the membrane defects [174] can also interfere with the membrane-ion association. Even noncharged species, such as highly concentrated sucrose and glucose in the membrane-bathing medium, can enhance ion binding to the membranes [202]. Uncharged anesthetics exhibit similar effects, as has been documented for the Ca 2÷ binding to the biological membranes [203,195]. One possible and probable explanation for this is that the noncharged but polar substances change the structure a n d / o r hydration of
336 the interfacial region. An indication for this is that the selectivity of binding is reversed upon the dilution of the negatively charged diacylglycerophosphoglycerol in the model lipid membranes by the uncharged glycolipid dimanosyldiacylglycerol [204]. Ion binding evokes structural changes both at the level of single molecules and of the whole membranes. Lipid headgroups with bound ions, for example, may turn out of the surface plane if the complex is sufficiently hydrophilic and charged, as witnessed by phosphorus nuclear magnetic resonance [172,205,206] and X-ray diffraction [207]. Surface charges increase the propensity for the molecular out-of-plane fluctuations [120] and (partial) elimination of the net charge leads to a reduced lipid headgroup mobility [208]. Headgroup deprotonation and ionization may lead to a hydrocarbon-chains tilt [211]; but also protonation of the headgroups which participate in hydrogen-bonding at very low pH may cause the lipids to form partly dehydrated bilayers with tilted chains [209]. Complete headgroup protonation decreases and maximal deprotonation, as a rule, increases the membrane hydration [210,211,123]. In final consequence, ions can force strongly hydrated lipids membranes in the ordered phase to form structures with interdigitated chains ([212,213], Kirchner and Cevc, unpublished). The major reason for such changes is that the membrane components or their parts are electrostatically and, even more so, hydrationally repelled from the interface. Salts always affect membrane polymorphism. Co-existing lamellar bilayer structures with different membrane-membrane separations are induced by Ca 2+ ions. This holds for pure [214,179,215] as well as for mixed [216], such as dipalmitoyl phosphatidylcholine/ cholesterol [217], membranes. In the latter, mixed systems the effects are somewhat smaller [218]. Ion binding, furthermore, affects the membrane phase transition temperatures, most notably - but not most strongly the chain melting transition temperature of the lipid bilayer (see section VII-A for a discussion of this effect). When the resulting process is associated with membrane dehydration the final phase can be of non-bilayer type [219,220]. Salt-induced phase transition temperature shift is in most cases [123] towards higher temperatures. For a one-molar monovalent salt solution it is on the order of 1 K or less, in the case of zwitterionic lipids, such as dimyristoylphosphatidylcholine *. Comparably concentrated divalent ions cause upward shifts of up to 7 degrees; the change of the phase transition temperature induced by a typical trivalent ion, such as La 3÷, is
* An exception is the lithium ion which produces similarly great shifts already in decimolar concentration range, owing to the strong phosphate-ion association.
similar [177]. But in diluted electrolytes the shift magnitude increases with the ion valency in the sequence: La 3+ >> Ca 2+ > Cd 2+ > Co 2+ >> Cs +-> Rb ~ > K * Na +, etc., below the saturation limit. Such dependence can not be described in the terms of mere electrostatics without invoking ion binding. Such binding, indeed, is very probable for the divalent and multivalent ions, but not inevitably necessarily for the monovalent ones; for the latter, other corrections discussed in Section II may also be important. The Poisson-Boltzmann equation suggests that the concentration dependence of the phase transition shift should increase approximately linearly with the square root of the bulk concentration. When this is not the case non-trivial ion-membrane interactions are probably a part of the game. Due to the gathering of counterions in the vicinity of the charged membranes the net membrane surface charge reduces the concentration required for an appreciable salt-induced shift of the bilayer transition temperature. The thermodynamic effects of the ion binding are therefore better expressed for the charged lipids. partly owing to the energetic contribution from the surface electrostatics and partly due to the changes in the membrane hydration. It would seem that the magnitude of the ordinary electrostatic phase transition shift is normally quite small, however, only a few degrees for the membranes composed from the biological lipids. By all means it is smaller than the corresponding hydration induced shift [20,123]. The reason for this is that the electrostatic membrane free energy is typically much smaller than the free energy of membrane hydration (cf. Fig. 24). Knowing this it is easy to comprehend the origin of the chain-melting transition temperature increase observed upon ion binding to the neutral phospholipids [177,221-223]. Such increase may exceed 50°C, if the interfacially bound ions have displaced most of the bound water from the interface. Essentially, anhydrous lipid-ion complexes in excess solution are no exception [224,143]. One example of this are metal-ion complexes of diacylphosphatidylserine bilayers [225]. These form highly ordered, essentially water-free bilayers with extremely high transition temperatures in the range between 151-155°C. Another example are (nearly) completely protonated lipid membranes and membranes with bound lithium which are also often nearly anhydrous [20,219]. As a rule, the highest chain-melting phase transition temperatures for diacylphospholipid membranes with bound monovalent ions or protons do not much exceed 100°C due to the lack of the strong intermolecular ionic coupling. Ions also induce lateral phase separation of phospholipids [226], acidic at least [227], gangliosides [228], and, perhaps, proteins in the membranes. Isothermal phase transitions upon ion binding also may occur [604].
337 1.2
B
DPPC C 0.5 m
0.8
A
o/°\
O.t,
ApH
v
/
0.3 0.2
=,. ..t)
0.4 k. ® rl
0
O
/
0.1
0.6
Tm 0.2
O o~
!
0
!
__L
6
2
° J"T"~ ' S
pH
0 20
i
i
i
i
i
25
50
55
40
45
50
T (*C)
Fig. 23. Effect of changing membrane properties on the ion binding to and transport through the lipid bilayers. (A) In the suspension of phosphatidic acid methyl-ester vesicles in 0.2 M NaCl-with a pK around 3.9-proton pulse is observed at the chain-melting phase transition (From Ref. 230.) (B) Similarly, a permeability maximum is caused by the hydrocarbon chain-melting in phosphatidylcholine bilayers which may be absent for other phospholipids.
Ion-induced phase transition shifts may go in either direction. When the membrane-ion complexes bind water more strongly than the membrane surface without bound ions the ion-induced shift of the bilayer main transition temperature is downwards, even for the charged membranes. This is the case with phosphatidylcholine in the presence of anions or with phosphatidylserine with bound organic counterions. The chain-melting phase transition temperature for such systems therefore decreases with the increasing bulk electrolyte concentration. Ion-lipid association is apt to change at all membrane phase transitions which change significantly the properties of the interfacial region. For zwitterionic lipids, such as phosphatidylcholine [229], and for certain anionic phospholipids, such as methyl ester of phosphatidic acid [230,167], calcium binding is stronger in the ordered gel phase; the affinity of phosphatidylserine bilayers for magnesium decreases with the degree of chain unsaturation and fluidity [170]. During the chain-melting phase transition, some of the bound protons [231] and nearly one half of the bound calcium ions consequently are expelled into the bulk solution [230] (Fig. 23A). The same is true for the interaction between Tl+-ion and dipalmitoylphosphatidylcholine, as witnessed by N M R [232]. Under proper conditions, membranes with ionizable groups may therefore function as a reservoir for protons and other ions: changes in the membrane structure can terminate with an ion pulse. Such peculiar behaviour is not observed for all membranes, however. The carboxylic group of phosphatidylserine, for example, which is located deep in the interfa-
cial region, is not fully accessible to divalent ions, such as calcium, unless membrane is in the fluid state. Binding pockets may also form with other headgroups if these interact strongly. Sensitivity of sodium binding to D- or L-phosphatidylglycerol head-group on the headgroup isomery corroborates this conclusion [233]. Permeability of membranes with bound ions differs from the leakiness of the corresponding virgin membranes in dilute salt solution. This is important because many transport studies are conducted in highly concentrated electrolytes or in solutions containing polyvalent ions. The latter may cause severe membrane perturbations and may even release the content of the membrane vesicles, probably by inducing membrane dehydration and lateral phase separation [183]. Ion-induced leakage is best established for phosphatidyserinecontaining lipid vesicles. It has also been observed with other anionic phospholipids, however. When it occurs during mutual membrane approach and partial surface dehydration it frequently terminates in a salt-induced fusion between the artificial lipid vesicles [234]. It is perhaps useful to remind that dimethonium, an organic cation with two charged quaternary nitrogens separated by approx. 0.3 nm, has been reported not to bind to the lipid membranes and yet to screen the electrostatic membrane potential [235]. This ion may therefore be used for studying certain aspects of the divalent-ion-membrane interactions. Dioalent and polyoalent ions normally bind to the charged membranes. Monooalent ions can but need not always do so. Deoiations from Gouy-Chapman theory, correspondingly, are sometimes indicatioe of the ion binding but may also be a consequence of other non-trioial
338
electrostatic, structural, or solvation effects. Association between membranes and ions always increases with increasing local ion concentration and ion valency. High ion polarizability also is often advantageous. The effective ion concentration near a membrane surface differs from that in the bulk, owing to the redistribution of the ions in the electrostatic, hydration, and steric-repulsion fields of the membrane. Consequently, increasing the net charge density on the membrane and decreasing the membrane hydration, which typically supports (counter)ion gathering in the interfacial region, enhances the ion binding to the membrane surface. Steric constraints are detrimental for the binding of monovalent ions," but intermolecular bridging or chelation of the membrane components in the case of the divalent and multivalent ions, as a rule, are only possible when the binding sites are mutually close. Membrane properties are strongly sensitive to the interfacial ion concentration and vice versa. Fine membrane structure often changes upon the ion binding. This is primarily owing to the changes in the membrane hydration and to the formation of the intermolecular bridges. The interfacial hydration typically increases and the membrane packing density decreases with the increasing polarity of the ion-membrane complexes. Membranes which bind small, especially polyvalent, inorganic ions at their main hydration sites become less hydrophilic and more rigid," sometimes they even form nearly anhydrous aggregates in excess solution. But larger bound ions, in particular the organic ones, often promote membrane solvation. Charged membranes expel small, interfacially bound ions during the phase transition. Ion pulses may be generated in this way. Organic ions, however, have a stronger affinity for the fluid than for the condensed bilayer phase. VI-C. Ion transport Transbilayer concentration gradient, which represents a sort of the chemical potential, or electrostatic transmembrane potentials, drive a flow of ions or other solute molecules across the membrane. This flow is either direct and diffusive, through the membrane core, or pores; otherwise, it involves ionophores or carrier molecules. Ionophore or carrier mediated flow is controlled by the properties of the ionophore or carrier itself; the interracial molecular concentration, which depends on the membrane surface electrostatics also plays some r61e. Such flow is normally modelled as a multi-step kinetic cascade, of which the first step, the interfacial ion-carrier association, is mostly the decisive one. In the simplest picture of the membrane permeability barrier the bilayer is envisaged to consist of the resistance of the hydrophobic membrane interior and of two interracial barriers (see Appendix F). For small ions the
former depends mainly on the membrane partition coefficient and is determined mainly by the large electrostatic (Born free) energy cost of charge transfer between the membrane interior and the solution [236], unless the ion permeation occurs via pores. The pore resistance to the ion permeation, as a rule, is much lower than the resistance of the bulk membrane core. as has been shown in the pioneering work by Parsegian [237]. This is true no matter whether or not such subtle electrostatic effects are important [238]. Ions thus normally trespass the membrane interior via channels, pores, or packing defects. Notwithstanding this, the overall membrane electrostatic may be important for the diffusive mass transport since, at least for the small solutes, the surface barrier frequently is substantially higher than the remaining transmembrane resistance [239]. Passive membrane conductance for ions controlled by the interracial permeability coefficient depends directly on the interracial barrier height. The latter can be evaluated by means of the Eyring absolute rate theory to be: P, nt = dpki, = ( d p R T / N ~ h ) e x p ( G * ( x ) / R T ) , where h is Planck's constant, and the total activation free energy is G*(x). In the first approximation, this activation energy can be set equal to the potential of mean force for the ion approaching a membrane (see Appendix E). Membrane permeability for ions increases with the ion size and decreases with the interfacial hydration, at least when the interfacial region is the rate-limiting zone and no specific ion-lipid association takes place. Direct ion binding to the membrane may compensate for this and the accumulation of ions in the interfacial region, brought about by the membrane electrostatics, as well as the membrane packing inhomogeneities are always advantageous for the ion transport [240]. Unstirred water layers also present a barrier for the transport of uncharged species across the membrane [241]. Ions which cross the membrane in the form of molecular complexes are less sensitive to the interfacial electrostatic permeability barrier than bare ions [242]. Perhaps this is the reason why halide ions permeate through the lipid bilayers relatively rapidly [243,244]. The other possible reason is that halides bind to the lipid heads. By all means, the halide ions, and possibly some other ions as well, cross the membrane in a bi-modal manner. The fast component, with a permeability of the order of 10-8_10 6 cm 2 s-1, is a function of the inter-ion or ion-lipid complex formation. It thus relies directly on the carrier activity of the lipid molecules. The slow component, for which permeability values on the order of 10 -H cm 2 s -~ are measured, is probably due to a single ion diffusion. The exchange transfer of H + / O H across the lipid membranes is also biphasic, but even faster than for the halide ions. The fast component with a permeability in the order of 1 0 - 6 "( P H , < 10 - 3 c m 2 s 1 has been pro-
339 posed [245,246,247], but not unambiguously proven [248] to occur via a network of pre-associated water molecules in the bilayer. The slower H÷/OH--flow depends on the counterion diffusion, however, and resembles that of other ions [246,249,250]. Ion transport across the membrane normally increases with the increasing temperature and sometimes exhibits a maximum near the chain melting phase transition [251,252] (see Fig. 23B). Critical fluctuations in the lateral membrane compressibility close to the bilayer phase transition [253,254], as well as the mismatch in molecular packing between coexisting gel and fluid regions at the phase transition [255] have been claimed to be responsible for such maxima. But other factors can be important as well. For example, stress in the lipid bilayer may open-up pores in the membrane interior [256]; the tendency of the lipid molecules to form highly curved surfaces, which is stronger for the more strongly charged and hydrophilic molecules [257,258], may also be beneficial [259]. * By all means, single ion channels have been reported to arise in pure phosphatidylcholine bilayers in the critical temperature region [260]. Moreover, the flip-flop rate upon binding of some divalent ions [261] or upon lipid chain fluidization in the presence of Na-ions [263] has also been shown to increase. Both latter effects could contribute to the ion permeability maxima in the case of lipidic ion-carriers [264]. In complex artificial lipid bilayer the permeability pattern is dominated by the short-chain lipids, especially by phosphatidylcholines [252], or by the charged lipids. It is unlikely, however, that the ion transport through the biological membranes goes directly through the hydrophobic bilayer core. It is more probable that it follows the paths along which the energy barrier for the permeation is lowered by the formation of the ion-lipid complexes and subsequent transmembrane movement of such complexes and/or is reduced by the lateral membrane density fluctuations. It may be, therefore, biologically relevant that the latter two phenomena are facilitated by the proton binding to the membrane and by the indirect surface charge effects [265]. The latter may also influence the ATP-dependent ion transport across membranes [266]. Charge transfer along the membrane surface has been rarely considered. But it is well worth concerning. In particular, proton transport is interesting from the
* The lack of an ion permeability maximum for phosphatidylethanolamines in the transition region, for example, might be caused by the inabilityof this lipid to form such curved areas. But the absence of a Rb+-permeation maximumat the transition of unilamellar phosphatidylcholinevesicles remains enigmatic [262] unless it is caused by the low capacity of this ion to bind to the lipids [230].
point of view of the membrane energetics and surface catalysis. Indeed, protons in the interracial region have first been theoretically hypothesized [267] and then claimed on the basis of measurements [268] to participate in the lateral membrane conductivity. There is some controversy about the interpretation of certain experimental data [269,270]. Anyhow, in order for such conductivity to be high, interfacial proton networks should exist. One possibility is that these involve the strongly bound, and thus well oriented and coordinated water molecules [271]. Such is the case with phosphatidylcholine membranes which, consequently, exhibit a very strong hydration dependence of the electric current along the membrane, reminiscent of Eqn. 40. Another, as yet tentative, possibility is that 'proton-wires' involve the membrane components which are mutually hydrogen bonded. Phosphatidylethanolamines and phosphatidylserine molecules are suitable candidates for the participation in the latter process. At the applicative, technological end of the electrostatic membrane research the development of solid-supported membranes attracts much attention. Bilayercoated semiconductors, through which charge transfer is induced, have been devised and proposed to act as a molecular energy device [272]. Other solid-supported membranes were used as bio-sensors [273-276]. Future applications could rely on the interfacial modulation of the enzyme activity, for example, or make use of the membrane transport system, the voltage dependent channels, etc. [277]. Many of the synthetic ionophores are also physiologically or therapeutically important. They act as antibiotics, presumably because they dissipate the transmembrane potential and concentration gradients. Calcium-transport behaviour of the channel-forming peptide alamethicin, for example, has been shown to be potential-dependent, in contrast to calcium transport by the ionophore A23187. The latter in one study has been found to be unaffected by the potential [278]; but in another report the depolarization of plasma membrane potential has been claimed to have a potent inhibitory effect also on A23187 and ionomycin mediated ion flux [279]. The reason why the membrane electrostatics is important for all such applications is that the activity of many membrane associated (macro)molecules is governed by the characteristics of the membrane-solution interface [281,282]. The conductance of many ion channels is determined by the electrolyte type and concentration at the two ends of the channel opening-up in the interfacial region. The membrane surface potential thus increases or decreases the interracial ion density, depending on its relative sign, and affects the ion flux through the channel. Gramicidin pores provide one [283], but not the only [284] example for this. The influence of the electrostatic effects on the conductance of gramicidin is greater for neutral channels embedded
340 in negatively charged bilayers than for related anionic channels in neutral lipid bilayers. For sufficiently low ion density or high transmembrane potential, the ion flux through all channels may be more sensitive to the electro-diffusion of ions through the electrolyte, in the close vicinity of the channel mouth, than to the intrinsic channel properties [285,286]. This does not hold for voltage-gated channels, of course. Transmembrane potentials, moreover, are prone to increase the rate of the membrane pore formation by enhancing the thermal membrane fluctuations [287] which then act as pore nuclei. Suppression of such fluctuations upon membrane rigidification therefore should lower the channel conductivity. But it also may be a consequence of the increased lateral pressure of the chains, which lower the probability for the internal opening-closing motions in proteins [288,289,291]. Channel conductivity may be thus indirectly, via a change in the membrane thickness [292], as well as directly regulated by the membrane surface electrostatics and polarity, and vice versa. Potential-induced changes of the channel conformation or orientation in the membrane enhance these effects further. The gap between simple electrostatic models of the ion permeation through the membrane channels [293-295] and more realistic descriptions of the transmembrane ion permeation process which include the structural details remains to be closed. Dramatic differences in the membrane permeability for organic anions and cations [296-299] have repeatedly been claimed to arise from the dipolar membrane potentials. It would seem, however, that the contribution from the interfacially bound water is at least comparably relevant. Two arguments speak in favour of such hypothesis. Firstly, the fact that phosphatidylethanolamines and phosphatidylcholines which have a similar dipolar headgroup moment [613,158,114,251,252] have entirely different permeation characteristics [251,252]. And secondly, the observation that the rate of transfer of (poly)ions across the bilayers of dialkyl (ether-bound) and diacyl (esterified) phospholipids are similar [300] despite the fact that the former lipids carry two extra, highly polarizable, carbonyl groups with a large dipole moment. When interpreting experimental data obtained with complex membranes one should be careful to consider different possible explanations. Valinomycin, for example, one of the most popular ionophores, also binds to the membrane proteins and thus may influence system properties in a non-trivial manner [290]. Gradients near or across the membrane cause a net flow of material along or through the membrane. The ion flow is governed by the electric potential difference and, in many cases, by the interfacial permeability barriers. Steric effects and hydration of the membrane surface contribute to the latter. Membrane permeability for an ion which has
not been dehydrated in the interfacial region is governed by the high energy cost of the ion-charge transfer from the highly polar bulk in the apolar membrane interior. Membrane pores or the formation of ion-carrier complexes can reduce this energy requirement dramatically. Consequently, such phases have a lions share in the ion transport across the biological membranes. The membrane surface electrostatics plays a role in transmembrane transport by affecting the interfacial ion concentration and by increasing the probability for the pore and ion-carrier complex formation.
VII. Thermodynamics of charged membranes Membrane and its surrounding aqueous subphase form a single thermodynamic entity. Regretfully, the basic thermodynamic quantities of such system are not directly amenable to experiments. Nevertheless, it is worthwhile to answer the question of their evaluation and relative magnitudes, since many properties of the charged membranes are affected by the surface thermodynamics (see also Appendix G). Lateral pressure of the membrane is believed to participate in the regulation of the membrane polymorphism; this pressure is nothing else but the surface density of the electrostatic membrane free energy. Similarly, the intermembrane pressure, which controls the interactions between the adjacent membranes, is, formally speaking, given by the variation of the membrane free energy with interfacial separation. VII-A. Free energy As long as the interfacial hydration is neglected the free energy of a charged surface [4,301,302] is identical to the electrical work done in charging up the surface (cf. Eqn. 53). * (In Appendix G it is briefly discussed how the electrostatic coulombic part of this work can be calculated.) In the linear regime of Gouy-Chapman approximation the dominant contribution to the electrostatic free energy comes from the internal energy, which is proportional to volume integral of the square of the electric field; in the nonlinear, high potential case the entropy term dominates [70]. As has been noted already, the surface charge density of the biological membranes is typically in the order of 0.025 C. m -2 or less. This means that the coulombic part of the electrostatic free energy for such membranes in a 0.1 molar, symmetrical, univalent solution is unlikely to be greater than 1 kJ mo1-1 (cf. Eqn. 54); this is below thermal energy and thus seldom significant. But the surface charge density of the artificial lipid bilayer
* F o r f o o t n o t e see n e x t p a g e .
341 membranes may exceed 0.25 C . m -2 ( = e / 0 . 7 nm2). The electrostatic free energy of such systems, correspondingly, could in principle be high, < 10 kJ mol- 1 (cf. Eqn. 56 and Fig. 24). I believe, however, that in reality the coulombic free energy of most membranes is only small, below 2 kJ mo1-1, owing to the interracial structure effects. Standard electrostatic membrane theory neither accounts for the interracial structure nor, and more importantly, does it allow for the membrane hydration effects. Especially the latter handicap may have far reaching consequences owing to the high value of the free energy of the membrane hydration. This is best seen if the integral free energy of a charged and hydrated membrane surface is written as a sum of the individual coulombic and hydration contributions (cf. Eqn. 57 in Appendix G) in analogy with the well-known result of classical electrostatics for coulombic free energy (Eqn. 53) [141,20]. For a given net surface charge density the membrane hydration causes the electrostatic part of the free energy to increase relative to the corresponding Gouy-Chapman result, the latter pertaining to a 'nonhydrated membrane'. This is because it includes the positive, 'coulombic' part of the total free energy of hydration, reminiscent of the increase of the electrostatic surface potential by the surface hydration. Correspondingly, it is the negative, 'chemical', contribution which minimizes the total free energy of the membrane-electrolyte system and guarantees the stability of hydrated membranes. This 'chemical' contribution, which is supplied
* If the electrostatic surface potential and not the net charge density at the membrane surface is fixed, which is rather improbable for the biological systems, a 'chemical part' of the double layer energy must also be included, which diminishes the total value somewhat. The reason for the difference between the cases of constant surface charge density and constant surface potential is the following [4]. If in the final state of equilibrium, an ionic excess of one species has been created in the double layer, there is obviously a chemical preference of these ions for the surface. During the transfer of ions from the bulk to the interfacial region there is, therefore, a gain in free energy, corresponding to the chemical free energy difference between the bulk and the bilayer surface. In the final state this free energy difference exactly compensates the electric potential difference due to the double layer and hence is equal to other part of the free energy change is associated with the build-up of the double layer. During the (hypothetical) charging process, the bilayer surface potential gradually increases from zero to the final potential value, ~k0, and the surface charge density correspondingly rises from zero to oct. The potential opposes ion transport from the bulk solution, where the electrical potential is zero, to the bilayer surface and hence the energy consumed in the whole charging process is that given by Eqn. 53. This may be called the electrical part of the free energy of the double layer in contrast to the chemical part, -
NAAcoetd/o.The
N.tA~oet~o.
40 c=0.1 M
.
s s
, ,,, "
o
E
~
-40 ( = 0.15 nm
0
0.1
%%% % %
0.2
"~.
0.3
Fig. 24. Electrostatic free energy (dashed curve) and hydration free energy (full curve) of a membrane as a function of the net surface charge density, or of the surface density of the local excess charge, ap. Curves were calculated for dilute electrolyte solution (c =10 -4 mol litre-1 ), where charge screening is essentially negligible, by means of Gouy-Chapman approximation and nonlocal electrostatic model, respectively.
act,
by the energy gain from the direct water binding, often is by more than one order of magnitude greater than the coulombic part. For phosphatidylcholines, for example, the absolute magnitudes of both hydration free energy components are > 10 and < 1 kJ tool -1, respectively. The nonlinear version of Eqn. 10 [122] and experiments [122,123] show this in detail. A feeling for the system characteristics is provided, however, already by the approximate linear expression for the total free energy of a hydrated charged membrane:
Gtot=NAAc{+ ~a2h[ 1 +
~(~-1)]
-t- c02` -~, (1- ~-~))
(10)
where the positive and negative signs refer to the cases of fixed surface charge density and potential, respectively. More general result is given in Appendix G. Eqn. 10 is valid for dilute electrolyte solutions within the framework of nonlocal electrostatics in the linear approximation (see Eqns. 37, 38 and 57). The underlaying approach is thus not exact and may result in 50% misjudgements. But it has the advantage that it permits estimates to be made about the effects of the ion screening on the total membrane free energy with the hydration being included.
342 VII-B. Electrostatic intermembrane pressure
Electrostatic intermembrane forces can be repulsive or attractive, depending on the interfacial separation, on the surface hydration, or on the sign and distribution of the surface charges. Electrostatic repulsion occurs between adjacent membranes because there is an excess of counterions in the region of overlapping double layers (Fig. 25). These ions are not at equilibrium. Therefore, a local excess osmotic pressure is set up between the membranes which drives solvent molecules between the interacting surfaces and tends to push the membranes apart. Standard coulombic repulsion between the charged surfaces is thus driven by the entropy of counterion distribution. The electrostatic repulsive force decays approximately exponentially on the length scale of DebyeHtickel length (see Appendix H). At separations between 2 and 5 nm this force for the model membranes is the prevailing repulsion (cf. Table III and [304]). Membranes with a thick, (partly) ionized surface-coat [303] or flexible membranes [305-307] may, however, exhibit a strong electrostatic repulsion at even greater separations, because the range of the electrostatic force is extended by the effects of the interracial structure and
I
hydration
~ 6
Ac [nm2]
k
0.7
~II.KX~I el. repuls,on
~,.0 A 7.0 o
5 I --__1__// , L|
0
t
4.0
,
I
8.0 dw[nrn]
I
I
12.0
I
I
16.0
Fig. 25. Electrostatic repulsive pressure between the lipid bilayer membranes of increasing surface charge density and decreasing area per charge (oct = e/A¢). Data stem from the osmotic pressure measurements of Rand and Parsegian (From Ref. 179).
fluctuations. In any case, suspensions of charged membranes attain a gel-like appearance, owing to the electrostatic coupling between the membranes, when the sah concentration is sufficiently low and when the concentration of the membrane vesicles is high enough. This effect, and the electrostatic repulsion in general, are more pronounced for the fluid membranes because of the dynamic, fluctuation-mediated force-enhancement. On the one hand, cations bound to the negatively charged membranes-predominantly to the lipid and to the glycolipid molecules-may diminish or nearly alleviate the consequences of the net membrane charge. Charged lipid vesicles therefore often aggregate and fuse after the addition of salts (for a review see Ref. 234). Increasing ion valency and surface charge density enhance such phenomena [383]. In complex electrolyte solutions the effects of the ion binding, especially for the polyvalent electrolytes, as well as the consequences of the ion-screening are simultaneously operative [308,309]. On the other hand, the ion binding to initially neutral membranes imposes a net electrostatic charge on the surface [179]. Consequently, even such membranes which are initialy per se uncharged may experience an electrostatic repulsion. For example, bound calcium or bound halide ions bring a net charge on the zwitterionic, such as phosphatidylcholine, membranes [310,311,175]. This creates a positive electrostatic membrane potential of 30-65 mV [179] in the millimolar or up to decimilar concentration range of this ion. Calcium and other bound ions thus cause appreciable electrostatic intermembrane repulsion in the appropriate concentration range. Only in relatively highly concentrated solutions of polyvalent ions such repulsion is screenedout or even replaced by a net, ion-dependent intermembrane attraction. Electrostatic attraction between membranes is a manifestation of van der Waals force [312] and of the higher-order electrostatic intermembrane correlations [20,313-319]. The latter reduce the repulsive entropic term and, moreover, give rise to an extra attractive energetic contribution. Membrane hydration decreases the former, close to the membrane by as much as 80% [320]; but it increases, which is more important, the latter contribution by more than one order of magnitude [321]. Electrostatic attraction arises, for example, between the membranes which attract dissolved ions as soon as the ionic distributions near the interacting surfaces are concerted. The resulting force depends sensitively on the interionic correlation function [313,315] but also on the dielectric properties of the interfacial region. The force magnitude, on the one hand, increases with the increasing density of the charges in the interfacial region and with the width of this region [69] (Fig. 26).
343 ,
,
7 -~~x~x ~ .~ ,,q-
Z
,
!
Ac=0/'*nm2 ~..
E 6-
"
-
"
-
4
0
t
0.5
I
I
1:0 1.5 dw [ nm]
2.0
Fig. 26. (Electrostatic) Attraction between two charged membranes caused by correlated distributions of interfacially compensated structural (full curve) or the ionic charges close to the membrane surface (dashed-and-dotted) in a 1.0 M monovalent electrolyte. The nominal surface charge densities are 0.4 and 0.23 C.m -2, respectively. The dashed lines refer to standard van der Waals force. (Modified from Refs. 61 and 316.)
Increasing salt concentration * and ion valency also enhances the intermembrane electrostatic attraction. Intermembrane attraction from the interion-correlations, moreover, is greater for the lower values of the dielectric constant. Such attraction, therefore, is relatively more important for the closdy apposed membranes since the dielectric constant in the interjacent region is then lower. This enhances the ion-correlation effects. Ultimately, the attractive component of the intermembrane force can be sufficiently strong to result in a net attraction even for equally charged membranes (cf. Fig. 26). (In other words: one can not squeeze out the solution if the two surfaces are not present simultaneously owing to the fact that the correlation effects cause the charge between membranes to be less than the charge on each separated membrane surface.) Numerical simulations [41,61,324-326] and analytical theories [69,327] prove this. A related 'electrostatic' attraction may arise between uncharged membranes immersed in water [313,330,20, 318,321]. A prerequisite for this is, that the distributions of the local atomic charges and/or of the dipoles on the interacting surfaces are correlated. I speculate that one of the reasons for the stronger attraction between the
* The reason for this is the following. Interion correlations near the membrane-solution interface cause the counterions to concentrate more towards the membrane, thus reducing the electrostatic repulsion between their associated electric double layers. Furthermore, correlated fluctuations in the ion clouds at the surfaces lead to an attraction of true van der Waals type. This type of interbilayer attraction is favoured in the 'strong-coupling' limit, i.e., with high surface charge density, low intedamellar dielectric permittivity, low temperature and with polyvalent ions [61].
artificial lipid membranes in the gel or crystalline phase is a correlation-dependent force of such an attractive type; excessive 'van der Waals' attraction between phosphatidylethanolamine bilayers might also have similar roots, as concluded from the fact that phosphatidylethanolamine headgroups have an intrinsic tendency for the formation of ribbon-like, interfacially correlated headgroup patterns [116]. Electrostatic attraction stemming from the intercharge correlations can exceed normal van der Waals attraction by more than one order of magnitude, at least when the interacting membranes are uniform and sufficiently close [61,316,321,324,322]. This attraction should be especially strong when the system contains multivalent or polyvalent ions. The correlations dependent force can thus be postulated to be one of the potential origins for the diminished lipid swelling in the solutions of divalent electrolytes, as compared to the monovalent electrolytes [217,179,70]. But unfortunately to date no simple analytic model of the correlations-dependent attraction has been devised. * For practical applications the attractive intermembrane electrostatic force, consequently, must be evaluated numerically from the detailed electrostatic surface models [69,328,321,331] or by using phenomenological expansion series [113]. Depending on whether the charges, which are the cause of anomalous attraction, are membrane-associated or ionic the correlationdependent force is expected to have a range of either less than 1.0 nm, or in excess of 1.5 nm, respectively [61,69]. At short separations from the surface such force is likely to be always appreciable. Consequently, the attractive force contribution should not be forgotten in the data analysis, especially when the results pertain to the specimen containing divalent and polyvalent ions. Electrostatic attraction between charged membranes has been measured directly for phosphatidylglycerol membranes [329]. Perhaps it is also the origin of the experimentally observed long range ' hydrophobic' interactions [335,334]. In sufficiently strong external electric fields the lipid vesicles, the membrane particles, and cells tend to aggregate and form long chains along the field direction (see, for example, [336,337]). This is probably caused by the induction of dipole moments on the (polyelectrolyte-like) membrane surface rather than being a consequence of the non-trivial intermembrane attraction. Such pearl-chain formation, consequently, is more pronounced in less concentrated electrolyte solutions. Field-induced membrane aggregation and fusion can be
* An exception is the 'counterions-only' model [332,333] in which the importance of correlations, and thus of the corresponding attraction between the charged surfaces, has been shown to be measured is by a dimensionless parameter oet(Ze)5/[dw(kTc%)3].
344 used for hybridoma production [338]; they are more efficient for the less charged cells because such cells repel each other relatively weakly. For the same reason electro-fusion efficiency varies with the cell cycle, the stage of cell differentiation, metabolic cell activity, etc. VII-C. Electrostatic lateral pressure
Surface density of the membrane free energy, or, strictly speaking, the derivative of this energy with regard to the surface (or molecular) area, gives an estimate of the lateral pressure within the membrane plane [301]. Within the framework of Gouy-Chapman theory this pressure is invariably found to be identical to twice the negative surface density of the internal energy, with an upper limit Ilet = -F~el, 0 in the high potential case [301]. The actual value of such pressure is probably < 18 mN • m -2, and thus less than the corresponding lateral pressure of hydration, 40-60 mN m-2 [143]. The main reason for this is the relatively slow decay of the electrostatic interactions compared to hydration. This causes the coulombic ('macroscopic electrostatic') interactions to prevail in the intermembrane interactions at large separations ( > 3 nm) but makes hydration ('quantum-mechanical electrostatics') the dominant factor of the short-range intermolecular forces. This explains why the lateral packing density and the molecular area of charged and uncharged lipids in the fluid phase are approx, the same, 0.75 < A,, < 0.85 nm 2 [339], as long as lipids are comparably hydrated. The curvature dependence of the electrostatic interaction, which has been proposed to determine the curvature elasticity of the fluid lipid bilayers [340], for the same reason is likely to be predominantly a function of the membrane surface structure and hydration [341]. VII-D. Transmembrane pressure
Transmembrane electric fields impose an electrostatic pressure across the hydrophobic membrane interior. High field strengths in excess of (1-3). 106 V. cm-1, which corresponds to the transmembrane potential jumps of 0.2-1 V, may dramatically diminish the membrane thickness and resistance. A high frequency field or a field pulse of sufficient strength may induce electric membrane breakdown within 10 -8 s. This rapid process is probably a manifestation of the field-induced changes in the lipid conformation and is less probable for the membranes stabilized with sterols [342,343]. Moreover, it reflects a concomitant local membrane expansion and defect formation. But the initial, reversible breakdown often leads to a mechanical breakdown during which a membrane ruptures. (In the case of cholesterol-free lipid bilayers the latter breakdown occurs within a period of 10-3 s or
longer [344,345].) The irreversible breakdown is an outcome of the membrane electrostriction and is theoretically discussed in Appendix I. Another possible source of the transmembrane pressure gradients is the membrane asymmetry. This is created and supported either by the asymmetric concentration or interactions between the transbilayer components and cell constituents or else, in the artificial systems, arises from the non-ideal mixing in the membrane. Membranes of certain small sonicated vesicles are asymmetric because of the latter reason [346]. The asymmetry of such unilamellar vesicles (r,, _< 30 nm) made of uncharged phospholipids phosphatidylcholine and phosphatidylethanolamine consequently decreases with increasing ion concentration [347]. Electrostatically triggered lateral phase separation or compartmentalization thus could be of relevance for the membrane structure and function, but this as yet has to be demonstrated experimentally. It has been hypothesized, furthermore, that the membrane asymmetry and transmembrane electrostatic potential participate in the processes of vesicle shedding, pinocytosis, and ameboid locomotion [348]. Asymmetric model membranes can be prepared [349,350] with the aid of nonspecific phospholipid exchange proteins [351] which catalize the lipid exchange. Excessive vesicle fragmentation is useful when lipids are sufficiently different. Asymmetric membranes are needed, for example, for the studies of the electrostatic transmembrane coupling and protein insertion. In the investigations involving proteins care must be taken to prepare low-defect vesicles in order to minimize the non-specific macromolecular adsorption. In thermodynamic terms each membrane and its bathing electrolyte are an inseparable system. Changing the properties of any of the system subcomponents therefore inevitably influences the characteristics of all the others. Especially the modifications in the region of the interfacially bound water are thermodynamically significant, owing to the fact that the hydration free energy is typically nearly one order of magnitude greater than the corresponding coulombic contribution, being 1 and < 10 kJ tool- 1, respectively. The intermembrane pressure, which is identical to the spatial derivative of the membrane-free energy, or the lateral membrane pressure, which corresponds to the free energy derivative with regard to the surface area, are both sensitive to this. In the membrane vicinity the repulsion and the lateral pressure of hydration are normally several orders of magnitude stronger than the corresponding coulombic contribution. But at separations greater than 2 - 3 nm the coulombic repulsion prevails over the hydration force, because the range of the former is normally larger than the reach of the former. When the electrostatic intermernbrane interaction is attractive owing to the charge-charge correlation effects
345 the resulting pressure in any case is stronger than standard van der Waals force, up to a distance of several nanometers.
VIII. Experimental studies Classical techniques for investigating the electrified surfaces by and large are based on the charge and potential measurements. Consequently, they mainly reveal the thermodynamic information about the solvent and the membrane surface. It is only with newer experimental techniques and models that the structure of the charged interface can be studied directly, often with a remarkable spatial and temporal resolution. Charged membranes offer an exceptionally promising and relevant experimental system not only for the biologically interested researcher but also for the electrochemists. This is chiefly owing to the fact that membranes typically provide a well defined, reproducible and easily modifiable surface which can be tailored to mimick the selected features of the living systems. Techniques for probing the electrostatic membrane properties are either direct or indirect, depending on whether or not they rely on the use of probes. In principle, direct methods are superior because they are non-perturbing. But in practice, indirect techniques frequently offer a much higher time-resolution and, sometimes, are far more sensitive. Often they are also more convenient and less costly to perform. VIII-A. Direct measurements
Direct studies of the electrostatic membrane properties by and large are based on the determination of the spatial variation of the ion concentration, its direct and indirect consequences, or on the ion transport studies (see further discussion). Alternatively, the mobility of the membrane vesicles (vesicle electrophoresis, zeta potential measurements) or the electrolyte flow past the membrane (electro-osmosis) in an electric field can be studied. For more direct, but classically low-resolution, experiments electrodes can be used directly. Ion distribution and binding studies. Visible or easily detectable ions offer the unique advantage of simultaneously acting as sensors of the electrostatic membrane properties as well as a natural system component. They thus provide direct insight into the electrostatically relevant structure of the membrane-solution interface without a need for foreign probes. Nuclear [352] and electron magnetic resonance [353,355], radioactive decay [357], and ion fluorescence upon optical [358,359], or X-ray excitation [54] can be used to make ions in the interfacial and in the bulk regions discernible, especially under the conditions of total reflection [360]. Replacement of the 'invisible' ions in the double layer by the
visible ones offers good experimental opportunities when the charged species under investigation can not be measured directly. * For the data analysis and the deduction of the electrostatic quantities of interest expression Wet(X)= Ziehbel(X ) o r related results can be used. The rationale for this is discussed in Section V.A and Appendix E. For example, if the signal pertaining to the probes at the interface and the signal stemming from the bulk are set in relation, the total potential of mean-force is found from the latter by using: Wtot( X = Xbind ) = R T
l n ( Cbulk / C ( X = Xbind ) )
(11)
Usually, in the studies of membrane electrostatics the coulombic membrane potential is segregated from the total membrane potential of mean-force. The assumption is then commonly made that such electrostatic potential is proportional to the logarithm of the relative probe concentration at the membrane surface Wel(X = Xbind ) ~
ZeJ/(x =
Xbirtd)
= ( R T / Z i ) In(Signalbulk/Signalmemb. . . . )
(12)
as is done in Gouy-Chapman approximation. Frequently this is a rather unrealistic oversimplification. Whilst it is clear that the coulombic electrostatic term is the leading term for each charged membrane other contributions should not be left out a priori. More general results, such as Eqn. 6, therefore offer a more reliable starting point for the electrostatic data analysis. The decision which terms should be included in the data analysis is neither an easy nor an unimportant one. As in any multi-parameter situation from a theoretical model it is relatively easy to find a solution. But it is far more difficult to prove that the obtained answer is the only, or at least the most plausible one. Wrong or too restrictive choice of the terms in Eqn. 6 may be fatal in this respect. In order to decide between various possibilities one must combine, therefore, the theoretical scrutiny with the time consuming, systematic experimental studies; various ions and closely spaced concentration intervals
* For example, binding of the trivalent, fluorescention Tb3+ to the plasma membrane can be used to study Ca2+ receptor sites on biological membranes [358]; electron paramagnetic resonance of manganese may be used to follow the substitution of this ion near the membrane surface by other divalent and monovalentions [353]. Deuteriated ions can be used for neutron experiments[354].
346 - - - l ---7 PA, MPA
F
7~
I
I
PS" PE" P G . P C ~'~ PS"
60
...... ~
50
""
~
~,
"'
£
k
I--
3 0,h \ \
',
pE ÷- " ,
X
',
\
PS PG
.PE ,PC*
-
',.
,,.
"\
~
I
,
~',,,
-,MPA
PA-
1.5 mol litre-~, where simple coulombic interactions are nearly completely screened out), for example, can largely compensate for the noncoulombic contribution to the total interaction potential. Approximate relation ~(x=xh.,d.,.)--(~)[ln(Signalbulk/Sig
z
hi ',.) ',/1 U.I n,"
1O0
J o/
/
J o/
o
tL
(15)
PS
=o i,i L)
5M]
therefore provides a far better basis for studying the electrostatic membrane properties than Eqn. 12.
1000
v
hal . . . . b. . . . ) l ,
--ln(Signalbulk/Signalmemb.... )l , ?
PC
I 0000
(14)
w . . ( x = xh, . d ) = R T In (Signal hulk/Signal ..... b..... )
/
10
z
1 0.001
0.01
0.1 VESICLE
0,001 CONCENTRATION
0.01
01
(mg/ml)
Fig. 30. Variation of the net fluorescence of a fluorescent probe (TNS = toluidino-naphthalene-sulphonate), as a measure of the membrane potential, for phosphatidylcholine (PC) and phosphatidylserine (PS) liposomes as a function of the vesicle concentration. TNS concentrations are 0.11, 0.22, 0.43, 0.82, 1.6, 3.5 and 7.0 #M. (From Ref. 595).
351 Calibration of the electrostatic membrane data obtained with charged probes is seldom possible without difficulties [429-435]. First of all, many probe characteristics, such as the quantum yield, strongly depend on the local properties of the binding site. The effective local polarity (dielectric constant) [436,437], or local (membrane) fluidity [437,438] play a rSle as well. The vicinity of other polarizable groups, on the proteins, say, is furthermore important. The calibration curve of cyanine and safranine dyes [439] or of diO-C6-3 [441], for example, is nonlinear at higher potential, mainly owing to the saturation effect or to the increased membrane permeability to protons [439]. Experimental conclusions are sensitive to the probe concentration. This concentration, consequently, can not be varied widely without side effects. Moreover, only at less than 5% molar concentration the commonly used optical potential probes dissolve ideally in the model lipid bilayers. At higher concentrations probe aggregation and m u t u a l i n t e r a c t i o n - b r o a d e n i n g or fluorescence-quenching become a problem. Lipophilic membrane probes frequently disturb the membrane quite strongly. They cause, among other things, marked changes in the lipid headgroup configuration and sometimes induce lipid vesicle fusion [432,433]. When interpreting the changes in the probe properties care must be exercised not to explain erroneously the consequences of the fluorescent dye destruction after (partial) lipid peroxidation, in terms of the potential changes [439]. In microscopic optical experiments, the analysis of the measured fluorescence intensities must be performed only after correction for the light collected from the outside plane of focus as well as for the non-potentiometric binding of the dye [434]. In biological systems, additional pitfalls open up. In the worst case, the potential sensing probes, carbocyanines in particular, monitor the potential of mitochondrial and not of plasma membranes [435]. More difficulties could be quoted. Optical Probes. Optically detectable probes include the safranine dyes [442], the fluorescent derivatives of anionic oxanol [443,A.~], the positively charged carbocyanines [435,445], the merocyanine containing fluorophores (see especially reviews [424] and [421]) and the more recently developed zwitterionic or positively charged styryl probes [446,447]. The mechanism and the rate of response, accumulation in cells, toxicity, potential sensitivity, etc, of all these probes vary widely. Nevertheless, such dyes have proved capable of functioning as optical indicators of charge separation on the biological (such as mitochondrial and related respiring preparations) erythrocytes, excitable squid tissues, heart tissue, cerebral cortex, as well as model membranes (see, for example, Ref. 433 for a brief survey). Certain fluorescent potential probes redistribute depending on the electrostatic membrane properties and also exhibit elec-
trochromic effect. Normally, however, just one of the two mechanism is exploited for the actual potential determination. The main strength of the fluorescent membrane probes are the kinetic potential studies. Their performance for the absolute membrane potential determination is much worse and requires cumbersome calibration protocols [423]. Carbocyanines and merocyanines, as well as the less toxic oxanols, offer a relatively low time resolution in the order of up to seconds. Their sensitivity to the membrane electric field is high as they mainly respond to the latter by a redistribution across the membrane. Cyanine dyes orient parallel to the hydrocarbon lipid chains as monomers and lay parallel to the surface if dimerized. Merocyanines exhibit a somewhat faster time response and have non-fluorescent dimers. This must be borne in mind, in particular, when measurements are performed with dyes at high concentrations. Oxanol dyes tend to aggregate less than the cyanine potential probes. They repartition in the membrane field and, moreover, respond to the latter by an intramolecular electronic redistribution which causes an electrochromic wavelength shift. The styryl probes apparently function by charge-shift mechanism. The aminostyryl dyes normally have their transition moments perpendicular to the membrane surface and thus are complementary to the class of long-chain cyanine dyes [448]. The response of styryl dyes to the changing membrane potential is fast, on the millisecond time scale. This makes such probes suitable for the imaging of rapid electrical phenomena in cells and on the membranes. Styryl dyes usually are non-fluorescent unless bound to membranes but have the disadvantage that their fluorescence changes upon field variation is relatively little, usually less than 5-10%. If suitable fluorescent probes are chosen, in principle at least, one can study both the overall membrane electrostatic properties as well as the space- and timedistribution of the electric activity on the membrane surface [449]. Moreover, the use of potential sensitive probes to estimate positive and negative electric transmembrane currents, may provide a methodology for monitoring changes in the membrane 'dipole potential' in vesicle systems [450]. But the analytic tools for extracting the electrostatically relevant information from measured data remain to be developed. A fluorescently labelled derivative of phosphoniumion has been reported to monitor the electrostatic potential deep in the interfacial region where no screening occurs [451]. It has been suggested that second harmonics signals could be obtained from living cells stained with fluorescent electrostatic potential probes [452]. Electrochromic effect. With the potential sensitive probes it is important to know where the fluorophores
352 are located. They all sense long-range electric fields, in which they redistribute. But the interfacial, intramembrane, or other local fields or gradients may dramatically change the probe properties and signal. The location of charge-shift electrochromic extrinsic probes in a bilayer membrane is particularly critical, since these indicators can respond only to that portion of the electrostatic potential difference which falls across the molecular dimension of the probe. In an electrochromic mechanism, molecules undergo an electronic redistribution upon excitation, and if the direction of the charge shift is parallel to the membrane field the energy of the electronic transition is altered. Because of the high anisotropy of many interfacial properties, such as the effective relative permittivity (dielectric constant, micropolarity), this may cause severe experimental problems. But it also ensures that the reaction is fast enough to follow essentially all physiologically relevant processes, the only movement upon excitation being that of an electron. * Electrochromic effect causes a shift of the fluorophore emisSion maximum. This shift, on the one hand, depends on all system characteristics which influence the polarizability of the valence electrons; variations in the solvent structure or changing electric fields of the membrane belong to these. On the other hand, it is decisive for the probe response whether or not the probe molecules participate in the intermolecular interactions. Changes in the protonation state of the probe, dimerization, or the formation of ion pairs are of paramount importance. Despite the complexity of phenomena underlying electrochromism, electrochromic probes are a valuable tool for the electrostatic membrane studies. With suitable calibration procedures they can provide simultaneous, albeit only semi-quantitative estimates of the effective dielectric constant at the binding site as well as of the electrostatic potential near the bound fluorophore [104,453-457]. Paramagnetic probes. An attractive technique for directly studying the electrostatic properties of strongly light absorbing, light scattering, and light sensitive specimen, including photo-active membranes, is to use electron spin resonance and paramagnetic ionic probes. Suitable natural paramagnetic probes are, for example, the common divalent ions such as manganese, cobalt, nickel, and copper. Synthetic probes have been devised, moreover, with attached paramagnetic spin-labels [459,418,460,297,461]. The advantage of the latter is that they can be tailored for a given application. Their disadvantage is the bulkiness of the paramagnetic group. * In practice, secondary, non-electrochromic,motion-dependent response mechanism may obscure the situation. In the best case, therefore, the sensitivityof such a method can reach 10%/100 mV I458].
20 13
T > E
). 0 ~0
20
40
60
Arginine addition (rnM)
80
20
b
> fo
O~
-~
. . . . . . . . . . . . . . . . . . . . .
I
I
I
I
40 60 80 NH4OAc oddition (raM) Fig. 31. Change of the membrane surface potential as a function of the bulk arginine concentration measured by the relaxationkinetics of a spin-label. Circles: outer surface potential changes; triangles: inner surface. All data were estimated for the phosphatidylcholinevesicles containing 8 mol% of phosphatidic acid. (From Ref. 461). 20
In typical electron spin resonance studies of the electrostatic membrane properties the distribution of paramagnetic [353] or spin-labelled ions [460], or of charged spin-labelled amphiphiles [459,418,375,451,65, 461,450,26] between the aqueous and the membrane subphase is followed. (Fig. 31). As long as the probe concentration is sufficiently low, so as not to change appreciably the membrane properties, the electrostatic membrane properties are determinable from Eqns. 14 or 15. Paramagnetic amphiphilic probes have been used to confirm that the discreteness of structural surface charge o n lipid bilayers does not significantly affect the electrostatic membrane potential [65]. Relaxation data recorded following the rapid mixing of a cationic spin probe with tetraphenylborate containing vesicles afforded simultaneous information about the electrostatic potentials at the outer and inner vesicles surfaces [461]. Nuclear magnetic resonance. This technique shares many of the nice features with the electron spin resonance method. Its additional advantage is that it does not depend on the use of perturbing labels; it may suffer, however, from the low natural abundance of the
353 detectable atomic species. With nuclear magnetic resonance, therefore, especially the model membranes which can easily be enriched with the latter, are frequently investigated. By and large the deuterium [466] or the narrow 31p signals from the sonicated vesicles and its variation with the concentration of shift reagents, such as paramagnetic ions, are measured [176,172,426,192]. The conclusions about the (electrostatic) interactions between the charged phospholipid headgroups and the ions are obtained after the data analysis, as in the case of other ionic labels, in terms of simplified relations such as Eqn. 14. Phosphorus resonance can be used also for the determination of the transmembrane Nernst diffusion potential [462]. From the 2H-nuclear magnetic resonance spectra of the molecules, which are specifically deuterated in the polar region, the molecular orientation can be apprehended. From this information conclusions can be drawn about the interracial membrane properties and, to a limited extent, about the membrane electrostatics. Examples with simple lipid [463,464,175,465, 466,419] or more complex [468,469] membranes are available. With nuclear magnetic resonance the main site of the phospholipid-ion association, the phosphate group, and its dependence on the electrolyte composition can be directly visualized. But other nuclei, such as 13C or 170 also are attractive and suitable for the electrostatic membrane NMR-studies, provided that a clear assignment of one part of the signal to the binding site or to the interfacial region is feasible. The latter goal is most readily achieved by measuring the specific resonances of various single ion nuclei, such as lithium [265,470], sodium or potassium [471,470,472], nitrogen [473], or others [474,475,232]. The main difficulty arises from masking of the bound-ion resonance peak by the dissolved ions; but this is a general trouble. Multi-nuclear resonance [476] under optimal conditions can thus unveil the distances between the membrane-bound monovalent and divalent cations. Owing to the sensitivity of the spin-lattice relaxation rate of the former on the distance between this ion and a divalent paramagnetic ion, such as Mn2÷, the relative separation between the binding sites is reflected in the nuclear magnetic resonance signals [352]. Seldom used, but potentially very powerful, for the studies of the membrane electrostatics is the nuclear Overhauser effect spectroscopy [477,478,479]. A recent such study with egg lecithin has revealed, for example, that hydrophobic ions (such as tetraphenylborate and tetraphenylphosphonium) at low concentrations partition preferentially in the depth of the hydrocarbon membrane core; conversely, at higher ion/lipid ratios specific interaction of tetrapehenylborate with the choline headgroup and ion insertion in the interfacial region occurs [4801.
Probe transport. Ion transport across a membrane depends on the transmembrane ion concentration and electric field gradients (cf. Eqn. 52). But it is also sensitive to the interfacial membrane potential which affects the interracial ion concentration and the on-off rate for the ion permeation (cf. Eqn. 51). Electrostatic membrane properties, therefore, can be studied by measuring the transmembrane flow of charged probes and by analyzing this in terms of Eqn. 52 or some other suitably generalized result. In membrane transport studies hydrophobic or amphiphilic ions are usually given preference over the hydrophilic ones, owing to their higher permeability. Such high permeability is mainly a result of the large effective ionic radius of the hydrophobic ions -and either is possible because the charge is surrounded by hydrophobic groups or because this charge is delocalized. This causes such ions to experience an electrostatic membrane barrier which is much lower than that felt by the smaller ions (cf. Eqn. 48). The difficulty with the hydrophobic ions is, however, that for the same reasons they are more susceptible to the non-trivial electrostatic or non-electrostatic effects. Ion polarization, direct binding, and other non-coulombic effects may lead to an undesired association, predominantly in the interracial region, and make quantitative data analysis difficult or impossible. For example, the suitability of the frequently used tetraphenyl phosphonium or triphenylmethylphosphonium ions as probes of transmembrane potential has been questioned [481] because of the binding of this ion to the membrane surface. Such binding mimics and masks the expected electrostatic phenomena. Results obtained with a series of organic ionic potential probes were reported to be in conflict with the corresponding data acquired with the charged fluorophores, the transmembrane potential values for the latter being in a rule higher [482]. With the organic cations as probes for the transmembrane potential of tonoplasts (vacuoles from higher plant cells) the potential value was found to be negative [483] whereas from the microelectrode studies it was concluded to be positive [484,485]. Most probably, these discrepancies are due to the binding of the lipophilic probes at the membrane-solution interface [486,487] and the resulting change of the local and overall electrostatic membrane characteristics. The teohniques used for monitoring the amount of transferred ions are similar to those used for the direct determination of ion concentration in general. Ion selective electrodes [274] or fluorescence measurements [488,489,490] are most popular. Radioisotopes. Owing to the low spatial resolution of the radiological techniques, radioactive ionic isotopes are of limited value for the investigations of the membrane electrostatics. Corresponding studies, to date, mainly deal with the binding or transport of simple
354 ions. They are either performed with the surface films and fl-sensitive detectors or with different versions of the dialysis technique with appropriate detection [357]. Other techniques. Uncommon, but potentially useful, techniques for studying the membrane surface electrostatics are (Fourier transform) infrared [491] and (surface enhanced) Raman spectroscopy [492-494]. Detection of the plasmon surface polaritons [495] is appealing as well. With any of these techniques the conclusions about the membrane electrostatic, in principle, can be deduced from the informations about the intramolecular conformational changes. Related but less direct are the second harmonic generation [496] and photo-acoustic spectroscopy [497]. Quartz microbalance technique [498] can directly monitor the mass of the interfacially adsorbed material. Lateral interfacial electrical conductivity is a sign, primarily, of the interfacial hydrogen bonding networks [271,499,268]. X-ray and neutron diffractometry [500,354] or spectroscopy, either in the normal mode [501], under the conditions of total external reflection (also in the combination with fluorescence detection [54]), or in the anomalous dispersion mode [502] can all potentially provide direct information about the ion distribution functions in the system. I trust that the scanning 'tunnel' microscope based on the ion conduction a n d / o r molecular force microscope in future will prove useful for monitoring the ion distribution near and on the surface of charged membranes. Last but not least, it should be realized that studies of the membrane electrostatics can not be conducted on their own. Different membrane properties are too intimately connected to allow separation of one aspect from the others. Detailed investigations of the membrane electrostatics should in my view be performed with keeping both the surface ionization and hydration in mind. The effects of membrane structure and polymorphism should also not be forgotten. Each of the techniques mentioned in this section, therefore, must be taken as a part of the experimental stream which leads us towards the ultimate membrane comprehension. A variety of experimental techniques can be used to probe the electrostatic membrane characteristics. Many are based on titrating the properties of intrinsic membrane components or of the whole membranes. Determinations of the molecular conformation, phase transition temperature, intermembrane force, mobility in external field, are a few examples. Experiments which rely on the use of (fluorescent, electron, nuclear, etc.) probes in the majority of cases are devised so as to monitor the adsorption or binding of these, suitably labelled, ionic substances to the membrane surface. Studying the probe transfer through the membrane can also be enlightening. In any case, the conclusions drawn from such studies are most relevant if the coulombic, as well as the non-trivial electrostatic effects are accounted for in the data analysis. Often, the accuracy can be increased by analyzing the differences
obtained after subtracting complementary sets of data measured under the conditions, which favor or disfavor certain effects, rather than by considering the original data. New methods for studying membrane electrostatics, such as surface-enhanced spectroscopy, microbalance methods, anomalous neutron and X-ray dispersion, etc. are being introduced These provide unprecedented spatial and occasionally temporal, resolution; some of them, moreover, can be tailored for optimal suitability for the scientific question assessed
IX. Biological significance Let me begin with a few 'macroscopic' examples, which are relatively remote to the topics discussed in this contribution, but demonstrate nicely the scope and wealth of the electrostatic phenomena in the biological systems. Electrostatic membrane potential changes in response to various odorants [503,504] and tasty substances [505]. Electric fields of the order of 4 V • cm 1 evolve in wounds [506] and electric field gradients along the cell may be expected to occur in moving or interacting cells, or cells in the external concentration or field gradients. Such electric fields may orient membranes or their parts [507] and induce the redistribution of charged cell surface components [508]. In consequence, membrane potential may change over minutes through to hours [509]. A spatially dependent change in the plasma membrane potential under the influence of such a field has been demonstrated for cells in an external, uniform electric field [510,387,434] and has been shown always to tend to oppose the change , l the field-induced transmembrane potential. Model calculations indicate that sodium pumps may produce a longitudinal electric field in the lateral intercellular spaces of renal proximal tubules, which should cause electromotive fluid flow in these tubules [511]. Longitudinal electric fields along neurons, moreover, can cause a flow of the charged ion channels along the membrane surface and might lead to the formation of distinct patterns on the cell-surface [512]. Net flow of charged intrinsic membrane proteins on other cells [509] as well as protein crowding on the plasma membrane [513] were already seen to occur in the external electric fields. Cells rotate in an outside revolving electric field (see, for example, ref. [389,514-516]) because of the different conductivities of the inner (e.g., cytoplasmatic) and outer (e.g., extracellular) compartments. Owing to the dependence of these conductivities on the ion concentration, membrane electrostatics can participate in this phenomenon indirectly. A direct contribution of the membrane potential to the interfacial instability at cell membranes has been
355 postulated to exist [517]. This, together with an increased capacity for the membrane aggregation, provides one possible explanation for the biological activity of certain ionic agents. For example, cations which bind hydrophobically to the lipid bilayers and thus increase the membrane charge necessitate a higher concentration of calcium for the induction of exocytosis of plasma membranes from sea urchin egg [518]. Surface potential or surface charge density also control the movement of synaptic vesicles inside the nerve cells, the vesicle-vesicle, as well as the vesicle-presinaptic membrane interactions [519]. The mechanism of regulation in all these cases probably is based on the variation of the interracial counterion and proton concentrations [520]. The electrostatic charge distribution is prone to play some rrle in proteins [521]. In a theoretical description of this problem it is important to allow for the low dielectric constant inside proteins, electrolyte screening effects and irregular protein surface topography [523,522]. But it is also essential to realize, that in certain cases, when specific effects are involved, such as ion transfer, exchange of a single amino acid residue can have dramatic consequences [524,525] and even may abolish the biological function. Under the influence of a trans-positive membrane potential many integral membrane proteins (such as glycophorin, the negatively charged asialoglycoprotein receptor, other cell coat proteins, immunoglobulins, and polypeptides, etc.) [526] appear to change their disposition with respect to the membrane. This implies that the electrostatic force can influence the location and perhaps the orientation of the integral membrane proteins. Thus, asymmetry in surface charge, which arises from the ionic or bilayer asymmetry, can be inferred to enhance the bias for correct protein orientation, at least with respect to the plasma membrane [527,528]. Moreover, the transmembrane electric fields may affect the protein conformation and folding in the membrane. For example, action of toxins has been reported to be potential-dependent [529]; colicin A, one of the E1 group types of bacterial toxins, can adopt two different conformation states within lipid bilayers depending on the bulk pH [530]; membrane-bound water soluble polypeptides, such as insulin, can trigger membrane fusion after a pH-drop. Many more examples could be given. Most of them have in common that an external electric field, irrespective of its.direction, decreases the a-helical fraction and increases the random fraction of the protein. This may be due to the unfolding of edges of helices as they submerge into the polar environment under the influence of the field [508]. (One practical aspect of the coupling between the electric field and membrane proteins is that protein-rich membrane fragments can be oriented in the external electric field [531].) Both charged peptide cationic groups and charged phospholipids, most notably phosphatidylglycerol, are
involved in the process of insertion and transport of the newly synthesized proteins across the inner bacterial membrane [532-534]. Binding of certain (cytoskeletal) proteins to the membrane surface seems to involve the charge-charge interactions as well [535]. Perhaps this observation has similar roots as the finding that upon transferring the macrophage-tumour cells from suspension onto a substratum the membrane potential increases from - 1 5 to - 7 0 mV over a period of 6-8 h [434]. Electrochemical phenomena, such as the variations in the cytoplasmic ion concentrations, sometimes are involved in the final steps of the cascade of activation of cells. As in the case of neurons, for example, the intracellular pH influences the resting membrane potential of isolated rat hepatocytes [536]. Intracellular alkalinization causes hyperpolarization and decreases the membrane resistance. Intracellular acidification causes just the opposite. Both these effects are probably a consequence of the change in the K+-channel conformation which is likely to depend on the variations of the overall electrostatic membrane properties. Physiological significance of the membrane electrostatics then can explain why the negative surface charge is a conserved feature of various Na-channel subtypes in the dog [537]; it also governs the binding of tetrodotoxin to sodium channels. Electric charges have been shown to participate in the binding of charged, predominantly anionic drugs [538] to a membrane. Certain second messengers, most notably calcium ions and phosphoinosites, are also charged. Electrostatic interactions, therefore, may play a rrle in their biological activities. The extracellular calcium concentration on excitable membranes, for example, is stabilized by such interactions [27]. Replacement of a specific arginine by a lysine, which reduces the local positive charge density on a molecule, increases the affinity constant for serine phosphorylation by phosphokinase C 20-fold. Addition of basic residues on either the N-terminus or on the COOH-terminal side of the phosphorylation site of the glycogen synthase peptide improves the kinetics of peptide phosphorylation [524,525]. Such effects are mimicked also by albumin, probably owing to its anionic character [539,540]. Anionic lipids are also beneficial in this respect [541]. A small net positive charge plays a role also as the targeting marker during the transport of proteins into thylakoids. This is recognized by the negative external membrane surface where the pool of proteins is located. Similar, (related?) processes probably exist for the membranes of the endoplasmatic reticulum [542] and for the export of bacterial proteins into the periplasmatic space and into the outer membrane [532,533]. Enzyme activity is affected by the interracial properties (such as diffusion resistance, local ion, or proton
356 concentration, etc.) for the membrane-bound [543,241, 544,545,281,102,546] as well as for the water-soluble proteins [547]. Electron transfer through membranes plays a central rSle in the energy transduction and utilization, in the cristae membranes of the mitochondria or in the thylakoid membranes of the chloroplasts, for example. It has been hypothesized that proton movement along the surface of such membranes, this is, in the region where the H + concentration is controlled by the surface electrostatic potential, is required for the coupling between the energy source and the ATP-synthase system [548]. By all means the interfacial charge is significant for the energy transduction [549,550,551] as well as for the protein binding [552]. Moreover, hormones, at least those which bind to the phospholipid (black) membranes, such as thyroxine and triiodothyronine [553], can strongly affect the transport of the hydrophilic ions across a bilayer. Electrostatic membrane potential and interracial polarity effects are influential in all these cases. It remains to be seen how successful this knowledge can be translated into the bio-technological and technological applications. Progress in the field of bio-sensors, for example, is encouraging [277]. In my view the membrane hydration is relevant for many, if not most, processes occurring near or at the membrane/solution interface. These include, among other things, ion binding and transport, (bio)chemical process at the interface, control of the membrane structure and phase behaviour by the electrolyte solution, as well as certain recognition phenomena at the membrane surface. Further from the membrane, however, coulombic phenomena prevail. As a final caveat I wish to remind that certain uncharged lipids, such as phosphatidylethanolamine, with special properties (lower hydrophilicity, or a tendency for the nonlamellar phase formation) may exhibit biological effects which commonly are attributed to the effect of the membrane surface charge, on the phosphatidylserine or phosphatidic acid molecules, for example. The reason for this is that such special properties are the same for all these lipids (see, for example, Ref. 554). Specific effects observed with the latter two lipids therefore may, but need not be of electrostatic origin.
X. Summaff In conclusion, charged membrane together with their adjacent electrolyte solution form a thermodynamic and physico-chemical entity. Their surfaces represent an exceptionally complicated interfacial system owing to intrinsic membrane complexity, as well as to the polarity and often large thickness of the interfacial region. Despite this, charged membranes can be described reasonably accurately within the framework of available theoretical models, provided that the latter are chosen
on the basis of suitable criteria, which are briefly discussed in Section A. Interion correlations are likely to be important for the regular a n d / o r rigid, thin membrane-solution interfaces. Lateral distribution of the structural membrane charge is seldom and charge distribution perpendicular to the membranes is nearly always electrostatically important. So is the interracial hydration, which to a large extent determines the properties of the innermost part of the interfacial region, with a thickness of 2-3 nm. Fine structure of the ion double-layer and the interfacial smearing of the structural membrane charge decrease whilst the surface hydration increases the calculated value of the electrostatic membrane potential relative to the result of common Gouy-Chapman approximation. In some cases these effects partly cancel-out; simple electrostatic models are then fairly accurate. Notwithstanding this, it is at present difficult to draw detailed molecular conclusions from a large part of the published data, mainly owing to the lack of really stringent controls or calibrations. Ion binding to the membrane surface is a complicated process which involves charge-charge as well as charge-solvent interactions. Its efficiency normally increases with the ion valency and with the membrane charge density, but it is also strongly dependent on the physico-chemical and thermodynamic state of the membrane. Except in the case of the stereospecific ion binding to a membrane, the relatively easily accessible phosphate and carboxylic groups on lipids and integral membrane proteins are the main cation binding sites. Anions bind preferentially to the amine groups, even on zwitterionic molecules. Membrane structure is apt to change upon ion binding but not always in the same direction: membranes with bound ions can either expand or become more condensed, depending on the final hydrophilicity (polarity) of the membrane surface. The more polar membranes, as a rule, are less tightly packed and more fluid. Diffusive ion flow across a membrane depends on the transmembrane potential and concentration gradients, but also on the coulombic and hydration potentials at the membrane surface. The hydration surface potential governs to a large extent the height of the interfacial permeability barrier and is often decisive for the overall rate of the ion permeation. But coulombic potential is the chief determinant of the number of ions approaching the membrane and thus affects the total transmembrane ion flow. In the presence of ion carriers or channels the diffusive transmembrane current normally makes out only a small contribution to the total flow. Numerous experimental methods can be used to study the membrane electrostatics but the evaluation of the specific electrostatic membrane parameters from the experimental data is never an easy, and frequently an
357 ambiguous task. In studies with charged probes and biological membranes, simple Gouy-Chapman theory is often a poor tool for the analysis of the experimental data. Suitable generalizations and control experiments are of help here, but for the model lipid membranes such theory works surprisingly well. New experimental techniques for studying the membrane electrostatics are emerging. These promise a significant increase in the depth and scope of our understanding of the electrostatic membrane properties. In this review I have attempted briefly to survey some of them. While doing so I have tried to point to the current consensi and conundrums in the field of electrostatic membrane research, especially with regard to the interracial structure and hydration. Some statements made in this review reflect my personal opinions which are not always the views of the majority. I hope that even in such cases they are a mirror of truth. By all means, I have made an effort to tear down the barrier separating the hitherto largely unconnected but closely related fields of classical electrochemistry and the studies with charged membranes. Both have grown rapidly and independently over the last years and would gain from mutual fertilization. Membrane researchers, on the one hand, should benefit from the new concepts, alternative experimental methods, and highly accurate analytic tools of the modern electrochemistry. And electrochemists, on the other hand, should get a chance of extending their area of interest to the challenging new realms of biology. It will be hoped that this will provide better insight into the molecular origin, physiological significance and practical applications of the electrostatic effects in the biological, biochemical, and biotechnological systems.
Appendix A Poisson-Boltzmann
approximation
= - Pel,i(X)//~CO
=-- -- ( P e t , i / / C O O )
peA x )
=
NAeY'.Ziq exp[ Zieq4 x ) / k T l i
=- F~_Z,c i e x p [ Z ~ F + ( x ) / R r ]
I
~(x)
(16)
with the boundary condition: d ~ ( i n t e r f a c e ) / d x = - o e t / c c , , . Parameters c, %, and co are the dielectric constants of the aqueous and membrane subphase and the permittivity of free space, respectively. Another key assumption of Gouy-Chapman approximation [30,31] is that the charge distribution near
(17)
i
e is the elementary electronic charge, Z~ is the ion valency, k is the Boltzmann constant, NA is the Avogadro number; F = e N A and R = k N A are Faraday and gas constant, respectively; T is the absolute temperature. In electrochemistry and in all electrostatic theories it is customary to introduce the Debye-Hiickel screening length by the general definition:
(
1
X = ~%kT/IO3NAe 2 Z c i
(18)
i
where the summation goes over all ionic species and the factor 103 appears because the bulk ion concentration is normally given in moles per litre. For a symmetrical electrolyte the Debye length X can be conveniently written as: .....
= ( c%k T/2O NAc ) ½/ l OZe
which after insertion of numerical values for T = 298 K (25°C) yields: h=0.304Z-lc 12nm Combination of Eqns. 16 and 17 affords PoissonBoltzmann equation for a planar charged surface. Specifically for the case of symmetrical ionic solution, this is for an A z+, B z- electrolyte, one obtains: d2~k ( x ) / d x 2 = ( 2"103NAZec / , % ) sinh[ Ze~ ( x ) / k T]
Traditionally, but not inevitably, the most popular electrostatic surface model, the Gouy-Chapman diffuse double layer theory, is derived from Poisson Eqn. 1 by neglecting the spatial distribution of the membrane associated charge. This is tantamount to writing: Pel,m(X) =--O~lS(interface) for the surface and P e l ( X ) = Pet,~(X) everywhere else. For a planar surface one then has: d2~b(x)/dx2
the membrane is governed solely by the electrostatic, Coulombic interactions according to the Boltzmann law:
(19)
Similar equation can be written for the membrane electric field. In terms of Debye length, the Poisson-Boltzmann equation for a symmetrical electrolyte reads: ~2d2~ ( x )//dx2 = ( k T / Z e ) sinh[ ZeqJ( x ) / k T ]
(20)
In principle, either the electrostatic surface potential, ~P0, or the surface density of the structural membrane charge, Oet= e / A o can be kept constant, say, by some metabolic process. In the majority of the biologically relevant cases the latter possibility is the more probable one. Therefore, the usual boundary conditions with which the solution to Eqn. 19 should be sought is oel = constant. For little charged surfaces, the relation between the electrostatic membrane potential and the surface charge
358 density from the Poisson-Boltzmann equation is found (by expanding Eqn. 2 in terms of the surface potential) to be:
roughness and the interfaoal hydration effects tend to diminish the significance of the correlation effects so that in many cases these, in fact, need not be included into the electrostatic membrane description.
% = %lh/cc0 > 2 R T / Z F --- - 5 1 . 4 Z -1 ln(0.36A~c~) mV, T= 25°C
(22)
In the numerical result the area per charge, A~, is in nm 2 and the concentration in moles litre -~. The result for spatial profile of the electrostatic potential near an isolated membrane in the linear approximation is given by Eqn. 3. The corresponding approximate nonlinear result is ~ ( x ) = (4kT/Ze) tanh l[exp( - x / X ) tanh(Ze%/4kT)]
(23)
The discrepancy between the latter approximation and the exact solution to the Poisson-Boltzmann equation is small [4], except in the region close to the membrane where the validity of Gouy-Chapman theory because of principal reasons anyhow breaks down. Much greater errors can be expected to result from the intrinsic over-simplifications of Gouy-Chapman model. The most obvious, but not the most severe one is the neglect of ion size. This can be improved in an ad hoc manner by postulating the existence of a distance of closest approach beyond which ions can not penetrate (the Stem layer). Calculations based on Eqns. 2 and 23, which neglect the steric, hard-core repulsion, therefore underestimate the double layer thickness. But cross correlations between the steric and electrostatic interactions give rise to an extra attractive potential for counterions and to an excessive repulsive potential for coions. Owing to the predominance of the former in the case of infinitely narrow interfaces, the net outcome of the ion size-contribution is an attractive correction to the potential of mean force. On the whole, the size effects thus cause the ionic double layer to be thinner and the electrostatic surface potential to be lower than predicted by Gouy-Chapman theory (see Fig. 9). Interionic correlations thus are more consequential than the ion size effects, at least for the simple surfaces with a narrow interface (see also Appendix B). GouyChapman theory which does not account for any correlations, therefore, can be qualitatively wrong for the description of charged membranes with a very regular surface especially when multivalent ions are present. Fortunately, however, for most membranes the surface
The modified Poisson-Boltzmann (MPB) approach represents the natural extension of standard electrostatic Gouy-Chapman approximation towards modern electrostatic theories [43-47,555,48-51]. It includes into the Poisson equation, via a fluctuation or self-atmosphere potential (~) but still within the framework of a mean field approximation, the most relevant higher order ion-surface and ion-ion effects, d 2 ~ ( x ) / d x 2 = -(Pet,,/C¢o) 1,¢~)+,~) = - ( NAe/C~o)~.~Z,2c, exp i
× [wi.m, ra ,.ore(x)/kT- Z , e + ( x ) / k T +(1/kT)
× lira fZ'el* r ~ 0"0
[
47rc~or
*--
~
bulk (24)
where r is the distance from the centre of the ion i and Wi.h~rd_core(X) represents the steric exclusion term. The fluctuation potential thus accounts for the difference between the actual mean-force potential (wi(x)) and Gouy-Chapman potential at any separation from the surface: ~ ~ ~j(x) - wi(x) - +et(X) (see Eqn. 45). In the above, typical form it is devised so as to allow for the higher order, such as volume exclusion, image force, and ion self-atmosphere (correlation), effects. * In the point-ion limit ion radius, and thus the hardcore term, is zero. MPB-equation then reduces to: d2q.,(x)/dx 2 = - ( N ~ e / c , o ) E Z ~ c ~e x p [ - Z~eq/(x)/kT i +(Z~e/8rrc¢o)× ( h - 1 - h ( x ) - l + [ ( c - ~ m ) /2x(~ + c,,,)] e x p ( - 2 x / ~ ( x ) ) ) k T ]
(25)
This shows that apart from the neglected ion-size improvement, which is also lacking in Eqn. 25, there are two essential-corrections to Gouy-Chapman point-ion theory. First, the local variation of the screening length,
* This is tantamount to replacing the ion profile, which corresponds to a singlet correlation function, by the pair correlation function in the calculation of the potential.
359 X(x); and secondly, the classical screened self-image term, which is proportional to the reflection coefficient (~ c,,)/(c + c=), where ~,, is the dielectric constant of the membrane core. For a planar surface and in the absence of images, the cavity term, which is a kind of cross-correlation between hard-core and electrostatic interactions [7], takes the simple form: -
~l'cavity = 0.5[ tp(x + ri) + ~p(x - ri) ]
This implies that the potential of mean force is no longer a local function of the mean electrostatic potential once the ion-size effects are included. An immediate consequence of this is that oscillations in the double layer may arise. The linearized modified Poisson-Boltzmann equation can be expanded in terms of higher order potential derivatives for all electrolytes and truncated [43,556,49,557]. This is equivalent to an alternative Poisson-Boltzmann equation of the approximate form: )~(x)2d2~(x)//dx
(26)
2 = ~(x)
with a solution, for an isolated surface: ~ ( x ) = (oelX(x)/c%) exp( - x / X ( x ) )
(27)
The effective, spatially variable screening length X(x) is given, in the first approximation [5], by: i
X(x)=(,%kT/lO3NAe2~_,Z2c,(x)) ~ i 1
- X[cosh(ZN~P(x,X(x))/kT)] -~
(28)
This reveals, after comparison with Eqn. 18, that the efficiency with which a charge in the interfacial region is screened is determined by the local, rather than by the bulk salt concentration. * Complete solution to the linearized modified Poisson-Boltzmann equation for a symmetrical electrolyte or for a mixture of such electrolytes has the form [555]: ~p(x) = ~ ~0.n e x p ( - x / X c ) n~l
where Xc are the roots, with positive real parts, of the transcendental equation: ( ri/Xc) cosh(ri/Xc) + ( ri/X ) sirth(ri/Xc) = (ri//)~c)3( 1 + ri//~.)//(ri//)~) 2
(29)
* When Eqn. 28 is used in order to evaluate the effective screening length, the agreement with more accurate models is the closest in the third iteration in terms of the corrected potential, the more approximate potential value being always too low. The discrepancy increases with decreasing salt concentration (cf. Fig. 11).
and r~ is the (hydrated) ion radius. Full expressions for nonlinear modified Poisson-Boltzmann equation and a thorough discussion of various aspects and implications of this approximation can be found in [43,45,555]. A brief discussion is given in Ref. 50. Effects of variable dielectric screening have been discussed by several russian scientists [558,148,138,139]. The first term in Eqn. 29 becomes identical to Gouy-Chapman result when the Debye screening length is far greater than ion radius. This leading term decays spatially on the length scale of XC> X until, for Debye lengths somewhat smaller than ion radius, all roots of Eqn. 29 are complex conjugates and MPB-theory implies a damped oscillatory behaviour (cf. Fig. 9). In comparison with MPB-result Gouy-Chapman theory predicts slightly too thick double layers. For 1 : 1 electrolyte solutions in contact with a planar charged surface the predictions of MPB model agree nearly quantitatively with the results of computer simulations for such a model system. This accuracy is greatly reduced for solutions of divalent ions, however, or for systems characterized by a low (interracial) dielectric constant. * * For a highly charged surface MPB approach suggests that nearly all of the counterion charges in the surface associated layer will condense into a very thin zone within the range of a few ion diameters from the wall. This should happen essentially independent of the excluded volume effects. The outcome is something like a dipole layer which contains anomalously high electrolyte concentrations; even for the colons this is approximately twice as high as in the bulk. Similar conclusions pertain to the effects of image charges. These arise because of the interfacial mismatch in dielectric properties ('constants') between the assumedly homogeneous solvent and the region behind the surface. A charge near such dielectric border can be taken to induce (via polarization) an image charge located at the equivalent distance 'behind' the interface. This induces its own image, etc, all of which are dependent on the precise interfacial dielectric profile. For membranes the image effects are likely to be less important than for solid electrified surfaces, owing to the continuum nature of the membrane-solution interface. Absolute magnitude of the image effects, however, is independent of the surface charge density and images are relatively more important for the little charged surfaces [149]. Electrostatic membrane potential is never strongly affected by the image interactions [48]. This is mainly because image force is independent of the sign of the
* * For membranes with a relatively hydrophilic surface, especially the latter, restriction is quite severe owing to the fact that solvent layers near the membrane-solution interface essentially behave as a system of low permittivity, c < 30.
360 charge on the ion and thus affects all ions in much the same way. For a realistic choice of the surface potential (< 100 mV) the image contribution to the electrostatic potential is less than 20%, leading in every case to a decreased surface potential, even for repulsive images and in the absence of specific adsorption [47]. But double-layer structure-and intermembrane interactions-may respond quite sensitively to image effects [48]. In the extreme case, when the ion concentration near a regular membrane surface is sufficiently low, all counterions near the interface are on the average separated far enough to experience a local surface charge determined by the self-image component. This can lead to local maxima or minima in the counterion and coion density profiles, respectively (cf. Fig. 5), and may cause the intermembrane repulsion to turn into attraction. Effects as dramatic as this, however, are expected for the closest membrane vicinity merely, which explains why image effects normally contribute so little to the value of the electrostatic membrane potential. For symmetrical 2 : 2 electrolytes the fractional change in the diffuse layer potential caused by image forces differs from the 1:1 case both by being much larger in the former system and by decreasing once the surface charge is increased beyond a certain point. For asymmetric, such as 2 : 1, electrolytes, moreover, a small charge separation is established even for uncharged surfaces owing to the stronger action of image forces on divalent ions relative to the monovalent ones. *
Appendix C Generalized Poisson-Boltzmann approximation When the structural surface charges are not confined all into one plane the spatial distribution of the membrane charge (Oel.,,(x)) as well as the distribution of ionic charges (Oel, i(x)) must be considered explicitly in the Poisson equation: d2~,( x ) / d x 2 = - ( Oel,i( X ) + pel.m( X ) ) /CCO
(30)
Under the assumption that the membrane charge distribution is unaffected by the electrolyte and if the
* It is noteworthy that potential of zero charge can arise also because of the size asymmetry. By solving the Poisson-Boltzmann equation for the case of coions and counterions of different size Valleau and Torrie have shown this [560,561]. They have, moreover, found that such ion-size effect gives rise to an asymmetrical charge-potential relationship. Identical positive and negative surface charge densities thus result in different electrostatic surface potentials. The latter conclusion is also in agreement with detailed Monte-Carlo simulations [562]. Corresponding analytic expressions, within the mean spherical approximation, for an arbitrary asymmetric electrolyte containing an unrestrained number of components with any diameter and charge, have been published by Khater and co-workers [563].
salt distribution is taken tO obey Boltzmann law, generalized Poisson-Boltzmann equation is obtained. For an isolated membrane, the solution to this equation is given [105] by the integral:
t~(X)=(~k/~o)(f0
Pel,m(X )exp(-x'/~)dx"
- foxPel'm(x , ) sinh[(x - x' ) / M d x ' )
(31)
Fortunately, a detailed knowledge of the membranecharge distribution is not really required. Different distribution profiles give approximately similar results, at least for the surface potential, provided that the average charge displacement perpendicular to the membrane, d C, remains the same [105]. For a hypothetical membrane with an exponentially decaying density of the structural charge density (Oel, m(X) = (ael/d C e x p ( - x / d c ) ), for example, one obtains in the linear approximation: ~k(x) = (o~)~/,Co) { (d,2/)~ 2 - 1) -~[(d,,/)~ ) exp( - x / d , . ) - e x p ( - x/)~)] }
(32)
This result contains explicit information about the spatial profile of the structural surface charge distribution and shows that the electrostatic potential profile near the membrane surface is governed by the membranecharge as well as by the ion-charge distribution. Far from the membrane the diffuse double layer has the form predicted by the ordinary Poisson-Boltzmann equation if the Debye length is greater than the interfacial thickness. In the opposite case the spatial variation of the electrostatic membrane properties are determined by the structural charge distribution. This limit corresponds to a special case of Donnan approximation. For a hypothetical membrane with a constant surface charge density distributed throughout a layer of thickness 2dp, which corresponds to a 'screened Donnan model', the electrostatic potential profile is found by straightforward integration of Eqn. 31 to be: ~ ( x ) = ( oei)~ / ~ o ) ( )~/ 2 d p ) e x p ( - x / )~ ) sinh(2dp/?~)
outside the charged region and again in the linear approximation.
Appendix D Nonlocal electrostatic model In classical electrostatics, valid for homogeneous dielectrics and structureless solvents, an external field of some form creates a polarization profile of similar shape. In dielectrically non-homogeneous media, which include
361 all structureable fluids and, in particular, aqueous solutions, the same applied field produces a polarization which is squewed, due to the correlations and coupling between the solvent molecules. The response to an applied field in such case is nonlocal. Its range is proportional to the space in which the water is correlated. (If these fluctuations are interpreted as a generalized form of 'polarization fluctuations' this corresponds formally to replacing the standard dielectric constant by the nonlocal dielectric response function, c(r,r'); the latter is especially important for the short wave-vectors k.) A very narrow field represents a local perturbation. This arises, for example, from individual polar residues on the membrane surface. The membrane surface field, therefore, must be assumed to create a response in water which is not limited to its site of origin, but rather extends over a finite distance of the order of the solvent correlation length ~. To account for the coulombic as well as for the hydration membrane properties from the electrostatic point of view a nonlocal electrostatic model proves useful. Within its framework two types of the 'membrane potential' are introduced in order to fully describe the membrane surface: the usual coulombic electrostatic potential, +et----~k, and the hydration potential, +h (or, alternatively to the latter, the polarization potential ~pp = --coo~ph/C ). Standard meaning of the Coulombic electrostatic potential has been discussed in previous sections and will be extended in the following. The hydration potential of the membrane originates primarily from the short-range fields associated with the polar membrane residues, this is, from the atomic charges on these residues and from their capacity to interact with the solution. Solvent 'polarization' and the corresponding membrane polarization potential therefore contain contributions from the electrical polarization as well as, and often more importantly, from the structural polarization described here in electrical terms. To model the membrane surface in nonlocal electrostatic approximation, consequently, either the surface values of the coulombic and hydration potentials, tk0 and tkh0, or else, the corresponding charge densities oel a n d op must be known or calculated (see further discussion). For many thermodynamic applications knowledge of the molecular area is also required. The distinctive water properties within the same model are described phenomenologicaUy by a static and a high frequency dielectric constant, c and 2 < coo < 6, respectively, and by some typical structural correlation length, ~. For water the latter is probably on the order of 0.1 nm or less, the effective value being sensitive to the electrolyte properties as well as to the interfacial properties and structure [113]. In the nonlinear approximation an effective 'charge of the water molecules', e~ -3.5.10 -20 C, which reflects the effects of the inter-
atomic charge-transfer in the hydrogen bond formation, is also needed [122]. Instead of a single Poisson-Boltzmann Eqn. (cf. 420) for the electrostatic potential, in the nonlocal electrostatic approximation two coupled equations x 2 [d2,~/dx 2 - (,/,oo - 1 ) d % , / d x ~ ] = ( c k T / , ~ h 2) s i n h ( Z e f / k T )
(33)
and I;2d2~p/dx 2 ~ ( , ~ k T / e w ) sinh([ew( ~ + t~p+ ¢/ho)]/kT }
(34)
for the electrostatic and for the 'polarization' or the hydration membrane potentials must therefore be solved simultaneously. The surface hydration potential is a measure of the membrane hydrophilicity (see further discussion). The main source of the membrane hydration potential are the atomic charges in the interracial region which give rise to the potential ~kh0. This is due to the fact that the chief mechanism of the membrane-water association is direct binding of the water molecules to the polar membrane residues and not the ordinary electrostatic water polarization. The sites and the strength of the water binding, consequently, are determined largely by the quantum-mechanical fields originating from the surface polar residues and much less so by the macroscopic fields from the net surface charges or from the interracial dipoles and higher multipoles. Water binding to a membrane is thus more chemical than electrical in nature. The use of the word polarization in this context must be interpreted correspondingly. This can be understood as follows. Many membrane components contain multiple charges of various signs or at least a few polarizable groups. Charged regions on these membrane components are typically separated by intramolecular barriers which prevent the charge exchange and charge neutralization within a single molecule. * Each membrane thus can be characterized by a surface density of the local excess charge, %, which provides a measure of the overall membrane capability for the direct binding of water. In the simplest approximation [371], the value of the phenomenological parameter op can be found by summing-up the corresponding individual charge density contributions from the appropriate atoms on the membrane surface (see Fig. 15). But the very first step is to weigh these contributions in order to account for the different
* Owing to the atomic origin of such charges the precise molecular configuration or organization(monomer or lattice) influencesonly little the intramoleculardistributionof excess charge [114].
362 accessibility, ct~, of the corresponding atoms or groups to water. Effective individual local charge density thus can be estimated from the relation: Op.~ = e r / A r. From this the total membrane local excess charge density is finally found by averaging [122] over the entire membrane surface:
potentials are found to be: % = { o.X[1 + ( ~ / c ~ ) ( ~ / X ) ]
+ oA(1 - ~ / ( ~ ) } Xl~%(X + 4)1 (37)
and -1 Op _- EOtrOp,r. EArAsurface
(35)
+~0 -- - [ o . ( , = X 2 - ~ )
Obviously, only such polar groups should be considered in this sum which are not compensated directly at the membrane surface; that is, only the polar residues which are not involved in the direct hydrogen bonds with other membrane constituents and, consequently, are free to bind water. If the availability of a given group or molecule for water decreases, in the present model, a lower value of the surface local excess charge density results from this as then: olr ~ 0 . * For mixed systems the corresponding expression is: Op.mix =
Y~l,pia ( Op,lipidA"pid ) ElipidAlipid
(36)
The dipolar and multipolar effects can be accounted for accordingly by introducing the appropriate surface densities ~e' ~'e' etc. Normally these contributions are relatively small, however. The main conclusion of nonlocal electrostatic approximation is that near a membrane surface both the ion as well as the hydration double layers should co-exist, owing to the simultaneous existence of the electrostatic and hydration interfacial fields [110,141,122,566, 20]. From such, admittedly oversimplified, approximation the surface values of these corresponding surface
* Protonation of any charged polar group on the m e m b r a n e surface, therefore, not only affects the ionization state of such a group but moreover-and sometimes more importantly-also decreases the capability of this group to bind water. When polar parts of the membrane constituents interact in the absence of water, the effective basic 'electroneutrality unit' of each molecular layer consists of the nearest-neighbour charged groups. For lipids, these are typically the phosphate and the a m m o n i u m groups. But the same groups also can form intermolecular hydrogen bonds and may partly exchange charges via the charge-transfer process along these bonds. This is the reason why in any realistic model of membrane hydration the intramolecular as well as the intermolecular charge neutralization must be considered. Moreover, even in the cases where no direct water-lipid hydrogen bonds exist, charge transfer between the lipid and water molecules takes place [564,565]. For the hydrated lipid bilayers-and surely also for other membrane types-local excess charges consequently are subject to the screening and charge-neutralization by the atomic local excess charges localized on the water molecules. This is mainly a consequence of the large strength with which the water molecules can participate in the hydrogen bonds and is also a consequence of the great separation between the positive and negative residues on the lipid headgroups.
+ o~(~X + ~)~][-o
, ~ ( x + ~)l
~(~/~) (38)
in the linear approximation. Implicit in Eqns. 37 and 38 is the assumption that ion screening length X is much greater than water structure length 4. Only if a given membrane does not interact with water (op = 0) and if the solvent structure is neglected (~ ~ 0) Eqns. 37 and 38 reduce to expression 22. Gouy-Chapman result is then recovered. In every other case the electrostatic potential at the hydrated membrane surface from the nonlocal model is implied to be greater than it would be near a comparable non-hydrated membrane (cf. Eqn. 5) being a 'superposition' of coulombic and hydration-dependent potential terms. This is also reflected in the expression for the electrostatic potential profile near an isolated, charged but nonpolar membrane surface (oe/~ O, Op = 0). In nonlocal electrostatic approximation [20] for the very dilute electrolyte solutions one has: + ( x , % = O) = (oe/?~/c%)[exp( - x / X ) + ( ~ / X ) ( c / c ~ - 1 ) exp( - x / ~ ) +Coulombic(X) 4- +hydration(X)
(39)
Recasting this in standard Gouy-Chapman form shows that the consequences of the membrane hydration resemble phenomenologically the effects of a low and variable interfacial dielectric constant: +(x) = oezX/c(x)%
This spatially variable dielectric response function in the expression for the electrostatic membrane potential of an infinitely narrow interface is approximately [148,20] given by: ~ ( x ) - c[1 + ( ~ / c ~ - 1) exp( - x / ~ ) ]
i ?~ >>
(40)
This suggests that the spatial profiles of the electrostatic and hydration membrane potentials are likely to depend on at least two characteristic decay lengths. The first pertains to the long ranged, primarily ionic, double layer. It is a combination of standard Debye screening length and of water order correlation length; from nonlocal electrostatic model its value is estimated [141] to be approximately: )~,,/-- )~[1 + (~/X)2(c - ~ ) / c ~ ] 1/2.
363 The second characteristic length approximately measures the thickness of the 'hydration double layer' near an infinitely narrow interface. In analogy with previous result its range is given approximately by: ~ei= ~[1 (~/)k)2(£ - - Eoc)/Coo]1/2. But this expression is model-dependent and thus of limited practical value. In reality, the spatial variation of the interfacial hydration potential, and consequently of all other hydration-dependent thermodynamic quantities, is apt to depend on the interfacial and not just on the solvent structure [113]. This should be true for the entire region accessible to or partly filled with the polar surface residues. To clarify this point, let me consider a simple uncharged membrane surface with an exponential distribution of the polar residues: pp(X) = (op/dp) e x p ( - x / d e). The spatial decay of the hydration variables, of the hydration potential, say, for such a model surface immersed in pure water is derived from a suitably generalized version of Poisson-Boltzmann equation [152] for hydration: Ch(X,~) = ~
( - - 0 . 5 f ~ ( X ') exp[(-- x ' ) / ~ l ~ '
+ ~Op(x') sinh[(x- x')/~ldx')
(41)
to be:
d~
~h( X'~'dP) = ¢~c0 dp-- ~ - exp( - x/~) + ~ e x p (
- x/dp)
]
(42) -- (ap~//,~Co)(~/dp) exp( - x/dp), dp >>~
(43)
and is bi-modal. Eqn. 41 can be derived in analogy to similar treatment of the double layer near a continuously charged surface (cf. Eqn. 31) [152]. The range of the first component is controlled by the solvent structure correlation length. The second component has the form of the interracial polarity profile. In the special, present case it falls-off exponentially on the length scale of the interfacial thickness (cf. Eqn. 43). Experimental aspects of the membrane hydration were recently reviewed in this journal by Rand and Parsegian [376] and previously by Rand [339].
Appendix E 1on distribution functions and membrane potential
mean-force
In the distribution function approach membrane vesicles are treated as one class of particles in the solution. Moreover, because the radius of these vesicles
is SO much greater than that of the other solutes each vesicle membrane is approximated with a planar bilayer situated at x = 0, the interactions with all other membranes being neglected. Concentration profiles. Each distribution function, gmi(X), normally corresponds to a density function. When multiplied by the appropriate bulk concentration values, c~, it thus gives the number density for any dissolved species i at a separation x from the membrane particle m. To determine the concentration profile of the dissolved molecules relatively to the membrane surface the radial distribution function for all dissolved species must be known. From the elementary theory of liquids [157] one learns that this function is related to the potential of mean force: gmi(X ) = exp[- w,m(x )/kT ]
(44)
where the mean-force potential, Wmi(X), is the integral potential associated with the force between a particle i and the membrane particle m, with the other particles canonically averaged over all positions. Eqn. 44 is a general result. It reduces only under certain restrictive conditions, discussed in this Appendix, to the common Boltzmann-factor approximation of Gouy and Chapman. To calculate the ion distribution profile from Eqn. 44 this result must be combined with Eqn. 6. Contributions to the mean-force potential are frequently assumed to be virtually independent. This leads to a superposition approximation: Wi( X ) = Wel,i( X ) q- d~( X ) -~- Whard . . . . . . i ( x ) -1- Whydrat,i( X ) + Wother,i(X)
(45)
where the index m for the membrane is omitted for simplicity. The choice and separation of individual contributions to the mean-force potential is largely a matter of convenience and of the characteristics of the membrane system under study. In Eqn. 45 the solute-surface mean-force potential has been chosen to consist of a coulombic term, wel,i(x), a fluctuation potential, ~(x), a term that depends solely on the steric, volume-exclusion, hard-core interactions, W h a r d _ c o r e , i ( X ) , and a hydration term, Whydrat,i(X ). In the membrane studies the first, the third and the fourth term are especially important; the last contribution, which accounts for all residual surface-solute interactions not otherwise included, Wother.i(X), is normally quite small. In all electrostatic models the coulombic contribution to the mean force potential is set equal to the ordinary electrostatic membrane potential: wet.i(x ) = Zie~bel( x )
(46)
364 where Z~ is the valency of the ith molecular species. From this result, together with Eqn. 6 and the PoissonBoltzmann equation, Gouy-Chapman electrostatic model is directly obtained. Additional allowance for the hard-core and fluctuation terms, which is described in some detail in Appendix B, yields the so-called 'modified Poisson-Boltzmann' approximation. The hard-core repulsion term, which is very important for the solid interfaces and in the immediate membrane vicinity, where it may give rise to notable volume effects, is probably relatively unimportant for the membranes, at least as far as the surface electrostatics is concerned [70]. On the one hand, this is owing to the mesh-like structure of the membrane-solution interface. On the other hand, the ion-packing constraints at the membrane surface are reduced by the interfacial mobility and the surface fluctuation effects. Steric terms in the membrane research therefore frequently can be omitted unless extended polar groups or the innermost interracial region are considered. Membrane hydration effects can be allowed for, in the simplest approximation, by assuming ad hoc that the dielectric properties of the aqueous sub-phase vary with the separation from the membrane surface. A connection between the mean-force potential and the dielectric 'constant' profile c(x) then can be made by identifying the potential Whydrat(X) with the Born [567] free energy of transfer of the solute charge between two sites with dielectric constants c and c(x) wB.... ,(x) = [Gh.i(~o)/NA][C/(c - 1)][1/c - 1/c(x)]
(47)
where Gh, i is the free energy of hydration for an isolated charge on particle i. This energy is tabulated and also can be calculated from suitable models [20,567]. The weakness of such ad hoc phenomenological model of the membrane hydration is that it gives no realistic description of the perturbed interfacial water structure and, in particular, provides no means for relating this with the membrane surface properties. Only if the large contributions from the interfacial ion and surface polarity effects are neglected, the effective dielectric profile caused by simple electric polarization can be calculated, for example, from the Booth [568] or some other [569] theory. Such, obviously unrealistic, assumption which allows only for the dielectric saturation effect leads to the conclusion that the interfacial counterion concentration are smaller by up to one order of magnitude compared to standard double layer theory results. The origin of this is the dielectric saturation of the interfacial region, this is the nonlinear electrostatic response. Such discrepancy should increase with decreasing ion size and become unimportant at separations larger than approximately twice the ion size [1501. A much better, but still only partly adequate, de-
scription of the membrane hydration is obtained if the hydration term in the mean-force potential is evaluated from the nonlocal electrostatic model. In brief, this is done by identifying the hydrational part of the meanforce interaction potential with the free energy of the solute-surface interaction. The latter is expressed in terms of the non-coulombic part of the electrostatic membrane potential, q~et(Oet,Op,~) and of Born hydration free energy, Ghj [151], to yield: Wh~,drat. i =
2(4~rco~/Zie )Gh,,/~t( oel,Op,~ )
(48)
in the absence of dielectric saturation. The potential value +el (Oct,Op, ~) corresponds to the short-range component of the electrostatic membrane potential, which is discussed in Appendix D. It can be evaluated, in the linear approximation and for a perfectly smooth membrane surface, from Eqn. 37, by retaining only the ~-dependent terms. Otherwise, complete solutions to Eqns. 33 and 34 must be used (see, for example, Refs. 141,20). A more general expression for the ion distribution profile which contains all relevant contributions to mean-force potential of the membrane is given as Eqn. 8 in the text.
Appendix F Permeability barrier and transmembrane currents
The direct, diffusive flow of charged particles can be calculated from the phenomenological relation: j =~~pi[ Ac, + (Zi~F/RT)A~,,, ]
(49)
i
where Ac, = c i ( x = O ) - c i ( x = - d i n ) is the transmembrane concentration gradient for the electrolyte component i, gi is the corresponding average ion concentration, and P~ the associated element of the permeability matrix. From the partial mass-flow the electric current across the membrane is calculated by summing-up all individual ion-flow components: I = Y'.~Z, Fji. In the steady state, the concentration driven diffusive flow is constant throughout the membrane. It is also proportional to the difference between the probability for the molecular species under investigation to be located at the two internal membrane interfaces. It is thus a function of the corresponding partition coefficients: j, = [(ci/par,)~_o--(ci/par,)x=
dm]Pm,i
(50)
Comparison with Eqn. 49 shows that the square bracket in previous result represents an effective concentration gradient between the internal sides of the two mere-
365 brane surfaces [239]. The inverse of the bilayer diffusion permeability: PZ.,] = - ~ D ( x )
-1 exp([Go,i(x) _ ZiF~ ( x ) l / R T }
Straightforward integration of Eqn. 53 by using the linear Gouy-Chapman result for the electrostatic potential, Eqn. 3, leads to:
(51)
G d --*oo × [ t a n h - l ( d w / 2 ) ~ ) - 2 ' % = c ° n s t a n t Get(dw) = et( ~ ) ttanh(dw/2)~), oel = constant
represents a sum of resistance elements d x / p a r ( x ) D ( x ) and gives the total resistance of the membrane hydrocarbon core (or pore) [239]. D ( x ) is the diffusion constant and G0.~(x) is the free energy profile for the solute i. The interracial partition coefficient can also be expressed in terms of the inward and outward interfacial rate constants, k~, and kou t [239]. The transmembrane permeability then becomes:
where dp is the interfacial thickness. From this, the height of membrane permeability barrier is seen to be determined either by the interfacial inward and outward rate constants, or else by the resistance of the hydrocarbon membrane interior, depending on the relative magnitudes of individual components. For ions crossing the lipid bilayers the former contribution, in the majority of cases, is the principal one. The electric current across the membrane from Eqns. 50 and 51 and from the basic ion-current definition is deduced to be: 1 =y'FZi(((i
cikout,i/ki..i ) Ix= -d~ exp(-- FZiA~bm/RT )
+ (ciko~,.i/ki,.i) l x=o } P,,,i)
(52)
But this relation may change, even qualitatively, if the surface or transmembrane potential values depend directly on the substratum concentration [570]. In such event the effects of ion binding on the (trans)membrane potential must be considered explicitly.
Appendix G Electrostatic free energy of the membrane
General result for the evaluation of the electrostatic membrane free energy is Get.° = NAAc fo°'~ ( o )do
(53)
and gives the electrical free energy part. The corresponding 'chemical part' is: -NAAcoe? k. It should be added to the electrical contribution in the biologically improbable case that the potential rather than the charge density at the membrane surface is fixed; this yields Gel,4, = Gel,o - NAAcoet~b.
(54)
with the maximal value 6,,( d, --, ~ ) = - ( NAA~o~X/ , o ) = O.O066(AeZcl/2 ) -1 j mol_ 1
