ION-MEMBRANE INTERACTIONS AS STRUCTURAL FORCES V. Adrian Parsegian National Znstitutes o f Health Division of Computer Research and Technology Physical Sciences Laboratory Bethesda, Maryland 20014
I would like to elaborate upon what I see as a inconsistency in the way we look at the passage of an ion across a low dielectric membrane. This process will be discussed in terms of the energy of interaction between ion and membrane. We know that a membrane poses an energetic barrier to the ions in the adjacent solutions. We know that the modulation of this barrier is the primary means by which a membrane controls ionic flow. But we tend to think of the interaction only as one imposed' by the membrane on the ion rather than a mutual interaction that might act to perturb the structure of the membrane. The cause of the ion-membrane interaction is the coulomb field that emanates from the ionic charge. This field acts to polarize the surrounding medium, to attract charge of opposite sign to the ion, to repel like charge and thereby to reduce ionic energy. Polarization is easy in an aqueous medium of high dielectric constant and hard in low dielectric lipid. Hence the hostility of a hydrocarbon medium such as a lipid membrane to ionic charge. But what do electric fields do to lipid membranes? Voltages of 100 or 200 mV applied to a 30- to 70-A lipid film will cause it to thin. Potentials of 300 mV or more will cause it to break. A potential of 300 mV across a 50-A layer corresponds to a breakdown field of 6 x lo5volts/cm. statcoulombs) in If one considers a single ionic charge (e = 4.8 x a lipid medium (dielectric constant = 2) there is an electric field greater than or equal to 6 x lo5 volts out to a distance of 35 A from the ionic charge ( e / d 2 6 x lo3 volts/cm, r _< 35 A ) . It turns out that while an ion is crossing a 50-A membrane, one or both faces of the membrane sufler an electric field ( f r o m the ion) greater than the known breakdown voltage. The direction of the electric stress is to pinch the membrane at the point of traverse. A thin film differs from a macroscopic medium in that the reach of the electric field from an ion exceeds the dimensions of the membrane. One need not think a priori of the membrane as a separate phase. In the following I will illustrate the magnitude of the electric stress on the membrane faces. I will also review some results for the electrostatic interaction between an ion and different forms of membrane defects created by peptides that facilitate transport. These peptides act to lower the energetic barrier to ion flow. It appears from some numerical examples that a low dielectric membrane might deform while an ion moves across. Probable energies for creating deformation are comparable to the lowering of ion-lipid interaction energy that would accompany deformation. Some experiments are suggested for detecting local changes in membrane structure. Several reasons are given for keeping aware of the structural conse161
162
Annals New York Academy of Sciences
quences of ion-lipid interactions. Recognizing the symmetry in which ion and membrane influence each other allows us to see that the deformative properties of membrane lipids, as well as membrane “fluidity” or “viscosity,” might be critical to the successful design of membrane systems.
Electrostatic Polarization Forces Exerted b y an Ion Moving across a Low Dielectric Membrane
One tends to think of the work required to move a charge from water to lipid primarily in terms of removing “water of hydration” from its immediate vicinity and replacing it with hydrocarbon. Neutral cyclic proteins-valinomycin, nonactin, enniatin B, diactin, etc.’-are said to act as “carriers” whose polar groups replace the hydration layers. The effective radius of the positively charged complex is much larger than that of the ion. Entry into a lipid is thereby facilitated. Once within the membrane the positive charge will induce a net negative charge on the interfaces between lipid and water. (Negative charge is pulled more easily to the interface from the water than away from the interface into the lipid.) If the membrane faces remain rigid, the ion must overcome attraction to this interfacial charge (conveniently modeled as an “image” force; see Appendix A ) in order to move away from the interface. The profile of the corresponding ion-interface interaction energy is shown in FIGURE 1. The pull on an ion at a distance of 10 A from the interface is about 3 x lo-’’ dynes. The stress on the interface is more than 200 atmospheres nearest to the ion. The local electric field at that part of the membrane is about 14 x 10” volts/cm. Incidentally, Eisenman’ has fitted carrier-transport data to a trapezoid potential which gives essentially the same result. In that mathematical model the ion climbs a hill about 10 kT high over a distance of the order of 10 A. The net force is 4 x 10-’’/10-’ = 4 x 10.‘‘ dynes between ion and membrane interface. At the middle of the membrane the net force on an ion is zero by symmetry. But the inward pressure on either face of a membrane 50 A thick is greater than one atmosphere over a circular path on the surface 50 A across.
Water Ew =
Interaction of ion with walls
80
FIGURE1. The “hill” an ionic charge must climb in order to go across a lipid membrane with fixed interfaces in water. The function S is described in Appendix A and has been used as a fixed barrier to ion transport.‘ (Compare Equation A2, Appendix A ) .
Parsegian:
Ion-Membrane Interactions
FIGURE 2. Profile of the dielectric stress on lipid-water interfaces due to an ionic charge, e, midway between them.6 Note that the spread of pressure is over a radius of the order of membrane thickness C and that the magnitude of stress is as the inverse foiirrlz power of thickness.
pressure in units (+)
,
31
, *
21
163
1:A P
0
distance from perpendiculor line through change e
-
We are told1.2 that lipid membranes have a static Young's modulus of the order of lo6 dynes/cm2 or 1 atrn. That is ad/C compressive pressure (atm). Electrostrictive stress from an ion is apparently significant on this scale. According to FIGURE 2 the stress profile is strongly localized above the point of ionic traverse. It extends over an area comparable in diameter to the distance of the charge from the interface. For the configuration of an ion in the middle of a SO-A thick membrane, there will be an electric field greater than the breakdown field, 6 X lo5 volts/cm, over a circle of 40-A radius. This comprises an area much bigger than the cross section of any carrier complex or of a membrane pore. A bilayer membrane might thin under the stab of the ionic electric field. Is there energy enough to do it? Electrostatic Znteraction Energies between Ionic Charge and Model Membrane
We may refer to three identical configurations of low dielectric membrane, high dielectric medium, and ion for illustrative calculation. These are: ( a ) A l , Appendix an ion in a low dielectric slab of finite thickness (FIGURE A ) ; (b) an ion in the center of a pore drilled through the slab (FIGURE B 1, Appendix B ) ; (c) an ion surrounded by a spherical jacket whose dielectric C1, Appendix polarizability differs from the surrounding medium (FIGURE C). In each case the energy of charging the ion, treated as a conducting sphere, has been found for various dimensions and dielectric va1ues.B
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Annals New York Academy of Sciences
(a) The ion’s energy differs from that in an infinite medium c by a term e4 --lncc
2d €
+
t’
where e’ is the dielectric outside the slab and G is slab thickness. For typical values of c, t’, and of ionic radius the energy is decreased only a few percent compared to 3). Given present estimates of artificial membrane the leading term e2/2ea (FIGURE thickness it is clear that finite width does not substantially change the barrier posed by low ion solubility in a lipid medium. (b) The barrier for passing through a high dielectric pore is much lower than that of a low dielectric region. The ion energy in a pore of radius b, dielectric tp is (Equation B5, Appendix B)
.!
+,,
where R is the ratio of low to high dielectric R = and P(R) is plotted in FIGURE B2. For water at 37 C, e p = 73; if E = 2, this energy is 50 kT/b(& or about 10 kT = 6.2 kcal/mol for b = 5 A. This is relative to the ion energy in a pore of infinite radius. We note that these numbers, assuming a water-filled pore, will not be accurate for a gramicidin A channel lined with flexibly mounted carbonyl groups. Judging from the low energetic barrier to ions posed by this pore’ and arguments regarding pore structure,” the effective polarizability of this channel is higher than for one containing only water. (c) A charge “solvated” by a high dielectric jacket (FIGURE C1) can have a substantially lower charging energy than one “bare” in a low dielectric medium. Reduction by a factor of 70% is reasonable even if the effective jacket dielectric is 20 and that of the medium is 2 for a charge of radius
t
40 20--
-e.o..i..
~
tr.r.i., t..r.i.r
-
1.05
T.J25
f;”.
= I0 =m =
-
.- 25kT
--
10 k1
u
Parsegian : Ion-Membrane Interactions
FIGURE 4. Scheme for possible indentation under electric stress (see FIGURE2 ) . If the shaded regions have the polarizability of water, then the difference in energies of (a) “carrier” and (b) “pore” give a measure of the electrostatic energy decrease in forming (c). Numbers for this case can 3. be taken from FIGURE
165
b
1.3 A and outer jacket radiu3 7 A (FIGURE3). These are sizes described for K’ ion complexed to valinomycin, which has been considered an ion carrier.’ The accuracy of these estimates, based on a macroscopic model, is open to valid question. As in the simple Born theory of ion solvation in water one treats all media as continuum dielectrics about a spherical charge. The ionic charge is given an “effective radius’’ (greater than its “crystal radius”) corresponding to the observed energy of solvation. Further, dielectric saturation (due to intense fields near the ion) is ignored. Although this is not the way to derive a complete description of an ionic charge interacting with its environment, it appears adequate for answers to questions of orders of magnitude required here. These questions deal with the effect of boundaries between dielectric media introduced a relatively large distance from the charge. One asks for the difference in energy between an ion in infinite medium and within slab walls many angstroms from the ion (FIGUREA l ) ; or between bulk medium and a cylindershaped medium where the effect is due to a field induced on the cylinder walls and is weak compared to the field near the ion; or between a charge in a cylinder and in a spherical shape dielectric where the difference in structure occurs several angstroms from the center charge. The resulting numbers give a rough estimate of the electrostatic energy decrease that would occur if the membrane pinched in at the place where an ion was crossing. We compare the energy of ion in carrier FIGURE4a with ion in pore FIGURE 4b to get an idea of the energy for indenting as in FIGURE 4c. From the computations of FIGURE 3 one expects energy decreases of the order of 5 to 15 kcal/mol or 10 to 30 kT/ion. This energy compares favorably with the “cost” of 5.5 kT computed by Haydon” to dimple a 48-A membrane down to 28 A at the site of gramicidin A or with the work of 1 kcal/mol per angstrom inferred by Urry8 from the same data. Energies of these magnitudes are also of the order as those interactions responsible for phase transitions in lipid.”-’s Comments and Consequence,s
The pressure pulse exerted by an ion on a membrane will last only for the time of ionic traverse and extend only over a region of hundreds or
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Annals New York Academy of Sciences
a few thousand square angstroms. But most of what we know about the response of membranes to applied fields is for indefinitely extended fields lasting for comparatively long times and causing concerted displacements by virtually all the membrane lipid molecules. An indentation that has the shape of the ionic stress (FIGURE 2 ) and that forms in the ion’s wake would not require displacement of more than a few lipid molecules. It would likely be difficult to detect directly. Even the relatively long-lived indentations of lipid near gramicidin A “pores” must be inferred indirectly. Certainly there is enough time for the lipid to respond to the ion. Probable displacements are not necessarily more than what is needed to accommodate a diffusing carrier. (And we do know carrier-coated ions get across membranes!) If membrane deformations cannot be seen directly, what kind of behavior would suggest its occurrence? The following questions might help: IS the membrane breakdown voltage ever a function of the ionic strength of the bathing medium or of the amount of assisting protein? (It is obviously impractical to speak of a breakdown current.) If individual ions deform a membrane, then the sum of many deformations may be what leads to breakdown. Will substances that cannot themselves cross a membrane aid in transport of ions? For example, small fatty alcohols or small amphophiles might stick to the lipid interface to lower its interfacial energy and deformability, but will not change the lipid interior (which determines the barrier posed by an immutable membrane). Facilitation of ion transport would suggest that the interfacial energy is important to ionic transit. Will substances that act like pores across thin membranes show some carrier properties across a macroscopically thick lipid barrier?“ For example, the channel-former gramicidin A interacts with ions via carbonyl groups, as does the carrier peptide valinomycin. If the membrane were much thicker than the 38-A length of gramicidin A, would this drug move as a mobile protein in association with a cation? Is the probability of an open channel formed by neutral peptides ever a function of salt concentration? The occurrence of alamethicin channels depends strongly on salt in the bathing solution and on applied voltages across the ~nembrane.~ Formation of gramicidin A pores appears to require some indentation by the host lipid membrane. It may be that presence of an ion in the pore protein creates an electric stress to promote indentation. The purpose behind asking these questions is to gather information for a physically plausible picture of transport. The lipids might be as important as proteins. Convenient lipid preparations may have anomalous physical properties that make the models qualitatively different from real membranes. On a large scale, solvent-containing phospholipids are probably less deformable than pure bilayers; their ability to conduct “carriers” or accommodate “pores” may also differ. Polar groups which change interfacial energy may be important in the design of real membranes; evidence of this might be seen in model systems. Rather than thinking of dissolution and diffusion across a membrane, it may be better to think of an instability occurring when an ion gets into a membrane.” Phenomenologically these mechanisms might possibly look similar under some model conditions, but physically the processes are very different. Under the actual conditions enjoyed by real membranes this difference may be essential.
Eldefrawi et al.: Acetylcholine Receptor
193
fluorescence of the protein we could detect as little as 1 pg/ml (FIGURE I), which was undectable by uv absorption or Lowry analysis. Unfortunately, there was no change in fluorescence emission spectra upon binding of carbamylcholine, d-tubocurarine, ACh, or decamethonium.” This eliminates the utilization of the protein’s native fluorescence in kinetic studies of its binding of ligands. The ACh-receptor protein contains 0.9-2.8 mol percent cysteic acid. In T . californica, we found that approximately 18% of the total (equivalent to 20 nmol of free SH groups per mg protein) was present as cysteine re~idues.~‘ The majority of these free SH groups were oxidized during purification of the receptor unless deaerated solutions were used, and the final dialysis was performed under nitr~gen.’~ There is a good possibility that the ACh-receptor is a glycoprotein since hexosamines were found in some of the purified preparations.*’*3‘. ” Based on the data of the interactions of the purified ACh-receptor with plant lectins, it was concluded that the receptor contained a carbohydrate moiety of at least D-mannose and N-acetyl-D-galacto~amine.’~N-acetyl-D-glucosamine was detected in another preparation3’ and by means of ion-exchange chromatography mannose, galactose and Dglucose were also found. TABLE3 FROM AMINOACIDCOMPOSITION (MOL % ) OF ACH-RECEPTORS ELECTRIC ORGANSOF SEVERAL FISH
THE
T.californica
T. marmorata T. nobiliana
E. electricus
Amino Acid
(35)
(37)
(27)
(34)
(59)
(26)
(33)
Lysine Histidine Arginine Aspartic acid Threonine Serine Glutamic acid F’roline Glycine Alanine Cysteic acid Valine Methionine Isoleucine Leuci n e Tyrosine Phenyhlanine Tryptophan Glucosamine
5.4 2.4 3.9 11.6 6.4 7.9 10.0 5.9 4.6 5 .o 1.2 7.1 2.0 8.2 9.5 3.7 4.5 2.4 -
6.1 2.7 4.1 11.9 6.3 6.6 10.2 5.9 4.9 5.1 0.9 7.0 1.8 7.5 9.7 3.8 4.6
6.1 2.1 3.5 11.8 6.3 7.1 10.7 6.2 6.4 6.0 2.0 5.5 1.7 5.2 9.3 3.6 4.4 2.1
5 .o 2.5 3.3 12.4 6.2 8.1 8.7 5.6 5.0 5 .o
-
-
4.5 2.5 3.7 12.2 6.8 6.4 9.7 7.1 5.0 4.5 2.8 6.2 1.6 6.2 10.2 4.2 4.2 1.5 2.0
4.6 2.2 4.2 11.4 5.6 6.2 10.2 5.7 5.9 5.8 2.0 8.6 2.0 6.4 10.5 4.0 5.7 0 -
6.3 2.5 4.2 9.8 6.0 8.2 9.0 6.7 4.8 5.4 1.7 6.9 3.4 8.1 10.7 3.8 5.1 2.4
Percentage of polar residuest
47
48
48
46
46
44
46
-
0.9
-
7.3 2.5 7.4 10.1 3.5 4.5
+*
* On the basis of the binding of the ACh-receptor to several plant lectins, it was suggested that the receptor carried a carbohydrate moiety containing at least D-mannose and N-acetylD-galactosamine. t The sum of Asp, Glu, Lys, Ser, Arg, Thr and His as classified by Capaldi and Vanderkooi.81 -
168
Annals New York Academy of Sciences
Discussion
DR. IBERALL:I would like to add another piece of information from solid-state theory. At extremely high strain rates, that is for rapid processes, there is perhaps some little-known material where in fact the Young's modulus approaches dynamically the order of bulk modulus, which means that the kina' of basic resiliances that you face in solid state is on the order of 20,000 atmospheres. DR. PARSEGIAN: Good point. DR. FINKELSTEIN: Do you have any estimates of the relaxation times it would take to bring about the pinching down of the membrane that you've mentioned? DR. PARSEGIAN: Yes, the kinds of deformations that one talks aboutthat much greater than what would because the stress is so localized-aren't have to be accommodated to get a carrier across anyway. I suspect that the time scale is not that different from what you'd expect from a diffusion, moving the chains apart by 10 or 20 A. In other words, I don't think that microseconds are too short a time for distances of movement of a few angstroms. Appendix A : Charge in a Finite Slab Consider a dielectric medium e which is planar, infinitely extended in the y and z directions but of finite thickness C in the x direction (FIGURE Al). The two faces are put at x = +C/2 and x = - d / 2 . The medium outside this slab has dielectric constant e'. One wants to know what will be the energy due to a charge, e, inside such a slab compared to the energy of that charge deep in the infinite medium or compared to its energy when immersed in an infinitely thick slab. It
t + 4 4 FIGURE Al. Charge e of radius a in a planar slab of dielectric thickness d. Outside dielectric is t'. The electrostatic potential is given by an infinite series of image charges uniformly spaced at a distance d along a line through the original charge and perpendicular to the plane of the slab.
Parsegian:
Ion-Membrane Interactions
169
is necessary to calculate the electrostatic potential about the charge in each of these cases, and then to find the energy of the system in this potential which it has created-that “self-energy.” For a spherical charge in a uniform medium, the self-energy is the wellknown Born charging energy ez/2ta
(-41)
The potential due to a point charge in a dielectric slab is the sum of the e/€r coulomb potential from interaction with the charge itself plus interaction with charge induced on the dielectric boundary surfaces. The latter is conveniently described as a set of image charges at positions which are reflections of the original charge through the dielectric boundary faces. The change in self-energy of a charge from that in a medium to one in a thin slab is the strength of interaction with the image charges induced (again with reference to a discharged state). When the point charge is a distance, x, from the center of the slab, the free energy of coulombic interaction with images is OD
where q = e =
(t
- d)/(t + d),
a = 2x/d. (As expected q = 0 (no image charge) if
el.)
When the charge is at the slab center x finite dielectric is concisely
=
0, and the extra energy of being in the
(A31 which is zero for t’
=
t
e2 (infinite medium) and goes to - - In 2 as
t‘
>> C.
In the general case for any a between f1 the sum becomes ef cosh a t
1
dt - In (1 - 4’)
(A41
which reduces to Equation A3 when a = 0. The energy is inverse to slab thickness 1 and to slab dielectric C. If the outside dielectric 6’ is less than C, q < 0 and (A2), (A3) and (A4) are negative. The self-energy is lowered by presence of the medium C’ and the x = 0 center point (A3) is a maximal energy position for the charge. The electrostatic stress perpendicular to the interface between slab and medium is the difference
> €. Let there be a cylindrical region of radius b, dielecB I ) . What is the charging energy of a charge, tric cP through the slab (FIGURE e, at the center of the slab and cylinder? For cylinder radius b much less than slab thickness 1 the potential in the pore due to a (point) charge e isla
Here r is the distance from the charge; R is the ratio e/cp; KO,Ko', loare modified Bessel functions of imaginary argument; p and z are cylindrical coordinates measuring distance from the pore axis ( p ) and distance along the axis from the charge (z). In this potential the ordinary coulomb term e/cpr is altered by a term due to the presence of the dielectric boundary at p = b. It is interaction of the charge with this second term that is the energy of being a distance b from the low-dielectric material outside the pore. Hence one is interested in the magnitude of Viadueed(O), which is
2e be,
= -f(R).
FIGUREB1. Cross section of cylindrical "pore" of dielectric ep radius b through the low dielectric slab >> thickness d. Treatment in text assumes b b. Inside the medium ec it is - + re:
i
c o (:
:)
---
.
The self-energy of the charge is then
and subtracting the energy of a chargewithout jacket, e2/2ch,a, one gets the difference
FIGURE C1.
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Annals New York Academy of Sciences
in free energy of charging due to presence of the material
r[(i-a)(&:)]=$[
(;-1)(1-9]
(C2)
The factor in square brackets is a convenient form for calculating the energetic effect of the layer ee relative to the energy of the charge in medium €llC alone. If Llle is not a constant but varies with distance, r, from the center of the spherical complex, we may write e, =
1 r*f(r)
For any convenient f(r) the self-energy is then
where
F(b) - F(a) =
/Lb f(r) dr.
I would like to elaborate upon what I see as a inconsistency in the way we look at the passage of an ion across a low dielectric membrane. This process will be discussed in terms of the energy of interaction between ion and membrane. We know that a membrane poses an energetic barrier to the ions in the adjacent solutions. We know that the modulation of this barrier is the primary means by which a membrane controls ionic flow. But we tend to think of the interaction only as one imposed' by the membrane on the ion rather than a mutual interaction that might act to perturb the structure of the membrane. The cause of the ion-membrane interaction is the coulomb field that emanates from the ionic charge. This field acts to polarize the surrounding medium, to attract charge of opposite sign to the ion, to repel like charge and thereby to reduce ionic energy. Polarization is easy in an aqueous medium of high dielectric constant and hard in low dielectric lipid. Hence the hostility of a hydrocarbon medium such as a lipid membrane to ionic charge. But what do electric fields do to lipid membranes? Voltages of 100 or 200 mV applied to a 30- to 70-A lipid film will cause it to thin. Potentials of 300 mV or more will cause it to break. A potential of 300 mV across a 50-A layer corresponds to a breakdown field of 6 x lo5volts/cm. statcoulombs) in If one considers a single ionic charge (e = 4.8 x a lipid medium (dielectric constant = 2) there is an electric field greater than or equal to 6 x lo5 volts out to a distance of 35 A from the ionic charge ( e / d 2 6 x lo3 volts/cm, r _< 35 A ) . It turns out that while an ion is crossing a 50-A membrane, one or both faces of the membrane sufler an electric field ( f r o m the ion) greater than the known breakdown voltage. The direction of the electric stress is to pinch the membrane at the point of traverse. A thin film differs from a macroscopic medium in that the reach of the electric field from an ion exceeds the dimensions of the membrane. One need not think a priori of the membrane as a separate phase. In the following I will illustrate the magnitude of the electric stress on the membrane faces. I will also review some results for the electrostatic interaction between an ion and different forms of membrane defects created by peptides that facilitate transport. These peptides act to lower the energetic barrier to ion flow. It appears from some numerical examples that a low dielectric membrane might deform while an ion moves across. Probable energies for creating deformation are comparable to the lowering of ion-lipid interaction energy that would accompany deformation. Some experiments are suggested for detecting local changes in membrane structure. Several reasons are given for keeping aware of the structural conse161
162
Annals New York Academy of Sciences
quences of ion-lipid interactions. Recognizing the symmetry in which ion and membrane influence each other allows us to see that the deformative properties of membrane lipids, as well as membrane “fluidity” or “viscosity,” might be critical to the successful design of membrane systems.
Electrostatic Polarization Forces Exerted b y an Ion Moving across a Low Dielectric Membrane
One tends to think of the work required to move a charge from water to lipid primarily in terms of removing “water of hydration” from its immediate vicinity and replacing it with hydrocarbon. Neutral cyclic proteins-valinomycin, nonactin, enniatin B, diactin, etc.’-are said to act as “carriers” whose polar groups replace the hydration layers. The effective radius of the positively charged complex is much larger than that of the ion. Entry into a lipid is thereby facilitated. Once within the membrane the positive charge will induce a net negative charge on the interfaces between lipid and water. (Negative charge is pulled more easily to the interface from the water than away from the interface into the lipid.) If the membrane faces remain rigid, the ion must overcome attraction to this interfacial charge (conveniently modeled as an “image” force; see Appendix A ) in order to move away from the interface. The profile of the corresponding ion-interface interaction energy is shown in FIGURE 1. The pull on an ion at a distance of 10 A from the interface is about 3 x lo-’’ dynes. The stress on the interface is more than 200 atmospheres nearest to the ion. The local electric field at that part of the membrane is about 14 x 10” volts/cm. Incidentally, Eisenman’ has fitted carrier-transport data to a trapezoid potential which gives essentially the same result. In that mathematical model the ion climbs a hill about 10 kT high over a distance of the order of 10 A. The net force is 4 x 10-’’/10-’ = 4 x 10.‘‘ dynes between ion and membrane interface. At the middle of the membrane the net force on an ion is zero by symmetry. But the inward pressure on either face of a membrane 50 A thick is greater than one atmosphere over a circular path on the surface 50 A across.
Water Ew =
Interaction of ion with walls
80
FIGURE1. The “hill” an ionic charge must climb in order to go across a lipid membrane with fixed interfaces in water. The function S is described in Appendix A and has been used as a fixed barrier to ion transport.‘ (Compare Equation A2, Appendix A ) .
Parsegian:
Ion-Membrane Interactions
FIGURE 2. Profile of the dielectric stress on lipid-water interfaces due to an ionic charge, e, midway between them.6 Note that the spread of pressure is over a radius of the order of membrane thickness C and that the magnitude of stress is as the inverse foiirrlz power of thickness.
pressure in units (+)
,
31
, *
21
163
1:A P
0
distance from perpendiculor line through change e
-
We are told1.2 that lipid membranes have a static Young's modulus of the order of lo6 dynes/cm2 or 1 atrn. That is ad/C compressive pressure (atm). Electrostrictive stress from an ion is apparently significant on this scale. According to FIGURE 2 the stress profile is strongly localized above the point of ionic traverse. It extends over an area comparable in diameter to the distance of the charge from the interface. For the configuration of an ion in the middle of a SO-A thick membrane, there will be an electric field greater than the breakdown field, 6 X lo5 volts/cm, over a circle of 40-A radius. This comprises an area much bigger than the cross section of any carrier complex or of a membrane pore. A bilayer membrane might thin under the stab of the ionic electric field. Is there energy enough to do it? Electrostatic Znteraction Energies between Ionic Charge and Model Membrane
We may refer to three identical configurations of low dielectric membrane, high dielectric medium, and ion for illustrative calculation. These are: ( a ) A l , Appendix an ion in a low dielectric slab of finite thickness (FIGURE A ) ; (b) an ion in the center of a pore drilled through the slab (FIGURE B 1, Appendix B ) ; (c) an ion surrounded by a spherical jacket whose dielectric C1, Appendix polarizability differs from the surrounding medium (FIGURE C). In each case the energy of charging the ion, treated as a conducting sphere, has been found for various dimensions and dielectric va1ues.B
164
Annals New York Academy of Sciences
(a) The ion’s energy differs from that in an infinite medium c by a term e4 --lncc
2d €
+
t’
where e’ is the dielectric outside the slab and G is slab thickness. For typical values of c, t’, and of ionic radius the energy is decreased only a few percent compared to 3). Given present estimates of artificial membrane the leading term e2/2ea (FIGURE thickness it is clear that finite width does not substantially change the barrier posed by low ion solubility in a lipid medium. (b) The barrier for passing through a high dielectric pore is much lower than that of a low dielectric region. The ion energy in a pore of radius b, dielectric tp is (Equation B5, Appendix B)
.!
+,,
where R is the ratio of low to high dielectric R = and P(R) is plotted in FIGURE B2. For water at 37 C, e p = 73; if E = 2, this energy is 50 kT/b(& or about 10 kT = 6.2 kcal/mol for b = 5 A. This is relative to the ion energy in a pore of infinite radius. We note that these numbers, assuming a water-filled pore, will not be accurate for a gramicidin A channel lined with flexibly mounted carbonyl groups. Judging from the low energetic barrier to ions posed by this pore’ and arguments regarding pore structure,” the effective polarizability of this channel is higher than for one containing only water. (c) A charge “solvated” by a high dielectric jacket (FIGURE C1) can have a substantially lower charging energy than one “bare” in a low dielectric medium. Reduction by a factor of 70% is reasonable even if the effective jacket dielectric is 20 and that of the medium is 2 for a charge of radius
t
40 20--
-e.o..i..
~
tr.r.i., t..r.i.r
-
1.05
T.J25
f;”.
= I0 =m =
-
.- 25kT
--
10 k1
u
Parsegian : Ion-Membrane Interactions
FIGURE 4. Scheme for possible indentation under electric stress (see FIGURE2 ) . If the shaded regions have the polarizability of water, then the difference in energies of (a) “carrier” and (b) “pore” give a measure of the electrostatic energy decrease in forming (c). Numbers for this case can 3. be taken from FIGURE
165
b
1.3 A and outer jacket radiu3 7 A (FIGURE3). These are sizes described for K’ ion complexed to valinomycin, which has been considered an ion carrier.’ The accuracy of these estimates, based on a macroscopic model, is open to valid question. As in the simple Born theory of ion solvation in water one treats all media as continuum dielectrics about a spherical charge. The ionic charge is given an “effective radius’’ (greater than its “crystal radius”) corresponding to the observed energy of solvation. Further, dielectric saturation (due to intense fields near the ion) is ignored. Although this is not the way to derive a complete description of an ionic charge interacting with its environment, it appears adequate for answers to questions of orders of magnitude required here. These questions deal with the effect of boundaries between dielectric media introduced a relatively large distance from the charge. One asks for the difference in energy between an ion in infinite medium and within slab walls many angstroms from the ion (FIGUREA l ) ; or between bulk medium and a cylindershaped medium where the effect is due to a field induced on the cylinder walls and is weak compared to the field near the ion; or between a charge in a cylinder and in a spherical shape dielectric where the difference in structure occurs several angstroms from the center charge. The resulting numbers give a rough estimate of the electrostatic energy decrease that would occur if the membrane pinched in at the place where an ion was crossing. We compare the energy of ion in carrier FIGURE4a with ion in pore FIGURE 4b to get an idea of the energy for indenting as in FIGURE 4c. From the computations of FIGURE 3 one expects energy decreases of the order of 5 to 15 kcal/mol or 10 to 30 kT/ion. This energy compares favorably with the “cost” of 5.5 kT computed by Haydon” to dimple a 48-A membrane down to 28 A at the site of gramicidin A or with the work of 1 kcal/mol per angstrom inferred by Urry8 from the same data. Energies of these magnitudes are also of the order as those interactions responsible for phase transitions in lipid.”-’s Comments and Consequence,s
The pressure pulse exerted by an ion on a membrane will last only for the time of ionic traverse and extend only over a region of hundreds or
166
Annals New York Academy of Sciences
a few thousand square angstroms. But most of what we know about the response of membranes to applied fields is for indefinitely extended fields lasting for comparatively long times and causing concerted displacements by virtually all the membrane lipid molecules. An indentation that has the shape of the ionic stress (FIGURE 2 ) and that forms in the ion’s wake would not require displacement of more than a few lipid molecules. It would likely be difficult to detect directly. Even the relatively long-lived indentations of lipid near gramicidin A “pores” must be inferred indirectly. Certainly there is enough time for the lipid to respond to the ion. Probable displacements are not necessarily more than what is needed to accommodate a diffusing carrier. (And we do know carrier-coated ions get across membranes!) If membrane deformations cannot be seen directly, what kind of behavior would suggest its occurrence? The following questions might help: IS the membrane breakdown voltage ever a function of the ionic strength of the bathing medium or of the amount of assisting protein? (It is obviously impractical to speak of a breakdown current.) If individual ions deform a membrane, then the sum of many deformations may be what leads to breakdown. Will substances that cannot themselves cross a membrane aid in transport of ions? For example, small fatty alcohols or small amphophiles might stick to the lipid interface to lower its interfacial energy and deformability, but will not change the lipid interior (which determines the barrier posed by an immutable membrane). Facilitation of ion transport would suggest that the interfacial energy is important to ionic transit. Will substances that act like pores across thin membranes show some carrier properties across a macroscopically thick lipid barrier?“ For example, the channel-former gramicidin A interacts with ions via carbonyl groups, as does the carrier peptide valinomycin. If the membrane were much thicker than the 38-A length of gramicidin A, would this drug move as a mobile protein in association with a cation? Is the probability of an open channel formed by neutral peptides ever a function of salt concentration? The occurrence of alamethicin channels depends strongly on salt in the bathing solution and on applied voltages across the ~nembrane.~ Formation of gramicidin A pores appears to require some indentation by the host lipid membrane. It may be that presence of an ion in the pore protein creates an electric stress to promote indentation. The purpose behind asking these questions is to gather information for a physically plausible picture of transport. The lipids might be as important as proteins. Convenient lipid preparations may have anomalous physical properties that make the models qualitatively different from real membranes. On a large scale, solvent-containing phospholipids are probably less deformable than pure bilayers; their ability to conduct “carriers” or accommodate “pores” may also differ. Polar groups which change interfacial energy may be important in the design of real membranes; evidence of this might be seen in model systems. Rather than thinking of dissolution and diffusion across a membrane, it may be better to think of an instability occurring when an ion gets into a membrane.” Phenomenologically these mechanisms might possibly look similar under some model conditions, but physically the processes are very different. Under the actual conditions enjoyed by real membranes this difference may be essential.
Eldefrawi et al.: Acetylcholine Receptor
193
fluorescence of the protein we could detect as little as 1 pg/ml (FIGURE I), which was undectable by uv absorption or Lowry analysis. Unfortunately, there was no change in fluorescence emission spectra upon binding of carbamylcholine, d-tubocurarine, ACh, or decamethonium.” This eliminates the utilization of the protein’s native fluorescence in kinetic studies of its binding of ligands. The ACh-receptor protein contains 0.9-2.8 mol percent cysteic acid. In T . californica, we found that approximately 18% of the total (equivalent to 20 nmol of free SH groups per mg protein) was present as cysteine re~idues.~‘ The majority of these free SH groups were oxidized during purification of the receptor unless deaerated solutions were used, and the final dialysis was performed under nitr~gen.’~ There is a good possibility that the ACh-receptor is a glycoprotein since hexosamines were found in some of the purified preparations.*’*3‘. ” Based on the data of the interactions of the purified ACh-receptor with plant lectins, it was concluded that the receptor contained a carbohydrate moiety of at least D-mannose and N-acetyl-D-galacto~amine.’~N-acetyl-D-glucosamine was detected in another preparation3’ and by means of ion-exchange chromatography mannose, galactose and Dglucose were also found. TABLE3 FROM AMINOACIDCOMPOSITION (MOL % ) OF ACH-RECEPTORS ELECTRIC ORGANSOF SEVERAL FISH
THE
T.californica
T. marmorata T. nobiliana
E. electricus
Amino Acid
(35)
(37)
(27)
(34)
(59)
(26)
(33)
Lysine Histidine Arginine Aspartic acid Threonine Serine Glutamic acid F’roline Glycine Alanine Cysteic acid Valine Methionine Isoleucine Leuci n e Tyrosine Phenyhlanine Tryptophan Glucosamine
5.4 2.4 3.9 11.6 6.4 7.9 10.0 5.9 4.6 5 .o 1.2 7.1 2.0 8.2 9.5 3.7 4.5 2.4 -
6.1 2.7 4.1 11.9 6.3 6.6 10.2 5.9 4.9 5.1 0.9 7.0 1.8 7.5 9.7 3.8 4.6
6.1 2.1 3.5 11.8 6.3 7.1 10.7 6.2 6.4 6.0 2.0 5.5 1.7 5.2 9.3 3.6 4.4 2.1
5 .o 2.5 3.3 12.4 6.2 8.1 8.7 5.6 5.0 5 .o
-
-
4.5 2.5 3.7 12.2 6.8 6.4 9.7 7.1 5.0 4.5 2.8 6.2 1.6 6.2 10.2 4.2 4.2 1.5 2.0
4.6 2.2 4.2 11.4 5.6 6.2 10.2 5.7 5.9 5.8 2.0 8.6 2.0 6.4 10.5 4.0 5.7 0 -
6.3 2.5 4.2 9.8 6.0 8.2 9.0 6.7 4.8 5.4 1.7 6.9 3.4 8.1 10.7 3.8 5.1 2.4
Percentage of polar residuest
47
48
48
46
46
44
46
-
0.9
-
7.3 2.5 7.4 10.1 3.5 4.5
+*
* On the basis of the binding of the ACh-receptor to several plant lectins, it was suggested that the receptor carried a carbohydrate moiety containing at least D-mannose and N-acetylD-galactosamine. t The sum of Asp, Glu, Lys, Ser, Arg, Thr and His as classified by Capaldi and Vanderkooi.81 -
168
Annals New York Academy of Sciences
Discussion
DR. IBERALL:I would like to add another piece of information from solid-state theory. At extremely high strain rates, that is for rapid processes, there is perhaps some little-known material where in fact the Young's modulus approaches dynamically the order of bulk modulus, which means that the kina' of basic resiliances that you face in solid state is on the order of 20,000 atmospheres. DR. PARSEGIAN: Good point. DR. FINKELSTEIN: Do you have any estimates of the relaxation times it would take to bring about the pinching down of the membrane that you've mentioned? DR. PARSEGIAN: Yes, the kinds of deformations that one talks aboutthat much greater than what would because the stress is so localized-aren't have to be accommodated to get a carrier across anyway. I suspect that the time scale is not that different from what you'd expect from a diffusion, moving the chains apart by 10 or 20 A. In other words, I don't think that microseconds are too short a time for distances of movement of a few angstroms. Appendix A : Charge in a Finite Slab Consider a dielectric medium e which is planar, infinitely extended in the y and z directions but of finite thickness C in the x direction (FIGURE Al). The two faces are put at x = +C/2 and x = - d / 2 . The medium outside this slab has dielectric constant e'. One wants to know what will be the energy due to a charge, e, inside such a slab compared to the energy of that charge deep in the infinite medium or compared to its energy when immersed in an infinitely thick slab. It
t + 4 4 FIGURE Al. Charge e of radius a in a planar slab of dielectric thickness d. Outside dielectric is t'. The electrostatic potential is given by an infinite series of image charges uniformly spaced at a distance d along a line through the original charge and perpendicular to the plane of the slab.
Parsegian:
Ion-Membrane Interactions
169
is necessary to calculate the electrostatic potential about the charge in each of these cases, and then to find the energy of the system in this potential which it has created-that “self-energy.” For a spherical charge in a uniform medium, the self-energy is the wellknown Born charging energy ez/2ta
(-41)
The potential due to a point charge in a dielectric slab is the sum of the e/€r coulomb potential from interaction with the charge itself plus interaction with charge induced on the dielectric boundary surfaces. The latter is conveniently described as a set of image charges at positions which are reflections of the original charge through the dielectric boundary faces. The change in self-energy of a charge from that in a medium to one in a thin slab is the strength of interaction with the image charges induced (again with reference to a discharged state). When the point charge is a distance, x, from the center of the slab, the free energy of coulombic interaction with images is OD
where q = e =
(t
- d)/(t + d),
a = 2x/d. (As expected q = 0 (no image charge) if
el.)
When the charge is at the slab center x finite dielectric is concisely
=
0, and the extra energy of being in the
(A31 which is zero for t’
=
t
e2 (infinite medium) and goes to - - In 2 as
t‘
>> C.
In the general case for any a between f1 the sum becomes ef cosh a t
1
dt - In (1 - 4’)
(A41
which reduces to Equation A3 when a = 0. The energy is inverse to slab thickness 1 and to slab dielectric C. If the outside dielectric 6’ is less than C, q < 0 and (A2), (A3) and (A4) are negative. The self-energy is lowered by presence of the medium C’ and the x = 0 center point (A3) is a maximal energy position for the charge. The electrostatic stress perpendicular to the interface between slab and medium is the difference
> €. Let there be a cylindrical region of radius b, dielecB I ) . What is the charging energy of a charge, tric cP through the slab (FIGURE e, at the center of the slab and cylinder? For cylinder radius b much less than slab thickness 1 the potential in the pore due to a (point) charge e isla
Here r is the distance from the charge; R is the ratio e/cp; KO,Ko', loare modified Bessel functions of imaginary argument; p and z are cylindrical coordinates measuring distance from the pore axis ( p ) and distance along the axis from the charge (z). In this potential the ordinary coulomb term e/cpr is altered by a term due to the presence of the dielectric boundary at p = b. It is interaction of the charge with this second term that is the energy of being a distance b from the low-dielectric material outside the pore. Hence one is interested in the magnitude of Viadueed(O), which is
2e be,
= -f(R).
FIGUREB1. Cross section of cylindrical "pore" of dielectric ep radius b through the low dielectric slab >> thickness d. Treatment in text assumes b b. Inside the medium ec it is - + re:
i
c o (:
:)
---
.
The self-energy of the charge is then
and subtracting the energy of a chargewithout jacket, e2/2ch,a, one gets the difference
FIGURE C1.
174
Annals New York Academy of Sciences
in free energy of charging due to presence of the material
r[(i-a)(&:)]=$[
(;-1)(1-9]
(C2)
The factor in square brackets is a convenient form for calculating the energetic effect of the layer ee relative to the energy of the charge in medium €llC alone. If Llle is not a constant but varies with distance, r, from the center of the spherical complex, we may write e, =
1 r*f(r)
For any convenient f(r) the self-energy is then
where
F(b) - F(a) =
/Lb f(r) dr.
