Bu1lerr11Cl/ M‘,rh‘wl“lu‘rl Biology. Vol. 41. pp. 365 385 Pergamon Press Ltd. 1979. Printed in Great Bn~am 6 Sociely lor Mathematical Biology

00074985/79/050160365

$02.@3/0

NERNST-PLANCK ANALOG EQUATIONS AND STATIONARY STATE MEMBRANE ELECTRIC POTENTIALS

W. S. VAIDHYANATHAN Department of Biophysical

Sciences,

State University of New York at Buffalo, 114B Carey Hall, Maincampus, New York, NY 14226, U.S.A.

A set of coupled nonlinear differential equations, determining the concentration profiles and electric potentials valid for isothermal transport of ions and molecules across a diffusion barrier are formulated, using a correction to the limiting expression for chemical potential gradients and the molecular expression for frictional force. These differential equations are similar to Nernst-Planck equations and reduce to these under appropriate approximations. Solutions of these equations valid under specified conditions are presented. Expressions for permeability, concentration protiles of many ion systems are included.

1. Introduction. Isothermal transport of solutions containing three or more permeant ions across a diffusion barrier, such as a biological membrane, is of significant biophysical interest. Though considerable amount of experimental information on electrical potentials, boundary concentrations of ions and influence of divalent ions on fluxes of monovalent ions and permeabilities are available (Blaustein et al., 1975a, b), the discussion is usually limited to analysis based on equilibrium Nernst Potential and sometimes involving activity coefficient correction (Chang, 1977). Available values of concentrations of sodium, potassium and chloride ions in solutions of axon membrane systems (Hurlbut, 1971) when utilized in equilibrium Nernst expression yield potentials not in agreement with observed membrane potentials. Our understanding of the theoretical basis for experimentally observed variation of permeability of membrane system to permeant ions, as a function of electric potential, fluxes of other ions and chemical reactions if any, is unsatisfactory (Vaidhyanathan, 1977a, b). 365

366

V. S. VAIDHYANATHAN

The central problem which must be solved is to compute the value of stationary state electric potential difference in terms of known physical parameters of the system on an acceptable physical basis. A large number of papers on this subject deal with the problem of finding solutions of familiar Nernst-Planck equations (Leuchtag and Swihart, 1977). As is well known, these equations are nonlinear and solutions of these equations in closed analytic form have not yet been obtained, in spite of attempts by many over a period of seven decades. In certain specialized circumstances, solutions associated with the names of Planck, Henderson, Schlogl are available (Lakshminarayaniah, 1964). The assumptions involved in obtaining these solutions are not applicable for real membrane systems of interest in biology. A widely accepted equation in membrane physiology is the Goldman equation (Goldman, 1943) which involves the assumption that the stationary state electric potential profile is linear. This assumption, however convenient and appealing, requires validity of microscopic electroneutrality and is not aesthetically attractive, when one must reconcile with Poisson equation. Agin (1971) has presented in adequate detail, the status of our current knowledge about the attempts and frustrations in dealing with Nernst-Planck equations. This manuscript deals with certain nonlinear differential equations applicable to membrane barrier systems. These equations, to be formulated, are similar to the Nernst-Planck equations and are expected to be of more general validity, since coupling effects between ions and molecules arising from intermolecular interactions are included. Though the set of our initial equations are less simple than Nernst-Planck equations, it is our belief that these equations will depict real situations better and that solutions may be obtained. Later in this manuscript are presented solutions for concentration profiles valid under specified conditions. The contribution of fluxes of ionic and neutral species to electric potential profile are explicitly presented. 2. Derivation of Basic Equations. The electric potential 4(x), felt by a unit charge placed at location x, satisfies the Poisson equation [d2$/dx2]

=4”(x) = - (4ne/c)C

Z,C,(x),

where the right-hand side is the local charge density. The potential includes the contributions of all charged species present in the system and any externally applied field. Z, is the signed valence charge number of ionic species r~, and e is the protonic charge. C,(x) denotes the concentration of ionic species 6, at location x, which is called the concentration profile. The

NERNST-PLANCK

ANALOG

EQUATIONS

367

x-axis is defined normal to the 4’2 plane of the diffusion barrier (membrane). In writing equation (l), the plausible assumption that the dielectric coefficient E(X), may be regarded as position independent is incorporated. The summation sign in (1) should be carried out over all Ionic species present in the system. In this paper, the Greek subscripts denote Ionic species and Roman subscripts are employed to denote non-Ionic species. The derivatives with respect to position variable x, are denoted by a single prime. The second and higher derivatives of functions with respect to x, will be denoted by appropriate number of primes. The generally accepted equations of electrodiffusion are, (X&t)

+ [dJ,/dx] = 0,

D, = W/L,).

(2)

(4)

Equation (2) is the one dimensional equation of continuity for permeant species cr, in the absence of a chemical reaction in which ITparticipates. t is the time variable. Equation (4) is the Einstein relation between diffusion coefficient of species 0, D,, and its frictional coefficient L’,,with the thermal energy kT. k is the Boltzmann constant and Tis the temperature in Kelvin scale. Equation (3) is the familiar one dimensional Nernst-Planck equation for species rr. J, denotes the flux of species Q, expressed in moles per unit area per unit time. In a system with n kinds of ionic species undergoing transport, there are (n + 1) unknown functions, namely the n concentration profiles and the electric potential profile 4(x). Equations (1) and n equations of kind (3) provide the required (n+ 1) independent equations, when one assumes that the diffusion coefficients are constants. One assumes that in principle these set of equations may be solved. Under stationary state conditions, one assumes validity of linear relations between forces and fluxes (Onsager and Fuoss, 1932), -J, = c Q,,Vp,; 0

(o=1,2...,n),

(5)

where R,, are the phenomenological coefficients. The forces responsible for flow of ions are the isothermal gradients of electrochemical potentials, VP,. To obtain (3) from (5) one neglects all cross-phenomenological coefficients, (Q,,, = 0, when q #a) and utilizes the approximate relations,

(7)

368

V. S. VAIDHYANATHAN

It is well known that (7) is an approximation of ideal solution expression for chemical potential. &! is a constant independent of x and composition. The experimentally observed deviation from the value of chemical potential of (7) is usually represented by the definition of activity coefficients. Even if the assumption of neglect of cross phenomenological coefficients is valid, from (6) it follows that if one regards diffusion coefficients D,,hence [,, to be independent of position, the phenomenological coefficient R,, will be a function of position, if C, is a function of position. The logarithmic dependence of chemical potential on concentration in solutions arises as well known from entropic terms. Deviation from unity of the value of activity coefficient of species 0, arises from existence in real systems of intermolecular interactions. Concentrations of ions in biological systems are of the order of OSmoles/liter, indicating that the activity coefficients are distinct from unity. In place of (7), one can utilize the approximate expression for chemical potential including the potential energy contributions (Kirkwood, 1967),

+I Cq(X)Hog +I

Cj(x)H,j,

(8)

j

rl

where H,, and Hcj are molecular integrals over the potential energy of interactions of Q with q-th and j-th kind of species. The last two terms of (8) represent the correction to (7), when activity coefficients do not equal unity. Under stationary state conditions of isothermal and isobaric nature, the Gibbs-Duhem relation,

cC,(X)Pu:, = 0, 0

where the summation should be performed over all species present in the system, should be satisfied. The expression for gradient of chemical potential, i.e., mean force acting on species c, as given by molecular theory (Bearman, 1958; Vaidhyanathan, 1971), satisfying the Gibbs-Duhem relation is

+

1 Jqiot,+ C Jjiaj v

j

(10)

NERNST-PLANCK

ANALOG EQUATIONS

The frictional coefficient [,, that a molecule of kind r~ experiences location x in the diffusion barrier can be written as 5,(x)=r,+cfl(x)i,~+

C

369

at

(11)

cq(xKq+Ccj(xKoj, j

qfa

where ra is the contribution to c,, arising from immobile molecules of membrane framework, through intermolecular interactions. co,, is known as the partial frictional coefficient and represents the contribution per molecule of kind q to [,. Assuming the validity of (8) and additional assumption that H,, terms may be regarded as position independent and that Ho,, can be represented as Z,Z,e’H and that H,j can be represented as Z,eH*, the expressions for gradients in electrochemical potential for ionic species 0, and for uncharged species j, may be expressed as ,u~(x)= kT[d In C,(x)/dx] + Z,e@ + He2

c Z,Z,C~

+ H*Z,e

c C;,

9

(12)

j

P:(X) = kT[d In Cj(,x)/dx] + H*e C Z,Cb +C HjkC;. 0 k

(13)

One obtains from (lo), (11) and (12), for a membrane system containing n kinds of permeant ions and one kind of permeant neutral species j, the (n + 1) concentration profiles which should satisfy the following set of coupled nonlinear differential equations C&+ [Z,eC,F’/kT]

+ (H*e/kT)C’,C3Zb

+R,

-S,,C,(X)+S,jCj(X)+C

‘I+a

S,qCq=O*

C;-+ (Hjj/kT)CjCS + Rj- SjjCj + C Sj,C, 0 =[H*1:/4nkT]C,(s)@“(x).

S,, = CJ,l,,lIkT; S,, = (l/k’)

C

sfa

J,i,,

(14)

(15)

+ Jji,j/kT

(16) R, = (J,r,/kT);

i2 = [H&/471],

(17)

310

V. S. VAIDHYANATHAN F’(x) = c$‘(x) -~2~“‘(X).

(18)

In obtaining these equations, the Poisson equation has been utilized. In a system with no chemical reactions, under stationary state conditions, the coefficients S,,, S,, and R, are constants independent of x. The parameter A2, representing ion-ion potential energy of interactions contribution to activity coefficient is also a constant. Since molecular integrals H,, of equation (8) represent integration over the whole volume space, of potential energy of interaction weighted by probability functions, their dependence on x, the position variable in membrane phase, is expected to be insignificant. This justifies representation of H,, by Z,Z,e2H, where H is a constant representing ion-ion interaction contribution to chemical potential of species cr. When H equals zero, the activity coefficient equals unity. The concentration dependence of diffusion coefficients and the position dependence of frictional coefficients is assumed to be sufficiently represented by (11) such that the partial frictional coefficients may be regarded as constants. Equations (14) and (15) may be written as C~+C,{(Z,e/kT)[F’+H*CJ]-I,~+(J,/D,(X))=O, C~+Cj~(HjjC~/kT)-[H*&/4~.kT]~“‘--j}

(19)

+ (Jj/‘Dj(X))=O,

~,=(I/kT)CJ,i,,+Jjiaj n ijj = ( 1/k T ) C Jaioj + Jjijj c-r

(GO)

(21)

(22)

.

If one sets A2, H* as zero and regards D, as independent is identical with (3).

of x, then (19)

3. C’errc~irtEswr Results. Under conditions of osmotic equilibrium, when fluxes of permeant species vanish, (1) (8) and (9) are valid. Substitution of (12) and ( 13) in Gibbs-Duhem relation (9), yields

+ (H*s/4~kT)(d/‘dX)[C~(X)~“(X)-C~(~)~”(~)]~ A(x)=C

0

C,(X).

(23) (24)

NERNST-PLANCK

Integration

EQUATIONS

371

+ lL2@‘(0)2;.

(25)

ANALOG

of (23) yields,

AA(x) + AC,(X) + (Hjj/2kT)CCj(x)’ - Cj(0)21

-I-

(&/8nkT)(@(x)2

- #+(0)2 -,2@‘(x)2

Equations (23), (24) and (25) are valid under conditions of osmotic equilibrium, when fluxes of permeant species vanish. Equation (23) is the generalization of Maxwell’s relation between osmotic pressure gradient and electric stress. When A’(x) and C3 vanish, under conditions of osmotic equilibrium, one has from equation (23) that the electric potential profile is a solution of nonlinear differential equation K, -K2@‘(x)

+ (c/8nkT)(@(x)2

- 12@‘(x)2) = 0.

(26)

It is interesting to note that a solution of (26) is simply,

4(x)=4(0)+ V%)Ce~“- 11-vQx.

(27) (28) (29)

K, = (.s/8nkT)Q2q2.

(30)

P, q and Q are constants characteristic of the system. When one neglects terms, one has the solution for electric potential profile under conditions of osmotic equilibrium as H*

@(x)=Pexp(qx)+Q*exp(-qx),

(31)

where P and Q* are constants. One also has the trivia1 solution that 4(x) is a linear function of position variable as solution of (26) (K, =O), which is constant field assumption result. The forma1 solutions of nonlinear differential (14) and (15) are easy to

372

V. S. VAIDHYANATHAN

write. The concentration are

profiles as given by formal solution of (19)-(22)

C,(x)=C,(O)expC-U,(x)]-Z,(x),

(32)

Cj(X)=Cj(0)eXp[Lrj(x)]-Ij(.u),

(33)

\

I,(x)=Joexp[-U,]

s ,I

;cxp [U,(x)]/D,(x))

dx,

(34)

.\ Zj(X)=JjeXp(Uj)

u I[c~PC -uj)]/oj(x)}dX>

(35)

s U,(x) = (Z,e/kT)[AF(.u) Uj(X)=

-

(HjjACj/kT)+

+ H*AC(x)] +&,x,

(36)

(H*~/471kT)A~“(~)+;l,iX, AF(x)=F(x)-F(0).

(37)

In a membrane system, under stationary state conditions, when the concentration gradients CJ and fluxes Jj of all neutral species vanish, provided that the ion-neutral molecule interaction term, H* is nonzero, it follows from (15) that 4”‘(x) must be a constant equal to - [4~nkTSjj/H*E]. Under these conditions, electric potential profile in membrane phase must be a cubic function of position variable, x. Multiplication by Z,e, each of the 11equations of kind (14) and summing over all ionic species, with the use of Poisson equation, one obtains 4”‘(x) = ~‘(x)[F’(x) + H*C;(x)]

CZ,(R,+Cjsnj)+CCC,Sq,(zq-z,)3 +(4ne/~) 1 i rJ 0 9 K2(x)=(47re2/dcT)C C,(x)Z,$ 0

(38)

The matrix S with elements S,, is singular. Thus, under stationary state conditions, with no chemical reactions, one obtains by summing the (n + 1)

NERNSI-PLANCK

ANALOG

373

EQUATIONS

equations (14) and (15), A’(X)+CS(X)(l+(HjjCj/kT))+CR,+Rj 0 = (&/471kT)~“(~)F’(X)+ [H*E/47tkT](d/dX){ Cj(X)~“(X)). For a system containing Z54(x)=~Z,zC0(x).

ions, i.e., Z, = *Z,

only symmetric

(39) one has

From (;2t(37), the expression for permeability of ionic species 0, P,, defined phenomenologically as [ -J,/AC,(h)], across a membrane of thickness h, can be obtained as h P,=(C,(h)exp(U,(h)J-C,(O)‘,

Ii

AC,

o

1

[lwW,WY&(x)l dx . 1

(40)

The influence of fluxes and concentrations of other species as well as the influence of electric potential on permeability of species (r, is thus formally expressed by (40) or a corresponding expression that one easily obtains from formal solution of (14). The Poisson equation can be expressed with the use of (32)-(37) as 4”(x)= (4ne/e) CZ,I,tX)-_CZ,C,(O) i Q

exp(- V,(X)) .

il

(41)

i

4. High Temperature Approximate Expression for Stationary State Electric Potential Profile. Equation (41) is a differential equation for 4(x) which is nonlinear even without the term ~OZolo(x). The similarity of (41) to familiar Poisson-Boltzmann equation is obvious. An approximate solution of (41) may be sought in the linear approximation. Upon expansion of the exponential and retention of the leading two terms and neglect of terms of order (kT)- 2 and higher order, one has

4”(x)

=

4”(O) + K~[AF + H*ACj] - (4ne/E) C Z,C,(O)&

1 rJ

1

X

+ (4Wc) C W, 0

(x 1,

(42)

374

V. S. VAIDHYANATHAN

The last equation is obtained from (32b(37) in the high temperature approximation and the assumption that H, term can be neglected. Elimination of Cj in (42) results in a differential equation, A@‘(x) = $A+@) - Mx + (4rcep2//;i ) ) C %,I, - k-($H*Zj . d

K$= (47re2/&T) 1

Z,2C,(O).

0

(43) Under stationary state conditions, with no chemical reactions M, p and ICY are constants. In the spirit of Picard’s method, if one temporarily ignores the terms ~oZalO(~) and rj(x), and obtains by twice differentiation, the linear differential equation (44) and its solution, designated as +r, @“I(X)

=/&b”(X),

~1(x)=Pexp(~x)+Qexp(-~x),

(44)

where P and Q are constants to be determined. The first order contribution of fluxes on electric potential profile is thus expressible by the relation,

M=p2@(0)+p(Q--I').

(45)

One obtains, Ar, (.x)=x(M/p2)+ (1/p2){P[e@AF,(x)=[l

l] + Q[evPX- i]},

-~2~2]~L-2{P(e~“-1)+Q(e-~X-l)}

+-$4’(O)+ (Q-f’Y/ll, F,(x)=&

-i’c$q;

AF,(x)=F~(x)-F1(0). (46)

NERNS-FPLANCK

Additional approximation

ANALOG

EQUATIONS

375

that AF, (x)=Krx,

K, = (Wp2)+

(1 -~2~2)(P-

QYPL,

and definition of K,, = (Z,e/kT)K, enables one to obtain approximate

-I,

(47)

expressions for I,(x) and Zj(x) as

~Ax)=CJ,I~~,~IC~-ev(-Kd17 ljtx)= - tJjlD&jIK1-exP WjlX)l, Kj, =~j+ (H*&~/4nkT)(PUj(X)=IjX+

(H*e/4nkT)(P(eRX-

Q),

l)+Q(e-““--

1))

zKjlx.

Substitution of the results of (48) in (43) and twice differentiation, the linear differential equation, @“‘(x)=~2@‘(x)-

(48) yields

1 (Z,Jbc,,lD,)exp(-K,,x) r 0 + (Kj,“~H*Jj/Dj) exp( -KjlX)

1

[4nep2/$J.

(49 1

Comparison of (49) with (44) indicates that the second-order contribution of fluxes to the stationary state electric potential profile occurs as inhomogeneous part of the differential equation. Particular solution of (49) is easy to obtain and the general solution of (49) can be written as @‘(x)=Pexp(px)+Qexp(-yx) +A,Kj, exp(-Kjlx)-CW,r

c

exp(-K,rx),

Aj= (4rcep2/E)[H*Jj,‘[Dj(p2 -K;l)]], A, = (4~e~21K~)Cz,J,ICD,(c12 -K,t,

III.

(50)

376

v.

S. VAIDHYANATHAN

Integration of (50) yields the second potential profile, &, as

improved

expression

for electric

(51)

+C (A,/K,,)(exp(-K,,.~)+K,,.u-lJ, 0

where A$i(x) is the result of (46). A, and A, are constants, whose values may be computed for a specified system. A second improved expression for AF(x) can now be obtained from (51) as

+ (Aj/‘Kj,)(l

-ll~‘K~~)[exp( -Kjlx)-

11,

(52)

where AF, (x) is the result presented in (46). AF,(x) can further be approximated as equal to K2x, with the definition of a constant Ka2, by the relations, K,, = (ZelkT)K2 -A,,

-Aj(l-1*K~~)-CA,(l-~*K~,). 0

(53)

I,(x) and rj(X) can be recomputed and the inhomogeneous part of (49) can be re-evaluated to obtain further correction to AF,(x) and A+(x). If one retains the approximations invoked in obtaining (49) and (53) in these iterations, the electric potential profile after such iterations will be of the same form as (50) with different computed values of constants.

5. Results interaction

when Jj= CJ=O. Provided that ion-neutral molecule term, H*, does not equal to zero, when the concentration

Edid

NERNS-I-PLANCK ANALOG

EQUATIONS

377

gradient and flux of any permeant neutral molecule (solvent, water) does vanish, from (15) it follows that 4”’ should be a constant independent of x, @” = - (472/H*&) 1 J,,iej,

(54)

0

under stationary state conditions, when there is no chemical involving permeant ionic species. Under these conditons one has

reaction

AF(x)=f,x-f,x2-f,x3, fi =4’(O)+ (H/H*)kTSjj, .fz = (2dc)

1 C, (0 )z,, I7

.f3= (2x/3H*E)C Joioj. (I

(55)

A(x) of (39) can thus be expressed as a fourth order polynomial Integration of (39), with the neglect of Hjj term yields,

in x.

AA(x)+ACj(u)+(RfRj)X

+ (.c/8nkT){@(x)2

-@(O)*

-A2@‘(x)2

Equations (55) and (56) yield, A(x)=A,+A,x+A,x2+A3X3+AqX4, 4 =c C,(O)7 d A, = -R-Cj

i.

C J,i,j/kT

1

+ (Ef,/4nkT)~“(O),

A, = [2& + 341& - 18L2&](c/4nkT)

+L*$I”(~)~).

(56)

178

V. S. VAIDHi’Ah:\ =

1 fl,\h

(~/87tk T)4”(0)’

- (f, /2H* k T) C J,i,j, 0

2

A4 = (9&/8nkT)4:

= (n/2kTEH*2)

.

(57)

From the definition of A(x) and 4”(x), one has for a three permeant ion system, (z,-z&(x)=Z,A(x)+

(~/47w)@‘(x)+ (Z,-z&(x),

(Z, -Z,)C,(x)=ZDA(x)+

(~/4ne)@‘+ (Z, -Z&‘,(x).

Substitution equation

(58)

of (58) in (14) for a-th kind of ion yields the differential

tz, -

Z,K, = .q& -Z&7

(Z/l -

z,IN, = s,;,- s,,,

(59)

and similar equations for C,(x) and C(x). From (54), (55) and (57), the solution of (59) can be expressed as a fifth order polynomial in x, for C,(x).

(11=

-CR, + C,(O)k,+ A&, + @‘(O)N,],

cl2= (Z,elkTK’,(O).f, - [@“(O)N,+ a1k,

+

A&1/2,

a,=(Z,e/3kT)C2~11f2+C,(O)f~]-(~1~k,

+A,&)/$

a4 = (.W4WC2~~,_L + 3nIf3] - (a3k, +A,&)/4, ~5 = (Zd5kT)C%f, kg = (Z,&‘kT)

+ 2n3f2] - (aek, + A,K,)/5, + M,.

(60)

NERNSI-PLANCK

ANALOG

Similar results can be obtained for the concentration valid when Ci=O=Jj.

EQUATIONS

379

profiles of /I and 7.

6. Permeubility Coefficient in Many Ion System. Under conditions when (54) and (55) are valid, when there is no chemical reactions, use of (32)(37) with the assumption that diffusion coefficients can be regarded as constants, one obtains the relation between difference in concentration of species D, on either side of the membrane, AC,(h), flux of species (r, J,, and electrical potential as AC,(h)= I,C,(O)+ (J,lD,k,,

)i[l%(h)-

G,(h) =exp[ -(Z,eAF/kT)

+ &Jr],

M,(h)=2k,Jl+

(k,,h-

I]

l)exp(k,,h)l

+3k,,[(k,ZihZ--2k,,h+2)exp(k,,h)-21,

knl = tZ,efl/W-

L

In obtaining (61) and the integrals Z,(h) of (32b(37), the term exp( - ko2x2 - k,,x3] has been approximated as [ 1 - kg2xZ - k,,x3]. The expression for permeability coefiicient of ionic species 0, across a membrane in many permeant ion system can be obtained as P, =[c,(O)G(h)-c,(h)l(D,k,,lCN(h)AC,(h)l), N(h)=

G(h){1 - [MWlk:,l;

- 1.

Certain comments are appropriate at this time. Our basic philosophy has been that the stationary state concentration profiles of permeant ionic and neutral species in a diffusion barrier are determined by the set of coupled nonlinear differential equations (14) and (15). The formal solutions of (14) and (15) are easy to write, bath when chemical reactions occur and when

380

v. S. VAIDHYANATHAN

chemical reactions do not occur in membrane phase. When chemical reactions occur, the fluxes of permeant species participating in any such reactions are functions of position variable x, with the result that the integrals of formal solutions will have fluxes inside the integral sign of Z,(x) and Zj(x). It is recognized generally that permeability coefficient is a phenomenological parameter and that in biological membrane systems, it is dependent on electric potential and influenced by chemical reactions. Given that there exists a specified stationary state concentration difference between the two adjacent solutions separated by the membrane diffusion barrier, of a pet-meant ionic or neutral species, it should be possible to account theoretically for any observed direction and magnitude of flux of this species. Historically, the permeability coefficient came in vogue in experimental biology, due to lack of knowledge of both the diffusion coefficient and thickness of membranes. Assuming Fick’s law and linear concentration profile, one defined that the flux of a specified species, J,, is given by the relations, J, = - D,C; = - (D,/h)AC,

= - P,AC,,

(631

where h is the thickness of membrane, P, is the phenomenoIogica1 permeability (per unit area) coefficient of c1 across the membrane. It became popular to measure permeability of different substances across a specified membrane and measure permeability of same species in different kinds of membranes. Implicit in these experimental endeavours are the notions that permeability is a constant coefficient characteristic of the permeant species and specified membrane, and that it is positive semidefinite. It is evident that only if the flux obeys Fick’s law and concentration profiles are linear functions of position variables and diffusion coefficients are constants, the permeability coefficient will be a constant. For a membrane system, the concentration profiles of permeant nonionic species, under stationary state conditions, will satisfy the set of coupled nonlinear differential (15) with H* term vanishing in the absence of charged species. The nonlinear character of these equations arise due to the presence of H, and Hj,terms. When these terms are negligible, the set of equations become linear, which can be solved. The concentration profiles of uncharged permeant species are then given by

cj(x)=cjCo)-pjlT,x+

i

k=2

k TSjk = Jjijk ;

(PjkYhhk)CeXP (VkX)-ll~ Sjj =

C Skj, kfj

(64)

NERNST-PLANCK

ANALOG

;s 1

EQUATIONS

where qk’s are the (n - 1) nonzero eigenvalues of the matrix S, whose diagonal and off-diagonal elements Sjj and Sj, are defined in (16). In writing down (64), it is assumed that there are n kinds of permeant uncharged species. Since S is singular, one of its eigenvalues is zero, and it is stipulated that ql =O. P, are the elements of a nonsingular matrix P, which diagonalizes the matrix S, by the operation P- ‘SP= n diagonal matrix. T is a constant vector (when there is no chemical reaction) obtained by the product, T=P-‘R, where the components of vector R are the R,‘s of (15). Thus, even in the linear approximation, it follows from (64) that concentration profiles in diffusion barrier of uncharged permeant species are not linear. The constants YLOof (44) can be evaluated from boundary concentrations under stationary state by an algebraic method.

77. Influence of Concentration of One Kind of Zen on Flux and Permeability of Another Kind. When diffusion coefficients in membrane phase may be regarded as constants and when AF(x) can be approximated as fix, one has from (32k(37) that

K, = (-&e/k T )fl

-J,

- L

= fAC,(h)/[exp ( -K,h))

- l] - C,(O))D&.

(65)

Thus, for a three permeant ion system, when Cj =O, J,j =O, one has kTK;. = Z,efl - J$,, +

-[i&&C,(O)

+ is&&Cp(0)l

AC,i,.,D$,/[exp +

( -K,h)

- l]

AC,ji,,,P,jK,,lCexp ( -K,h) - 13.

(66)

The permeability coefficient PT.,of ionic species 7, is given by the relation P,= -(J,/AC,)={AC,(h)+C,(O)[I-exp(-&Jr)1 x [D,Xj./[AC,texp(

-K,h)

- 1 )I]

(67)

Equations (65), (66) and (67) explicitly express the influence of AC, and AC,, on the flux of y.

382

V. S. VAIDHYANATHAN

8. Discussion. It is generally stated that nature, with scant regard for the comfort and convenience of theorists, has devised problems in biology, with significant nonlinear character. The Nernst-Planck equations are too simple to be justified on the basis of physical arguments to be applied to biological membrane systems and too complicated to be solved mathematically, as evidenced by endeavours by many for the last seven decades. The contents of this manuscript was initiated by the philosophy that maybe a more complicated set of equations than contained in the derivation of Nernst-Planck equations may be simpler to solve and may lead to fruitful and useful results. In some regard, the material presented in this paper justifies this expectation. The electric potential profile problem is thus solvable exactly under conditions of osmotic equilibrium as presented in (27t(30) valid when ion-neutral molecule interaction term, H* does not vanish. The results of (23), (27), (32)(37) (38) and (39) are exact, subject only to the validity of assumed basic differential (14) and (15). In Section 4, the results obtained in the spirit of Debye-Huckel theory are presented. The results of Section 5, indicate that a Taylor series expression can be utilized, for electric potential profile and concentration profiles, when flux of neutral solvent (water) molecule is negligible. As presented in Section 5, when flux and concentration gradient of water molecule is zero, A(x), 4(x) and C,(x) are polynomials in x. Knowledge of lower order Taylor expansion coefficients enable one to compute higher order coefficients. It is interesting at this point, to present a method of computation of resting membrane potentials of nerve axon systems, from the properties of surrounding solutions and resistance property of membrane. If one denotes the concentration of ion of kind (T, in axoplasm by C,,(Z) and in bathing solution by CJl), the concentration of water by C,(Z) and C,(II), from the values for various axon systems listed by Hurlbut (1971) one obtains that AA(d)+AC,(d)=O, where

A(d)=CC,(IZ)-CC,(Z).

C,(d)=C,(ZZ)-C,.(Z).

The concentrations of ions listed by Hurlbut are the macroscopic bulk average concentrations which will equal concentrations at locations in solutions far removed from membrane-solution interfaces. At distances far from membrane solution interfaces, electroneutrality condition will be valid and thus, @‘(cl)=O, &‘(O)=O. d is the extent of solution and membrane in which microscopic electroneutrality is not valid. One has from the integral of (39) the relation, R=xR,;

d

R O=

CJ,lD,(m)).

(68)

NERNST-PLANCK

ANALOG

EQUATIONS

383

From the values of fluxes observed, one estimates that R is of the order of about 200moles/cm4 for giant squid axon systems, while C,S,, terms of (38) are of the order lo- 5. From (,38), one has (1 + k-~A2)~“‘(d)= &4’(d) + (4ne/&)c Z,R,, d (1 + tigi’)$“‘(O)

= I&‘(O) + (4ne/&) c Z,R,, m

K; =

(47re2/skT) 1 Z,2C,(O),

tig = (47ce2/&T)

1

ZzC,(ZZ),

d

C,(O)= c,m.

(69)

From considerations of Section 5, when H* does not vanish, 4”’ is a constant, if C~=O=Jj. One cannot satisfy the condition that 4”(d) =0 =4”(O), under these conditions unless 4”‘=0. Thus, Rjf 0, Cj#O, in general for nerve axon membrane system. It should be noted that (68) is a direct result of (14) and (15) and could not be derived from Nernst-Planck equations. If now one assumes that 4(.x) can be expanded in a Taylor series, with the assumption that coefficients of x7 and higher order can be neglected, one has AqQd)=~,d+&d3+~qd4+&d5+&,d6, (b”(d)=d(6$,

+ 12~4d2+20~5d3+30~6d4)=0,

$2=f#+‘(o)/2=o. (70)

If one assumes that F’(d)=F’(O), it may be shown that d6 =0, and A2= - (dZ/12)= 8.333 x lo-l4 cm, if d = 1 x 10m6 cm. From (69) thus, from the knowledge of Debye-Huckel parameters, Z, and R,, d1 can be expressed in terms of +3. In addition, one has 4”‘(d) - 4”‘(O) = 12d[2$, + 5&d],

(1++2)$““(d)= k.;@(d)

(1 +I&~)~#?‘(O) - &f(o).

(71 I

384

V. S. VAIDHYANATHAN

Knowledge of (R +Rj) enables one to express 4’(d) in terms of 4’(O)= 41. Thus, we have sufficient number of equations to evaluate the Taylor expansion coefficients of (70) and one can compute the resting potential from knowledge of boundary concentrations and R,, Rj. Values of membrane potentials calculated in this manner yield values for all axon systems listed by Hurlbut in the right order of magnitude of observed membrane potentials. Discussion of details of calculation of electric potentials from knowledge of boundary solution concentrations and fluxes, as well as comparison with experimental data are not presented in this paper. The case of Loligo forbesii axon system is presented in another paper (Vaidhyanathan, 1978). The values of partial frictional coefficients obtained from experimental literature is of the order 1O-8 erg set cm/mole. The term (47ce/&)x,,Z,R, for most axon systems is of the order 1016esu/cm4.

LITERATURE In Found~rtior~s of Mrrthemutical Agin, D. 1971. “Excitability Phenomena in Membranes.” Biology, Vol. 1. (Ed. R. Rosen). New York: Academic Press. Blaustein, M. P. and J. M. Russell. 1975. “Sodium-Calcium Exchange and CalciumCalcium Exchange in Internally Dialyzed Squid Giant Axons.” J. Memb. Biol., 22, 285 312. Bearman, R. J. 1958. “Statistical Mechanical Theory of Thermal Conductivity in Binary Liquid Solutions.” J. Chem. Phys.. 21, 1278-1286. Chang, D. C. 1977. A Physical Model of Nerve Axons.” Bull. Math. Biol.. 39, l-22. Goldman, D. E. 1943. “Potential, Impedance and Rectification in Membranes.” J. Gen. Phvsiol., 27, 37-60. Hurlbut, W. P. 1971. Membrtrnes und Ion Transport, Vol. 2 (Ed. E. Bittar). New York: Wiley Interscience. Theory of‘ Solutions. p. 26. New Kirkwood. J. G. 1967. “Collected Papers of Kirkwood.” York: Gordon B Breach. Lakshminarayaniah, N. 1969. Trtrnsport Phenomena in Membrunes. New York: Academic Press. Onsager, L. and R. M. Fuoss. 1931. “Irreversible Process in Electrolytes.” J. Phys. Chem., 36, 2689-2778.

Leuchtag, H. R. and Solution for ions of TyrrelL H. J. v. 1961. Vaidhyanathan. V. S.

J. C. Swihart.

1977. “Steady-state

Electrodiffusion,

Scaling.

Exact

one charge, and the phase plane.” Biophys. J., 17, 27. D&Gon

and Heut Flow in Liquids.

1971. “Influence

London:

Butterworth.

01 Chemical ReactIons on Fluxes.” J. T/Ic,o~. flit)/

33. I- 27. Vaidhyanathan. V. S. 1977. “Philosophy and Phenomenology of Ion Transport Chemical Reactions in Membrane Systems.” In Topics in Bioelectrochemistr~. Bioenergetics, Vol. 1 (Ed. G. Milazzo). England: John Wiley & Sons.

Vaidhyanathan, Electrical

;111d

V. S. 1977b. “Permeability

Phenomena

ut Biologiccd

8

and Related Phenomena of Membranes.” In (Ed. R. Roux). New York: Elsevier. State Membrane Potential of Loligo Forbesii Axon

Membranes

Vaidhyanathan, V. S. 1978. “Stationary System.” J. Bioelectrochem. Bioenergetics,

00, OO&OOO.

RECEIVED REVISED

2-23-77 l-23-78

NERNSTPPLANCK

ANALOG

EQUATIONS

385

APPENDIX The values of partial frictional coefficients are needed to compare predictions in future, of theory with experimental results of specified membrane system. In this appendix are presented values of partial frictional coefficients for ions of most interest in biological systems. The limiting values of tracer diffusion coefftcients at infinite dilution are related to mobility by the Nernst equation,

0,” = l&R T/I z, 1F, (A.1) where a,, is the mobility, IZ, 1 is magnitude of valence charge number of ion cr. R is gas constant and F is Faraday. Tis temperature in Kelvin scale. At‘25”C, for univalent ions, 0,” =2.662x 10-71~cmzsec-1, where ,I: is the limiting equivalent ion conductivity. These values are available in literature. Assuming that, at infinite dilution, most of the contribution to frictional coefficient [,, arises due to resistance proferred by water, the ion-water interaction partial frictional coefficients can be computed using the relation, cz =(kT/D,O) One = C,L,,, where k is the Boltzmann constant and C,. equals 55.555 x 10e3 molescm-3. obtains, at 25-C values of 7.19685, 5.55417, 3.78746, 3.64747, 4.68273 and 5.24802 ( x lo-’ erg cm set mole- ’ ) respectively for Li, Na. K. Cl, Ca, and Mg ions. Values for ion-ion partial frictional coefficients can be evaluated from experimental data of diffusion coefficients summarised in the book of Tyrrell (1961). In solutions of concentrations lower than 0.25 moles/liter, the NaCl or KCI systems, Tyrell’s figures indicate in this uncertainties. An extreme point of view is that D,, remains essentially constant concentration range, which enables one to approximate that iNaN== -INsCI An approximate relation between partial frictional coefficients of different species, that

i,, = c&CD;+ D;l/(D,o + @?),

(A.2)

where 0,” is the self-diffusion coefficient at infinite dilution of species r~, of molecular theory within 20% by (A.2). The can be utilized. The values of