20 Influence of Fluxes on Stationary-State
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Electric Potential and Concentration Profiles V. S. V A I D H Y A N A T H A N Department of Biophysical Sciences, 114 B, Cary Hall, School of Medicine, State University of New York at Buffalo, Main Campus, Buffalo, N Y 14214 The influence of fluxes on concentration and electrical potential profile, as expressed by a set of coupled nonlinear differential equations, is analyzed. These equations are derived using a correction to the limiting expressions for chemical potentials and the molecular expressions for frictional force. From general theory and numerical considerations a simplified set of expressions are obtained enabling one to compute stationary-state membrane potential from fluxes and properties of surrounding electrolyte solutions. Application of the theory to various giant axon systems yield reasonable values of resting potentials.
Tsothermal transport of solutions containing three or more permeant ions across a diffusion barrier, such as a biological membrane, is of significant biophysical interest ( I ) . The central problem which must be solved is to compute the value of the electric potential difference between the two solutions adjacent to the membrane phase i n 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 ( 2 ) . These equations are nonlinear and solutions of these equations i n closed analytic form have not been obtained yet, i n spite of attempts by many over a period of seven decades. In certain specialized circumstances, solutions associated with the names of Planck, Henderson, and Schlogl are available ( 3 ) . I n the opinion of the author, the assumptions involved in obtaining these solutions are not applicable for real membrane systems of interest i n biology. (One of the referees has expressed the opinion that transport 0-8412-0473-X/80/33-188-313$06.00/l © 1980 American Chemical Society
Blank; Bioelectrochemistry: Ions, Surfaces, Membranes Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
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BIOELECTROCHEMISTRY:
IONS, SURFACES, M E M B R A N E S
in biological membranes is accomplished by discrete macromolecular complexes of one kind or another, and therefore any continuous treatment is unrelated to biology and should not be done. In spite of this remark, the continuous nature of present treatment has some merits in the opinion of the author.) A widely accepted equation i n membrane physiology is the Goldman equation (4), which involves the assumption that electric potential profile is linear. This assumption enables one to evaluate the formal integral of Equation 7 of this chapter. However, this assumption requires validity of microscopic electroneutrality and is not aesthetically attractive when one must reconcile with the Poisson Equation. In spite of these, the discussion of observed phenomena i n biological literature usually is limited to analysis based on equilibrium Nernst potential and sometimes involving activity coefficient correction. (The literature i n this field is vast (see e.g. Ref. 6 ) , and specific references are avoided consciously in this chapter.) In spite of a considerable amount of available experimental information on electrical potential difference, boundary concentrations of ions, and the influence of divalent ions on fluxes of monovalent (5) ions and permeabilities, our understanding of the theoretical basis for experimentally observed variation of fluxes and electrical potential as a function of the properties of a membrane barrier and surrounding solutions is far from satisfactory ( 6 , 7 ) . Attempts to explain the uneven distribution of monovalent ions in solutions of either side of a biological membrane on the basis of the Donnan Effect also is present in the literature. The well-studied excitation phenomena of nerve axon systems (8) clearly indicate that the state of affairs present i n biological systems are definitely not i n an equilibrium state. Under nonequilibrium conditions, fluxes that are present alter the equilibrium concentration profiles and the electric potential profile through the Poisson Equation. Thus, any theory that attempts to explain the influence of fluxes must explicitly exhibit the flux dependence of electrical and concentration profile gradients (9). I n this chapter, the influence of fluxes on concentration profiles of both ionic and nonionic species are assumed to be described by the basic set of coupled nonlinear differential equations (see Equations 6a-6e.) These equations are sufficiently general enough to include the influence of any chemical reactions occurring in the diffusion barrier, though analysis of this chapter is restricted to stationary states i n the absence of chemical reactions. The justification for validity of assumed basic differential equations is presented later i n this chapter. This chapter focuses on the inclusion of ion-neutral molecule interaction contribution to chemical potentials of permeant ionic species, thus reducing the problem to be solved by the Taylor Series. Certain exact results such as Equations 8, 9, and 10 can be obtained. Equation 10 for example is the generalization of Maxwell's well-known osmotic balance equation. Blank; Bioelectrochemistry: Ions, Surfaces, Membranes Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
20.
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Electric Potential and Concentration Profiles
315
Certain results of mathematical interest based on these equations have been published elsewhere (10). These technical details are not very interesting to the experimenter. W i t h this i n view, a simplified theory is presented i n the section dealing w i t h theory which enables one to compute membrane potential from D e b y e - H u c k e l parameters of the surrounding solution and the observed stationary-state fluxes of permeant ions and neutral molecules. As shown i n the section on theory, Equation 17, which relates membrane potential difference to gradients of electric potential profile at specified locations, is derived. Methods of evaluating the various quantities are presented. Application of the theory to various axon systems leading to the calculation of resting potentials is presented i n the "Sample Calculation' section. Though admittedly they are approximate and somewhat ambitious, calculated values are i n reasonable agreement with observed values. Provided that one has values of stationary-state fluxes of individual permeant ionic and neutral species—independently measured—properties of solutions and the value of the electric potential which also is measured independently, it is possible to apply the theory to a clean physical system and to verify the extent to which theory agrees with experimentation. Such flux data are scant, and the data available for axon systems do not satisfy the zero net electric current condition. In spite of these deficiencies and the approximate validity of simplified equations, the reasonable agreement between theory and experiments for various axon systems is evidently not the result of coincidence. Certain symmetry arguments and roots of polynomial considerations enable one to compute charge-density profiles. One computes membrane thickness to be ~ 57 A and the thickness of diffuse double-layer regions to be ~ 21 A , if the region lacking microscopic electroneutrality is of the order of 100 A . A measure of the activity coefficient correction factor, denoted by a constant parameter, A , (assumed to be constant i n this chapter), plays an important role i n determining the stationary-state membrane potential, as determined by the ionic flux term, ZR. W h e n A equals zero, the activity coefficient of all solute ions equals unity. The ion-neutral molecule interaction energy contribution to the chemical potential of permeant ions is denoted by H * . A n approximate expression for A as equalling ~~bi + V2 ]/(vi ?72 ), where rji and rj are the D e b y e - H u c k e l parameters of Solutions 1 and 2 surrounding the membrane phase, is derived. 2
2
2
2
2
2
2
2
2
2
This chapter shows that the stationary-state membrane potential evolves i n a natural self-consistent manner, determined by bulk solution properties of electrolytes and by the fluxes as determined by membrane properties. These also determine the extent of the region lacking microscopic electroneutrality. It is worthwhile recalling that the classic D e b y e H u c k e l approach assumes the existance of regions lacking electroneuBlank; Bioelectrochemistry: Ions, Surfaces, Membranes Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
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IONS, S U R F A C E S ,
MEMBRANES
trality of a spherical shape having a radius of I / 7 . This chapter is for physical chemists who are interested i n various aspects of membranetransport problems. The results presented i n this chapter indicate that nature is simple because of its complexity. Experimentally it is known that the permeability coefficient of a specified ion and molecule across a specific membrane is affected by electrical potential difference and chemical reactions. Experimental observations that the flux of one kind of species is influenced by forces on another kind of molecules is understood by the formal expression (see Equation 8) of this chapter. In D e b y e - H i i c k e l theory, the limiting expression for the chemical potential of ionic species a is given by the relation (11) ra — M*° (T, P) + kT In C + Z e a
(eVc)
a
(1)
where C is the concentration of ion a i n solution, c is the dielectric coefficient of solution, Z is the signed valence charge number of ion a, e is the electronic charge, k is the Boltzmann constant, and T is absolute temperature. The expression fia° is the composition-independent part of the chemical potential and 17 is the D e b y e - H i i c k e l ionic atmosphere parameter. The approximations involved regarding the potential of mean force in the derivation of Equation 1 are well known. a
a
Derivation
of Basic
Equations
The experimental verification of Equation 1 for extremely dilute electrolyte solutions clearly indicates the need for the last term in Equation 1, while the usual derivation of Nernst-Planck Equations contain only the first three terms. (However, the last term is incorporated when one expresses the Nernst-Planck Equation with activity). Assume that the expressions for chemical potentials of ionic species cr and nonionic species 7 in an inhomogeneous diffusion medium may be written as (12) — Ma*(r,P) +
Z e(x) +
kTlnC'(x) -\ "'{x) 2
A = 2
"(X) —
(ffc/4ir) -
(Wc)
E
ZaCaix)
(3)
In Equation 3, as well as in the following equations, the first derivative with respect to position variable x is denoted by a single prime. Higherorder derivatives are denoted by the appropriate number of primes. The approximation that the positional dependence of the dielectric coefficient, c, may be neglected has been incorporated. This is a serious assumption whose validity to biological membrane systems is questionable. A n empirical expression stating that c(x) = c(o) — e dx + C2X , where c(o) is the dielectric constant of the bulk solution, d is the distance between the two solutions where inhomogeneity exists, and c is the second Taylor coefficient of c(x), can be utilized. In this chapter, for simplicity, c is regarded as x-independent. 2
2
2
The frictional coefficient of ionic species a and &r and of the nonionic species ; and £/ in the diffusion barrier can be written as t„(x) — [kT/D„(x)]
=
r + C* U a
+
Z
C„(x)U
+ E C,(x)C, i
Blank; Bioelectrochemistry: Ions, Surfaces, Membranes Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
(4a)
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BIOELECTROCHEMISTRY: IONS, SURFACES, M E M B R A N E S
t,(x)
—
[kT/D (x)] j
= r, + Cfa + +
Z
HC Ui k
(4b)
C £(x) =
()
0 for v a l u e s of 0 < x
^
J
323
.
;
\
A
0
\
\
i /
x
Figure 1. Charge-density profile computed from membrane potential (60 mV = 1.98824 X 10~ esu/cm) and the value of d = 10~ cm for a membrane system using Taylor expansion coefficients. The y axis is in 10' esu/cm . Plot A is schematic; Plot B is obtained when X = 0; and Plot C is obtained using X = -0.193 X 10 cm . 4
8
6
3
2
2
14
2
The terms rj and are the values of D e b y e - H u c k e l parameters of the axoplasm and bathing solutions of axon systems, respectively. Computed values of these and the values of A ( o ) and A(d) from the concentrations of ions reported by Hurlbut (16) for various axon systems are presented in Table I. The assumption of the validity of Equation 13 with the additional assumption that terms of the order of and higher can be neglected, lead to an extremely simple expression for the stationary-state membrane potential difference (d) 2
0
T J
2
2
7
(d) — (d/2)8*' + (d /120)8tf>"'
(17a)
3
8 * ' - * ' ( d ) +*'(o)
(17b)
« * ' " — *'"( + 5 ^ d + 9 ^ed ] 2
-
2
4
2
5
12 k cP 2
(20)
6
If in a membrane system under the stationary state, the condition that A F ' ( d ) = 0 is satisfied and, in addition, "'(d)] +
ZR/[ "'(d)] 2
V2
(21b) (21c)
Calculation
From analysis to be presented in the section dealing w i t h a chargedensity profile, one could assume that under the stationary state, A' +
V
"' (o) — ArfZR
(23)
If i n the membrane system, the conditions where £ — 0 and A = 0, fa =
2
(Zfl/lOd ),
c/> =-(Z#/4d)
2
6
^ _
4
(ZB/6)
3
(24)
Since A F ' ( d ) equals zero when A"{d) = - 1 2 fad and d = 2x . If fa and higher-order terms of Equation 13 are negligible, the inflection point * i and the extremum points x of the function "(x) are given by 2
2
2
*i= x
2
2
(fa/5fa)
+ (2fa/5fa) x + (fa/10 fa) = 0 2
x = x {l ± (3)" } 2
1/2
1
since 5fad = — 2 fa fa = — 2 fad/3 when x = d / 2
(31)
x
Thus, when the inflection point of "(x) occurs at midpoint, x equals 0.2113 and 0.7887 d. If extremum i n charge density occurs at membranesolution interfaces, one has the membrane thickness h, as the distance between two extreme values of "(x) = 0.5774 d . W h e n d equals 1 X 10" cm on this premise, the membrane should have a thickness of about 57 X 10~ c m and the diffuse double layers should have a thickness of 21 X I O cm on both sides. 2
6
8
-8
W h e n the fa, fa, fa, fa, and fa terms of Equation 13 are not negligible, while fa and higher-order terms are negligible, and "(*) has one unique inflection point at x = x one has u
xi + (fa/3fa) x + (fa/15fa) — 0 2
xi — -
x
(fa/6fa);
5fa — 12*4*6 2
(32)
Generally, when there are two real inflection points of "(*) at x = Xx the relation is
Blank; Bioelectrochemistry: Ions, Surfaces, Membranes Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
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BIOELECTROCHEMISTRY: IONS, SURFACES, M E M B R A N E S
5'"F" + "F"' 1
{3AV"(o) -*'(o)}*""(o) +
(4irfcrA)A"'(o) — 0
Ay"(o)*""(o) — -(4irfcrA)A'"(o)
(42)
Such equations together with the constraint t h a t ^ t ( t — l ) A i d " ( i
can
2 )
1 =2
be used to evaluate the Taylor coefficients of A ( J C ) . F o r example, when ZR = 0 and A — 0, and if A A ( x ) = A x + A x? + A 3 X , then A — —R; A — - [ A A ^ V W and A d equals (2/3)AA /(S ). 2
2
x
3
2
3
2 2 VO V2
1
2
V
Discussion
In biological literature, the resting potentials of various axon systems often are computed w i t h the Nernst equilibrium expression, which
Blank; Bioelectrochemistry: Ions, Surfaces, Membranes Advances in Chemistry; American Chemical Society: Washington, DC, 1980.
20.
VAiDHYANATHAN
Electric Potential and Concentration Profiles
333
invariably does not tally with the observed potentials (16). Abundant support can be given for the statement that the state of affairs of the axon systems at resting state is a stationary-state condition with nonvanishing individual fluxes and is not an equilibrium condition. Historically, Teorell ( I ) defined these dynamic effects as a diffusion effect on the ionic distributions. Our objective has been to crystalize the qualitative ideas expressed by Teorell. Many schools of thought have recognized these problems, and have attempted to solve the problem of the influence of fluxes on stationary-state membrane potentials by using the familiar Nernst-Planck Equations. The only satisfactory solution of the Nernst-Planck Equation is the result of the work by Planck and is restricted to a two-ion symmetrical electrolyte (2). In spite of many attempts, a solution of the Nernst-Planck Equations i n closed analytic form valid for many ion systems is not available. In the opinion of the author, the Nernst-Planck Equations are too simple to be justified on the basis of physical arguments and too complicated to be solved mathematically. The approach that this chapter took was initiated with the philosophy that a more complicated set of equations than the one presented i n this chapter may be simpler to solve and may be applied realistically to biological membrane systems. As may be noted from Equation 17, which is a simple though evidently approximate result, one concludes that nature is simple because of its complexity. The results of Equation 10 may be recognized as a generalization of Maxwells osmotic balance relation with strain owing to the electric field (15). Subject only to the validity of assumed Equation Sets 2, 4, and 6, the results of Equation Sets 8, 9, and 11 are exact. The results of Table II indicate the applicability of the approach of this chapter to biological membrane transport problems. The conclusion that A ' ( o ) = A ' ( d ) = — R, obtained from Equation 37 (which is exact), can be reconciled with the basic Equation 6a, neglecting the H * term i n a primitive manner by stating that (Zae/kT) Ca (x)F' (x) =
C
a
- £
(43)
A three-permeant-ion system, together with the zero net electric current condition, ^ Z
