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Electrostatic coupling across a membrane with titratable surface groups
J. theor. Biol. (1975) 50, 317-325
Electrmtic
Coupling Across a Membrane with Titratable Sarface Group@
A. P. NELSON~, P. COLONOMO@
AND
D. A. MCQUARRIB
Department of Chemistry, Indkana University, Bloomington, Indiana 47401, U.S.A. (Received 14 January 1974, and in revisedform
1 May 1974)
The potential drop across a membrane is calculated for the case of ionizable groups on both membrane surfaces. The presence of both acid and amine grotips on the membrane surfaces is considered. The membrane surface potential is obtained from the non-linear Poisson-Roltzmann equation by
treating the fraction of dissociated ionizable surface groups as a selfconsistent functional of the electrostatic potential. A discussion of the error due to ignoring the electrostatic coupling of the potential across the membrane is presented. The error turns out to be quite small for most membrane problems of biological interest. Finally, the conductance data of Mozhayeva & Naumov (1970) for the frog #nodeare reanalyzed within the context of the diffuse double layer theory. It is shown to be m
to invoke a specific binding of divalent cations to the membrane.
Most of the excitation phenomena of excitable cells are readily explained in terms of the theory of Hodgkin & Huxley (1952). This model describes the action potential in terms of the voltage-dependent membrane permeabilities of the several ions in the system. The essentials of this model have been confirmed for a wide variety of excitable cells. One of the major problems yet to be solved is the determination of the molecular mechanism responsible for the ionic permeability changes described by the Hodgkin-Huxley model (Ehrenstein & Lecar, 1972). One approach has been to change the concentration of various ions inside and outside an axon and observe the effect on the nerve conduction process. Many of these experimental data can be underWood in terms of the diffuse double layer theory of a membrane with a uniform surface charge density. Chandler, Hodgkin $ Meves (1965) treated the surface charge as a constant and were able to explain the ionic strength dependence of the sodium t Contribution No. 2569 from the Chadstry Depmnent of Indiana university. ; ~.D.E.~v’re&wtoral Fellow. ~de~eaascka(iBcruRaQctoralFeEow. 317 T.k
21
318
A.
P.
NELSON
ET
AL.
threshold and inactivation curves of the giant axon. Gilbert & Ehrenstein (1969, 1970) allowed the surface charge to be neutralized by binding with ions from solution. They explained the effect of the external divalent cation concentration on the potassium conductance and sodium conductance, and the effect of external pH on the sodium conductance of a frog node (Hille, 1968). Mozhayeva & Naumov (1970) have explained the effect of pH and divalent ions on the potassium conductance of a frog node by assuming the presence on the membrane surface of three different groups with different binding capabilities, The calculations presented in the literature for the membrane with a uniform charge density on its surfaces have been incomplete in several respects. A common assumption is that the electrostatic potentials at the two surfaces are independent. In other words, the potential in the membrane does not couple the two surface potentials to any significant extent. A discussion of the errors inherent in this assumption has not been presented. Another common assumption is that divalent cations are bound to the membrane surface. Several conductance studies on thin lipid membranes, however, have shown that the divalent cations can be treated completely within the context of diffuse double layer theory (McLaughlin, Szabo & Eisenman, 1971; Muller & Finkelstein, 1972). In this paper we present calculations which are complete within the limitations of the model. Our model is that used by Chandler et af. (1965), a dielectric layer of infinite lateral extent with a uniform charge density on each surface. The membrane is bounded by two ionic solutions and the electrostatic potential in each phase is assumed to be governed by the nonlinear Poisson-Boltzmann equation. In using an equilibrium theory like the Poisson-Boltzmann equation, we are assuming that the rate at which ions diffuse across the membrane is slow so that the equilibrium structure of the double layer is not disturbed to any sign&ant degree. We ignore effects due to the presence of dipoles in the membrane. The possibility that the surface charge density is not uniform but may be clumped around membrane pores is not considered. Our approach to the problem follows that of Ninham & Parsegian (1971). As they have pointed out, the membrane surface charge density is not a reIevant biological parameter. Instead, the surface density of ionizable groups and their dissociation characteristics should be specified. The surface charge density and potential then follow automatically and depend on the Zocul environment of the ionizable groups. Thus the potential drop across the membrane is calculated by treating the fraction of dissociated ionizable surface groups as a self-consistent functional of the electrostatic potential. The dissociation constant for the ionizable groups is taken to be a constant
ELECTROSTATIC
COUPLlNO
ACROSS
A MEMBRANE
319
and does not depend ‘on the electric field at the surface (thus we are ignoring the Wien effect recently discussed by Bass & Moore, 1968). Our membrane model is solved for two cases. In the first, all groups on the membrane surface are acid groups of a single type. The second case is a generalization of the first, and we derive equations for the presence of two different acid groups and one amine group on the surface. The error caused by ignoring the coupling of the surface potentials is discussed for the first case. We then apply the equations for the second case to the data of Mozhayeva & Naumov (1970) for the steady-state potassium conductance of a frog node. 2 Theory A useful membrane model and that dielectric layer of infinite lateral extent The three phases are denoted by I, II, &OSe,, co, respectively. The thickness Phase
I
Phase
x-0
1. Diekctric layer with charge potential is Y and the potential
II
Phase
(membmne)
(inside)
Fro. resting
used by Chandler et al. (1965) is a bounded by aqueous phases (Fig. 1). III, and their dielectric constants by of the dielectric layer is d, and the
(outside)
x=d
.
densities
III
u and d on its surfaces. membrane is !&,.
drop acrossthe
The membrane
320
A.
P.
NELSON
ET
AL.
surfaces have surface charge densities Q and 6’. The electrostatic potential of each of the three phases, 3-Z”,,YZ and Y, can be calculated by solving the non-linear Poisson-Boltzmann equation (see Hill, 1960, for a discussion of the statistical mechanical limitations of this equation). d2Y1 -~ 47vlW XSO (1) s= Eo d2% -= o Osxld dx2 d2Y3 4XPdX) m=-x>d dx2 so where p(x) is the charge density and is specified later. The system of equations must satisfy the following boundary conditions lim dm = 0 x+m dx lim dy’,(x) lim Yi(x) = I/ (5) x+-m dx . x-+--m The potentials at the surfaces are continuous and the normal components of the electric displacements are connected by the boundary conditions lim Y3(x) = 0 x+00
(6)
For simplicity we consider a solution containing only monovalent ions and divalent cations. The ionic concentrations in the reservoir are denoted by n-(r)
= n
n+(r)
= 41-v)
n++(r)
= ni
(8)
where 0 5 q s 1. The ion concentrations are governed by a Boltzmann distribution, so that pl(x) = ne[-~~S’Yl-~)+(1-~)e-ea(Yi-Y)+r,-2eacY1-v,] (9) p3(x) = n’e[ - ee@P3 + (I- q’)e-‘fl’3 + q’e-2e~ya]
(10)
where e is the unit charge, /3 = l/kT, k is Boltzmann’s constant, T is the absolute temperature, and primed quantities denote phase III. The solution to the equations for the case when the surface charge density 0 is constant, the divalent ion concentration is zero, and the outer surface potential Y”,(d) is zero has been given by Chandler et al. (1965). The surface
ELECTROSTATIC
potcotidisgiven
COUPLING
ACROSS
A
MEMBRANE
321
by
ym 2b+zeox
sinh eY?,(U)flkT eYl(0)/2kT
-
1
= @’
(11)
where K is the usual Debye screeninglength de&d by 8nne’ x2 =Qz* One of the ditliculties with using equation (11) is that the surface charge density is due to ionizable groups and changes as the environment of the membraneis changed. For example,if the surface charge is from acid groups, a change in the pH of the solution will change the surface charge density. This problem has been solved with the assumption of a small electric field in the membrane (Gilbert & Ehrenstein, 1969, 1970), but a complete solution has not beenpresented. The assumption of a small electric field in the membrane leads to the result that the two membrane surface potentials are completely independent of each other. In the general case,the two surface potentials are not independent but are connected by the boundary conditions, equations (6) and (7). The general solution to the problem for the case of acid groups on the membrane surface is presented in the next section. (A) ACID GROUPS OF ONE TYPE ON THE SURPACE
Consider a model in which the surface charge is due entirely to ionizable acid groups and all the acid groups are identical. If S and S’ are the surface areas per acid group, and a, ~1’are the fraction of ionized surface groups, then or -- ea Q’=: --. ea’ S’ As The fraction of ionixable groups depends on the local environment of the groups and thus on the surface potential. The surface potential depends on the surface charge. Thus a and a’ must be determined in a self consistent manner. The surface groups have a known dissociation constant for the reaction AH # H+ +A- given by L (14) Ol-a where [H+ 1,is the hydrogen ion concentration at the surface.In keeping with the Poisson-Boltxmann equation, [H+ 1, is related to the hydrogen ion concentration in the reservoir, H, by the following equation [H+]* = ~e-ti(~~(0)-Y)s (1%
322
A.
P.
NELSON
ET
AL.
Equations completely analogous to equations (14) and (15) can be written for the outer membrane surface. Equations for !P1(0) and !Ps(d) are obtained by combining equations (1) and (3) with (9) and (10). It is now convenient to define several new variables (pl(x) = eem-u &(x) = eeSv3. (16) After performing
several straightforward
integrations,
we obtain
(449-O ( 41@I+z)t--@-$){ln$$-e/W} ‘I + 4ne2j3K &oKS (4,(d)-1)
(
C#J(d)+ ‘1 *+E143(4 3
2)
Tjz-{ln($$)
dm) H+Kc$,(O)
= O (17)
-eflq
4xe2/K’ -t-----
-___4<(d)
H’-tK’&(d)
EOK’S
= 0. (18)
Equations (17) and (18) are two coupled algebraic equations for the surface potentials. They can be solved by iterative techniques with the aid of a computer. Then the potential drop across the membrane, Y, can be calculated : Y, = yY,(d) - yY,(W (19) Also, the tetms coupling the two surface potentials are proportional to EJE,, which is usually small. Thus these terms can often be ignored, in which case equations (17) and (18) reduce to the equation used by Gilbert & Ehrenstein (1970). The results of the complete set of equations will be presented in the next section. (B)
DIFFERENT
IONIZABLE
GROUPS
ON SURFACE
The theory presented above can be generalized to include the presence of several different ionizable groups on the membrane surface. For example, consider the presence of two different acid groups and one amine group on the surface. The two dissociation constants for the acid reaction AH # A- + H+ are given by K1 = [H+1,fi
K2 = [H+],*. -
-
1
For the base reaction, BH # B+H+, K3 = CH+]o[B]
[BH+I
2
we have = [H’]
lz
’ a3
(21)
ELECTROSTATIC
COUP.LING
ACROS,S
323
A MEMBRANE
and so the surface charge is given -by a1 a*’ a3 ----+(22) Sl s2 s3 where Sr, S, and S3 are the surface areas per ionizable group. Equations (20) through (22) are then combined with equations (I), (6) and (9), and if the coupling of the surface potentials across the membrane is ignored [e.g. drop the term proportional to e, in equation (6)], then one obtains CI
4~e2BK34iW + &$3K(H
+K34i(O))
-
(23)
Since the coupling of the potentials across the membrane is being ignored, an equation identical with equation (23) is obtained for the outer surface potential, Y3(d). and Discussion The term coupling the two surface potentials in equations (17) and (18) is proportional to eJe,,. If this term is ignored, the surface potentials are independent and equations (17) and (18) reduce to the one used by Gilbert & Ehrenstein (1970). Due to the highly nonlinear nature of the equations, the actual error introduced by ignoring the coupling term can be determined only by solving the equations for a specified set of parameters. We have solved the equations with the membrane thickness,d, set equal to 100 A, the dielectric constant of the solution equal to 78.5, and a wide range of ionic concentrations and surface charge densities.Several generalizrttions can be made. The error due to ignoring the coupling term depends mainly on the potential drop acrossthe membrane, !?“=,and on the dielectric constant of the membrane. A change in the concentration of the ions in solution seemsto a&t the error in an indirect mannex by changing the size of the potential drop acrossthe membrane. The maximum percentageerrors for Y,,,that one finds for various values of thesetwo parameters are listed in Table 1. One notes that the errors are quite small and usually can be ignored in most biological membrane problems. Moxhayeva 8c Naumov (1970) have studied the effect of changesin the external concentrations of H+, Ca*+, Mg*+, Ni*+ and external ionic strength on the steady-state potassium conductance of a frog node. They found that the curve relating membrane conductanceto membrane potential 3. Caladations
324
A. P. NELSON
ET AL.
TABLB 1 Approximate errors in Y,,, Y&nV)
.!?I= 2
e1 = 10
El = 20
0.1%
0.2% 1.0
0.8% 2.0
40 :
8:; ;:;
1.8 25
;:;
100
1.0
iti
shifts along the voltage axis with changes in the concentration of the various ions. A parameter convenient to define is V,, the membrane potential at which the membrane conductance is half the sum of the minimum and maximum conductance. They were able to correlate shifts in V, with a calculated membrane surface potential by assuming that there are groups of three types on the membrane surface. Type I groups are negatively charged and capable
- 50
‘/ I 4
-40 t
I, 5
I 6
I
I 7
I
I 6
I
I 9
I
I IO
II II
PH
FIG. 2. Change in V+ and !?‘~a@)with change in pH and calcium concentration. (a) 2 nw Cap+, (b) 10 rmt Cap+, (c) 20 rmn Caa+ . The extcanal KC1 concentration is 100 mu. The points are the experimental value from Mozhayeva & Naumov (1970). Figmw to the left of the V, axis are for the set of experiments from pH 4-7. Those to the right of the V, axis are for the set of experifients f’rom pH 7-103.
ELECTROSTATIC
COUPLING
ACROSS
325
A MEMBRANE
of binding H+ and Ca’+. T ype II groups are neutral and bind H+. Type III groups are negative and are not capable of binding ions. But we feel it is not necessary to introduce an equilibrium constant for the binding of Ca’+ to the membrane. Due to their large charge, divalent cations are concentrated into the double layer much more readily than monovalent cations (see Fig. 2 of Ninham & Parsegian, 1971, for a numerical example) and are thus more effective in “screening” the surface charge than monovalent ions. Such a treatment of divalent cations is the same as that used by McLaughlin, Sxabo & Eisenman (1971) and Muller & Finkelstein (1972) in conductance calculations on thin lipid membranes, The data of Mozhayeva 8c Naumov (1970) can be analyzed by a model of two acid groups and one amine group on the membrane surface. The surface potential can then be calculated from equation (23) with the parameters chosen to give the best fit to the data. The results are shown in Fig. 2. The two acid groups have parameters of pK, = 3.7, pK, = 2.15, S1 = 300 A2, and S, = 100 A2. The parameters for the amine group are pK, = 9-O and s, = 143 AZ. At pH 7 and 2m~ Ca these parameters give a net surface charge of -6.3 x 1013 electronic charges cmB2 compared with a value of - l-7 x 1013 electronic charges cmW2 calculated by Mozhayeva & Naumov (1970). Due to the brevity of their paper, the origin of this discrepancy is not apparent. The authors would like to thank Walter J. Moore for introducing them to this problem and Ludvik Bass and Alfred Strickholm for helpful encouragement and discussion.
REFERENCES Bm, L. & MOORE, W. J. (1968). In Structural Chembtry and Molecular Biology (A. Rich C N. Davidson, cds) p. 356. San Francisco: W. H. Freeman. CHANDLER, W. K., HOJXXIN, A. L. & MEWS, H. (1965). J. Physiol., Land. 180, 821. %REBSEIN, G. & LECAR,H. (1972). A. Rev, Biophys. Bioettg. 1, 347. GILBERT, D. L. & EARBNSTEIN, G. (1969). Biophys. J. 9,447. GILBERT, D. L. & JImimmm, G. (1970). J. gen. Physiol. 55,822. HILL, T. L. (1960). Introduction to Statistical l7aermodjwamics. Reading, Mass.: AddisonHUE, B. (1968). J. gm. Physkd 51,221. HO~KIN, A. L. & Hm. A. F. (1952). J. Physiol., Land. 117,500. MCL.AUGL.RIN, G. G. A., SZABO, G. & EISENMAN, G. (1971). J. gen. Physiol. MOZHA~A, G. N. & NAUMOV, A. P. (1970). Nature, Land. 228,164. MULLER, R. V. & FINEEISIEIN, A. (1972). J. gen. Physkd. 60,285. NINHAM, B. W. & P-IAN, V. A. (1971). J. theor. Biol. 31,405.
58,667.
Electrmtic
Coupling Across a Membrane with Titratable Sarface Group@
A. P. NELSON~, P. COLONOMO@
AND
D. A. MCQUARRIB
Department of Chemistry, Indkana University, Bloomington, Indiana 47401, U.S.A. (Received 14 January 1974, and in revisedform
1 May 1974)
The potential drop across a membrane is calculated for the case of ionizable groups on both membrane surfaces. The presence of both acid and amine grotips on the membrane surfaces is considered. The membrane surface potential is obtained from the non-linear Poisson-Roltzmann equation by
treating the fraction of dissociated ionizable surface groups as a selfconsistent functional of the electrostatic potential. A discussion of the error due to ignoring the electrostatic coupling of the potential across the membrane is presented. The error turns out to be quite small for most membrane problems of biological interest. Finally, the conductance data of Mozhayeva & Naumov (1970) for the frog #nodeare reanalyzed within the context of the diffuse double layer theory. It is shown to be m
to invoke a specific binding of divalent cations to the membrane.
Most of the excitation phenomena of excitable cells are readily explained in terms of the theory of Hodgkin & Huxley (1952). This model describes the action potential in terms of the voltage-dependent membrane permeabilities of the several ions in the system. The essentials of this model have been confirmed for a wide variety of excitable cells. One of the major problems yet to be solved is the determination of the molecular mechanism responsible for the ionic permeability changes described by the Hodgkin-Huxley model (Ehrenstein & Lecar, 1972). One approach has been to change the concentration of various ions inside and outside an axon and observe the effect on the nerve conduction process. Many of these experimental data can be underWood in terms of the diffuse double layer theory of a membrane with a uniform surface charge density. Chandler, Hodgkin $ Meves (1965) treated the surface charge as a constant and were able to explain the ionic strength dependence of the sodium t Contribution No. 2569 from the Chadstry Depmnent of Indiana university. ; ~.D.E.~v’re&wtoral Fellow. ~de~eaascka(iBcruRaQctoralFeEow. 317 T.k
21
318
A.
P.
NELSON
ET
AL.
threshold and inactivation curves of the giant axon. Gilbert & Ehrenstein (1969, 1970) allowed the surface charge to be neutralized by binding with ions from solution. They explained the effect of the external divalent cation concentration on the potassium conductance and sodium conductance, and the effect of external pH on the sodium conductance of a frog node (Hille, 1968). Mozhayeva & Naumov (1970) have explained the effect of pH and divalent ions on the potassium conductance of a frog node by assuming the presence on the membrane surface of three different groups with different binding capabilities, The calculations presented in the literature for the membrane with a uniform charge density on its surfaces have been incomplete in several respects. A common assumption is that the electrostatic potentials at the two surfaces are independent. In other words, the potential in the membrane does not couple the two surface potentials to any significant extent. A discussion of the errors inherent in this assumption has not been presented. Another common assumption is that divalent cations are bound to the membrane surface. Several conductance studies on thin lipid membranes, however, have shown that the divalent cations can be treated completely within the context of diffuse double layer theory (McLaughlin, Szabo & Eisenman, 1971; Muller & Finkelstein, 1972). In this paper we present calculations which are complete within the limitations of the model. Our model is that used by Chandler et af. (1965), a dielectric layer of infinite lateral extent with a uniform charge density on each surface. The membrane is bounded by two ionic solutions and the electrostatic potential in each phase is assumed to be governed by the nonlinear Poisson-Boltzmann equation. In using an equilibrium theory like the Poisson-Boltzmann equation, we are assuming that the rate at which ions diffuse across the membrane is slow so that the equilibrium structure of the double layer is not disturbed to any sign&ant degree. We ignore effects due to the presence of dipoles in the membrane. The possibility that the surface charge density is not uniform but may be clumped around membrane pores is not considered. Our approach to the problem follows that of Ninham & Parsegian (1971). As they have pointed out, the membrane surface charge density is not a reIevant biological parameter. Instead, the surface density of ionizable groups and their dissociation characteristics should be specified. The surface charge density and potential then follow automatically and depend on the Zocul environment of the ionizable groups. Thus the potential drop across the membrane is calculated by treating the fraction of dissociated ionizable surface groups as a self-consistent functional of the electrostatic potential. The dissociation constant for the ionizable groups is taken to be a constant
ELECTROSTATIC
COUPLlNO
ACROSS
A MEMBRANE
319
and does not depend ‘on the electric field at the surface (thus we are ignoring the Wien effect recently discussed by Bass & Moore, 1968). Our membrane model is solved for two cases. In the first, all groups on the membrane surface are acid groups of a single type. The second case is a generalization of the first, and we derive equations for the presence of two different acid groups and one amine group on the surface. The error caused by ignoring the coupling of the surface potentials is discussed for the first case. We then apply the equations for the second case to the data of Mozhayeva & Naumov (1970) for the steady-state potassium conductance of a frog node. 2 Theory A useful membrane model and that dielectric layer of infinite lateral extent The three phases are denoted by I, II, &OSe,, co, respectively. The thickness Phase
I
Phase
x-0
1. Diekctric layer with charge potential is Y and the potential
II
Phase
(membmne)
(inside)
Fro. resting
used by Chandler et al. (1965) is a bounded by aqueous phases (Fig. 1). III, and their dielectric constants by of the dielectric layer is d, and the
(outside)
x=d
.
densities
III
u and d on its surfaces. membrane is !&,.
drop acrossthe
The membrane
320
A.
P.
NELSON
ET
AL.
surfaces have surface charge densities Q and 6’. The electrostatic potential of each of the three phases, 3-Z”,,YZ and Y, can be calculated by solving the non-linear Poisson-Boltzmann equation (see Hill, 1960, for a discussion of the statistical mechanical limitations of this equation). d2Y1 -~ 47vlW XSO (1) s= Eo d2% -= o Osxld dx2 d2Y3 4XPdX) m=-x>d dx2 so where p(x) is the charge density and is specified later. The system of equations must satisfy the following boundary conditions lim dm = 0 x+m dx lim dy’,(x) lim Yi(x) = I/ (5) x+-m dx . x-+--m The potentials at the surfaces are continuous and the normal components of the electric displacements are connected by the boundary conditions lim Y3(x) = 0 x+00
(6)
For simplicity we consider a solution containing only monovalent ions and divalent cations. The ionic concentrations in the reservoir are denoted by n-(r)
= n
n+(r)
= 41-v)
n++(r)
= ni
(8)
where 0 5 q s 1. The ion concentrations are governed by a Boltzmann distribution, so that pl(x) = ne[-~~S’Yl-~)+(1-~)e-ea(Yi-Y)+r,-2eacY1-v,] (9) p3(x) = n’e[ - ee@P3 + (I- q’)e-‘fl’3 + q’e-2e~ya]
(10)
where e is the unit charge, /3 = l/kT, k is Boltzmann’s constant, T is the absolute temperature, and primed quantities denote phase III. The solution to the equations for the case when the surface charge density 0 is constant, the divalent ion concentration is zero, and the outer surface potential Y”,(d) is zero has been given by Chandler et al. (1965). The surface
ELECTROSTATIC
potcotidisgiven
COUPLING
ACROSS
A
MEMBRANE
321
by
ym 2b+zeox
sinh eY?,(U)flkT eYl(0)/2kT
-
1
= @’
(11)
where K is the usual Debye screeninglength de&d by 8nne’ x2 =Qz* One of the ditliculties with using equation (11) is that the surface charge density is due to ionizable groups and changes as the environment of the membraneis changed. For example,if the surface charge is from acid groups, a change in the pH of the solution will change the surface charge density. This problem has been solved with the assumption of a small electric field in the membrane (Gilbert & Ehrenstein, 1969, 1970), but a complete solution has not beenpresented. The assumption of a small electric field in the membrane leads to the result that the two membrane surface potentials are completely independent of each other. In the general case,the two surface potentials are not independent but are connected by the boundary conditions, equations (6) and (7). The general solution to the problem for the case of acid groups on the membrane surface is presented in the next section. (A) ACID GROUPS OF ONE TYPE ON THE SURPACE
Consider a model in which the surface charge is due entirely to ionizable acid groups and all the acid groups are identical. If S and S’ are the surface areas per acid group, and a, ~1’are the fraction of ionized surface groups, then or -- ea Q’=: --. ea’ S’ As The fraction of ionixable groups depends on the local environment of the groups and thus on the surface potential. The surface potential depends on the surface charge. Thus a and a’ must be determined in a self consistent manner. The surface groups have a known dissociation constant for the reaction AH # H+ +A- given by L (14) Ol-a where [H+ 1,is the hydrogen ion concentration at the surface.In keeping with the Poisson-Boltxmann equation, [H+ 1, is related to the hydrogen ion concentration in the reservoir, H, by the following equation [H+]* = ~e-ti(~~(0)-Y)s (1%
322
A.
P.
NELSON
ET
AL.
Equations completely analogous to equations (14) and (15) can be written for the outer membrane surface. Equations for !P1(0) and !Ps(d) are obtained by combining equations (1) and (3) with (9) and (10). It is now convenient to define several new variables (pl(x) = eem-u &(x) = eeSv3. (16) After performing
several straightforward
integrations,
we obtain
(449-O ( 41@I+z)t--@-$){ln$$-e/W} ‘I + 4ne2j3K &oKS (4,(d)-1)
(
C#J(d)+ ‘1 *+E143(4 3
2)
Tjz-{ln($$)
dm) H+Kc$,(O)
= O (17)
-eflq
4xe2/K’ -t-----
-___4<(d)
H’-tK’&(d)
EOK’S
= 0. (18)
Equations (17) and (18) are two coupled algebraic equations for the surface potentials. They can be solved by iterative techniques with the aid of a computer. Then the potential drop across the membrane, Y, can be calculated : Y, = yY,(d) - yY,(W (19) Also, the tetms coupling the two surface potentials are proportional to EJE,, which is usually small. Thus these terms can often be ignored, in which case equations (17) and (18) reduce to the equation used by Gilbert & Ehrenstein (1970). The results of the complete set of equations will be presented in the next section. (B)
DIFFERENT
IONIZABLE
GROUPS
ON SURFACE
The theory presented above can be generalized to include the presence of several different ionizable groups on the membrane surface. For example, consider the presence of two different acid groups and one amine group on the surface. The two dissociation constants for the acid reaction AH # A- + H+ are given by K1 = [H+1,fi
K2 = [H+],*. -
-
1
For the base reaction, BH # B+H+, K3 = CH+]o[B]
[BH+I
2
we have = [H’]
lz
’ a3
(21)
ELECTROSTATIC
COUP.LING
ACROS,S
323
A MEMBRANE
and so the surface charge is given -by a1 a*’ a3 ----+(22) Sl s2 s3 where Sr, S, and S3 are the surface areas per ionizable group. Equations (20) through (22) are then combined with equations (I), (6) and (9), and if the coupling of the surface potentials across the membrane is ignored [e.g. drop the term proportional to e, in equation (6)], then one obtains CI
4~e2BK34iW + &$3K(H
+K34i(O))
-
(23)
Since the coupling of the potentials across the membrane is being ignored, an equation identical with equation (23) is obtained for the outer surface potential, Y3(d). and Discussion The term coupling the two surface potentials in equations (17) and (18) is proportional to eJe,,. If this term is ignored, the surface potentials are independent and equations (17) and (18) reduce to the one used by Gilbert & Ehrenstein (1970). Due to the highly nonlinear nature of the equations, the actual error introduced by ignoring the coupling term can be determined only by solving the equations for a specified set of parameters. We have solved the equations with the membrane thickness,d, set equal to 100 A, the dielectric constant of the solution equal to 78.5, and a wide range of ionic concentrations and surface charge densities.Several generalizrttions can be made. The error due to ignoring the coupling term depends mainly on the potential drop acrossthe membrane, !?“=,and on the dielectric constant of the membrane. A change in the concentration of the ions in solution seemsto a&t the error in an indirect mannex by changing the size of the potential drop acrossthe membrane. The maximum percentageerrors for Y,,,that one finds for various values of thesetwo parameters are listed in Table 1. One notes that the errors are quite small and usually can be ignored in most biological membrane problems. Moxhayeva 8c Naumov (1970) have studied the effect of changesin the external concentrations of H+, Ca*+, Mg*+, Ni*+ and external ionic strength on the steady-state potassium conductance of a frog node. They found that the curve relating membrane conductanceto membrane potential 3. Caladations
324
A. P. NELSON
ET AL.
TABLB 1 Approximate errors in Y,,, Y&nV)
.!?I= 2
e1 = 10
El = 20
0.1%
0.2% 1.0
0.8% 2.0
40 :
8:; ;:;
1.8 25
;:;
100
1.0
iti
shifts along the voltage axis with changes in the concentration of the various ions. A parameter convenient to define is V,, the membrane potential at which the membrane conductance is half the sum of the minimum and maximum conductance. They were able to correlate shifts in V, with a calculated membrane surface potential by assuming that there are groups of three types on the membrane surface. Type I groups are negatively charged and capable
- 50
‘/ I 4
-40 t
I, 5
I 6
I
I 7
I
I 6
I
I 9
I
I IO
II II
PH
FIG. 2. Change in V+ and !?‘~a@)with change in pH and calcium concentration. (a) 2 nw Cap+, (b) 10 rmt Cap+, (c) 20 rmn Caa+ . The extcanal KC1 concentration is 100 mu. The points are the experimental value from Mozhayeva & Naumov (1970). Figmw to the left of the V, axis are for the set of experiments from pH 4-7. Those to the right of the V, axis are for the set of experifients f’rom pH 7-103.
ELECTROSTATIC
COUPLING
ACROSS
325
A MEMBRANE
of binding H+ and Ca’+. T ype II groups are neutral and bind H+. Type III groups are negative and are not capable of binding ions. But we feel it is not necessary to introduce an equilibrium constant for the binding of Ca’+ to the membrane. Due to their large charge, divalent cations are concentrated into the double layer much more readily than monovalent cations (see Fig. 2 of Ninham & Parsegian, 1971, for a numerical example) and are thus more effective in “screening” the surface charge than monovalent ions. Such a treatment of divalent cations is the same as that used by McLaughlin, Sxabo & Eisenman (1971) and Muller & Finkelstein (1972) in conductance calculations on thin lipid membranes, The data of Mozhayeva 8c Naumov (1970) can be analyzed by a model of two acid groups and one amine group on the membrane surface. The surface potential can then be calculated from equation (23) with the parameters chosen to give the best fit to the data. The results are shown in Fig. 2. The two acid groups have parameters of pK, = 3.7, pK, = 2.15, S1 = 300 A2, and S, = 100 A2. The parameters for the amine group are pK, = 9-O and s, = 143 AZ. At pH 7 and 2m~ Ca these parameters give a net surface charge of -6.3 x 1013 electronic charges cmB2 compared with a value of - l-7 x 1013 electronic charges cmW2 calculated by Mozhayeva & Naumov (1970). Due to the brevity of their paper, the origin of this discrepancy is not apparent. The authors would like to thank Walter J. Moore for introducing them to this problem and Ludvik Bass and Alfred Strickholm for helpful encouragement and discussion.
REFERENCES Bm, L. & MOORE, W. J. (1968). In Structural Chembtry and Molecular Biology (A. Rich C N. Davidson, cds) p. 356. San Francisco: W. H. Freeman. CHANDLER, W. K., HOJXXIN, A. L. & MEWS, H. (1965). J. Physiol., Land. 180, 821. %REBSEIN, G. & LECAR,H. (1972). A. Rev, Biophys. Bioettg. 1, 347. GILBERT, D. L. & EARBNSTEIN, G. (1969). Biophys. J. 9,447. GILBERT, D. L. & JImimmm, G. (1970). J. gen. Physiol. 55,822. HILL, T. L. (1960). Introduction to Statistical l7aermodjwamics. Reading, Mass.: AddisonHUE, B. (1968). J. gm. Physkd 51,221. HO~KIN, A. L. & Hm. A. F. (1952). J. Physiol., Land. 117,500. MCL.AUGL.RIN, G. G. A., SZABO, G. & EISENMAN, G. (1971). J. gen. Physiol. MOZHA~A, G. N. & NAUMOV, A. P. (1970). Nature, Land. 228,164. MULLER, R. V. & FINEEISIEIN, A. (1972). J. gen. Physkd. 60,285. NINHAM, B. W. & P-IAN, V. A. (1971). J. theor. Biol. 31,405.
58,667.