Membrane solvation as a basis for ionic selectivity

J. Theoret. Biol. (1967) 17, 213-228

Membrane Solvatlon as a Basis far Ionic Selectivity BARRY D. LINDLEY Department

of Physiology,

Western Reserve University, Cleveland, Ohio, U.S.A.

(Received 22 March 1967)

Replacement of hydration energy by solvation energy in the membrane matrix offers a reasonable alternative to chargecharge interaction as a basis for ionic discrimination by cell membranes. Both the order of magnitude of selectivities and the activation energies are suitable. However, agreement with experimental selectivity orders requires some additional postulate, such as the adjustment of partition terms by mobility ratios. In this sense, further work on the model is required. 1. Introduction

A large number of studies reported in the literature indicate that the ionic selectivity of a membrane can be characterized by parameters extracted from the relationship between the potential difference and the ionic concentrations of the bathing solutions (for example, Lindley & Hoshiko, 1964; Leb, Hoshiko & Lindley, 1965). The object of theoretical studies of ionic selectivity is to produce models which give quantitative values agreeing with the experimentally obtained parameters. It is obvious that a major criterion of an adequate model is whether it fits the experimental data at more points than there are adjustable parameters in the model. It is with the idea of obtaining a number of independent values by using different ions that ionic comparison studies are undertaken. The aims of an adequate theory of membrane selectivity may be stated as follows (some points suggested by Teorell, 1953): (1) to predict the relationship between ion fluxes and the electrochemical potential gradients across the membrane; (2) to predict the relationship between the potential difference across the membrane and the ionic concentrations in the phases adjoining the membrane ; (3) to predict the variation with temperature of the flux and potential selectivities ; (4) for excitable membranes, to predict the variation with the membrane potential and the ionic concentrations of the flux and potential selectivities; T.B.

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(5) to account for the major modes of pharmacological (and endocrine) alterations of selectivity; (6) to account for osmotic phenomena associated with the membrane, such as the nature of water movement and the reflection coefficients for non-electrolytes. These properties should ideally be predictable from the properties of the ions measured in free solution, in the crystalline salt, etc., rather than simply being fit by quasi-empirical equations. For the present, my aim is the more modest one of simply looking at the stable (i.e. not time- or voltage-dependent) discrimination among ionic species with respect to their effect on the potential difference. The model chosen for development is largely an extension of the notions of Mullins (1956). Without data on many more aspects of membrane behavior than can be studied with the techniques commonly used, there remains a great deal of freedom in the formulation of the model. Measurements of the potential difference alone could not even easily distinguish between two concepts so apparently unlike as that of membrane diffusion potentials and phase boundary potentials. However, one basic principle emerges clearly: at some point in the system, ionic discrimination must be based upon the partition of ions among various energy states. This partition may be a true equilibrium partition, as between an ion-exchange resin and a solution, or it may be a quasiequilibrium partition, as occurs in the models treated by absolute reaction rate theory, where the substance passes through a high energy “activated state”. Notice also that “thick” membrane penetration could involve two possible rate-limiting steps-partition between solution and membrane or movement through membrane matrix. Selection could occur at either step. The basic question suggested by this basic principle is whether there is some general intermediate stage which would appear likely to be common to all types of mechanisms. If there is such an “intermediate common path”, the first goal of selectivity theory should be to calculate the energy of ions in that stage. The process of penetration of ions through membranes must involve the transition solution-membrane solution. Thus the three stages that must be investigated are: (1) the state of ions in solution; (2) the transition state or activated state between solution and membrane; (3) the state of ions in the membrane. It should be pointed out that it would be possible to think of penetration as

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largely a “one-jump” process, with only a single large energy barrier between the two solutions. In such a case, the state of ions “in the membrane” would be close to that in solution. 2. Ion!4 in solatioll The Grst task is thus to investigate the state of ions in aqueous solution. For this purpose I shall follow the review of Ccmway & Bockris (1954). The primary feature of such ions is that there has been a substantial decrease in free energy as a result of the interaction of the charged ion with the polar water; values of ionic hydration energies are given in Table 1. These energies represent that required to transfer the ion from a vacuum into solution at unit TABLE 1 Ionic radii and hydration energies (after Conway & Bockris, 1954) Ion

FclBrILi+ Na+ K+ Rb+ cs+ Mg++ ca++ Sr++ Ba++

Pauling radius A 1.36 1.81 l-95 2.16 0.60 0.95 1.33 1.48 1.69 0.65 0.99 1.13 1.35

Goldschmidt radius A 1.33 1.81 1.96 220 0.78 0.98 1.33 1.49 1.65 0.78 E

“Empirical” radius A 1.16 1.64 1.80 2.05 o-94 1.17 149 1.63 1.86

Eley & Evans

A1G” Blandamer kcal/g-ion

- 81 - 52 - 47 - 26t -123 -107 -84 - 75 - 61$ -479 -410

& Symons -103 - 76 - 69 - 62 -123 - 98 - 81 - 76 - 67

1 a43

t Laidler 8c Pegis, 1957. $ Latimer, Pitzer and Slansky, cited in Conway and Bock&. The “empirical” radii are from the paper of Blandamer & Symons (1963).

activity. It is to be noted that the impossibility of achieving a gas or solution of pure ions renders “non-thermodynamic” the discussion of single ion hydration energies, activities, etc. Nevertheless, one can calculate on the basis of models that which cannot be independently measured. The critical test of a model for hydration energies is thus whether the combination of anion and cation hydration energies gives the correct (measurable) hydration energy for the salt. Free energies of hydration are quite large, being on the order of at least 100 RT at room temperature. Consequently one can expect practically no ions

216

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(10e4’) to exist in the unhydrated state. There are considerable differences (usually at least 10%) among the values reported for hydration energies by various workers. Especially variable are the results for anions. Thus, it is apparent that one cannot look for extreme accuracy in calculations of this sort. The values in the last column of the Table, taken from Blandamer & Symons (1963), combine to give values for salt hydration energies that agree much better with experimental values than do the data from the next to last column. Notice that the two sets of values agree reasonably well for cations, but are quite different for anions. The one feature which is most apparent from the data in the Table is that as the naked size (crystal ionic radius) of the ion increases, the hydration energy decreases (becomes less negative). From studies of mobility, etc., it is also obvious that the apparent radius of the hydrated ion decreases as the radius of the naked ion increases. This phenomenon can be roughly explained in terms of changing charge density. Eisenman (1961) and Mullins (1956) have both seized upon this inversion of order between hydrated and naked conditions in constructing their selectivity models. Although agreement is not precise, the major portion of the free energy of hydration can be accounted for on rather simple terms by considering the charging of a sphere in a homogeneous dielectric. This is the so-called “Born charging” model. The major discrepancies in this approach are related to the facts that (1) water would not have its macroscopic dielectric constant in the intense field found in the vicinity of an ion (“dielectric saturation”) and (2) ordering and disordering effects on the water structure are neglected. In addition, short-range interactions between the ion and the immediately adjacent water molecules cause other alterations in the energy. Models seeking to correct for such effects are discussed by Conway & Bockris (1954). It seems clear, however, that a good first approximation to the free energy of hydration beyond the jirst water shell might be obtained by a consideration of the simple electrostatic interaction. Furthermore, in biological systems it would seem unreasonable to expect an ion to lose all its water of hydration, since the free energy change involved is so great. To evaluate the hydration free energy of such a complex of an ion with one shell of water, we consider a gas at unit activity of ion-water complexes. The complex is discharged in the vacuum (AGvac,disch), then introduced into solution at unit activity. The neutral complex is then recharged (AG,,,,, ch). Since the activities are unity in both solution and gas, the standard free energy of hydration is thus AGyd

=

AGso~n,

ch+

AGvac,

disch

+ 8.

Notice that this is not the usual ionic free energy of hydration,

since the

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217

species being hydrated is the ion plus the first water shell. 6 represents additional terms due to water-water interactions and the removal of water from the solution to form a cavity into which the complex is placed. At distances beyond a few A from the charge center, the macroscopic dielectric constant should hold for the water, and we use the Born charging process is the quantity which is most difficult to to eva1uate AGsoIn, ch’ AGvac, disch evaluate in the usual theories of ionic hydration (or rather a quantity of like magnitude but opposite sign); however, this quantity need not be explicitly evaluated for the present purpose. The work of placing the charge on the complex in solution can be determined according to the method discussed in Lewis t Randall (1961, see also Gurney, 1953). The electric field around a sphere with a charge q at a distance r from the center of the sphere is:

The work of introduction

of the charge gives:

AG = $~4m2drze’~(l

- F)D.dD. 0

No is the Avogadro numbez e. the dielectric constant of a vacuum, z the valence, e the electronic charge, and e the dielectric constant of the medium. r. is the radius of the ion (in the present case, of the ion-water complex). Thus, 00 zc

The coefficient e2No/8xeo has the value of 166 kcal A/mole. 3. Ions in the Intermediate State It next is necessary to build a model of the intermediate state of the ion and to calculate the free energy of transferring the ion from the gaseous ion-water complex to this intermediate state. It is at this point that the basic postulate of the present model enters; the suggestion of Mullins is the basis for this postulate. It is assumed that the intermediate state (whether equilibrium or quasiequilibrium) is that of an ion within a cluster of water molecules in a largely lipid matrix. That is, there is a cavity within the membrane, normally occupied by water, and a certain amount of water is removed from this cavity and replaced by an ion-water complex. It is further assumed that the free energy of removing water combines with the free energy of water-water

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interactions between the complex and the residual water of the cavity to give a term 6. The complex is discharged in the vacuum, then transferred into the membrane cavity and recharged. The free energy change for this process of membrane solvation (at unit activities) is AGknb,

so~v= AGmemb, ch + AGwc, discb f 6.

We next construct a cycle: Ion-water complex VACUUM :

I

Neutral ion-water complex

Neutral ion-water

AGiL,

AGLb.

ci,

‘\

complex

ch 1

I

Charged ion-water complex

Charged ion-water complex

f

I AGZ-,

= -AGL,

ch- a-AGv,c,

disch+AGvac, disch+ 6 +AGk,,,

A%,,

= -A%,,

eh -I- AG:,,,,t., ch’

ch

Thus, evaluation of AG,!&,,, Cr.,allows the estimation of the standard free energy change for the transfer of the ion from solution to intermediate state in the membrane, for the ion-water complex is in a sense a convenient fiction, entering only because we stipulate that the ion need never be naked for the process we imagine. The importance of this restriction is that the activation energy would be too large for the ion to go through the naked state. In the process pictured here, the activation energy need be no larger than AG:-,, whose approximate value will be calculated below. AGkn~, ch can be calculated by using again the Born charging process, this time for an inhomogeneous membrane whose dielectric constant is a function of the distance from the charge center.

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Here the region between r. and ri has dielectric constant Q, and that from rl out has dielectric constant e2. Over regions in which the dielectric constant

is constant, integration AG:emb,

ch =

-

2

is reasonably simple. [ s’(l-~4rr’~]~‘dr+ D2

+

=-2

SC

1 -t

4nr2 >

Ze/4%?=

[1

dr

20

[j$(l-z)dr+J’$(l-z)dr]

In a similar fashion, for an ion-water complex of radius r,,

A&n, .A =e z2e2N 0

In the cavity, the region between r. and ri is considered to be a vacuum, i.e. E, =

Eg

1-3

1.2

122 6

E,+e,---r.

-52 rl

This last expression gives the standard change in free energy for the transfer of one mole of ions from aqueous solution to a membrane of the description given. Notice that for the sake of simplicity rotational terms, etc., have been assumed to cancel out.? t Recently a treatment of ionic hydration has appeared (Stokes, 1964) which employs a similar notion of using an intermediate region of low dielectric constant along with an “infinite” outer region of normal dielectric constant for water. Stokes suggested that the energy in the vacuum should be based on van der Waals radii, rather than crystal ionic with experimental energies of hydration. Thus, his radii, and found rather good mt treatment involves the electrostatic energy in the vacuum of an ion with the van der Waals radius and the electrostatic energy in a “two-region” solution with the crystal ionic radius. He says that the ion “shrinks in the wash”.

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4. Selectivity

Pammeters

The next part of the problem is to show the way in which this energy term enters the selecti&y parameter. The explicit nature of the physical model now assumes increasing importance. In the simplest model, there is unlimited “room” in the intermediate state (i.e. solution very dilute), and partition of ions between membrane and solution may be looked at as a classical twosolvent distribution problem. ion (soln) z$ ion (membrane) K = ay/ai = exp (- AG’/RT). However, membrane ions can be present in any of a number of forms (i.e. with one, two, etc., hydration shells). K1 = a; I/af, K2 = a? Jaf, K, = a: Jaf,

etc. The subscript here indicates the number of water shells. If the concentration is taken as a good approximation to the activity, K1 = C$/C:,

etc. The total concentration of ions of all degrees of hydration in the membrane is C~=C”(Ki,,+Ki,z+Ki,3+...).

Here, Ki, j = exp (-AC: j/RT). This free energy change is approximately that for the transfer from solution to membrane of a complex of the ith ion with j water shells. An apparent overall equilibrium constant is thus given by Ri=Ki,1+Ki,2+Ki,3+...

=C;lCs.

Notice that this treatment prescribes additivity of equilibrium constants; this is to be distinguished from a series of steps in which the free energies would add. Here each membrane complex species is assumed to be in equilibrium with the aqueous ion, and the “membrane concentration” of the ion is taken as the sum of the concentrations of the complexes. The free energy of formation of the complex in aqueous solution is assumed negligible compared to that for the transfer into the membrane phase (intermediate state). This simple model does not allow for saturation effects or for competition among various species. It is assumed that the membrane penetration is proportional to Cy. Thus, in the treatment of Parlin & Eyring (1954), Ii; = I(k,T/h)R,.

SELECTIVITY

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This value is calculated for a single pore radius. In a real membrane one would expect not a single radius, but rather a distribution. It is possible to define an overall specific rate constant,

where g(r) is the probability density function for the cavity radii. This integral is simply the normal expression for the average of a property. In the treatment of Hodgkin & Katz, the Ri defined above is equivalent to /I, the distribution coefficient. It is possible to treat slightly more sophisticated models, in which the possibility of saturation and competition is allowed. In this case, the number of cavities is not very large compared to the number of ions. The likelihood of this model can be assessed by calculating the number of ions in the membrane expected on the basis of the simplest model and comparing this to the likely number of cavities. Such a calculation is based on comparing I(k,T//z) exp (- AG’/RT) for reasonable values of AGO calculated from the present model with the measured PN. from toad node of Ranvier and squid axon (Frankenhaeuser, 1960). The calculation indicates that 0401 to 0.1 of the membrane area is available for diffusion with the activation energy described by the model; even a smaller fraction is available in other tissues such as skeletal muscle. Pores of radius 6.6A (chosen to give the sodium selectivity observed for the nerves) would be present in a number of 10” to 1Ol3 per cm2. In a 0.1 M solution, ions of the same sign should be about 29A apart on the average, yet the pore centers would be about 30 to 1000 A apart on the average. It would thus seem that there is a substantial possibility of saturation or competition with membranes of physiological permeabilities and solutions of physiological concentrations. Looked at in another way, there should be on the order of one collision per pore per molecular vibration (h/kT) based on the frequency of collision of particles with the membrane and the number of pores per unit area. The equilibrium between a limited capacity intermediate state and solution can be described from a number of different approaches (see, e.g., Sjodin, 1961). The expression for the number of ions of species i with j water shells in the cavities is %=

NKijCi l+xK,,C,’ ij

where K,j = exp (- AG$RT),

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and N is the total number of cavities. The total number of ions of species i is thus, NCiCKij

Under this model,

N C Kij k,O= l(k,T/h)

. 1+C~iCKij’ i

j

It is seen that the same quantities

appear as appeared in the simple model. However, in this case the function to be averaged over the radius is of a different form. In view of the difficulty of averaging kz as it appears here, it will be replaced by m I

IE; = A(k,T/h)N

’ 1 +cici

Ki(r)dr)

dr

m 1 K(r)dr)

df

0

Because of its appearance in both models, we replace cc s K(r)dr) dr 0

by the symbol Q, and proceed to its evaluation. g(r) is taken as a normal distribution with mean p and standard deviation CJ. Apart from the factor I(kJ’/h), the expression for K,(r) was derived above and is given by c exp (- AGG/RT), where

r. is the radius of the ion-water complex and is given by

r. = r,+jd, , where r, is the naked ionic radius, d, is the diameter of a water molecule, and

j is the number of shells.

g(r) = .jG ~ ew [-(I - d2/2a21-

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Consequently,

{-g$t[;;;]}

[exp(a[s]

-$Z$)dr.

5. Numerical Method This integral is of the form m

s 0

exp---a [x

(X-W2

& C

I

.

It cannot be integrated in closed form and must be evaluated by a numerical procedure. The method adopted was that of Monte Carlo integration with importance sampling (Kahn, 1956). (Dr Glenn E. Bartsch and Dr A. Cicchinelli were instrumental in developing the procedure.) This method is based on having to evaluate an integral of the form,

If h(x) is a probability density function, this expression is simply the mean value of f(x). Thus the expected value of the integral can be obtained by the usual statistical methods (and the probable error evaluated) by evaluating the functionj(x) at a series of values of x drawn at random from the population described by the probability density function h(x). The advantage of this method is mainly that the error can be reliably assessed; also, given the proper form it becomes an economical method of numerical integration. The problem then is to cast the integral given above into the form of the product of one function, to be evaluated, and a probability density function. Wise choice of the probability function will result in substantial reduction in the variance of the estimate. In this case it was decided to use a normal distribu-

224

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tion chosen so as to approximate very closely the integrand. problem becomes one of evaluating the integral

Thus the

al

I -ew [:-(@?;I exp [--(x-U)21 dx ! Iexp

[-‘C!I!J!-]

2w2

’

Notice that the new integral is the same as the old one, but written in a different form. The factor Z normalizes the probability density function exp [-(X- u)~/~w’] over the range sought. Now the correspondence to the terms above is f(x)

= : exp [“; - (xcb)z

h(x)=Zexp

+ ‘s],

[-(s].

u and w are to be chosen so that the normal distribution has its mean close to the maximum of the function K(r) and follows closely the curve elsewhere. If this is so, f(x) will be approximately constant, and the estimate will have a small variance. The estimate of the integral is N

where the X, are a random sample from the population described by the function h(x). In the present studies the sample size was usually 15, giving a standard error of the mean which was less than 5% of the value Q. The same sample population of radii was used for all ions, thus affording greater precision of comparison (as in using paired animals in an experiment concerning two variables). The quantity -RT In Q’ is the “equivalent free energy change”. All calculations were done on the IBM 1620 for a temperature of 298°K and a pore radius standard deviation of O-1 A. The dielectric constant of the membrane matrix was taken as 3.0. The major effect of a larger dielectric constant would have been to lower the energies and shift the abscissa slightly.

SELECTIVITY

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6. Results and Discu~on Figures 1, 2 and 3 give sample curves plotting the “equivalent free energy change” versus the mean pore radius for a number of naked ionic radii. The lower an ion, the more favored its penetration. For any ion, there are periodic minima in the free energy change, resulting from the opposing effects of allowing more water in as the pore radius increases, but keeping a larger region of vacuum (lower dielectric constant) present in between radii which allow extra water shells. Furthermore, at any given mean pore radius a selectivity order can be established for the ions, and of the ions studied, a mean pore radius can be found which selects any one ion over all the others. However, not all permutations occur, and it is the permutations and quantitative values that are critical for evaluating the model. Of special significance is the fact that selection among anions occurs in addition to that among cations-thus, this model, which has essentially two parameters (,D and EJ, can predict a large number of testable independent values. Notice also that there are regions of pore radii where large shifts in selectivity occur rapidly, whereas in other regions there are no changes.

T1290.15 +=I30

c*=3 d,=2.76 0 =O*l&

‘O

I 4*0

I 50

I

6.0

I 7.0 Mean

1. 8.0 pore

I 9.0 radius

I 10~0

8,

I Il.0

I 12.0

0

(8)

FIO. 1. Equivalent free energy change for membrane solvation of cations. The data used are given in the figure, except for the Pauling radii (see Table 1). All lines were drawn to co~ect points at O-2 A intervals cakuked by computer. The sektivity in terms of CQ,.would be given by exp [(-Ace + AGt)/RT). Thus, it is the difference in ordinates here that is proportional to the logarithm of the selectivity on the basis of the simplest sort of reasoning. The lower an ion on the graph, the more favored its entrance into the membrane cavities. (-*-) Li; (-) Na; (- -) K; (. . . .) Rb; (-- -) Cs.

B. D. LINDLBY

226 25

‘1’ : ,

,

I

I

I

I

I

I

I

I 11’0

I 12.0

T= 298.15

5-

0

I 5.0

’ 4’0

I 6.0

1 7.0

I 6.0

Mean

pore

I 9.0

radius

I IO.0

13.0

(A)

FIG. 2. Equivalent free energy change for membrane solvation of anions. The data used are given in the figure, except for the Paul@ radii (see Table 1). See legend to Fig. 1 forfurthercomments.(-)F;(....)Cl;(--)Br;(---)I.

25,

,

I

I

I

I

I

I

T= 298.15 +,=80 677= 3 d,=2,76 a m =O.I a

0

' 4.0

I 5.0

I 6.0

I 7.0

Mean

pore

I 9.0

I 8.0

radius

I

i

I 10'0

IWO

(1)

Fro. 3. Equivalent free energy change for membrane solvation with Blandamer & Symons’ radii. The symbols for Li and Cs indicate the positions of the maxima and minima in their curves; thus, approximate curves can be easily visualixed. The Rb curve is very close to that for K (see Fig. 1). See legend to Fig. 1 for further comments. (. , . .) Cl; (-) Na; (- -) K.

SELECTIVITY

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This model produces all the orders suggested by the original model of Mullins, and in addition several others. Not all of Eisenman’s orders appear to be present, but a number of orders not predicted by Eisenman are. When comparing the results of this model with those of other models, it is important to note that there is considerable formal resemblance between the calculation of solvation according to the Born charging method and the calculation of charge-charge interaction. The major differences are in the sign of the energy change and in the significance attached to the parameters. In terms of the probable quantitative reliability of these calculations one should think mainly of orders of magnitude since many drastic simplifications have entered. Thus it would seem that one could not discard the model without finding qualitative discrepancies or disagreement on relative selectivities by some orders of magnitude. A further consideration is that one must be sure that the experimental parameter and the theoretical parameter correspond in significance. Thus it would be wrong to compare ~x~x/u&3~~ from the Hodgkin & Katz formulation with &/&. from the model unless u, = u,,. In the Parlin t Eyring formulation A occupies a position similar to that of u. The problem now is one of making a decision on which set of naked ionic radii to use and then comparing available values with the model. However, an examination of the literature shows that mobility ratios different from unity are necessary to correct the estimates. Thus, Sjodin (1961) found a different order for the parameter corresponding to R, here from the apparent order of permeabilities. With the present model, the order for R, of K > Rb > Cs > Na > Li does not occur. Yet this order and the similar one of K > Rb > Cs > Li > Na do occur experimentally in frog skin inside (Lindley & Hoshiko, 1964) and frog muscle resting membrane (Sjodin, 1961). In addition, anion selectivities as predicted do not coincide particularly well with the data for the outside of the frog skin. Thus, the model as it stands cannot be considered as giving a better fit to experiment than other models. Future work might go in one of two directions. The membrane solvation hypothesis might be abandoned, or one might go to “cleaner” systems. Even the best systems investigated still are subject to many complications, such as possibly “mosaic” membranes. There does seem to be some virtue in the membrane solvation model, for good orders of selectivity are predicted with reasonable activation energies at pore sizes large enough that the use of the Born charging method should not be too gross an approximation. A recent thermodynamic analysis of salt and water movement across membranes capable of active transport (Hoshiko & Lindley, 1967) indicates

228

B. D. LINDLEY

that the experimental description of ionic selectivity requires several sorts of measurements. With two different cations and a common anion as the only penetrating species, we have in the traditional treatments only four coefficients required to specify the selectivity-the permeability coefficients for each of the two salts and the transference numbers for each of the two cations. In a full treatment based on non-equilibrium thermodynamics it appears that nine coefficients are necessary-in addition to the four above we have a linkage coefficient, two reflection coefficients, and two active transport coefficients. Only with all these coefficients measured can one evaluate parameters relating to molecular interactions. The present molecular model for membrane solvation thus offers only one aspect of a unifying theory of ionic selectivity in biological membranes. This study was supported by United States Public Health Service GM09310, PHS 26-899 and 5Tl GM-17 and by National ScienceFoundation grant GB5197. In part, it appearedin a dissertationIonic Selectivity of the Isolated Frog Skin, Western ReserveUniversity, 1964. REFERENCES BLANDAMER, M. J. & SYMONS, M. C. R. (1963). J. phys. Chem. 67, 1304. CONWAY, B. E. & BOCKIUS, J. O’M. (1954). In “Modem Aspects of Electrochemistry”, (J. O’M. Bockris, ed.). London: Butterworths Scientific. ELTENMAN, G. (1961). In “Membrane Transport and Metabolism”, (A. Kieinzeher and A. Kotyk, eds). New York: Academic Press. FRANKENHAEUSER,B. (1960). J. Physiol. 152, 159. GURNEY, R. W. (1953). “Ionic Processes in Solution”. New York: Dover Publications, Inc. HOSHIKO, T. & LINDLEY, B. D. (1967). J. gen. Physiol. 50,729. KAHN, H. (1956). In “Symposium on Monte Carlo Methods”, (H. A. Meyer, ed.). New York: John Wiley & Sons, Inc. LAIDLER, K. J. & PEGIS, C. (1957). Proc. R. Sot. A 241, 60. LEE, D. E., Hoswto, T. & LINDLEY, B. D. (1965). J. gen. Physiol. 48,527. LEWIS, G. N. & RANDALL, M. (1961). “Thermodynamics”. 2nd edition, revised by K. S. Pitzer and L. Brewer. New York: McGraw-Hill Book Company. LINDLEY, B. D. & HOSHIKO, T. (1964). J. gen. Physiol. 47, 749. MULLINS, L. J. (1956). “Molecular Structure and Functional Activity of Nerve Cells”. Publication No. 1 of American Institute of Biological Science. PARLIN, R. B. & EYR~NG, H. (1954). In “Ion Transport across Membranes”, (H. T. Clarke and D. Nachmanson, eds). New York: Academic Press. SJODIN, R. A. (1961). J. gem Physiol. 44,929. STOKES, R. H. (1964). J. Am. them. Sot. 86, 979. TE~RELL, T. (1953). Prog. Biophys. biophys. Chem. 3, 305.