On the criteria of active transport

3. Theoret. Biol. (1965) 9,489-501

On the Criteria of Active Transport V. S. VAIDHYANATHAN Department of Medicine, University of Arkansas and V.A. Hospital, Little Rock, Arkansas, U.S.A. (Received 25 January, and in revised form 20 April 1965) The phenomenological aspects of inclusion of effects of chemical reactions on diffusion currents are discussed. Formal statistical expressions for systems with chemical reactions inside an anisotropic membrane consistent with hydrodynamic equations are derived. The properties of a multicomponent system with chemical reactions are expressed in terms of the properties of an identical system with no chemical reaction, A critical appraisal of the state of the theories including Kedem’s criteria of active transport is presented. The relationship between metabolic energy and active transport is analyzed and an explanation based on partial heat of transport is presented. The considerations are shown to be consistent with our proposed explanation of active transport, which was based on consideration of intercorrelation of fluxes, changes in iso-thermal perturbations of pair interactions, and the effects of localized charges on the magnitude and direction of flux of a specified kind of ion.

1. Introduction Recently, on the basis of molecular theory of transport of multicomponent electrolyte mixture, we demonstrated the significant role played by localized charges in the determination of the direction of flux of a specified kind of ion i (Vaidhyanathan, 1965a, b, 1966). It was derived that an ion of kind i will appear to be actively transported (flux of ion i being in a direction opposite to that suggested by considerations of chemical potential gradients) if certain inequalities hold (equations (36) and (37), Vaidhyanathan, 1965b). Our explanation is the result of statistical mechanical theory and the arguments for active transport were based on the existence of plane regions normal to the direction of transport inside the membrane where such inequalities hold. Considerations of metabolic reactions and accompanied energy changes were avoided due to lack of development of statistical theory of multicomponent fluid mixtures including chemical reactions. It was remarked that the effect of metabolic chemical reactions could be the alteration, either directly or indirectly the ratio of concentration of these localized charges, to the con489

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centration of mobile ions. The need for quantitative inclusion of the effect of metabolic reactions cannot be neglected. A most general molecular theory of active transport should account for energy requirement and possibly a stoichiometric relation between active transport and metabolic energy. On the other hand, Kedem (1960) has attempted to formulate criteria of active transport within the framework of the formalism of irreversible thermodynamics. She considers transport of multicomponent fluid mixtures across a membrane with the additional restriction that a chemical reaction takes place inside the membrane. The driving force for the flows of solvent and solutes is the difference in electro-chemical potentials at both sides of the membrane. Kedem contends that her formalism is in accordance with the belief that a stoichiometric relation between transport and metabolism will be an important criterion of active transport. An interesting point which Kedem makes is that the phenomenological resistance coefficient Ri, must be a vector, the significance of which is that the vectorial cross coefficient creates a directed flow. These points raise the question whether Kedem’s formulation is correct, and if they are valid, whether they are consistent with our molecular approach or uicr cersu. The neglect of energy requirement considerations does not preclude in any way the merits of our conclusions. We adopted this philosophy due to lack of know-how about systems with chemical reactions undergoing transport. In this connection it may be stated that the treatment of chemical reactions either alone or in conjunction with other irreversible processes has not reached a satisfactory state (Fitts, 1962). In spite of a blind man’s concept of the elephant, one may venture to make some progress by a critical appraisal of the state of the theories. Bernhard (1964) has recently presented a lucid survey of some biological aspects of irreversible thermodynamics. It is difficult to comprehend the meaning of his equations (1.3-2) and (1.3-3) since JR is a scalar and JA is a vector. Hence any argument based on his equation (1.3-7) is not valid. In the next section, we review what we consider to-be a correct interpretation of the results of nonequilibrium thermodynamics as applied to our present problem. In section 2, we present the problem regarding treatment of chemical reactions and coupled transport processes. In section 3, we present a molecular approach to the problem and first derive statistical expressions for hydrodynamic equations. In this way we get precise interpretation for the molecular parameters of statistical theory of transport for a multicomponent system including chemical reactions. The procedure is identical with the method of Bearman & Kirkwood (1958) and we have been guided and probably influenced by their philosophy. Having formulated the equations which yield the correct form of hydrodynamical equations, we venture to present the phenomenological equations for systems including

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chemical reactions. We suggest possible methods of including energy considerations and chemical reactions in determination of the direction and magnitude of flux of a specified component. The existence of some unsolved fundamental problems of statistical mechanics of transport excludes our formulations of section 4 to be exact. The contents of the following sections are consistent with the present knowledge of the state of theories. 2. Considerations

of Nonequilibrium

Thermodynamics

The general theory of irreversible processes involves the validity of three postulates: (1) all thermodynamic functions of state exist for each element of the system; (2) Jhe fluxes J, are linear homogeneous functions of the forces Y,; (3) the matrix of the phenomenological coefficients (defined as a result of validity of postulate (2)) for the set of conjugated fluxes and forces is symmetric. As a corollary to the second postulate there is the Curie’s theorem that entities whose tensorial characters differ by an odd integer cannot interact in isotropic systems (Bray & White, 1957). An elegant analysis of the above postulates is presented by Fitts (1962). For the problem at hand one might state outright that it is not satisfactory to write linear relations between Gibbs free energy and the reaction fluxes. In other words, the validity of postulate (2) for a system in which chemical reactions occur is highly questionable. Postulate (1) does not imply postulate (2) although postulate (2) is dependent on postulate (1) for a precise definition of the fluxes and forces. The nearer a system is close to equilibrium the more valid is postulate (1) and it is usually assumed that postulate (2) also approximates the true situation. Any extensive property G of a system obeys the equation of conservation [a(pc)/af]

+ v .J, - f$G = 0

(2.1)

where G is defined as G per unit mass. Jc is the current density (flux) of G and & is the internal source of G per unit volume and time. When G is identified with specific entropy S and the assumption of local equilibrium made, one obtains p(dS/dt) = q&-V. j, (2.2) where j, is the entropy flux due to diffusion and heat flow. If T is the local temperature, one may define 4 by the relation (Fitts, 1962)

(2.3) 4 = 4sT = d1+42+43+44* & refers to the viscous term, q5z the contribution from chemical reactions, & the diffusion current part and +4 the heat flow contribution in an equivalent equilibrium system. In 4I both flux and force are dyadics, in & both are scalars and in & and r$4 both are vectors. The definition of & and 44

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permits one to define phenomenological

coefficients by the relations

(2.4) In an analogous manner, the definition ficients Kik by the relation, (dl,/dt)

of & suggests the definition

=f

YkjAFj*

j=l

of coef(2.5)

L, is the progress variable for the kth reaction and defined as grams of reaction and defined as grams of reaction occurred per gram of original reactants. AF, = i V.P. a=1

is the Gibbs free energy charge for reaction k. From Curie’s theorem it follows that coupling of chemical reactions and diffusion currents or heat flow is prohibited in an isotropic system. If the membrane is anisotropic (with respect to chemica1 reactions), Curie’s theorem need not apply and one may write & in the form, h = -&AF,e,.

e, WklW.

(2.6)

The linear relations between conjugated force and flux of chemical reactions may now be defined

(2.7) where Tk, are the new phenomenological coefficients. These coefficients are scalar. The phenomenological coefficients must be scalar and should be functions of state variables like temperature, pressure and composition and should not be functions of gradients of these local state variables. Therefore, Kedem’s definition of vectorial cross coefficient R, cannot be correct. However, it is usually a poor approximation to write the reaction fluxes (d&/dt) as linear homogeneous functions of the forces AF,as written in equations (2.5) and (2.6). In addition, the phenomenological coefficients thus defined will be constants indepehdent of the departure of the system from equilibrium and should not be functions of the progress variable A,. If the system is assumed to be in a state very close to equilibrium one may assume that postulates (1) and (2) hold. If one assumes microscopic reversibility, then postulate (3) may also hold, which imply reciprocal relations. In the next section, we present further considerations along these lines, following the procedure of Kirkwood & Crawford

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(1952). Considerations, presented in the following sections, forbids formulation of cross coefficients between chemical reactions and diffusion fluxes even when Curie’s theorem is not obeyed. On the other hand, they lend support to our formulation of the theory of active transport. The effect of chemical reactions is to alter the local concentrations of components in the system. The’concentrations are related to the singlet space probabilities of statistical mechanics. Hence, the perturbation functions of nonequilibrium system, appropriately defined, also undergo change. These in turn determine the isothermal diffusion flux (neglecting detailed considerations of resultant temperature gradients) and hence the effect of chemical reactions inside the membrane on the diffusion flux of a specified component is highly complicated though not intractable. 3. Method of Kirkwood and Crawford In order to overcome the difficulty of writing down equation (2.5) which is a poor approximation, Kirkwood and Crawford have suggested the following improvement. They factor the progress variable I for the kth chemical reaction as sum of an equilibrium part and a perturbation part. At = g++r,.

(3.1)

The parameter A0 is the equilibrium value of the progress variable for the kth reaction for the instantaneous local temperature, pressure and composition. The variable & is a measure of the deviation of the chemical reaction from equilibrium. The Gibbs free energy change for the reaction k, AF, vanishes when I, equals ,I:. On this basis, chemical lag may be treated by expansion of the rate function Jr about the equilibrium point A,” (k = I,2 . . . . s) and consider only terms linear in &. Now we may write the linear relations (3.3)

where the coefficients pkn are given by (3.4)

Where hk is a formal expression for each chemical reaction k. h, (TlcIc~ . . . . c) - dAJdt (k = I,2 . . . . s). <,, is related to the concentration of the components that take part in reaction n. If the membrane is anisotropic, one might visualize that the chemical reaction n taking place inside the membrane is at various stages of departure from equilibrium as one travels from one side of the membrane to the other.

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The equation for & now becomes d2 = - pk$AFke,.

e,(dll,/W

(3.5)

or even better approximation & = - i E,e,.e,.(dA,/dt). (3.6) n= 1 In this manner, the phenomenological coefficients will remain scalars and wiI1 be functions of state variables like pressure, temperature and composition alone and will not include gradients. One should not be misled to believe that equation (3.6) for & supports the validity of Kedem’s formulation. In fact, it does not. The precise form in which the fluxes are influenced by chemical reactions inside the membrane can be obtained only from molecular theory. The set of equations of constraints which completely describes the hydrodynamic behavior of a multicomponent fluid mixture with chemical reactions is given by Fitts. In the next section, we develop the statistical theory which yield the partial and total hydrodynamic equations of motion and relate the effects of chemical reactions on the molecular parameters of statistical theory of transport. On the basis of these results one can obtain the phenomenological relations in section 5. The analysis presented in the next two sections clearly indicate inadequacy of Kedem’s equations. On the other hand, they support our explanation of active transport as resulting from the effect of localized charges and changes in isothermal perturbation functions. 4. Molecular Development The hydrodynamic equations for multicomponent fluid mixtures containing Y different chemical species and in which s different chemical reactions take place are: (aP,/at)

+ v. (P,U,) = P i

YIkd*

k=l

@p/&)+V.(pu) Equation

= 0.

(4.2)

of Motion (ap,u,/at)

= v . oa- p,(u,u + uu, - uu) + c,F:l)*+ a(pu)/at

+ v . (puu) = cx + v . (T

p,x,

= cx

: cu.

pm = m,c, a :

c,X,

(4.3) (4.4)

= 0 1 c, = C.

(4.5) II Equations (4.1) and (4.3) are the partial equation of motion and equations (4.2) and (4.4) are the total equation of motion for the system. Equation l$ P,qP

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(4.3) is the partial equation of motion of each component’ in a multicomponent fluid mixture derived by Bearman and Kirkwood. 6, is the partial stress tensor of component a and c,l?Ji)’ a term which arises from considerations of intermolecular forces. It may be shown that (4.6) When there are no chemical reactions taking place inside the system, the right-hand side of equation (4.1) will be zero and the remaining equations are the same for both system with chemical reactions and system with no chemical reactions. The statistical mechanical expressions for hydrodynamic equations for multicomponent system with no chemical reactions have been obtained by Bearman and Kirkwood, Their treatment is based on the postulation that the time-smoothed probability function obeys Liouville equation and that the expectation value of a dynamical variable cp,which is a function of all degrees of freedom, is given by (cp; f) where f is the timesmoothed probability function. The brackets indicate integration over the entire phase space. They make extensive use of the theorem of Irving and Kirkwood which enables one to express (a ( cp; f) /at) in terms of time independent terms. In order to derive various partial and total equations of hydrodynamics they choose appropriate definitions of this dynamical variable. Irving and Kilkwood’s theorem for multicomponent system is valid so long as the dynamical variable does not have an explicit dependence on time. In the following we shall consider two systems with identical components and state variables and allow s types of chemical reactions to take place in one system (denoted as (2)) while none in the other (1). We shall be concerned with derivation of statistical expressions of hydrodynamical equations for system (2) in close analogy with the expressions for system (1). If there are a total of N molecules in system (1) and consider only the three degrees of freedom of position and three degrees in velocity for each molecule, the time smoothed distribution function f for this system will have 6N variables (Irving & Kirkwood, 1950). In addition to these 6N variables, allowing s chemical reactions introduces additional degrees of freedom in system (2). Let us denote these additional degrees of freedom by wi and their conjugate momenta by ylt. Therefore, the time smoothed distribution function!* in phase space for system (2) is of higher dimension than f of system (l), and the expectation value of a dynamical variable cp is given for system (2) by < 40;f * ). For system (2) the Liouville equation is now obeyed by f * and this can be expressed by f (of system (1)) by amending the Liouville equation by a difference term denoted by F. Thus the expectation value of a dynamical variable in system (2) may be expressed in terms of its K

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expectation value in system (1) plus a correction term as

= +i$l
(4.7)

In order to derive the equation of continuity for system (1) with no chemical reactions, Bearman and Kirkwood identify the dynamical variable as (4.8) where 6(R,, - ri) is the appropriate Dirac delta function and m, the mass of molecule of kind a. Substituting these in the theorem of Irving and Kirkwood they obtain the partial equation of continuity as (4.9) m/w + v,, Pa”, = 0. I shall use the same definition for these quantities in system (2). Therefore

Adopting a procedure quite identical with that of Bearman and Kirkwood one obtains the equation of continuity for the system with chemical reactions as (4.11) Comparing

this with equation (1) it follows that (4.12)

Since molecules in systems (1) and (2) are identical, any neglected internal degrees of freedom as rotation and vibration will be the same for both systems. Irving and Kirkwood point out that the expression for the rate of change of the expectation value of a dynamical variable independent of internal degrees of freedom is unchanged by the existence of these internal degrees of freedom. They also point out that their theorem is equally valid for central or non-central forces depending upon rotational or other degrees of freedom. Thus, the reaction coordinates wi and their conjugate momenta vi which are present only in system (2) need concern us. I do not propose to indulge in obtaining physical meaning for these degrees of freedom and assume that equation (4.10) is reasonable. In obtaining the partial and total equation of motion, Bearman and Kirkwood define the dynamical variable as

Pai is the momentum

of molecule j of kind CI.

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Their derivation of partial equation of continuity is more complicated than for single component system. We do not propose to write down the procedure and shall be concerned only with the final form of results that one expects on the basis of equation (4.10). Our results are purely conjectural in nature and it may be recalled that conjecture played an important role in Bearman and Kirkwood’s derivation of equations of motion for system (1). In obtaining the following relation for systemf(l), $2

:f > = b%%~,/~t> = V~r[O,-p~(U,U+uU.-UU)]+CorF~l)~+c,X,.

Bearman and Kirkwood such that

anticipated

the validity

(4.14)

of their end results to be

c m,%/w= fQu)Pt

(4.15)

= qp,u,>/at = V.[o,-‘4+lq+c,x,. (4.16) They also anticipated that the term A will be analogous to pu and that the term B will comprise components which sum to zero and will lead to the phenomenological equations. In a like manner, utilization of the same expression for dynamical variable cpZ the partial and total equations of motion for system (2) may be written as @P, %)/at = v,, . [%-P&U

+ uu, - uu)] - c, X, + c, F:“’ + ; 8 (cp2 : F) f

1

(4.17)

CG4/~tl = VY C~-Puul+ cx +#$g c <(P2: 0

(4.18)

I I Since this should be in agreement with equation (4), it follows that (4.19) and already

Therefore, one might include the effect of chemical reactions $ $ (opt : F) i-l

in the expression for mean force and write the partial equation of motion for system (2) as (4.20) [a(P#uJ/at] = V,iC~~-p,(U,U+UU,-UU)]+C~"*+C,X,

T caF:*)**= o

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Summing overall components, one obtains equation (4.4) with the definitions of c,Fb(l)** given by

(4.21) If we define the frictional force experienced by a molecule (or ion) of kind cc in the two systems as the difference in force experienced in the systems at nonequilibrium and the force at equilibrium, F(l)* = F~l.l)*+F;l.O)* F’b* (4.22) a = F;‘.‘)“+F~‘.O)* F (l*ol’ (rl) is the equ‘l’b I I rium contribution to the average intermolecular f&ce acting on a molecule OLat a point r 1. The double starred quantities refer to system (2) and the single starred quantities refer to system (1). The above analysis helps us to identify the effect of chemical reactions as the change in frictional force experienced by a molecule of type LXin the system compared to its average frictional force in an identical system with no chemical reactions. Thus F(l-‘f**-F~l.‘)* = -[V,p*-V,c(m] a

=-q

~c;c:L7-cpimpl&-ua)

(4.23)

~2,~= C1/6(@ + $>I i (d%&W lI$$ + ~~a1d~~0’d3r (4.24) <,, is the partial frictional coefficient due to interaction between a molecule of type M with all molecules of the type j?, in system (1). r,*, is the same quantity in system (2). In writing down equation (4.26), we have lumped the effect of chemical reactions into changes that perturbation functions undergo in the system. The perturbation functions t,Gmpare determined by the pair correlation functions in nonequilibrium and equilibrium states which are functions of concentrations. The effect of chemical reactions is to alter these concentrations which in turn affects the pair probability of simultaneous observations of two speciJied kinds of molecdes at two specified locations.

The perturbation functions are defined by equations (3.1) and (3.3) in the previous paper (Vaidhyanathan, 1965a). The explanation of active transport of ions across a charged biological membrane involved essentially considerations of these perturbation functions for mobile ion-localized ion interactions. In the next section we present the relationship that exists between chemical reactions and fluxes which are consistent with the above development.

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Relations

The object of the previous section has been to find a proper way of inclusion of the effect of chemical reactions at the molecular level, such that the equations are consistent with hydrodynamics, statistical theory, thermodynamics and phenomenological equations. In this section, we shall write down equations defining the phenomenological coefficients for systems with chemical reactions. Contrary to the usual procedure of deriving statistical mechanical equations from phenomenological theory, we adopted the procedure of deriving phenomenological equations from statistical theory. In this way they are consistent with thermodynamics of non-equilibrium systems including chemical reactions. The phenomenological theory correlates the phenomenological coefficients R,, with the gradients of chemical potential of molecule of type with fluxes of all kinds of molecules in the system. Thus (5.1) - V,,P. = c 4, J, ; J, = C,q P From molecular theories of Bearman and co-workers, we have isothermally, c cp~&-ug). P Therefore, for system (l), with no chemical reaction,

(5.2)

c Ra,J,= L11,- c LgJg

(5.3)

c R,*,JB= L$, - c Ca*gJB.

(5.4)

-V,,,u,

Similarly,

= -Ff’)‘=

P P for system (2), we have, inside the membrane,

b P From equations (5.3) and (5.4) one obtains

c R,*,J,= F R,BJp+K,*-Lh -F K.*-L&Jp

‘(5.5)

P Equation (5.5) may be written as

- Vr,d = 1 R,*B Jp = c L,,,J+ f&J, B P L, = Rx,+ imp-C$ Kz =
(5.6) It is interesting to compare equation (5.6) with equation (3.13) of a recent paper by Frisch (1964). Thus, the chemical reactions influence the values of the phenomenological coefficients Las and K, in a rather complicated manner. The chemical reactions inside the membrane alter the concentrations cy and hence, r,* <,* and r,*,. These effects may be quantitatively computed using equations (4.23) and (4.26). The concentration parameters are related to the coefficients <, of equation (3.5).

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If one defines the cross coefficients RaKby the relation

-VA = i R,& + ik RakerWkldO

a = 1,2, . . . r (5.7) 0 it is seen that the cross coefficients Rcrkshould be scalars and independent of R,,. Equations (5.5) and (5.6) show that Rrrkso defined is dependent on R,,. Therefore, the definition of these cross coefficients by the linear relation (5.7) is incorrect, in spite of the fact that equation (5.7) refers to an anisotropic medium in which Curie’s theorem need not apply. 6. Relationship between Metabolic Energy and Diiusion Currents In the preceding section, the effect of chemical reactions on diffusion fluxes was deduced as due to changes in perturbation functions and partial frictional coefficients. In this manner, one may consider that the presence of metabolic reactions alters the magnitude of the phenomenological coefficients from R,, to R,*B.The phenomenological coefficients thus defined are scalars, and are functions of state variables like temperature, pressure and chemical potentials of all the components. They cannot be functions of gradients of these state variables, and hence the definition of vectorial phenomenological coefficients within the spirit of linear phenomenological theory is meaningless. In this section, we shall suggest the manner in which the metabolic energy is related to diffusion flow. The heat of transport of component a, Q: may be defined in two equivalent ways (Helfand & Kirkwood, 1960). Vrp. = - QfV In T (6.1)

d Q.*i,.

q= 1

For system with no chemical reaction, one has the additional ; since Gibbs-Duhem

c,Q.* = 0

restriction that (6.2)

relation

c C,V,P, = 0 (6.3) a should be obeyed. At constant temperature, pressure and composition, the corresponding Gibbs-Duhem relation for the system with chemical reactions (2) does not vanish and may be related to affinities (Frisch, 1964) ; C.V*Pa z 0.

(6.4)

It follows therefore that for system (2),

(6.5)

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related to the energy of the

chemical reactions taking’ place inside the system. Therefore, the heat of transport of component OL,in system (2), Q:*, computed by the prescriptions of Bearman, Kirkwood, Helfand & Vaidhyanathan (AX. cir.) will be different from the same quantity Q.* computed for system (1). The origin of this difference may be traced back to the difference in pair space diffusion currents, j$, in the two systems. j$, will have different values in the two systems if, and only if, perturbation functions #a@ are different in systems (1) and (2). Thus, equation (6.5) supports the formulation that the effect of chemical reactions is to alter the perturbation functions $.s. These in turn determine the magnitude and direction of diffusion currents. Equation (6.1) suggests that this relation between metabolic energy and diffusion flux will be stoichiometric as a first approximation. One may describe the transport of multicomponent fluid mixture across a biological membrane in the following converse manner. Given certain properties of the membrane, constituents of the fluid mixture and concentration difference (gradient) on either side of the membrane for the solute components, under steady state, the pair correlation functions remain time independent. Therefore, if change in magnitude and direction of the flux of one of the specified component results and is experimentally observed, this should involve change in perturbation functions I+%,~which can be caused only by the nonvanishing character of ~c,Q~. resulting from chemical reactions. Thus, experimentally active transport is observed to occur accompanied by energy changes resulting from some metabolic processes. REFERENCES BEARMAN, R. J. & KIRKWOOD, J. G. (1958). J. them. Phys., 28, 136. BEARMAN, R. J. & VAIDHYANATHAN, V. S. (1963). J. them. Phys. 39, 3411. BERNHARD, R. (1964). J. Theoret. Bioi. 7, 532. BRAY, H. G. & WHITE, K. (1957). “Kinetics and Thermodynamics in

Biochemistry”. Amsterdam : North-Holland. FITTS, D. D. (1962). “Nonequilibrium Thermodynamics”, chapters 4 and 11. New York: McGraw-Hill. FRISCH, H. L. (1964). J. cllem. Phys. 41, 3679. HELFAND, E. & KIRKWOOD, J. G. (1960). J. clzem. Pbyx. 32, 857. IRVING, J. H. & KIRKWOOD, J. G. (1950). J. c/rem. Phys. 18, 817. KEDEM, 0. (1960). Proc. Symp. “Membrane Transport and Metabolism”, p. 87. (Kleinzeller & Kotyk, eds). New York: Academic Press. KIRKWOOD, J. G. & CRAWFORD, B. (1952). J. phys. Chem. 56, 1048. VAIDHYANATHAN, V. S. (1965a). J. Theoret. Biol. 8, 344. VAWHYANATHAN, V. S. (19656). J. Theoret. Biol. 9, 478. VAIDHYANATHAN, V. S. (1966). J. Theoret. Biol. (in press).