Thermodynamics, mechanical and electrical properties of biomembranes

Z theor. Biol. (1989) 136, 295-307

Thermodynamics, Mechanical and Electrical Properties of Biomembranes VALERY V. MALEV AND ANATOLY I. RUSANOV

Institute of Cytology, Academy of Sciences of the USSR, Leningrad •94064, USSR, and Chemical Department, Leningrad State University, Leningrad •99004, USSR (Received 6 November 1987, and accepted in revised form 2 September 1988) A set of thermodynamic fundamental equations has been derived, including chemical potentials of immobile components of a biomembrane. The results obtained are compatible with those known in the theory of shells, but the approach developed is more general. The influence of an outer electric field on the isotropic tension of a biomembrane has been discussed. The thermodynamic relations derived are used for the treatment of intracellular pressure changes recently observed by Terakawa (1985, J. Physiol. 369, 229) in the squid giant axon at potential variations.

Introduction From the physical point of view, biomembranes should be considered as thin solid films possessing " i m m o b i l e " components (e.g. proteins, etc.). The latter form a stable framework which provide membranes with elastic properties. Naturally, this does not exclude the presence in a system of some " m o b i l e " components which may be exchanged between a membrane and the adjacent solutions. Besides, the difference between the mobile and immobile components may disappear if the observation time is long enough. These circu~,astances have not been taken into account in the classical theory of elasticity since it is usually assumed that surface properties of a solid do not contribute essentially in its energy. This assumption becomes incorrect when the thickness of an object is comparable with that of interfacial layers. The principles of the thermodynamics of interfaces were elaborated by Gibbs (1878), but in both Gibbs' p a p e r and in later investigations (Rusanov, 1978b; Evans & Skalak, 1980) surface thermodynamics was less developed for solids than for liquids. Up to the present time the mechanical behavior of biomembranes has been mainly treated from the position of the theory of shells (Evans & Skalak, 1980). The results obtained recently on the thermodynamics of a solid (Rusanov, 1978a; Rusanov, 1986) show that it is possible to rigorously describe the b i o m e m b r a n e mechanics. This is the aim of the paper. We will pay attention mainly to the derivation of fundamental equations for a thin solid film, and will then consider the dependence of its tension on the transmembrane potential difference. We will also apply our results to the treatment of Terakawa's measurements on the intra-axonal pressure changes at potential variations (Terakawa, 1985). In the next paper the theory will be used for the analysis of dynamic strain in a nerve fiber under the action potential (Terakawa, 1985; Levin & Golfand, 1980; Levin et al., 1984; Tasaki et al., 1980; Tasaki & Iwasa, 1982). 295

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Theory F U N D A M E N T A L E Q U A T I O N S O F T H I N S O L I D FILMS

Let a system have the volume V, consisting of two bulk liquid phases, a and fl, and a planar film, f. Directing the z-axis normally to the film surface from phase a to phase fl, we can introduce two arbitrarily dividing surfaces with co-ordinates z~ and z~, respectively, which divide the volume V into the parts V~, Vs and Vs. Let us also suppose that the local pressure tensor/3 is homogeneous at any point inside the volumes V,~ and Vs. In other words, the interracial layers are supposed to be included into the film volume, Vy. Then one can write Gibbs' fundamental equation for the energy of the system under consideration, U, at constant mass of immobile components and in the absence of an external field: dU=TdS-PdV+A(~,:d~Y))+

~. ~ l " ) d M l m),

m = a , fl, f,

(1)

i,m

according to which a change in the free energy of the system, d F = d U - T dS, at constant temperature, T, includes the deformation work, d W = - P d V + A(~ : d~f)), /~f(m) and the chemical term, ~./~im d----i , connected with changes in the mole numbers of mobile components, M~. Here S is entropy, P, pressure, A, the film area, (~/: d~ Cf)) the scalar productf of the stress tensor and the film strain tensor d~ of) (for the definition of ~ see Appendix), /~, the chemical potential of the ith mobile component, and summation is applied to all mobile components of the system. The deformation is supposed to be uniform in the membrane area. The superscript m is introduced to take into account the possible absence of equilibrium between the phases with respect to some mobile components. If we now apply eqn (1) to the phase volumes V~ and V~ (excluding the term A ( ~ : d ~ f ) ) ) and then subtract both the results from eqn (1), we shall obtain the first fundamental eqn for a thin film: d U ~ = T d S ~y) - P d V/+ A( ~ : d~ (f)) + ~ /~r,) dMl,,f),

(2)

i, m

where every quantity with the superscript f is the sum of the corresponding quantities for the film and the adjacent layers of phases a and fl, since the thickness z~ - z~ defined above exceeds the real (physical) thickness of the film, h0. It should be noted here that the energy of a solid film is also dependent on the quantity of immobile components, and, consequently, is not a homogeneous function of the variables given in eqn (2). That is why an integral expression for the energy of a film cannot be obtained directly from eqn (2). Strictly speaking, the problem is reduced to introduction of the chemical potentials for immobile components. This can be done by two equivalent ways. After the paper by Rusanov (1986) we can define the chemical potential of the jth immobile component as a tensor,/2j, with

f Choosing the x- and y-axes parallel to the principal directions for the pressure tensor/3, the product ('~:d~ ej}) can be written as (-~:d# ~y)) = y~ de~ + 3'2 de2, where 7~, Y2 are the principal tensions and e,, e 2 are the principal extension ratios (Evans & Skalak, 1980; Rusanov, 1978a).

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components which are given by the partial derivatives from the energy, U (or U(Y)), on the mass changes of substance j along a chosen direction under the constancy of all other extensive variables. This approach had been used in (Rusanov & Malev, 1988). On the other hand, we can use the procedure applied by Rusanov (1978) for the case of a solid bulk phase. The procedure is based on the local description of interfacial layers in accordance with the main postulate of quasi-thermodynamics (Ono & Kondo, 1960). A reader can find the necessary details of such a procedure in the Appendix. On the way chosen, the following expressions generalizing the Gibbs adsorption equation for a thin film can be derived (3"~-y2) d e ~ = d T 2 + S ~ ) d T - ( z a - z , ~ ) d P +

Y, m,--(mf) dl.~Im)+ ~ mj d/.tj2 ,' i,m

j

( T 2 - Ti) de2 = d3", + stAf) d T - ( za - za ) d P + ~. o7 i-(mf) d/3.~rn)-]- ~. r~ dp.jl, ' i. rn j

(3)

or in the other form: d T = % d(el - e2) + (z, - z~) d P - S ~ ) d T -

Y~ i.m

tti--(rnf)i

ill'6"4"tm)i

- ½ E mj d ( / / . ~ 2 - bl,~l),

(4)

J

dT.=½gm~d(~.'.-~j,)-T.d(e,+e~), J

m=~,fl.

In these eqns S ~ I ) = s t f ) / A , m~ = M J A , ,,,-~"Y)=M~"Y)/A,, T=(T~+3,2)/2 is the isotropic tension of the film, % = ( Y l - 3'2)/2 is the shear tension, h4j is the mole number of immobile component j, and /z~, /zj2 are certain mean values of its chemical potential for the principal directions 1 and 2, respectively (see Appendix). Since the values of U ~y), Sty), P,/zl ") do not depend on the direction chosen, there is also invariance of the sum 3'~ + ~ j m j / ~ , where v = 1, 2, so that the shear tension, %, is given by % =½~ m~(/*~2-/z~,),

(5)

J

as one can see from eqn (A2.3) and in accordance with the second eqn (4). RELATIONS A M O N G TENSIONS A N D C H E M I C A L POTENTIALS

Considering now the free energy of the system, F = U - TS, as a function of e~ and e2 at constant T, V and mass of mobile components, we have /din a~ [olna~ (03"2~ _ ( 0 3 " ~ "Yl ~ - ~ e 2 J e , -- T 2~ - ~ e l ) e2 = k cge l / e2 \ Oe 2/ , , '

(6)

according to eqn (1). Using the equalities d A / a e l = aA/ae2 = A [see eqn (A1.4)] and the tensions, T and %, the last equation is reduced to __

2ys L" a~ _l,~-L Tee2 J,,

(7)

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The partial derivatives included in eqn (7) can be expressed with the help o f eqn

(4)

as

(o.,,) O~ek.,:(-1 (0%~

\ O e j ~t

l

~ =L Oek J.,~ -Y.+,~m.s[o('.
P, T,/~I '~) = Constant; k, 1 = 1, 2; k # l.

(8)

The substitution of these relations into eqn (7) leads to the expression for %:

r:0 :2

Y~=½~miL\ael/e~

\ae2/d

],

P, T,/~}m)=Constant,

(9)

which is additional to eqn (5). Equations (5), (8) and (9) relate the tensions y and % to the chemical potentials /~jl and/~s2,' and the extension ratios, el and e2. E X P R E S S I O N S FOR THE E L A S T I C I T Y A N D S H E A R M O D U L E S

One can also derive analogous relations for the modules of the theory of elasticity. For the sake of brevity, we restrict ourselves only to the case o f isotropic (in the z = Constant cross-section) films. Let us represent the tangential potentials, #zjl arid #~j2, in the form of the Taylor double expansions in powers of el - eo, and e 2 - eo: tzj,( e, , e2) = #~ji(eo) + b{( e , - eo) + b~2(ek -- eo) + a{( e, -

eo) 2

+a~2(et-eo)(ek--eo)+aS3(ek--eo)2+...,

k,l=l,2;k#l,

(10)

where a~, ~ are constant coefficients at P, ~, "" #zi(') - constant and eo is the equilibrium extension existing for a film with free borders. It is obvious that #zjl(eo) = ~jk(eo) = #~s(eo) because the directions 1 and 2 are equivalent at the initial equilibrium state of the film. Returning the second order terms in eqn (10) and inserting them into eqns (5) and (9), we obtain the following results: % = ½(el - e2) ~'. m j [ b ~ - b{ + (el + e2 - 2eo)(a~ - a~)], J

% = ½(el - e2) E ms(2aJ3 - a~).

(11)

J

From a comparison of these equations we find the constraints a~=a {

and

aY2=2a~-(bS2-bO.

(12)

Therefore the isotropic tension and the shear one are given by y = -½(e~ + e2 - 2Co) E m°Cb~ + b{ + (el + e2 -2eo)(2a~ - b~)], i %=½(el_e2) E mO[l_(el+e2_2eo)](bJ_b{), msO= mff Ao, i

(13)

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with the accuracy to the third order terms of the strains. By comparing these expressions with the corresponding results o f the linear theory of shells (Evans & Skalak, 1980):

y=K(el+e2-2eo),

7s = ] z ( e l - e2)

(14)

we obtain the following definitions of the isothermic area modulus of compressibility, K, K = - ½ E mj(b2+b{) o j (15) J

and the shear modulus,/z,

= ½y m jo( jb 2 - b{)

(16)

J

RELATIONS

AMONG

TENSIONS

AND

ELECTRIC

POTENTIAL

The above equations relating tensions and modules to chemical potentials are very convenient for introducing the electric potential, which is related to the chemical potential in a trivial way. Therefore we are in a position to complement the consideration performed with the analysis of the influence of the transmembrane potential on the isotropic tension. If we set d

(m)

I~i

f0,

m = t~

=).zl~)FdC},

m=/3,

where ~ is the electric potential difference between the bulk phases/3 and a; zla)F the molar charge of the ith mobile component in phase/3 ( F is the Faraday number), it follows from eqns (4) at el, e2, P, T = constant,

7 ( * ) -- 7 ( ' o ) - F E

L

zl°), l

dO

o

-½ E mj [/z~-t(~) + p~;2(~) - Jz;,(dPo) -/z~-2(qbo)],

(17)

J

where r~l ay) is the Gibbs excess of a charged mobile component on the/3-side of the film, so that the quantity SO*oQ dO = F Y., S**ozl ~). rnl ay) dqb is the electric work per unit film area that accompanies the potential variation from the initial value, dPo, to the resultant one, qb. At the same time, we have, with the accuracy to the second order of e o - e o°, 7(qb) - 7(qbo) = - 2 K ( e o - eo°); = t 0 = tzjI (eo) - lzj(eo) - (b~+ bO(eo- eo°),

(18)

where eo° is the equilibrium extension of a film with free borders at t} = Do; coefficients, b~ and b j, being considered as independent of the D-potential. In the same approximation we have, from eqn (10), /~.(eo) - tzj(' eo) ° = (b J+ bJ)( eo - e°),

(19)

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if changes in p.~ are referred to the transition from the initial extension, e~, to the resultant one, Co. This means that chemical potentials ~ ) a n d / ~ 2 practically retain their values during the potential variation from Oo to • [see eqn (19) and the second eqn (18)]. Thus we finally get

3'(o) = 3'(o0)- F

zl ), l i

dO.

(20)

o

Since the difference eo-e ° is usually o f a small value, we retained only the first order terms with respect to this difference, so that eqn (20) is approximate. A more general consideration had been performed in Rusanov & Malev (1988).

Comparison with Terakawa's Experiment As was shown by Terakawa (1985), the intracellular pressure of the squid giant axon, Ap, depends on the transmembrane potential difference, O. Terakawa connected this phenomenon to the influence of the electric potential on the membrane tension and established the empiric formula: AP(O) = --9.15.10 -1 . [(O - 0 . 0 6 1 ) 2 - 0 . 1 0 9 2 ]

(21)

where Ap and • are expressed in newtons/m 2 and volts, respectivelyt. The perfused axon, investigated by Terakawa (1985), can be represented as a thin cylindric tube which is filled up with a perfusion solution, and surrounded by an outer solution (sea water). The essential difference between such a system and that discussed above consists of the presence of the axon curvature, I/R, where R is the axon radius. This circumstance can be easily taken into account in the approach developed. In particular, providing the axon radius R is much larger than the thickness of axon's shell (i.e., the fiber membrane plus the Schwann cell layer), one can show the following expressions to be valid: 3"~=0,

AP=

T2/R.

(22)

In these equations the bending moment effects are neglected and Y~, 72 are the above principal tensions. The validity of eqns (22) is obvious because they are just the conditions o f force equilibria on a cylindric shell with free borders (Evans & Skalak, 1980). In accordance with the experimental results, we may consider the fiber strains appearing at potential variations to be sufficiently small. Therefore, we can use the above equations for 3' and 3's to give

k(e,+e2-2e~)+~(el-e2)-F~

•

I; .(o~_-(os~.~,~_ /'i

lrl i

u'.x." - - 0

o

k(e,+e2_2eO)_tz(e _e2)_F~.

(23)

~,'~°)'z"~°n'~'~-,~...,i RoAP o

1"The dependence was represented by Terakawa in the normalized form, A P ( d P ) / A P ( O . I ) , where AP(0-1) is the intracellular pressure that corresponds to the depolarizing voltage, 0-1 V. As follows from the reference, this factor is about 10 mPa, which is taken into account in eqn (21).

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Here we have not taken into account the pressure influence, AP, on the principal tensions, since the corresponding effects are negligible due to the smallness of A p < 100 mPa. Deformation of an axon is considered to start from a resting state so that ~o is the resting potential, R0 is the resting radius and, consequently, eo° = 1 is the equilibrium extension ratio at q~ = ~o- Extension ratios el and e2 are not independent in the case under consideration, and are related to each other by the equation el - e ° = -2(e2 - e°),

(24)

that is the condition of the axon volume constancy, which is practically fulfilled in Terakawa's measurements (see Appendix). Inserting eqn (24) into eqns (23) yields the following results: AP=

R o ( 61.*F K+3p.) '

fo,

~i ,,,i

dqb,

0

(25)

e 2 - e °~- ( R - Ro)/ Ro = RoAP/61z. Thus it remains to find the charge, Q = F ~.i z~)m~ zf), as a function of the potential difference, q~, in order to get the results in an explicit form. The equations for an equilibrium double layer should be applied for this purpose. Bearing in mind the well-known complexity of the double layer structure on biomembranes (Pastushenko & Donat, 1976; Ohki, 1978) we will, nevertheless, use the simplest G o u y - C h a p m e n ' s model for such a layer. Moreover, for the sake of simplicity, we shall not take into consideration the possible specific adsorption of the mobile components at the membrane surfaces. Under these restrictions one can derive, by solving Poisson's equation for the potential distribution in aqueous solutions, the following expressions for the membrane surface potentials, q),~ and q~: ~

2RT 1-A,~ = - ' - F -In I + A ~ '

¢p~ =

q~+2RT l 1-A~ T n I+A~

(26)

where the integration constants, A~ and As, are determined by the equations: A ~ , / 1 - A 2=

¢rF r E m

qo

]

.

(27) AI3/1 - Ag - eKdR-----T

(¢Pa - d~, ) - qo ,

reflecting the discontinuity of the electrical polarization vector (Delahay, 1965) at the membrane surfaces. Here Kd = [81rF%o/ekT] ~/2 is the inverse of the Debye length, e and em are the dielectric permeabilities of an aqueous solution and the membrane, respectively, Co is a 1:1 electrolyte concentration (mainly NaCI in the outer solution and KCI in the axoplasma), h0 is the membrane thickness, and qk is the charge density on the kth membrane surface (k = a,/3). After substituting eqns (26) into eqns (27) the latter can be linearized with respect to their deviations from some values A ° and A~. It is convenient to determine these values from the condition

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of the absence of the intramembrane electric field ~ the relation

AOk_ 0

(02+q2k)'/2,

qk

=~

0 =eRTKd

qk

= ~ o , so that A ° obeys

k = a , fl

(28)

2~'F

and the corresponding transmembrane potential difference qb = ~(o) is ~(o) = 2RTIn (1 + A~)(1 - A °) F (1 - A ~ ) ( I + A ° ) "

(29)

The solving of the linearized eqns (27) leads to the expression • ~-~

1-

=

1/ / , ~ ,~o~ 2era(,~ _ AO2AO~ '~ ~J o2 2 o2 ~ ' ~ - ' ~ s eKahoD(1 - A,~ ) ( 1 - AO ) J

(30)

where

[2Em(1 -A~Ao)+(I+A°~2)(I+A°~ °2 °2 ]/ D=ceKdho 2)

[(1-a°2)(1-a~2)] 2.

The accuracy of this expression is better, the high the ionic strength (i.e. Co) of aqueous solutions, or, alternatively, the higher the charge density, qk. For example, under physiological conditions (Co-~0.4 M), or Iqkl ~ 103-104 electrostatic units of charge (e.s.u.), eqn (30) is valid up to I ~ - ~ 1 ~ (5-7). R T / F . Since the charge Q = F ~i zlt3)m~t3r) multiplied by 4zr/e coincides with the electric field at the contact of phase /3 (axoplasma) with the membrane, we derive after simple transformations em Q=~ho

02

1

02

2e~ (1-A~A~) ~ ~ _ ~o)). KdehoD [(l_AO:)( I _ A ~ ) ] 2 J ( ~

(31)

Therefore eqns (20) and (25) take the forms:

v(o) = ~,(Oo) - ~ [ ( o - o~°~): - (COo-~o~):], AP(O) --

3/.tcm r(qb or°))2- ( ~ o - or°))2], Ro(K+3p.) L. .

(32)

(33)

where c,. = e,./4~rho is the specific capacitance of the membrane, and the terms proportional to the usually small (for biological systems) quantity e,./eKah o are omitted. Now we see that the theoretical dependence A p ( ~ ) is of the same form as was established by Terakawa. By comparing eqns (33) and (21), we find oto) = 2RTIn (1 + A~)(1 - A °) = 0.061 V, F ( 1 - A ~ ) ( I + A °)

(34)

which shows the existence of the inequalities qu < q~ < 0. Besides, it is a convincing fact that the charge densities, qk, of the squid giant axon are high enough

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( > 103 e.s.u./cm2). Unfortunately, the estimation of these parameters is now impossible since there is only one equation [eqn (34)] for two magnitudes. But the principal possibility of such an estimation would a p p e a r if dynamic strains (Terakawa, 1985; Levin et al., 1984; Tasaki & Iwasa, 1982) of the same axons would be treated with an adequate method. Using the values Cm ~-- I I ~ F and R0 ~- 3 . 1 0 -2 cm (Terakawa, 1985), we can also find, from eqns (33) and (21), that the modulus K o f a perfused squid axon is much higher than the shear modulus K -~ 102 ./~.

(35)

A similar relation takes a place for red blood cells (Evans & Skalak, 1980). Therefore, one can think about the mechanical similarity of the shells of the objects compared. The data on the radial extension ratio, e2-e °, are absent for the clamp conditions from Terakawa (1985), so we can not apply the second relation eqn (25) in order to calculate exactly the modulus/~. The u p p e r limit is estimated in the Appendix. Conclusions In this paper, we have obtained thermodynamic relationships that provide a general a p p r o a c h to the description of thin films: biomembranes, in particular. As has been shown, the results obtained are compatible with those of the theory of isotropic shells, but if the latter were derived under the constant mass of immobile components, the above approach is valid in the more general case of constant chemical (or electrochemical) potentials of mobile components. Nevertheless, we consider our results only as an initial step in the study of biomembranes. A word should be said about the potential dependence of the m e m b r a n e tension, y(qb). Obviously, liquid films (e.g. lipid bilayers) may be considered as similar to solid ones provided the rate of the transmembrane potential change is high enough. In such conditions, equilibrium between a lipid bilayer and the adjacent G i b b s Plateau border seems to be absent, and lipid particles become equivalent to immobile components of a solid film. Since the surface tension, o-, of a lipid bilayer depends on the transmembrane potential, ~ , in the same manner as in eqn (28) (Rusanov, 1972), there will be no change in a value of cr after a potential drop if the m e m b r a n e electrocompressibility is neglected. This is due to the constancy of the chemical potentials of lipids, in both cases. For lipid bilayers there is also a possibility for the analysis of capacitance changes at potential variations. However, this is beyond the scope of the paper. REFERENCES DELAHEY, P. (1965). Double Layer and Electrode Kinetics. New York; London; Sydney: Wiley. EVANS, E. A. & SKALAK, R. (1980). Mechanics and Thermodynamics of Biomembranes. Boca Raton,

Florida: CRC Press, Inc. Glaas, J. W. (1878). Trans. Conn. Acad. Arts Sci. 3, 343. LEVIN, S. V. ,g" GOLFAND, K. A. (1980). Tzitologia, 22, 717 (in Russian). LEVlN, S. V., MALEV,V. V., GOLFAND, K. A. & TROSH]N,A. S. (1984). Dokl. Akad. Nauk SSSR, 275, 1246 (in Russian).

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OHKI, S. (1978). Bioelectrochent Bioenerg. 5, 204. ONO, S. & KONDO, S. (1960). In: Handbuch der Physik, Band X. (Fliigge S. ed.) Berlin; Grttingen; Heidelberg: Springer-Verlag. PASTUSHENKO,V. F. & DONATH, E. (1976). $tudia biophysica, B.56, 7. RUSANOV, A. I. (1972). DokL Akad. Nauk SSSR 203, 387 (in Russian). RusANov, A. I. (1978a). Z Coil, Interface Sci. 63, 330. RUSANOV,A. I. (1978b). Phasengleichgewichte und Grenzlachenerscheinungen. Berlin: Akademie-Verlag. RUSANOV, A. I. (1986). In: Adhesion of Meltings and Soldering of Materials. 17, p. 3. Kiev: Naukowa dumka (in Russian). RUSANOV, A. 1. & MALEV,V. V. (1988). DokLL Akad. Nauk SSSR 298, 407 (in Russian). TASAKt, I., IWASA,K. & GmaoNs, R. C. (1980), Jpn. Z PhysioL 30, 897. TASAKt, I. & IWASA,K. (1982). Jpn. J. Physiol. 32, 69. TERAKAWA,S. (1985). J. Physiol. 369, 229.

APPENDIX 1. Definition of the Stress Tensor For the state of mechanical equilibrium, the local pressure tensor o f the system, /3(x, y, z), must have uniform values,/3(~) and/3(a) for the bulk phases, and is only a function of the z-co-ordinate inside the film and the adjacent surface layers, according to the symmetry conditions. I f iso-osmolarity and isotropy of phases a a n d / 3 are suggested additionally, we shall have/3(°) =/3(~) = P~ where P is hydrostatic pressure and I, the unit tensor (its components are given by Kroneker's symbol 8ik). The general expression for the elementary work o f deformation is

(AI.I) (v)

(v)

where 8~ is the strain tensor and the integrand is written as the scalar product of tensors /3 and 8~. Let the co-ordinates of the dividing surfaces, z~ and za, be so determined that the equality P i - / 3 = 0 is fulfilled at any point inside the volumes V~ and Va. Then one may define the two-dimensional tensor .~ =

( p ~ _ / 3 ) dz

(A1.2)

fz a'~

and rewrite eqn (AI.1) as 8 W = - P d V + A ( ~ : d~(f)),

(A1.3)

where #(f) is also a two-dimensional tensor obeying the condition (f) + 8eyy (f) ) A = 8 A , (Se~x

(A1.4)

and A is the film area. It should be emphasized that the tensor of film tension, 9, does not depend on values z~ and z a if the condition P I - fi = 0 is valid inside the volumes V~ and V~ up to the co-ordinates z, and z~, respectively.

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2. Generalizations of the Gibbs Adsorption Equation for the Case of a Solid Film

According to the p a p e r by Rusanov (1978a), for every cross-section given by vector if, the film under consideration is imagined to be in contact with a hypothetical solid phase with respect to which all components of the film (including immobile ones) are mobile. The hypothetical phase may be a liquid for the principal directions in the film corresponding the main pressure tensor values. Considering an equilibrium local phase transition between the film and the hypothetical phase, the following expression for the local density of the film energy, ~t/)(z), can be derived from Gibbs' equilibrium principle: fi(f)(z) = T g ( f ) ( z ) - P . . +

Y'. tz'/")plmf'(z)+Y, txj.(z)pj(z), i, rn

(A2.1)

j

where v = x, y; g(f~(z) is the local density of entropy; P.~, a diagonal pressure tensor component in the film; p~i"f)(z) and pj(z) are the concentrations of the ith mobile and t h e j t h immobile component, respectively; izj~(z) is the usual chemical potential of the j t h component in a hypothetical equilibrium phase corresponding to the ~, cross-section of the film, which is treated as the local chemical potential of the j t h immobile component of the film along the v-direction. If we multiply eqn (A2.1) by the film area A and then integrate over z from z. up to zt3, the expression will be derived

U (f~ = TS ~f) - A

1:',,,, d z + ~ tz c,,)i ~vl"'~"¢)i :o

+ A ~j

i, rn

I:°.

tzj,,(z)pj(z) dz.

(A2.2)

The last term on the right may be arranged as follows

A

~i,,(z)pj(z) dz =/zj,,A

pj(z) dz = I~j,,(z)Mj

7.a

2~,

where Mj is the mole number of an immobile component j and ~z.},. is a certain mean value of its chemical potential Izj,,(z). Using also the tensions % = I~° (P,,,,- P) dz, we can rewrite eqn (A2.2) in the form

u ( j ~ _- T S ( J ) _ P V r + A y , , + ~ txit,,,)..(.,f)-r. , lvli 1- L tzj.Mj. i,m

(A2.3)

j

Equation (A2.3) plays the role of the second fundamental equation for a thin film. By differentiating eqn (A2.3), at constant mass of immobile components and also at constant values of z. and zt~, and comparing with eqn (2), we obtain one more equation: d y . = ('~ - y J):d~cr)+(zr~ - z,~) d P - S~ r) d T -

~ ,,,~l"'f) d / z ~ " ) - ~ rnj dlzj,.; i, m

~, = x, y

j

(A2.4)

where ~,¢(Y)=S(Y)/A, m(/'¢~-- ...~tt4~''" ,/A, rn/ = M j / A . Choosing the x- and y-axes parallel to the principal directions (1 and 2) for the pressure tensor, eqns (A2.4) can be written in the form of eqns (3).

306

v.v.

MALEV

AND

A.

I.

RUSANOV

3. Incompressibility of an Axon in Terakawa's Experiment

In Terakawa's experiments, the intracellular by measuring the meniscus curvature, 1/r, at the put in an axon. Since there was neither ejection the relative change of the axon volume, 6V/Vo,

pressure responses were recorded end of a glass capillary which was nor suction at pressure variations, should be approximately

4 - 8Vl vo-

<,,3.,>

4LoR~ L r

where 1~to is the initial curvature of the meniscus; po-~200 i~m, the inner radius of the capillary; R o = 3 0 0 Ixm, and Lo = 1 cm, the radius of the axon and its initial length, respectively (Terakawa, 1985). The difference 1/r-1~to can be expressed through the pressure change, AP, according to the Laplace eqn

A,,==<,.p-'_!l, L r roJ so that eqn (A3.1) takes the form - ~v/Vo

/A3.2)

= (P°~'R~P/8Lo~

\ Ro/

where t r = 7 2 - 8 m N / m is the surface tension of water. Since there may exist water filtration through the fiber membrane, as the result of the pressure drop across the axolemma, Ap, it could be supposed that the relation derived was incorrect. But the relative volume change due to water filtration, A V/Vo, is equal to

2PaV,, A V~ Vo- Ro~ 7"

AP(t) dt=2PaV~,APzo/RoNT

where Pa = 10-3 c m / s e c is the water permeability of the fiber membrane; vw = t8 cm3/mol, the partial molar volume of water; r0 = 10msec, the duration of Terakawa's measurements. Therefore, it is possible to compare this change with the previous one given by eqn (A3.2). As can be seen, their ratio, AV/~3V~-I6Pav,,Loroo-/R3~T, is a very small value so that the water filtration effect should be negligible at the conditions discussed. Continuing our estimations, we also use the relationship

6V/ Vo ~- 6L/ Lo+ 26R/ Ro= e, - e° + 2(ez - e °) provided the fiber strains are sufficiently small. This eqn and eqn (A3.2) permit us to relate the extension ratios e t and e2 to each other. As a general result, we have

t'

zi<,,,.<,,j., m, d~/Ro(K+31~) 1+ "

o

\Roi

1

e 2 - e ° - R ° A P [1 + ( 0 0 ) 4 Ro/z 6#z " f R o / 4---~o~.J"

o.K

l

2(K+~)Lo~d

(A3.3)

BIOMEMBRANE

307

THERMODYNAMICS

By comparing eqns (A3.3) with those derived previously [eqns (25)], we see that the results compared coincide if the inequality

RolzK <(Po~ 4 Roll ~-4 10-Sp.<< 1, Ro/ 2 ( K + 3 / z ) L o t 7 \Ro/ 2Loot "

(P0~ 4

(A3.4)

is fulfilled. From the first eqn (A3.3) we find the upper limit for/z-values: Ro 4 4Loo.

<(R°142L°(K+3tz) (-~o) P" \ Po/ RoK <

Ro ~ '

.

104dynes/cm'

At this limiting value of p. the pressure change Ap--~ 10 mPa should correspond to an increase in the fiber radius, 6R = 6 . 1 0 - 4 nm, as follows from the second eqn (A3.3). This value, being maximal at excitation, is three orders smaller than that measured by Terakawa under these conditions. So we have to think the shear modulus /~ of perfused squid axons does not exceed 50 dynes/cm. Such values of /z obey the inequality eqn (A3.4). Therefore, the axon in Terakawa's measurements may be considered as an incompressible body. But there is also another question to be answered. We have supposed that the situation of an axon with free borders (two or one) was realized in the experiment. However, the author wrote nothing about it. One can easily investigate the opposite case with fixed borders (i.e. e~-eo° = 0) and show that such a situation is incompatible with the experimental results. One has also to take into account the fact that the pressure changes measured, APexp, were essentially smaller than the maximal possible ones, APm,x, in this case

I Io F

laP¢~i<< laPm~xl= -Ro

_(~)

~'

.z."(~r~ ddp

i)

"

I

as has also been pointed out by Terakawa. Hence, the ratings performed above justify the validity of eqns (22) and (25).