J . Phys. Chem. 1987, 91, 5103-5106 hydroxyl groups of the amphotericin molecule, or, more likely a distortion of the bilayer geometry. 4. Cyclosporin AIDPPC Multilayers. Cyclosporin A exhibits only a small effect on the C=O stretching mode spectra of the lipid bilayer. Incorporation of CSA into the bilayer has a relatively small effect, compared to cholesterol, on the line widths of the C=O modes, and no effect on the ability to detect the transition by measuring these line widths. This lack of a significant spectral effect parallels the absence of major perturbation to either the width or the temperature of the phase transition in DPPC/ cyclosporin A multilayers.20 In addition, this observation indicates that the effect of bilayer components such as cyclosporin A and cholesterol on the interface region is not simply that of an inert “spacer” molecule which separates lipid molecules and inhibits carbonyl-carbonyl interactions, but is intimately related to the effect of the added component on the packing interactions of the lipid bilayer as a whole. A simple spacer effect would presumably be exerted by cyclosporin A as well as cholesterol; the dramatically different effects on the geometry of the bilayer as a whole are reflected by the differences in calorimetric behavior which parallel the spectral differences in the C=O stretching mode spectra.
Summary Using infrared spectroscopy of 2- [ 1-I3C]DPPC liposomes we have investigated the effects of hydration, cholesterol, amphotericin B, and cyclosporin A on the structure of the bilayer interface region. Our results suggest that hydration causes a significant change in the conformation of the interface region of one of the crystal forms of anhydrous DPPC. Once hydrated, the conformation of the interface region of the bilayer is similar in lipids having the acyl chains arranged in either a quasi-hexagonal or an orthorhombic geometry. The frequencies of sn-1 and sn-2 chain C-0 stretching modes change slightly upon melting or upon addition of cholesterol, suggesting that these frequencies are
5103
sensitive to the conformation of the bilayer as a whole. The change in frequency is too small, however, to be accounted for by hydrogen bonding between the cholesterol hydroxyl moiety and either acyl chain C=O group. Whereas at 50 mol % cholesterol both sn-1 and sn-2 chain C=O bandwidths are significantly increased over those seen for the pure lipid, at 20 mol % cholesterol the effect on the sn-2 chain is relatively greater than the effect on the sn-1 chain feature. This suggests that cholesterol has a greater effect in disordering the sn-2 chain interface region at this concentration. Addition of amphotericin B to the cholesterol/DPPC bilayer causes a dramatic increase in the widths of the C=O stretching mode features and an even more dramatic splitting of the sn-1 chain C=O mode feature than that seen with cholesterol alone. These changes indicate that the formation of the amphotericin/cholesterol/DPPC transmembrane complex affects the charge distribution of the region surrounding the lipid carbonyl group more that does cholesterol alone. This effect on the dielectric constant may result from changes in bilayer geometry or water penetration, or from a direct effect of amphotericin B hydroxyl groups. In contrast to cholesterol or cholesterol/amphotericin complexes, cyclosporin A has little effect on the C=O stretching mode spectra. This finding, together with those above, indicates that the addition of bilayer components known to penetrate the lipid matrix alters the acyl chain C = O stretching mode spectra through their effects on packing interactions within the entire bilayer, rather than by simply separating the interface regions of adjacent lipid molecules.
Acknowledgment. We thank Dr. Richard Dluhy for sharing with us prepublication infrared spectra of isotopically substituted DMPC and DMPC/cholesterol bilayers. Registry No. 2-[ 1-’3C]DPPC, 109614-02-0;DPPC,2644-64-6; cholesterol, 57-88-5; amphotericin B, 1397-89-3;cyclosporin A, 59865-13-3.
The Kinetic Mass Action Law Revisited by Thermodynamics G . Lebon,* Institute of Physics, Li2ge University, B 4000 Lidge, Belgium
M. Torrisi, and A. Valenti Dipartimento di Matematica, Universita di Catania, Catania. Italy (Received: December 6, 1986)
A simple thermodynamical model is proposed in order to derive the nonlinear kinetic mass action law. The model makes use of the same set of variables as in the classical theory of nonequilibrium thermodynamics but follows closely the guidelines provided by extended irreversible thermodynamics. It is shown that the mass action law is recovered without appealing to ad hoc hypotheses; only the classical definitions and expressions for the chemical potential and the affinity are utilized.
1. Introduction Recently, there have been several attempts to interpret the laws of kinematic chemistry in terms of nonequilibrium thermodynamics. The classical theory of irreversible processes as proposed by Onsager,’ Prigogine? Haase? De Groot and Mazur4 has known the widest audience. Nevertheless, the results of the classical formalism concerning chemistry have recently been criticized by Garcia-Colin et al.5-7311who revisited kinetics chemistry in the framework of the so-called extended irreversible thermodynamics
(EIT) .8-11 *Also at the Department of Thermodynamics, Louvain University, B 1348 Louvain-la-Neuve, Belgium, and author to whom correspondence should be
addressed.
0022-365418712091-5 103$01.50/0
In this article, we adopt a position intermediate between the classical theory and EIT. We use the same variables as in ordinary (1) Onsager, L. Phys. Reu. 1931, 37. 405. (2) Prigogine, I. Introduction to Thermodynamics of Irreversible Processes; Interscience: New York, 1961; Chapters 3-5. (3) Haase, R. Thermodynamics of Irreversible Processes; Addison-Wesley: Reading, 1969; Chapter 2. (4) De Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; Dover: New York, 1985; Chapter 10. ( 5 ) Garcia-Colin, L.; de la Selva, S. Non-Equilib. Thermodyn. 1983, 8, 277. (6) Garcia-Colin, L.; de la Selva, S.;Pina, E . J. Phys. Chem. 1986, 90, 953. (7) De la Selva, M.; Garcia-Colin, L. J . Chem. Phys. 1986, 85, 2140. (8) Muller, I. Z Phys. 1967, 198, 329. (9) Lebon, G. Bull. CI.Sci., Acad. R . Belg. 1978, 64, 456.
0 1987 American Chemical Society
5104
The Journal of Physical Chemistry, Vol. 91, No. 19, 1987
thermodynamics, but we follow closely the formalism of EIT. To fix the notation, let us consider a single chemical reaction VI'XI
i
+ v*/x*+ ... v;
Xk q
Vk+'rXk+'
k
+ ... V*/XN (1.1)
between the species X I , X,, ... X,; vir, Q / , ... 9' are the corresponding stoichiometric coefficients and k and k the forward and backward rate constants. Since we work with mass fractions, we introduce the modified stoichiometric coefficient^:^ (1.2)
v, = MeV,/
where M, is the molecular weight; moreover, it is admitted that the coefficients are counted positive when referring to products of the reaction and negative when related to reactants. With this convention in mind, the total mass balance equation associated with the chemical reaction reads as 4
xu, = 0
(1.3)
,=I
The rate of production of species a , in the absence of transport processes (homogeneous mixtures), is given by the law of definite proportions
e,
= Val
(1.4)
wherein an overhead dot means derivation with respect to the time, ,$ is the degree of advancement of the reaction, and c, is the mass fraction of species a. It is also convenient to introduce the affinity A defined by h
A = -CV,C(,
(1.5)
a= I
with at equilibrium From now on, superscript e denotes a quantity evaluated at equilibrium; the quantity p, is the chemical potential of constituent a and is classically defined as (1.6) = U, - Ts, + PV, where un, s,, and u, represent respectively the specific internal energy, the specific entropy, and the specific volume of species a and T and p are the temperature and the pressure of the system. The chemical potential is related to the total specific Gibbs free energy g through n g= (1.7) pa
cc,cL,
,=l
After these general considerations, we review briefly the essential contents of the classical theory of irreversible processes.'-4 The main assumption is the local equilibrium hypothesis stating that the local state of the system is defined by the same set of variables as in equilibrium; for a chemical reaction, the basic variables are the absolute temperature, T, the pressure p , and the N mass fractions c,, c2, ... cN. The thermodynamic potential associated to this set of variables is the Gibbs free energy, and the corresponding Gibbs equation is written as
+ p-'
dp
os = p T I A &2 0
+
x p , dc, ,=I
(1.8)
designates the total density. With (1.4) and (1.6), relation I .8 can be cast into the form
p
dg = -S d T + p-' d p - A d,$
(1.9) The standard procedure of classical nonequilibrium thermodynamics allows calculation of the rate of entropy production d produced per unit volume by the chemical reaction; this quantity is given by ~~~
(10) Lebon, G , Jou, D , Casas-Vazquez, J J Phys A Math Gen 1980, 1 3 , 215 ( I I ) Casas-Vazquez, J , Jou, D , Lebon, G Lecr Notes Phys 1984, 199, 32, 12, 1 4 1
(1.10)
According to the postulated extension of the second law of thermodynamics, d is a positive definite quantity. The bilinear expression (1 .IO) suggests a derivation of a phenomenological law between the "thermodynamic flux" & and the "thermodynamic force" p T 1 A:
& = IpT'A
(1.11)
the phenomenological coefficient 1 is a positive quantity as directly checked by substitution of (1.11) in (1.10). These results appeal to two comments still formulated among others by Garcia-Colin et a1.5-73" Obviously, expressions 1.8 and 1.9 are not consistent since in equilibrium state, or for a reversible process, A = 0 and the term A d l should be absent from eq 1.9. Despite this observation, eq 1.9 is generally used as the standard form of the local equilibrium hypothesis. The second criticism addressed against the classical theory is the linearity between & and A. Such a result is indeed contradicted by chemical kinetics, which predicts a nonlinear law of the form
& = i c , c , ... [ I
-~X~(-A/RT)~
(1.12)
wherein R designates the gas constant. Clearly, it is only in the close vicinity of equilibrium for which A
