Equilibrium Inside Small Vesicles

In accordance with the aforesaid, there may arise the problem of the correct calculation of chemical potential difference between inside and outside bulk phases of small vesicles. For macroscopic vesicles this problem is trivial, and can be easily solved using the well-known thermodynamic formula of the Gibbs-Nernst type for transmembrane difference in chemical potentials, Afip = k Tln(Cin/Cout) where and are the concentrations of a component P inside and outside the vesicle, respectively. Under conditions of chemical equilibrium these concentrations can be calculated on the basis of the mass action law. For small enough vesicles, however, the problem of the adequate estimation of mean concentrations of particles inside a vesicle becomes more complicated. There are three reasons why the conventional thermodynamic approach to the calculation of particles concentrations could be misleading  

The conventional approach of equilibrium thermodynamics is applicable only in the case of the so-called thermodynamic limit N/V = const, at F 00, where iV is a number of particles in the volume V, This approach ignores all the factors mentioned above and being applied to small systems may therefore lead to erroneous results. For example, considering within the formalism of the thermodynamic approach the transmembrane transfer of neutral P particles from closed vesicles loaded with the reaction mixture of P and Q particles (P -h Q PQ), we have obtained one paradoxical result for small enough vesicles the transfer of even one P particle along its concentration gradient can be thermodynamically unfavorable under certain conditions 

In our further work performed in collaboration with Grosberg [76] we have considered the same hypothetical model system using the approach of equilibrium statistical mechanics that takes into account the discrete properties of the system. It was demonstrated that the puzzling result obtained in [75] is a consequence of two factors  

A similar problem of the apparent violation of the Second Law of Thermodynamics had faced Westerhoff and his colleagues, who used the nonequilibrium thermodynamics and kinetic approaches to analyze the problems of energy coupling in small vesicles [77-79] and the reaction yield in the so-called channeled systems [80, 81]. 

Model Description. Let us consider, following [75, 76], a simplified model system consisting of identical closed vesicles of inside volume V (Fig. 3.8). The vesicles and their surroundings contain neutral particles P and Q which participate in the reversible dissociation/association reaction, PQ P + Q. For the sake of simplicity we assume that in all other respects the system is ideal. We also suppose that the vesicle s envelope is impenetrable for P, Q, and PQ particles, at least for the time intervals sufficient to reach the dynamic equilibrium in the reaction PQ P -h Q. The state of thermodynamic equilibrium is defined by giving the following parameters the total number of particles P and Q inside the vesicle (parameters P and Q which include the numbers of free as well as of associated, particles in PQ form), a vesicle volume V, and equilibrium constant K. 

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It follows from (3.15) that the Gibbs-Nernst equation for the AP value is exact and holds true regardless of the system s volume V, We have to emphasize, however, that the correct value of the equilibrium free P particles concentration inside the vesicle, =

/K, calculated within the framework of statistical mechanics, in general, may differ from that value formally calculated by the conventional thermodynamic approach, i.e., from the mass action law. This is the case for the reaction mixture of P, Q, and PQ confined within sufficiently small vesicle. 

In model systems for bilayers, one typically considers systems which are composed of one type of phospholipid. In these systems, vesicles very often are observed. The size of vesicles may depend on their preparation history, and can vary from approximately 50 nm (small unilamellar vesicles or SUVs) up to many pm (large unilamellar or LUV). Also one may find multilamellar vesicular structures with more, and often many more than, one bilayer separating the inside from the outside. Indeed, usually it is necessary to follow special recipes to obtain unilamellar vesicles. A systematic way to produce such vesicles is to expose the systems to a series of freeze-thaw cycles [20]. In this process, the vesicles are repeatedly broken into fragments when they are deeply frozen to liquid nitrogen temperatures, but reseal to closed vesicles upon thawing. This procedure helps the equilibration process and, because well-defined vesicles form, it is now believed that such vesicles represent (close to) equilibrium structures. If this is the case then we need to understand the physics of thermodynamically stable vesicles. 

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