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A theory of ion binding and phase equilibria in charged lipid membranes. I. Proton binding
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31.see also H. L. Friedman, Ionic Solution Theory (Interscience, New York, 1962)] provides a justification for a description of solutions in which only the solute particles are explicitly considered. The interionic interactions that appear in such a description are actually interionic potentials of mean force at infinite dilution in solvent, and the free energy that is obtained from the logarithm of the partition function is related to an excess free energy relative to infinite dilution at constant chemical potential of solvent. Very precise calculations using McMillan‐Mayer theory involve corrections that take into account the variation in solvent chemical potential with solute concentration at constant pressure. These corrections can be important for concentrated solutions, but we shall neglect them for the low concentrations of interest in this paper. We also make the usual assumption that, except at very short distances, the interionic potential of mean force is the Coulomb potential divided by the dielectric constant of pure solvent.
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33.The potential (or for that matter ) is a function of T and the five extensive variables M, T, A, V, and θ, in addition to γ. Since the potential is an intensive property, it can also be written as a function of T and four intensive variables that are appropriate combinations of M, T, A, V, and θ. We take these four variables to be σ, α, c, and In the limit where there is a large excess of solution, we expect that the variable should be unimportant. The potential is therefore a function of σ, α, c, T, and γ.
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42.Since the charges on the surface of a membrane are localized on particular sites and since Gouy‐Chapman theory assumes that the surface charge is continuously distributed on the surface, it is necessary to consider carefully the way in which the Gouy‐Chapman theory is used in theories of membranes. In the present theory, the membrane electrostatic potential of interest is the potential at the site of a lipid with partial charge caused by all the other charges in the system. To assume that this is the Gouy‐Chapman would be equivalent to assuming that the discrete nature of the partial charge has no effect on the value of Such an assumption is especially inaccurate for small surface charge densities; moreover, it is not necessary to make such an assumption in order to obtain simple results. Instead we have assumed that is equal to i.e., at an uncharged site on the membrane surface the potential is the same as that calculated by smearing the membrane charge uniformly over the surface of the membrane. We believe that this is a more accurate way to introduce the Gouy‐Chapman potential. Moreover, in Eq. (3.17) we have assumed that the potential at the partially charged site is the sum of the potential when the partial charge is zero (i.e., and the potential at the partial charge in the absence of charge on the other lipids [i.e., This is certainly an oversimplification. However, it is correct in the limit of low surface charge density.
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47.See the subsequent paper: B. R. Copeland and H. C. Andersen, J. Chem. Phys. 74, 2547 (1981).
48.The effects of the activity coefficients and which appear in Eq. (4.3), have been completely neglected in the discussion of transition temperature. At low (full protonation in both phases), and are each unity, and these coefficients contribute nothing to Eq. (4.3). At high (full deprotonation in both phases) these coefficients are both quite different from unity, but their ratio is unity—at least in the high potential limit—and these coefficients again contribute nothing to Eq. (4.3). At intermediate where both phases are partially deprotonated, is somewhat greater than and according to Eq. (4.3) these coefficients contribute to a decrease in relative to This contribution is typically 5 K or less.
49.B. R. Copeland and H. C. Andersen, Biochemistry (submitted for publication).
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