Pore-size Distribution in a Biomembrane Model

In a recent paper the author has suggested a model which each lipid has more space available. This leads for a two-dimensional lipid bilayer in which both shortto an increase in the energy of the configuration. range repulsive forces and long-range attractive forces Because of the very large number of degrees of freeplay a role, and for which the equilibrium statistical dom the outcome of this competitive process is very mechanics could be worked out in detail [1]. In this hard to forecast unless one can calculate the contribumodel the hydrocarbon chains of the lipids are restricted tions of the various parts of the bilayer’s phase space to the bonds of a square lattice; the short-range repulexactly. When this is done it is found that the most sion between the chains is simulated by the constraint probable configurations of the bilayer are those in that two chains cannot intersect, and the long-range which the lipids are locally packed in long stretches attraction between two chains separated by a distance of relatively high density, separated by short holes. Ix —x’I derives from a potential of the form: Hence the density fluctuations in the membrane are highly non-linear. V(x—x)’—~m7w exp(—’ylx—xI). (1) The statistical properties of the pores can be found Here ~ is the range of the attractive force, w0 from the representation of the grand canonical partidenotes a positive constant and 2m1 denotes the thicktion function as a Wiener integral: ness of the membrane. (1sJ2 equals the distance between L 2 nearest neighbours in the underlying square lattice.) Zz ~ L~= N— 1 ~ m r The potential (1) represents the total attractive inter~, , ~, exp — 2~w 07 2~ action between two chains (mainly van der Waals 0 (2) interactions), but for mathematical convenience it had —m i B(cb)dx d[Ø(x)]. to be assumed that this total interaction acts between the head groups of the lipids only. 0 In the present note we would like to point out that Here L denotes the length of the membrane,N a norin thermal equilibrium under the influence of the heat malization constant, j3 = kB T with kB denoting Boltzmotion small holes will form in this bilayer, and we mann’s constant and T the absolute temperature and shall calculate their statistical characteristics. The for~ mation of these pores can be understood physically B(~)= — ~j3p 0 (~e~). (3) in the following way. If the bilayer is in a configuration in which in a certain region the lipids are packed The function j3p0 is the grand canonical partition funcvery closely, the energy is very low, corresponding to tion of the model without the attractive interaction, a large Boltzmann factor. However, closely packed and ~ is related to the fugacity z by ~ = z exp(—~~3c.07 2). hydrocarbon chains do not have many available confiIn [1] the Wiener integral (2) was calculated straightgurations due to steric hindrance, so that the correforwardly, which leads to the thermodynamic funcsponding entropy is small. On the other hand, the tions of the bilayer. It was found that the isotherm bilayer can gain entropy by assuming a configuration in showed a phase transition at a density p = p 5(fl. The