Statistical Mechanical\nTheories of Membrane Permeability

A popular theoretical framework to compute the permeability\ncoefficient\nof a molecule is provided by the classic Smoluchowski-Kramers treatment\nof the steady-state diffusive flux across a free-energy barrier. Within\nthis framework, commonly termed “inhomogeneous solubility-diffusion”\n(ISD), the permeability, <i>P</i>, is expressed in closed\nform in terms of the potential of mean force and position-dependent\ndiffusivity of the molecule of interest along the membrane normal.\nIn principle, both quantities can be calculated from all-atom MD simulations.\nAlthough several methods exist for calculating the position-dependent\ndiffusivity, each of these is at best an estimate. In addition, the\nISD model does not account for memory effects along the chosen reaction\ncoordinate. For these reasons, it is important to seek alternative\ntheoretical formulations to determine the permeability coefficient\nthat are able to account for the factors ignored by the ISD approximation.\nUsing Green-Kubo linear response theory, we establish the familiar\nconstitutive relation between the flux density across the membrane\nand the difference in the concentration of a permeant molecule, <i>j</i> = <i>P</i>Δ<i>C</i>. On this\nbasis, we derive a time-correlation function expression for the nonequilibrium\nflux across a membrane that is reminiscent of the transmission coefficient\nin the reactive flux formalism treatment of transition rates. An analysis\nbased on the transition path theory framework is exploited to derive\nalternative expressions for the permeability coefficient. The different\nstrategies are illustrated with stochastic simulations based on the\ngeneralized Langevin equation in addition to unbiased molecular dynamics\nsimulations of water permeation of a lipid bilayer.