Title: A methodology of quantifying membrane permeability based on returning probability theory and molecular dynamics simulation

Abstract: We propose a theoretical approach to estimate the permeability coefficient of substrates (permeants) for crossing membranes from donor (D) phase to acceptor (A) phase by means of molecular dynamics (MD) simulation. A fundamental aspect of our approach involves reformulating the returning probability (RP) theory, a rigorous bimolecular reaction theory, to describe permeation phenomena. This reformulation relies on the parallelism between permeation and bimolecular reaction processes. In the present method, the permeability coefficient is represented in terms of the thermodynamic and kinetic quantities for the reactive (R) phase that exists within the inner region of membranes. One can evaluate these quantities using multiple MD trajectories starting from phase R. We apply the RP theory to the permeation of ethanol and methylamine at different concentrations (infinitely dilute and 1 mol% conditions of permeants). Under the 1 mol% condition, the present method yields a larger permeability coefficient for ethanol ($0.12 \pm 0.01 ~\mathrm{cm~s^{-1}}$) than for methylamine ($0.069\pm 0.006~\mathrm{cm~s^{-1}}$), while the values of the permeability coefficient are satisfactorily close to those obtained from the brute-force MD simulations [$0.18\pm 0.03 ~\mathrm{cm~s^{-1}}$ and $0.052 \pm 0.005 ~\mathrm{cm~s^{-1}}$ for ethanol and methylamine, respectively]. Moreover, upon analyzing the thermodynamic and kinetic contributions to the permeability, we clarify that a higher concentration dependency of permeability for ethanol, as compared to methylamine, arises from the sensitive nature of ethanol's free-energy barrier within the inner region of the membrane against ethanol concentration.