  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.
