IEPDYN: Integral-equation formalism of population dynamics

We propose the integral-equation formalism of population dynamics (IEPDYN) to describe the population dynamics of distinct configurational states. According to classical reaction dynamics theory, the probability density associated with a given state obeys the Liouville equation, including influx from and efflux to neighboring states. By introducing a Markov approximation for the crossing of boundaries separating the states, tractable integral equations governing the state populations are derived. Once the time-dependent quantities appearing in these equations are evaluated, the population dynamics on long timescales can be obtained. Because these quantities depend only on a few states in the local neighborhood of a given state, they can be computed using a set of short-timescale molecular dynamics (MD) simulations. The IEPDYN method is formulated in continuous time and therefore does not rely on a coarse-grained timescale (lag time). Consequently, kinetic quantities obtained from IEPDYN are free from lag-time dependence, which has been discussed as a limitation in other approaches. We apply the IEPDYN method to the binding and unbinding kinetics of CH$_4$/CH$_4$, Na$^+$/Cl$^-$, and 18-crown-6-ether (crown ether)/K$^+$ in water. For both kinetics, the time constants estimated from the IEPDYN method are almost comparable to those obtained from brute-force MD simulations. The required timescale of each MD trajectory in the IEPDYN method is approximately two orders of magnitude shorter than that in the brute-force MD approach in the crown ether/K$^+$ system. This reduction in the trajectory timescale enables applications to complex binding and unbinding systems whose characteristic timescales are far beyond those directly accessible by brute-force MD simulations.