Application of the Path Integration Formalism to Analysis of Activated Processes In Complex Molecular Systems

One of the challenges in modeling of dynamics of complex molecular systems is prediction of rates of rare evens, i.e. events involving activated transitions between metastable states. The timescale of such events is often inaccessible to direct molecular dynamics simulations. Therefore, these events are analyzed using a reduced stochastic model, such as the Langevin equation, which explicitly accounts for dynamics of a small number of key degrees of freedom (reaction coordinates) while modeling all other degrees of freedom as a thermal bath. The rates of rare events can then be obtained by solving the obtained stochastic equation(s). A common method of analysis of rare events in systems with multiple reaction coordinates is based on the assumption that the system follows a minimum energy path (MEP), i.e. a path such that the free energy is minimized in all directions transversal to the path direction. This assumption effectively reduces the multi-dimensional stochastic process to a quasi-one-dimensional process confined to the MEP.

However, the MEP assumption is not always valid. In particular, our group has demonstrated that substantial deviations from the MEP may occur even in processes involving motion on relatively simple free energy landscapes, such as molecular transport across a liquid-liquid interface or a lipid membrane. These deviations from the MEPs occur due to non-adiabatic coupling between the translational motion of a solute molecule and thermal fluctuations of the interface.

In this talk, we present a theoretical analysis of such non-adiabatic processes. This analysis is based on applying the path integration formalism to solution of the Langevin equation. This formalism yields a distribution of paths connecting metastable states. In particular, this approach enables us to identify the most likely path (MLP) to be taken by the system. We demonstrate that the MLP may significantly deviate from the MEP. Once the MLP is identified, we obtain the rate of the activated events by computing the path integral. The path integral is computed using a harmonic (WKB-like) approximation in the neighborhood of the MLP.