Integrative Generalized Master Equation: A Theory to Study Long-timescale Biomolecular Dynamics via the Integrals of Memory Kernels

The generalized master equation (GME) provides a powerful approach to study biomolecular dynamics via non-Markovian dynamic models built from molecular dynamics (MD) simulations. Previously, we have implemented the GME for biomolecular dynamics, namely the quasi Markov State Model (qMSM), where we explicitly calculate the memory kernel and propagate protein dynamics using a discretized GME. qMSM can be constructed with much shorter MD simulation trajectories than the Markov State Model (MSM). However, since qMSM needs to explicitly compute the time-dependent memory kernels, it is heavily affected by the numerical fluctuations of simulation data when applied to study complicated conformational changes of biomolecules. This can lead to numerical instability of predicted long-time dynamics, greatly limiting the applicability of the qMSM in complicated molecules. In this paper, we propose a new theory, the Integrative GME (IGME), to overcome the challenges of the qMSM by using the time integrations of memory kernels. The IGME avoids the numerical instability induced by explicit computation of time-dependent memory kernel functions, giving more robust predictions of long-time dynamics. Using our analytical solution of the IGME, we propose a new approach to compute memory kernels and long-time dynamics in a numerically stable, accurate and efficient way. To demonstrate its effectiveness, we have applied the IGME in three biomolecules: the alanine dipeptide, the FIP35 WW domain, and Taq RNAP. In each system, the IGME achieves significantly smaller fluctuations for both memory kernels and long-time dynamics compared to the qMSM. We anticipate that the IGME can be widely applied to investigate complex conformational changes of biomolecules.