Neural Stochastic Forecasting for Lateral Molecular Diffusion Simulations

Abstract

Predicting the long-time diffusion of membrane proteins is challenging because molecular dynamics (MD) simulations are computationally expensive, while experimental single-particle tracking is often limited by finite observation times. Here, we introduce a neural stochastic (NS) model for forecasting the collective diffusion of membrane proteins from coarse-grained MD trajectories. The model combines a transformer-based representation of protein configurations, recurrent networks operating at multiple temporal scales, and a stochastic latent process with learned memory. A cross-attention decoder then generates correlated protein displacements while accounting for the instantaneous system geometry and periodic boundary conditions. We evaluate the model in two membrane systems: seven interacting PIEZO channels and a mixed system containing seven PIEZO and ten TREK channels. For the seven PIEZO system, the NS model reproduces the anomalous diffusion exponent measured in MD simulations $\alpha \sim 1$, substantially improving on a previously developed variational autoregressive model that exhibits superdiffusive behavior. The model also generates trajectories extending to 40,000 frames and captures the distinct diffusion characteristics of PIEZO and TREK populations in the mixed system. Comparisons of time-averaged mean-square displacements, Van Hove displacement distributions, and non-Gaussian parameters show that the model preserves key statistical features of the simulated dynamics. These results establish neural stochastic modeling as a promising approach for generating long trajectories of multi-protein membrane diffusion and extending the statistical analysis of molecular dynamics simulations.