Deep learning of position-dependent diffusivity from umbrella sampling molecular dynamics simulations

A widely used computational framework to calculate the membrane permeability coefficient of small molecules is the inhomogeneous solubility-diffusion (ISD) model. It requires two ingredients that can be calculated using molecular dynamics simulations: the potential of mean force, which is well-defined, and the position-dependent diffusivity, which is often problematic and challenging. Two methods (Woolf-Roux and Hummer) have been proposed to determine the position-dependent diffusivity profiles using biased umbrella sampling simulations. While both are constructed from similar time-correlation functions, they can disagree quantitatively. Here, a reconciliation of these methods is achieved through deep learning memory functions in the time domain. The diffusivity extracted through this analysis is shown to be in best agreement with the equilibrium counting permeability for the same membrane system compared to both the Woolf-Roux and Hummer diffusivities. The effect of memory on the rate of barrier crossing is assessed through numerical simulations of the generalized Langevin equation (GLE). The GLE is efficiently simulated via Markovian embedding, which relies on the positive, decaying exponential form of the memory functions extracted by deep learning. Our results based on the ISD permeability, the known permeability from equilibrium molecular dynamics, and the numerical GLE simulations indicate that memory effects most likely do not have a significant effect on the permeation of water.