Applied Mathematics Colloquium

Abstract: The stochastic, diffusive motion of molecules influences the rate of molecular encounters and hence many biochemical processes in living cells. Tiny time steps are typically required to accurately resolve the probabilistic encounter events in a simulation of cellular signaling, creating the dilemma that only simplistic models can be studied at cell-biologically relevant timescales. As a remedy, a number of algorithms employ analytical representations of Green's functions of the diffusion equation to allow for larger time steps. However, in these approaches, position updates require sampling from solutions that are difficult to evaluate numerically, thus introducing a major bottleneck. 

In my talk, I will discuss an algorithm that accurately propagates molecule pairs using large time steps without the need to invoke the full analytical solutions. Because the proposed method only uses uniform and Gaussian random numbers, it allows for position updates that are two to three orders of magnitude faster than those of a corresponding scheme based on full solutions. Neither simplifying nor ad hoc assumptions that are foreign to the underlying Smoluchowski theory are employed. Instead, the algorithm faithfully implements a space-time path-decomposition representation of the diffusion propagator, which is inspired by an analogue factorization of the quantum mechanical Schroedinger kernel. The method is flexible and applicable in 1, 2 and 3 dimensions, suggesting that it may find broad usage in various stochastic simulation algorithms.

This research was supported by the Intramural Research Program of the NIH.