We have developed and implemented a numerical evolution scheme for a class of stochastic problems in which the temporal evolution occurs on widely separated time scales and for which the slow evolution can be described in terms of a small number of moments of an underlying probability distribution. We demonstrate this method via a numerical simulation of chemotaxis in a population of motile, independent bacteria swimming in a prescribed gradient of a chemoattractant. The microscopic stochastic model, which is simulated using a Monte Carlo method, uses a simplified deterministic model for excitation/adaptation in signal transduction, coupled with a realistic, stochastic description of the flagellar motor. We show that projective time integration of "coarse" variables can be carried out on time scales long compared to those of microscopic dynamics. Our coarse description is based on the spatial cell density distribution. Thus we are assuming that the system "closes" on this variable so that it can be described on long time scales solely by the spatial cell density. Computationally, the variables are the components of the density distribution expressed in terms of a few basis functions, given by the singular vectors of the spatial density distribution obtained from a sample Monte Carlo time evolution of the system. We present numerical results and analysis of errors in support of the efficacy of this time-integration scheme.
Bibliographical noteFunding Information:
We thank Michael P. Friedlander, Matilde Gargiani, Sai Praneeth Karimireddy, Jakub Konečný, Ching-pei Lee, and Peter Richtárik for their help and for fruitful discussions. We are additionally grateful to the reviewers for their valuable comments. We wish to acknowledge support from the U.S. National Science Foundation, under award number NSF:CCF:1618717, NSF:CMMI:1663256 and NSF:CCF:1740796; the Swiss National Science Foundation, under grant number 175796; and the Mathematical Data Science program of the Office of Naval Research, under grant number N00014-15-1-2670.
- Bacterial chemotaxis
- Coarse integration
- Monte Carlo simulation