Application of Coarse Integration to Bacterial Chemotaxis

We have developed and implemented a numerical evolution scheme for a class ofstochastic problems in which the temporal evolution occurs on widely-separatedtime scales, and for which the slow evolution can be described in terms of asmall number of moments of an underlying probability distribution. Wedemonstrate this method via a numerical simulation of chemotaxis in apopulation of motile, independent bacteria swimming in a prescribed gradient ofa chemoattractant. The microscopic stochastic model, which is simulated using aMonte Carlo method, uses a simplified deterministic model forexcitation/adaptation in signal transduction, coupled to a realistic,stochastic description of the flagellar motor. We show that projective timeintegration of ``coarse'' variables can be carried out on time scales longcompared to that of the microscopic dynamics. Our coarse description is basedon 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 scalessolely by the spatial cell density. Computationally the variables are thecomponents of the density distribution expressed in terms of a few basisfunctions, given by the singular vectors of the spatial density distributionobtained from a sample Monte Carlo time evolution of the system. We presentnumerical results and analysis of errors in support of the efficacy of thistime-integration scheme.