Roger Sidje | Department of Mathematics, University of Alabama
An adaptive Magnus expansion method for solving the chemical master equation with time-dependent rates
There has been increasing interest in recent years to analyze the stochasticity of biochemical reaction systems by directly solving the chemical master equation (CME), thereby providing an alternative to Monte Carlo methods such as the stochastic simulation algorithm (SSA) or its tau-leaping variants. Many of those CME solvers build on the finite state projection (FSP) and assume that reaction rates remain constant. They are therefore not applicable for biological problems where reaction rates change over time, for instance due to cell volume or temperature. In such cases, a possible approach is to use the Magnus expansion to represent the solution as a matrix exponential for which Krylov-based methods such as EXPOKIT can be applied. We discuss various Magnus schemes stemming from the analysis of the error and residual. Of particular focus is an adaptive time-stepping Magnus-SSA algorithm with not only a variable time-step but also a variable state space that changes at each step via the SSA. The talk will include a comparison with some other approaches such as Adams-Bashforth, Runge-Kutta and Backward-differentiation formula (BDF).