Solving the chemical master equation by aggregation and Krylov approximations

The chemical master equation (CME) is often difficult to solve directly due to the curse of dimensionality. Aggregation was among the earliest ideas for addressing this challenge. It consists in coarsening the state space of the CME to make it more tractable. This reduction inevitably introduces a numerical error that is not trivial to account for, consequently only a few implementations deal with the estimation and control of this aggregation error. Here, we implement an error control by using an adaptive aggregation strategy and the concept of defect (or residual) in the numerical solution of ordinary differential equations. We embed the aggregation approach in the step-by-step matrix exponential implementation of Expokit that uses Krylov subspace approximations. We conduct numerical experiments on test problems that model the toggle switch, receptor oligomerization, and repressilator. This work is an important step toward solving problems that arise in the stochastic modeling of larger biological networks.