Adaptive coarse-graining for transient and quasi-equilibrium analyses of stochastic gene regulation

Intracellular populations of genes, RNA and proteins are often described by continuous-time, discrete-state Markov processes, whose time-varying probability distributions evolve according to the large or infinite dimensional linear ordinary differential equation known as the chemical master equation (CME). Numerical integration and stochastic simulation of the CME are often impossible or time consuming. We introduce new methods to project the full CME onto a lower dimensional space, while retaining the transient and equilibrium statistics of the original process. First, we investigate three complementary sets of coarse-graining rules: (i) The previously described finite state projection approach; (ii) A modification of existing coarse-graining approaches to reduce the system dimension while capturing the processes equilibrium distribution; and (iii) New time-scale correction terms to recapture transient dynamics of the original system. Next, we explore different iterative algorithms that automatically adapt the projection resolution to improve accuracy and efficiency of the CME solution. We test these projection and refinement strategies on several gene regulatory processes, and we comment on the efficiency and accuracy of the coarse-graining rules and refinement strategies.