Stochastic analysis of retroactivity in transcriptional networks through singular perturbation

The input/output dynamic behavior of a biomolecular system is affected by interconnection to other downstream systems through impedance-like effects called retroactivity. In this paper, we study the effects of retroactivity at the interconnection between two transcriptional modules, focusing on stochastic behavior. In particular, we describe the system through the Master equation and develop a singular perturbation theory to obtain a reduced Master equation. We prove that the solution of the original Master equation converges fast to an e neighbor of the solution of the reduced Master equation, in which e is the singular perturbation parameter. Our analysis shows that the upstream system and the downstream one are statistically independent at the steady state. However, the interconnection slows down the dynamics of both the expectation and the variance of the output of the upstream transcriptional module.