A control theoretical approach to gene regulation reveals quantitative constraints for dynamic homeostasis in stochastic gene expression

Cell phenotype dynamic homeostasis contrasts with the inherent randomness of intracellular reactions. Although feedback control of regulator genes (RG) is a key strategy for limiting the range of downstream gene expression, understanding the quantitative constraints and corresponding mechanisms enabling such a dynamic stability under noise remains elusive. Here we model RG expression as a stochastic process and downstream genes as sensors whose responses conditionally induce RG activity. We show that at homeostatic regime: i. the trajectories of the RG expression levels can be adjusted towards specific ranges using both the exact solutions of the stochastic model and the exact stochastic simulation algorithm (SSA); ii. there exists a sampling rate which optimizes the feedback control of the RG activity, and non-optimal controls resulting in alternative homeostatic dynamics; iii. the feedback control of RG activity leads to updates whose intensities and time intervals are non-linearly related; iv. the ON state probability of an RG promoter has dynamics confined within a narrow domain. Our results help to understand the quantitative constraints underpinning dynamic homeostasis despite randomness, the mechanisms underlying alternative, non-optimal, homeostatic regimes, and may be useful for theoretically prototyping therapies aiming at gene network modulation.