In-Silico Proportional-Integral Moment Control of Stochastic Gene Expression

TL;DR

This paper develops in-silico proportional-integral control strategies for regulating mean and variance in stochastic gene expression, including systems with protein dimerization, without relying on moment closure techniques.

Contribution

It introduces robust multivariable PI controllers for mean and variance regulation in gene expression, extending control to systems with protein dimerization and providing explicit bounds and stability conditions.

Findings

Global and robust tracking of protein mean levels achieved.

Simultaneous control of mean and variance demonstrated.

Explicit bounds on controller gains and variance bounds provided.

Abstract

The problem of controlling the mean and the variance of a species of interest in a simple gene expression is addressed. It is shown that the protein mean level can be globally and robustly tracked to any desired value using a simple PI controller that satisfies certain sufficient conditions. Controlling both the mean and variance however requires an additional control input, e.g. the mRNA degradation rate, and local robust tracking of mean and variance is proved to be achievable using multivariable PI control, provided that the reference point satisfies necessary conditions imposed by the system. Even more importantly, it is shown that there exist PI controllers that locally, robustly and simultaneously stabilize all the equilibrium points inside the admissible region. The results are then extended to the mean control of a gene expression with protein dimerization. It is shown that the moment closure problem can be circumvented without invoking any moment closure technique. Local stabilization and convergence of the average dimer population to any desired reference value is ensured using a pure integral control law. Explicit bounds on the controller gain are provided and shown to be valid for any reference value. As a byproduct, an explicit upper-bound of the variance of the monomer species, acting on the system as unknown input due to the moment openness, is obtained. The results are illustrated by simulation.