On the gain of entrainment in a class of weakly contractive bilinear control systems with applications to the master equation and the ribosome flow model

TL;DR

This paper analyzes how weakly contractive bilinear control systems respond to periodic inputs, deriving formulas for their gain of entrainment, and demonstrates applications in biological and physical models like the master equation and ribosome flow.

Contribution

It provides a closed-form formula for the gain of entrainment in weakly contractive bilinear systems and shows this gain is a higher-order effect, with applications to biological and physical models.

Findings

The gain of entrainment is always a higher-order phenomenon.

Derived a first-order formula for the gain of entrainment.

Validated results with applications to the master equation and ribosome flow model.

Abstract

We consider a class of bilinear weakly contractive systems that entrain to periodic excitations. Entrainment is important in many natural and artificial processes. For example, in order to function properly synchronous generators must entrain to the frequency of the electrical grid, and biological organisms must entrain to the 24h solar day. A dynamical system has a positive gain of entrainment (GOE) if entrainment also yields a larger output, on average. This property is important in many applications from the periodic operation of bioreactors to the periodic production of proteins during the cell cycle division process. We derive a closed-form formula for the GOE to first-order in the control perturbation. This is used to show that in the class of systems that we consider the GOE is always a higher-order phenomenon. We demonstrate the theoretical results using two applications: the master equation and a model from systems biology called the ribosome flow model, both with time-varying and periodic transition rates.