Maximizing average throughput in oscillatory biological synthesis systems: an optimal control approach

TL;DR

This paper formulates and analyzes an optimal control problem to maximize average throughput in oscillatory biological systems, revealing that constant inputs are often optimal despite the presence of periodic signals.

Contribution

It introduces a novel optimal control framework for resource allocation in biological oscillatory systems and demonstrates conditions under which constant inputs are optimal.

Findings

Constant inputs are optimal in various biological architectures.

Periodic signals may indicate an underlying time-varying objective.

The framework can be analyzed using Pontryagin maximum principle.

Abstract

A dynamical system entrains to a periodic input if its state converges globally to an attractor with the same period. In particular, for a constant input the state converges to a unique equilibrium point for any initial condition. We consider the problem of maximizing a weighted average of the system's output along the periodic attractor. The gain of entrainment is the benefit achieved by using a non-constant periodic input relative to a constant input with the same time average. Such a problem amounts to optimal allocation of resources in a periodic manner. We formulate this problem as a periodic optimal control problem which can be analyzed by means of the Pontryagin maximum principle or solved numerically via powerful software packages. We then apply our framework to a class of occupancy models that appear frequently in biological synthesis systems and other applications. We show that, perhaps surprisingly, constant inputs are optimal for various architectures. This suggests that the presence of non-constant periodic signals, which frequently appear in biological occupancy systems, is a signature of an underlying time-varying objective functional being optimized.