# Phenotypic switching of populations of cells in a stochastic environment

## TL;DR

This paper models how cell populations adapt their phenotypic switching strategies in randomly changing environments, revealing that stochastic switching often leads to higher growth rates and more heterogeneity than periodic environments.

## Contribution

It derives stochastic differential equations for phenotypic switching in random environments and analyzes optimal responses, extending previous models focused on periodic changes.

## Key findings

- Optimal phenotypic responses are complex for slow/intermediate environmental changes.
- Stochastic environments favor more heterogeneous phenotypic responses.
- Net growth rates are generally higher under stochastic environmental dynamics.

## Abstract

In biology phenotypic switching is a common bet-hedging strategy in the face of uncertain environmental conditions. Existing mathematical models often focus on periodically changing environments to determine the optimal phenotypic response. We focus on the case in which the environment switches randomly between discrete states. Starting from an individual-based model we derive stochastic differential equations to describe the dynamics, and obtain analytical expressions for the mean instantaneous growth rates based on the theory of piecewise deterministic Markov processes. We show that optimal phenotypic responses are non-trivial for slow and intermediate environmental processes, and systematically compare the cases of periodic and random environments. The best response to random switching is more likely to be heterogeneity than in the case of deterministic periodic environments, net growth rates tend to be higher under stochastic environmental dynamics. The combined system of environment and population of cells can be interpreted as host-pathogen interaction, in which the host tries to choose environmental switching so as to minimise growth of the pathogen, and in which the pathogen employs a phenotypic switching optimised to increase its growth rate. We discuss the existence of Nash-like mutual best-response scenarios for such host-pathogen games.