Phenotypic switching of populations of cells in a stochastic environment

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.

Model reduction methods for population dynamics with fast-switching environments: Reduced master equations, stochastic differential equations, and applications

“…Figure 13(b) shows the probability that a given component is still reliable at time t. The black line is obtained through Monte Carlo simulations, whereas the dashed line is the prediction of Eq. (82). For the specified parameters, the two lines show agreement.…”

Bet-hedging strategies in expanding populations

“…The 83 growth rate of the risky phenotype is s a in environment (a) and s b in environment (b) 84 ( Fig. 1B) [55]. The two environments occur with the same probability, p a = p b = 1/2.…”

“…In particular, defining the 111 normalized growth ratess a ≡ s a /s s ands b ≡ s b /s s , we find that the optimal strategy is 112 α * = 1 whens b > 2 −s a and α * = 0 otherwise. This means that no bet-hedging 113 strategy is possible in this model in the well-mixed case [55]. finite switching rates among strategies [32,57], or c) a delta-correlated environment [52].…”

“…We 302 have assumed that the fraction of individuals adopting each phenotype is fixed by the 303 phenotypic switching rates. To understand the evolution of bet-hedging, it could be 304 interesting to study scenarios in which the phenotypic switching rates are slower, so that 305 phenotypes can be selected, and/or are themselves subject to evolution and 306 selection [55,69]. Another potentially relevant extension would be to consider 307 two-dimensional habitats.…”

Exploration-exploitation tradeoffs dictate the optimal distributions of phenotypes for populations subject to fitness fluctuations

“…the persistence of a sub-population of resistant but slow-growing bacteria within a popula-tion subject to high doses of antibiotics [15,16]. Starting with [17], several mathematical models have shown that switching between different phenotypes at the individual cell level can be advantageous in rapidly changing conditions, depending essentially on (i) the statistics of environmental fluctuations and (ii) the specific coupling between the environment and the allowed phenotypes [18][19][20][21][22][23][24][25][26][27][28]. Such models capture the physical and mathematical complexity of these systems starting from minimal assumptions about the environment and/or the space of feasible phenotypes.…”