Pareto-optimal trade-off for phenotypic switching of populations in a stochastic environment

Abstract

Finding optimal survival strategies of living systems embedded in fluctuating environments generally involves a balance between phenotypic diversification and sensing. If we neglect sensing mechanisms, it is known that slow, resp. fast, environmental transitions favor a regime of heterogeneous, resp. homogeneous, phenotypic response. We focus here on the simplest non-trivial case, i.e. two randomly switching phenotypes subjected to two stochastically switching environments. The optimal asymptotic (long term) growth rate of this model was studied elsewhere; we further expand these results by discussing finite time growth rate fluctuations. An exact asymptotic expression for the variance, alongside with approximations valid in different regimes, are tested numerically in details. Our simulations of the dynamics suggest a close connection between this variance and the extinction probability, understood as risk for the population. Motivated by an earlier trade-off analysis between average capital growth rate and risk in Kelly’s gambling model, we study the trade-off between the average growth rate and the variance in the present model. Despite considerable differences between the two models, we find similar optimal trade-off curves (Pareto fronts), suggesting that our conclusions are robust, and broadly applicable in various fields ranging from biology/ecology to economics.

Optimal metabolic strategies for microbial growth in stationary random environments

“…To better visualize the robustness of these outcomes with respect to β, we display in figure 3(e) the re-scaled fitness F defined in (18) as a function of the 'inverse temperature' . As anticipated, F has a maximum at intermediate value of β, related to the cost-per-bit c of encoding information as shown in (19) (bottom panel): higher costs imply lower optimal values of β. Noticeably, though, with the realistic parameters we used, the maximum of the fitness function is extremely flat.…”

“…maintaining a fraction of slow-growing cells even in rich media or sustaining a lower short-term growth to ensure faster long-term growth) can yield significant fitness gains in a wide variety of situations [16]. Biological implications of these results have been explored against several backdrops [17][18][19], albeit never specifically in the context of metabolism.…”

The trade-off between growth and risk in Kelly’s gambling and beyond

“…In previous work, we have studied this trade-off in a version of Kelly model with a risk constraint [16]. In subsequent work, we found a similar trade-off in the context of a biological population with phenotypic switching in a fluctuating environment [17] by building on a piece-wise Markov model introduced earlier [18]. We have also proposed an adaptive version of Kelly’s gambling based on Bayesian inference [19].…”

Adaptive strategies in Kelly’s horse races model

“…In a recent work, we have studied the properties of this trade-off inspired by ideas in stochastic thermodynamics [8]. We then studied another manifestation of this trade-off this time in the context of a biological population with phenotypic switching in a fluctuating environment [9].…”