Delayed bet-hedging resilience strategies under environmental fluctuations

Many biological populations, such as bacterial colonies, have developed through evolution a protection mechanism, called bet hedging, to increase their probability of survival under stressful environmental fluctuation. In this context, the concept of preadaptation refers to a common type of bet-hedging protection strategy in which a relatively small number of individuals in a population stochastically switch their phenotypes to a dormant metabolic state in which they increase their probability of survival against potential environmental shocks. Hence, if an environmental shock took place at some point in time, preadapted organisms would be better adapted to survive and proliferate once the shock is over. In many biological populations, the mechanisms of preadaptation and proliferation present delays whose influence in the fitness of the population are not well understood. In this paper, we propose a rigorous mathematical framework to analyze the role of delays in both preadaptation and proliferation mechanisms in the survival of biological populations, with an emphasis on bacterial colonies. Our theoretical framework allows us to analytically quantify the average growth rate of a bet-hedging bacterial colony with stochastically delayed reactions with arbitrary precision. We verify the accuracy of the proposed method by numerical simulations and conclude that the growth rate of a bet-hedging population shows a nontrivial dependency on their preadaptation and proliferation delays. Contrary to the current belief, our results show that faster reactions do not, in general, increase the overall fitness of a biological population.

Figure 1

Bet-hedging population ($n=2$).

Figure 2

Growth rates $\rho$ of the dynamical population model (2) for $p=0.5$, 1, 1.5, and 2 (from the bottom). Solid lines: Growth rates predicted by the proposed method. Circles: Growth rates estimated from the sample trajectories of the model.

Figure 3

Growth rates $\rho$ versus $d$ and $p$ for various global fitness parameter $\tau$: (a) $\tau =0$, (b) $\tau =0.1$, (c) $\tau =0.5$. The figures show a nontrivial dependency of the growth rates on the parameters $d, p$, and $\tau$. Specifically, when $\tau =0$ [Fig. 3], we observe that, the larger the delay, the higher the growth rates; while for $\tau =0.5$ [Fig. 3], we observe the opposite tendency.

Figure 4

Growth rates $\rho$ versus $\kappa$ and $p$ for various exponent $\varphi$ on $p$: (a) $\varphi =0.25$, (b) $\varphi =0.5$, (c) $\varphi =0.95$. The figures show that the exponent $\varphi$ drastically changes the dependency of the growth rates on the delayed growth factor $p$. For example, in the regime of small $\varphi$ [Fig. 4], we observe that a good strategy for a population to increase its growth rate is to bet on a delayed growth factor $p$. On the other hand, in the case of relatively large $\varphi$ [Fig. 4], we see that keeping $p$ small results in a higher growth rate.

Figure 5

State-transition diagram for delayed adaptation under environmental fluctuations. Suppose that the environment is of the first type and the population's knowledge is currently updated, i.e., $\epsilon =\sigma =1$ (the top-left regime in the figure). Once the environment $\epsilon$ changes from 1 to 2 (the top-right regime), it takes $T^{12}$ units of time for the population's knowledge $\sigma$ to be updated to 2, which results in the transition to the bottom-right regime.

Figure 6

State transition diagram of the Coxian distribution $C(\alpha ,\beta )$. The starting and absorbing states are 1 and $s+1$, respectively.

Figure 7

Markov chain for the dynamics of the environment-knowledge pair $(\epsilon ,\sigma )$. The thin arrows represent the dynamics of phase-type distributions, while the thick arrows represent changes in the environment.

Figure 8

Growth rates versus $g_1^2$ and $\mu (=E\left[T^{12}\right]=E\left[T^{21}\right])$ when $g_2^1=0$: (a) $k=1$, (b) $k=4$, (c) $k=16$. Dashed lines indicate the values of $g_1^2$ at which the optimal value of $\mu$ changes. $\mu =0$ is the optimal on the right of the dashed lines, while letting $\mu$ as large as possible yields the optimal growth rates on the left of the dashed lines.

Figure 9

Growth rates versus $E\left[T^{12}\right]$ and $E\left[T^{21}\right]$ when $g_2^1=0$ and $k=4$: (a) $g_1^2=-1$, (b) $g_1^2=-0.23$, (c) $g_1^2=-0.15$.

Figure 10

Phase diagram of the optimal length of expected adaptation delays $(E\left[T^{12}\right],E\left[T^{21}\right])$ versus $g_2^1$ and $g_1^2$.