Optimality and adaptation of phenotypically switching cells in fluctuating environments

Stochastic switching between alternative phenotypic states is a common cellular survival strategy during unforeseen environmental fluctuations. Cells can switch between different subpopulations that proliferate at different rates in different environments. Optimal population growth is typically assumed to occur when phenotypic switching rates match environmental switching rates. However, it is not well understood how this optimum behaves as a function of the growth rates of phenotypically different cells. In this study, we use mathematical and computational models to test how the actual parameters associated with optimal population growth differ from those assumed to be optimal. We find that the predicted optimum is practically always valid if the environmental durations are long. However, the regime of validity narrows as environmental durations shorten, especially if subpopulation growth rate differences differ from each other (are asymmetric) in two environments. Furthermore, we study the fate of mutants with switching rates previously predicted to be optimal. We find that mutants which match their phenotypic switching rates with the environmental ones can only sweep the population if the assumed optimum is valid, but not otherwise.

Figure 1

A two-state Markov chain model of phenotypic switching in an environment fluctuating between two different states, ${\epsilon }_1$ and ${\epsilon }_2$. Cells switch phenotypes between type I and type II. (a) Environment ${\epsilon }_1$ favors the gray type I cells and (b) environment ${\epsilon }_2$ favors the green type II cells. (c) The cells grow with growth rates $g_i\left({\epsilon }_1\right)$ and switch from phenotype $j$ to phenotype $i$ randomly with rate ${\omega }_{ij}$ in environment ${\epsilon }_1$ that lasts for a time ${\tau }_1$. Then the cells grow with growth rate $g_i\left({\epsilon }_2\right)$ and switch phenotypes as described above in environment ${\epsilon }_2$ that has a time duration ${\tau }_2$. (d) Averaging the population dynamics over time in fluctuating environments: the cells grow in environment ${\epsilon }_1$ for duration ${\tau }_1$ and then the environment switches to environment ${\epsilon }_2$, imposing different growth rates for each phenotype for a duration ${\tau }_2$. The switching of environments was carried out for sufficiently long time so that the distribution of cells within states is identical for two consecutive periods. One period lasts for ${\tau }_1+{\tau }_2$ duration of time. The time-varying population fitness, $G(\epsilon ,t)$, is averaged over one such period after the distribution is stabilized.

Figure 2

Optimal switching rates do not always match the environmental switching rates for ${\tau }_1={\tau }_2=10$ $(•$ and $+$ indicate locations of the assumed and actual fitness optima, respectively). The color bar indicates the population growth rate (fitness). Panels (a) correspond to symmetric growth rates $[[g_1\left({\epsilon }_1\right),g_2\left({\epsilon }_1\right)]=(0.50,0.0001),[g_1\left({\epsilon }_2\right),g_2\left({\epsilon }_2\right)]=(0.0001,0.50)]$. Panels (b) correspond to asymmetric growth rates with parameter values $[[g_1\left({\epsilon }_1\right),g_2\left({\epsilon }_1\right)]=(0.50,0.0001),[g_1\left({\epsilon }_2\right),g_2\left({\epsilon }_2\right)]=(0.0001,0.3250)]$, where the actual optimum shifts away from the assumed one, as indicated by both stochastic (upper row) and deterministic (lower row) models. We refer to this situation as inaccurate assumption. Finally, panel (c) shows a case of invalid assumption for $[[g_1\left({\epsilon }_1\right),g_2\left({\epsilon }_1\right)]=(0.50,0.0001),[g_1\left({\epsilon }_2\right),g_2\left({\epsilon }_2\right)]=(0.0001,0.1251)]$, where the actual optimum is on the edge of the fitness landscape.

Figure 3

Validity and accuracy of assumed optimum in the parameter space. The smooth white region indicates that the assumption is invalid. The hashed region indicates that the optimum occurs for switching rates equal to zero. The color indicates the relative shift in the position of the optimum relative to the assumed position. The rows of panels (a)–(c) correspond to different environmental durations with growth rates $g_1\left({\epsilon }_1\right)=0.50$ and $g_2\left({\epsilon }_1\right)=\left[0.00010.3750.425\right]$, respectively. (a) Validity of the assumed optimum for environmental duration ${\tau }_1={\tau }_2=10$. The point marked “P” corresponds to Figs. 2 and 4 where the assumed optimum occurs practically as predicted. The point marked by “S” corresponds to Figs. 2 and 4 where the optimum is shifted. Finally, point “N” corresponds to Figs. 2 and 4 where the assumed optimum is invalid. (b) Validity of the assumed optimum for environmental durations ${\tau }_1={\tau }_2=25$. (c) Validity of the assumed optimum for environmental durations ${\tau }_1={\tau }_2=100$. We kept fixed the growth rate of phenotype I in environment 1 at $[g_1\left({\epsilon }_1\right)=0.50]$ and varied the other growth rates $\mathrm{[}{g}_2\left({\epsilon }_1\right),g_1\left({\epsilon }_2\right)$, and $g_2\left({\epsilon }_2\right)]$ in each of the plots from rows (a)–(c). (d) The phase space where the optimum exists for $[g_1\left({\epsilon }_1\right)=0.5,g_1\left({\epsilon }_2\right)=0.0001]$ for environmental durations of ${\tau }_1={\tau }_2=5$, 10, and 20, respectively. (e) The $g_2\left({\epsilon }_2\right)$ intercepts correspond to the point on the horizontal axis where the optimum becomes invalid in panels (a)–(c). The first intercept decreases roughly linearly over narrow ranges of growth rates, whereas over a broader range it decreases symmetrically with respect to the second intercept, which decays exponentially (parameter values are given in Table 1 in the Appendix). The solid lines are least-squares fits and the dots are the actual points extracted from the phase space in panel (d).

Figure 4

The fate of mutants that match their phenotypic switching rates with the rates of environmental fluctuations. We initiate a mutant whose switching rates are equal to the environmental switching rates, that is, $\left({\omega }_{ij}=\frac{1}{\tau }\right)$ in ancestral populations with various phenotypic switching rates. Points N, S, and P in Fig. 3 are the conditions of the growth rates where mutant propagation was tested. (a)–(c) Snapshot of the fraction of mutant cell population with growth rates located at points P, S, and N in Fig. 3, respectively, at time periods of $T=25,T=30,$ and $T=35$ in deterministic models. As time progresses, the fraction of phenotype switching rate mutants expands to sweep almost all ancestral populations if the optimum occurs as assumed (a). (c), (b) The same is not true if the assumed optimum is inaccurate or invalid. Here the mutants cannot sweep the ancestral population for many parameter combinations. (d) The probability that a mutant invades the ancestral population in stochastic simulations relative to $1/N$, the probability of invasion for neutral genetic drift, where $N=100$ is the population size. (e) Probability of invading the ancestral population for a neutral mutant as a function of the population size [25].