Dynamics and evolution of stochastic bistable gene networks with sensing in fluctuating environments

We study how cells can optimize fitness in variable environments by tuning the internal fluctuations of protein expression of a bistable genetic switch. We model cells as bistable toggle switches whose dynamics are governed by a delayed stochastic simulation algorithm. Each state of the toggle switch makes the cell more fit in one of two environmental conditions. Different noise levels in protein expression yield different fitness values for cells in an environment that randomly switches between the two conditions. We compare the behavior of two cell types, one that can sense the environmental condition and one that cannot. In fast changing environments both cell types evolve to be as noisy as possible while maintaining bistability of the toggle switch. In slowly changing environments, evolved nonsensing cells are less noisy while sensing cells evolve the same noise level as in fast changing environments. Sensing removes the need of genotypic changes to adapt to changes in the environment fluctuation rate, providing an evolutionary advantage in unpredictable environments.

Figure 1

Time series of $P_1$ and $P_2$ $(⟨P_1+P_2⟩\sim 150)$ of a cell of population 2 (unable to sense the environment state).

Figure 2

Time series of $P_1$ and $P_2$ $(⟨P_1+P_2⟩\sim 150)$ of a cell of population 9 (unable to sense the environment state).

Figure 3

Noise in the time series of $P_1+P_2$ of individual cells, averaged over 1000 cells per cell population. Cell populations of the two cell types (sensing and nonsensing cells) differ in the values of $k_t$ and $k_d$. $W_{\mathit{fr}}=10000\mathrm{s}$.

Figure 4

Fitness of individual cells at the end of their lifetime, averaged over 1000 cells per cell population. Cell populations of the two cell types (sensing and nonsensing cells), differ in the values of $k_t$ and $k_d$. $W_{\mathit{fr}}=10000\mathrm{s}$.

Figure 5

The fraction of lifetime of individual cells during which both proteins are present (averaged over 1000 cells per cell population) for sensing and nonsensing cells. $W_{\mathit{fr}}=10000\mathrm{s}$.

Figure 6

Average noise level of $P_1+P_2$ time series of individual cells for sensing and nonsensing cells. 30 generations, 100 cells per generation. $W_{\mathit{fr}}=10000\mathrm{s}$. Changes in the mean levels of $P_1+P_2$ are below $\sim 10\%$ in both cell types.

Figure 7

Average fitness of sensing and nonsensing cells. 30 generations, 100 cells per generation. $W_{\mathit{fr}}=10000\mathrm{s}$.

Figure 8

Fitness at the end of the lifetime of nonsensing cells (averaged over 1000 cells per population) of cell populations with distinct values of $k_t$ and $k_d$ subject to fast $(W_{\mathit{fr}}=10000\mathrm{s})$ and slowly switching environments $(W_{\mathit{fr}}=500000\mathrm{s})$.

Figure 9

Fitness at the end of the lifetime of sensing cells (averaged over 1000 cells per population) of cell populations with distinct values of $k_t$ and $k_d$ subject to fast $(W_{\mathit{fr}}=10000\mathrm{s})$ and slowly switching environments $(W_{\mathit{fr}}=500000\mathrm{s})$.

Figure 10

Normalized fitness at the end of the lifetime of sensing and nonsensing cells (averaged over 1000 cells per population) of cell populations with distinct values of $k_t$ and $k_d$ subject to rapidly $(W_{\mathit{fr}}=10000\mathrm{s})$ and slowly switching environments $(W_{\mathit{fr}}=500000\mathrm{s})$.

Figure 11

Average of 100 simulations of the ratio of sensing cells in populations of 100 cells over ten generations in a slowly switching environment $(W_{\mathit{fr}}=500000\mathrm{s})$ and in a rapidly switching environment $(W_{\mathit{fr}}=10000\mathrm{s})$.