Extrinsic Noise Driven Phenotype Switching in a Self-Regulating Gene

Analysis of complex gene regulation networks gives rise to a landscape of metastable phenotypic states for cells. Heterogeneity within a population arises due to infrequent noise-driven transitions of individual cells between nearby metastable states. While most previous works have focused on the role of intrinsic fluctuations in driving such transitions, in this Letter we investigate the role of extrinsic fluctuations. First, we develop an analytical framework to study the combined effect of intrinsic and extrinsic noise on a toy model of a Boolean regulated genetic switch. We then extend these ideas to a more biologically relevant model with a Hill-like regulatory function. Employing our theory and Monte Carlo simulations, we show that extrinsic noise can significantly alter the lifetimes of the phenotypic states and may fundamentally change the escape mechanism. Finally, our theory can be readily generalized to more complex decision making networks in biology.

Figure 1

${\tau }_{\mathrm{hi}\to \mathrm{low}}$ versus the relative strength for white ${\tau }_c={10}^{-2}$ (a) and long-correlated ${\tau }_c={10}^3$ (b),(c) EN. MC simulations (the symbols) with noise in the production (circles) and decay (crosses) rates (that are indistinguishable) are compared with theory (lines): Eq. (8) (a) and Eq. (10) (b),(c). Here $N=5000$, ${\alpha }_0=0.01$, and $x_0=0.93$ (a),(b) or $x_0=0.915$ (c). The values of ${\tau }_{\mathrm{hi}\to \mathrm{low}}$ are measured in units of cell cycles.

Figure 2

${\tau }_{\mathrm{hi}\to \mathrm{low}}$ as function of ${\tau }_c$ for various EN strengths: ${\sigma }_{\mathrm{EX}}/\mu =0.01$ (top left), 0.1 (top right), and 0.2 (bottom left). The lines are the analytical predictions of Eq. (8) (solid) and Eq. (10) (dotted). The lower right panel (see also inset) shows ${\tau }_c^{\mathrm{opt}}$ versus EN strength: MC results (symbols) confirm the functional dependence on ${\sigma }_{\mathrm{EX}}$ predicted by Eq. (11) (line). Here $N=5000$, ${\alpha }_0=0.01$, and $x_0=0.93$.

Figure 3

(a),(b) The location of the stochastic fixed points of Eq. (1) with $f(x)$ given by Eq. (12), for EN with ${\tau }_c={10}^3$ and ${\sigma }_{\mathrm{EX}}/\mu =0.01$ (a) and ${\sigma }_{\mathrm{EX}}/\mu =0.05$ (b). The solid lines show the deterministic fixed points. (c)–(e) The steady-state 2D PDFs of finding protein number $n$ and instantaneous EN magnitude ${\xi }_i$ for ${\sigma }_{\mathrm{EX}}/\mu =0.05$ and $x_0=0.42$ (c), 0.48 (d), and 0.56 (e). Here $N=300$, and ${\alpha }_0=0.05$.

Figure 4

(a) ${\tau }_{\mathrm{low}\to \mathrm{hi}}$ and (b) ${\tau }_{\mathrm{hi}\to \mathrm{low}}$ for various EN strengths and ${\tau }_c={10}^3$. The population fraction $P_{\mathrm{low}}$ (c) and $P_{\mathrm{hi}}$ (d) in the low and hi states in steady state.