Regime shifts driven by dynamic correlations in gene expression noise

Gene expression is a noisy process that leads to regime shifts between alternative steady states among individual living cells, inducing phenotypic variability. The effects of white noise on the regime shift in bistable systems have been well characterized, however little is known about such effects of colored noise (noise with nonzero correlation time). Here, we show that noise correlation time, by considering a genetic circuit of autoactivation, can have a significant effect on the regime shift between distinct phenotypic states in gene expression. We demonstrate this theoretically, using stochastic potential, stationary probability density function, and first-passage time based on the Fokker-Planck description, where the Ornstein-Uhlenbeck process is used to model colored noise. We find that an increase in noise correlation time in the degradation rate can induce a regime shift from a low to a high protein concentration state and enhance the bistable regime, while an increase in noise correlation time in the basal rate retains the bimodal distribution. We then show how cross-correlated colored noises in basal and degradation rates can induce regime shifts from a low to a high protein concentration state, but reduce the bistable regime. We also validate these results through direct numerical simulations of the stochastic differential equation. In gene expression understanding the causes of regime shift to a harmful phenotype could improve early therapeutic intervention in complex human diseases.

Figure 1

A schematic of the autoactivating genetic switch. The expression of gene leads to protein monomers (TF-A) and after oligomerization they bind to the upstream regulatory site (TF-RE), activating production of monomers. Degradation of protein and mRNA are denoted by the slashed circles ($⌀$).

Figure 2

Stochastic potential $\varphi \left(x\right)$ and SPDF $P_s\left(x\right)$ of the system (10): (a) $\varphi \left(x\right)$ for the noise intensity ${\sigma }_1=0.7$ with ${\tau }_1=0.5, r=0.1$, and $a=3$, and (b) $P_s\left(x\right)$ for three different values of ${\sigma }_1$: ${\sigma }_1=0.3$ (blue curve), ${\sigma }_1=0.5$ (green curve), and ${\sigma }_1=0.7$ (red curve) with fixed ${\tau }_1=0.5, r=0.1$, and $a=3$. (c) $\varphi \left(x\right)$ for the correlation time ${\tau }_1=0.9$ with ${\sigma }_1=0.5, r=0.1$, and $a=3.5$, and (d) $P_s\left(x\right)$ for three different values of ${\tau }_1$: ${\tau }_1=0.009$ (blue curve), ${\tau }_1=0.4$ (green curve), and ${\tau }_1=0.9$ (red curve) with fixed ${\sigma }_1=0.5, r=0.1$, and $a=3.5$. The increase in ${\sigma }_1$ induces regime shift from high to low protein concentration state, whereas increase in ${\tau }_1$ induces shift from low to high protein concentration state.

Figure 3

The evolution of SPDF $P_s\left(x\right)$ of the system (10) for continuously changing correlation time ${\tau }_1$. The other parameters are ${\sigma }_1=0.5, r=0.1$, and $a=3.5$. As the $P_s\left(x\right)$ corresponding to the right potential well has increased with increase in ${\tau }_1$, the system experiences a regime shift from low to high protein concentration state.

Figure 4

Sample paths of the protein concentration $x$ obtained from the direct numerical simulation of Langevin equation (4). The protein concentration $x$ stays: (a) in the vicinity of the low concentration state, and (b) in the vicinity of the high concentration state. In (a) the parameter values are same as in Fig. 2 and in (b) the parameter values are same as in Fig. 2. The results resemble the theoretical calculation as presented in Fig. 2 for the same set of parameter values.

Figure 5

Extrema of the SPDF $P_s\left(x\right)$ of the model (10) coupled with noise only in the degradation rate, as a function of $a$: (a) for a fixed ${\tau }_1=1.9$ and increasing values of ${\sigma }_1$, and (b) for a fixed ${\sigma }_1=0.2$ and increasing values ${\tau }_1$. The other parameters are $r=0.1, {\sigma }_2=0, {\tau }_2=0, {\tau }_3=0$, and $\lambda =0$. The increase in ${\sigma }_1$ reduces the bistability regime, whereas increase in ${\tau }_1$ increases the bistability regime.

Figure 6

$\varphi \left(x\right)$ and $P_s\left(x\right)$ when noise is present only in the basal rate: (a) $\varphi \left(x\right)$ for the noise intensity ${\sigma }_2=0.29$ with ${\tau }_2=1$, and (b) $P_s\left(x\right)$ for three different values of ${\sigma }_2$: ${\sigma }_2=0.25$ (blue curve), ${\sigma }_2=0.27$ (green curve), and ${\sigma }_2=0.29$ (red curve) with fixed ${\tau }_2=1$. (c) $\varphi \left(x\right)$ for the correlation time ${\tau }_2=1$ with ${\sigma }_2=0.5$, and (d) $P_s\left(x\right)$ for three different of values of ${\tau }_2$: ${\tau }_2=0.05$ (blue curve), ${\tau }_2=0.5$ (green curve), and ${\tau }_2=1$ (red curve) with fixed ${\sigma }_2=0.5$. Other parameters are $r=0.1$ and $a=1.9$. Increase in ${\sigma }_2$ and ${\tau }_2$ has not much effect on $\varphi \left(x\right)$ and $P_s\left(x\right)$, however, both the valleys and tops of $P_s\left(x\right)$ decay in height.

Figure 7

Sample paths of the protein concentration $x$ obtained from the direct numerical simulation of Langevin Eq. (4). In both (a) and (b), the protein concentration $x$ successively switches in between the low and the high protein concentration states. In (a) the parameter values are same as in Fig. 6 and in (b) the parameter values are same as in Fig. 6. The results resemble the theoretical prediction as depicted in Fig. 6.

Figure 8

$\varphi \left(x\right)$ and $P_s\left(x\right)$, when the model (4) driven by cross-correlated noises: (a) $\varphi \left(x\right)$ for the cross-correlation strength $\lambda =0.9$ with ${\sigma }_1=0.01, {\sigma }_2=0.004, {\tau }_1=0.01, {\tau }_2=0.08$, and ${\tau }_3=0.03$, and (b) $P_s\left(x\right)$ for three different values of $\lambda$: $\lambda =0.3$ (blue curve), $\lambda =0.7$ (green curve), and $\lambda =0.9$ (red curve) with ${\sigma }_1=0.01, {\sigma }_2=0.004, {\tau }_1=0.01, {\tau }_2=0.08$, and ${\tau }_3=0.03$. (c) $\varphi \left(x\right)$ for the correlation time ${\tau }_3=0.9$ with ${\sigma }_1=0.01, {\sigma }_2=0.004, \lambda =0.3, {\tau }_1=0.01$, and ${\tau }_2=0.08$, and (d) $P_s\left(x\right)$ for three different values of ${\tau }_3$: ${\tau }_3=0.009$ (blue curve), ${\tau }_3=0.09$ (green curve), and ${\tau }_3=0.9$ (red curve) with ${\sigma }_1=0.01, {\sigma }_2=0.004, \lambda =0.3, {\tau }_1=0.01$, and ${\tau }_2=0.08$. Other parameters are $r=0.1$ and $a=1.9$. Increase in both the $\lambda$ and ${\tau }_3$ induces regime shifts from low to high protein concentration state.

Figure 9

The evolution of SPDF $P_s\left(x\right)$ of the system (10) with continuous changes: (a) in the cross-correlation strength $\lambda$ for ${\tau }_3=0.03$, and (b) in the correlation time ${\tau }_3$ for $\lambda =0.3$. The other parameters are $r=0.1, a=1.9, {\sigma }_1=0.01, {\sigma }_2=0.004, {\tau }_1=0.01$, and ${\tau }_2=0.08$.

Figure 10

Sample paths of the protein concentration $x$. In both (a) and (b), the protein concentration $x$ shifts to the high protein concentration state and stays there. In (a) the parameter values are same as in Fig. 8 and in (b) the parameter values are same as in Fig. 8. The results resemble the theoretical calculation as presented in Fig. 8.

Figure 11

Extrema of the SPDF $P_s\left(x\right)$ of the gene regulation model (4) driven by cross-correlated noises, as a function of $a$: (a) for increasing values of the cross-correlation strength $\lambda$ with other parameter values $r=0.1, {\sigma }_1=0.2, {\sigma }_2=0.5, {\tau }_1=0.01, {\tau }_2=0.01$, and ${\tau }_3=0.1$, and (b) for increasing values of the correlation time ${\tau }_3$ with other parameter values $r=0.1, {\sigma }_1=0.2, {\sigma }_2=0.5, {\tau }_1=0.5, {\tau }_2=0.5$, and $\lambda =0.1$. The bistability regime reduces with increase in both $\lambda$ and ${\tau }_3$.

Figure 12

The effect of correlated noise in the degradation rate and the basal rate on the MFPT. (a) The MFPT $\langle T_{x_l^{st}\to x_u^{st}}\rangle$ decreases and $\langle T_{x_u^{st}\to x_l^{st}}\rangle$ increases with the increase in ${\tau }_1$. The other parameters are same as in Fig. 3. (b) MFPTs for both the states are behaving same with the variations in ${\tau }_2$, as also seen from the stochastic potential and SPDF in Figs. 6 and 6. The other parameters are same as in Fig. 6.

Figure 13

The effect of $\lambda$ and ${\tau }_3$ on the MFPT. (a) The MFPT $\langle T_{x_l^{st}\to x_u^{st}}\rangle$ decreases and $\langle T_{x_u^{st}\to x_l^{st}}\rangle$ increases with the increase of $\lambda$ for ${\tau }_3=0.03$. (b) Similar situation arises with the increase of ${\tau }_3$ for $\lambda =0.3$. The other parameters are same as in Fig. 9.