Optimal transition paths of phenotypic switching in a non-Markovian self-regulation gene expression

Gene expression is a complex biochemical process involving multiple reaction steps, creating molecular memory because the probability of waiting time between consecutive reaction steps no longer follows exponential distributions. What effect the molecular memory has on metastable states in gene expression remains not fully understood. Here, we study transition paths of switching between bistable states for a non-Markovian model of gene expression equipped with a self-regulation. Employing the large deviation theory for this model, we analyze the optimal transition paths of switching between bistable states in gene expression, interestingly finding that dynamic behaviors in gene expression along the optimal transition paths significantly depend on the molecular memory. Moreover, we discover that the molecular memory can prolong the time of switching between bistable states in gene expression along the optimal transition paths. Our results imply that the molecular memory may be an unneglectable factor to affect switching between metastable states in gene expression.

Figure 1

A sketch map of gene expression with a self-regulation circuit.

Figure 2

The stationary probability distributions and the Fano factors of gene expression. (a) The probability distributions for the different molecular memories $k$. $\alpha =0.5$, $\beta =1$, $K=5$, $m=2$, $\delta =0.2,$ and $\mathrm{\Omega }=5$. (b) The Fano factors in the protein copy number for the different molecular memory parameter $k$. $\alpha =0.5$, $\beta =1$, $K=5$, $m=2$, $k=2,$ and $\mathrm{\Omega }=5$. (c) The Fano factors for the different feedback strength $\beta$. (d) The Fano factor versus $k$ and $\beta$. $\alpha =0.5$, $K=5$, $m=2$, $\delta =0.2,$ and $\mathrm{\Omega }=$ 5.

Figure 3

(a) The curves of the functions in the system (21). The black line is $\delta x$. The red curve is the feedback function $f\left(x\right)$ without the molecular memory, the green and blue curves corresponding to the feedback functions with $k=5$ and $k=3$, respectively. (b) Equilibrium points versus the strength of the molecular memory $k$, where the low, intermediate and high states correspond to the points $x_0$ (stable), $x_1$ (unstable) and $x_2$ (stable), respectively. The parameters are taken as $\alpha =0.2$, $\beta =5$, $K=10$, $m=10$, $\delta =0.2$ in (a), and $\delta =0.35$ in (b).

Figure 4

The phase diagram ($x,\lambda$) of the Hamilton system (19). The points ($x_0,0$), ($x_1,0$) and ($x_2,0$) are all unstable saddle points. The parameters are taken as $\alpha =0.2$, $\beta =5$, $K=10$, $m=10$, $k=1,$ and $\delta =2$.

Figure 5

The optimal transition paths and the minimal actions versus the strength of the molecular memory $k$. (a) The optimal transition paths of switching from the point ($0,0$) to the point ($2,0$). (b) The optimal transition paths of switching from the point ($2,0$) to the point ($0,0$). (c) The minimal actions along the optimal transition paths in the subplot A. (d) The minimal actions along the optimal transition paths in the subplot B. The parameters are set as $\alpha =0.2$, $\beta =5$, $K=10$, $m=10$, and $\delta =0.35$.

Figure 6

The switching time versus the molecular memory $k$. (a) The time of switching from the low state to the high state. (b) The time of switching from the high state to the low state. The parameters are set as $\alpha =0.2$, $\beta =5$, $K=10$, $m=10$, $\mathrm{\Omega }=50,$ and $\delta =0.35$.