Stochastic simulations of genetic switch systems

Genetic switch systems with mutual repression of two transcription factors are studied using deterministic methods (rate equations) and stochastic methods (the master equation and Monte Carlo simulations). These systems exhibit bistability, namely two stable states such that spontaneous transitions between them are rare. Induced transitions may take place as a result of an external stimulus. We study several variants of the genetic switch and examine the effects of cooperative binding, exclusive binding, protein-protein interactions, and degradation of bound repressors. We identify the range of parameters in which bistability takes place, enabling the system to function as a switch. Numerous studies have concluded that cooperative binding is a necessary condition for the emergence of bistability in these systems. We show that a suitable combination of network structure and stochastic effects gives rise to bistability even without cooperative binding. The average time between spontaneous transitions is evaluated as a function of the biological parameters.

Figure 1

Schematic illustrations of (a) the general switch circuit, that includes two transcription factors, $A$ and $B$, which negatively regulate each other’s synthesis; (b) the exclusive switch, in which there is an overlap between the promoter sites of $A$ and $B$ proteins, so they cannot be bound simultaneously.

Figure 2

(Color online) The probability distribution $P(N_A,N_B)$ for the general switch, under conditions of (a) weak repression $(k=0.005)$ where there is one symmetric peak; and (b) strong repression $(k=50)$ where three peaks appear, one dominated by $A$, the second dominated by $B$ and the third in which both species are mutually suppressed. The weights of the three peaks are about the same.

Figure 3

(Color online) The population sizes of free $A$ and $B$ proteins vs time obtained from a Monte Carlo simulation (a) for the general switch, where the system exhibits fast transitions between its three states; (b) for the exclusive switch. The bistable behavior is clearly observed, where the population size of the dominant specie is between $20–60$ and the other specie is nearly diminished. Failed switching attempts are clearly seen. The typical switching time is in the order of ${10}^5$ (s) or roughly 1 day. Bound proteins are also shown. Their fast binding and unbinding events cannot be resolved on the time scale that is presented. In both cases, $g=0.2$, $d=0.005$, ${\alpha }_0=0.2$, and ${\alpha }_1=0.01$ $\left({\mathrm{s}}^{-1}\right)$.

Figure 4

(Color online) The probability distribution $P(N_A,N_B)$ for the exclusive switch, under conditions of (a) weak repression $(k=0.005)$ where there is one symmetric peak (b) intermediate repression $(k=1)$ where two distinct peaks begin to emerge but are still connected, and (c) strong repression $(k=50)$, where bistability is observed.

Figure 5

(Color online) Scaling properties of the switching time $\tau$ for the exclusive switch vs the protein synthesis rate $g$, the degradation rate $d$ and the repression strength $k$.

Figure 6

Population sizes of the free $A$ and $B$ proteins vs $k$ for the BRD switch obtained from the rate equations. The parameters are $g=0.05$, $d=d_r=0.005$, ${\alpha }_1=0.01$, and ${\alpha }_0$ is varied. Here $k_c\approx 1.92$. Stable solutions are shown by solid lines and unstable solutions by dashed lines.

Figure 7

(Color online) The switching time $\tau$ vs $k$ for the exclusive $(\times )$, BRD $(\circ )$ and PPI $(▵)$ switch systems. The parameters used are $g=0.05$, $d=d_r=0.005$, and $\gamma =0.1$ $\left({\mathrm{s}}^{-1}\right)$.

Figure 8

The range of parameters in the $(\gamma ,k)$ plane in which bistability takes place in the PPI switch (solid line) and in the exclusive-PPI switch (dashed line), using rate equations (a) and using the master equation (b). The other parameters are $g=0.05$ and $d=0.005$ $\left({\mathrm{s}}^{-1}\right)$.

Figure 9

(Color online) The dependence of the switching time $\tau$ for the dimers exclusive switch on the dimers degradation rate $d_D$ (a) and the dimerization rate ${\gamma }_D$ (b).

Figure 10

Probability for the exclusive switch to flip during 1800 (s) after the initiation of the signal, as a function of the external signal duration. The parameters used were $g=0.2$, $d=0.005$, ${\alpha }_1=0.01$, and ${\alpha }_0=0.2$ or zero during the signal.