Improving the speed of the genetic toggle switch without sacrificing its dynamic stability

Determinants of the switching speed of the genetic toggle switch remain unknown. Analysis shows that the decay rate of proteins predominantly sets the speed limit, but its modification introduces a trade-off between increased speed and decreased bistability. Incorporating protein-modifying enzymes into the switch gives extra degrees of freedom to address this trade-off. The condition for bistability when such enzymes are incorporated is derived. Under this condition, speed increases with the maximal rate of gene expression.

Figure 1

Trade-off between speed and bistability in the genetic toggle switch. (a) Common design of the genetic toggle switch ([5, 6, 7]). (b) Flow field in the phase plane of the genetic toggle switch. Magnitude of speed is indicated by the length of marks. (c) Speed of the system increases as $\eta$ increases. Simulation was done with the following parameters: $X\left(0\right)=2.0\%$ of the maximal level, $Y\left(0\right)=1.0\%$ of the maximal level, $\epsilon =5.0$, $\gamma =5.0$, $\kappa =1.5$, and $\eta =0.2$ (—), 0.4 (– – –), 0.6 $(–\bullet –)$, 0.8 $(\cdots )$. Time step $\mathrm{\Delta }t=0.005$. Both $X\left(t\right)$ and $Y\left(t\right)$ were normalized by their maximal levels. (d) The system loses bistability as the decay rate of the repressors $\eta$ increases. Shown is the ratio between the steady-state concentrations of the two repressors as a function of $\eta$ (system parameters: $\epsilon =5.0$, $\gamma =5.0$, $\kappa =1.5$). Dotted line $(\cdots )$ is the approximation given by Eq. (A1).

Figure 2

Incorporation of protein-modifying enzymes into the toggle switch. (a) Topology of the system described by Eq. (2). (b) Speed of the system increases when enzymes are incorporated. Solid lines (—) represent the system described by Eq. (2) [simulation parameters: $X\left(0\right)=0.2$, $Y\left(0\right)=0.1$, $Z\left(0\right)=0$, $W\left(0\right)=0$, $\epsilon =5.0$, $\gamma =5.0$, $\kappa =1.5$, $\eta =1.0$, $\lambda =2.0$, $K_m=2.0$, $\mathrm{\Delta }t=0.005$]. Dotted lines $(\cdots )$ represent the system described by Eq. (1) (simulation parameters same as above, except $\lambda =0$ to account for the lack of enzymatic action).

Figure 3

(Color) Bifurcation diagram for the parameter $\lambda$. As $\lambda$ increases from zero, the system first transforms from bistability into tristability, and then into monostability. Theoretical prediction for the boundary between bistability and its breakdown [Eq. (D1)] is $\lambda =3.8$ when $K_m=1.0$, and $\lambda =7.7$ when $K_m=2.0$. Increasing $K_m$ expands the territory of bistability. Values of other parameters: $\epsilon =2.0$, $\gamma =5.0$, $\kappa =1.5$, and $\eta =1.0$. Unstable branches are drawn in light colors, and stable branches in dark. Note that $\lambda =0$ corresponds to the system described by Eq. (1).

Figure 4

(Color) (a) Bifurcation diagram for the parameter ${\lambda }_2$ at different values of $K_{m2}$. Values of other parameters: $\epsilon =2.0$, $\gamma =5.0$, $\kappa =1.5$, $\eta =1.0$, $K_{m1}=1.0$, and ${\lambda }_1=3.0$. (b) Bifurcation diagram for the parameter ${\lambda }_2$ at different values of $\epsilon$, $\gamma$, and $\kappa$. Values of other parameters: $\eta =1.0$, $K_{m1}=K_{m2}=1.0$, and ${\lambda }_1=3.0$. Unstable branches are drawn in light colors, and stable branches in dark. Plots of different colors are slightly vertically translated to prevent superposition. See Appendix for the mathematical details.