Multistability of synthetic genetic networks with repressive cell-to-cell communication

We investigate an experimentally feasible synthetic genetic network consisting of two phase repulsively coupled repressilators, which evokes multiple coexisting stable attractors with different features. We perform a bifurcation analysis to determine and classify the dynamical structure of the system. Moreover, some of the dynamical regimes found, such as inhomogeneous steady states and inhomogeneous limit cycles can further be associated with artificial cell differentiation. We also report and characterize the emergence of chaotic dynamics resulting from the intercell coupling.

Figure 1

(Color online) Scheme of the repressilator with quorum sensing cell-to-cell communication: with reinforcing coupling [5] (top) and repressive coupling (bottom).

Figure 2

(Color online) Typical time series of the $a_i$ mRNA concentration for the four stable regimes: (a) $Q=0.1$, oscillatory; (b) $Q=0.3$, inhomogeneous limit cycle; (c) $Q=0.4$, inhomogeneous steady state; and (d) $Q=0.4$, homogeneous steady state. The common parameters are $N=2$, $n=2.6$, $\alpha =216$, ${\beta }_a=0.85$, ${\beta }_b=0.1$, ${\beta }_c=0.1$, $\kappa =25$, $k_{s0}=1.0$, $k_{s1}=0.01$, $\eta =2.0$.

Figure 3

(Color online) Bifurcation diagram obtained by variation of $Q$, illustrating the stable-steady-state regimes (HSS and IHSS). For parameters values see Fig. 2. Here, thin solid lines denote the HSS, thick solid lines the IHSS, and dashed lines denote unstable steady states. The same bifurcation diagram is valid for the second repressilator.

Figure 4

(Color online) Bifurcation diagram versus coupling $Q$, with a focus on the IHLC and the IHSS. The stable IHSS is represented by a thick blue line, the stable IHLC with a thick orange line, and the unstable IHLC is represented with a dashed yellow line. Parameters are those of Fig. 2.

Figure 5

(Color online) Bifurcation diagram versus coupling $Q$, focusing on the stable antiphase oscillations (thick yellow line). Parameters are those of Fig. 2.

Figure 6

(Color online) Bifurcation diagram versus the coupling $Q$ where the periodic branch of the unstable synchronous oscillations (red dashed lines) is shown. Parameters as in Fig. 2.

Figure 7

(Color online) Comparison between the bifurcation analysis (top) and the direct calculation with random initial conditions (bottom). Note the logarithmic scale of both ordinates in the two plots. The oscillatory regime is represented by a yellow solid line (top) and a yellow area (bottom); the IHLC by solid orange lines (top) and a orange-white chess board pattern (bottom); the IHSS by solid blue lines (top) and a small blue striped area (bottom); and finally the HSS is illustrated by a solid black line (top) and a grey area (bottom). Parameters are those of Fig. 2.

Figure 8

Phase diagrams of the self-oscillations in the $A_1\text{−}{B}_1$ plane for large coupling $Q$: (a) $Q=0.5$, (b) $Q=0.6$, (c) $Q=0.7$, and (d) $Q=0.96$. Other parameters are those in Fig. 2.

Figure 9

Maximal Lyapunov exponent (top) and the corresponding bifurcation diagram (bottom) versus coupling $Q$. Parameters as in Fig. 2.

Figure 10

(Color online) Correlation sum $C\left(\epsilon \right)$ for (a) the biologically relevant range $Q∊[0,1)$, and (b) $Q$ outside of the biologically relevant range (only for theoretical completeness and justification of Fig. 9).