A genetic timer through noise-induced stabilization of an unstable state

Abstract

Stochastic fluctuations affect the dynamics of biological systems. Typically, such noise causes perturbations that can permit genetic circuits to escape stable states, triggering, for example, phenotypic switching. In contrast, studies have shown that noise can surprisingly also generate new states, which exist solely in the presence of fluctuations. In those instances noise is supplied externally to the dynamical system. Here, we present a mechanism in which noise intrinsic to a simple genetic circuit effectively stabilizes a deterministically unstable state. Furthermore, this noise-induced stabilization represents a unique mechanism for a genetic timer. Specifically, we analyzed the effect of noise intrinsic to a prototypical two-component gene-circuit architecture composed of interacting positive and negative feedback loops. Genetic circuits with this topology are common in biology and typically regulate cell cycles and circadian clocks. These systems can undergo a variety of bifurcations in response to parameter changes. Simulations show that near one such bifurcation, noise induces oscillations around an unstable spiral point and thus effectively stabilizes this unstable fixed point. Because of the periodicity of these oscillations, the lifetime of the noise-dependent stabilization exhibits a polymodal distribution with multiple, well defined, and regularly spaced peaks. Therefore, the noise-induced stabilization presented here constitutes a minimal mechanism for a genetic circuit to function as a timer that could be used in the engineering of synthetic circuits.

Fig. 1.

Continuous and discrete stochastic simulations of a simple activator–repressor circuit generate qualitatively identical excitable dynamics. (A) Schematic diagram of the activator–repressor circuit. Activator and repressor proteins are denoted as A and R with corresponding genes a, r and promoters P_a and P_r, respectively. B and D as well as corresponding C and E compare continuous and discrete stochastic simulations of the repressor–activator circuit (k_r = 0.12). (B) Nullcline portrait of the activator–repressor circuit obtained from the two-dimensional continuous model described in the text. Activator and repressor nullclines are depicted in blue and green, respectively. One stable and two unstable fixed points at the three nullcline intersections are shown in filled red circle (stable node), open red circle (unstable saddle), and open red diamond (unstable focus), respectively. Black line denotes the continuous excitable trajectory of the system with gray arrows indicating its direction. (C) Continuous time traces of activator A (blue) and repressor R (green) molecule numbers during the transient excitable event shown in B. (D) The same nullcline portrait shown in B overlaid with five sample excitable trajectories (black) of excitable dynamics obtained from discrete stochastic simulations, which inherently account for biochemical noise within the repressor–activator circuit. Shown in E are stochastic time traces of activator A and repressor R molecule numbers during one of the excitable events shown in D.

Fig. 2.

Near a bifurcation, noise induces qualitative differences in activator–repressor circuit dynamics. (A) Enlarged view around the upper fixed point (in red) shown in Fig. 1 B and D, of activator and repressor nullclines in blue and green, respectively. For increasing values of the k_r parameter (which accounts for the binding affinity of the repressor promoter for activator protein), the blue activator nullcline shifts from left to right. At the critical value of k_r = 0.1559 (indicated with black-yellow dotted line) the upper fixed point undergoes a Hopf bifurcation and switches from an unstable focus (open red diamond) to a stable focus (filled red circle). Thus, for k_r values smaller than the critical value, the activator–repressor circuit constitutes an excitable system, whereas for higher k_r values the system is bistable. (B and D) Shown in black are excitable trajectories of continuous and discrete stochastic simulations, respectively, for a value of k_r = 0.14 for which the upper fixed point is deterministically unstable. Note that in D, the three sample trajectories obtained from discrete stochastic simulations orbit the upper deterministically unstable fixed point, whereas no such behavior is observed in the continuous simulation shown in B. (C) Continuous simulation time traces of activator and repressor molecule numbers are consistent with excitable dynamics as shown in Fig. 1C. (E) Discrete stochastic simulation time traces exhibit small oscillations at high molecule numbers that clearly differ from continuous simulation results shown in C obtained for identical k_r values.

Fig. 3.

Noise effectively stabilizes a deterministically unstable fixed point. (A) Phase portrait of the system for k_r = 0.14, with activator and repressor nullclines shown in blue and green, respectively. The deterministic stable manifold of the saddle point (open red circle) is shown in magenta. The stable fixed point is shown as a filled red circle, and the unstable focus as an open red diamond. (B) Enlarged view of phase portrait for the region around the upper unstable fixed point, as indicated by gray dotted line in A. Black lines are two deterministic trajectories with two different starting points with respect to the stable manifold (yellow-highlighted region). One of these trajectories (dashed black line) starts just outside of the stable manifold and returns directly to the stable fixed point shown in A. The other trajectory (solid back line) starts just inside the stable manifold and orbits the unstable fixed point once before returning to the stable fixed point. The red line is a trajectory obtained from discrete stochastic simulations. Noise inherent to the stochastic simulations causes the trajectory to cross the stable manifold (yellow-highlighted region). This results in the system orbiting the unstable fixed point at least once before ultimately returning back to the lower stable fixed point. This extends the time the system spends in the high-activity state and thus the duration of the trajectory.

Fig. 4.

Noise-induced stabilization generates quantized durations of activator–repressor circuit dynamics. A and B show results from discrete stochastic simulations obtained for k_r = 0.1475, which is smaller than the deterministically critical bifurcation value. (A) Sample time traces aligned in time with respect to maximum of first high molecule number peak. Depicted in gray, magenta, green, and blue are sample traces that orbit the deterministically unstable fixed point once, twice, three, or four times, respectively. Transitions from high- to low-molecule-number states are restricted to discretized time windows. (B) Histogram of duration times in high-activity state obtained from 5,000 trajectories including sample traces depicted in A. Note the polymodal nature of the distribution. (C) Histograms of high-activity-state durations (as shown in B) for indicated k_r values, normalized to its maximum for each k_r value, and color-coded with respect to the histogram of count values (in logarithmic scale). Note the polymodal distribution of duration times.