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Stochastic Stabilization of Phenotypic States: The Genetic Bistable Switch as a Case Study

Figure 3

Bifurcation diagram of the simple genetic switch.

Deterministic system (blue line), white-noise stochastic system (Langevin: green line; Exact solution: red line). The bifurcation diagram of a system with colored fluctuations in the limit is also depicted (cyan line). In all cases . The results from stochastic simulations are in agreement with the analytical results, as can be seen by the detected peaks (orange circles) of the probability distribution of at steady state (color code, logarithmic scale). The numerical simulations for a non-null correlation time noisy sytem, , indicate that the effect is lessened when memory is considered (purple circles). The top inset reveals that the probability distributions obtained in numerical simulations (orange histogram) are in perfect agreement with the exact solution (red line) and the Langevin description (green line), . For that value of the control parameter the deterministic system only have one stable solution and the probability distribution corresponds to a Dirac delta (blue arrow). When the correlation time of the noise is not null, , the stability of the low state decreases with respecto to the white noise case (purple line). The circles in the inset denote the maxima as detected by the Gaussian peak detection algorithm. Bottom inset: increasing noise (decreasing volume) clearly extends the stable branch of the low state, an effect that we call stochastic stabilization: , 50, 30 and 12.5.

doi: https://doi.org/10.1371/journal.pone.0073487.g003