Lévy noise induced transition and enhanced stability in a gene regulatory network

We investigate a quantitative bistable two-dimensional model (MeKS network) of gene expression dynamics describing the competence development in the Bacillus subtilis under the influence of Lévy as well as Brownian motions. To analyze the transitions between the vegetative and the competence regions therein, two dimensionless deterministic quantities, the mean first exit time (MFET) and the first escape probability, are determined from a microscopic perspective, as well as their averaged versions from a macroscopic perspective. The relative contribution factor $λ$⁠, the ratio of non-Gaussian and Gaussian noise strengths, is adopted to identify an optimum choice in these transitions. Additionally, we use a recent geometric concept, the stochastic basin of attraction (SBA), to exhibit a pictorial comprehension about the influence of the Lévy motion on the basin stability of the competence state. Our main results indicate that (i) the transitions between the vegetative and the competence regions can be induced by the noise intensities, the relative contribution factor $λ$ and the Lévy motion index $α$⁠; (ii) a higher noise intensity and a larger $α$ with smaller jump magnitude make the MFET shorter, and the MFET as a function of $λ$ exhibits one maximum value, which is a signature of the noise-enhanced stability phenomenon for the vegetative state; (iii) a larger $α$ makes the transition from the vegetative to the adjacent competence region to occur at the highest probability. The Lévy motion index $α0≈0.5$ (a larger jump magnitude with a lower frequency) is an ideal choice to implement the transition to the non-adjacent competence region; (iv) there is an expansion in SBA when $α$ decreases.