Most probable dynamics of a genetic regulatory network under stable Lévy noise

Abstract

Numerous studies have demonstrated the important role of noise in the dynamical behaviour of a complex system. The most probable trajectories of nonlinear systems under the influence of Gaussian noise have recently been studied already. However, there has been only a few works that examine how most probable trajectories in the two-dimensional system (MeKS network) are influenced under non-Gaussian stable Lévy noise. Therefore, we discuss the most probable trajectories of a two-dimensional model depicting the competence behaviour in B. subtilis under the influence of stable Lévy noise. On the basis of the Fokker-Planck equation, we describe the noise-induced most probable trajectories of the MeKS network from the low ComK protein concentration (vegetative state) to the high ComK protein concentration (competence state) under stable Lévy noise.We demonstrate choices of the non-Gaussianity index α and the noise intensity ǫ which generate the ComK protein escape from the low concentration to the high concentration. We also reveal the optimal combination of both parameters α and ǫ making the tipping time shortest. Moreover, we find that different initial concentrations around the low ComK protein concentration evolve to a metastable state, and provide the optimal α and ǫ such that the distance between the deterministic competence state and the metastable state is smallest.

The first passage problem for stable linear delay equations perturbed by power law Lévy noise

“…Furthermore Bódai and Franzke (2017) give evidence for physicality of Lévy noises in particular due to the predictability of fat-tail extremes while in the limit of small noise intensity, the (re-normalized) exit times are exponentially distributed and hence memoryless or unpredictable. Heavy-tailed noise has also been found present in many physical systems, for instance in works by Ditlevsen (1999a,b), Chen et al (2019) and Gairing et al (2017).…”

State transitions in the Morris-Lecar model under stable Lévy noise

“…The stochastic pitchfork bifurcation for a system under multiplicative stable Lévy noise by the maximal likely trajectory is studied in [38]. Similarly, this indicator is also used in gene regulatory systems under stable Lévy noise [39]. The definition of the maximal likely trajectory is as follows:…”

Nonlocal Dynamics for Non-Gaussian Systems Arising in Biophysical Modeling

“…This review article follows our earlier works [3,4] about gene transcriptions in a two-dimensional genetic regulatory model, as proposed by Süel et al [5]. Comparing with [3], we make a rescaling of this model when considering the mean exit time and escape probability.…”

“…The Fokker-Planck equation for the distribution of the conditional probability density p(k, s, t) = p(k, s, t|k 0 , s 0 , 0) , i.e., the probability of the process (k(t), s(t)) has value (k 0 , s 0 ) at time t given it had value (k 0 , s 0 ) at time 0, is p t = A * p given by [1,2], where A * is the adjoint operator for A . Thus we have the Fokker-Planck equation [4]:…”

“…The probability density function of the stochastic solution, with a given initial state, for the system (2.2) is governed by the corresponding Fokker-Planck equation [4,21]. On the basis of the Fokker-Planck equation, we can get the maximal likely trajectory, with the given initial state, by computing the maximizer of the probability density function p(k, s, t) at every time t. This offers the maximal likely trajectory of a stochastic dynamic system [22].…”