Data-driven approximation for extracting the transition dynamics of a genetic regulatory network with non-Gaussian Lévy noise

Abstract

In the study of biological systems, several methods based on statistical physics or machine learning have been developed for inference or prediction in the presence of complicated nonlinear interactions and random noise perturbations. However, there have been few studies dealing with the stochastic non-Gaussian perturbation case, which is more natural and universal than Gaussian white noise. In this manuscript, for a two-dimensional biological model (the MeKS network) perturbed by non-Gaussian stable Lévy noise, we use a data-driven approach with theoretical probabilistic foundation to extract the rare transition dynamics representing gene expression. This involves theories of non-local Kramers–Moyal formulas and the non-local Fokker–Planck equation, as well as the corresponding numerical algorithms, aimed at extracting the maximum likelihood transition path. The feasibility and accuracy of the method are checked. Furthermore, several dynamical behaviors and indicators are investigated. In detail, the investigation shows a bistable transition probability state of the ComK protein concentration and bifurcations in the learned transition paths from vegetative state to competence state. Analysis of the tipping time illustrates the difficulty of the gene expression. This method will serve as an example in the study of stochastic systems with non-Gaussian perturbations from biological data, and provides some insights into the extraction of other dynamical indicators, such as the mean first exit time and the first escape probability with respect to their own biological interpretations.