Transitions in a genetic transcriptional regulatory system under Lévy motion

Abstract

Based on a stochastic differential equation model for a single genetic regulatory system, we examine the dynamical effects of noisy fluctuations, arising in the synthesis reaction, on the evolution of the transcription factor activator in terms of its concentration. The fluctuations are modeled by Brownian motion and α-stable Lévy motion. Two deterministic quantities, the mean first exit time (MFET) and the first escape probability (FEP), are used to analyse the transitions from the low to high concentration states. A shorter MFET or higher FEP in the low concentration region facilitates such a transition. We have observed that higher noise intensities and larger jumps of the Lévy motion shortens the MFET and thus benefits transitions. The Lévy motion activates a transition from the low concentration region to the non-adjacent high concentration region, while Brownian motion can not induce this phenomenon. There are optimal proportions of Gaussian and non-Gaussian noises, which maximise the quantities MFET and FEP for each concentration, when the total sum of noise intensities are kept constant. Because a weaker stability indicates a higher transition probability, a new geometric concept is introduced to quantify the basin stability of the low concentration region, characterised by the escaping behaviour.

The Onsager–Machlup function as Lagrangian for the most probable path of a jump-diffusion process

“…However, random fluctuations in complex biological and physical systems are often non-Gaussian rather than Gaussian [34,32,31]. Lévy motions are appropriate models for a class of important non-Gaussian processes with jumps or bursts [5].…”

The maximum likelihood climate change for global warming under the influence of greenhouse effect and Levy noise

“…The optimization problem related to the most probable transition path and time t f are employed by the theory of large deviations [34]. Here, the arrival time t f can be determined by Monte Carlo simulations, which calculate the average time of arriving at state x f , or by the first mean exit time starting at state x 0 from an interval D as in our earlier work [35]. We emphasize here that the time t ∈ [0, 10] is dimensionless.…”

Geometric Methods for Stochastic Dynamical Systems

“…The low concentration and high concentration of specific protein correspond to different cell activities. Recent studies [Xu et al (2013), Zheng et al (2016), ] have recognized that Lévy motion can induce switches between different protein concentrations. Multiple phenotypic states often arise in a single cell with different gene expression states.…”