Most Probable Evolution Trajectories in a Genetic Regulatory System Excited by Stable L\'evy Noise

TL;DR

This paper investigates the most probable concentration trajectories in a genetic regulation system influenced by non-Gaussian stable Le9vy noise, revealing counter-intuitive phenomena and potential control strategies for gene transcription.

Contribution

It introduces a method to compute the most probable trajectories under Le9vy noise and uncovers novel effects of noise intensity and asymmetry on gene transcription.

Findings

Smaller noise intensity can trigger transcription, larger noise may not.

Symmetric Le9vy noise always induces transition to high concentration.

Asymmetric Le9vy noise can fail to trigger transcription.

Abstract

We study the most probable trajectories of the concentration evolution for the transcription factor activator in a genetic regulation system, with non-Gaussian stable L\'evy noise in the synthesis reaction rate taking into account. We calculate the most probable trajectory by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker-Planck equation. We especially examine those most probable trajectories from low concentration state to high concentration state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period); (ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including (a) a smaller noise intensity may trigger the transcription process, while a larger noise intensity can not, under the same asymmetric L\'evy noise. This phenomenon does not occur in the case of symmetric L\'evy noise; (b) the symmetric L\'evy motion always induces transition to high concentration, but certain asymmetric L\'evy motions do not trigger the switch to transcription. These findings provide insights for further experimental research, in order to achieve or to avoid specific gene transcriptions, with possible relevance for medical advances.