On the Equivalence of Langevin and Fokker-Planck Approaches in Stochastic Gene Expression

Shreshtha, Mayank (2016) On the Equivalence of Langevin and Fokker-Planck Approaches in Stochastic Gene Expression. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Mayank Shreshtha (11MS053)) Mayank_thesis.pdf - Submitted Version Restricted to Repository staff only Download (2MB)

Abstract

Stochasticity in gene expression is an inherent property and it plays key role in determining cellular fate(lysis-lysogeny), boosting up the survival chances in dynamically changing environment and affects growth rate. Thus, the quantitative details of life in a single cell is entirely dependent on the mathematical description of corresponding stochastic processes. We first study all established stochastic models(without feedback) to quantify the noise strength(Fano factor) using Langevin approach. Equivalence of analytical results obtained earlier with Langevin description is first established. Langevin technique assumes high copy-number of the involved quantity and yields steady-state distributions even for the most realistic three-stage model(eukaryotes and mammalian cells) without putting any restriction on the chemical rate constants involved and is in agreement with the previously given results in steady state limit. Exploiting the stochastic differential equation(SDE)<->Fokker-Planck equation(FPE) correspondence, we obtained the propagator of mRNA and protein for all the existing stochastic models and quantfied transient dynamics as well as applied it to calculate mean first passage time and its dependence on the phenomena of transcriptional and translational bursting. Steady state Fano factor showed a non-monotonic behaviour whereas FPT and its other moments displayed a monotonic decay with increasing burst size. This study gives new mechanism which may underlie lysis-time variation and viral-latency and highlight the importance of bursting phenomena. Our analytical results are supported by various experimental findings especially in the case of viral-latency in HIV-1 and is relevant to both biological and clinical studies. Gene expression forms the basis of all-life processes and is very vital to understand how biological systems are constructed, function and evolve.

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