A Fokker-Planck approach to the study of robustness in gene expression

Pierre Degond; Maxime Herda; Sepideh Mirrahimi; Pierre Degond; Maxime Herda; Sepideh Mirrahimi

doi:10.3934/mbe.2020338

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 6459-6486. doi: 10.3934/mbe.2020338

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A Fokker-Planck approach to the study of robustness in gene expression

1.
Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK
2.
Inria, University Lille, CNRS, UMR 8524 Laboratoire Paul Painlev, F-59000 Lille, France
3.
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

Received: 26 June 2020 Accepted: 15 September 2020 Published: 24 September 2020

We study several Fokker-Planck equations arising from a stochastic chemical kinetic system modeling a gene regulatory network in biology. The densities solving the Fokker-Planck equations describe the joint distribution of the mRNA and μRNA content in a cell. We provide theoretical and numerical evidence that the robustness of the gene expression is increased in the presence of μRNA. At the mathematical level, increased robustness shows in a smaller coefficient of variation of the marginal density of the mRNA in the presence of μRNA. These results follow from explicit formulas for solutions. Moreover, thanks to dimensional analyses and numerical simulations we provide qualitative insight into the role of each parameter in the model. As the increase of gene expression level comes from the underlying stochasticity in the models, we eventually discuss the choice of noise in our models and its influence on our results.
- gene expression,
- microRNA,
- Fokker-Planck equations,
- inverse gamma distributions
Citation: Pierre Degond, Maxime Herda, Sepideh Mirrahimi. A Fokker-Planck approach to the study of robustness in gene expression[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6459-6486. doi: 10.3934/mbe.2020338

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Abstract

We study several Fokker-Planck equations arising from a stochastic chemical kinetic system modeling a gene regulatory network in biology. The densities solving the Fokker-Planck equations describe the joint distribution of the mRNA and μRNA content in a cell. We provide theoretical and numerical evidence that the robustness of the gene expression is increased in the presence of μRNA. At the mathematical level, increased robustness shows in a smaller coefficient of variation of the marginal density of the mRNA in the presence of μRNA. These results follow from explicit formulas for solutions. Moreover, thanks to dimensional analyses and numerical simulations we provide qualitative insight into the role of each parameter in the model. As the increase of gene expression level comes from the underlying stochasticity in the models, we eventually discuss the choice of noise in our models and its influence on our results.

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