×

A Fokker-Planck approach to the study of robustness in gene expression. (English) Zbl 1471.92124

Summary: 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 \(\mu\)RNA content in a cell. We provide theoretical and numerical evidence that the robustness of the gene expression is increased in the presence of \(\mu\)RNA. At the mathematical level, increased robustness shows in a smaller coefficient of variation of the marginal density of the mRNA in the presence of \(\mu\)RNA. 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.

MSC:

92C40 Biochemistry, molecular biology
92C42 Systems biology, networks
35Q84 Fokker-Planck equations

Software:

DLMF
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] L, Synthetic incoherent feedforward circuits show adaptation to the amount of their genetic template, Molecular systems biology, 7, 519 (2011) · doi:10.1038/msb.2011.49
[2] R. Blevins, L. Bruno, T. Carroll, J. Elliott, A. Marcais, C. Loh, et al., Micrornas regulate cellto-cell variability of endogenous target gene expression in developing mouse thymocytes, PLoS genetics, 11 (2015).
[3] M, Roles for micrornas in conferring robustness to biological processes, Cell, 149, 515-524 (2012) · doi:10.1016/j.cell.2012.04.005
[4] H, Micrornas and gene regulatory networks: managing the impact of noise in biological systems, Genes & development, 24, 1339-1344 (2010)
[5] M. Osella, C. Bosia, D. Corá, M. Caselle, The role of incoherent microrna-mediated feedforward loops in noise buffering, PLoS Comput Biol, 7 (2011), e1001101.
[6] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981. · Zbl 0511.60038
[7] D, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22, 403-434 (1976) · doi:10.1016/0021-9991(76)90041-3
[8] C, Gene autoregulation via intronic micrornas and its functions, BMC Syst. Biol., 6, 131 (2012) · doi:10.1186/1752-0509-6-131
[9] P. Degond, S. Jin, Y. Zhu, An uncertainty quantification approach to the study of gene expression robustness, preprint, arXiv: 1910.07188.
[10] B. Perthame, Parabolic Equations in Biology, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, Cham, 2015. · Zbl 1333.35001
[11] P, Dimensional reduction of the Fokker-Planck equation for stochastic chemical reactions, Multiscale Modeling & Simulation, 5, 593-614 (2006) · Zbl 1126.82027
[12] P. Degond, M. Herda, S. Mirrahimi, FPmuRNA, 2020. Available from: https://gitlab.inria.fr/herda/fpmurna.
[13] D, The chemical langevin equation, The Journal of Chemical Physics, 113, 297-306 (2000) · doi:10.1063/1.481811
[14] W, Steady states of Fokker-Planck equations: Existence I., J. Dynam. Differential Equations, 27, 721-742 (2015) · Zbl 1339.35322 · doi:10.1007/s10884-015-9454-x
[15] W, Integral identity and measure estimates for stationary FokkerPlanck equations, Ann. Probab., 43, 1712-1730 (2015) · Zbl 1319.35268 · doi:10.1214/14-AOP917
[16] V. I. Bogachev, N. V. Krylov, M. Röckner, S. V. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, American Mathematical Society, Providence, RI, 2015. · Zbl 1342.35002
[17] R. Z. Has’minskiĭ, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Teor. Verojatnost. i Primenen., 5 (1960), 196-214. · Zbl 0093.14902
[18] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, SpringerVerlag, Berlin, 2001. · Zbl 1042.35002
[19] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, NIST Handbook of Mathematical Functions Hardback and CD-ROM, Cambridge University Press, Cambridge, 2010. · Zbl 1198.00002
[20] M, Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations, Math. Comp., 89, 1093-1133 (2020) · Zbl 1440.65108
[21] J, A practical difference scheme for Fokker-Planck equations, Journal of Computational Physics, 6, 1-16 (1970) · Zbl 0221.65153 · doi:10.1016/0021-9991(70)90001-X
[22] C, Finite-volume schemes for noncoercive elliptic problems with Neumann boundary conditions, IMA J. Numer. Anal., 31, 61-85 (2011) · Zbl 1211.65144 · doi:10.1093/imanum/drp009
[23] H. J. Brascamp, E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis, 22 (1976), 366-389. · Zbl 0334.26009
[24] S, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal., 10, 1028-1052 (2000) · Zbl 0969.26019 · doi:10.1007/PL00001645
[25] D. Bakry, Michel Émery, Diffusions hypercontractives, in Séminaire de Probabilités, XIX, 1983/84, Springer, Berlin, (1985), 177-206. · Zbl 0561.60080
[26] D. Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes, in Lectures on Probability Theory (Saint-Flour, 1992), Springer, Berlin, (1994), 1-114. · Zbl 0856.47026
[27] A, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differ. Equations, 26, 43-100 (2001) · Zbl 0982.35113 · doi:10.1081/PDE-100002246
[28] A, Refined convex Sobolev inequalities, J. Funct. Anal., 225, 337-351 (2005) · Zbl 1087.35018 · doi:10.1016/j.jfa.2005.05.003
[29] B. Arras, Y. Swan, A stroll along the gamma, Stochastic Process. Appl., 127 (2017), 3661-3688. · Zbl 1381.60058
[30] D, Remarques sur les semigroupes de Jacobi, Astérisque, 236, 23-39 (1996) · Zbl 0859.47026
[31] M, Exponential concentration for first passage percolation through modified Poincaré inequalities, Ann. Inst. Henri Poincaré Probab. Stat., 44, 544-573 (2008) · Zbl 1186.60102 · doi:10.1214/07-AIHP124
[32] L. Miclo, Sur l’inégalité de Sobolev logarithmique des opérateurs de Laguerre à petit paramètre, in Séminaire de Probabilités, XXXVI, Springer, Berlin, (2003), 222-229. · Zbl 1053.60014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.