Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 16 (2019) open journal systems 


Mathematical models of gene expression

Philippe S. Robert, INRIA de Paris


Abstract
In this paper we analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, the production process of proteins. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.

AMS 2000 subject classifications: Primary 60G55; Secondary 92C40.

Keywords: Stochastic models, gene expression, marked Poisson point processes.

Creative Common LOGO

Full Text: PDF


Robert, Philippe S., Mathematical models of gene expression, Probability Surveys, 16, (2019), 277-332 (electronic). DOI: 10.1214/19-PS332.

References

[1]    Anderson, D. F. and Kurtz, T. G. (2015). Stochastic analysis of biochemical systems. Mathematical Biosciences Institute Lecture Series. Stochastics in Biological Systems, Vol. 1. Springer, Cham; MBI Mathematical Biosciences Institute, Ohio State University, Columbus, OH.

[2]    Berg, O. G. (1978). A model for the statistical fluctuations of protein numbers in a microbial population. Journal of Theoretical Biology 71, 4 (Apr.), 587–603.

[3]    Bokes, P., King, J. R., Wood, A. T. A., and Loose, M. (2012). Exact and approximate distributions of protein and mRNA levels in the low-copy regime of gene expression. Journal of Mathematical Biology 64, 5 (Apr.), 829–854.

[4]    Bremer, H. and Dennis, P. (2008). Modulation of chemical composition and other parameters of the cell at different exponential growth rates. EcoSal Plus.

[5]    Brenner, S., Jacob, F., and Meselson, M. (1961). An unstable intermediate carrying information from genes to ribosomes for protein synthesis. Nature 190, 4776, 576–581.

[6]    Bressloff, P. C. (2014). Stochastic processes in cell biology. Interdisciplinary Applied Mathematics. Springer, Cham [u.a.].

[7]    Burrill, D. R. and Silver, P. A. (2010). Making cellular memories. Cell 140, 1, 13–18.

[8]    Chen, H., Shiroguchi, K., Ge, H., and Xie, X. (2015). Genome-wide study of mRNA degradation and transcript elongation in escherichia coli. Mol. Syst. Biol. 11, 1 (Jan.), 781. Published correction appears in Mol. Syst. Biol. 11, (May), 808 (2015).

[9]    Dawson, D. A. (1993). Measure-valued Markov processes. In École d’Été de Probabilités de Saint-Flour XXI—1991. Lecture Notes in Math., Vol. 1541. Springer, Berlin, 1–260.

[10]    Eldar, A. and Elowitz, M. B. (2010). Functional roles for noise in genetic circuits. Nature 467, 7312, 167.

[11]    Feller, W. (1971). An introduction to probability theory and its applications, 2nd ed. Vol. II. John Wiley & Sons Ltd, New York.

[12]    Friedman, N., Cai, L., and Xie, X. S. (2006). Linking stochastic dynamics to population distribution: an analytical framework of gene expression. Physical Review Letters 97, 16, 168302.

[13]    Fromion, V., Leoncini, E., and Robert, P. (2013). Stochastic gene expression in cells: A point process approach. SIAM Journal on Applied Mathematics 73, 1 (Jan.), 195–211. arXiv:1206.0362.

[14]    Gupta, A., Briat, C., and Khammash, M. (2014). A scalable computational framework for establishing long-term behavior of stochastic reaction networks. PLoS Computational Biology 10, 6, e1003669.

[15]    Jacob, F. and Monod, J. (1961). Genetic regulatory mechanisms in the synthesis of proteins. Journal of Molecular Biology 3, 3, 318–356.

[16]    Johnson, N. L., Kemp, A. W., and Kotz, S. (2005). Univariate discrete distributions, Third ed. Wiley Series in Probability and Statistics. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ.

[17]    Karp, P. D., Kothari, A., Fulcher, C. A., Keseler, I. M., Latendresse, M., Krummenacker, M., Subhraveti, P., Ong, Q., Billington, R., Caspi, R., Paley, S. M., Ong, W. K., and Midford, P. E. (2017). The BioCyc collection of microbial genomes and metabolic pathways. Briefings in Bioinformatics.

[18]    Kelly, F. P. (1979). Reversibility and stochastic networks. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Ltd., Chichester.

[19]    Kingman, J. F. C. (1993). Poisson processes. Oxford Studies in Probability.

[20]    Kurtz, T. (1992). Averaging for martingale problems and stochastic approximation. In Applied Stochastic Analysis, US-French Workshop. Lecture Notes in Control and Information Sciences, Vol. 177. Springer Verlag, 186–209.

[21]     Leoncini, E. (2013). Ph.D. document. Ph.D. thesis, École Polytechnique. https://pastel.archives-ouvertes.fr/pastel-00924232.

[22]    Lestas, I., Vinnicombe, G., and Paulsson, J. (2010). Fundamental limits on the suppression of molecular fluctuations. Nature 467, 7312, 174.

[23]    Levin, D. A., Peres, Y., and Wilmer, E. L. (2009). Markov chains and mixing times. American Mathematical Society, Providence, RI.

[24]    Loynes, R. (1962). The stability of queues with non independent inter-arrival and service times. Proc. Cambridge Ph. Soc. 58, 497–520.

[25]    Mackey, M. C., Santillán, M., Tyran-Kamińska, M., and Zeron, E. S. (2016). Simple mathematical models of gene regulatory dynamics. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham.

[26]    Murray, J. D. (2002). Mathematical biology. I, Third ed. Interdisciplinary Applied Mathematics, Vol. 17. Springer-Verlag, New York. An introduction.

[27]    Neveu, J. (1977). Processus ponctuels. In École d’été de Probabilités de Saint-Flour, P.-L. Hennequin, Ed. Lecture Notes in Mathematics, Vol. 598. Springer-Verlag, Berlin, 249–445.

[28]    Norman, T. M., Lord, N. D., Paulsson, J., and Losick, R. (2013). Memory and modularity in cell-fate decision making. Nature 503, 7477, 481.

[29]    Paulsson, J. (2005). Models of stochastic gene expression. Physics of Life Reviews 2, 2 (June), 157–175.

[30]    Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures & Algorithms 9, 1-2, 223–252.

[31]    Raj, A., Peskin, C. S., Tranchina, D., Vargas, D. Y., and Tyagi, S. (2006). Stochastic mRNA synthesis in mammalian cells. PLoS Biology 4, 10, e309.

[32]    Rigney, D. and Schieve, W. (1977). Stochastic model of linear, continuous protein synthesis in bacterial populations. Journal of Theoretical Biology 69, 4 (Dec.), 761–766.

[33]    Robert, P. (2003). Stochastic networks and queues. Stochastic Modelling and Applied Probability Series. Springer-Verlag, New York.

[34]    Russell, J. B. and Cook, G. M. (1995). Energetics of bacterial growth: balance of anabolic and catabolic reactions. Microbiol. Mol. Biol. Rev. 59, 1, 48–62.

[35]    Shahrezaei, V. and Swain, P. S. (2008). Analytical distributions for stochastic gene expression. Proceedings of the National Academy of Sciences 105, 45, 17256–17261.

[36]    Swain, P. S., Elowitz, M. B., and Siggia, E. D. (2002). Intrinsic and extrinsic contributions to stochasticity in gene expression. Proceedings of the National Academy of Sciences 99, 20, 12795–12800.

[37]    Taniguchi, Y., Choi, P. J., Li, G.-W., Chen, H., Babu, M., Hearn, J., Emili, A., and Xie, X. S. (2010). Quantifying e. coli proteome and transcriptome with single-molecule sensitivity in single cells. Science 329, 5991, 533–538.

[38]    Thattai, M. and van Oudenaarden, A. (2001). Intrinsic noise in gene regulatory networks. Proceedings of the National Academy of Sciences 98, 15, 8614–8619.

[39]    Valgepea, K., Adamberg, K., Seiman, A., and Vilu, R. (2013). Escherichia coli achieves faster growth by increasing catalytic and translation rates of proteins. Molecular BioSystems 9, 9, 2344–2358.

[40]    Van Kampen, N. G. (1992). Stochastic processes in physics and chemistry. Vol. 1. Elsevier.

[41]    Watson, J. D., Baker, T. A., Bell, S. P., Gann, A., Levine, M., Losick, R., and CSHLP, I. (2007). Molecular biology of the gene, 6th ed. Pearson/Benjamin Cummings; Cold Spring Harbor Laboratory Press, San Francisco; Cold Spring Harbor, N.Y.

[42]    Watson, J. D., Crick, F. H., and others. (1953). Molecular structure of nucleic acids. Nature 171, 4356, 737–738.

[43]    Whittaker, E. T. and Watson, G. N. (1996). A course of modern analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge. Reprint of the fourth (1927) edition.

[44]    Williams, D. (1991). Probability with martingales. Cambridge University Press, Cambridge.




Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787