×

Noise slows the rate of Michaelis-Menten reactions. (English) Zbl 1382.92153

Summary: Microscopic randomness and the small volumes of living cells combine to generate random fluctuations in molecule concentrations called “noise”. Here I investigate the effect of noise on biochemical reactions obeying Michaelis-Menten kinetics, concluding that substrate noise causes these reactions to slow. I derive a general expression for the time evolution of the joint probability density of chemical species in arbitrarily connected networks of non-linear chemical reactions in small volumes. This equation is a generalization of the chemical master equation (CME), a common tool for investigating stochastic chemical kinetics, extended to reaction networks occurring in small volumes, such as living cells. I apply this equation to a generalized Michaelis-Menten reaction in an open system, deriving the following general result: \(\langle p\rangle\leq\overline p\) and \(\langle s\rangle\geq\overline s\), where \(\overline s\) and \(\overline p\) denote the deterministic steady-state concentration of reactant and product species, respectively, and \(\langle s\rangle\) and \(\langle p\rangle\) denote the steady-state ensemble average over independent realizations of a stochastic reaction. Under biologically realistic conditions, namely when substrate is degraded or diluted by cell division, \(\langle p\rangle\leq\overline p\). Consequently, noise slows the rate of in vivo Michaelis-Menten reactions. These predictions are validated by extensive stochastic simulations using Gillespie’s exact stochastic simulation algorithm. I specify the conditions under which these effects occur and when they vanish, therefore reconciling discrepancies among previous theoretical investigations of stochastic biochemical reactions. Stochastic slowdown of reaction flux caused by molecular noise in living cells may have functional consequences, which the present theory may be used to quantify.

MSC:

92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
92C40 Biochemistry, molecular biology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Software:

Cellerator
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ariga, Katsuhiko; Ji, Qingmin; Mori, Taizo; Naito, Masanobu; Yamauchi, Yusuke; Abe, Hideki; Hill, JonathanP., Enzyme nanoarchitectonics: organization and device application, Chem. Soc. Rev., 42, 15 (2013), Royal Society of Chemistry: 6322-45
[2] Berg, OttoG., A model for the statistical fluctuations of protein numbers in a microbial population, J. Theor. Biol., 71, 4, 587-603 (1978)
[3] Briggs, GeorgeEdward; Haldane, JohnBurdonSanderson, A note on the kinetics of enzyme action, Biochem. J., 19, 2, 338 (1925)
[4] Cantrell, RobertStephen; Cosner, Chris, Spatial Ecology via Reaction-Diffusion Equations (2004), John Wiley & Sons
[5] Colomb, Warren; Sarkar, SusantaK., Extracting physics of life at the molecular level: a review of single-molecule data analyses, Phys. Life Rev., 13, 107-137 (2015), Elsevier
[6] Desai, MichaelM.; Fisher, DanielS., Beneficial mutation-selection balance and the effect of linkage on positive selection, Genetics, 176, 3, 1759-1798 (2007), Genetics Soc America
[7] Elf, J.; Ehrenberg, M., Fast evaluation of fluctuations in biochemical networks with the linear noise approximation, Genome Res., 13, 11, 2475-2484 (2003)
[8] Elowitz, MichaelB.; Levine, ArnoldJ.; Siggia, EricD.; Swain, PeterS., Stochastic gene expression in a single cell, Science, 297, 5584, 1183-1186 (2002)
[9] English, BrianP.; Min, Wei; Van Oijen, AntoineM.; Lee, KangTaek; Luo, Guobin; Sun, Hongye; Cherayil, BinnyJ.; Kou, S. C.; Sunney Xie, X., Ever-fluctuating single enzyme molecules: Michaelis-Menten equation revisited, Nat. Chem. Biol., 2, 2, 87-94 (2006)
[10] Fisher, RonaldAylmer, The wave of advance of advantageous genes, Ann. Eugenics, 7, 4, 355-369 (1937), Wiley Online Library · JFM 63.1111.04
[11] Friedman, Nir; Cai, Long; Xie, X. Sunney, Linking stochastic dynamics to population distribution: an analytical framework of gene expression, Phys. Rev. Lett., 97, 16 (2006), APS: 168302
[12] Gillespie, DanielT., Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81, 25, 2340-2361 (1977)
[13] Grima, R., Noise-induced breakdown of the michaelis-menten equation in steady-state conditions, Phys. Rev. Lett., 102, 21 (2009), doi: ARTN21810310.1103/PhysRevLett.102.218103
[14] Grima, R., An effective rate equation approach to reaction kinetics in small volumes: theory and application to biochemical reactions in nonequilibrium steady-state conditions, J. Chem. Phys., 133, 3 (2010), doi: Artn03510110.1063/1.3454685
[15] Grima, R., Intrinsic biochemical noise in crowded intracellular conditions, J. Chem. Phys., 132, 18 (2010), doi: Artn18510210.1063/1.3427244
[16] Hallatschek, Oskar, The noisy edge of traveling waves, Proc. Natl. Acad. Sci., 108, 5, 1783-1787 (2011), National Acad Sciences
[17] Hallatschek, Oskar; Korolev, K. S., Fisher waves in the strong noise limit, Phys. Rev. Lett., 103, 10 (2009), APS: 108103
[18] Ingalls, BrianP., Mathematical Modeling in Systems Biology: An Introduction (2013), The MIT Press: The MIT Press Cambridge, Massachusetts · Zbl 1312.92003
[19] Kacser, H.; Burns, J. A., The control of flux, (Symp. Soc. Exp. Biol., 27 (1973)), 65-104
[20] Kondo, Shigeru; Miura, Takashi, Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329, 5999, 1616-1620 (2010), American Association for the Advancement of Science · Zbl 1226.35077
[21] Levine, Erel; Hwa, Terence, Stochastic fluctuations in metabolic pathways, Proc. Natl. Acad. Sci., 104, 22, 9224-9229 (2007), National Acad Sciences · Zbl 1156.92019
[22] McAdams, HarleyH.; Arkin, Adam, Stochastic mechanisms in gene expression, Proc. Natl. Acad. Sci., 94, 3, 814-819 (1997)
[23] Michaelis, Leonor; Menten, MaudL., Die Kinetik Der Invertinwirkung, Biochem. Z, 49, 333-369, 352 (1913)
[24] Ochab-Marcinek, Anna, Extrinsic noise passing through a Michaelis-Menten reaction: a universal response of a genetic switch, J. Theor. Biol., 263, 4, 510-520 (2010), Elsevier · Zbl 1406.92274
[25] Ochab-Marcinek, Anna; Tabaka, Marcin, Bimodal gene expression in noncooperative regulatory systems, Proc. Natl. Acad. Sci., 107, 51, 22096-22101 (2010), National Acad Sciences
[26] Paulsson, Johan; Berg, OttoG.; Ehrenberg, Måns, Stochastic focusing: fluctuation-enhanced sensitivity of intracellular regulation, Proc. Natl. Acad. Sci., 97, 13, 7148-7153 (2000)
[27] Pedraza, JuanM.; Paulsson, Johan, Effects of molecular memory and bursting on fluctuations in gene expression, Science, 319, 5861, 339-343 (2008)
[28] Pulkkinen, Otto; Metzler, Ralf, Variance-corrected Michaelis-Menten equation predicts transient rates of single-enzyme reactions and response times in bacterial gene-regulation, Sci. Rep., 5, 17820 (2015), Nature Publishing Group
[29] Qian, Hong; Elson, ElliotL., Single-molecule enzymology: stochastic Michaelis-Menten kinetics, Biophys. Chem., 101, 565-576 (2002), Elsevier
[30] Raj, Arjun; Peskin, CharlesS.; Tranchina, Daniel; Vargas, DianaY.; Tyagi, Sanjay, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4, 10 (2006)
[31] Rao, ChristopherV.; Arkin, AdamP., Stochastic chemical kinetics and the quasi-steady-state assumption: application to the gillespie algorithm, J. Chem. Phys., 118, 11, 4999-5010 (2003)
[32] Raser, JonathanM.; O’Shea, ErinK., Control of stochasticity in Eukaryotic gene expression, Science, 304, 5678, 1811-1814 (2004)
[33] Raser, JonathanM.; O’Shea, ErinK., Noise in gene expression: origins, consequences, and control, Science, 309, 5743, 2010-2013 (2005)
[34] Sanft, KevinR.; Gillespie, DanielT.; Petzold, LindaR., Legitimacy of the stochastic Michaelis-Menten approximation, Syst. Biol IET, 5, 1, 58-69 (2011)
[35] Shahrezaei, Vahid; Swain, PeterS., Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci., 105, 45, 17256-17261 (2008)
[36] Shapiro, BruceE.; Levchenko, Andre; Meyerowitz, ElliotM.; Wold, BarbaraJ.; Mjolsness, EricD., Cellerator: extending a computer algebra system to include biochemical arrows for signal transduction simulations, Bioinformatics, 19, 5, 677-678 (2003)
[37] Taniguchi, Yuichi; Choi, PaulJ.; Li, Gene-Wei; Chen, Huiyi; Babu, Mohan; Hearn, Jeremy; Emili, Andrew; Xie, X. Sunney, Quantifying E. coli Proteome and Transcriptome with single-molecule sensitivity in single cells, Science, 329, 5991, 533-538 (2010)
[38] Thattai, Mukund; van Oudenaarden, Alexander, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci., 98, 15, 8614-8619 (2001)
[39] van Kampen, N. G., Stochastic Processes in Physics and Chemistry (2007), Elsevier: Elsevier New York · Zbl 0511.60038
[40] Wang, Zhi; Zhang, Jianzhi, Impact of gene expression noise on organismal fitness and the efficacy of natural selection, Proc. Natl. Acad. Sci., 108, 16, E67-E76 (2011)
[41] Xie, X. Sunney; Lu, H. Peter, Single-molecule enzymology, J. Biol. Chem., 274, 23, 15967-15970 (1999), ASBMB:
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.