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A stochastic analysis of first-order reaction networks

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Abstract

A stochastic model for a general system of first-order reactions in which each reaction may be either a conversion reaction or a catalytic reaction is derived. The governing master equation is formulated in a manner that explicitly separates the effects of network topology from other aspects, and the evolution equations for the first two moments are derived. We find the surprising, and apparently unknown, result that the time evolution of the second moments can be represented explicitly in terms of the eigenvalues and projections of the matrix that governs the evolution of the means. The model is used to analyze the effects of network topology and the reaction type on the moments of the probability distribution. In particular, it is shown that for an open system of first-order conversion reactions, the distribution of all the system components is a Poisson distribution at steady state. Two different measures of the noise have been used previously, and it is shown that different qualitative and quantitative conclusions can result, depending on which measure is used. The effect of catalytic reactions on the variance of the system components is also analyzed, and the master equation for a coupled system of first-order reactions and diffusion is derived.

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References

  • Arazi, A., Ben-Jacob, E., Yechiali, U., 2004. Bridging genetic networks and queuing theory. Physica A 332, 585–616.

    Article  MathSciNet  Google Scholar 

  • Athreya, K., Ney, P., 1972. Branching Processes. Springer-Verlag.

  • Austin, R.H., Beeson, K.W., Eisenstein, L., Frauenfelder, H., Gunsalus, I.C., 1975. Dynamics of ligand binding to myoglobin. Biochemistry 14(24), 5355–5373.

    Article  Google Scholar 

  • Bartholomay, A.F., 1958. Stochastic models for chemical reactions: I. theory of the unimolecular reaction process. Math. Biophys. 20, 175–190.

    MathSciNet  Google Scholar 

  • Bartholomay, A.F., 1959. Stochastic models for chemical reactions: II. the unimolecular rate constant. Math. Biophys. 21, 363–373.

    MathSciNet  Google Scholar 

  • Blake, W.J., Kaern, M., Cantor, C.R., Collins, J.J., 2003. Noise in eukaryotic gene expression. Nature 422(6932), 633–637.

    Article  Google Scholar 

  • Bodewig, E., 1959. Matrix Calculus. Interscience Publishers, Inc., New York.

    MATH  Google Scholar 

  • Bokinsky, G., Rueda, D., Misra, V.K., Rhodes, M.M., Gordus, A., Babcock, H.P., Walter, N.G., Zhuang, X., 2003. Single-molecule transition-state analysis of RNA folding. Proc. Natl. Acad. Sci. USA 100(16), 9302–9307.

    Article  Google Scholar 

  • Brown, F.L.H., 2003. Single-molecule kinetics with time-dependent rates: a generating function approach. Phys. Rev. Lett. 90(2), 028302.

    Google Scholar 

  • Darvey, I.G., Ninham, B.W., Staff, P.J., 1966. Stochastic models for second-order chemical reaction kinetics, the equilibrium state. J. Chem. Phys. 45(6), 2145–2155.

    Article  Google Scholar 

  • Darvey, I.G., Staff, P.J., 1966. Stochastic approach to first-order chemical reaction kinetics. J. Chem. Phys. 44(3), 990.

    Article  MathSciNet  Google Scholar 

  • Delbruck, M., 1940. Statistical fluctuations in autocatalytic reactions. J. Chem. Phys. 8, 120–124.

    Article  Google Scholar 

  • Durrett, R., 1999. Essentials of Stochastic Processes. Springer-Verlag, New York.

    MATH  Google Scholar 

  • Elowitz, M.B., Levine, A.J., Siggia, E.D., Swain, P.S., 2002. Stochastic gene expression in a single cell. Science 297(5584), 1183–1186.

    Article  Google Scholar 

  • Fredrickson, A.G., 1966. Stochastic triangular reactions. Chem. Engg. Sci. 21, 687–691.

    Article  Google Scholar 

  • Gani, J., 1965. Stochastic models for bacteriophage. J. Appl. Prob. 2, 225–268.

    Article  MATH  MathSciNet  Google Scholar 

  • Gans, P.J., 1960. Open first-order stochastic processes. J. Chem. Phys. 33(3), 691.

    Article  MathSciNet  Google Scholar 

  • Gardiner, C.W., 1983. Handbook of Stochastic Methods. Springer, Berlin, Heidelberg.

    MATH  Google Scholar 

  • Gillespie, D.T., 1976. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434.

    Article  MathSciNet  Google Scholar 

  • Harris, T., 1963. The Theory of Branching Processes. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Horn, F., Jackson, R., 1972. General mass action kinetics. Arch. Ration. Mech. Anal. 48, 81.

    MathSciNet  Google Scholar 

  • Iorio, E.E.D., Hiltpold, U.R., Filipovic, D., Winterhalter, K.H., Gratton, E., Vitrano, E., Cupane, A., Leone, M., Cordone, L., 1991. Protein dynamics. comparative investigation on heme-proteins with different physiological roles. Biophys. J 59(3), 742–754.

    Google Scholar 

  • Kelly, F.P., 1979. Reversibility and Stochastic Networks. In: Wiley Series in Probability and Mathematical Statistics. John Wiley and Sons, New York, NY, USA, London, UK, Sydney, Australia.

    Google Scholar 

  • Kendall, D.G., 1948. On the generalized “birth-and-death” process. Ann. Math. Stat. 19(1), 1–15.

    MATH  MathSciNet  Google Scholar 

  • Kepler, T.B., Elston, T.C., 2001. Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations. Biophys. J. 81(6), 3116–3136.

    Article  Google Scholar 

  • Kim, S.K., 1958. Mean first passage time for a random walker and its application to chemical kinetics. J. Chem. Phys. 28(6), 1057–1067.

    Article  Google Scholar 

  • Klein, M.J., 1956. Generalization of the Ehrenfest urn model. Phys. Rev. 103(1), 17–20.

    Article  MATH  MathSciNet  Google Scholar 

  • Krieger, I.M., Gans, P.J., 1960. First-order stochastic processes. J. Chem. Phys. 32(1), 247.

    Article  MathSciNet  Google Scholar 

  • Kuthan, H., 2001. Self-organisation and orderly processes by individual protein complexes in the bacterial cell. Prog. Biophys. Mol. Biol. 75(1–2), 1–17.

    Article  Google Scholar 

  • Laurenzi, I.J., 2000. An analytical solution of the stochastic master equation for reversible biomolecular reaction kinetics. J. Chem. Phys. 113(8), 3315–3322.

    Article  Google Scholar 

  • Levsky, J.M., Singer, R.H., 2003. Gene expression and the myth of the average cell. Trends Cell Biol. 13(1), 4–6.

    Article  Google Scholar 

  • Mayor, U., Guydosh, N.R., Johnson, C.M., Grossmann, J.G., Sato, S., Jas, G.S., Freund, S.M., Alonso, D.O., Daggett, V., Fersht, A.R., 2003. The complete folding pathway of a protein from nanoseconds to microseconds. Nature 421(6925), 863–867.

    Article  Google Scholar 

  • McQuarrie, D.A., 1963. Kinetics of small systems. J. Chem. Phys. 38(2), 433–436.

    Article  Google Scholar 

  • McQuarrie, D.A., Jachimowski, C.J., Russell, M.E., 1964. Kinetics of small systems. II. J. Chem. Phys. 40(10), 2914.

    Article  Google Scholar 

  • Montroll, E.W., Shuler, K.E., 1958. The application of the theory of stochastic processes to chemical kinetics. Adv. Chem. Phys. 1, 361–399.

    Google Scholar 

  • Nicolis, G., Prigogine, I., 1977. Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. John Wiley and Sons, New York, NY, USA, London, UK, Sydney, Australia, A Wiley-Interscience Publication.

    MATH  Google Scholar 

  • Othmer, H.G., 1969. Interactions of reaction and diffusion in open systems. Ph.D. Thesis, University of Minnesota, Minneapolis.

    Google Scholar 

  • Othmer, H.G., 1979. A graph-theoretic analysis of chemical reaction networks, Lecture Notes, Rutgers University.

  • Othmer, H.G., 1981. The interaction of structure and dynamics in chemical reaction networks. In: Ebert, K.H., Deuflhard, P., Jager, W. (Eds.), Modelling of Chemical Reaction Systems. Springer-Verlag, New York, pp. 1–19.

    Google Scholar 

  • Othmer, H.G., Scriven, L.E., 1971. Instability and dynamic pattern in cellular networks. J. Theor. Biol. 32, 507–537.

    Article  Google Scholar 

  • Ozbudak, E.M., Thattai, M., Kurtser, I., Grossman, A.D., van Oudenaarden, A., 2002. Regulation of noise in the expression of a single gene. Nat. Genet. 31(1), 69–73.

    Article  Google Scholar 

  • Rao, C.V., Arkin, A.P., 2003. Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. J. Chem. Phys. 118(11), 4999–5010.

    Article  Google Scholar 

  • Shuler, K.F., 1960. Relaxation processes in multistate systems. Phys. Fluids 2(4), 442–448.

    Article  MathSciNet  Google Scholar 

  • Siegert, A.J.F., 1949. On the approach to statistical equilibrium. Phys. Rev. 76(11), 1708–1714.

    Article  MATH  Google Scholar 

  • Singer, K., 1953. Application of the theory of stochastic processes to the study of irreproducible chemical reactions and nucleation processes. J. Roy. Stat. Soc. Ser. B 15(1), 92–106.

    MATH  Google Scholar 

  • Spudich, J.L., Koshland, D.E., 1976. Non-genetic individuality: chance in the single cell. Nature 262(5568), 467–471.

    Article  Google Scholar 

  • Stundzia, A.B., Lumsden, C.J., 1996. Stochastic simulation of coupled reaction-diffusion processes. J. Comput. Phys. 127(0168), 196–207.

    Article  MATH  Google Scholar 

  • Swain, P.S., Elowitz, M.B., Siggia, E.D., 2002. Intrinsic and extrinsic contributions to stochasticity in gene expression. Proc. Natl. Acad. Sci. USA 99(20), 12795–12800.

    Google Scholar 

  • Thattai, M., van Oudenaarden, A., 2001. Intrinsic noise in gene regulatory networks. Proc. Natl. Acad. Sci. USA 98(15), 8614–8619.

    Article  Google Scholar 

  • Turing, A.M., 1952. The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B 237, 37–72.

    Google Scholar 

  • Tyson, J.J., Othmer, H.G., 1978. The dynamics of feedback control circuits in biochemical pathways. Prog. Theor. Biol. 5, 1–62.

    Google Scholar 

  • Wei, J., Prater, C.D., 1962. The structure and analysis of complex reaction systems. Adv. Catal. 13, 203.

    Article  Google Scholar 

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Author notes

  1. Hans G. Othmer

    Present address: School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Chetan Gadgil & Chang Hyeong Lee

  2. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, Leipzig, Germany

    Hans G. Othmer

Authors
  1. Chetan Gadgil
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  2. Chang Hyeong Lee
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  3. Hans G. Othmer
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Correspondence to Hans G. Othmer.

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All authors contributed equally to this work.

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Gadgil, C., Lee, C.H. & Othmer, H.G. A stochastic analysis of first-order reaction networks. Bull. Math. Biol. 67, 901–946 (2005). https://doi.org/10.1016/j.bulm.2004.09.009

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  • DOI: https://doi.org/10.1016/j.bulm.2004.09.009

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