Abstract
Having exhaustively studied deterministic models of reactions, the aim of the present chapter is to introduce and discuss the induced kinetic Markov process endowed with stochastic kinetics, also called the usual stochastic model of reactions. The two main differences compared to the previous models are that the state space is now discrete and the nature of determination is probabilistic. First we deal with the master equation, i.e., the evolution equation for the probability of the process being in a particular state. We formulate the analogue of “mass action type kinetics” and outline various characterizations of the underlying process which also serve as the basis for simulation methods. Then further evolution equations are discussed; in particular, the characterization of short- and long-term behavior of the induced kinetic Markov process and properties of the stationary states are covered. At the end of the chapter, we broadly discuss the relationship between the induced kinetic Markov process and the solution of the induced kinetic differential equation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abbasi S, Diwekar UM (2014) Characterization and stochastic modeling of uncertainties in the biodiesel production. Clean Techn Environ Policy 16(1):79–94
Anderson DF (2007) A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J Chem Phys 127(21):214, 107
Anderson DF (2008) Incorporating postleap checks in tau-leaping. J Chem Phys 128(5):054, 103
Anderson DF, Kurtz TG (2015) Stochastic analysis of biochemical systems. Mathematical Biosciences Institute and Springer, Columbus and Berlin
Anderson DF, Craciun G, Kurtz TG (2010) Product-form stationary distributions for deficiency zero chemical reaction networks. Bull Math Biol 72:1947–1970
Arakelyan VB, Simonyan AL, Gevorgyan AE, Sukiasyan TS, Arakelyan AV, Grigoryan BA, Gevorgyan ES (2004) Fluctuations of the enzymatic reaction rate. Electron J Nat Sci 1(2):43–45
Arakelyan VB, Simonyan AL, Kintzios S, Gevorgyan AE, Sukiasyan TS, Arakelyan AV, Gevorgyan ES (2005) Correlation fluctuations and spectral density of the enzymatic reaction rate. Electron J Nat Sci 2(5):3–7
Arányi P, Tóth J (1977) A full stochastic description of the Michaelis–Menten reaction for small systems. Acta Biochim Biophys Hung 12(4):375–388
Arnold L (1980) On the consistency of the mathematical models of chemical reactions. In: Haken H (ed) Dynamics of synergetic systems. Springer, Berlin, pp 107–118
Arnold L, Theodosopulu M (1980) Deterministic limit of the stochastic model of chemical reactions with diffusion. Adv Appl Probab 12(2):367–379
Athreya KB, Ney PE (2004) Branching processes. Courier Corporation, Chelmsford
Atkins P, Paula JD (2013) Elements of physical chemistry. Oxford University Press, Oxford
Atlan H, Weisbuch G (1973) Resistance and inductance-like effects in chemical reactions: influence of time delays. Isr J Chem 11(2-3):479–488
Barabás B, Tóth J, Pályi G (2010) Stochastic aspects of asymmetric autocatalysis and absolute asymmetric synthesis. J Math Chem 48(2):457–489
Bartholomay AF (1958) Stochastic models for chemical reactions: I. Theory of the unimolecular reaction process. Bull Math Biol 20(3):175–190
Bartis JT, Widom B (1974) Stochastic models of the interconversion of three or more chemical species. J Chem Phys 60(9):3474–3482
Becker N (1973a) Carrier-borne epidemics in a community consisting of different groups. J Appl Prob 10(3):491–501
Becker NG (1970) A stochastic model for two interacting populations. J Appl Prob 7(3):544–564
Becker NG (1973b) Interactions between species: some comparisons between deterministic and stochastic models. Rocky Mountain J Math 3(1):53–68
Bibbona E, Sirovich R (2017) Strong approximation of density dependent markov chains on bounded domains. arXiv preprint arXiv:170407481
Cao Y, Petzold L (2005) Trapezoidal τ-leaping formula for the stochastic simulation of biochemical systems. Proceedings of foundations of systems biology in engineering, pp 149–152
Cao Y, Gillespie DT, Petzold L (2005) Avoiding negative populations in explicit Poisson τ-leaping. J Chem Phys 123(5):054, 104, 8
Cao Y, Gillespie DT, Petzold LR (2006) Efficient step size selection for the τ-leaping simulation method. J Chem Phys 124(4):044, 109, 11 pp
Cappelletti D, Wiuf C (2016) Product-form Poisson-like distributions and complex balanced reaction systems. SIAM J Appl Math 76(1):411–432
Chatterjee A, Vlachos DG, Katsoulakis MA (2005) Binomial distribution based τ-leap accelerated stochastic simulation. J Chem Phys 122(2):024, 112
Chibbaro S, Minier JP (2014) Stochastic methods in fluid mechanics. Springer, Wien
CombustionResearch (2011) Chemical-kinetic mechanisms for combustion applications. http://combustion.ucsd.edu, San Diego Mechanism web page, version 2011-11-22
Dambrine S, Moreau M (1981) On the stationary distribution of a chemical process without detailed balance. J Stat Phys 26(1):137–148
Darvey IG, Staff PJ (2004) Stochastic approach to first-order chemical reaction kinetics. J Chem Phys 44(3):990–997
Edman L, Rigler R (2000) Memory landscapes of single-enzyme molecules. Proc Natl Acad Sci USA 97(15):8266–8271
English BP, Min W, van Oijen AM, Lee KT, Luo G, Sun H, Cherayil BJ, Kou SC, Xie XS (2006) Ever-fluctuating single enzyme molecules: Michaelis–Menten equation revisited. Nat Chem Biol 2:87–94
Érdi P, Lente G (2016) Stochastic chemical kinetics. Theory and (mostly) systems biological applications. Springer series in synergetics. Springer, New York
Érdi P, Ropolyi L (1979) Investigation of transmitter-receptor interactions by analyzing postsynaptic membrane noise using stochastic kinetics. Biol Cybern 32(1):41–45
Érdi P, Tóth J (1976, in Hungarian) Stochastic reaction kinetics “nonequilibrium thermodynamics” of the state space? React Kinet Catal Lett 4(1):81–85
Érdi P, Tóth J (1989) Mathematical models of chemical reactions. Theory and applications of deterministic and stochastic models. Princeton University Press, Princeton
Érdi P, Sipos T, Tóth J (1973) Stochastic simulation of complex chemical reactions by computer. Magy Kém Foly 79(3):97–108
Ethier SN, Kurtz TG (2009) Markov processes: characterization and convergence. Wiley, Hoboken
Feller W (2008) An introduction to probability theory and its applications, vol 2. Wiley, Hoboken
Frank FC (1953) On spontaneous asymmetric synthesis. Biochim Biophys Acta 11:459–463
Gadgil C (2008) Stochastic modeling of biological reactions. J Indian Inst Sci 88(1):45–55
Gadgil C, Lee CH, Othmer HG (2005) A stochastic analysis of first-order reaction networks. Bull Math Biol 67(5):901–946
Gans PJ (1960) Open first-order stochastic processes. J Chem Phys 33(3):691–694
Gardiner CW (2010) Stochastic methods: a handbook for the natural and social sciences, 4th edn. Springer series in synergetics. Springer, Berlin
Gardiner CW, Chaturvedi S (1977) The Poisson representation. I. A new technique for chemical master equations. J Stat Phys 17(6):429–468
Gibson MA, Bruck J (2000) Efficient exact stochastic simulation of chemical systems with many species and many channels. J Phys Chem A 104(9):1876–1889
Gikhman II, Skorokhod AV (2004a) The theory of stochastic processes I. Springer, Berlin
Gikhman II, Skorokhod AV (2004b) The theory of stochastic processes II. Springer, Berlin
Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361
Gillespie DT (2001) Approximate accelerated stochastic simulation of chemically reacting systems. J Chem Phys 115(4):1716–1733
Gillespie CS (2009) Moment-closure approximations for mass-action models. IET Syst Biol 3(1):52–58
Gillespie DT, Petzold LR (2003) Improved leap-size selection for accelerated stochastic simulation. J Chem Phys 119(16):8229–8234
Goss PJE, Peccoud J (1998) Quantitative modeling of stochastic systems in molecular biology using stochastic Petri nets. Proc Natl Acad Sci USA 95:6750–6755
Grima R, Walter NG, Schnell S (2014) Single-molecule enzymology à la Michaelis–Menten. FEBS J 281(2):518–530
Hárs V (1976) A sztochasztikus reakciókinetika néhány kérdéséről (Some problems of stochastic reaction kinetics). Msc, Eötvös Loránd University, Budapest
Hong Z, Davidson DF, Hanson RK (2011) An improved H2O2 mechanism based on recent shock tube/laser absorption measurements. Combust Flame 158(4):633–644. https://doi.org/10.1016/j.combustflame.2010.10.002
Iosifescu M, Tăutu P (1973) Stochastic processes and applications in biology and medicine. II. Models. Editura Academiei, New York
Jahnke T, Huisinga W (2007) Solving the chemical master equation for monomolecular reaction systems analytically. J Math Biol 54(1):1–26
Joshi B (2015) A detailed balanced reaction network is sufficient but not necessary for its Markov chain to be detailed balanced. Discret Contin Dyn Syst Ser B 20(4):1077–1105
Juette MF, Terry DS, Wasserman MR, Zhou Z, Altman RB, Zheng Q, Blanchard SC (2014) The bright future of single-molecule fluorescence imaging. Curr Opin Chem Biol 20:103–111
Kelly FP (1979) Reversibility and stochastic networks. Wiley, New York
Kingman JFC (1969) Markov population processes. J Appl Prob 6(1):1–18
Kolmogoroff A (1935) Zur Theorie der Markoffschen Ketten. Math Ann 112:155–160
Krieger IM, Gans PJ (1960) First-order stochastic processes. J Chem Phys 32(1):247–250
Kurtz TG (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Prob 7(1):49–58
Kurtz TG (1972) The relationship between stochastic and deterministic models for chemical reactions. J Chem Phys 57(7):2976–2978
Kurtz TG (1976) Limit theorems and diffusion approximations for density dependent Markov chains. In: Stochastic systems: modeling, identification and optimization, I. Springer, Berlin, pp 67–78
Kurtz TG (1978) Strong approximation theorems for density dependent Markov chains. Stoch Process Appl 6(3):223–240
Lai JYW, Elvati P, Violi A (2014) Stochastic atomistic simulation of polycyclic aromatic hydrocarbon growth in combustion. Phys Chem Chem Phys 16:7969–7979
Lánský P, Rospars JP (1995) Ornstein–Uhlenbeck model neuron revisited. Biol Cybern 72(5):397–406
Lee NK, Koh HR, Han KY, Lee J, Kim SK (2010) Single-molecule, real-time measurement of enzyme kinetics by alternating-laser excitation fluorescence resonance energy transfer. Chem Commun 46:4683–4685
Lente G (2004) Homogeneous chiral autocatalysis: a simple, purely stochastic kinetic model. J Phys Chem A 108:9475–9478
Lente G (2005) Stochastic kinetic models of chiral autocatalysis: a general tool for the quantitative interpretation of total asymmetric synthesis. J Phys Chem A 109(48):11058–11063
Lente G (2010) The role of stochastic models in interpreting the origins of biological chirality. Symmetry 2(2):767–798
Leontovich MA (1935) Fundamental equations of the kinetic theory of gases from the point of view of stochastic processes. Zhur Exper Teoret Fiz 5:211–231
Li G, Rabitz H (2014) Analysis of gene network robustness based on saturated fixed point attractors. EURASIP J Bioinform Syst Biol 2014(1):4
Liggett TM (2010) Continuous time Markov processes: an introduction, vol 113. American Mathematical Society, Providence.
Lipták G, Hangos KM, Pituk M, Szederkényi G (2017) Semistability of complex balanced kinetic systems with arbitrary time delays. arXiv preprint arXiv:170405930
Matis JH, Hartley HO (1971) Stochastic compartmental analysis: model and least squares estimation from time series data. Biometrics, pp 77–102
McAdams HH, Arkin A (1997) Stochastic mechanisms in gene expression. Proc Natl Acad Sci USA 94(3):814–819
Mode CJ (1971) Multitype branching processes: theory and applications, vol 34. American Elsevier, New York.
Mozgunov P, Beccuti M, Horvath A, Jaki T, Sirovich R, Bibbona E (2017) A review of the deterministic and diffusion approximations for stochastic chemical reaction networks. arXiv preprint arXiv:171102567
Munsky B, Khammash M (2006) The finite state projection algorithm for the solution of the chemical master equation. J Chem Phys 124(4):044, 104
Nagypál I, Epstein IR (1986) Fluctuations and stirring rate effects in the chlorite-thiosulphate reaction. J Phys Chem 90:6285–6292
Nagypál I, Epstein IR (1988) Stochastic behaviour and stirring rate effects in the chlorite-iodide reaction. J Chem Phys 89:6925–6928
Norris JR (1998) Markov chains. Cambridge University Press, Cambridge
Øksendal B (2003) Stochastic differential equations, 5th edn. Springer, Berlin
Paulevé L, Craciun G, Koeppl H (2014) Dynamical properties of discrete reaction networks. J Math Biol 69(1):55–72
Pokora O, Lánský P (2008) Statistical approach in search for optimal signal in simple olfactory neuronal models. Math Biosci 214(1–2):100–108
Qian H, Elson EL (2002) Single-molecule enzymology: stochastic Michaelis–Menten kinetics. Biophys Chem 101:565–576
Rathinam M, Petzold LR, Cao Y, Gillespie DT (2003) Stiffness in stochastic chemically reacting systems: the implicit tau-leaping method. J Phys Chem A 119(24):12,784, 11 pp
Rathinam M, Petzold LR, Cao Y, Gillespie DT (2005) Consistency and stability of tau-leaping schemes for chemical reaction systems. Multiscale Model Simul 4(3):867–895
Reddy VTN (1975) On the existence of the steady state in the stochastic Volterra–Lotka model. J Stat Phys 13(1):61–64
Rényi A (1954, in Hungarian) Treating chemical reactions using the theory of stochastic processes. Magyar Tudományos Akadémia Alkalmazott Matematikai Intézetének Közleményei 2:83–101
Robertson HH (1966) In: Walsh JE (ed) The solution of a set of reaction rate equations, Thompson Book, Toronto, pp 178–182
Sakmann B, Neher E (eds) (1995) Single-channel recording, 2nd edn. Plenum Press, New York
Samad HE, Khammash M, Petzold L, Gillespie D (2005) Stochastic modeling of gene regulatory networks. Int J Robust Nonlinear Control 15:691–711
Siegert AJF (1949) On the approach to statistical equilibrium. Phys Rev 76(11):1708
Singer K (1953) Application of the theory of stochastic processes to the study of irreproducible chemical reactions and nucleation processes. J R Stat Soc Ser B 15(1):92–106
Sipos T, Tóth J, Érdi P (1974a) Stochastic simulation of complex chemical reactions by digital computer, I. The model. React Kinet Catal Lett 1(1):113–117
Sipos T, Tóth J, Érdi P (1974b) Stochastic simulation of complex chemical reactions by digital computer, II. Applications. React Kinet Catal Lett 1(2):209–213
Smith G, Golden D, Frenklach M, Moriary N, Eiteneer B, Goldenberg M, Bowman C, Hanson R, Song S, Gardiner W, Lissianski V, Qin Z (2000) Gri-mech 3.0. http://www.me.berkeley.edu/gri_mech
Soai K, Shibata T, Morioka H, Choji K (1995) Asymmetric autocatalysis and amplification of enantiomeric excess of a chiral molecule. Nature 378:767–768
Šolc M (2002) Stochastic model of the n-stage reversible first-order reaction: relation between the time of first passage to the most probable microstate and the mean equilibrium fluctuations lifetime. Z Phys Chem 216(7):869–893
Stoner CD (1993) Quantitative determination of the steady state kinetics of multi-enzyme reactions using the algebraic rate equations for the component single enzyme reactions. Biochem J 291(2):585–593
Tóth J (1981, in Hungarian) A formális reakciókinetika globális determinisztikus és sztochasztikus modelljéről (On the global deterministic and stochastic models of formal reaction kinetics with applications). MTA SZTAKI Tanulmányok 129:1–166
Tóth J (1981) Poissonian stationary distribution in a class of detailed balanced reactions. React Kinet Catal Lett 18(1–2):169–173
Tóth J (1988a) Contribution to the general treatment of random processes used in chemical reaction kinetics. In: Transactions of the Tenth Prague Conference on information theory, statistical decision functions, random processes, held at Prague, from July 7 to 11, 1986, Academia (Publ. House of the Czechosl. Acad. Sci.), Prague, vol 2, pp 373–379
Tóth J (1988b) Structure of the state space in stochastic kinetics. In: Grossmann V, Mogyoródi J, Vincze I, Wertz W (eds) Probability theory and mathematical statistics with applications, Springer, pp 361–369
Tóth J, Érdi P (1992) A sztochasztikus kinetikai modellek nélkülözhetetlensége (The indispensability of stochastic kinetical models). In: Bazsa G (ed) Nemlineáris dinamika és egzotikus kinetikai jelenségek kémiai rendszerekben (Nonlinear dynamics and exotic kinetic phenomena in chemical systems), Jegyzet, Pro Renovanda Cultura Hungariae–KLTE Fizikai Kémiai Tanszék, Debrecen–Budapest–Gödöllő, chap 3, pp 117–143
Tóth J, Rospars JP (2005) Dynamic modelling of biochemical reactions with applications to signal transduction: principles and tools using Mathematica. Biosystems 79:33–52
Tóth J, Török TL (1980) Poissonian stationary distribution: a degenerate case of stochastic kinetics. React Kinet Catal Lett 13(2):167–171
Tóth J, Érdi P, Török TL (1983, in Hungarian) Significance of the Poisson distribution in the stochastic model of complex chemical reactions (A Poisson-eloszlás jelentősége összetett kémiai reakciók sztochasztikus modelljében). Alkalmazott Matematikai Lapok 9(1–2):175–196
Turányi T (1990) Sensitivity analysis of complex kinetic systems. Tools and applications. J Math Chem 5(3):203–248
Turányi T, Tomlin AS (2014) Analysis of kinetic reaction mechanisms. Springer, Berlin
Turner TE, Schnell S, Burrage K (2004) Stochastic approaches for modelling in vivo reactions. Comput Biol Chem 28(3):165–178
Urzay J, Kseib N, Davidson DF, Iaccarino G, Hanson RK (2014) Uncertainty-quantification analysis of the effects of residual impurities on hydrogen–oxygen ignition in shock tubes. Combust Flame 161(1):1–15
Van Kampen NG (2006) Stochastic processes in physics and chemistry, 4th edn. Elsevier, Amsterdam
Vellela M, Qian H (2007) A quasistationary analysis of a stochastic chemical reaction: Keizer’s paradox. Bull Math Biol 69(5):1727–1746
Vellela M, Qian H (2009) Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited. J R Soc Interface 6:925–940
Velonia K, Flomenbom O, Loos D, Masuo S, Cotlet M, Engelborghs Y, Hofkens J, Rowan AE, Klafter J, Nolte RJM, de Schryver FC (2005) Single-enzyme kinetics of CALB catalyzed hydrolysis. Angew Chem Int Ed 44(4):560–564
Wadhwa RR, Zalányi L, Szente J, Négyessy L, Érdi P (2017) Stochastic kinetics of the circular gene hypothesis: feedback effects and protein fluctuations. Math Comput Simul 133:326–336
Weber J, Celardin F (1976) A general computer program for the simulation of reaction kinetics by the Monte Carlo technique. Chimia 30(4):236–237
Weiss S (1999) Fluorescence spectroscopy of single biomolecules. Science 283(5408):1676–1683
Whittle P (1975) Reversibility and acyclicity. In: Perspectives in probability and statistics. Applied probability trust
Whittle P (1986) Systems in stochastic equilibrium. Wiley, Hoboken
Yan CCS, Hsu CP (2013) The fluctuation-dissipation theorem for stochastic kinetics—implications on genetic regulations. J Chem Phys 139(22):224, 109
Zhang J, Hou Z, Xin H (2005) Effects of internal noise for calcium signaling in a coupled cell system. Phys Chem Chem Phys 7(10):2225–2228
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Science+Business Media, LLC, part of Springer Nature
About this chapter
Cite this chapter
Tóth, J., Nagy, A.L., Papp, D. (2018). Stochastic Models. In: Reaction Kinetics: Exercises, Programs and Theorems. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8643-9_10
Download citation
DOI: https://doi.org/10.1007/978-1-4939-8643-9_10
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-8641-5
Online ISBN: 978-1-4939-8643-9
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)