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On the roles of the Bessel and Poisson distributions in chemical kinetics

Published online by Cambridge University Press:  14 July 2016

Peter Hall
Peter Hall*
Affiliation:
The Australian National University
*
∗ Postal address: Department of Statistics, The Faculties, The Australian National University, G.P.O. Box 4, Canberra, ACT 2601, Australia.
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Abstract

We consider the Darvey, Ninham and Staff model for reversible chemical reactions, in the case where the ratio of the rate constants is either very large or very small. It is shown that the distribution of the number of molecules at equilibrium may sometimes be closely approximated by the Poisson distribution, and on other occasions, by a distribution with much smaller tails than the Poisson. The second type of approximating distribution is termed a Bessel distribution, and its properties are studied.

Keywords

APPROXIMATING DISTRIBUTIONNORMAL DISTRIBUTIONREVERSIBLE REACTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 3 , September 1983 , pp. 585 - 599
Copyright
Copyright © Applied Probability Trust 1983 

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References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Buchholz, H. (1969) The Confluent Hypergeometric Function. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Darvey, I. G., Ninham, B. W. and Staff, P. J. (1966) Stochastic models for second-order chemical reaction kinetics. The equilibrium state. J. Chem. Phys. 45, 2145–2155.CrossRefGoogle Scholar
Darvey, I. G. and Staff, P. J. (1967) The application of the theory of Markov processes to the reversible one substrate-one intermediate-one product enzymic reaction. J. Theoret. Biol. 14, 157–172.CrossRefGoogle Scholar
Delbrück, M. (1940) Statistical fluctuations in autocatalytic reactions. J. Chem. Phys. 8, 120–124.Google Scholar
Dunstan, F. D. J. and Reynolds, J. F. (1981) Normal approximation for distributions arising in the stochastic approach to chemical reaction kinetics. J. Appl. Prob. 18, 263–267.Google Scholar
Erdélyi, A. (1960) Asymptotic forms for Laguerre polynomials. Indian Math. J. 24, 235–250.Google Scholar
Formosinho, S. J. and Miguel, M. Da G. M. (1979) Markov chains for plotting the course of chemical reactions. J. Chem. Education 56, 582–585.Google Scholar
Hatlee, M. D. and Kozak, J. J. (1980) A stochastic approach to the theory of intramicellar kinetics, I. Master equation for irreversible reactions. J. Chem. Phys. 72, 4358–4367.CrossRefGoogle Scholar
Hatlee, M. D. and Kozak, J. J. (1981) A stochastic approach to the theory of intramicellar kinetics, II. Master equation for reversible reactions. J. Chem. Phys. 74, 1098–1109.Google Scholar
Hawkins, R. and Rice, S. A. (1971) Study of concentration fluctuations in model systems. J. Theoret. Biol. 30, 579–596.Google Scholar
Hertzberg, R. C. and Gallucci, V. F. (1980) First-order stochastic chemical reactions and oscillations in the variance. J. Appl. Prob. 17, 1087–1093.CrossRefGoogle Scholar
Heyde, C. C. and Heyde, E. (1969) A stochastic approach to a one substrate, one product enzyme reaction in the initial velocity phase. J. Theoret. Biol. 25, 159–167.Google Scholar
Heyde, C. C. and Heyde, E. (1971) Stochastic fluctuations in a one substrate one product enzyme system: are they ever relevant? J. Theoret. Biol. 30, 395–404.Google Scholar
Irwin, J. O. (1937) The frequency distribution of the difference between two independent variates following the same Poisson distribution. J. R. Statist. Soc. A 100, 415–416.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Discrete Distributions. Houghton Mifflin, Boston.Google Scholar
Kirby, M. R. (1969) Stochastic method for the simulation of biochemical systems on a digital computer. Nature, London 222, 298–299.Google Scholar
Mcquarrie, D. A. (1967) Stochastic approach to chemical kinetics. J. Appl. Prob. 4, 413–478.Google Scholar
Noack, A. (1950) A class of random variables with discrete distributions. Ann. Math. Statist. 21, 127–132.Google Scholar
Oppenheim, I., Shuler, K. E. and Weiss, G. H. (1969) Stochastic and deterministic formulation of chemical rate equations. J. Chem. Phys. 50, 460–466.Google Scholar
Orriss, J. (1969) Equilibrium distributions for systems of chemical reactions with applications to the theory of molecular adsorption. J. Appl. Prob. 6, 505–515.Google Scholar
Saito, N. (1974) Fluctuations in chemical reactions around the steady state. J. Chem. Phys. 61, 3644–3650.Google Scholar
Smeach, S. C. and Smith, W. (1973) A comparison of stochastic and deterministic models for cell membrane transport. J. Theoret. Biol. 42, 157–167.CrossRefGoogle ScholarPubMed
Staff, P. J. (1967) Approximation method for equilibrium distributions in second-order chemical reaction kinetics. J. Chem. Phys. 46, 2209–2212.CrossRefGoogle Scholar
Szegö, G. (1959) Orthogonal Polynomials. American Mathematical Society, New York.Google Scholar
Tallis, G. M. and Leslie, R. T. (1969) General models for r-molecular reactions. J. Appl. Prob. 6, 74–87.Google Scholar
Thakur, A. K., Rescigno, A. and De Lisi, C. (1978) Stochastic theory of second order chemical reactions. J. Phys. Chem. 82, 552–558.Google Scholar