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September  2012, 17(6): 1775-1794. doi: 10.3934/dcdsb.2012.17.1775

Slow manifold reduction of a stochastic chemical reaction: Exploring Keizer's paradox

Parker Childs 1, and  James P. Keener 1,

1. 

Department of Mathematics, University of Utah, Salt Lake City, UT 84112, United States, United States

Received  May 2011 Revised  November 2011 Published  May 2012

Keizer's paradox refers to the observation that deterministic and stochastic descriptions of chemical reactions can predict vastly different long term outcomes. In this paper, we use slow manifold analysis to help resolve this paradox for four variants of a simple autocatalytic reaction. We also provide rigorous estimates of the spectral gap of important linear operators, which establishes parameter ranges in which the slow manifold analysis is appropriate.
Keywords: stochastic chemical reactions, Markov processes, autocatalytic reactions, slow manifold reduction, Chemical master equations, spectral gap theory..
Mathematics Subject Classification: Primary: 60J27, 34E13; Secondary: 80A3.
Citation: Parker Childs, James P. Keener. Slow manifold reduction of a stochastic chemical reaction: Exploring Keizer's paradox. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1775-1794. doi: 10.3934/dcdsb.2012.17.1775
References:
[1]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,'', Pearson Education, (2003).   Google Scholar

[2]

J. Beischke, P. Weber, N. Sarafoff, M. Beekes, A. Giese and H. Kretzschmar, Autocatalytic self-propagation of misfolded prion protein,, Proc. Natl. Acad. Sci., 101 (2004), 12207.  doi: 10.1073/pnas.0404650101.  Google Scholar

[3]

C. W. Gardiner, "Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences,'' Second edition,, Springer Series in Synergetics, 13 (1985).   Google Scholar

[4]

B. L. Granovsky and A. I. Zeifman, The decay function of nonhomogeneous birth-death processes, with application to mean-field models,, Stoch. Process. Appl., 72 (1997), 105.  doi: 10.1016/S0304-4149(97)00085-9.  Google Scholar

[5]

B. L. Granovsky and A. I. Zeifman, The $N$-limit of spectral gap of a class of birth-death Markov chains,, Appl. Stoch. Models Bus. Ind., 16 (2000), 235.  doi: 10.1002/1526-4025(200010/12)16:4<235::AID-ASMB415>3.3.CO;2-J.  Google Scholar

[6]

J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes,'', Springer-Verlag, (1987).   Google Scholar

[7]

T. G. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential processes,, J. Appl. Prob., 8 (1971), 344.   Google Scholar

[8]

T. G. Kurtz, The relationship between stochastic and deterministic models for chemical reactions,, J. Chem. Phys., 57 (1972), 2976.  doi: 10.1063/1.1678692.  Google Scholar

[9]

I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model,, Journal of Theoretical Biology, 211 (2001), 11.  doi: 10.1006/jtbi.2001.2328.  Google Scholar

[10]

H. Qian and L. M. Bishop, The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: Linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks,, Int. J. Mol. Sci., 11 (2010), 3472.  doi: 10.3390/ijms11093472.  Google Scholar

[11]

N. van Kampen, "Stochastic Processes in Physics and Chemistry,'', Lecture Notes in Mathematics, 888 (1981).   Google Scholar

[12]

M. Vellela and H. Qian, A quasistationary analysis of a stochastic chemical reaction: Keizer's paradox,, Bull. Math. Biol., 69 (2007), 1727.  doi: 10.1007/s11538-006-9188-3.  Google Scholar

show all references

References:
[1]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,'', Pearson Education, (2003).   Google Scholar

[2]

J. Beischke, P. Weber, N. Sarafoff, M. Beekes, A. Giese and H. Kretzschmar, Autocatalytic self-propagation of misfolded prion protein,, Proc. Natl. Acad. Sci., 101 (2004), 12207.  doi: 10.1073/pnas.0404650101.  Google Scholar

[3]

C. W. Gardiner, "Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences,'' Second edition,, Springer Series in Synergetics, 13 (1985).   Google Scholar

[4]

B. L. Granovsky and A. I. Zeifman, The decay function of nonhomogeneous birth-death processes, with application to mean-field models,, Stoch. Process. Appl., 72 (1997), 105.  doi: 10.1016/S0304-4149(97)00085-9.  Google Scholar

[5]

B. L. Granovsky and A. I. Zeifman, The $N$-limit of spectral gap of a class of birth-death Markov chains,, Appl. Stoch. Models Bus. Ind., 16 (2000), 235.  doi: 10.1002/1526-4025(200010/12)16:4<235::AID-ASMB415>3.3.CO;2-J.  Google Scholar

[6]

J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes,'', Springer-Verlag, (1987).   Google Scholar

[7]

T. G. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential processes,, J. Appl. Prob., 8 (1971), 344.   Google Scholar

[8]

T. G. Kurtz, The relationship between stochastic and deterministic models for chemical reactions,, J. Chem. Phys., 57 (1972), 2976.  doi: 10.1063/1.1678692.  Google Scholar

[9]

I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model,, Journal of Theoretical Biology, 211 (2001), 11.  doi: 10.1006/jtbi.2001.2328.  Google Scholar

[10]

H. Qian and L. M. Bishop, The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: Linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks,, Int. J. Mol. Sci., 11 (2010), 3472.  doi: 10.3390/ijms11093472.  Google Scholar

[11]

N. van Kampen, "Stochastic Processes in Physics and Chemistry,'', Lecture Notes in Mathematics, 888 (1981).   Google Scholar

[12]

M. Vellela and H. Qian, A quasistationary analysis of a stochastic chemical reaction: Keizer's paradox,, Bull. Math. Biol., 69 (2007), 1727.  doi: 10.1007/s11538-006-9188-3.  Google Scholar

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