# American Institute of Mathematical Sciences

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

 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.
Citation: Parker Childs, James P. Keener. Slow manifold reduction of a stochastic chemical reaction: Exploring Keizer's paradox. Discrete and 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, Inc., 2003. [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), 12207-12211. doi: 10.1073/pnas.0404650101. [3] C. W. Gardiner, "Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences,'' Second edition, Springer Series in Synergetics, 13, Springer-Verlag, Berlin, 1985. [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), 105-120. doi: 10.1016/S0304-4149(97)00085-9. [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), 235-248. doi: 10.1002/1526-4025(200010/12)16:4<235::AID-ASMB415>3.3.CO;2-J. [6] J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes,'' Springer-Verlag, 1987. [7] T. G. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential processes, J. Appl. Prob., 8 (1971), 344-356. [8] T. G. Kurtz, The relationship between stochastic and deterministic models for chemical reactions, J. Chem. Phys., 57 (1972), 2976-2978. doi: 10.1063/1.1678692. [9] I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, Journal of Theoretical Biology, 211 (2001), 11-27. doi: 10.1006/jtbi.2001.2328. [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), 3472-3500. doi: 10.3390/ijms11093472. [11] N. van Kampen, "Stochastic Processes in Physics and Chemistry,'' Lecture Notes in Mathematics, 888, North-Holland Publishing Co., Amsterdam-New York, 1981. [12] M. Vellela and H. Qian, A quasistationary analysis of a stochastic chemical reaction: Keizer's paradox, Bull. Math. Biol., 69 (2007), 1727-1746. doi: 10.1007/s11538-006-9188-3.

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##### References:
 [1] L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,'' Pearson Education, Inc., 2003. [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), 12207-12211. doi: 10.1073/pnas.0404650101. [3] C. W. Gardiner, "Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences,'' Second edition, Springer Series in Synergetics, 13, Springer-Verlag, Berlin, 1985. [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), 105-120. doi: 10.1016/S0304-4149(97)00085-9. [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), 235-248. doi: 10.1002/1526-4025(200010/12)16:4<235::AID-ASMB415>3.3.CO;2-J. [6] J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes,'' Springer-Verlag, 1987. [7] T. G. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential processes, J. Appl. Prob., 8 (1971), 344-356. [8] T. G. Kurtz, The relationship between stochastic and deterministic models for chemical reactions, J. Chem. Phys., 57 (1972), 2976-2978. doi: 10.1063/1.1678692. [9] I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, Journal of Theoretical Biology, 211 (2001), 11-27. doi: 10.1006/jtbi.2001.2328. [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), 3472-3500. doi: 10.3390/ijms11093472. [11] N. van Kampen, "Stochastic Processes in Physics and Chemistry,'' Lecture Notes in Mathematics, 888, North-Holland Publishing Co., Amsterdam-New York, 1981. [12] M. Vellela and H. Qian, A quasistationary analysis of a stochastic chemical reaction: Keizer's paradox, Bull. Math. Biol., 69 (2007), 1727-1746. doi: 10.1007/s11538-006-9188-3.
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