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 & 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

[1]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[2]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[3]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[4]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[5]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[6]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[7]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[8]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[9]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[10]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[11]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[12]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[13]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[14]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[15]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[16]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[17]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[18]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

[19]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[20]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]