Bulletin of Mathematical Biology

, Volume 69, Issue 5, pp 1727–1746 | Cite as

A Quasistationary Analysis of a Stochastic Chemical Reaction: Keizer’s Paradox

  • Melissa VellelaEmail author
  • Hong Qian
Original Article


For a system of biochemical reactions, it is known from the work of T.G. Kurtz [J. Appl. Prob. 8, 344 (1971)] that the chemical master equation model based on a stochastic formulation approaches the deterministic model based on the Law of Mass Action in the infinite system-size limit in finite time. The two models, however, often show distinctly different steady-state behavior. To further investigate this “paradox,” a comparative study of the deterministic and stochastic models of a simple autocatalytic biochemical reaction, taken from a text by the late J. Keizer, is carried out. We compute the expected time to extinction, the true stochastic steady state, and a quasistationary probability distribution in the stochastic model. We show that the stochastic model predicts the deterministic behavior on a reasonable time scale, which can be consistently obtained from both models. The transition time to the extinction, however, grows exponentially with the system size. Mathematically, we identify that exchanging the limits of infinite system size and infinite time is problematic. The appropriate system size that can be considered sufficiently large, an important parameter in numerical computation, is also discussed.


Stochastic models Nonlinear Chemical kinetics Quasistationary Uniform convergence 


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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of Washington SeattleSeattleUSA

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