Vellela Melissa, Qian Hong
Department of Applied Mathematics, University of Washington Seattle, Seattle, WA 98195, USA.
Bull Math Biol. 2007 Jul;69(5):1727-46. doi: 10.1007/s11538-006-9188-3. Epub 2007 Feb 23.
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.
对于一个生化反应系统，从T.G.库尔茨的研究工作[《应用概率杂志》8, 344 (1971)]可知，基于随机公式的化学主方程模型在有限时间内，于无限系统规模极限下趋近于基于质量作用定律的确定性模型。然而，这两个模型通常表现出明显不同的稳态行为。为了进一步研究这个“悖论”，我们对一个简单的自催化生化反应的确定性模型和随机模型进行了比较研究，该反应取自已故的J.凯泽的一篇文献。我们计算了随机模型中的灭绝预期时间、真实随机稳态和准稳态概率分布。我们表明，随机模型在合理的时间尺度上预测了确定性行为，这可以从两个模型中一致地得到。然而，灭绝的过渡时间随系统规模呈指数增长。在数学上，我们发现交换无限系统规模和无限时间的极限是有问题的。我们还讨论了在数值计算中可被视为足够大的合适系统规模，这是一个重要参数。
