ON CONSTRAINED LANGEVIN EQUATIONS AND (BIO)CHEMICAL REACTION NETWORKS
ON CONSTRAINED LANGEVIN EQUATIONS AND (BIO)CHEMICAL REACTION NETWORKS
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DOI：
10.1137/18m1190999
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发表时间：
2019-01-01
影响因子：
1.6
通讯作者：
Williams, Ruth J.
中科院分区：
文献类型：
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作者：
Anderson, David F.;Higham, Desmond J.;Williams, Ruth J.
随机效应在系统生物学等领域的化学反应系统的时间演化建模中发挥着重要作用，其中某些组成分子的浓度可能很低。这些系统最常见的随机模型是连续时间马尔可夫链，它跟踪每种化学物质的分子丰度。通常，这些随机模型是通过计算机模拟来研究的，这很快就会变得计算成本高昂。减少计算量的常见方法是通过连续值扩散过程来近似离散值马尔可夫链。然而，现有的扩散近似要么不遵守化学浓度永远不会为负的约束（线性噪声近似），要么通常仅在某些化学物质的浓度首先变为零之前才有效（化学朗之万方程）。在本文中，我们提出（倾斜）反射扩散，它尊重化学浓度的非负性，作为化学反应网络的马尔可夫链模型的近似。这些反射扩散满足“约束朗之万方程”，因为它们的行为类似于正正交体内部的化学朗之万方程的解，并且通过边界处的瞬时倾斜反射被约束到正交体。为了激发它们的形式，我们首先用两个简单的例子来说明我们的约束朗之万近似。然后我们描述我们提出的近似的一般形式。我们通过将两个示例的平稳分布与马尔可夫链模型的平稳分布进行比较以及通过模拟更复杂的示例来说明我们的近似值的性能。
Stochastic effects play an important role in modeling the time evolution of chemical reaction systems in fields such as systems biology, where the concentrations of some constituent molecules can be low. The most common stochastic models for these systems are continuous time Markov chains, which track the molecular abundance of each chemical species. Often, these stochastic models are studied by computer simulations, which can quickly become computationally expensive. A common approach to reduce computational effort is to approximate the discrete valued Markov chain by a continuous valued diffusion process. However, existing diffusion approximations either do not respect the constraint that chemical concentrations are never negative (linear noise approximation) or are typically only valid until the concentration of some chemical species first becomes zero (chemical Langevin equation). In this paper, we propose (obliquely) reflected diffusions, which respect the nonnegativity of chemical concentrations, as approximations for Markov chain models of chemical reaction networks. These reflected diffusions satisfy "constrained Langevin equations," in that they behave like solutions of chemical Langevin equations in the interior of the positive orthant and are constrained to the orthant by instantaneous oblique reflection at the boundary. To motivate their form, we first illustrate our constrained Langevin approximations for two simple examples. We then describe the general form of our proposed approximation. We illustrate the performance of our approximations through comparison of their stationary distributions for the two examples with those of the Markov chain model and through simulations of more complex examples.