生物系统的混合开关跳跃扩散近似分析
Approximate analysis of biological systems by hybrid switching jump diffusion
摘要：
在本文中，我们考虑大状态空间连续时间马尔可夫链（MC）中出现的系统生物学领域。对于代表大群相同物体相互作用的密度相关MC族，Kurtz提出了两种近似。一个是基于常微分方程，而另一个使用扩散过程。确定性近似的计算成本明显较低，但扩散近似保留了随机性，并且能够再现相关的随机特征，如方差，双峰性和尾部行为。在最近的一篇论文中，对于特定的随机Petri网模型，我们提出了一个跳跃扩散近似，其目的是适用于超出库尔茨扩散近似的限制，即当过程以不可忽略的概率到达边界时。扩散近似在其原始形式的其他限制是，它可以提供不准确的结果时，在某些群体中的对象的数量往往或不断低，它只能适用于纯密度依赖马尔可夫链。为了克服这些缺点，在本文中，我们建议只适用于那些组件的模型是在密度依赖的形式，并与高人口水平的跳跃扩散近似。其余分量被视为离散量。由此产生的过程是一个混合开关跳跃扩散。我们表明，随机微分方程的特点，这个过程可以自动从原始马尔可夫链的描述或从一个更高层次的描述语言，如随机Petri网。所提出的方法是说明三个模型：一个建模所谓的疯狂时钟反应，一个描述病毒感染动力学和最后考虑转录调控。
英文摘要：
In this paper we consider large state space continuous time Markov chains (MCs) arising in the field of systems biology. For density dependent families of MCs that represent the interaction of large groups of identical objects, Kurtz has proposed two kinds of approximations. One is based on ordinary differential equations, while the other uses a diffusion process. The computational cost of the deterministic approximation is significantly lower, but the diffusion approximation retains stochasticity and is able to reproduce relevant random features like variance, bimodality, and tail behavior. In a recent paper, for particular stochastic Petri net models, we proposed a jump diffusion approximation that aims at being applicable beyond the limits of Kurtz's diffusion approximation, namely when the process reaches the boundary with non-negligible probability. Other limitations of the diffusion approximation in its original form are that it can provide inaccurate results when the number of objects in some groups is often or constantly low and that it can be applied only to pure density dependent Markov chains. In order to overcome these drawbacks, in this paper we propose to apply the jump-diffusion approximation only to those components of the model that are in density dependent form and are associated with high population levels. The remaining components are treated as discrete quantities. The resulting process is a hybrid switching jump diffusion. We show that the stochastic differential equations that characterize this process can be derived automatically both from the description of the original Markov chains or starting from a higher level description language, like stochastic Petri nets. The proposed approach is illustrated on three models: one modeling the so called crazy clock reaction, one describing viral infection kinetics and the last considering transcription regulation.
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