Wu Jialiang, Vidakovic Brani, Voit Eberhard O
Deparment of Mathematics, Bioinformatics Program, Georgia Institute of Technology, Atlanta, GA30332, USA.
BMC Syst Biol. 2011 Nov 8;5:187. doi: 10.1186/1752-0509-5-187.
Gillespie's stochastic simulation algorithm (SSA) for chemical reactions admits three kinds of elementary processes, namely, mass action reactions of 0th, 1st or 2nd order. All other types of reaction processes, for instance those containing non-integer kinetic orders or following other types of kinetic laws, are assumed to be convertible to one of the three elementary kinds, so that SSA can validly be applied. However, the conversion to elementary reactions is often difficult, if not impossible. Within deterministic contexts, a strategy of model reduction is often used. Such a reduction simplifies the actual system of reactions by merging or approximating intermediate steps and omitting reactants such as transient complexes. It would be valuable to adopt a similar reduction strategy to stochastic modelling. Indeed, efforts have been devoted to manipulating the chemical master equation (CME) in order to achieve a proper propensity function for a reduced stochastic system. However, manipulations of CME are almost always complicated, and successes have been limited to relative simple cases.
We propose a rather general strategy for converting a deterministic process model into a corresponding stochastic model and characterize the mathematical connections between the two. The deterministic framework is assumed to be a generalized mass action system and the stochastic analogue is in the format of the chemical master equation. The analysis identifies situations: where a direct conversion is valid; where internal noise affecting the system needs to be taken into account; and where the propensity function must be mathematically adjusted. The conversion from deterministic to stochastic models is illustrated with several representative examples, including reversible reactions with feedback controls, Michaelis-Menten enzyme kinetics, a genetic regulatory motif, and stochastic focusing.
The construction of a stochastic model for a biochemical network requires the utilization of information associated with an equation-based model. The conversion strategy proposed here guides a model design process that ensures a valid transition between deterministic and stochastic models.
吉莱斯皮化学反应随机模拟算法（SSA）允许三种基本过程，即零阶、一阶或二阶质量作用反应。所有其他类型的反应过程，例如那些包含非整数动力学阶数或遵循其他类型动力学定律的过程，都假定可以转换为这三种基本类型之一，以便可以有效地应用SSA。然而，转换为基本反应通常很困难，甚至不可能。在确定性背景下，经常使用模型简化策略。这种简化通过合并或近似中间步骤并省略诸如瞬态复合物等反应物来简化实际反应系统。将类似的简化策略应用于随机建模将是有价值的。实际上，已经致力于操纵化学主方程（CME），以实现简化随机系统的适当倾向函数。然而，对CME的操纵几乎总是很复杂，并且成功仅限于相对简单的情况。
我们提出了一种将确定性过程模型转换为相应随机模型的相当通用的策略，并描述了两者之间的数学联系。确定性框架假定为广义质量作用系统，随机类似物采用化学主方程的形式。分析确定了以下情况：直接转换有效的情况；需要考虑影响系统的内部噪声的情况；以及倾向函数必须在数学上进行调整的情况。通过几个代表性示例说明了从确定性模型到随机模型的转换，包括具有反馈控制的可逆反应、米氏酶动力学、遗传调控基序和随机聚焦。
生化网络随机模型的构建需要利用与基于方程的模型相关的信息。这里提出的转换策略指导了一个模型设计过程，确保在确定性模型和随机模型之间进行有效的转换。
