Department of Electrical and Computer Engineering, and Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois, United States of America.
Department of Biology, California State University, Northridge, Northridge, California, United States of America.
PLoS One. 2020 Jun 30;15(6):e0235070. doi: 10.1371/journal.pone.0235070. eCollection 2020.
A gene regulatory network can be described at a high level by a directed graph with signed edges, and at a more detailed level by a system of ordinary differential equations (ODEs). The former qualitatively models the causal regulatory interactions between ordered pairs of genes, while the latter quantitatively models the time-varying concentrations of mRNA and proteins. This paper clarifies the connection between the two types of models. We propose a property, called the constant sign property, for a general class of ODE models. The constant sign property characterizes the set of conditions (system parameters, external signals, or internal states) under which an ODE model is consistent with a signed, directed graph. If the constant sign property for an ODE model holds globally for all conditions, then the ODE model has a single signed, directed graph. If the constant sign property for an ODE model only holds locally, which may be more typical, then the ODE model corresponds to different graphs under different sets of conditions. In addition, two versions of constant sign property are given and a relationship between them is proved. As an example, the ODE models that capture the effect of cis-regulatory elements involving protein complex binding, based on the model in the GeneNetWeaver source code, are described in detail and shown to satisfy the global constant sign property with a unique consistent gene regulatory graph. Even a single gene regulatory graph is shown to have many ODE models of GeneNetWeaver type consistent with it due to combinatorial complexity and continuous parameters. Finally the question of how closely data generated by one ODE model can be fit by another ODE model is explored. It is observed that the fit is better if the two models come from the same graph.
基因调控网络可以用带符号边的有向图在高层次上描述，也可以用常微分方程（ODE）系统在更详细的层次上描述。前者定性地模拟基因对之间有顺序的因果调控相互作用，而后者定量地模拟 mRNA 和蛋白质的时变浓度。本文澄清了这两种模型之间的联系。我们为一类一般的 ODE 模型提出了一个称为常数符号属性的属性。常数符号属性表征了一组条件（系统参数、外部信号或内部状态），在这些条件下，ODE 模型与有符号、有向图一致。如果 ODE 模型的常数符号属性在所有条件下都全局成立，那么 ODE 模型只有一个有符号、有向图。如果 ODE 模型的常数符号属性仅局部成立，这可能更为典型，那么 ODE 模型在不同的条件下对应于不同的图。此外，给出了两种常数符号属性的版本，并证明了它们之间的关系。作为一个例子，详细描述了基于 GeneNetWeaver 源代码中模型的、捕获涉及蛋白质复合物结合的顺式调控元件影响的 ODE 模型，并显示它们满足具有唯一一致基因调控图的全局常数符号属性。即使是一个单一的基因调控图，由于组合复杂性和连续参数，也显示出有许多与它一致的 GeneNetWeaver 类型的 ODE 模型。最后，探讨了一个 ODE 模型生成的数据与另一个 ODE 模型拟合得有多紧密的问题。观察到，如果两个模型来自同一个图，则拟合效果更好。
