Reis Matthias, Kromer Justus A, Klipp Edda
Institute of Biology, Theoretical Biophysics, Humboldt-Universität zu Berlin, 10117, Berlin, Germany.
Center for Advancing Electronics Dresden, Technische Universität Dresden, 01062, Dresden, Germany.
J Math Biol. 2018 Aug;77(2):377-419. doi: 10.1007/s00285-018-1205-2. Epub 2018 Jan 20.
Multimodality is a phenomenon which complicates the analysis of statistical data based exclusively on mean and variance. Here, we present criteria for multimodality in hierarchic first-order reaction networks, consisting of catalytic and splitting reactions. Those networks are characterized by independent and dependent subnetworks. First, we prove the general solvability of the Chemical Master Equation (CME) for this type of reaction network and thereby extend the class of solvable CME's. Our general solution is analytical in the sense that it allows for a detailed analysis of its statistical properties. Given Poisson/deterministic initial conditions, we then prove the independent species to be Poisson/binomially distributed, while the dependent species exhibit generalized Poisson/Khatri Type B distributions. Generalized Poisson/Khatri Type B distributions are multimodal for an appropriate choice of parameters. We illustrate our criteria for multimodality by several basic models, as well as the well-known two-stage transcription-translation network and Bateman's model from nuclear physics. For both examples, multimodality was previously not reported.
多峰性是一种使仅基于均值和方差的统计数据分析变得复杂的现象。在此，我们提出了由催化反应和分裂反应组成的层次一阶反应网络中多峰性的判据。那些网络的特征在于独立子网和相关子网。首先，我们证明了此类反应网络的化学主方程（CME）的一般可解性，从而扩展了可解CME的类别。我们的一般解在允许对其统计特性进行详细分析的意义上是解析的。给定泊松/确定性初始条件，然后我们证明独立物种服从泊松/二项分布，而相关物种呈现广义泊松/Khatri B型分布。对于适当的参数选择，广义泊松/Khatri B型分布是多峰的。我们通过几个基本模型以及著名的两阶段转录 - 翻译网络和核物理中的贝特曼模型来说明我们的多峰性判据。对于这两个例子，以前都没有报道过多峰性。
