Abstract
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.
Highlights
The development of single-molecule methods such as Fluorescence in situ Hybridization (FISH) has resulted in considerable advances in fundamental research in biology (Trcek et al 2011, 2012)
Time-dependent, analytical solutions only exist for monomolecular reaction networks (Gans 1960; Jahnke and Huisinga 2007), consisting exclusively of conversion Si ↔ S j, degradation Si → ∅ and production ∅ → S j reactions, where Si defines a species of molecules
We show that the independent part of the network exhibits Poissonian and Binomial marginal distributions, while the dependent part is described by Discrete Compound Poisson (DCP) and Kathri Type B (KTB) marginal distributions
Summary
The development of single-molecule methods such as Fluorescence in situ Hybridization (FISH) has resulted in considerable advances in fundamental research in biology (Trcek et al 2011, 2012). In contrast to monomolecular networks, first-order networks result in marginal distributions of individual species that are not unimodal in general, as we show in Theorems 3 and 4. These distributions are poorly characterized by mean and variance, two essential measures used to describe statistical data in natural sciences. DCP/KTB marginal distributions from the dependent part can lead to multimodality under quite general conditions (Theorems 3 and 4) We illustrate these general results by several basic models The derivation of probability mass functions from generating functions is reviewed in Appendix A.1
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