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General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks
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 indep...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Springer Berlin Heidelberg
2018
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6061068/ https://www.ncbi.nlm.nih.gov/pubmed/29353313 http://dx.doi.org/10.1007/s00285-018-1205-2 |
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author | Reis, Matthias Kromer, Justus A. Klipp, Edda |
author_facet | Reis, Matthias Kromer, Justus A. Klipp, Edda |
author_sort | Reis, Matthias |
collection | PubMed |
description | 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. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (10.1007/s00285-018-1205-2) contains supplementary material, which is available to authorized users. |
format | Online Article Text |
id | pubmed-6061068 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2018 |
publisher | Springer Berlin Heidelberg |
record_format | MEDLINE/PubMed |
spelling | pubmed-60610682018-08-09 General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks Reis, Matthias Kromer, Justus A. Klipp, Edda J Math Biol Article 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. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (10.1007/s00285-018-1205-2) contains supplementary material, which is available to authorized users. Springer Berlin Heidelberg 2018-01-20 2018 /pmc/articles/PMC6061068/ /pubmed/29353313 http://dx.doi.org/10.1007/s00285-018-1205-2 Text en © The Author(s) 2018 Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
spellingShingle | Article Reis, Matthias Kromer, Justus A. Klipp, Edda General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks |
title | General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks |
title_full | General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks |
title_fullStr | General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks |
title_full_unstemmed | General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks |
title_short | General solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks |
title_sort | general solution of the chemical master equation and modality of marginal distributions for hierarchic first-order reaction networks |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6061068/ https://www.ncbi.nlm.nih.gov/pubmed/29353313 http://dx.doi.org/10.1007/s00285-018-1205-2 |
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