Delay chemical master equation: direct and closed-form solutions
The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equati...
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| Published in: | Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Vol. 471; no. 2179; p. 20150049 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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08.07.2015
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| ISSN: | 1364-5021 |
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| Abstract | The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived. |
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| AbstractList | The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived. The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived. |
| Author | Leier, Andre Marquez-Lago, Tatiana T |
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| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/26345616$$D View this record in MEDLINE/PubMed |
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| Keywords | delay stochastic simulation algorithm delay chemical master equation closed-form solution direct solution |
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| References | 21210242 - Bull Math Biol. 2011 Sep;73(9):2231-47 16199522 - Proc Natl Acad Sci U S A. 2005 Oct 11;102(41):14593-8 17411109 - J Chem Phys. 2007 Mar 28;126(12):124108 18067349 - J Chem Phys. 2007 Dec 7;127(21):214107 16953443 - J Math Biol. 2007 Jan;54(1):1-26 19044893 - J Chem Phys. 2008 Sep 7;129(9):095105 20202198 - BMC Syst Biol. 2010 Mar 04;4:19 17228945 - J Chem Phys. 2007 Jan 14;126(2):024109 19355717 - J Chem Phys. 2009 Apr 7;130(13):134107 16460146 - J Chem Phys. 2006 Jan 28;124(4):044104 23940327 - Proc Natl Acad Sci U S A. 2013 Aug 27;110(35):14261-5 23514472 - J Chem Phys. 2013 Mar 14;138(10):104114 23679462 - Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Apr;87(4):042720 19154084 - IET Syst Biol. 2009 Jan;3(1):52-8 24694895 - J R Soc Interface. 2014 Apr 02;11(95):20140108 16965175 - PLoS Comput Biol. 2006 Sep 8;2(9):e117 |
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