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Chemical Master Equation and Langevin regimes for a gene transcription model

Raya Khanin, Desmond J. Higham
2008 Theoretical Computer Science  
Gene transcription models must take account of intrinsic stochasticity.  ...  To analyse the Chemical Master Equation we use some recent work of Gadgil, Lee and Othmer, and to analyse the Chemical Langevin Equation we use Ito's Lemma.  ...  Fig. 3 . 3 One path from the Chemical Master Equation (CME) and Chemical Langevin Equation (CLE) for the dimerization model (36)-(39) expectations leads to an ODE for E[P ] that involves E[P 2 ].  ... 
doi:10.1016/j.tcs.2008.07.007 fatcat:axpgb5zw3jdwnhlx6ede7735ha

Chemical Master Equation and Langevin Regimes for a Gene Transcription Model [chapter]

Raya Khanin, Desmond J. Higham
Lecture Notes in Computer Science  
Gene transcription models must take account of intrinsic stochasticity.  ...  To analyse the Chemical Master Equation we use some recent work of Gadgil, Lee and Othmer, and to analyse the Chemical Langevin Equation we use Ito's Lemma.  ...  Fig. 3 . 3 One path from the Chemical Master Equation (CME) and Chemical Langevin Equation (CLE) for the dimerization model (36)-(39) expectations leads to an ODE for E[P ] that involves E[P 2 ].  ... 
doi:10.1007/978-3-540-75140-3_1 fatcat:nwjfgckgazh3pinvubtbmgqe7m

The chemical Langevin equation for biochemical systems in dynamic environments [article]

Lucy Ham, Megan Coomer, Michael P.H. Stumpf
2021 bioRxiv   pre-print
We develop an extrinsic chemical Langevin equation - a physically motivated extension of the chemical Langevin equation - to model intrinsically noisy reaction networks embedded in a stochastically fluctuating  ...  The extrinsic CLE is a continuous approximation to the Chemical Master Equation (CME) with time-varying propensities.  ...  THE EXTRINSIC CHEMICAL LANGEVIN EQUATION A.  ... 
doi:10.1101/2021.12.19.473404 fatcat:rv7ydd6ffzalth2npasze3y65m

Chemical Master versus Chemical Langevin for First-Order Reaction Networks

Desmond J. Higham, Raya Khanin
2008 Open Applied Mathematics Journal  
Hence, in terms of means, variances and correlations, the Chemical Langevin Equation is an excellent proxy for the Chemical Master Equation.  ...  We also show that the moment matching result extends to the case where a gene regulation model of Raser and O'Shea (Science, 2004) is replaced by a hybrid model that mixes elements of the Master and Langevin  ...  Acknowledgements This work was supported by EPSRC grant GR/S62383/01 (DJH) and by a Synergy grant from the Universities of Strathclyde and Glasgow (RK and DJH).  ... 
doi:10.2174/1874114200802010059 fatcat:ahh2amvejbbqhce3axokwha3fe

Modeling stochastic noise in gene regulatory systems

Arwen Meister, Chao Du, Ye Henry Li, Wing Hung Wong
2014 Quantitative Biology  
Expansion of the Master equation shows that for systems with a single stable steady-state, the stochastic model reduces to a deterministic model in a first-order approximation.  ...  The Master equation is considered the gold standard for modeling the stochastic mechanisms of gene regulation in molecular detail, but it is too complex to solve exactly in most cases, so approximation  ...  Since gene regulation depends on a series of chemical reactions (including the binding of TFs and RNAP to the promoter, transcription, translation, and degradation), it can be modeled with chemical equations  ... 
doi:10.1007/s40484-014-0025-7 pmid:25632368 pmcid:PMC4306437 fatcat:s2rsekzqszagxes4toozklrumu

Exact time-dependent solutions for a self-regulating gene

A. F. Ramos, G. C. P. Innocentini, J. E. M. Hornos
2011 Physical Review E  
The exact time-dependent solution for the stochastic equations governing the behavior of a binary selfregulating gene is presented.  ...  Using the generating function technique to rephrase the master equations in terms of partial differential equations, we show that the model is totally integrable and the analytical solutions are the celebrated  ...  The master equations for the spin-boson model are equivalent to a linear partial differential equation for the time-dependent generating function and consequently present a single equilibrium regime reached  ... 
doi:10.1103/physreve.83.062902 pmid:21797428 fatcat:zn7jh4zqorbclakor5x36glayi

Gene regulation and noise reduction by coupling of stochastic processes

Alexandre F. Ramos, José Eduardo M. Hornos, John Reinitz
2015 Physical Review E  
Here we characterize the low noise regime of a stochastic model for a negative self-regulating binary gene. The model has two stochastic variables, the protein number and the state of the gene.  ...  We consider the chemical reaction rates necessary for Poisson and sub-Poisson processes in prokaryotes and eucaryotes.  ...  JR thanks the hospitality of the Graduate School on Modeling of Complex Systems, EACH, USP.  ... 
doi:10.1103/physreve.91.020701 pmid:25768447 pmcid:PMC4476401 fatcat:uhf66h24jrcetgi74ijxdn5fgi

Chemical Master Equation Closure for Computer-Aided Synthetic Biology [chapter]

Patrick Smadbeck, Yiannis N. Kaznessis
2014 Msphere  
A closure scheme solution is also presented for the master equation of chemical reaction networks. The master equation is a complete model of randomly evolving molecular populations.  ...  With the first complete solution of chemical master equations, a wide range of experimental observations of biomolecular interactions may be mathematically conceptualized.  ...  ACKNOWLDGEMENTS This work was supported by a grant from the National Institutes of Health (American Recovery and Reinvestment Act grant GM086865) and a grant from the National Science Foundation (CBET-  ... 
doi:10.1007/978-1-4939-1878-2_9 pmid:25487098 pmcid:PMC4820257 fatcat:gnvveo3onzhtfefhgb5icihb3e

A spectral methods-based solution of the Chemical Master Equation for gene regulatory networks

Michael Nip, Joao P. Hespanha, Mustafa Khammash
2012 2012 IEEE 51st IEEE Conference on Decision and Control (CDC)  
We present a new method to approximate the time evolution of the probability density function (PDF) for molecule counts in gene regulatory networks, modeled by the Chemical Master Equation (CME).  ...  We observe that the CME is amenable to the use of spectral methods adapted from partial differential equations and our method results from expanding the solution using carefully selected basis functions  ...  In this framework, the evolution of the probability density of the system's chemical populations is governed by the Chemical Master Equation (CME).  ... 
doi:10.1109/cdc.2012.6425804 dblp:conf/cdc/NipHK12 fatcat:trlzx4lhfbbmxcfy6hwiighp6i

Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems

Stefanie Winkelmann, Christof Schütte
2017 Journal of Chemical Physics  
Well-mixed stochastic chemical kinetics are properly modelled by the chemical master equation (CME) and associated Markov jump processes in molecule number space.  ...  We derive a novel general description of such hybrid models that allows to express various forms by one type of equation.  ...  Also the spatiotemporal chemical Master equation (ST-CME) is depicted as a model extension for non well-mixed reaction systems.  ... 
doi:10.1063/1.4986560 pmid:28938803 fatcat:4ai7gclcsbbsjdrhix4pj3aoee

Noise in Gene Regulatory Networks

Ioannis Lestas, Johan Paulsson, Nicholas E. Ross, Glenn Vinnicombe
2008 IEEE Transactions on Automatic Control  
It is shown in this paper how a noise-filtering setup with an operator theoretic interpretation can be relevant for analyzing the intrinsic stochasticity in jump processes described by master equations  ...  It is discussed that bistability in the deterministic mass action kinetics and bimodality in the steady-state solution of the master equation neither always imply one another nor do they necessarily lead  ...  The "stochastic chemical kinetics" that arise due to random births and deaths of individual molecules give rise to jump Markov processes, which can be analyzed by means of master equations and simulated  ... 
doi:10.1109/tac.2007.911347 fatcat:ofqwnvivlvc5bd72i2qepvbhqe

Noise in Gene Regulatory Networks

I. Lestas, J. Paulsson, N. E. Ross, G. Vinnicombe
2009 IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications  
It is shown in this paper how a noise-filtering setup with an operator theoretic interpretation can be relevant for analyzing the intrinsic stochasticity in jump processes described by master equations  ...  It is discussed that bistability in the deterministic mass action kinetics and bimodality in the steady-state solution of the master equation neither always imply one another nor do they necessarily lead  ...  The "stochastic chemical kinetics" that arise due to random births and deaths of individual molecules give rise to jump Markov processes, which can be analyzed by means of master equations and simulated  ... 
doi:10.1109/tcsi.2007.911347 fatcat:z6nw2zxxwnb4vdrl5uxv5quruu

Hybrid simulation of cellular behavior

T. R. Kiehl, R. M. Mattheyses, M. K. Simmons
2004 Bioinformatics  
Thus in modeling such systems, we are working within the regime of statistical 316 Bioinformatics 20 (3)  ...  Hybrid simulation provides a new method for exploring the sources and nature of stochastic behavior in cells.  ...  ACKNOWLEDGEMENTS We thank Michael Peters for his contributions to the development of the simulation engine.  ... 
doi:10.1093/bioinformatics/btg409 pmid:14960457 fatcat:5qwxgaahqfgfvnjlgyww6lbaem

Biochemical Network Stochastic Simulator (BioNetS): software for stochastic modeling of biochemical networks

David Adalsteinsson, David McMillen, Timothy C Elston
2004 BMC Bioinformatics  
For the continuous random variables, BioNetS constructs and numerically solves the appropriate chemical Langevin equations.  ...  BioNetS has a graphical user interface that allows models to be entered in a straightforward manner, and allows the user to specify the type of random variable (discrete or continuous) for each chemical  ...  In the Implementation section, the mathematical background for the Gillespie method, chemical Langevin equations and hybrid models is presented, along with a discussion of the numerical algorithms used  ... 
doi:10.1186/1471-2105-5-24 pmid:15113411 pmcid:PMC408466 fatcat:5ogemi2zgjab5cacrchlkqyyby

Intrinsic fluctuations in stochastic delay systems: Theoretical description and application to a simple model of gene regulation

Tobias Galla
2009 Physical Review E  
Gene regulatory systems in particular often contain only a small number of molecules, leading to significant fluctuations in mRNA and protein concentrations.  ...  As an application we therefore study a minimalistic model of the expression levels of hes1 mRNA and Hes1 protein, representing the simple motif of an auto-inhibitory feedback loop and motivated by its  ...  Acknowledgments TG is an RCUK Fellow (RCUK reference EP/E500048/1), and would like to thank R. Schlicht for useful discussions on stochastic delay systems, and A. J. McKane and R. P.  ... 
doi:10.1103/physreve.80.021909 pmid:19792153 fatcat:f5q57gxfhnhlnffbjcpc7pdesm
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