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Hybrid method for the chemical master equation

JOURNAL OF COMPUTATIONAL PHYSICS 2007年第1期227卷 100-122页

作者： Hellander, Andreas Lotstedt, Per Uppsala Univ Dept Informat Technol Div Comp Sci S-75105 Uppsala Sweden

The chemical master equation is solved by a hybrid method coupling a macroscopic, deterministic description with mesoscopic, stochastic model. The molecular species are divided into one subset where the expected values of the number of molecules are computed and one subset with species with a stochastic variation in the number of molecules. The macroscopic equations resemble the reaction rate equations and the probability distribution for the stochastic variables satisfy a master equation. The probability distribution is obtained by the stochastic simulation algorithm due to Gillespie. The equations are coupled via a summation over the mesoscale variables. This summation is approximated by Quasi-Monte Carlo methods. The error in the approximations is analyzed. The hybrid method is applied to three chemical systems from molecular cell biology. (C) 2007 Elsevier Inc. All rights reserved.

关键词： master equation reaction rate equations- stochastic simulation algorithm Monte Carlo method Quasi-Monte Carlo me hod stochastic chemical kinetics

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Preconditioned Bayesian Regression for stochastic Chemical Kinetics

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JOURNAL OF SCIENTIFIC COMPUTING 2014年第3期58卷 592-626页

作者： Alexanderian, Alen Rizzi, Francesco Rathinam, Muruhan Le Maitre, Olivier P. Knio, Omar M. Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA Duke Univ Dept Mech Engn & Mat Sci Durham NC 27708 USA Univ Maryland Baltimore Cty Dept Math & Stat Baltimore MD 21250 USA LIMSI CNRS F-91403 Orsay France

We develop a preconditioned Bayesian regression method that enables sparse polynomial chaos representations of noisy outputs for stochastic chemical systems with uncertain reaction rates. The approach is based on the definition of an appropriate multiscale transformation of the state variables coupled with a Bayesian regression formalism. This enables efficient and robust recovery of both the transient dynamics and the corresponding noise levels. Implementation of the present approach is illustrated through applications to a stochastic Michaelis-Menten dynamics and a higher dimensional example involving a genetic positive feedback loop. In all cases, a stochastic simulation algorithm (SSA) is used to compute the system dynamics. Numerical experiments show that Bayesian preconditioning algorithms can simultaneously accommodate large noise levels and large variability with uncertain parameters, and that robust estimates can be obtained with a small number of SSA realizations.

关键词： Polynomial chaos Bayesian regression Preconditioner stochastic simulation algorithm Chemical kinetics

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SWITCHING AND DIFFUSION MODELS FOR GENE REGULATION NETWORKS

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MULTISCALE MODELING & simulation 2009年第1期8卷 30-45页

作者： Intep, Somkid Higham, Desmond J. Mao, Xuerong Univ Strathclyde Dept Math Glasgow G1 1XH Lanark Scotland Univ Strathclyde Dept Stat & Modelling Sci Glasgow G1 1XH Lanark Scotland

We analyze a hierarchy of three regimes for modeling gene regulation. The most complete model is a continuous time, discrete state space, Markov jump process. An intermediate "switch plus diffusion" model takes the form of a stochastic differential equation driven by an independent continuous time Markov switch. In the third "switch plus ODE" model the switch remains, but the diffusion is removed. The latter two models allow for multiscale simulation where, for the sake of computational efficiency, system components are treated differently according to their abundance. The switch plus ODE regime was proposed by Paszek [Bull. Math. Biol., 69 (2007), pp. 1567-1601], who analyzed the steady state behavior, showing that the mean was preserved, but the variance only approximated that of the full model. Here, we show that the tools of stochastic calculus can be used to analyze first and second moments for all time. A technical issue to be addressed is that the state space for the discrete-valued switch is infinite. We show that the new switch plus diffusion regime preserves the biologically relevant measures of mean and variance, whereas the switch plus ODE model uniformly underestimates the variance in the protein level. We also show that, for biologically relevant parameters, the transient behavior can differ significantly from the steady state, justifying our time-dependent analysis. Extra computational results are also given for a protein dimerization model that is beyond the scope of the current analysis.

关键词： diffusion hybrid model Gillespie's algorithm Ito lemma Markov chain slow scale simulation stochastic simulation algorithm transcription transition rate translation

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A hybrid multiscale Monte Carlo algorithm (HyMSMC) to cope with disparity in time scales and species populations in intracellular networks

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BMC BIOINFORMATICS 2007年第1期8卷 1-23页

作者： Samant, Asawari Ogunnaike, Babatunde A. Vlachos, Dionisios G. Univ Delaware Dept Chem Engn Newark DE 19716 USA

Background: The fundamental role that intrinsic stochasticity plays in cellular functions has been shown via numerous computational and experimental studies. In the face of such evidence, it is important that intracellular networks are simulated with stochastic algorithms that can capture molecular fluctuations. However, separation of time scales and disparity in species population, two common features of intracellular networks, make stochastic simulation of such networks computationally prohibitive. While recent work has addressed each of these challenges separately, a generic algorithm that can simultaneously tackle disparity in time scales and population scales in stochastic systems is currently lacking. In this paper, we propose the hybrid, multiscale Monte Carlo ( HyMSMC) method that fills in this void. Results: The proposed HyMSMC method blends stochastic singular perturbation concepts, to deal with potential stiffness, with a hybrid of exact and coarse- grained stochastic algorithms, to cope with separation in population sizes. In addition, we introduce the computational singular perturbation ( CSP) method as a means of systematically partitioning fast and slow networks and computing relaxation times for convergence. We also propose a new criteria of convergence of fast networks to stochastic low- dimensional manifolds, which further accelerates the algorithm. Conclusion: We use several prototype and biological examples, including a gene expression model displaying bistability, to demonstrate the efficiency, accuracy and applicability of the HyMSMC method. Bistable models serve as stringent tests for the success of multiscale MC methods and illustrate limitations of some literature methods.

关键词： Probability Distribution Function Fast Mode stochastic simulation algorithm Participation Index Fast Network

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Bacteriophage Infection Dynamics: Multiple Host Binding Sites

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MATHEMATICAL MODELLING OF NATURAL PHENOMENA 2009年第6期4卷 109-134页

作者： Smith, H. L. Trevino, R. T. Arizona State Univ Sch Math & Stat Sci Tempe AZ 85287 USA

We construct a stochastic model of bacteriophage parasitism of a host bacteria that accounts for demographic stochasticity of host and parasite and allows for multiple bacteriophage adsorption to host. We analyze the associated deterministic model, identifying the basic reproductive number for phage proliferation, showing that host and phage persist when it exceeds unity, and establishing that the distribution of adsorbed phage on a host is binomial with slowly evolving mean. Not surprisingly, extinction of the parasite or both host and parasite can occur for the stochastic model.

关键词： bacteriophage stochastic simulation algorithm phage adsorption persistence basic reproductive number

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BINNING FOR EFFICIENT stochastic MULTISCALE PARTICLE simulationS

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MULTISCALE MODELING & simulation 2013年第4期11卷 1071-1096页

作者： Michelotti, M. D. Heath, M. T. West, M. Univ Illinois Dept Comp Sci Urbana IL 61801 USA Univ Illinois Dept Mech Sci & Engn Urbana IL 61801 USA

Gillespie's stochastic simulation algorithm (SSA) is an exact procedure for simulating the evolution of a collection of discrete, interacting entities, such as coalescing aerosol particles or reacting chemical species. The high computational cost of SSA has motivated the development of more efficient variants, such as Tau-Leaping, which sacrifices the exactness of SSA. For models whose interacting entities can be characterized by a continuous parameter, such as a measure of size for aerosol particles, we analyze strategies for accelerating these algorithms by aggregating particles of similar size into bins. We show that for such models an appropriate binning strategy can dramatically enhance efficiency, and in particular can make SSA computationally competitive without sacrificing exactness. These strategies are especially effective for highly multiscale problems. We formulate binned versions of both the SSA and Tau-Leaping algorithms and analyze and demonstrate their performance.

关键词： stochastic simulation algorithm continuous-time Markov process Monte Carlo atmospheric aerosol coagulation

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Modeling and simulating chemical reactions

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SIAM REVIEW 2008年第2期50卷 347-368页

作者： Higham, Desmond J. Univ Strathclyde Dept Math Glasgow G1 1XH Lanark Scotland

Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain large-scale limit of a sequence of finer-scale probabilistic models. In studying this hierarchy of models, students can be exposed to a range of modern ideas in applied and computational mathematics. This article introduces some of the basic concepts in an accessible manner and points to some challenges that currently occupy researchers in this area. Short, downloadable MATLAB codes are listed and described.

关键词： birth-and-death process chemical master equation chemical Langevin Euler-Maruyamal Gillespie kinetic Monte Carlo law of mass action Michaelis-Menten reaction rate equation stochastic simulation algorithm stoichiometric vector tau-leaping

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What is special about autocatalysis?

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MONATSHEFTE FUR CHEMIE 2019年第5期150卷 763-775页

作者： Schuster, Peter Univ Vienna Inst Theoret Chem Vienna Austria Santa Fe Inst Santa Fe NM 87501 USA

The single-step autocatalytic reaction A+X2X in different environmentsbatch reactor, flow reactor, logistic equationis studied by means of conventional deterministic kinetics and as a stochastic process. Self-enhancement requires an initial concentration of the autocatalyst Xat least in seeding amountsfor starting the reaction. Deterministic solution curves have sigmoid shapes. At small concentrations, three stochastic phenomena are observed: (1) thermal fluctuations, (2) stochastic delay, and (3) stochastic bifurcations and anomalous fluctuations in case of multiple final states. The introduction of heterogeneous populations containing subspecies with different fitness values gives rise to natural selection in all three environments investigated here. In large populations, survival of the fittest is observed, whereas random fluctuations may result in selection of each of the subspecies. Then, the fitness values determine only probabilities of selection. The fittest subspecies, of course, has the largest probability of selection. There is a smooth transition to neutral evolution where the probabilities of selection are the same for all subspecies. [GRAPHICS] .

关键词： Autocatalysis Evolution Master equation Optimization Reaction kinetics Replication stochastic simulation algorithm

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Approximate Exponential algorithms to Solve the Chemical Master Equation

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MATHEMATICAL MODELLING AND ANALYSIS 2015年第3期20卷 382-395页

作者： Mooasvi, Azam Sandu, Adrian Virginia Polytech Inst & State Univ Dept Comp Sci Sci Computat Lab Blacksburg VA 24060 USA

This paper discusses new simulation algorithms for stochastic chemical kinetics that exploit the linearity of the chemical master equation and its matrix exponential exact solution. These algorithms make use of various approximations of the matrix exponential to evolve probability densities in time. A sampling of the approximate solutions of the chemical master equation is used to derive accelerated stochastic simulation algorithms. Numerical experiments compare the new methods with the established stochastic simulation algorithm and the tau-leaping method.

关键词： stochastic simulation algorithm exact solution tau-leap stochastic chemical kinetics chemical master equation

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MULTISCALE MODELING OF CHEMICAL KINETICS VIA THE MASTER EQUATION

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MULTISCALE MODELING & simulation 2008年第4期6卷 1146-1168页

作者： Macnamara, Shev Burrage, Kevin Sidje, Roger B. Univ Queensland Adv Computat Modelling Ctr Brisbane Qld 4072 Australia Univ Queensland Dept Math Brisbane Qld 4072 Australia

We present numerical methods for both the direct solution and simulation of the chemical master equation (CME), and, compared to popular methods in current use, such as the Gillespie stochastic simulation algorithm (SSA) and tau-Leap approximations, this new approach has the advantage of being able to detect when the system has settled down to equilibrium. This improved performance is due to the incorporation of information from the associated CME, a valuable complementary approach to the SSA that has often been felt to be too computationally inefficient. Hybrid methods, that combine these complementary approaches and so are able to detect equilibrium while maintaining the efficiency of the leap methods, are also presented. Amongst CME-solvers the recently suggested finite state projection algorithm is especially well suited to this purpose and has been adapted here for the task, leading to a type of "exact tau-Leap." It is also observed that a CME-solver is often more efficient than an SSA or even a tau-Leap approach for computing moments of the solution such as the mean and variance. These techniques are demonstrated on a test suite of five biologically inspired models, namely, stochastic models of the genetic toggle, receptor oligomerization, the Schlogl reactions, Goutsias' model of regulated gene transcription, and a decaying-dimerizing reaction set. For the gene toggle it is observed that important experimentally measurable traits such as the percentage of cells that undergo so-called switching may also be more efficiently approximated via a CME-based approach.

关键词： chemical master equation stochastic simulation algorithm systems biology

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