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COMPARING HITTING TIME BEHAVIOR OF MARKOV JUMP PROCESSES AND THEIR DIFFUSION APPROXIMATIONS

MULTISCALE MODELING & SIMULATION 2010年第2期8卷 605-621页

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

Markov jump processes can provide accurate models in many applications, notably chemical and biochemical kinetics, and population dynamics. Stochastic differential equations offer a computationally efficient way to approximate these processes. It is therefore of interest to establish results that shed light on the extent to which the jump and diffusion models agree. In this work we focus on mean hitting time behavior in a thermodynamic limit. We study three simple types of reactions where analytical results can be derived, and we find that the match between mean hitting time behavior of the two models is vastly different in each case. In particular, for a degradation reaction we find that the relative discrepancy decays extremely slowly, namely, as the inverse of the logarithm of the system size. After giving some further computational results, we conclude by pointing out that studying hitting times allows the Markov jump and stochastic differential equation regimes to be compared in a manner that avoids pitfalls that may invalidate other approaches.

关键词： birth and death process chemical Langevin equation finite difference method gillespie algorithm mean exit time square root process stochastic differential equation thermodynamic limit

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Three-dimensional cytoplasmic calcium propagation with boundaries

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Communications in Theoretical Physics 2021年第1期73卷 87-98页

作者： Han-Yu Jiang Jun He Department of Physics and Institute of Theoretical Physics Nanjing Normal UniversityNanjing210097China Sino-U.S.Center for Grazingland Ecosystem Sustainability/Pratacultural Engineering Laboratory of Gansu Province/Key Laboratory of Grassland Ecosystem Ministry of Education/College of Pratacultural ScienceGansu Agricultural UniversityLanzhou730070China

Ca^(2+)plays an important role in cell signal *** intracellular propagation is the most basic process of Ca^(2+)signaling,such as calcium wave and double messenger *** this work,with both numerical simulation and mean field ansatz,the three-dimensional probability distribution of Ca^(2+),which is read out by phosphorylation,is studied in two scenarios with *** coverage of distribution of Ca^(2+)is found at an order of magnitude ofμm,which is consistent with experimental observed calcium spike and *** results suggest that the double messenger system may occur in the ER-PM junction to acquire great *** buffer effect of kinase is also discussed by calculating the average position of phosphorylations and free Ca^(2+).The results are helpful to understand the mechanism of Ca^(2+)signaling.

关键词： Ca^(2+)signaling diffusion gillespie algorithm mean field ansatz

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Modeling and simulation of intracellular dynamics: Choosing an appropriate framework

IEEE TRANSACTIONS ON NANOBIOSCIENCE

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IEEE TRANSACTIONS ON NANOBIOSCIENCE 2004年第3期3卷 200-207页

作者： Wolkenhauer, O Ullah, M Kolch, W Cho, KH Univ Rostock Syst Biol & Bioinformat Grp D-18051 Rostock Germany Seoul Natl Univ Coll Med Seoul 110799 South Korea Seoul Natl Univ Korea Bio MAX Ctr Seoul 110799 South Korea

Systems biology is a reemerging paradigm which, among other things, focuses on mathematical modeling and simulation of biochemical reaction networks in intracellular processes. For most simulation tools and publications, they are usually characterized by either preferring stochastic simulation or rate equation models. The use of stochastic simulation is occasionally accompanied with arguments against rate equations. Motivated by these arguments, we discuss in this paper the relationship between these two forms of representation. Toward this end, we provide a novel compact derivation for the stochastic rate constant that forms the basis of the popular gillespie algorithm. Comparing the mathematical basis of the two popular conceptual frameworks of generalized mass action models and the chemical master equation, we argue that some of the arguments that have been put forward are ignoring subtle differences and similarities that are important for answering the question in which conceptual framework one should investigate intracellular dynamics.

关键词： chemical master equation (CME) generalized mass action models gillespie algorithm intracellular dynamics systems biology

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ERROR BOUND FOR PIECEWISE DETERMINISTIC PROCESSES MODELING STOCHASTIC REACTION SYSTEMS

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MULTISCALE MODELING & SIMULATION 2012年第4期10卷 1119-1147页

作者： Jahnke, Tobias Kreim, Michael Karlsruhe Inst Technol Inst Appl & Numer Math Dept Math D-76133 Karlsruhe Germany

Biological processes involving the random interaction of d species with integer particle numbers are often modeled by a Markov jump process on N-0(d). Realizations of this process can, in principle, be generated with gillespie's classical stochastic simulation algorithm, but for very reactive systems this method is usually inefficient. Hybrid models based on piecewise deterministic processes offer an attractive alternative which can decrease the simulation time considerably in applications where species with rather low particle numbers interact with very abundant species. We investigate the convergence of the hybrid model to the original one for a class of reaction systems with two distinct scales. Our main result is an error bound which states that, under suitable assumptions, the hybrid model approximates the marginal distribution of the discrete species and the conditional moments of the continuous species up to an error of O(M-1), where M is the scaling parameter of the partial thermodynamic limit.

关键词： stochastic reaction systems gillespie algorithm hybrid models piecewise deterministic processes error bounds chemical master equation thermodynamic limit

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Binomial leap methods for simulating stochastic chemical kinetics

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JOURNAL OF CHEMICAL PHYSICS 2004年第21期121卷 10356-10364页

作者： Tian, TH Burrage, K Univ Queensland Adv Computat Modelling Ctr Brisbane Qld 4072 Australia

This paper discusses efficient simulation methods for stochastic chemical kinetics. Based on the tau-leap and midpoint tau-leap methods of gillespie [D. T. gillespie, J. Chem. Phys. 115, 1716 (2001)], binomial random variables are used in these leap methods rather than Poisson random variables. The motivation for this approach is to improve the efficiency of the Poisson leap methods by using larger stepsizes. Unlike Poisson random variables whose range of sample values is from zero to infinity, binomial random variables have a finite range of sample values. This probabilistic property has been used to restrict possible reaction numbers and to avoid negative molecular numbers in stochastic simulations when larger stepsize is used. In this approach a binomial random variable is defined for a single reaction channel in order to keep the reaction number of this channel below the numbers of molecules that undergo this reaction channel. A sampling technique is also designed for the total reaction number of a reactant species that undergoes two or more reaction channels. Samples for the total reaction number are not greater than the molecular number of this species. In addition, probability properties of the binomial random variables provide stepsize conditions for restricting reaction numbers in a chosen time interval. These stepsize conditions are important properties of robust leap control strategies. Numerical results indicate that the proposed binomial leap methods can be applied to a wide range of chemical reaction systems with very good accuracy and significant improvement on efficiency over existing approaches. (C) 2004 American Institute of Physics.

关键词： gillespie algorithm REACTING SYSTEMS GENERATION

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Multiple epidemic waves as the outcome of stochastic SIR epidemics with behavioral responses: a hybrid modeling approach

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NONLINEAR DYNAMICS 2023年第1期111卷 887-926页

作者： Ochab, Magdalena Manfredi, Piero Puszynski, Krzysztof d'Onofrio, Alberto Silesian Tech Univ Dept Syst Biol & Engn 16 Akad St PL-44100 Gliwice Poland Univ Pisa Dept Econ & Management Via Ridolfi 10 I-5612 Pisa Italy Strathclyde Univ Dept Math & Stat Glasgow Lanark Scotland Int Prevent Res Inst 95 Cours Lafayette F-69006 Lyon France Univ Claude Bernard Lyon 1 Inst Camille Jordan 21 Ave Claude Bernard F-69100 Villeurbanne France

In the behavioral epidemiology (BE) of infectious diseases, little theoretical effort seems to have been devoted to understand the possible effects of individuals' behavioral responses during an epidemic outbreak in small populations. To fill this gap, here we first build general, behavior implicit, SIR epidemic models including behavioral responses and set them within the framework of nonlinear feedback control theory. Second, we provide a thorough investigation of the effects of different types of agents' behavioral responses for the dynamics of hybrid stochastic SIR outbreak models. In the proposed model, the stochastic discrete dynamics of infection spread is combined with a continuous model describing the agents' delayed behavioral response. The delay reflects the memory mechanisms with which individuals enact protective behavior based on past data on the epidemic course. This results in a stochastic hybrid system with time-varying transition probabilities. To simulate such system, we extend gillespie's classic stochastic simulation algorithm by developing analytical formulas valid for our classes of models. The algorithm is used to simulate a number of stochastic behavioral models and to classify the effects of different types of agents' behavioral responses. In particular this work focuses on the effects of the structure of the response function and of the form of the temporal distribution of such response. Among the various results, we stress the appearance of multiple, stochastic epidemic waves triggered by the delayed behavioral response of individuals.

关键词： Multiple epidemic waves Social distancing Hybrid systems Stochastic epidemic models Human behavior Memory effects gillespie algorithm

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Intracellular viral infection kinetics using a stochastic approach

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PROGRESS IN REACTION KINETICS AND MECHANISM 2013年第4期38卷 359-376页

作者： Taherkhani, Farid Taherkhani, Fariborz Rezania, Hamed Akbarzadeh, Hamed Razi Univ Dept Phys Chem Kermanshah Iran Sharif Univ Technol Dept Comp Engn Tehran Iran Razi Univ Dept Phys Kermanshah Iran Hakim Sabzevari Univ Dept Chem Sabzevar Iran

Stochastic simulation is carried out to investigate intracellular viral reaction kinetics and the time evolution of the average particle number ((N) over bar) and coefficient variation (CV) for genome, template, and structural protein. The coefficient variation of these components is found to be ordered as: CVtemplate > CVstructural protein > CVgenome. The average particle number is also calculated via a deterministic approach. The magnitude value of the difference between the stochastic and deterministic approaches is found to be (N) over bar (template) approximate to (N) over bar (structural protein) > (N) over bar (genome). The Poisson algorithm has been used to investigate the number of particles in the dynamics of intracellular viral reaction kinetics. Our results show that the average particle number ((N) over bar) obtained using the Poisson algorithm for both dynamics and stationary states is less than that obtained using the mean field approach relative to the quantity ((N) over bar) found via the gillespie algorithm. Also, the standard deviation of (N) over bar calculated via the Poisson algorithm is more than that obtained via the gillespie algorithm. The equilibrium time for the population of particles via the Poisson algorithm is less than that via the gillespie algorithm and the mean field approach. In the present study, the gillespie algorithm is more reliable than the Poisson algorithm.

关键词： stochastic approach Poisson algorithm gillespie algorithm intracellular viral reaction kinetics

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Molecular computing for Markov chains

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NATURAL COMPUTING 2020年第3期19卷 593-608页

作者： Zhang, Chuan Shen, Ziyuan Wei, Wei Zhao, Jing Zhang, Zaichen You, Xiaohu Southeast Univ Lab Efficient Architectures Digital Commun & Sign Quantum Informat Ctr Southeast Univ Natl Mobile Commun Res Lab Nanjing Peoples R China Nanjing Univ State Key Lab Coordinat Chem Sch Chem & Chem Engn State Key Lab Pharmaceut BiotechnolSch Life Sci Nanjing Peoples R China

In this paper, it is presented a methodology for implementing arbitrarily constructed time-homogenous Markov chains with biochemical systems. Not only discrete but also continuous-time Markov chains are allowed to be computed. By employing chemical reaction networks as a programmable language, molecular concentrations serve to denote both input and output values. One reaction network is elaborately designed for each chain. The evolution of species' concentrations over time well matches the transient solutions of the target continuous-time Markov chain, while equilibrium concentrations can indicate the steady state probabilities. Additionally, second-order Markov chains are considered for implementation, with bimolecular reactions rather than unary ones. An original scheme is put forward to compile unimolecular systems to DNA strand displacement reactions for the sake of future physical implementations. Deterministic, stochastic and DNA simulations are provided to enhance correctness, validity and feasibility.

关键词： Molecular computing DNA strand displacement Markov chain Mass action kinetics gillespie algorithm

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Identifiability of stochastically modelled reaction networks

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EUROPEAN JOURNAL OF APPLIED MATHEMATICS 2021年第5期32卷 865-887页

作者： Enciso, German Erban, Radek Kim, Jinsu Univ Calif Irvine Dept Math Irvine CA 92697 USA Univ Oxford Math Inst Radcliffe Observ Quarter Woodstock Rd Oxford OX2 6GG England Univ Calif Irvine NSF Simons Ctr Multiscale Cell Fate Res Irvine CA 92697 USA

Chemical reaction networks describe interactions between biochemical species. Once an underlying reaction network is given for a biochemical system, the system dynamics can be modelled with various mathematical frameworks such as continuous-time Markov processes. In this manuscript, the identifiability of the underlying network structure with a given stochastic system dynamics is studied. It is shown that some data types related to the associated stochastic dynamics can uniquely identify the underlying network structure as well as the system parameters. The accuracy of the presented network inference is investigated when given dynamical data are obtained via stochastic simulations.

关键词： Chemical reaction networks identifiability stochastic simulation network inference gillespie algorithm

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On the stochastic modelling of surface reactions through reflected chemical Langevin equations

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COMPUTERS & CHEMICAL ENGINEERING 2018年 117卷 145-158页

作者： Pineda, M. Stamatakis, M. UCL Dept Chem Engn Roberts BldgTorrington Pl London WC1E 7JE England

Modelling of small-scale heterogeneous catalytic systems with master equations captures the impact of molecular noise, but can be computationally expensive. On the other hand, the chemical Fokker-Planck approximation offers an excellent alternative from an efficiency perspective. The Langevin equation can generate stochastic realisations of the Fokker-Planck equation;yet, these realisations may violate the conditions 0 <= theta <= 1 (where theta is surface coverage). In this work, we adopt Skorokhod's formulations to impose reflective boundaries that remedy this issue. We demonstrate the approach on a simple system involving a single species and describing adsorption, desorption, reaction and diffusion processes on a lattice. We compare different numerical schemes for the solution of the resulting reflected Langevin equation and calculate rates of convergence. Our benchmarks should guide the choice of appropriate numerical methods for the accurate and efficient simulation of chemical systems in the catalysis field. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Molecular noise Chemical master equation Fokker-Planck equation Langevin equation Reflective boundary gillespie algorithm

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