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153
Efficient, correct simulation of biological processes in the stochastic picalculus
 Gilmore (Eds.), Proc. Int. Conf. Computational Methods in Systems Biology (CMSB’07
, 2007
"... Abstract. This paper presents a simulation algorithm for the stochastic πcalculus, designed for the efficient simulation of biological systems with large numbers of molecules. The cost of a simulation depends on the number of species, rather than the number of molecules, resulting in a significant ..."
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Cited by 42 (13 self)
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Abstract. This paper presents a simulation algorithm for the stochastic πcalculus, designed for the efficient simulation of biological systems with large numbers of molecules. The cost of a simulation depends on the number of species, rather than the number of molecules, resulting in a significant gain in efficiency. The algorithm is proved correct with respect to the calculus, and then used as a basis for implementing the latest version of the SPiM stochastic simulator. The algorithm is also suitable for generating graphical animations of simulations, in order to visualise system dynamics. 1
Rules for Modeling SignalTransduction Systems
 Science’s STKE
, 2006
"... Formalized rules for proteinprotein interactions have recently been introduced to represent the binding and enzymatic activities of proteins in cellular signaling. Rules encode an understanding of how a system works in terms of the biomolecules in the system and their possible states and interactio ..."
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Cited by 38 (11 self)
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Formalized rules for proteinprotein interactions have recently been introduced to represent the binding and enzymatic activities of proteins in cellular signaling. Rules encode an understanding of how a system works in terms of the biomolecules in the system and their possible states and interactions. A set of rules can be as easy to read as a diagrammatic interaction map, but unlike most such maps, rules have precise interpretations. Rules can be processed to automatically generate a mathematical or computational model for a system, which enables explanatory and predictive insights into the system’s behavior. Rules are independent units of a model specification that facilitate model revision. Instead of changing a large number of equations or lines of code, as may be required in the case of a conventional mathematical model, a protein interaction can be introduced or modified simply by adding or changing a single rule that represents the interaction of interest. Rules can be defined and visualized by using graphs, so no specialized training in mathematics or computer science is necessary to create models or to take advantage of the representational precision of rules. Rules can be encoded in a machinereadable format to enable electronic storage and exchange of models, as well as basic knowledge about proteinprotein interactions. Here, we review the motivation for rulebased modeling; applications of the approach; and issues that arise in model specification, simulation, and testing. We also discuss rule visualization and exchange and the software available for rulebased modeling.
Stochastic models for chemically reacting systems using polynomial stochastic hybrid systems
 Int. J. Robust Nonlinear Control
, 2005
"... Abstract. A stochastic model for chemical reactions is presented, which represents the population of various species involved in a chemical reaction as the continuous state of a polynomial Stochastic Hybrid System (pSHS). pSHSs correspond to stochastic hybrid systems with polynomial continuous vecto ..."
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Cited by 31 (16 self)
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Abstract. A stochastic model for chemical reactions is presented, which represents the population of various species involved in a chemical reaction as the continuous state of a polynomial Stochastic Hybrid System (pSHS). pSHSs correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. We show that for pSHSs, the dynamics of the statistical moments of its continuous states, evolves according to infinitedimensional linear ordinary differential equations (ODEs), which can be approximated by finitedimensional nonlinear ODEs with arbitrary precision. Based on this result, a procedure to build this types of approximation is provided. This procedure is used to construct approximate stochastic models for a variety of chemical reactions that have appeared in literature. These reactions include a simple bimolecular reaction, for which one can solve the master equation; a decayingdimerizing reaction set which exhibits two distinct time scales; a reaction for which the chemical rate equations have a continuum of equilibrium points; and the bistable Schögl reaction. The accuracy of the approximate models is investigated by comparing with Monte Carlo simulations or the solution to the Master equation, when available. 1
Consistency and stability of tau leaping schemes for chemical reaction systems
 SIAM Multiscale Modeling
, 2005
"... Abstract. We develop a theory of local errors for the explicit and implicit tauleaping methods for simulating stochastic chemical systems, and we prove that these methods are firstorder consistent. Our theory provides local error formulae that could serve as the basis for future stepsize control t ..."
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Cited by 22 (6 self)
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Abstract. We develop a theory of local errors for the explicit and implicit tauleaping methods for simulating stochastic chemical systems, and we prove that these methods are firstorder consistent. Our theory provides local error formulae that could serve as the basis for future stepsize control techniques. We prove that, for the special case of systems with linear propensity functions, both tauleaping methods are firstorder convergent in all moments. We provide a stiff stability analysis of the mean of both leaping methods, and we confirm that the implicit method is unconditionally stable in the mean for stable systems. Finally, we give some theoretical and numerical examples to illustrate these results.
Analysis of explicit tauleaping schemes for simulating chemically reacting systems
 Multiscale Model. Simul
"... Abstract. This paper builds a convergence analysis of explicit tauleaping schemes for simulating chemical reactions from the viewpoint of stochastic differential equations. Mathematically, the chemical reaction process is a pure jump process on a lattice with statedependent intensity. The stochast ..."
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Cited by 18 (4 self)
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Abstract. This paper builds a convergence analysis of explicit tauleaping schemes for simulating chemical reactions from the viewpoint of stochastic differential equations. Mathematically, the chemical reaction process is a pure jump process on a lattice with statedependent intensity. The stochastic differential equation form of the chemical master equation can be given via Poisson random measures. Based on this form, different types of tauleaping schemes can be proposed. In order to make the problem wellposed, a modified explicit tauleaping scheme is considered. It is shown that the mean square strong convergence is of order 1/2 and the weak convergence is of order 1 for this modified scheme. The novelty of the analysis is to handle the nonLipschitz property of the coefficients and jumps on the integer lattice.
Algorithms and software for stochastic simulation of biochemical reacting systems
, 2007
"... Traditional deterministic approaches for simulation of chemically reacting systems fail to capture the randomness inherent in such systems at scales common in intracellular biochemical processes. In this article we briefly review the state of the art in discrete stochastic and multiscale algorithms ..."
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Cited by 15 (5 self)
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Traditional deterministic approaches for simulation of chemically reacting systems fail to capture the randomness inherent in such systems at scales common in intracellular biochemical processes. In this article we briefly review the state of the art in discrete stochastic and multiscale algorithms for simulation of biochemical systems and we present the StochKit software toolkit.
Modeling and simulating chemical reactions
 SIAM Review
, 2007
"... 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 largescale limit of a sequence of finerscale probabilistic models. In studying this hierarchy of models, st ..."
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Cited by 14 (0 self)
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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 largescale limit of a sequence of finerscale 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. 1
CONTINUOUS TIME MARKOV CHAIN MODELS FOR CHEMICAL REACTION NETWORKS
"... A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transition ..."
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Cited by 14 (9 self)
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A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions of the chain. This chapter is devoted to the mathematical study of such stochastic models. We begin by developing much of the mathematical machinery we need to describe the stochastic models we are most interested in. We show how one can represent counting processes of the type we need in terms of Poisson processes. This random timechange representation gives a stochastic equation for continuoustime Markov chain models. We include a discussion on the relationship between this stochastic equation and the corresponding martingale problem and Kolmogorov forward (master) equation. Next, we exploit
Time series analysis via mechanistic models. In review; prepublished at arxiv.org/abs/0802.0021
, 2008
"... The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consi ..."
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Cited by 13 (5 self)
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The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plugandplay property. Our work builds on recently developed plugandplay inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae. 1. Introduction. A
The adaptive explicitimplicit tauleaping method with automatic tau selection
 J. Chem. Phys
, 2007
"... The existing tauselection strategy, which was designed for explicit tauleaping, is here modified to apply to implicit tauleaping, allowing for longer steps when the system is stiff. Further, an adaptive strategy is proposed that identifies stiffness and automatically chooses between the explicit ..."
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Cited by 12 (2 self)
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The existing tauselection strategy, which was designed for explicit tauleaping, is here modified to apply to implicit tauleaping, allowing for longer steps when the system is stiff. Further, an adaptive strategy is proposed that identifies stiffness and automatically chooses between the explicit and the (new) implicit tauselection methods to achieve better efficiency. Numerical testing demonstrates the advantages of the adaptive method for stiff systems. ∗ Author to whom correspondence should be addressed.