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21
Approximate Moment Dynamics for Chemically Reacting Systems
 IEEE Trans. Automatic Control
, 2010
"... In the stochastic formulation of chemical kinetics, the differential equation that describes the time evolution of the lowerorder statistical moments for the number of molecules of the different species involved, is generally not closed, in the sense that the righthand side of this equation depend ..."
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In the stochastic formulation of chemical kinetics, the differential equation that describes the time evolution of the lowerorder statistical moments for the number of molecules of the different species involved, is generally not closed, in the sense that the righthand side of this equation depends on higherorder moments. Recent work has proposed a moment closure technique based on derivativematching, which closes the moment equations by approximating higherorder moments as nonlinear functions of lowerorder moments. We here provide a mathematical proof of this moment closure technique, and highlight its performance through comparisons with alternative methods. These comparisons reveal that this moment closure technique based on derivativematching provides more accurate estimates of the moment dynamics, especially when the population size is small. Finally, we show that the accuracy of the proposed moment closure scheme can be arbitrarily increased by incurring additional computational effort.
Moment closure for biochemical networks
"... Abstract—Moment closure is a technique used to construct systems of differential equations to approximately compute means, standard deviations, and correlations between molecule counts of species involved in chemical reactions. These techniques are especially useful when the number of molecules exhi ..."
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Cited by 11 (0 self)
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Abstract—Moment closure is a technique used to construct systems of differential equations to approximately compute means, standard deviations, and correlations between molecule counts of species involved in chemical reactions. These techniques are especially useful when the number of molecules exhibit large stochasticity, which is not uncommon in biochemical reactions. We discuss several approaches to moment closure that have been proposed in the literature and that have been recently implemented in a Matlab toolbox. I. INTRODUCTION TO MOMENT CLOSURE Consider a set of chemical species X1,X2,...,Xn involved in a set of chemical reactions and let us denote by x:= (x1,x2,...,xn) a vector containing their molecule counts. Given a vector of integers m: = (m1,m2,...,mn), we use the
Fluid model checking
 in: Proceedings of CONCUR 2012
, 2012
"... In this paper we investigate a potential use of fluid approximation techniques in the context of stochastic model checking of CSL formulae. We focus on properties describing the behaviour of a single agent in a (large) population of agents, exploiting a limit result known also as fast simulation. In ..."
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Cited by 10 (2 self)
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In this paper we investigate a potential use of fluid approximation techniques in the context of stochastic model checking of CSL formulae. We focus on properties describing the behaviour of a single agent in a (large) population of agents, exploiting a limit result known also as fast simulation. In particular, we will approximate the behaviour of a single agent with a timeinhomogeneous CTMC, which depends on the environment and on the other agents only through the solution of the fluid differential equation, and model check this process. We will prove the asymptotic correctness of our approach in terms of satisfiability of CSL formulae. We will also present a procedure to model check timeinhomogeneous CTMC against CSL formulae.
Stochastic Analysis of Gene Regulatory Networks Using Moment Closure
, 2007
"... Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of chemical reactions and the small populations of proteins, mRNAs present inside cells. These fluctuations are usually reported in terms of the first and second order statistical moments of the protein po ..."
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Cited by 7 (4 self)
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Random fluctuations in gene regulatory networks are inevitable due to the probabilistic nature of chemical reactions and the small populations of proteins, mRNAs present inside cells. These fluctuations are usually reported in terms of the first and second order statistical moments of the protein populations. If the birthdeath rates of the mRNAs or the proteins are nonlinear, then the dynamics of these moments generally do not form a closed system of differential equations, in the sense that their timederivatives depends on moments of order higher than two. Recent work has developed techniques to obtain the two lowestorder moments by closing their dynamics, which involves approximating the higher order moments as nonlinear functions of the two lowest ones. This paper uses these moment closure techniques to quantify noise in several gene regulatory networks. In gene expression mechanisms in which a protein inhibits its own transcription, the resulting negative feedback reduces stochastic variations in the protein populations. Often the protein itself is not active and combines with itself to form an active multimer, which them inhibits the transcription. We demonstrate that this more sophisticated form of negative feedback (using multimerization) is more effective in suppressing noise. We also consider a twogene cascade activation network in which the protein expressed by one gene activates another gene to express a second protein. Analysis shows that the stochastic fluctuations in the population of the activated protein increases with the degree of multimerization in the activating protein.
2012. Models of stochastic gene expression and Weyl algebra
 In Algebraic and Numeric Biology
"... Abstract. This paper presents a symbolic algorithm for computing the ODE systems which describe the evolution of the moments associated to a chemical reaction system, considered from a stochastic point of view. The algorithm, which is formulated in the Weyl algebra, seems more efficient than the co ..."
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Abstract. This paper presents a symbolic algorithm for computing the ODE systems which describe the evolution of the moments associated to a chemical reaction system, considered from a stochastic point of view. The algorithm, which is formulated in the Weyl algebra, seems more efficient than the corresponding method, based on partial derivatives. In particular, an efficient method for handling conservation laws is presented. The output of the algorithm can be used for a further investigation of the system behaviour, by numerical methods. Relevant examples are carried out.
Moment closures for performance models with highly nonlinear rates
 in 9th European Performance Engineering Workshop (EPEW
, 2012
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The finite state projection approach for the solution of the master equation and its applications to stochastic gene regulatory networks
, 2008
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Hybrid Behaviour of Markov Population Models
, 2012
"... We investigate the behaviour of population models written in Stochastic Concurrent Constraint Programming (sCCP), a stochastic extension of Concurrent Constraint Programming. In particular, we focus on models from which we can define a semantics of sCCP both in terms of Continuous Time Markov Chains ..."
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Cited by 2 (0 self)
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We investigate the behaviour of population models written in Stochastic Concurrent Constraint Programming (sCCP), a stochastic extension of Concurrent Constraint Programming. In particular, we focus on models from which we can define a semantics of sCCP both in terms of Continuous Time Markov Chains (CTMC) and in terms of Stochastic Hybrid Systems, in which some populations are approximated continuously, while others are kept discrete. We will prove the correctness of the hybrid semantics from the point of view of the limiting behaviour of a sequence of models for increasing population size. More specifically, we prove that, under suitable regularity conditions, the sequence of CTMC constructed from sCCP programs for increasing population size converges to the hybrid system constructed by means of the hybrid semantics. We investigate in particular what happens for sCCP models in which some transitions are guarded by boolean predicates or in the presence of instantaneous transitions.