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Lognormal Moment Closures for Biochemical Reactions
"... In the stochastic formulation of chemical reactions, the dynamics of the the first Morder moments of the species populations generally do not form a closed system of differential equations, in the sense that the timederivatives of first Morder moments generally depend on moments of order higher t ..."
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Cited by 21 (7 self)
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In the stochastic formulation of chemical reactions, the dynamics of the the first Morder moments of the species populations generally do not form a closed system of differential equations, in the sense that the timederivatives of first Morder moments generally depend on moments of order higher thanM. However, for analysis purposes, these dynamics are often made to be closed by approximating the needed derivatives of the first Morder moments by nonlinear functions of the same moments. These functions are called the moment closure functions. Recent results have introduced the technique of derivativematching, where the moment closure functions are obtained by first assuming that they exhibit a certain separable form, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. However, for multispecies reactions these results have been restricted to second order truncations, i.e, M = 2. This paper extends these results by providing explicit formulas to construct moment closure functions for any arbitrary order of truncation M. We show that with increasing M the closed moment equations provide more accurate approximations to the exact moment equations. Striking features of these moment closure functions are that they are independent of the reaction parameters (reaction rates and stoichiometry) and moreover the dependence of higherorder moment on lower order ones is consistent with the population being jointly lognormally distributed. To illustrate the applicability of our results we consider a simple bimolecular reaction. Moment estimates from a third order truncation are compared with estimates obtained from a large number of Monte Carlo simulations.
Modeling and analysis of stochastic hybrid systems
 IEE Proc — Control Theory & Applications, Special Issue on Hybrid Systems 153(5
, 2007
"... The author describes a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events. The rate at which these transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Several examples of SHSs arising fr ..."
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Cited by 17 (7 self)
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The author describes a model for Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events. The rate at which these transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Several examples of SHSs arising from a varied pool of application areas are discussed. These include modeling of the Transmission Control Protocol’s (TCP) algorithm for congestion control both for longlived and onoff flows; stateestimation for networked control systems; and the stochastic modeling of chemical reactions. These examples illustrate the use of SHSs as a modeling tool. Attention is mostly focused on polynomial stochastic hybrid systems (pSHSs) that generally correspond to stochastic hybrid systems with polynomial continuous vector fields, reset maps, and transition intensities. For pSHSs, the dynamics of the statistical moments of the continuous states evolve according to infinitedimensional linear ordinary differential equations (ODEs). We show that these ODEs can be approximated by finitedimensional nonlinear ODEs with arbitrary precision. Based on this result, a procedure to build this type of approximations for certain classes of pSHSs is provided. This procedure is applied to several examples and the accuracy of the results obtained is evaluated through comparisons with Monte Carlo simulations. I.
Direct solution of the chemical master equation using quantized tensor trains. PLoS Comput Biol
"... The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other me ..."
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Cited by 16 (2 self)
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The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements in the number of problem dimensions. We present a novel approach that has the potential to ‘‘lift’ ’ this curse of dimensionality. The approach is based on the use of the recently proposed Quantized Tensor Train (QTT) formatted numerical linear algebra for the low parametric, numerical representation of tensors. The QTT decomposition admits both, algorithms for basic tensor arithmetics with complexity scaling linearly in the dimension (number of species) and sublinearly in the mode size (maximum copy number), and a numerical tensor rounding procedure which is stable and quasioptimal. We show how the CME can be represented in QTT format, then use the exponentiallyconverging hpdiscontinuous Galerkin discretization in time to reduce the CME evolution problem to a set of QTTstructured linear equations to be solved at each time step using an algorithm based on Density Matrix Renormalization Group (DMRG) methods from quantum chemistry. Our method automatically adapts the ‘‘basis’ ’ of the solution at every time step guaranteeing that it is large enough to capture the dynamics of interest but no larger than necessary, as this would increase the computational
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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Cited by 15 (5 self)
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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.
Subtilin production by Bacillus subtilis: Stochastic hybrid models and parameter identification
 IEEE Transactions on Circuits and Systems I – IEEE Transactions on Automatic Control
, 2008
"... This paper presents methods for the parameter identification of a model of subtilin production by Bacillus subtilis. Based on a stochastic hybrid model, identification is split in two subproblems: estimation of the genetic network regulating subtilin production from gene expression data, and estima ..."
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Cited by 7 (1 self)
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This paper presents methods for the parameter identification of a model of subtilin production by Bacillus subtilis. Based on a stochastic hybrid model, identification is split in two subproblems: estimation of the genetic network regulating subtilin production from gene expression data, and estimation of population dynamics based on nutrient and population level data. Techniques for identification of switching dynamics from sparse and irregularly sampled observations are developed and applied to simulated data. Numerical results are provided to show the effectiveness of our methods.
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.
Safety Analysis of Sugar Cataract Development Using Stochastic Hybrid Systems
 HYBRID SYSTEMS: COMPUTATION AND CONTROL 2007 LNCS 4416
, 2007
"... Modeling and analysis of biochemical systems are critical problems because they can provide new insights into systems which can not be easily tested with real experiments. One such biochemical process is the formation of sugar cataracts in the lens of an eye. Analyzing the sugar cataract developmen ..."
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Cited by 6 (5 self)
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Modeling and analysis of biochemical systems are critical problems because they can provide new insights into systems which can not be easily tested with real experiments. One such biochemical process is the formation of sugar cataracts in the lens of an eye. Analyzing the sugar cataract development process is a challenging problem due to the highlycoupled chemical reactions that are involved. In this paper we model sugar cataract development as a stochastic hybrid system. Based on this model, we present a probabilistic verification method for computing the probability of sugar cataract formation for different chemical concentrations. Our analysis can potentially provide useful insights into the complicated dynamics of the process and assist in focusing experiments on specific regions of concentrations. The verification method employs dynamic programming based on a discretization of the state space and therefore suffers from the curse of dimensionality. To verify the sugar cataract development process we have developed a parallel dynamic programming implementation that can handle large systems. Although scalability is a limiting factor, this work demonstrates that the technique is feasible for realistic biochemical systems.
New Insights on Stochastic Reachability
"... Abstract—In this paper, we give new characterizations of the stochastic reachability problem for stochastic hybrid systems in the language of different theories that can be employed in studying stochastic processes (Markov processes, potential theory, optimal control). These characterizations are fu ..."
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Cited by 6 (5 self)
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Abstract—In this paper, we give new characterizations of the stochastic reachability problem for stochastic hybrid systems in the language of different theories that can be employed in studying stochastic processes (Markov processes, potential theory, optimal control). These characterizations are further used to obtain the probabilities involved in the context of stochastic reachability as viscosity solutions of some variational inequalities.
Modelling and analysis of the sugar cataract development process using stochastic hybrid systems
 IET SYSTEMS BIOLOGY
, 2008
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