Approximate Moment Dynamics for Chemically Reacting Systems

Singh, Abhyudai; Hespanha, João P.

doi:10.1109/tac.2010.2088631

Cited by 224 publications

(240 citation statements)

References 19 publications

Supporting

Mentioning

236

Contrasting

Order By: Relevance

“…45 Often solutions of the equilibrium RRE corresponding to the asymptotic VFP will suffice, 47 although more sophisticated moment closure approximations for the asymptotic VFP will usually be more accurate. 48,49 In some circumstances it may be easier to estimate the required averages by making brief SSA runs of the VFP. 50 A software implementation of the ssSSA which automatically and adaptively partitions the system and efficiently computes the modified slow propensity functions for general mass action models is available.…”

mentioning

confidence: 99%

Perspective: Stochastic algorithms for chemical kinetics

Gillespie

Hellander²,

Petzold³

2013

The Journal of Chemical Physics

297

341

View full text Add to dashboard Cite

We outline our perspective on stochastic chemical kinetics, paying particular attention to numerical simulation algorithms. We first focus on dilute, well-mixed systems, whose description using ordinary differential equations has served as the basis for traditional chemical kinetics for the past 150 years. For such systems, we review the physical and mathematical rationale for a discretestochastic approach, and for the approximations that need to be made in order to regain the traditional continuous-deterministic description. We next take note of some of the more promising strategies for dealing stochastically with stiff systems, rare events, and sensitivity analysis. Finally, we review some recent efforts to adapt and extend the discrete-stochastic approach to systems that are not wellmixed. In that currently developing area, we focus mainly on the strategy of subdividing the system into well-mixed subvolumes, and then simulating diffusional transfers of reactant molecules between adjacent subvolumes together with chemical reactions inside the subvolumes. © 2013 AIP Publishing LLC.[http://dx

show abstract

mentioning

confidence: 99%

Perspective: Stochastic algorithms for chemical kinetics

Gillespie

Hellander²,

Petzold³

2013

The Journal of Chemical Physics

297

341

View full text Add to dashboard Cite

show abstract

“…There have been numerous attempts to define F, either by assuming an underlying distribution (12,15,16) or through numerical approximation (17)(18)(19). However, closure schemes thus far exhibit limited accuracy and uncertain utility.…”

mentioning

confidence: 99%

A closure scheme for chemical master equations

Smadbeck

Kaznessis

2013

Proc. Natl. Acad. Sci. U.S.A.

107

106

View full text Add to dashboard Cite

Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70 years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higherorder moments by maximizing the information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kinetic Monte Carlo simulations, which merely sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time.stochastic models | information theory | entropy maximization | statistical mechanics T he fabric of all things living is discrete and noisy, individual molecules in perpetual random motion. However, humans, in our effort to understand and manipulate the biological cosmos, have historically perceived and modeled nature as large collections of molecules with behaviors not far from an expected average. Mathematical models, founded on such determinism, may be excellent approximations of reality when the number of molecules is very large, approaching the limit of an infinitely sized molecular population (1-5). Of course, the size of biomolecular systems is far from infinite. And we know that the behavior of a few molecules fluctuating from the average in unexpected ways may forever seal the fate of a living organism. It has thus been commonly recognized that models of small, evolving molecular populations better account for the noisy, probabilistic nature of outcomes (6-8).The most complete model of stochastically evolving molecular populations is one based on the master probability equation (9). The "master" in the name reflects the all-encompassing nature of an equation that purports to govern all possible outcomes for all time. Because of its ambitious character, the master equation has remained unsolved for all but the simplest of molecular interaction networks, even though it is now over seven decades since the first master equations were set up for chemical systems (10,11). Herein we ...

show abstract

“…Azunre et al 39 showed that for very small molecule numbers, using only two moments can lead to unstable results. Singh and Hespanha 16 developed a derivative-matching approach in the context of polynomial rate equations, which proved to work very well in particular for small molecule numbers. For the examples described in this paper, the proposed method shows differences in behaviour for very low molecule numbers.…”

Section: Discussionmentioning

confidence: 99%

A general moment expansion method for stochastic kinetic models

Ale

Kirk

Stumpf

2013

The Journal of Chemical Physics

View full text Add to dashboard Cite

Moment approximation methods are gaining increasing attention for their use in the approximation of the stochastic kinetics of chemical reaction systems. In this paper we derive a general moment expansion method for any type of propensities and which allows expansion up to any number of moments. For some chemical reaction systems, more than two moments are necessary to describe the dynamic properties of the system, which the linear noise approximation is unable to provide. Moreover, also for systems for which the mean does not have a strong dependence on higher order moments, moment approximation methods give information about higher order moments of the underlying probability distribution. We demonstrate the method using a dimerisation reaction, Michaelis-Menten kinetics and a model of an oscillating p53 system. We show that for the dimerisation reaction and MichaelisMenten enzyme kinetics system higher order moments have limited influence on the estimation of the mean, while for the p53 system, the solution for the mean can require several moments to converge to the average obtained from many stochastic simulations. We also find that agreement between lower order moments does not guarantee that higher moments will agree. Compared to stochastic simulations, our approach is numerically highly efficient at capturing the behaviour of stochastic systems in terms of the average and higher moments, and we provide expressions for the computational cost for different system sizes and orders of approximation. We show how the moment expansion method can be employed to efficiently quantify parameter sensitivity. Finally we investigate the effects of using too few moments on parameter estimation, and provide guidance on how to estimate if the distribution can be accurately approximated using only a few moments.

show abstract

Approximate Moment Dynamics for Chemically Reacting Systems

Cited by 224 publications

References 19 publications

Perspective: Stochastic algorithms for chemical kinetics

Perspective: Stochastic algorithms for chemical kinetics

A closure scheme for chemical master equations

A general moment expansion method for stochastic kinetic models

Contact Info

Product

Resources

About