Novel bivariate moment-closure approximations

Krishnarajah, Isthrinayagy; Marion, Glenn; Gibson, Gavin J.

doi:10.1016/j.mbs.2006.12.002

Cited by 17 publications

(16 citation statements)

References 25 publications

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“…Later they also described multivariate methods for an extended model. 17 In parallel, several approaches have been developed which do not proceed from a particular distribution. The most straightforward approach is the truncation of the system applied by Lee et al 18 and similar to what we will refer to as zero closure in the manuscript.…”

Section: Introductionmentioning

confidence: 99%

Multivariate moment closure techniques for stochastic kinetic models

Lakatos

Ale

Kirk

et al. 2015

The Journal of Chemical Physics

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A study of the accuracy of moment-closure approximations for stochastic chemical kinetics J. Chem. Phys. 136, 154105 (2012) Stochastic effects dominate many chemical and biochemical processes. Their analysis, however, can be computationally prohibitively expensive and a range of approximation schemes have been proposed to lighten the computational burden. These, notably the increasingly popular linear noise approximation and the more general moment expansion methods, perform well for many dynamical regimes, especially linear systems. At higher levels of nonlinearity, it comes to an interplay between the nonlinearities and the stochastic dynamics, which is much harder to capture correctly by such approximations to the true stochastic processes. Moment-closure approaches promise to address this problem by capturing higher-order terms of the temporally evolving probability distribution. Here, we develop a set of multivariate moment-closures that allows us to describe the stochastic dynamics of nonlinear systems. Multivariate closure captures the way that correlations between different molecular species, induced by the reaction dynamics, interact with stochastic effects. We use multivariate Gaussian, gamma, and lognormal closure and illustrate their use in the context of two models that have proved challenging to the previous attempts at approximating stochastic dynamics: oscillations in p53 and Hes1. In addition, we consider a larger system, Erk-mediated mitogen-activated protein kinases signalling, where conventional stochastic simulation approaches incur unacceptably high computational costs. C 2015 AIP Publishing LLC.[http://dx

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Section: Introductionmentioning

confidence: 99%

Multivariate moment closure techniques for stochastic kinetic models

Lakatos

Ale

Kirk

et al. 2015

The Journal of Chemical Physics

View full text Add to dashboard Cite

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“…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.…”

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confidence: 99%

A closure scheme for chemical master equations

Smadbeck

Kaznessis

2013

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

107

106

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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 ...

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“…Its use however is limited to closed reaction networks (i.e., reaction networks with fixed total population) that contain only two species. Finally, using mixture distributions for deriving a moment closure scheme shows great promise but has only been employed in few limited cases (Krishnarajah et al, 2005;Krishnarajan et al, 2007).…”

Section: Moment Approximationmentioning

confidence: 99%

Markovian dynamics on complex reaction networks

2013

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Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multi-agent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples.

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Novel bivariate moment-closure approximations

Cited by 17 publications

References 25 publications

Multivariate moment closure techniques for stochastic kinetic models

Multivariate moment closure techniques for stochastic kinetic models

A closure scheme for chemical master equations

Markovian dynamics on complex reaction networks

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