Simplification of Mathematical Models of Chemical Reaction Systems

Okino, Miles S.; Mavrovouniotis, Michael L.

doi:10.1021/cr950223l

Cited by 295 publications

(224 citation statements)

References 91 publications

(210 reference statements)

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“…The constraint that only a part of the network can be observed is generic, on the other hand. In protein interaction networks, for example, only a few molecular species can typically be tagged biochemically in such a way that their concentrations can be tracked with reasonable accuracy [16,17].…”

Section: Extended Plefka Expansion With Hidden Nodesmentioning

confidence: 99%

Inference for dynamics of continuous variables: the extended Plefka expansion with hidden nodes

Bravi¹,

Sollich²

2017

J. Stat. Mech.

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Abstract. We consider the problem of a subnetwork of observed nodes embedded into a larger bulk of unknown (i.e. hidden) nodes, where the aim is to infer these hidden states given information about the subnetwork dynamics. The biochemical networks underlying many cellular and metabolic processes are important realizations of such a scenario as typically one is interested in reconstructing the time evolution of unobserved chemical concentrations starting from the experimentally more accessible ones. We present an application to this problem of a novel dynamical mean field approximation, the Extended Plefka Expansion, which is based on a path integral description of the stochastic dynamics. As a paradigmatic model we study the stochastic linear dynamics of continuous degrees of freedom interacting via random Gaussian couplings. The resulting joint distribution is known to be Gaussian and this allows us to fully characterize the posterior statistics of the hidden nodes. In particular the equal-time hidden-to-hidden variance -conditioned on observations -gives the expected error at each node when the hidden time courses are predicted based on the observations. We assess the accuracy of the Extended Plefka Expansion in predicting these single node variances as well as error correlations over time, focussing on the role of the system size and the number of observed nodes.

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Section: Extended Plefka Expansion With Hidden Nodesmentioning

confidence: 99%

Inference for dynamics of continuous variables: the extended Plefka expansion with hidden nodes

Bravi¹,

Sollich²

2017

J. Stat. Mech.

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“…In the example applications presented in the following, the software package MUSCOD-II 29,30 originally developed for solving large scale optimal control problems for nonlinear dynamical systems is used for the numerical solution of problem (1). In MUSCOD-II the direct multiple shooting method 29 is implemented.…”

Section: General Methodologymentioning

confidence: 99%

Approximation of Slow Attracting Manifolds in Chemical Kinetics by Trajectory-Based Optimization Approaches

Reinhardt

Winckler

Lebiedz³

2008

J. Phys. Chem. A

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Many common kinetic model reduction approaches are explicitly based on inherent multiple time scales and often assume and directly exploit a clear time scale separation into fast and slow reaction processes. They approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones. The corresponding restrictive assumption of full relaxation of fast modes often renders the resulting approximation of slow attracting manifolds inaccurate as a representation of the reduced model and makes the numerical solution of the nonlinear "reduction equations" particularly difficult in many cases where the gap in intrinsic time scales is not large enough. We demonstrate that trajectory optimization approaches can avoid such severe restrictions by computing numerical solutions that correspond to "maximally relaxed" dynamical modes in a suitable sense. We present a framework of trajectory-based optimization for model reduction in chemical kinetics and a general class of reduction criteria characterizing the relaxation of chemical forces along reaction trajectories. These criteria can be motivated geometrically exploiting ideas from differential geometry and fundamental physics and turn out to be highly successful in example applications. Within this framework, we provide results for the computational approximation of slow attracting low-dimensional manifolds in terms of families of optimal trajectories for a six-component hydrogen combustion mechanism.

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“…However, lumping analysis can also eliminate fast variables, as the quasi-equilibrium or quasi-steady-state approximations do 14,16 . Given a U, whether A described by (1) can be exactly lumped into A ′ by U is decided by whether A ′ has an autonomous RE (3), as discussed above.…”

Section: Lumping Rate Equationsmentioning

confidence: 99%

“…Here, f is a Gaussian white noise with f (t) = 0, f (t ′ )f T (t) = Γδ(t − t ′ ), and f (t ′ )N T (t) = 0 for t < t ′ , where the covariance matrix Γ is symmetric, positive semi-definite, and generally time-dependent. The solution of (14), N(t) = e MtN (0) + t 0 e Mτ f (t − τ ) dτ , is also a Gaussian random variable. The conditional covariance of δN is σ ≡ δNδN T = t 0 e Mτ Γ e Mτ T dτ , which is symmetric and has the time derivative dσ/dt = Mσ + σM T + Γ.…”

Section: Lumping Stochastic Differential Equationsmentioning

confidence: 99%

Stochastic lumping analysis for linear kinetics and its application to the fluctuation relations between hierarchical kinetic networks

Deng

Chang

2015

The Journal of Chemical Physics

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Conventional studies of biomolecular behaviors rely largely on the construction of kinetic schemes. Since the selection of these networks is not unique, a concern is raised whether and under which conditions hierarchical schemes can reveal the same experimentally measured fluctuating behaviors and unique fluctuation related physical properties. To clarify these questions, we introduce stochasticity into the traditional lumping analysis, generalize it from rate equations to chemical master equations and stochastic differential equations, and extract the fluctuation relations between kinetically and thermodynamically equivalent networks under intrinsic and extrinsic noises. The results provide a theoretical basis for the legitimate use of low-dimensional models in the studies of macromolecular fluctuations and, more generally, for exploring stochastic features in different levels of contracted networks in chemical and biological kinetic systems.

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Simplification of Mathematical Models of Chemical Reaction Systems

Cited by 295 publications

References 91 publications

Inference for dynamics of continuous variables: the extended Plefka expansion with hidden nodes

Inference for dynamics of continuous variables: the extended Plefka expansion with hidden nodes

Approximation of Slow Attracting Manifolds in Chemical Kinetics by Trajectory-Based Optimization Approaches

Stochastic lumping analysis for linear kinetics and its application to the fluctuation relations between hierarchical kinetic networks

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