Mesoscale Analysis of the Equation-Free Constrained Runs Initialization Scheme

Leemput, Pieter Van; Vanroose, Wim; Roose, Dirk

doi:10.1137/07069403x

Cited by 22 publications

(24 citation statements)

References 21 publications

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“…The ZDP was pioneered by Kreiss and coworkers [25][26][27] in the applied mathematics/computational physics community. It has been employed to obtain accurate, yet simplified descriptions of complex models arising in meteorology [28], computational physics [29,30] and more general multiscale systems [31,32], but not yet in the current biochemical context. We apply the ZDP to two systems: first, to a prototypical example with a reversible enzymatic reaction and, second, to the substantially more complex phosphotransferase system (PTS), comprising a signal transduction pathway regulating and catalyzing glucose uptake in enteric bacteria.…”

Section: Introductionmentioning

confidence: 99%

Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations

Härdin

Zagaris

Krab³

et al. 2009

The FEBS Journal

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Much of enzyme kinetics builds on simplifications enabled by the quasi‐steady‐state approximation and is highly useful when the concentration of the enzyme is much lower than that of its substrate. However, in vivo, this condition is often violated. In the present study, we show that, under conditions of realistic yet high enzyme concentrations, the quasi‐steady‐state approximation may readily be off by more than a factor of four when predicting concentrations. We then present a novel extension of the quasi‐steady‐state approximation based on the zero‐derivative principle, which requires considerably less theoretical work than did previous such extensions. We show that the first‐order zero‐derivative principle, already describes much more accurately the true enzyme dynamics at enzyme concentrations close to the concentration of their substrates. This should be particularly relevant for enzyme kinetics where the substrate is an enzyme, such as in phosphorelay and mitogen‐activated protein kinase pathways. We illustrate this for the important example of the phosphotransferase system involved in glucose uptake, metabolism and signaling. We find that this system, with a potential complexity of nine dimensions, can be understood accurately using the first‐order zero‐derivative principle in terms of the behavior of a single variable with all other concentrations constrained to follow that behavior.

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Section: Introductionmentioning

confidence: 99%

Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations

Härdin

Zagaris

Krab³

et al. 2009

The FEBS Journal

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“…For the stochastic part E 2 , we again consider a forward-Euler scheme for the diffusion equation with estimated diffusion D; see Eq. (22). We obtain…”

Section: Error Propagationmentioning

confidence: 88%

“…For the deterministic case, solutions have been proposed for some specific settings [10,22], but in general, lifting is a difficult problem [8].…”

Section: Drift and Diffusion Estimationmentioning

confidence: 99%

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An analysis of noise propagation in the multiscale simulation of coarse Fokker-Planck equations

Frederix¹,

Samaey²,

Roose³

2010

ESAIM: M2AN

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We consider multiscale systems for which only a fine-scale model describing the evolution of individuals (atoms, molecules, bacteria, agents) is given, while we are interested in the evolution of the population density on coarse space and time scales. Typically, this evolution is described by a coarse Fokker-Planck equation. In this paper, we investigate a numerical procedure to compute the solution of this Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale, individualbased system. As these parameters might be solution-dependent, the estimation is performed in every spatial discretization point and at every time step. If the fine-scale model is stochastic, the estimation procedure introduces noise on the coarse level. We investigate stability conditions for this procedure and present an analysis of the propagation of the estimation error in the numerical solution of the coarse Fokker-Planck equation. The results show that for decreasing spatial discretization error, the total error grows rapidly due to the use of estimated coefficients. This effect can be avoided by increasing the quality of the estimates when the spatial discretization decreases. Although the procedure is illustrated for a specific class of multiscale stochastic systems, it is devised so that it can easily be generalized to other stochastic or particle models. Abstract We consider multiscale systems for which only a fine-scale model describing the evolution of individuals (atoms, molecules, bacteria, agents) is given, while we are interested in the evolution of the population density on coarse space and time scales. Typically, this evolution is described by a coarse Fokker-Planck equation. In this paper, we investigate a numerical procedure to compute the solution of this Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale, individual-based system. As these parameters might be solution-dependent, the estimation is performed in every spatial discretization point and at every time step. If the fine-scale model is stochastic, the estimation procedure introduces noise on the coarse level. We investigate stability conditions for this procedure and present an analysis of the propagation of the estimation error in the numerical solution of the coarse Fokker-Planck equation. The results show that for decreasing spatial discretization error, the total error grows rapidly due to the use of estimated coefficients. This effect can be avoided by increasing the quality of the estimates when the spatial discretization decreases. Although the procedure is illustrated for a specific class of multiscale stochastic systems, it is devised so that it can easily be generalized to other stochastic or particle models.

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“…Several approaches have been suggested to address this problem. We refer to [9,10,30] for an algorithm that only uses a time-stepper for the original fine-scale system.…”

Section: The Coarse Time-steppermentioning

confidence: 99%

An Analysis of Equivalent Operator Preconditioning for Equation-Free Newton–Krylov Methods

Samaey¹,

Vanroose²

2010

SIAM J. Numer. Anal.

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We consider the computation of a fixed point of a time-stepper using NewtonKrylov methods, and propose and analyze equivalent operator preconditioning for the resulting linear systems. For a linear, scalar advection-reaction-diffusion equation, we investigate in detail how the convergence rate depends on the choice of preconditioner parameters and on the time discretization. The results are especially valuable when computing fixed points of a coarse time-stepper in the equation-free multiscale framework, in which one simulates an unavailable coarse-scale model by wrapping a set of computational routines around appropriately initialized fine-scale simulations. Both analytical results and numerical experiments are presented, showing that one can speed up the convergence of iterative methods significantly for a wide range of parameter values in the preconditioner.

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Mesoscale Analysis of the Equation-Free Constrained Runs Initialization Scheme

Cited by 22 publications

References 21 publications

Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations

Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations

An analysis of noise propagation in the multiscale simulation of coarse Fokker-Planck equations

An Analysis of Equivalent Operator Preconditioning for Equation-Free Newton–Krylov Methods

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