Microscopic Simulations of Chemical Instabilities

Baras, F.; Malek-Mansour, Mohammad-Ali

doi:10.1002/9780470141595.ch5

Cited by 38 publications

(50 citation statements)

References 239 publications

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“…In a stochastic formulation, it is useful to categorize the origin of the fluctuations into two broad classes (van Kampen 1992;Swain et al 2002): intrinsic (or demographic) noise arising from the discrete stochastic interactions of the individual constituents, and extrinsic (or environmental) noise arising from un-modelled dynamics outside the system of interest. For a spatially varying stochastic model, fluctuations are included in the governing equations in one of two ways: either as multiplicative noise in a stochastic partial differential equation (Garcia-Ojalvo & Sancho 1999), or by representing the dynamics in terms of a probability distribution for the state using a reaction-transport master equation (van Kampen 1992;Baras & Mansour 1998;Gardiner 2004).…”

Section: Introductionmentioning

confidence: 99%

Approximating intrinsic noise in continuous multispecies models

2010

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In small-scale chemical reaction networks, the local density of molecules is changed by discrete jumps owing to reactive collisions, and through transport. A systematic perturbation scheme is developed to analytically characterize these non-equilibrium intrinsic fluctuations in a multispecies spatially varying system. The method is illustrated on a variety of model systems. In all cases, the continuous approximation method is corroborated with extensive stochastic simulation. As an example of our technique applied to a spatially varying steady state, we demonstrate that a model for embryonic patterning mediated by regulatory mRNA is surprisingly robust to intrinsic fluctuations.

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Section: Introductionmentioning

confidence: 99%

Approximating intrinsic noise in continuous multispecies models

2010

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“…The Fokker-Planck equation (FPE) is a partial differential equation (PDE) approximating the master equation [2,16,23]. The state of the system is assumed to change continuously and common numerical methods for PDEs can be invoked for solution of the FPE.…”

Section: Introductionmentioning

confidence: 99%

Numerical method for coupling the macro and meso scales in stochastic chemical kinetics

Ferm

Lötstedt

2007

Bit Numer Math

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A numerical method is developed for simulation of stochastic chemical reactions. The system is modeled by the Fokker-Planck equation for the probability density of the molecular state. The dimension of the domain of the equation is reduced by assuming that most of the molecular species have a normal distribution with a small variance. The numerical approximation preserves properties of the analytical solution such as nonnegativity and constant total probability. The method is applied to a nine dimensional problem modelling an oscillating molecular clock. The oscillations stop at a fixed point with a macroscopic model but they continue with our two dimensional, mixed macroscopic and mesoscopic model.

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“…With (ψ (35) and (32) we have derived the macroscopic reaction rate equations (2). The same equations are obtained directly with the assumption…”

Section: M and |δ| = J δ J Let The Expected Value Of W Bementioning

confidence: 99%

“…This model is accurate for reactions with many participating molecules of each species. If the number of molecules is small or if the reactions occur far from thermodynamic equilibrium, then a stochastic model on a mesoscale such as the master equation [2,9,10,16,24] is a much more precise description. This is often the case in molecular biology where the majority of the reacting molecules are found in copy numbers less than one hundred in cells such as Escherichia coli [14].…”

Section: Introductionmentioning

confidence: 99%

Dimensional Reduction of the Fokker–Planck Equation for Stochastic Chemical Reactions

Lötstedt¹,

Ferm²

2006

Multiscale Model. Simul.

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The Fokker-Planck equation models chemical reactions on a mesoscale. The solution is a probability density function for the copy number of the different molecules. The number of dimensions of the problem can be large making numerical simulation of the reactions computationally intractable. The number of dimensions is reduced here by deriving partial differential equations for the first moments of some of the species and coupling them to a Fokker-Planck equation for the remaining species. With more simplifying assumptions, another system of equations is derived consisting of integro-differential equations and a Fokker-Planck equation. In this way, the simulation of the chemical networks is possible without the exponential growth in computatational work and memory of the original equation and with better modelling accuracy than the macroscopic reaction rate equations. Some terms in the equations are small and are ignored. Conditions are given for the influence of these terms to be small on the equations and the solutions. The difference between different models is illustrated in a numerical example.

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Microscopic Simulations of Chemical Instabilities

Cited by 38 publications

References 239 publications

Approximating intrinsic noise in continuous multispecies models

Approximating intrinsic noise in continuous multispecies models

Numerical method for coupling the macro and meso scales in stochastic chemical kinetics

Dimensional Reduction of the Fokker–Planck Equation for Stochastic Chemical Reactions

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