Asymptotic properties of markovian master equations

Mansour, M. Malek; Broeck, C. Van den; Nicolis, Grégoire; Turner, J. W.

doi:10.1016/0003-4916(81)90033-6

Cited by 115 publications

(31 citation statements)

References 44 publications

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“…Figure 1 shows plots of the ODE, dx/dt versus x, for an example of chemical equilibrium and for a bistable set of the parameters. When there is only one real root, the steady-state behaviour of the stochastic system will match that of the deterministic system ( Mansour et al 1981). However, as the parameters move away from the chemical detailed balance condition, the shape of the probability steady-state function will deform from its Poissonian shape.…”

Section: Chemical Flux and Steady-state Behaviourmentioning

confidence: 99%

“…Diffusion (Fokker-Planck) approximations to the master equation were first developed by Van Kampen (1981) and shown by Kurtz (1976Kurtz ( , 1978 to match the solution to the master equation in the thermodynamic limit for some finite time. However, unless the steady state is unique in the macroscopic description (Mansour et al 1981), the two models can disagree in the infinite time limit (see appendix A). Microscopic simulations have validated the master equation as the most accurate description of a reactive process; see Baras & Mansour (1997) for an up-to-date review.…”

Section: Stochastic Modellingmentioning

confidence: 99%

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Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited

Vellela

Qian

2008

J. R. Soc. Interface.

212

221

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Schlögl's model is the canonical example of a chemical reaction system that exhibits bistability. Because the biological examples of bistability and switching behaviour are increasingly numerous, this paper presents an integrated deterministic, stochastic and thermodynamic analysis of the model. After a brief review of the deterministic and stochastic modelling frameworks, the concepts of chemical and mathematical detailed balances are discussed and non-equilibrium conditions are shown to be necessary for bistability. Thermodynamic quantities such as the flux, chemical potential and entropy production rate are defined and compared across the two models. In the bistable region, the stochastic model exhibits an exchange of the global stability between the two stable states under changes in the pump parameters and volume size. The stochastic entropy production rate shows a sharp transition that mirrors this exchange. A new hybrid model that includes continuous diffusion and discrete jumps is suggested to deal with the multiscale dynamics of the bistable system. Accurate approximations of the exponentially small eigenvalue associated with the time scale of this switching and the full time-dependent solution are calculated using MATLAB. A breakdown of previously known asymptotic approximations on small volume scales is observed through comparison with these and Monte Carlo results. Finally, in the appendix section is an illustration of how the diffusion approximation of the chemical master equation can fail to represent correctly the mesoscopically interesting steady-state behaviour of the system.

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Section: Chemical Flux and Steady-state Behaviourmentioning

confidence: 99%

Section: Stochastic Modellingmentioning

confidence: 99%

Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited

Vellela

Qian

2008

J. R. Soc. Interface.

212

221

View full text Add to dashboard Cite

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“…It has then been shown that a sufficient condition for the validity of the Kurtz theorem in the limit of infinitely long time (i.e. stationary state) is the uniqueness of the macroscopic attractor, which thus includes systems exhibiting sustained oscillatory regimes (limit cycle; [61]). Molecular dynamic simulations based on Newtonian hard spheres have shown that it is the master equation that provides the correct answer [62].…”

Section: Diffusion Approximation For the Cmementioning

confidence: 99%

Non-equilibrium phase transition in mesoscopic biochemical systems: from stochastic to nonlinear dynamics and beyond

Qian

2010

J. R. Soc. Interface.

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A theory for an non-equilibrium phase transition in a driven biochemical network is presented. The theory is based on the chemical master equation (CME) formulation of mesoscopic biochemical reactions and the mathematical method of large deviations. The large deviations theory provides an analytical tool connecting the macroscopic multi-stability of an open chemical system with the multi-scale dynamics of its mesoscopic counterpart. It shows a corresponding non-equilibrium phase transition among multiple stochastic attractors. As an example, in the canonical phosphorylation -dephosphorylation system with feedback that exhibits bistability, we show that the non-equilibrium steadystate (NESS) phase transition has all the characteristics of classic equilibrium phase transition: Maxwell construction, a discontinuous first-derivative of the 'free energy function', Lee-Yang's zero for a generating function and a critical point that matches the cusp in nonlinear bifurcation theory. To the biochemical system, the mathematical analysis suggests three distinct timescales and needed levels of description. They are (i) molecular signalling, (ii) biochemical network nonlinear dynamics, and (iii) cellular evolution. For finite mesoscopic systems such as a cell, motions associated with (i) and (iii) are stochastic while that with (ii) is deterministic. Both (ii) and (iii) are emergent properties of a dynamic biochemical network.

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“…a Taylor expansion around the corresponding jump states  ±  to transfer the contributions from the transition probabilities to the transition moments in the balance equation of the probability distribution evolution), the master eq. (2.1) can be reduced to the following FPE [1,[10][11][12] …”

Section: Broadening Exponent Of the Critical Fluctuation In Chemical mentioning

confidence: 99%

A critical transition rule for broadening exponent of fluctuation and its effect on dissipation in chemical reaction-heat conduction coupling systems

2011

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A stochastic model of chemical reaction-heat conduction-diffusion for a one-dimensional gaseous system under Dirichlet or zero-fluxes boundary conditions is proposed in this paper. Based on this model, we extend the theory of the broadening exponent of critical fluctuations to cover the chemical reaction-heat conduction coupling systems as an asymptotic property of the corresponding Markovian master equation (ME), and establish a valid stochastic thermodynamics for such systems. As an illustration, the non-isothermal and inhomogeneous Schlögl model is explicitly studied. Through an order analysis of the contributions from both the drift and diffusion to the evolution of the probability distribution in the corresponding Fokker-Planck equation(FPE) in the approach to bifurcation, we have identified the critical transition rule for the broadening exponent of the fluctuations due to the coupling between chemical reaction and heat conduction. It turns out that the dissipation induced by the critical fluctuations reaches a deterministic level, leading to a thermodynamic effect on the nonequilibrium physico-chemical processes.stochastic model of chemical reaction-heat conduction-diffusion systems, stochastic thermodynamics of chemical reaction-heat conduction coupling processes, broadening exponent of critical fluctuation, entropy production of fluctuations, fluctuation-dissipation effect

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Asymptotic properties of markovian master equations

Cited by 115 publications

References 44 publications

Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited

Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited

Non-equilibrium phase transition in mesoscopic biochemical systems: from stochastic to nonlinear dynamics and beyond

A critical transition rule for broadening exponent of fluctuation and its effect on dissipation in chemical reaction-heat conduction coupling systems

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