Search PubMed⌕ Search

SEARCH · Search PubMed

Results for “Master equation”

Search indexed PubMed citations on genomics, clinical trials, systematic reviews and public health. Explore titles, authors and supplied subject terms, then open the PubMed record.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 379 records · Page 21Linked to original sources
PubMed1987

Asymmetry and external noise-induced free energy transduction.

The effect of fluctuations in rate constants on the kinetic behavior of cyclic reacting systems, caused by a fluctuating external parameter, is studied. It is shown (i) that the stochastic properties of the system can be analyzed analytically by using the usual master equation approach when the external parameter is fluctuating with discrete square pulses and (ii) that the system is equivalent to an expanded chemical kinetic system with no fluctuation in rate constants. When applied to a linear four-state cyclic enzyme system, the formalism can be used to prove analytically the finding that an enzyme can extract the free energy from an externally applied fluctuating membrane potential and perform active free energy transduction. The formalism also can be used to assess the asymmetry constraints imposed on the values of the rate constants in order for the model to work.

Biological Transport↗
PubMedDate not supplied

Using motion planning to study RNA folding kinetics.

We propose a novel, motion planning based approach to approximately map the energy landscape of an RNA molecule. A key feature of our method is that it provides a sparse map that captures the main features of the energy landscape which can be analyzed to compute folding kinetics. Our method is based on probabilistic roadmap motion planners that we have previously successfully applied to protein folding. In this paper, we provide evidence that this approach is also well suited to RNA. We compute population kinetics and transition rates on our roadmaps using the master equation for a few moderately sized RNA and show that our results compare favorably with results of other existing methods.

Computational Biology↗
PubMed2005

Stochastic reaction-diffusion simulation with MesoRD.

UNLABELLED: MesoRD is a tool for stochastic simulation of chemical reactions and diffusion. In particular, it is an implementation of the next subvolume method, which is an exact method to simulate the Markov process corresponding to the reaction-diffusion master equation. AVAILABILITY: MesoRD is free software, written in C++ and licensed under the GNU general public license (GPL). MesoRD runs on Linux, Mac OS X, NetBSD, Solaris and Windows XP. It can be downloaded from http://mesord.sourceforge.net. CONTACT: johan.elf@icm.uu.se; johan.hattne@embl-hamburg.de SUPPLEMENTARY INFORMATION: 'MesoRD User's Guide' and other documents are available at http://mesord.sourceforge.net.

Algorithms↗
PubMed2006

Dynamic partitioning for hybrid simulation of the bistable HIV-1 transactivation network.

MOTIVATION: The stochastic kinetics of a well-mixed chemical system, governed by the chemical Master equation, can be simulated using the exact methods of Gillespie. However, these methods do not scale well as systems become more complex and larger models are built to include reactions with widely varying rates, since the computational burden of simulation increases with the number of reaction events. Continuous models may provide an approximate solution and are computationally less costly, but they fail to capture the stochastic behavior of small populations of macromolecules. RESULTS: In this article we present a hybrid simulation algorithm that dynamically partitions the system into subsets of continuous and discrete reactions, approximates the continuous reactions deterministically as a system of ordinary differential equations (ODE) and uses a Monte Carlo method for generating discrete reaction events according to a time-dependent propensity. Our approach to partitioning is improved such that we dynamically partition the system of reactions, based on a threshold relative to the distribution of propensities in the discrete subset. We have implemented the hybrid algorithm in an extensible framework, utilizing two rigorous ODE solvers to approximate the continuous reactions, and use an example model to illustrate the accuracy and potential speedup of the algorithm when compared with exact stochastic simulation. AVAILABILITY: Software and benchmark models used for this publication can be made available upon request from the authors.

Algorithms↗
PubMed2002

Quality and irreversibility: constraints on ecosystem development.

We discuss one of the most general mathematical tools for analysing dynamical systems: the master equation (ME). The ME is used to derive models for entropy production in closed and open systems. Due to dissipation in open systems, the direction of evolution of important characteristics can be opposite to those imposed on closed systems. When applying these models to soil organic matter it can be shown that the principle of minimum entropy production necessitates that more and more recalcitrant organic matter is produced the further the decomposition proceeds. The necessity to dissipate entropy can also impose a limit on the degree to which litters can decompose, but interaction between litters of differing ages can remove this constraint. This is an example of the 'priming' effect.

Ecosystem↗
PubMed2003

Fast evaluation of fluctuations in biochemical networks with the linear noise approximation.

Biochemical networks in single cells can display large fluctuations in molecule numbers, making mesoscopic approaches necessary for correct system descriptions. We present a general method that allows rapid characterization of the stochastic properties of intracellular networks. The starting point is a macroscopic description that identifies the system's elementary reactions in terms of rate laws and stoichiometries. From this formulation follows directly the stationary solution of the linear noise approximation (LNA) of the Master equation for all the components in the network. The method complements bifurcation studies of the system's parameter dependence by providing estimates of sizes, correlations, and time scales of stochastic fluctuations. We describe how the LNA can give precise system descriptions also near macroscopic instabilities by suitable variable changes and elimination of fast variables.

Artifacts↗
PubMed1999

Effect of symmetry on volume conserving surface models.

We study the effect of symmetry on volume conserving models without deposition and evaporation. By using the master equation approach, we identify two types of stochastic continuum equation with a conservative noise, depending on the symmetry of hopping rate in diffusion rules. In the model with symmetric hopping rate, a Laplacian term is essentially absent from the continuum equation. The dynamic scaling of this model is thus determined by the nonlinear fourth order equation with a conservative noise. When the symmetry is broken, a Laplacian term may be present, so the asymptotic scaling behavior is governed by the Laplacian term with nonzero coefficient. We verify this result by investigating a simple discrete model analytically.

Journal Article↗
PubMed1999

Reaction-controlled cooperative desorption in a one-dimensional lattice: a dynamical approach.

The spinlike dynamics of immobile reactants in a one-dimensional lattice is analyzed for two representative systems involving cooperative desorption. An exact combinatorial approach is worked out. Its failure to reproduce the results of microscopic simulations is shown to be associated with the lack of sufficiently strong ergodic properties, as a result of which the final state depends strongly on the initial conditions. A dynamical approach to the problem based on the Master equation description is subsequently developed, leading to full agreement with the microscopic simulations.

Journal Article↗
PubMed1999

Metastable states of an Ising-like thermally bistable system.

The lifetimes of the metastable states are investigated in an Ising-like model associated with thermally bistable systems. A discrete mesoscopic Markovian dynamic is established using an optimized version of the previously presented Monte Carlo entropic sampling method. This is well suited to an extensive study of the role of the physical parameters: temperature, interaction parameter, electronic energy gap. By combining a discrete Markovian mesoscopic dynamic and the absorbing Markov chain technique, we obtain an analytical access to the average lifetime of the metastable state. One-variable and two-variable approximations for the original microscopic master equation are presented and discussed. A typical difference in the thermal dependence of the lifetime of the low- and the high-temperature metastable states is found, and explained as a consequence of the temperature-dependent field associated with the Ising-like model. The validity, the advantages, and the limits of the method are discussed, as well as the possible consequences on the behavior of spin transition systems. A prospective for a possible phenomenological finite-size scaling is presented.

Journal Article↗
PubMed2000

Potential energy landscape and long-time dynamics in a simple model glass

We analyze the properties of a Lennard-Jones system at the level of the potential energy landscape. After an exhaustive investigation of the topological features of the landscape of the systems, obtained by studying small size samples, we describe the dynamics of the systems in multidimensional configurational space by means of a simple model. This considers the configurational space as a connected network of minima where the dynamics proceeds by jumps described by an appropriate master equation. Using this model we are able to reproduce the long-time dynamics and the low temperature regime. We investigate both the equilibrium regime and the off-equilibrium one, finding those typical glassy behaviors usually observed in the experiments such as (i) a stretched exponential relaxation, (ii) a temperature-dependent stretching parameter, (iii) a breakdown of the Stokes-Einstein relation, and (iv) the appearance of a critical temperature below which one observes a deviation from the fluctuation-dissipation relation as a consequence of the lack of equilibrium in the system.

Journal Article↗
PubMed2000

Relation between coupled map lattices and kinetic ising models

A spatially one-dimensional coupled map lattice possessing the same symmetries as the Miller-Huse model is introduced. Our model is studied analytically by means of a formal perturbation expansion which uses weak coupling and the vicinity to a symmetry breaking bifurcation point. In parameter space four phases with different ergodic behavior are observed. Although the coupling in the map lattice is diffusive, antiferromagnetic ordering is predominant. Via coarse graining the deterministic model is mapped to a master equation which establishes an equivalence between our system and a kinetic Ising model. Such an approach sheds some light on the dependence of the transient behavior on the system size and the nature of the phase transitions.

Journal Article↗
PubMed2000

Quasistationary distributions of dissipative nonlinear quantum oscillators in strong periodic driving fields

The dynamics of periodically driven quantum systems coupled to a thermal environment is investigated. The interaction of the system with the external coherent driving field is taken into account exactly by making use of the Floquet picture. Treating the coupling to the environment within the Born-Markov approximation one finds a Pauli-type master equation for the diagonal elements of the reduced density matrix in the Floquet representation. The stationary solution of the latter yields a quasistationary, time-periodic density matrix which describes the long-time behavior of the system. Taking the example of a periodically driven particle in a box, the stationary solution is determined numerically for a wide range of driving amplitudes and temperatures. It is found that the quasistationary distribution differs substantially from a Boltzmann-type distribution at the temperature of the environment. For large driving fields it exhibits a plateau region describing a nearly constant population of a certain number of Floquet states. This number of Floquet states turns out to be nearly independent of the temperature. The plateau region is sharply separated from an exponential tail of the stationary distribution which expresses a canonical Boltzmann-type distribution over the mean energies of the Floquet states. These results are explained in terms of the structure of the matrix of transition rates for the dissipative quantum system. Investigating the corresponding classical, nonlinear Hamiltonian system, one finds that in the semiclassical range essential features of the quasistationary distribution can be understood from the structure of the underlying classical phase space.

Journal Article↗
PubMed2000

Fluctuation-response relations for multitime correlations

We show that time-correlation functions of arbitrary order for any random variable in a statistical dynamical system can be calculated as higher-order response functions of the mean history of the variable. The response is to a "control term" added as a modification to the master equation for statistical distributions. The proof of the relations is based upon a variational characterization of the generating functional of the time correlations. The same fluctuation-response relations are preserved within moment closures for the statistical dynamical system, when these are constructed via the variational Rayleigh-Ritz procedure. For the two-time correlations of the moment variables themselves, the fluctuation-response relation is equivalent to an "Onsager regression hypothesis" for the small fluctuations. For correlations of higher order, there is a further effect in addition to such linear propagation of fluctuations present instantaneously: the dynamical generation of correlations by nonlinear interaction of fluctuations. In general, we discuss some physical and mathematical aspects of the Ansatze required for an accurate calculation of the time correlations. We also comment briefly upon the computational use of these relations, which is well suited for automatic differentiation tools. An example will be given of a simple closure for turbulent energy decay, which illustrates the numerical application of the relations.

Journal Article↗
PubMed2000

Collective diffusion and a random energy landscape

Starting from a master equation in a quantum Hamiltonian form and a coupling to a heat bath, we derive an evolution equation for a collective hopping process under the influence of a stochastic energy landscape. Different equations result for an arbitrary occupation number per lattice site or in a system under exclusion. Based on scaling arguments it will be demonstrated that both systems belong below the critical dimension d(c) to the same universality class, leading to anomalous diffusion in the long time limit. The dynamical exponent z can be calculated by an epsilon=d(c)-d expansion. Above the critical dimension we discuss the differences in the diffusion constant for sufficient high temperatures. For a random potential we find a higher mobility for systems with exclusion.

Journal Article↗
PubMed2000

Phenomenological glass model for vibratory granular compaction

A model for weakly excited granular media is derived by combining the free volume argument of Nowak et al. [Phys. Rev. E 57, 1971 (1998)] and the phenomenological model for supercooled liquids of Adam and Gibbs [J. Chem. Phys. 43, 139 (1965)]. This is made possible by relating the granular excitation parameter Gamma, defined as the peak acceleration of the driving pulse scaled by gravity, to a temperaturelike parameter eta(Gamma). The resulting master equation is formally identical to that of Bouchaud's trap model for glasses [J. Phys. I 2, 1705 (1992)]. Analytic and simulation results are shown to compare favorably with a range of known experimental behavior. This includes the logarithmic densification and power spectrum of fluctuations under constant eta, the annealing curve when eta is varied cyclically in time, and memory effects observed for a discontinuous shift in eta. Finally, we discuss the physical interpretation of the model parameters and suggest further experiments for this class of systems.

Journal Article↗
PubMed2000

Bulk dynamics for interfacial growth models

We study the influence of the bulk dynamics of a growing cluster of particles on the properties of its interface. First, we define a general bulk growth model by means of a continuum Master equation for the evolution of the bulk density field. This general model just considers an arbitrary addition of particles (though it can be easily generalized to consider subtraction) with no other physical restriction. The corresponding Langevin equation for this bulk density field is derived where the influence of the bulk dynamics is explicitly shown. Finally, when a well-defined interface is assumed for the growing cluster, the Langevin equation for the height field of this interface for some particular bulk dynamics is written. In particular, we obtain the celebrated Kardar-Parisi-Zhang equation. A Monte Carlo simulation illustrates the theoretical results.

Journal Article↗
PubMed2000

Vector k small middle dotp approach for photonic band structures

We point out that k small middle dotp treatments of photonic band gap materials based on the usual master equation must employ not only the physical photonic band solutions of that equation, but also unphysical solutions, in order to form a complete set. Nonetheless, it is possible to construct correct k small middle dotp expressions for the group velocity and its dispersion in terms of matrix elements involving only the photonic band solutions.

Journal Article↗