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 181 records · Page 10Linked to original sources
PubMed2004

Trajectory approach to dissipative quantum phase space dynamics: Application to barrier scattering.

The Caldeira-Leggett master equation, expressed in Lindblad form, has been used in the numerical study of the effect of a thermal environment on the dynamics of the scattering of a wave packet from a repulsive Eckart barrier. The dynamics are studied in terms of phase space trajectories associated with the distribution function, W(q,p,t). The equations of motion for the trajectories include quantum terms that introduce nonlocality into the motion, which imply that an ensemble of correlated trajectories needs to be propagated. However, use of the derivative propagation method (DPM) allows each trajectory to be propagated individually. This is achieved by deriving equations of motion for the partial derivatives of W(q,p,t) that appear in the master equation. The effects of dissipation on the trajectories are studied and results are shown for the transmission probability. On short time scales, decoherence is demonstrated by a swelling of trajectories into momentum space. For a nondissipative system, a comparison is made of the DPM with the "exact" transmission probability calculated from a fixed grid calculation.

Journal Article↗
PubMed2006

Kinetic theory of binary nucleation based on a first passage time analysis.

The binary classical nucleation theory (BCNT) is based on the Gibbsian thermodynamics and applies the macroscopic concept of surface tension to nanosize clusters. This leads to severe inconsistencies and large discrepancies between theoretical predictions and experimental results regarding the nucleation rate. We present an alternative approach to the kinetics of binary nucleation which avoids the use of classical thermodynamics for clusters. The new approach is an extension to binary mixtures of the kinetic theory previously developed by Narsimhan and Ruckenstein and Ruckenstein and Nowakowski [J. Colloid Interface Sci. 128, 549 (1989); 137, 583 (1990)] for unary nucleation which is based on molecular interactions and in which the rate of emission of molecules from a cluster is determined via a mean first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution of a "surface" molecule moving in a potential field created by the cluster. The starting master equation is a Fokker-Planck equation for the probability distribution of a surface molecule with respect to its phase coordinates. Owing to the hierarchy of characteristic time scales in the evolution of the molecule, this equation can be reduced to the Smoluchowski equation for the distribution function involving only the spatial coordinates. The new theory is combined with density functional theory methods to determine the density profiles. This is essential for nucleation in binary systems particularly when one of the components is surface active. Knowing these profiles, one can determine the potential fields created by the cluster, its rate of emission of molecules, and the nucleation rate more accurately than by using the uniform density approximation. The new theory is illustrated by numerical calculations for a model binary mixture of Lennard-Jones monomers and rigidly bonded dimers of Lennard-Jones atoms. The amphiphilic character of the dimer component (i.e., its surface activity) is induced by the asymmetry in the interaction between a monomer and the two different sites of a dimer. The inconsistencies of the BCNT are avoided in the new theory.

Journal Article↗
PubMed1995

Simple model of protein folding kinetics.

A simple model of the kinetics of protein folding is presented. The reaction coordinate is the "correctness" of a configuration compared with the native state. The model has a gap in the energy spectrum, a large configurational entropy, a free energy barrier between folded and partially folded states, and a good thermodynamic folding transition. Folding kinetics is described by a master equation. The folding time is estimated by means of a local thermodynamic equilibrium assumption and then is calculated both numerically and analytically by solving the master equation. The folding time has a maximum near the folding transition temperature and can have a minimum at a lower temperature.

Kinetics↗
PubMed2005

Kinetics of model energy landscapes: an approach to complex systems.

An idealized potential energy surface (PES), simply a PES-like network of stationary points, is demonstrated to be a useful tool to study kinetic relaxation of complex energy landscapes. Combined with a master equation, we show that if constructed with proper regularity, the kinetics of the PES is easy to predict and understand by carefully examining the eigenmodes of the master equation. By modifying the idealized PES model to make it more and more complicated, we demonstrate a systematic method to study the complex kinetics on large PES. The idealized PES model is used to explore the feasibility and the robustness of statistical sampling of large PES. We develop several sampling strategies, such as the "rough topography sampling" and the "low barrier saddle sampling" in the idealized PES model and find they are also applicable to a realistic PES of the 13-atom Morse cluster with range parameter rho= 6. To measure the robustness of the sampling methods, we compare the eigenvalue spectra, the eigenvector similarity and the relaxation times of the total energy of the full and sample PESs.

Algorithms↗
PubMed1985

Complex membrane transport systems. A non-Markovian approach.

This paper suggests a method of how to deal with complex membrane transport systems such as ion channels or ion pumps formed by proteins. The complexity of these systems results from the fact that proteins may undergo an internal dynamics of conformational changes and may thereby affect the transmembrane transport. Usually, complex transport systems are mapped into multi-state graphs and couched in terms of Markovian master equations. It is shown in this paper how the dimensionality of such multi-state systems can be reduced. The resulting description may be expressed in the form of a generalized master equation with a memory function as integral kernel. The memory function reflects the protein's own dynamics and its overall effect on the transport. This formalism, non-Markovian in nature, is applied to describe the time-dependent action of ion pumps. A general model is constructed on the basis of the rate theory which contains all the essential parts of ion pumps such as a catalytic unit and a channel-like conduit for ion translocation and which is still analytically tractable. The short-time behaviour of the pumping process turns out to be of particular interest, since it reveals the dynamics of the catalytic unit itself. A strong correlation of the particle's motion over times less than a certain correlation time has been found. This result is compared with experimental findings on the proton pump of Halobacterium halobium. It is concluded that such a perfect short-time memory could be a generic property of active transport systems.

Adenosine Triphosphatases↗
PubMed2005

Critical dynamics of dimers: implications for the glass transition.

The Adam-Gibbs view of the glass transition relates the relaxation time to the configurational entropy, which goes continuously to zero at the so-called Kauzmann temperature. We examine this scenario in the context of a dimer model with an entropy-vanishing phase transition and stochastic loop dynamics. We propose a coarse-grained master equation for the order parameter dynamics which is used to compute the time-dependent autocorrelation function and the associated relaxation time. Using a combination of exact results, scaling arguments, and numerical diagonalizations of the master equation, we find nonexponential relaxation and a Vogel-Fulcher divergence of the relaxation time in the vicinity of the phase transition. Since in the dimer model the entropy stays finite all the way to the phase transition point and then jumps discontinuously to zero, we demonstrate a clear departure from the Adam-Gibbs scenario. Dimer coverings are the "inherent structures" of the canonical frustrated system, the triangular Ising antiferromagnet. Therefore, our results provide a new scenario for the glass transition in supercooled liquids in terms of inherent structure dynamics.

Journal Article↗
PubMed2006

Fewest switches adiabatic surface hopping as applied to vibrational energy relaxation.

In this contribution quantum/classical surface hopping methodology is applied to vibrational energy relaxation of a quantum oscillator in a classical heat bath. The model of a linearly damped (harmonic) oscillator is chosen which can be mapped onto the Brownian motion (Caldeira-Leggett) Hamiltonian. In the simulations Tully's fewest switches surface hopping scheme is adopted with inclusion of dephasing in the adiabatic basis using a simple decoherence algorithm. The results are compared to the predictions of a Redfield-type quantum master equation modeling using the classical heat bath force correlation function as input. Thereby a link is established between both types of quantum/classical approaches. Viewed from the latter perspective, surface hopping with dephasing may be interpreted as "on-the-fly" stochastic realization of a quantum/classical Pauli master equation.

Half-Life↗
PubMed2004

Theory of photon statistics and optical coherence in a multiple-scattering random-laser medium.

We derive the photon-number probability distribution and the resulting degree of second-order optical coherence for light emission from a uniformly distributed active species within a multiple-light-scattering medium. This is obtained from a master equation describing the probability distribution for photons in the vicinity of position r, traveling with a wave vector k, related, in turn, to a coarse-grained average of the optical Wigner coherence function. Using a simple model for isotropic, spatially uncorrelated scatterers, this reduces to a generalization of the master equation of a conventional laser in which the medium behaves like a random collection of low-quality factor cavities that are coupled by photon diffusion between a given cavity and its neighbors. Laserlike coherence, on average, is obtained in the random laser above a specific pumping threshold. Photon-number statistics above and below the lasing threshold are computed by first assuming that the atomic response to the local electromagnetic fields is nearly instantaneous. Corrections to this simple model, arising from nonadiabatic atomic dynamics, are then estimated. The dependence of the photon statistics on scatterer density, gain concentration, and position within a sample reveal that, on average, increase of the scattering strength (decrease of the photon transport mean free path) in the medium leads to a sharper peak in the local photon-number distribution, characteristic of increased local coherence in the optical field. We also evaluate the coherence of the output field at points outside the random-laser medium. This is a weighted average of radiation emitted at different positions in the sample, exhibiting varying degrees of coherence due to variations in the local pumping intensity.

Journal Article↗
PubMed2006

Model for proton transport coupled to protein conformational change: application to proton pumping in the bacteriorhodopsin photocycle.

A modeling method is presented for protein systems in which proton transport is coupled to conformational change, as in proton pumps and in motors driven by the proton-motive force. Previously developed methods for calculating pKa values in proteins using a macroscopic dielectric model are extended beyond the equilibrium case to a master-equation model for the time evolution of the system through states defined by ionization microstate and a discrete set of conformers. The macroscopic dielectric model supplies free energy changes for changes of protonation microstate, while the method for obtaining the energetics of conformational change and the relaxation rates, the other ingredients needed for the master equation, are system dependent. The method is applied to the photoactivated proton pump, bacteriorhodopsin, using conformational free energy differences from experiment and treating relaxation rates through three adjustable parameters. The model is found to pump protons with an efficiency relatively insensitive to parameter choice over a wide range of parameter values, and most of the main features of the known photocycle from very early M to the return to the resting state are reproduced. The boundaries of these parameter ranges are such that short-range proton transfers are faster than longer-range ones, which in turn are faster than conformational changes. No relaxation rates depend on conformation. The results suggest that an "accessibility switch", while not ruled out, is not required and that vectorial proton transport can be achieved through the coupling of the energetics of ionization and conformational states.

Bacteriorhodopsins↗
PubMed2001

Test particle in a quantum gas.

A master equation with a Lindblad structure is derived, which describes the interaction of a test particle with a macroscopic system and is expressed in terms of the operator valued dynamic structure factor of the system. In the case of a free Fermi or Bose gas the result is evaluated in the Brownian limit, thus obtaining a single generator master equation for the description of quantum Brownian motion in which the correction due to quantum statistics is explicitly calculated. The friction coefficients for Boltzmann and Bose or Fermi statistics are compared.

Journal Article↗
PubMed2001

Stochastic effects in a thermochemical system with Newtonian heat exchange.

We develop a mesoscopic description of stochastic effects in the Newtonian heat exchange between a diluted gas system and a thermostat. We explicitly study the homogeneous Semenov model involving a thermochemical reaction and neglecting consumption of reactants. The master equation includes a transition rate for the thermal transfer process, which is derived on the basis of the statistics for inelastic collisions between gas particles and walls of the thermostat. The main assumption is that the perturbation of the Maxwellian particle velocity distribution can be neglected. The transition function for the thermal process admits a continuous spectrum of temperature changes, and consequently, the master equation has a complicated integro-differential form. We perform Monte Carlo simulations based on this equation to study the stochastic effects in the Semenov system in the explosive regime. The dispersion of ignition times is calculated as a function of system size. For sufficiently small systems, the probability distribution of temperature displays transient bimodality during the ignition period. The results of the stochastic description are successfully compared with those of direct simulations of microscopic particle dynamics.

Journal Article↗
PubMed1978

Current noise around steady states in discrete transport systems.

Subject of this paper is the transport noise in discrete systems. The transport systems are given by a number (n) of binding sites separated by energy barriers. These binding sites may be in contact outer reservoirs. The state of the systems is characterized by the occupation numbers of particles (current carriers) at these binding sites. The change in time of the occupation numbers is generated by individual "jumps" of particles over the energy barriers, building up the flux matter (for charged particles: the electric current). In the limit n leads to infinity continuum processes as e.g. usual diffusion are included in the transport model. The fluctuations in occupation numbers and other quantities linearly coupled to the occupation numbers may be treated with the usual master equation approach. The treatment of the fluctuation in fluxes (current) makes necessary a different theoretical approach which is presented in this paper under the assumption of vanishing interactions between the particles. This approach may be applied to a number of different transport systems in biology and physics (ion transport through porous channels in membranes, carriers mediated ion transport through membranes, jump diffusion e.g. in superionic conductors). As in the master equation approach the calculation of correlations and noise spectra may be reduced to the solution of the macroscopic equations for the occupation numbers. This result may be regarded as a generalization to non-equilibrium current fluctuations of the usual Nyquist theorem relating the current (voltage) noise spectrum in thermal equilibrium to the macroscopic frequency dependent admittance. The validity of the general approach is demonstrated by the calculation of the autocorrelation function and spectrum of current noise for a number of special examples (e.g. pores in membranes, carrier mediated ion transport).

Binding Sites↗
PubMed2006

Homogeneity and Markovity of electronic dephasing in liquid solutions.

The electronic dephasing dynamics of a solvated chromophore is formulated in terms of a non-Markovian master equation. Within this formulation, one describes the effect of the nuclear degrees of freedom on the electronic degrees of freedom in terms of a memory kernel function, which is explicitly dependent on the initial solvent configuration. In the case of homogeneous dynamics, this memory kernel becomes independent of the initial configuration. The Markovity of the dephasing process is also the most conveniently explored by comparing the results obtained via the non-Markovian master equation to these obtained via its Markovian counterpart. The homogeneous memory kernel is calculated for a two-state chromophore in liquid solution, and used to explore the sensitivity of photon echo signals to the heterogeneity and non-Markovity of the underlying solvation dynamics.

Journal Article↗
PubMed2004

Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases.

Bi-stable chemical systems are the basic building blocks for intracellular memory and cell fate decision circuits. These circuits are built from molecules, which are present at low copy numbers and are slowly diffusing in complex intracellular geometries. The stochastic reaction-diffusion kinetics of a double-negative feedback system and a MAPK phosphorylation-dephosphorylation system is analysed with Monte-Carlo simulations of the reaction-diffusion master equation. The results show the geometry of intracellular reaction compartments to be important both for the duration and the locality of biochemical memory. Rules for when the systems lose global hysteresis by spontaneous separation into spatial domains in opposite phases are formulated in terms of geometrical constraints, diffusion rates and attractor escape times. The analysis is facilitated by a new efficient algorithm for exact sampling of the Markov process corresponding to the reaction-diffusion master equation.

Adaptation, Physiological↗
PubMed2005

Zero-range model of traffic flow.

A multicluster model of traffic flow is studied, in which the motion of cars is described by a stochastic master equation. Assuming that the escape rate from a cluster depends only on the cluster size, the dynamics of the model is directly mapped to the mathematically well-studied zero-range process. Knowledge of the asymptotic behavior of the transition rates for large clusters allows us to apply an established criterion for phase separation in one-dimensional driven systems. The distribution over cluster sizes in our zero-range model is given by a one-step master equation in one dimension. It provides an approximate mean-field dynamics, which, however, leads to the exact stationary state. Based on this equation, we have calculated the critical density at which phase separation takes place. We have shown that within a certain range of densities above the critical value a metastable homogeneous state exists before coarsening sets in. Within this approach we have estimated the critical cluster size and the mean nucleation time for a condensate in a large system. The metastablity in the zero-range process is reflected in a metastable branch of the fundamental flux-density diagram of traffic flow. Our work thus provides a possible analytical description of traffic jam formation as well as important insight into condensation in the zero-range process.

Journal Article↗
PubMed2000

Class of integrable diffusion-reaction processes

We consider a process in which there are two types of particles, A and B, on an infinite one-dimensional lattice. The particles hop to their adjacent sites, like the totally asymmetric exclusion process (ASEP), and have also the following interactions: A+B-->B+B and B+A-->B+B, which all occur with equal rate. We study this process by imposing four boundary conditions on the ASEP master equation. It is shown that this model is integrable, in the sense that its N-particle S matrix is factorized into a product of two-particle S matrices and, more importantly, the two-particle S matrix satisfies the quantum Yang-Baxter equation. Using the coordinate Bethe-ansatz, the N-particle wave functions and the two-particle conditional probabilities are found exactly. Further, by imposing four reasonable physical conditions on two-species diffusion-reaction processes (where the most important ones are the equality of the reaction rates and the conservation of the number of particles in each reaction), we show that among the 4096 types of interactions which have these properties and can be modeled by a master equation and an appropriate set of boundary conditions, there are only 28 independent interactions which are integrable. We find all these interactions and also their corresponding wave functions. Some of these may be new solutions of the quantum Yang-Baxter equation.

Journal Article↗
PubMed2005

Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions.

The dynamical solution of a well-mixed, nonlinear stochastic chemical kinetic system, described by the Master equation, may be exactly computed using the stochastic simulation algorithm. However, because the computational cost scales with the number of reaction occurrences, systems with one or more "fast" reactions become costly to simulate. This paper describes a hybrid stochastic method that partitions the system into subsets of fast and slow reactions, approximates the fast reactions as a continuous Markov process, using a chemical Langevin equation, and accurately describes the slow dynamics using the integral form of the "Next Reaction" variant of the stochastic simulation algorithm. The key innovation of this method is its mechanism of efficiently monitoring the occurrences of slow, discrete events while simultaneously simulating the dynamics of a continuous, stochastic or deterministic process. In addition, by introducing an approximation in which multiple slow reactions may occur within a time step of the numerical integration of the chemical Langevin equation, the hybrid stochastic method performs much faster with only a marginal decrease in accuracy. Multiple examples, including a biological pulse generator and a large-scale system benchmark, are simulated using the exact and proposed hybrid methods as well as, for comparison, a previous hybrid stochastic method. Probability distributions of the solutions are compared and the weak errors of the first two moments are computed. In general, these hybrid methods may be applied to the simulation of the dynamics of a system described by stochastic differential, ordinary differential, and Master equations.

Algorithms↗
PubMed2002

Extinction dynamics in mainland-island metapopulations: an N-patch stochastic model.

A generalization of the well-known Levins' model of metapopulations is studied. The generalization consists of (i) the introduction of immigration from a mainland, and (ii) assuming the dynamics is stochastic, rather than deterministic. A master equation, for the probability that n of the patches are occupied, is derived and the stationary probability Ps(n), together with the mean and higher moments in the stationary state, determined. The time-dependence of the probability distribution is also studied: through a Gaussian approximation for general n when the boundary at n = 0 has little effect, and by calculating P(0, t), the probability that no patches are occupied at time t, by using a linearization procedure. These analytic calculations are supplemented by carrying out numerical solutions of the master equation and simulations of the stochastic process. The various approaches are in very good agreement with each other. This allows us to use the forms for Ps(0) and P(0,t) in the linearization approximation as a basis for calculating the mean time for a metapopulation to become extinct. We give an analytical expression for the mean time to extinction derived within a mean field approach. We devise a simple method to apply our mean field approach even to complex patch networks in realistic model metapopulations. After studying two spatially extended versions of this nonspatial metapopulation model--a lattice metapopulation model and a spatially realistic model--we conclude that our analytical formula for the mean extinction time is generally applicable to those metapopulations which are really endangered, where extinction dynamics dominates over local colonization processes. The time evolution and, in particular, the scope of our analytical results, are studied by comparing these different models with the analytical approach for various values of the parameters: the rates of immigration from the mainland, the rates of colonization and extinction, and the number of patches making up the metapopulation.

Animals↗