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 235 records · Page 13Linked to original sources
PubMed1981

Random walk theory of a trap-controlled hopping transport process.

A random walk theory of hopping motion in the presence of a periodic distribution of traps is presented. The solution of the continuous-time random walk equations is exact and valid for arbitrary intersite interactions and trap concentration. The treatment is shown to be equivalent to an exact solution of the master equation for this trapping problem. These interactions can be a general function of electric field and are not restricted to nearest neighbors. In particular, with the inclusion of trap-to-trap interactions, as well as trap-to-host interactions, an exact treatment of the change from one hopping channel to another has been obtained. The trap-modulated propagator has been derived in terms of a type of Green's function that is introduced. The results are specialized to spatial moments of the propagator, from which expressions for the drift velocity and diffusion coefficient are obtained. Numerical results for the drift velocity are presented and shown to account for the change in hopping channels in recent transport measurements in mixed molecularly doped polymers.

Journal Article↗
PubMed2005

Theory of nonlinear dispersive waves and selection of the ground state.

A theory of time-dependent nonlinear dispersive equations of the Schrödinger or Gross-Pitaevskii and Hartree type is developed. The short, intermediate and large time behavior is found, by deriving nonlinear master equations (NLME), governing the evolution of the mode powers, and by a novel multitime scale analysis of these equations. The scattering theory is developed and coherent resonance phenomena and associated lifetimes are derived. Applications include Bose-Einstein condensate large time dynamics and nonlinear optical systems. The theory reveals a nonlinear transition phenomenon, "selection of the ground state," and NLME predicts the decay of excited state, with half its energy transferred to the ground state and half to radiation modes. Our results predict the recent experimental observations of Mandelik et al. in nonlinear optical waveguides.

Journal Article↗
PubMed1978

Stochastic theory of ligand migration in biomolecules.

When ligand binding to proteins involves the presence of more than one ligand inside a given biomolecule, linear deterministic rate equations become useless. A stochastic approach, however, permits a treatment of the migration and binding of small molecules to proteins even at high ligand concentrations. An appropriate linear master equation and its analytic solution are given. As an example, the binding of carbon monoxide to myoglobin at partial pressures from 1 to 10(3) bars (0.1 to 100 MPa) is treated.

Binding Sites↗
PubMed2006

Coarse-graining a restricted solid-on-solid model.

A procedure suggested by Vvedensky for obtaining continuum equations as the coarse-grained limit of discrete models is applied to the restricted solid-on-solid model with both adsorption and desorption. Using an expansion of the master equation, discrete Langevin equations are derived; these agree quantitatively with direct simulation of the model. From these, a continuum differential equation is derived, and the model is found to exhibit either Edwards-Wilkinson or Kardar-Parisi-Zhang exponents, as expected from symmetry arguments. The coefficients of the resulting continuum equation remain well-defined in the coarse-grained limit.

Journal Article↗
PubMed2004

Accurate discretization of advection-diffusion equations.

We present an exact mathematical transformation which converts a wide class of advection-diffusion equations into a form allowing simple and direct spatial discretization in all dimensions, and thus the construction of accurate and more efficient numerical algorithms. These discretized forms can also be viewed as master equations which provide an alternative mesoscopic interpretation of advection-diffusion processes in terms of diffusion with spatially varying hopping rates.

Journal Article↗
PubMed2006

The interplay between discrete noise and nonlinear chemical kinetics in a signal amplification cascade.

We used various analytical and numerical techniques to elucidate signal propagation in a small enzymatic cascade which is subjected to external and internal noises. The nonlinear character of catalytic reactions, which underlie protein signal transduction cascades, renders stochastic signaling dynamics in cytosol biochemical networks distinct from the usual description of stochastic dynamics in gene regulatory networks. For a simple two-step enzymatic cascade which underlies many important protein signaling pathways, we demonstrated that the commonly used techniques such as the linear noise approximation and the Langevin equation become inadequate when the number of proteins becomes too low. Consequently, we developed a new analytical approximation, based on mixing the generating function and distribution function approaches, to the solution of the master equation that describes nonlinear chemical signaling kinetics for this important class of biochemical reactions. Our techniques work in a much wider range of protein number fluctuations than the methods used previously. We found that under certain conditions the burst phase noise may be injected into the downstream signaling network dynamics, resulting possibly in unusually large macroscopic fluctuations. In addition to computing first and second moments, which is the goal of commonly used analytical techniques, our new approach provides the full time-dependent probability distributions of the colored non-Gaussian processes in a nonlinear signal transduction cascade.

Cytosol↗
PubMed2001

Derivation of continuum stochastic equations for discrete growth models.

We present a formalism to derive the stochastic differential equations (SDEs) for several solid-on-solid growth models. Our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion. We then write the master equations for these corresponding particle systems and find the SDEs for the particle densities. Finally, by connecting the particle densities with the growth heights, we derive the SDEs for the height variables. Applying this formalism to discrete growth models, we find the Edwards-Wilkinson equation for the symmetric body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation for the asymmetric BCSOS model and the generalized restricted solid-on-solid (RSOS) model, and the Villain-Lai-Das Sarma equation for the conserved RSOS model. In addition to the consistent forms of equations for growth models, we also obtain the coefficients associated with the SDEs.

Journal Article↗
PubMed2004

Quantitative characterization of pore-scale disorder effects on transport in "homogeneous" granular media.

Breakthrough curves (BTC) of a passive tracer in macroscopically homogeneous granular materials (well-sorted, unconsolidated sands or glass beads) were measured in a series of column experiments. The early and late arrival times are observed to differ systematically from theoretical predictions based on solution of the advective-dispersion equation for uniform porous media. We propose that subtle and residual pore-scale disorder effects in the porous media can account for these observations. We determine an ensemble-averaged distribution of particle transfer rates (based on a master equation for the local flux-averaged concentration) which incorporates these effects, and utilize it to calculate BTC that are in excellent agreement with the entire series of observations. Theoretical prediction of the dependence of the effective macroscopic parameters on measurable quantities is also in excellent agreement with the observations.

Journal Article↗
PubMed2006

Diffusion equations for a Markovian jumping process.

We consider a Markovian jumping process which is defined in terms of the jump-size distribution and the waiting-time distribution with a position-dependent frequency, in the diffusion limit. We assume the power-law form for the frequency. For small steps, we derive the Fokker-Planck equation and show the presence of the normal diffusion, subdiffusion, and superdiffusion. For the Lévy distribution of the step size, we construct a fractional equation, which possesses a variable coefficient, and solve it in the diffusion limit. Then we calculate fractional moments and define the fractional diffusion coefficient as a natural extension to the cases with the divergent variance. We also solve the master equation numerically and demonstrate that there are deviations from the Lévy stable distribution for large wave numbers.

Journal Article↗
PubMed2002

Nonequilibrium coupled Brownian phase oscillators.

A model of globally coupled phase oscillators under equilibrium (driven by Gaussian white noise) and nonequilibrium (driven by symmetric dichotomic fluctuations) is studied. For the equilibrium system, the mean-field state equation takes a simple form and the stability of its solution is examined in the full space of order parameters. For the nonequilbrium system, various asymptotic regimes are obtained in a closed analytical form. In a general case, the corresponding master equations are solved numerically. Moreover, the Monte Carlo simulations of the coupled set of Langevin equations of motion is performed. The phase diagram of the nonequilibrium system is presented. For the long time limit, we have found five regimes. Three of them can be obtained from the mean-field theory. One of them, the oscillating regime, cannot be predicted by the mean-field method and has been detected in the Monte Carlo numerical experiments.

Journal Article↗
PubMed2004

Simple jumping process with memory: transport equation and diffusion.

We present a stochastic jumping process, defined in terms of jump-size probability density and jumping rate, which is a generalization of the well-known kangaroo process. The definition takes into account two process values: after and before the jump. Therefore, the process is able to preserve memory about its previous values. It possesses a simple stationary limit. Its master equation is interpreted as the kinetic equation with variable collision rate. The process can be easily applied to model systems which relax to distributions other than Maxwellian. The case of a constant jumping rate corresponds to the diffusion process, either normal or ballistic.

Journal Article↗
PubMed2004

Modeling and simulation of intracellular dynamics: choosing an appropriate framework.

Systems biology is a reemerging paradigm which, among other things, focuses on mathematical modeling and simulation of biochemical reaction networks in intracellular processes. For most simulation tools and publications, they are usually characterized by either preferring stochastic simulation or rate equation models. The use of stochastic simulation is occasionally accompanied with arguments against rate equations. Motivated by these arguments, we discuss in this paper the relationship between these two forms of representation. Toward this end, we provide a novel compact derivation for the stochastic rate constant that forms the basis of the popular Gillespie algorithm. Comparing the mathematical basis of the two popular conceptual frameworks of generalized mass action models and the chemical master equation, we argue that some of the arguments that have been put forward are ignoring subtle differences and similarities that are important for answering the question in which conceptual framework one should investigate intracellular dynamics.

Algorithms↗
PubMed2003

Fractional dynamics from the ordinary Langevin equation.

We consider the usual Langevin equation depending on an internal time. This parameter is substituted by a first passage time of a self-similar Markov process. Then the Gaussian process is parent, and the hitting time process is directing. The probability to find the resulting process at the real time is defined by the integral relationship between the probability densities of the parent and directing processes. The corresponding master equation becomes the fractional Fokker-Planck equation. We show that the resulting process has non-Markovian properties, all its moments are finite, the fluctuation-dissipation relation and the H-theorem hold.

Journal Article↗
PubMed2006

Calibration of HbA1c and its measurement in the presence of variant haemoglobins: report on questionnaire to manufacturers.

AIMS: To review 'Diabetes Control and Complications Trial (DCCT)-aligned' HbA(1c) reporting in UK, use of individual/network equations relating IFCC calibration to 'DCCT alignment', and whether HbA(1c) in the presence of variant haemoglobins is, according to manufacturers, suitable for current, clinical guidelines. METHODS: Questionnaire sent to nine manufacturers and responses analysed. RESULTS: All methods were certified as 'DCCT-aligned' by National Glycohemoglobin Standardization Program (NGSP); UK EQA schemes reported 95% of results 'DCCT-aligned' in December 2004. The master equation relating networks was used by six manufacturers and specific equations for individual methods by three. HbA(1c) results from laboratory/point of care testing analysers can be affected by variant haemoglobins including elevated HbF; only IE HPLC (and LPLC) detect their presence. If chromatographic separation is ideal in heterozygous patients, laboratories either choose not to report HbA(1c) and propose another strategy for monitoring glycaemia, or report HbA(1c) and issue a caution that it may not be appropriate for guidelines. HbA(1c) reported from immunochemistry or affinity chromatography in presence of variant haemoglobins, may not be reliable for use with clinical guidelines. CONCLUSIONS: For clinical care, HbA(1c) must reflect its relationship to glycaemia in clinical trials underpinning national guidelines. A flowchart to establish if HbA(1c) measurement is appropriate has been produced for use in a clinical setting.

Blood Chemical Analysis↗
PubMed2003

Towards deterministic equations for Lévy walks: the fractional material derivative.

Lévy walks are random processes with an underlying spatiotemporal coupling. This coupling penalizes long jumps, and therefore Lévy walks give a proper stochastic description for a particle's motion with broad jump length distribution. We derive a generalized dynamical formulation for Lévy walks, in which the fractional equivalent of the material derivative occurs. Our approach is expected to be useful for the dynamical formulation of Lévy walks in an external force field or in phase space, for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited.

Journal Article↗
PubMedDate not supplied

Dynamics of excitions created by a single picosecond pulse.

A theoretical analysis of bimolecular annihilation in finite domains is presented. A Pauli master equation is formulated for the case of varying incident delta function excitation sources. Expressions for the quantum fluorescence yield and its time dependence are derived. The relationship between the fluorescence yield and the number of hits per domain depends on two parameters: the rate constant of bimolecular exciton annihilation and the dimension of the domain in which this annihilation occurs. Recent experimental results imply that the exciton diffusion constant (D) is large (D approximately to greater than 10(-3) cm2 S-1) and that the photosystem II domains may contain as many as five photosynthetic units. An analysis of the time decay of the fluorescence indicates that, for a few hits per domain, the decay may be considered as exponential but for many hits it becomes non-exponential. Thus the fluorescence decay depends on the intensity of the excitation source and/or on the dimension of the domains. Conditions which change the effective size of the domain may change the shape of the fluorescence decay. Some biological consequences and experimental applications of this theory are presented.

Kinetics↗
PubMed2003

Computing the transition state populations in simple protein models.

We describe the master equation method for computing the kinetics of protein folding. We illustrate the method using a simple Go model. Presently most models of two-state fast-folding protein folding kinetics invoke the classical idea of a transition state to explain why there is a single exponential decay in time. However, if proteins fold via funnel-shaped energy landscapes, as predicted by many theoretical studies, then it raises the question of what is the transition state. Is it a specific structure, or a small ensemble of structures, as is expected from classical transition state theory? Or is it more like the denatured states of proteins, a very broad ensemble? The answer that is usually obtained depends on the assumptions made about the transition state. The present method is a rigorous way to find transition states, without assumptions or approximations, even for very nonclassical shapes of energy landscapes. We illustrate the method here, showing how the transition states in two-state protein folding can be very broad ensembles.

Kinetics↗
PubMed2006

Multipeak distributions of first passage times in bistable dynamics in a model of a thermochemical system.

A master equation is used to study transitions between the stable limit cycle and stable focus in the two-variable bistable system. The distribution function of the mean first passage time between these attractors and the relative dispersion of the mean first return time from the stable focus to itself as a function of the intensity of fluctuations are calculated and discussed. A coherence resonance is observed for the return time from the focus to itself.

Journal Article↗