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Chemical master equation simulation

Gillespie s algorithm numerically reproduces the solution of the chemical master equation, simulating the individual occurrences of reactions. This type of description is called a jump Markov process, a type of stochastic process. A jump Markov process describes a system that has a probability of discontinuously transitioning from one state to another. This type of algorithm is also known as kinetic Monte Carlo. An ensemble of simulation trajectories in state space is required to accurately capture the probabilistic nature of the transient behavior of the system. [Pg.297]

To use a computer to simulate a stochastic trajectory of the chemical master equation such as described in Figure 11.4, one must establish the rules of how to move the system from one grid point to its neighboring points. The essential idea is to draw random moves from the appropriate distribution and to assign random times (also drawn from the appropriate distribution) to each move. Thus each simulation step in the simulation involves two random numbers, one to determine the associated time step and one to determine the grid move. [Pg.276]

It is widely appreciated that chemical and biochemical reactions in the condensed phase are stochastic. It has been more than 60 years since Delbriick studied a stochastic chemical reaction system in terms of the chemical master equation. Kramers theory, which connects the rate of a chemical reaction with the molecular structures and energies of the reactants, is established as a central component of theoretical chemistry [77], Yet study of the dynamics of chemical and biochemical reaction systems, in terms of either deterministic differential equations or the stochastic CME, is not the exclusive domain of chemists. Recent developments in the simulation of reaction systems are the work of many sorts of scientists, ranging from control engineers to microbiologists, all interested in the dynamic behavior of biochemical reaction systems [199, 210],... [Pg.280]

Given the simplicity of the current system, it was possible to analytically solve the resulting chemical master equation. However, this is not always the case and one is limited to simulating individual realizations of the stochastic process in order to reconstruct the probability distributions out from several simulations. Below, we introduce the celebrated Gillespie algorithm (Gillespie 1977) to simulate the stochastic evolution of continuous-time discrete-state stochastic processes, like the one analyzed in the present chapter. [Pg.21]

MC is also successful in far from equilibrium processes encountered in the areas of diffusion and reaction. It is precisely this class of non-equilibrium reaction/diffusion problems that is of interest here. Chemical engineering applications of MC include crystal growth (this is probably one of the first areas where physicists applied MC), catalysis, reaction networks, biology, etc. MC simulations provide the stochastic solution to a time-dependent master equation... [Pg.10]

Direct solution of the master equation is impractical because of the huge number of equations needed to describe all possible states (combinations) even of relatively small-size systems. As one example, for a three-step linear pathway among 100 molecules, 104 such equations are needed. As another example, in biological simulation for the tumor suppressor p53, 211 states are estimated for the monomer and 244 for the tetramer (Rao et al., 2002). Instead of following all individual states, the MC method is used to follow the evolution of the system. For chemically reacting systems in a well-mixed environment, the foundations of stochastic simulation were laid down by Gillespie (1976, 1977). More... [Pg.10]

Figure 11.7 A schematic illustration of the Monte Carlo simulation method for computing the stochastic trajectories of a chemical reaction system following the CME. Two random numbers, r and r2, are sampled from a uniform distribution to simulate each stochastic step r determines when to move, and r2 determines where to move. For a given state of a master equation graph shown in the upper panel, there are four outward reactions, labeled 1-4, each with their corresponding rate constants q, (i = 1, 2, , 4). The upper and lower panels illustrate, respectively, the calculation of the random time T associated with a stochastic move, and the probability pm of moving to state m. Figure 11.7 A schematic illustration of the Monte Carlo simulation method for computing the stochastic trajectories of a chemical reaction system following the CME. Two random numbers, r and r2, are sampled from a uniform distribution to simulate each stochastic step r determines when to move, and r2 determines where to move. For a given state of a master equation graph shown in the upper panel, there are four outward reactions, labeled 1-4, each with their corresponding rate constants q, (i = 1, 2, , 4). The upper and lower panels illustrate, respectively, the calculation of the random time T associated with a stochastic move, and the probability pm of moving to state m.
Then numerical methods of matrix diagonalization are used to find the eigenvalues of the matrix operator 0)(P —I) — K, which are the time constants that determine both the chemical kinetics and the energy relaxation. Part three of this work deals in detail with the formulation of the Master Equation for a number of different systems, for example termolecular association reactions and reversible reactions. It then deals with methods for finding the time constants and simulating the kinetics. The Master Equation is the method of choice at present for modelling the competition between energy transfer and reaction. [Pg.51]

It is often stated that MC methods lack real time and results are usually reported in MC events or steps. While this is immaterial as far as equilibrium is concerned, following real dynamics is essential for comparison to solutions of partial differential equations and/or experimental data. It turns out that MC simulations follow the stochastic dynamics of a master equation, and with appropriate parameterization of the transition probabilities per unit time, they provide continuous time information as well. For example, Gillespie has laid down the time foundations of MC for chemical reactions in a spatially homogeneous system.f His approach is easily extendable to arbitrarily complex computational systems when individual events have a prescribed transition probability per unit time, and is often referred to as the kinetic Monte Carlo or dynamic Monte Carlo (DMC) method. The microscopic processes along with their corresponding transition probabilities per unit time can be obtained via either experiments such as field emission or fast scanning tunneling microscopy or shorter time scale DFT/MD simulations discussed earlier. The creation of a database/lookup table of transition... [Pg.1718]

Not being able to solve the master equation in the more general cases we are often satisfied by the determination of the first and second moments. Furthermore, different techniques can be applied to approximate the jump processes by continuous processes, which are more easily solvable. The clear structure of the stochastic model of chemical reactions allows the possibility of simulating the reaction. By simulation procedures realisations of the processes can be obtained. The methods for obtaining solutions will be illustrated by discussing particular examples. [Pg.105]

As was mentioned earlier the master equation (5.37) generally cannot be solved. To get some experience of the behaviour of chemical systems we might do stochastic simulation experiments using Monte-Carlo techniques (Introductions to Monte-Carlo methods are given in Hammersby Hand-scomb 1964, and Srejder 1965. Their applications in chemical physics are discussed in Binder (1979.)... [Pg.112]

As we have seen in the previous sectimis, the definition of a reasonably accurate kinetic master equation from the microscopic structure can be achieved by combining atomistic simulations, quantum chemical calculatirais, and microelectrostatic... [Pg.169]

As already mentioned, there is at present no direct derivation of Equation (2) from first principles. We shall therefore adopt the master equation description as a working model and justify, whenever possible, its predictions through the comparison with microscopic simulations. We notice that the reduction of chaotic dynamics to a Markov process can be justified rigorously in certain classes of discrete time mappings [12,13]. It would certainly be interesting to extend this line of approach to chaotic mappings of more direct chemical relevance. [Pg.577]

The integral in equation (6.2) was evaluated numerically for the retardation spectra obtained from master curves, together with ar from the time-temperature superposition. A typical result is demonstrated in Fig. 6.3(a), where recovery is shown for the DBDI/PTHFeso polymer, for a simulated constant rate of heating r = 0.1 K/s. This Figure also illustrates the definitions of two parameters used to compare the responses the temperature of maximum recovery rate Tmax and the width of the recovery window AT. These parameters of the SMPs were compared with respect to chemical composition and crosslink density ric, expressed as the number of moles of 3-point star crosslinks, per 100 g of polymer. [Pg.224]


See other pages where Chemical master equation simulation is mentioned: [Pg.51]    [Pg.66]    [Pg.29]    [Pg.148]    [Pg.241]    [Pg.739]    [Pg.7]    [Pg.92]    [Pg.348]    [Pg.192]    [Pg.3141]    [Pg.445]    [Pg.236]    [Pg.298]    [Pg.77]    [Pg.798]    [Pg.253]   
See also in sourсe #XX -- [ Pg.276 ]




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