Gillespie equation

A final comment on the interpretation of stochastic simulations We are so accustomed to writing continuous functions—differential and integrated rate equations, commonly called deterministic rate equations—that our first impulse on viewing these stochastic calculations is to interpret them as approximations to the familiar continuous functions. However, we have got this the wrong way around. On a molecular level, events are discrete, not continuous. The continuous functions work so well for us only because we do experiments on veiy large numbers of molecules (typically 10 -10 ). If we could experiment with very much smaller numbers of molecules, we would find that it is the continuous functions that are approximations to the stochastic results. Gillespie has developed the stochastic theory of chemical kinetics without dependence on the deterministic rate equations. 

Ghiorso, Albert, 515 Giauque, William, 174 Gibbs, J. Willard, 459 Gibbs-Helmholtz equation The relation AG = AH - TAS, 459,461,474q Gillespie, R. J., 175 Glucose... 

RNApolymerase molecules are involved in the process. If the system is well stirred so that spatial degrees of freedom play no role, birth-death master equation approaches have been used to describe such reacting systems [33, 34]. The master equation can be simulated efficiently using Gillespie s algorithm [35]. However, if spatial degrees of freedom must be taken into account, then the construction of algorithms is still a matter of active research [36-38]. 

Gillespie D.T. Gillespie, Markov Processes (Academic Press, San Diego 1992). coffey, kalmykov and waldron W.T. Coffey, Yu.P. Kalmykov, and J.T. Waldron, The Langevin Equation (2nd edition, World Scientific, 2004). 

Gillespie Mid GrahMn [42] have carried out a cryometric exMnination of solutions of nitric acid in oleum. The results obtained Me in agreement with the following equation, postulating formation of the nitronium ion ... 

A computational method was developed by Gillespie in the 1970s [381, 388] from premises that take explicit account of the fact that the time evolution of a spatially homogeneous process is a discrete, stochastic process instead of a continuous, deterministic process. This computational method, which is referred to as the stochastic simulation algorithm, offers an alternative to the Kolmogorov differential equations that is free of the difficulties mentioned above. The simulation algorithm is based on the reaction probability density function defined below. 

It has been shown from cryoscopic (Dean et al., 1970) and conducti-metric (Gillespie and Moss, 1966) measurements that SbF6 is a strong acid in HF ionizing in dilute solutions according to equation (13). 

A major difficulty is that such hierarchies of molecular models are not exactly known. Recent work by Gillespie (2000, 2002) has established such a hierarchy for stochastic models of chemical reactions in a well-mixed batch reactor. This hierarchy is depicted in Fig. 3b. In particular, it was shown that the chemical master equation is deduced to a chemical Langevin equation when the population sizes are relatively large. Finally, the deterministic behavior can be... 

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... 

It is usually not possible (and never easy ) to solve equations such as Equation (11.27) analytically. So computational simulation of the stochastic trajectories are necessary. The numerical method to obtain stochastic trajectories by Monte Carlo sampling, which we shall discuss in Section 11.4.4, is known as the Gillespie algorithm [68], However, it happens that the steady state of Equation (11.27) can be obtained in closed form. This is because in steady state, the probability of leaving state 0, v0po has to exactly balance the probability of entering state 0 from state 1, wopi. Similarly, since v0po = vjqp, we have v p = w p2, and so on ... 

Figure 11.8 An example of the stochastic trajectory from Monte Carlo simulation according to the Gillespie algorithm for reaction system given in Equation (11.19) and corresponding master equation graph given in Figure 11.4. Here we set Ns = 100 and Nes = 0 at time zero and total enzyme number Ne = 10. (A) The fluctuating numbers of S and ES molecules as functions of time. (B) The stochastic trajectory in the phase space of (m, n).

In this fashion, the set of rate equations of any simple pathway (unless it is part of a network) can be reduced to a single rate equation and the algebraic equations expressing the stoichiometry. To illustrate how much work can be saved in this way, let us return to the Gillespie-Ingold mechanism of nitration of aromatics, for which a repeated application of the Bodenstein approximation provided a rate equation in Example 4.4 in Section 4.3. 

Gillespie and Beattie [89] (see also [33]) were by for the most successful experimentally in establishing a firm basis for an analytical expression of the equilibrium constant in the range of industrial interest. The values in Tables 10 and 11 were calculated using their equation. A detailed description, with literature data and many tables, appears in [33]. A description of the equilibrium using the Redlich-Kwong equation of state is given in [90]. 

Heat of Reaction. Haber investigated the heat of reaction at atmospheric pressure [91]. Numerous authors have estimated the pressure dependence under various assumptions. Today, most people use the Gillespie-Beattie equation [92]. This equation was used in calculating the values in Table 12. For further data see, for example, [33]. Reference [93] contains test results for the range 120-200 MPa (1200 - 2000 bar) and 450-525 °C. Additional literature can be found in [94]. 

Fig. 6.3. Distributions of x in between two steady states (A, C), and dynamic fluctuations between these two states (B, D). The steady-state distributions (A and C) were calculated using (6.4) in the text. The fluctuations in x were calculated using a Gillespie-type Monte Carlo algorithm to the chemical master equation (Beard and Qian, 2008). Parameters Panels (a) and (b), fci = 2.7, = 0.6, ks = 0.25, fc4 =...

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... 

In a second report Gillespie described in detail his procedure, to which the following equation applies pH = pKi -f- log Drop Ratio. By using aqueous indicator solutions, he found the following j)Ki values ... 

Gillespie (8) on the other hand developed an equation of the Lucas-Washburn type without specific reference to an explicit pore model on the basis of D Arcy s law (6). Assuming that AP was constant Gillespie derived the following equation for two dimensional radial spreading of a liquid drop... 

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