Stochastic simulation equation

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

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. 

V, ip, x, and t) in the PDF transport equation makes it intractable to solve using standard discretization methods. Instead, Lagrangian PDF methods (Pope 1994a) can be used to express the problem in terms of stochastic differential equations for so-called notional particles. In Chapter 7, we will discuss grid-based Eulerian PDF codes which also use notional particles. However, in the Eulerian context, a notional particle serves only as a discrete representation of the Eulerian PDF and not as a model for a Lagrangian fluid particle. The Lagrangian Monte-Carlo simulation methods discussed in Chapter 7 are based on Lagrangian PDF methods. 

Only deterministic models for cellular rhythms have been discussed so far. Do such models remain valid when the numbers of molecules involved are small, as may occur in cellular conditions Barkai and Leibler [127] stressed that in the presence of small amounts of mRNA or protein molecules, the effect of molecular noise on circadian rhythms may become significant and may compromise the emergence of coherent periodic oscillations. The way to assess the influence of molecular noise on circadian rhythms is to resort to stochastic simulations [127-129]. Stochastic simulations of the models schematized in Fig. 3A,B show that the dynamic behavior predicted by the corresponding deterministic equations remains valid as long as the maximum numbers of mRNA and protein molecules involved in the circadian clock mechanism are of the order of a few tens and hundreds, respectively [128]. In the presence of molecular noise, the trajectory in the phase space transforms into a cloud of points surrounding the deterministic limit cycle. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

The lowest-lying potential energy surfaces for the 0(3P) + CH2=C=CH2 reaction were theoretically characterized using CBS-QB3, RRKM statistical rate theory, and weak-collision master equation analysis using the exact stochastic simulation method. The results predicted that the electrophilic O-addition pathways on the central and terminal carbon atom are dominant up to combustion temperatures. Major predicted end-products are in agreement with experimental evidence. New H-abstraction pathways, resulting in OH and propargyl radicals, have been identified.254... 

For linear systems, the differential equation for the jth cumulant function is linear and it involves terms up to the jth cumulant. The same procedure will be followed subsequently with other models to obtain analogous differential equations, which will be solved numerically if analytical solutions are not tractable. Historically, numerical methods were used to construct solutions to the master equations, but these solutions have pitfalls that include the need to approximate higher-order moments as a product of lower moments, and convergence issues [383]. What was needed was a general method that would solve this sort of problem, and that came with the stochastic simulation algorithm. 

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. 

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... 

As another example of hybrid simulation touched upon above, Haseltine and Rawlings (2002) treated fast reactions either deterministically or with Langevin equations and slow reactions as stochastic events. Vasudeva and Bhalla (2004) presented an adaptive, hybrid, deterministic-stochastic simulation scheme of fixed time step. This scheme automatically switches reactions from one type to the other based on population size and magnitude of transition probability. 

The stochastic Liouville equation is highly useful when applied at high field, as techniques exist to reduce in size the typically large matrices it produces, and it has thus been used to simulate electron and nuclear spin polarizations in magnetic resonance experiments.A relatively recent book describes the approach in detail. However, for determining field dependences, such reductions are not possible, meaning that the sizes of the matrices are too large for even modern computers, and so this approach is seldom used for the simulation of field effects. 

Both vibrational and rotovibrational relaxation can be described analyti-caDy as multiplicative stochastic processes. For these processes, RMT is equivalent to the stochastic Liouville equation of Kubo, with the added feature that RMT takes into account the back-reaction from the molecule imder consideration on the thermal bath. The stochastic Liouville equation has been used successfully to describe decoupling in the transient field-on condition and the effect of preparation on decay. When dealing with liquid-state molecular dynamics, RMT provides a rigorous justification for itinerant oscillator theory, widely applied to experimental data by Evans and coworkers. This implies analytically that decoupling effects should be exhibited in molecular liquids treated with strong fields. In the absence of experimental data, the computer runs described earlier amount to an independent means of verifying Grigolini s predictions. In this context note that the simulation of Oxtoby and coworkers are semistochastic and serve a similar purpose. 

Similar to fluorescence depolarization and NMR, two limiting cases exist in which the molecular motion becomes too slow or too fast to further effect the ESR lineshape (Fig. 8) (35). At the fast motion limit, one can observe a narrow triplet centered around the average g value igxx + gyy + giz with a distance between lines of aiso = Axx- -Ayy- -A2,z)l3, where gu and Ajj are principal values of the g-tensor and the hyperflne splitting tensor A, respectively. At the slow motion limit, which is also referred to as the rigid limit, the spectrum (shown in Fig. 8) is a simple superposition of spectra for all possible spatial orientations of the nitroxide with no evidence of any motional effects. Between these limits, the analysis of the ESR lineshape and spectral simulations, which are based on the Stochastic Liouville Equation, provide ample information on lipid/protein dynamics and ordering in the membrane (36). 

Steric requirements, hydrogen and deuterium, 299 Stem-Volmer plot, 181 Stiff differential equations, 109 Stochastic simulation, 109 Stoichiometric coefficients, 11 Stokes-Einstein equation, 135 Stopped flow, 179 Stmetured water, 395 Structure-reactivity relationships, 311 Sublimation energy, 403 Substituent, 313 Substituent constant, 323 alkyl group, 341 electrophilic, 322 Hammett, 316 inductive, 325, 338 normal. 324 polar, 339 primary, 324 resonance, 325... 

Recently Calderon introduced a surrogate process approximation (SPA) to improve the sampling in calculation of the JE. The scheme is applied to the study of the unravelling of deca-alanine at constant temperature in a steered molecular dynamics simulation. The distribution of the work is approximated by developing a model for the dynamics using a relatively small number of real trajectories in conjunction with stochastic differential equations selected to model the process. The... 

In a previous work [56] we deduced an alternative expression for the error (11) by performing extensive numerical simulations of the stochastic differential equation (19). The metadynamics parameters, w/tq, Sa, and the system-dependent parameters, f3, D and S were systematically varied, and for each choice of the parameters the error (11) was computed by repeating several metadynamics reconstructions. Fitting the results, we obtained that the data were reproduced within an accuracy of 20% by... 

A. Cardenas and R. Elber (2003) Atomically detailed simulations of helix formation with the stochastic difference equation. Biophysical Journal, 85, pp. 2919-2939... 

Pardoux, E., and Talay, D., Discretization and Simulation of Stochastic Differential-Equations, Acta Applicandae Mathematicae 3 (1) 23 7 (1985). 

Stochastic dynamics The stochastic dynamics (SD) method is a further extension of the original molecular dynamics method. A space-time trajectory of a molecular system is generated by integration of the stochastic Langevin equation which differs from the simple molecular dynamics equation by the addition of a stochastic force R and a frictional force proportional to a friction coefficient g. The SD approach is useful for the description of slow processes such as diffusion, the simulation of electrolyte solutions, and various solvent effects. 

Instead of mobilizing large scale Monte Carlo simulations of the visitation probability P as a function of walklength n, Pk can be evaluated (as a function of time) using the stochastic master equation [60]. Suppose at time f = 0 a random walker is positioned with unit probability at a site m in the interior of the lattice (away from the boundary of the system). For t > 0 this probability evolves among the lattice sites as determined by Eq. (4.3). An entropy-like quantity... 

The above approach, based on the solution of Eq. (4.3), is numerically superior and more accurate than one based on conventional Monte Carlo simulations. For comparison, Pitsianis et al. [59b] performed Monte Carlo simulations using 100,000 random walks on a Sierpinski gasket of 29,526 sites and obtained a value of 1.354 for dj (the exact value is 1.365). In the approach elaborated above, the value 1.367 was obtained by solving the stochastic master equation on a gasket of only 366 sites. 

Modeling the dynamics of an IP3R on the basis of its subunits leads to various consequences for a cluster of N IP3RS. As long as every IP3R is treated individually and subunits are assigned to individual channels -as has been done in stochastic simulations [6] - the state of the cluster is uniquely determined by the states of its subunits. However, an approach based on a population of subunits not grouped into individual channels is more suitable for the derivation of master equations and Fokker-Planck equations which we would like to use. That requires to determine the number of open channels from the total number of activatable subunits in the subunit population. We assume that the activatable subunits are randomly scattered across the channels. The distribution of the mo activatable subunits on the 4A subunits of a cluster decides upon the value of rio and hence the Ca " concentration. We show in the appendix that this distribution is sharply peaked around its mean value. Therefore, we set m = (rio) = Ua-ria is defined in equation (11.52). 

Transient EMR has also been reported on the triplet state of retinal dissolved in liquid crystalline phase (Munzenmaier et al., 1992). The simulation of the transients with the stochastic Liouville equation provides the motional and order parameters of the pigment. The anisotropy ofmotional correlation times is high as expected for such an extended linear molecule and the correlation times couldbe followed with temperature over a range of two orders of magnitude in the nematic and smectic phase. 

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