Stochastic approximations

Spontaneous absolute asymmetric synthesis is described in the formation of enantiomerically enriched pyrimidyl alkanol from the reaction of pyrimidine-5-car-baldehyde and /-Pr2Zn without adding chiral substance in combination with asymmetric autocatalysis. The approximate stochastic distribution of the absolute conhgurations of the product pyrimidyl alkanol strongly suggests that the reaction is a spontaneous absolute asymmetric synthesis. 

Indeed, when 2-alkynylpyrimidine-5-carbaldehyde was reacted with z-Pr2Zn in a mixed solvent of ether and toluene, the subsequent one-pot asymmetric autocatalysis with amplification of ee gave enantiomerically enriched pyrimidyl alkanol 12 well above the detection level [103]. The absolute configurations of the pyrimidyl alkanol exhibit an approximate stochastic distribution of S and R enantiomers (formation of S 19 times and R 18 times) (Fig. 6). 

We have demonstrated the stochastic formation of (S)- and (,R)-5-pyrimidyl alkanol 12 from pyrimidine-5-carbaldehyde 11 and i-V Zn without the intervention of a chiral auxiliary. Even in the reactions performed in toluene alone, stochastic behavior of the formation of (S)- and (A)-12 was observed in the presence of achiral silica gel. We believe that the approximate stochastic behavior in the formation of alkanols fulfils one of the conditions necessary for chiral symmetry breaking by spontaneous absolute asymmetric synthesis. 

H. Meirovitch,/. Phys. A, 15, 2063 (1982). An Approximate Stochastic Process for Computer Simulation of the Ising Model at Equilibrium. 

It can be shown that ARCH and GARCH models are able to approximate stochastic differential processes if the latter fulfil certain properties. Albeit the goodness of fit is limited, both types of methods are related and can be converted into each other. Moreover, simple stochastic processes show quite simple auto-correlation structures similar to basic ARMA models. For instance, the Ornstein-Uhlenbeck process can be seen as the continuous equivalent of the AR(1) process. In other words, an Ornstein-Uhlenbeck process measured in discrete intervals can be interpreted/modeUed as an AR(1) process (see also (2.23), (2.60), and (2.61)). ... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

Vd satisfying PdVd = Vd, which means that Vd is an approximation of an invariant measure. For an invariant measure, any numerical discretization may be interpreted as a stochastic perturbation of the original problem. 

In LN, the bonded interactions are treated by the approximate linearization, and the local nonbonded interactions, as well as the nonlocal interactions, are treated by constant extrapolation over longer intervals Atm and At, respectively). We define the integers fci,fc2 > 1 by their relation to the different timesteps as Atm — At and At = 2 Atm- This extrapolation as used in LN contrasts the modern impulse MTS methods which only add the contribution of the slow forces at the time of their evaluation. The impulse treatment makes the methods symplectic, but limits the outermost timestep due to resonance (see figures comparing LN to impulse-MTS behavior as the outer timestep is increased in [88]). In fact, the early versions of MTS methods for MD relied on extrapolation and were abandoned because of a notable energy drift. This drift is avoided by the phenomenological, stochastic terms in LN. 

Abstract. A stochastic path integral is used to obtain approximate long time trajectories with an almost arbitrary time step. A detailed description of the formalism is provided and an extension that enables the calculations of transition rates is discussed. 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

The errors in the present stochastic path formalism reflect short time information rather than long time information. Short time data are easier to extract from atomically detailed simulations. We set the second moment of the errors in the trajectory - [Pg.274]

Yun-Yu S, W Lu and W F van Gunsteren 1988. On the Approximation of Solvent Effects on Conformation and Dynamics of Cyclosporin A by Stochastic Dynamics Simulation Teclmiqi Molecular Simulation 1 369-383. 

In finite boundary conditions the solute molecule is surrounded by a finite layer of explicit solvent. The missing bulk solvent is modeled by some form of boundary potential at the vacuum/solvent interface. A host of such potentials have been proposed, from the simple spherical half-harmonic potential, which models a hydrophobic container [22], to stochastic boundary conditions [23], which surround the finite system with shells of particles obeying simplified dynamics, and finally to the Beglov and Roux spherical solvent boundary potential [24], which approximates the exact potential of mean force due to the bulk solvent by a superposition of physically motivated tenns. 

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. 

The relative fluctuations in Monte Carlo simulations are of the order of magnitude where N is the total number of molecules in the simulation. The observed error in kinetic simulations is about 1-2% when lO molecules are used. In the computer calculations described by Schaad, the grids of the technique shown here are replaced by computer memory, so the capacity of the memory is one limit on the maximum number of molecules. Other programs for stochastic simulation make use of different routes of calculation, and the number of molecules is not a limitation. Enzyme kinetics and very complex oscillatory reactions have been modeled. These simulations are valuable for establishing whether a postulated kinetic scheme is reasonable, for examining the appearance of extrema or induction periods, applicability of the steady-state approximation, and so on. Even the manual method is useful for such purposes. 

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. 

Both of the numerical approaches explained above have been successful in producing realistic behaviour for lamellar thickness and growth rate as a function of supercooling. The nature of rough surface growth prevents an analytical solution as many of the growth processes are taking place simultaneously, and any approach which is not stochastic, as the Monte Carlo in Sect. 4.2.1, necessarily involves approximations, as the rate equations detailed in Sect. 4.2.2. At the expense of... 

After planetary accretion was complete there remained two groups of surviving planetesimals, the comets and asteroids. These populations still exist and play an important role in the Earth s history. Asteroids from the belt between Mars and Jupiter and comets from reservoirs beyond the outer planets are stochastically perturbed into Earth-crossing orbits and they have collided with Earth throughout its entire history. The impact rate for 1 km diameter bodies is approximately three per million years and impacts of 10 km size bodies occur on a... 

In the stochastic theory of branching processes the reactivity of the functional groups is assumed to be independent of the size of the copolymer. In addition, cyclization is postulated not to occur in the sol fraction, so that all reactions in the sol fraction are intermolecular. Bonds once formed are assumed to remain stable, so that no randomization reactions such as trans-esterification are incorporated. In our opinion this model is only approximate because of the necessary simplifying assumptions. The numbers obtained will be of limited value in an absolute sense, but very useful to show patterns, sensitivities and trends. 

The methods discussed so far are essentially limited to isolated ion-pairs or, in the admittedly crude approximation, to cases when a multiple ion-pair spur can be considered to be a collection of single ion-pairs. Additionally, it is difficult to include an external field, as that will destroy the spherical symmetry of the problem. Stochastic treatments can incorporate both multiple ion-pairs and the effects of an external field. 

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