Stochastic boundary approach

The most common boundary representation is periodic boundary conditions which assumes that the system consists of a periodic array (or a crystal ) of identical systems [1], Another common method, developed for the simulation of biomacromolecules, is the stochastic boundary approach, in which the influences of the atoms outside the boundary are replaced by a simple boundary force [78, 79, 80], Warshel uses a Langevin dipoles model in which the solvent is explicitly replaced by a grid of polarizable dipoles. The energy is calculated in a similar way to the polarization energy in a molecular mechanics force field (see above) [15]. 

The stochastic boundary approach in conjunction with molecular dynamics is an approximate technique for studying such localized events in many-body systems.99 The method was developed initially to study nonequilibrium phenomena98 (e.g., chemical reactions and atomic diffusion across thermal gradients) and hence is well suited for some of the problems of interest here. The approach has been used to treat simple fluids,100101 as well as more complex fluids, including water,102 and solvated biomolecules.103... 

Brooks III, C.L., Briinger, A.T. and Karplus, M. (1985) Active Site Dynamics in Protein Molecules A Stochastic Boundary Molecular Dynamics Approach,... 

Clearly, the present approach precludes the detailed study of the dynamics of explicit solute-solvent interactions because the solvent (bath) degrees of freedom have been eliminated. Also, as in the stochastic boundary model,... 

To illustrate the solvent effect on the average structure of a protein, we describe results obtained from conventional molecular dynamics simulations with periodic boundary conditions.92,193 This method is well suited for a study of the global features of the structure for which other approaches, such as stochastic boundary simulation methods, would not be appropriate. We consider the bovine pancreatic trypsin inhibitor (BPTI) in solution and in a crystalline environment. A simulation was carried out for a period of 25 ps in the presence of a bath of about 2500 van der Waals particles with a radius and well depth corresponding to that of the oxygen atom in ST2 water.193 The crystal simulation made use of a static crystal environment arising from the surrounding protein molecules in the absence of solvent. These studies, which were the first application of simulation methods to determine the effect of the environment on a protein, used simplified representations of the surround-... 

To characterize structural waters and to follow the dynamics of reactions involving waters, it is necessary to be able to treat in detail the motions of these molecules. A methodology which includes solvents explicitly is required. Both conventional molecular dynamics techniques and the stochastic boundary molecular dynamics approaches can be used. When the region of... 

To overcome problems arising fi-om the finite system size used in MC or MD simulation, boundary conditions are imposed using periodic-stochastic approximations or continuum models. In particular, in stochastic boundary conditions the finite system is not duplicated but a boundary force is applied to interact with atoms of the system. This force is set as to reproduce the solvent regions that have been neglected. Anyway, in general any of the methods used to impose boundary conditions in MC or MD can be used in the QM/MM approach. 

An alternative approach is to employ stochastic boundary conditions where the finite molecular system employed in the simulation is not duplicated, but rather, a boundary force is applied to interact with atoms of the sys-tem. 2-67 Yhe boundary force is chosen to mimic the bulk solvent regions that have been neglected. However, a difficulty with the use of this method is due to the ambiguity associated with the definition and parameterization of the boundary forces. Thus, there is sometimes the impression that the method is not rigorous. Recently, Beglov and Roux provided a formal definition and suggested a useful implementation of the boundary forces. Their results are very promising. 

Any of the methods used in classical Monte Carlo and molecular dynamics simulations may be borrowed in the combined QM/MM approach. However, the use of a finite system in condensed phase simulations is always a severe approximation, even when appropriate periodic or stochastic boundary conditions are employed. A further complication is the use of potential function truncation schemes, particular in ionic aqueous solutions where the long-range Coulombic interactions are significant beyond the cutoff distance.Thus, it is alluring to embed a continuum reaction field model in the quantum mechanical calculations in addition to the explicit solute—solvent interaaions to include the dielectric effect beyond the cutoff distance. - uch an onion shell arrangement has been used in spherical systems, whereas Lee and Warshel introduced an innovative local reaction field method for evaluation of long-... 

Several reviews have focussed on stochastic dynamics. Harris and Schiitz discuss a range of fluctuation theorems (the Jarzynski equality, the ES FR and the GC FR) in the context of stochastic, Markov systems, but in a widely applicable context. They investigate the conditions under which they apply, including an analysis of the conditions under which the GC FR is valid. In 2006, Gaspard reviewed studies where a stochastic approach to the treatment of boundaries is used to obtain FRs for the current in nanosystems, with a focus on work from his group, and also published a review on Hamiltonian systems that includes a discussion on FRs and the JE. 

This means that larger scale motions can be explicitly resolved and deterministically forecasted. The smaller scale motions, namely turbulence, are not explicitly resolved. Rather, the effects of those sub-grid scales on the larger scales are approximated by turbulence models. These smaller size motions are said to be parameterized by sub-grid scale stochastic (statistical) approximations or modes. The referred experimental data analyzes of the flow in the atmospheric boundary layer determine the basis for the large eddy simulation (LES) approach developed by meteorologists like Deardorff [27] [29] [30]. Large-Eddy Simulation (LES) is thus a relatively new approach to the calculation of turbulent flows. 

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