Dynamical simulation methods algorithms

The theory of Brownian motion was developed to describe the dynamics of a massive particle in a medium of small particles. Ermak and McCammon have formulated a Brownian dynamics simulation method suitable for concentrated dispersions. The method follows the traditional deterministic molecular dynamics algorithm used to simulate fluids, except that the set of 3N Newtonian equations is replaced by a set of 3N coupled Langevin equations ... 

In this section the theory behind Brownian dynamics simulation methods that are used to study diffusion-controlled biochemical processes is presented. In the first subsection two Brownian dynamics algorithms are described. The second subsection gives an overview of the theory for computing encounter rates. A discussion of electrostatics is in the third subsection followed by subsection four on hydrodynamic forces. The fifth subsection discusses the inclusion of flexibility into Brownian dynamics simulations. 

Other methods which are applied to conformational analysis and to generating multiple conformations and which can be regarded as random or stochastic techniques, since they explore the conformational space in a non-deterministic fashion, arc genetic algorithms (GA) [137, 1381 simulation methods, such as molecular dynamics (MD) and Monte Carlo (MC) simulations 1139], as well as simulated annealing [140], All of those approaches and their application to generate ensembles of conformations arc discussed in Chapter II, Section 7.2 in the Handbook. 

The input to a minimisation program consists of a set of initial coordinates for the system. The initial coordinates may come from a variety of sources. They may be obtained from an experimental technique, such as X-ray crystallography or NMR. In other cases a theoretical method is employed, such as a conformational search algorithm. A combination of experimenfal and theoretical approaches may also be used. For example, to study the behaviour of a protein in water one may take an X-ray structure of the protein and immerse it in a solvent bath, where the coordinates of the solvent molecules have been obtained from a Monte Carlo or molecular dynamics simulation. 

There are many variants of the predictor-corrector theme of these, we will only mention the algorithm used by Rahman in the first molecular dynamics simulations with continuous potentials [Rahman 1964]. In this method, the first step is to predict new positions as follows ... 

Computational methods have played an exceedingly important role in understanding the fundamental aspects of shock compression and in solving complex shock-wave problems. Major advances in the numerical algorithms used for solving dynamic problems, coupled with the tremendous increase in computational capabilities, have made many problems tractable that only a few years ago could not have been solved. It is now possible to perform two-dimensional molecular dynamics simulations with a high degree of accuracy, and three-dimensional problems can also be solved with moderate accuracy. 

An algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

To overcome the limitations of the database search methods, conformational search methods were developed [95,96,109]. There are many such methods, exploiting different protein representations, objective function tenns, and optimization or enumeration algorithms. The search algorithms include the minimum perturbation method [97], molecular dynamics simulations [92,110,111], genetic algorithms [112], Monte Carlo and simulated annealing [113,114], multiple copy simultaneous search [115-117], self-consistent field optimization [118], and an enumeration based on the graph theory [119]. 

This section does not contain any fundamentals or mathematics bur tries to describe the basic energy flows and the methods used in thermal building-dynamics simulation codes to model these. Also, the methods are described without stating the underlying algorithms and equations, for which the reader is referred to the literature and references. A short outline of how these models affect the application possibilities and limits is given at the end of this section and also in Section 11.3.7. 

The MCFT algorithm is the most favorable of the studied simulation methods. The only draw-back of the method is that it cannot be used to simulate the effects of the mass accumulation in the fines removal system. In this paper it has been shown however, that the effects of the mass accumulation in the fines removal system on the process dynamics can not be neglected, unless low a outsize for the fines removal and low fines recycle rates are used. 

One increasingly popular method which lead to non-dynamical trajectories is replica-exchange MC or MD [11-13], which employs parallel simulations at a ladder of temperatures. The "trajectory" at any given temperature includes repeated visits from a number of (physically continuous) trajectories wandering in temperature space. Because the continuous trajectories are correlated in the usual sequential way, their intermittent — that is, non-sequential — visits to the various specific temperatures produce non-sequential correlations when one of those temperatures is considered as a separate ensemble or "trajectory" [14]. Less prominent examples of non-dynamical simulations occur in a broad class of polymer-growth algorithms (e.g., refs. 15-17). 

The advent of high-speed computers, availability of sophisticated algorithms, and state-of-the-art computer graphics have made plausible the use of computationally intensive methods such as quantum mechanics, molecular mechanics, and molecular dynamics simulations to determine those physical and structural properties most commonly involved in molecular processes. The power of molecular modeling rests solidly on a variety of well-established scientific disciplines including computer science, theoretical chemistry, biochemistry, and biophysics. Molecular modeling has become an indispensable complementary tool for most experimental scientific research. 

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