MC sampling methods

Perform simulation either employ MD to determine trajectories and, provided the simulation is long enough, calculate equilibrium properties, or employ molecular mechanics to probe the structure of low energy configurations and subsequently use MC sampling methods to compute equilibrium properties. 

In particular, the upper and lower bounds are found resorting to a MC sampling method implemented by the following procedure (Zio, 2013) ... 

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

A successful application of GA to conformation sampling is, for example, as a part of flexible docking [12-14], It should be noted, however, that none of the three sampling methods discussed above, MD, MC, and GA, was shown to outperform the other two in any general way. In fact, a comparison of the three methods in the context of flexible docking showed similar efficiency for all three [12], although specific advantages are likely to exist for particular apphcations. 

Let us leave, for a moment, the domain of discrete torsion angles and consider torsion-angle moves in a torsion continuum. Much effort has been directed to the development of efficient phase-space sampling methods for MC [17,23,37,128-138]. The simplest off-lattice move - being a pivotal move [128,131] of a single torsion angle at a time - hardly samples configuration... 

The discrete protonation states methods have been tested in pKa calculations for several small molecules and peptides, including succinic acid [4, 25], acetic acid [93], a heptapeptide derived from ovomucoid third domain [27], and decalysine [61], However, these methods have sofar been tested on only one protein, the hen egg lysozyme [16, 61, 71], While the method using explicit solvent for both MD and MC sampling did not give quantitative agreement with experiment due to convergence difficulty [16], the results using a GB model [71] and the mixed PB/explicit... 

An overreaching theme of the present chapter, besides broken ergodicity, has to do with the fact that most of the enhanced sampling methods that we shall discuss address situations in which one cannot clearly identify a reaction coordinate that can be conveniently used to describe the kinetic evolution of the system of interest. While methods for enhanced sampling are designed to yield accurate results faster than regular molecular dynamics or Monte Carlo (MC) methods, it is our belief that there is no perfect method, but that, rather, there are methods that perform better for particular applications. Moreover, it should be noted that, while in instances when a proper reaction coordinate can be identified methods described in other chapters are probably more efficient, they could still benefit by sampling in conformational directions perpendicular to the reaction coordinate. 

In the MC/MD method, the stochastic sampling by MC steps permit one to restrict the sampling of chemical space, i.e., the space of A. For example, Kollman and co-workers6,7 limit their chemical sampling only to transitions between the end points in their CMC/MD simulations... 

Where necessary, determine system properties as ensemble expectation values (for the data in Table 13.2, a MC sampling scheme was employed, but MD methods are equally applicable). Every time the coordinates of any atom in the system change, i.e., at each time step in a MD trajectory or following an accepted MC move of either a QM or MM atom, recompute die QM wave function (since at least one term in the operator involving the relative positions of the QM electrons and the MM atoms must be different). Note the contrast widi the AOC method, where only internal moves in the QM system s coordinates necessitate a recomputation of the wave function. 

To study protein folding theoretically, simulation methods have proved indispensable. The folding transition is ultimately governed by statistical thermodynamics and hence it is paramount to use sampling methods that are able to reproduce the canonical Boltzmann distribution. Common sampling techniques are molecular dynamics (MD), Langevin or Brownian dynamics (BD) and Monte Carlo (MC). 

Population models describe the relationship between individuals and a population. Individual parameter sets are considered to arise from a joint population distribution described by a set of means and variances. The conditional dependencies among individual data sets, individual variables, and population variables can be represented by a graphical model, which can then be translated into the probability distributions in Bayes theorem. For most cases of practical interest, the posterior distribution is obtained via numerical simulation. It is also the case that the complexity of the posterior distribution for most PBPK models is such that standard MC sampling is inadequate leading instead to the use of Markov Chain Monte Carlo (MCMC) methods... 

The Metropolis MC [41] was originally developed as a method suited to electronic computers for calculating statistical mechanical configurational integrals. Since the MC sampling is a Markovian process, if we introduce a time scale t that actually labels the order of subsequent configurations X, the dynamic evolution of the probability distribution function P(X, t) is governed by the master equation... 

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