Enhanced sampling

Using this expression, the standard 5=1 equilibrium average properties may be calculated over a trajectory which samples the generalized statistical distribution for 5 7 1 with the advantage of enhanced sampling for g > 1. 

The Langevin model has been employed extensively in the literature for various numerical and physical reasons. For example, the Langevin framework has been used to eliminate explicit representation of water molecules [22], treat droplet surface effects [23, 24], represent hydration shell models in large systems [25, 26, 27], or enhance sampling [28, 29, 30]. See Pastor s comprehensive review [22]. 

Though LI failed for general biomolecular applications [50], it has been found to be a useful ingredient in two other contexts macroscopic separable models, and enhanced sampling. 

Fiber R and M Karplus 1990. Enhanced Sampling in Molecular Dynamics Use of the Time-Dependent Hartree Approximation for a Simulation of Carbon Monoxide Diffusion through Myoglobin. Journal of the American Chemical Society 112 9161-9175. 

Simmerling C, T Fox and P A Kollman 1998. Use of LocaUy Enhanced Sampling in Free Energ Calculations Testing and Application to the o —> /3 Anomerisation of Glucose. Journal of tl American Chemical Society 120 5771-5782. 

A powerful and general technique to enhance sampling is the use of umbrella potentials, discussed in Section IV. In the context of alchemical free energy simulations, for example, umbrella potentials have been used both to bias the system toward an experimentally determined conformation [26] and to promote conformational transitions by reducing dihedral and van der Waals energy terms involving atoms near a mutation site [67]. 

Feig M, Karanicolas J, Brooks CL, III (2004) MMTSB tool set Enhanced sampling and multiscale modeling methods for applications in structure biology. J Mol Graph Model 22 377—395. 

Also in the early 1990s, a somewhat related method for calculating free energy differences was proposed by Ron Elber and coworkers [92, 93]. It relies on simulating multiple, noninteracting replicas that differ only locally. As a result, the method is applicable to systems that undergo only local modification - e.g., point mutations in proteins. For this reason, it has been called the locally enhanced sampling (LES) technique. 

Roitberg, A. Elber, R., Modeling side chains in peptides and proteins application of the locally enhanced sampling technique and the simulated annealing methods to find minimum energy conformations, J. Chem. Phys. 1991, 95, 9277-9287... 

Verkhivker, G. Elber, R. Nowak, W., Locally enhanced sampling in free energy calculations application of mean held approximation to accurate calculation of free energy differences, J. Chem. Phys. 1992, 97, 7838-7841... 

If the two important regions do not overlap, or overlap only partially, it is usually necessary to use the enhanced sampling techniques introduced in Sect. 1.4. This is schematically illustrated in Fig. 2.4d. One of these techniques, stratification, has... 

Fig. 2.4. Schematic representation of the different relationships between the important regions in phase space for the reference (0) and the target (1) systems, and their possible interpretation in terms of probability distributions - it should be clarified that because AU can be distributed in a number of different ways, there is no obvious one-to-one relation between P0(AU), or Pi (AU), and the actual level of overlap of the ensembles [14]. (a) The two important regions do not overlap, (b) The important region of the target system is a subset of the important region of the reference system, (c) The important region of the reference system overlaps with only a part of the important region of the target state. Then enhanced sampling techniques of stratification or importance sampling that require the introduction of an intermediate ensemble should be employed (d)...

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. 

As one of the possible ways to alter the sampling distribution in a manner that is conducive to enhanced sampling, we present a strategy based on probability distributions that arise in a generalization of statistical mechanics proposed by Tsallis [31]. In this... 

Enhanced sampling in conformational space is not only relevant to sampling classical degrees of freedom. An additional reason to illustrate this particular method is that the delocalization feature of the underlying distribution in Tsallis statistics is useful to accelerate convergence of calculations in quantum thermodynamics [34], We focus on a related method that enhances sampling for quantum free energies in Sect. 8.4.2. 

This section is used to introduce the momentum-enhanced hybrid Monte Carlo (MEHMC) method that in principle converges to the canonical distribution. This ad hoc method uses averaged momenta to bias the initial choice of momenta at each step in a hybrid Monte Carlo (HMC) procedure. Because these average momenta are associated with essential degrees of freedom, conformation space is sampled effectively. The relationship of the method to other enhanced sampling algorithms is discussed. 

One general approach to enhancing sampling, which is the focus of this section, is based on the fact that both fast and slow dynamical modes contribute to the time evolution of biomolecular systems, but in most cases the motions of primary interest are the slow ones, which typically correspond to the largest structural changes [72, 73],... 

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