Overlap Sampling Techniques

It would be valuable if one could proceed with a reliable free energy calculation without having to be too concerned about the important phase space and entropy of the systems of interest, and to analyze the perturbation distribution functions. The OS technique [35, 43, 44, 54] has been developed for this purpose. Since this is developed from Bennett s acceptance ratio method, this will also be reviewed in this section. That is, we focus on the situation in which the two systems of interest (or intermediates in between) have partial overlap in their important phase space regions. The partial overlap relationship should represent the situation found in a wide range of real problems. 

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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)...

This approach is one of the oldest techniques for improving FEP calculations [36]. It is often called the simple overlap sampling (SOS) method and is usually markedly more accurate than simple averaging. It requires that one forward and one backward calculation be performed at every intermediate state. It is worth noting that no sampling is performed from the ensemble characterized by Xi+AX/2, so that the number of stages is the same as in the pure forward, or backward calculation. 

A detailed presentation of the overlap sampling approach will be given in Sect. 6.6 of Chap. 6. In the present chapter, we merely note that applying this scheme, or any other similar technique that will be discussed extensively later on in the book, almost always improves the quality of the results. It is, therefore, highly recommended that they be routinely used in FEP calculations, perhaps in combination with other techniques. 

As the initial and final states are set by the problem under study, their important phase space relationship could be any one of the cases illustrated in Fig. 6.1. For cases Fig. 6.1c, d, it is impossible to construct a funnel path from 0 to 1 directly. To satisfy the funnel requirement, similar to the MFEP calculation, a staged NEW calculation can be performed. For example, in the case Fig. 6.1c, one can first construct an intermediate in the common region of / ,[ and /), then perform two separate NEW calculations following the paths 0 —> M and 1 —> M, respectively. This NEW-overlap sampling (NEW-OS) technique will be discussed in detail in Sect. 6.6. 

Many real-world samples consist of a large number of substances whose spectra are relatively broad and relatively featureless. This limits the ability to discriminate between compounds with similar or substantially overlapping spectra. There are two techniques that can be used to discriminate between compounds that are spectrally overlapping. One technique is discrimination based on fluorescent lifetime. Two compounds that spectrally overlap can be distinguished by their fluorescence lifetimes, provided that their fluorescent lifetimes are sufficiently different. The second technique is chromatographic discrimination i.e. the use of a separation technique. 

The relationship in Eq. [28] can be used for a single solute pair if simulations are performed using the umbrella sampling technique.In this method, the solute-solute distance is constrained to lie within a small window, allowing for the efficient calculation of g r) over that restricted distance range. The effect of the constraint is subtracted from the calculated u> r) after the fact. Results from several overlapping windows may be combined to yield the full potential of mean force. 

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