Umbrella sampling developments

What has been developed within the last 20 years is the computation of thermodynamic properties including free energy and entropy [12, 13, 14]. But the ground work for free energy perturbation was done by Valleau and Torrie in 1977 [15], for particle insertion by Widom in 1963 and 1982 [16, 17] and for umbrella sampling by Torrie and Valleau in 1974 and 1977 [18, 19]. These methods were primarily developed for use with Monte Carlo simulations continuous thermodynamic integration in MD was first described in 1986 [20]. 

Note that for both FEP and TI, the umbrella restraint introduces a term that depends on e+VblaJRT, which may (since Vbias is always > 0) fluctuate widely, especially if the biasing function is attempting to restrain the system to a conformation far from a local minimum. As a result, use of the umbrella term Equations 19 and 20 is often problematic. This has led to the development of alternate (but related) approaches to Umbrella Sampling.41,51,52 Many of these derive from the following equation, which relates the work function W to the probability of states, corrected for use of the biasing potential ... 

It would be hard to write a definitive account of the development of extended sampling methods we will not attempt to do so. The seminal ideas are probably correctly attributed to Torrie and Valleau [22], who coined the terminology umbrella sampling. The huge literature of subsequent advances and rediscoveries may be rationalized a little by dividing it into two, according to how the macrostates to be weighted are defined. 

Originally the technique of umbrella sampling was developed to derive onedimensional potentials of mean force. Beutler et al. extended the method to compute the two-dimensional potential of mean force for the two degrees of freedom and of the four phenylalanine side chains in antamanide [84]. They obtained good agreement between computed and experimental NMR 3j-coupling constants. 

In this section, a new sampling scheme called REPSWA or reference systems spatial warping algorithm will be developed [38]. It will be contrasted to standard approaches such as umbrella sampling and guiding potentials so that the advantages of the new technique can be underscored. 

The author appreciates the ongoing support of this research by the Natural Sciences and Engineering Research Council of Canada. He has also enjoyed the helpful advice of many colleagues, and especially Glenn Torrie, who shared in much of the early development of umbrella-sampling techniques, and Stu Whittington, always a source of insight and enthusiasm. 

Further developments of these ideas took place in computational structural biology, where nonphysical local transformations were implemented within the framework of thermodynamic cycles. These nonphysical transformations were introduced in 1981 by Warshel, who studied ionization in acidic residues in proteins pK calculations). Although the cycle included nonphysical transformations, they were not carried out by the perturbation technique. A year later Warshel used the perturbation method together with umbrella sampling to study the solvation free energy contribution to an electron transfer reaction coordinate, using two spheres for donor and acceptor in water the perturbation, however, was performed along a physical path. Warshel also modeled some enzymatic reactions that involve nonphysical processes. ... 

Dynamics. There are calculations in which the metal is modeled as an Einstein solid with harmonic vibrations[33j. When surface molecules and ions are strongly adsorbed molecular dynamics becomes an inefEcient way to study surface processes due to the slow exchange between surface and solution. In this case it is possible to use umbrella sampling to compute distribution profiles[34, 35]. Recently the idea underlying Car-Parrinello was used for macroion dynamics[36, 37] in which the solvent surrounding charged macroions is treated as a continuum in a self consistent scheme for the potential controlling ion dynamics. Dynamical corrections from the solvent can be added. There is a need to develop statistical methods to treat the dynamics of complex objects that evolve on several different time scales. 

Umbrella sampling is a biased-sampling technique developed by Torrie and Valleau [26] that enhances the sampling by modifying the potential energy function, forcing the system to stay in the desired states. The basic idea of umbrella sampling is presented below for further details, one can refer to the recent review paper by Kastner [27]. 

The utility of the approach was much enhanced through the development of umbrella sampling by Valleau and co-wor-kers. An artificial biasing function ( ) can be added to the potential energy to extend the range of sampling of e.g., by flattening the problematic barriers. The true distribution can subsequently be recovered from the biased one, g K), via... 

Given the inherent difficulty of making such a choice, much effort in the past decade has been focused on developing CVs suitable to describe correctly a variety of conformational changes. We review here a few of the most popular choices used with metadynamics but which can also be used in other schemes like umbrella sampling, thermodynamic integration, and steered molecular dynamics. 

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