PMF function

The PMF function by Muegge " consists of a set of distance-dependent atom-pair potentials Eij (r) that are written as... 

Besides differences in the functional form and reference state, from a more practical point of view, the knowledge-based potentials differ also with respect to scope of atom type definitions and the amount of structural data used for their derivation. The number of different atom types ranges from 17 in Drug-Score to 40 nonmetal atom types in BLEEP. In all cases, the Protein Data Bank (321) was the source of the solved crystal structures. For BEEEP 351 selected complexes were used, whereas the PMF function was extracted from 697 complexes, and DrugScore was derived using 1376 complexes. In the latter case, the data have been extracted from Relibase (322,323). 

The calculation of the binding affinity with con.sidcration of all tbc.se effects for virtual screening is not possible. In order to circumvent thus difficulty, scoring functions arc used instead, c.g., the Liidi scoring function [80, or consensus scoring functions derived from FlevX score, DOCK score, GOLD score, ChemScore, or PMF score [81 ]. 

The function W(X) is called the potential of mean force (PMF). The fundamental concept of the PMF was first introduced by Kirkwood [4] to describe the average structure of liquids. It is a simple matter to show that the gradient of W(X) in Cartesian coordinates is related to the average force. 

A tractable example is the pmf between two particular particles in a macromolecule as a function of their separation q. The free energy to increase q by 8q becomes... 

As we pointed out earlier, calculating the derivative of the free energy appears to require a full set of generalized coordinates. However, this may seem quite surprising. Assume that we want to define the PMF as a function of the distance between two molecules. This force is clearly independent of the particular choice of generalized coordinates made to calculate it. In fact, we are now going to prove that an equation can be derived which does not require an explicit definition of generalized coordinates other than . 

As an example, this approach was applied to the calculation of the PMF for alanine dipeptide as a function of the two torsion angles

resulting free energy surface is shown in Fig. 4.5. Bilinear Qi elements were used to approximate the free energy. Control points were chosen such that there are four of them around each data point. This was done in order to increase the smoothness and quality of the reconstructed free energy. The position of the Q i nodes and control points is shown in Fig. 4.6. 

Several techniques exist to compute AA. Following our earlier discussion for the PMF we will discuss TI. In this approach a parameterized Hamiltonian J a(x, Pa,) iS defined such that when A = 0, Mx = M and when A = 1, Mx = M. MA(x, p ) interpolates smoothly between the two Hamiltonian functions. The free energy A becomes itself a function of A and we have... 

The function U fXj is called the PMF it was first introduced by Kirkwood to describe the structure of liquids [61]. It plays the role of a free energy surface for the solute. Notice that the dynamics of the solute on the free energy surface W(X) do not correspond to the true dynamics. Rather, an MD simulation on 1T(X) should be viewed as a method to sample conformational space and to obtain equilibrium, thermally averaged properties. 

The second general method for performing PMF calculations relies on the use of Umbrella Sampling.50 In its simplest form, Umbrella Sampling adds a bias restraint (umbrella) term to the standard potential function... 

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