Free Energies from Histograms

Note that the expression in (3.1) is a continuous probability distribution in that p(U T)dU gives the probability of macrostates with energy U dU/2. In an NVT simulation, we measure this distribution to a finite precision by employing a nonzero bin width All. Letting f(U) be the number of times an energy within the range [U,U I All] is visited in the simulation, the normalized observed energy distribution 

Notice that this equation allows us to calculate ft(U) from a probability distribution measured from a simulation at temperature T0. We do not know the value of Z(T0), but it is a constant independent of U. Furthermore, since 12 has no dependence on T, measurement of p at any temperature should in principle permit its complete determination. In practice, however, the potential energies in a canonical simulation are sharply distributed around their average, away from which the statistical quality of p and hence 12 in (3.3) becomes extremely poor. 

Proceeding conceptually for a moment without these logistical difficulties, once we have determined the density of states we can calculate thermodynamic properties at any temperature of interest. The average potential energy is 

A more general version of the canonical reweighting scheme in (3.5), in which the value of any order parameter is reweighted to different temperatures, is given by  

As alluded to previously, numerical issues actually create a more complex situation than that just been described. For starters, the density of states is almost never calculated directly, as it typically spans many orders of magnitudes. This, in turn, would quickly overwhelm standard double-precision calculations in personal computers. This is easily remedied by working instead with the dimensionless entropy / = In Q, which for the purposes of this chapter will inherit all of the same notation used for the density of states in Chap. 1 - subscripts tot, ex, etc. 

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The next three chapters deal with the most widely used classes of methods free energy perturbation (FEP) [3], methods based on probability distributions and histograms, and thermodynamic integration (TI) [1, 2], These chapters represent a mix of traditional material that has already been well covered, as well as the description of new techniques that have been developed only recendy. The common thread followed here is that different methods share the same underlying principles. Chapter 5 is dedicated to a relatively new class of methods, based on calculating free energies from nonequilibrium dynamics. In Chap. 6, we discuss an important topic that has not received, so far, sufficient attention - the analysis of errors in free energy calculations, especially those based on perturbative and nonequilibrium approaches. 

The environmental (i.e., solvent and/or protein) free energy curves for electron transfer reactions can be generated from histograms of the polarization energies, as in the works of Warshel and coworkers [79,80]. 

These bounds are the nonequilibrium equivalents of the Gibbs-Bogoliubov bounds discussed in Chap. 2. Having the free energy now bounded from above and below already demonstrates the power of using both forward and backward transformations. Moreover, as was shown by Crooks [18, 19], the distribution of work values from forward and backward paths satisfies a relation that is central to histogram methods in free energy calculations... 

Although these equations are expressions for the density of states as a function of both energy and reaction coordinate, the free energies are identical to those obtained from the standard one-dimensional multiple histogram equation. 

Figure 10. Proline jf PMF of Ala-cwPro-Tyr extracted from the )/ histogram of 33.6 ns free MD simulation of Ala-cisPro-Tyr in aqueous solution at 278 K. High free energy regions are truncated (indicated by dotted lines) because the histogram statistics are not accurate enough to evaluate the free energy in these regions. The al, bl, and b2 labels indicate the position of the local free energy minima of the respective conformations.

Once a suitable weight function has been determined, a long simulation is performed in the course of which both phases are visited many times. During this run, the biased form of the order parameter distribution P(M 7) is accumulated in the form of a histogram. The applied bias can be unfolded from this distribution in the usual fashion to yield the true equilibrium distribution P(M), from which the relative free energy difference of the two phases can be read off as the logarithm of the ratio of peak weights in P(M). 

Fig. 4.8 (a) Histograms of pull-off force values obtained with an unmodified Si3N4 tip on untreated and oxyfluorinated iPP films in ethanol. The total surface free energy y of the polymer film is shown, (b) Mean values of pull-off force measured with COOH-terminated tips on modified polyolefin surfaces (iPP, isotactic polypropylene LDPE, low-density polyethylene) in ethanol (top) and with OH-terminated tips on oxyfluorinated iPP in water (pH 3.8, bottom) as a function of cos 0 (contact angle measured with water). (Reprinted in part/adapted with permission from [26, 27]. Copyright 1998, 2000, American Chemical Society.)... 

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