Free energy calculations error analysis

Understanding and Improving Free Energy Calculations in Molecular Simulations Error Analysis and Reduction Methods... 

Consider the forward calculation as an example. Earlier in this chapter, we concluded that the most important contribution to a forward free energy calculation comes from the low-x tail of f(x)9 and its poor sampling results in the major systematic error of the calculation. To simplify further the analysis we assume that there is a limit-perturbation x f such that x > Xf is well sampled and x < Xf is never sampled. This is illustrated in Fig. 6.5. 

What level of inaccuracy can be expected for a simulation with a certain sample size N1 This question can be transformed to another one what is the effective limit-perturbation Xf or xg in the inaccuracy model [(6.22) or (6.23)] To assess the error in a free energy calculation using the model, one may histogram / and g using the perturbations collected in the simulations, and plot x in the tail of the distribution. However, if Xf is taken too small the accuracy is overestimated, and the assessed reliability of the free energy is therefore not ideal. In the following, we discuss the most-likely analysis, which provides a more systematic way to estimate the accuracy of free energy calculations. 

In practice it is helpful to know the order of magnitude of the sample size N needed to reach a reasonably accurate free energy. The inaccuracy model described above presents an effective way to relate the sample size N and the finite sampling error through perturbation distribution functions. Alternatively, one can develop a heuristic that does not involve distribution functions and is determined by exploring the common behavior of free energy calculations for different systems [25]. Although only FEP calculations are considered in this section, the analysis extends to NEW calculations. 

From (9.27), we see that this approach will work nicely if the variance is always small Taylor s theorem with remainder tells us that the error of the first-derivative - mean-field - contribution is proportional to the second derivative evaluated at an intermediate A. That second derivative can be identified with the variance as in (9.27). If that variance is never large, then this approach should be particularly effective. For further discussion, see Chap. 4 on thermodynamic integration, and Chap. 6 on error analysis in free energy calculations. 

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. 

Until recently, advances in calculating the free energy were not accompanied by comparable progress in rigorous error analysis and reduction. Although a variety of methods to estimate the error in calculated free energies were proposed [32, 106], they were usually somewhat heuristic or involved approximations that were not always sufficiently well supported. Only recently, considerable progress has been made on this front, in particular by Daniel Zuckerman and Thomas Woolf [107]. 

Perhaps the most challenging part of analyzing free energy errors in FEP or NEW calculations is the characterization of finite sampling systematic error (bias). The perturbation distributions / and g enable us to carry out the analysis of both the finite sampling systematic error (bias) and the statistical error (variance). 

An analysis of the effect of sampling errors on free energy difference calculations has demonstrated a definite relation between the calculated error... 

The sensitivity analysis approach has also been shown to be useful for studying error propagations due to the use of nonoptimal parameters in biomolecular simulations and for examining how error cancellations may occur in free energy difference calculations. The sensitivity analysis approach can also suggest how potential functions could be simplified and how the parameters of these functions can be effectively refined. 

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