To free or not to free?

Abstract

Refinement of an atomic model is the optimization step that makes the model accurate enough for its structural interpretation. As for any optimization problem, the choice of the target is one of three key decisions to take (two others are the model parameterization and optimization method).From the early days (Hughes, 1941) a straightforward choice was minimizing the leastsquares (LS) discrepancy between the experimental structure-factor amplitudes F obs hkl È É ; hkl ð Þ 2 S, and the corresponding values F calc hkl È É calculated from the atomic model, where S is the complete set of diffraction data. However, macromolecular models may contain errors that cannot be corrected by any choice of parameter values; one example is atoms missing from the model, which is always the case for macromolecules. LS-based refinement of incomplete models may make them worse.The errors that cannot be removed by changing model parameters can be accounted for statistically. This comes down to the question: how large is the probability (statistical likelihood) to obtain calculated amplitudes equal to the observed ones after necessary corrections are made randomly, for example after the model has been completed randomly with 'atoms' to compensate for the missing ones? In crystallographic refinement, the likelihood is usually approximated by the product of probabilities corresponding to individual structure factors, implying their independence. Defining the maximum-likelihood (ML) target as a negative logarithm of the likelihood, results in a sum over reflections where each term is expressed through the corresponding F obs hkl , F calc hkl and two parameters, hkl and hkl (or a single parameter A, hkl ) that reflect model errors.For the purpose of refinement one needs first to obtain the shape of the ML target, i.e. to find the parameter values that define the target, and then search for a better model that minimizes this target. The values of the parameters hkl , hkl , or A, hkl may be assumed to be the same for reflections in thin resolution shells and can be obtained from comparison of F obscorresponding to the available model. This approach gives accurate parameter estimates for models that have never been refined but not for those refined previously. The reason for this is that structure factors F calc hkl È É are calculated through atomic parameters, refinement of which against the full set of structure-factor amplitudes makes the calculated structure factors in this data set S mutually dependent.A solution was to estimate hkl , hkl , or A, hkl , using F calc hkl È É from a subset of diffraction data excluded from refinement (Lunin & Skovoroda, 1995;Pannu & Read, 1996;Murshudov et al., 1997). The original goal to split the whole data set into work and test subsets (Brü nger, 1992) was to use the former for refinement and the latter only for model validation. The test subset S test S usually contains reflections chosen randomly and uniformly in space, 5-10% of the total set, and the crystallographic R factor calculated over S test ...