Bulk-solvent and overall scaling revisited: faster calculations, improved results. Corrigendum.

Equations in Sections 2.3 and 2.4 of the article by Afonine et al. [Acta Cryst. (2013). D69, 625-634] are corrected.

Bulk-solvent and overall scaling revisited: faster calculations, improved results.

A fast and robust method for determining the parameters for a flat (mask-based) bulk-solvent model and overall scaling in macromolecular crystallographic structure refinement and other related calculations is described. This method uses analytical expressions for the determination of optimal values for various scale factors. The new approach was tested using nearly all entries in the PDB for which experimental structure factors are available. In general, the resulting R factors are improved compared with previously implemented approaches. In addition, the new procedure is two orders of magnitude faster, which has a significant impact on the overall runtime of refinement and other applications. An alternative function is also proposed for scaling the bulk-solvent model and it is shown that it outperforms the conventional exponential function. Similarly, alternative methods are presented for anisotropic scaling and their performance is analyzed. All methods are implemented in the Computational Crystallography Toolbox (cctbx) and are used in PHENIX programs.

On a fast calculation of structure factors at a subatomic resolution.

In the last decade, the progress of protein crystallography allowed several protein structures to be solved at a resolution higher than 0.9 A. Such studies provide researchers with important new information reflecting very fine structural details. The signal from these details is very weak with respect to that corresponding to the whole structure. Its analysis requires high-quality data, which previously were available only for crystals of small molecules, and a high accuracy of calculations. The calculation of structure factors using direct formulae, traditional for 'small-molecule' crystallography, allows a relatively simple accuracy control. For macromolecular crystals, diffraction data sets at a subatomic resolution contain hundreds of thousands of reflections, and the number of parameters used to describe the corresponding models may reach the same order. Therefore, the direct way of calculating structure factors becomes very time expensive when applied to large molecules. These problems of high accuracy and computational efficiency require a re-examination of computer tools and algorithms. The calculation of model structure factors through an intermediate generation of an electron density [Sayre (1951). Acta Cryst. 4, 362-367; Ten Eyck (1977). Acta Cryst. A33, 486-492] may be much more computationally efficient, but contains some parameters (grid step, 'effective' atom radii etc.) whose influence on the accuracy of the calculation is not straightforward. At the same time, the choice of parameters within safety margins that largely ensure a sufficient accuracy may result in a significant loss of the CPU time, making it close to the time for the direct-formulae calculations. The impact of the different parameters on the computer efficiency of structure-factor calculation is studied. It is shown that an appropriate choice of these parameters allows the structure factors to be obtained with a high accuracy and in a significantly shorter time than that required when using the direct formulae. Practical algorithms for the optimal choice of the parameters are suggested.

Likelihood-based refinement. I. Irremovable model errors.

In conventional structure refinement, the discrepancy between the calculated magnitudes and those observed in X-ray experiments is attributed to errors inherent in preliminary assigned values of the model parameters. However, the chosen set of model parameters may not be adequate to describe the structure factors precisely. For example, if some atoms are not included in the current model, then the structure factors calculated from such a partial model contain 'irremovable errors'. These errors cannot be eliminated by any choice of the parameters of the partial structure. Probabilistic modelling suggests a way to take irremovable errors into account. Every trial set of values of the model parameters is now associated with the joint probability distribution of the calculated magnitudes, rather than with a particular set of magnitudes. The new goal of the refinement is formulated as the search for the distribution that is the most consistent with the observed data. The statistical likelihood is a possible measure of the consistency. The suggested quadratic approximation of the likelihood function allows the likelihood-based refinement to be considered as a kind of least-squares refinement that uses appropriate weights and modified targets for the calculated magnitudes. This in turn enables the analysis of tendencies of the likelihood-based refinement in comparison with the classical least-squares refinement.