Facing the phase problem

A note on the Hendrickson-Lattman phase probability distribution and its equivalence to the generalized von Mises distribution

Hendrickson & Lattman [Acta Cryst. (1970), B26, 136-143] introduced a method for representing crystallographic phase probabilities defined on the unit circle. Their approach could model the bimodal phase probability distributions that can result from experimental phase determination procedures. It also provided simple and highly effective means to combine independent sources of phase information. The present work discusses the equivalence of the Hendrickson-Lattman distribution and the generalized von Mises distribution of order two, which has been studied in the statistical literature. Recognizing this connection allows the Hendrickson-Lattman distribution to be expressed in an alternative form which is easier to interpret, as it involves the location and concentration parameters of the component von Mises distributions. It also allows clarification of the conditions for bimodality and access to a simplified analytical method for evaluating the trigonometric moments of the distribution, the first of which is required for computing the best Fourier synthesis in the presence of phase, but not amplitude, uncertainty.

Systematic enhancement of protein crystallization efficiency by bulk lysine-to-arginine (KR) substitution

Structural genomics consortia established that protein crystallization is the primary obstacle to structure determination using x-ray crystallography. We previously demonstrated that crystallization propensity is systematically related to primary sequence, and we subsequently performed computational analyses showing that arginine is the most overrepresented amino acid in crystal-packing interfaces in the Protein Data Bank. Given the similar physicochemical characteristics of arginine and lysine, we hypothesized that multiple lysine-to-arginine (KR) substitutions should improve crystallization. To test this hypothesis, we developed software that ranks lysine sites in a target protein based on the redundancy-corrected KR substitution frequency in homologs. This software can be run interactively on the worldwide web at https://www.pxengineering.org/. We demonstrate that three unrelated single-domain proteins can tolerate 5-11 KR substitutions with at most minor destabilization, and, for two of these three proteins, the construct with the largest number of KR substitutions exhibits significantly enhanced crystallization propensity. This approach rapidly produced a 1.9 Å crystal structure of a human protein domain refractory to crystallization with its native sequence. Structures from Bulk KR-substituted domains show the engineered arginine residues frequently make hydrogen-bonds across crystal-packing interfaces. We thus demonstrate that Bulk KR substitution represents a rational and efficient method for probabilistic engineering of protein surface properties to improve crystallization.

Commentary on `Facing the phase problem' by Wayne Hendrickson

A commentary on Wayne Hendrickson's article `Facing the phase problem'.