Bayesian Correlated MAD Phasing

Abstract

A Bayesian treatment for phase calculation in the multiwavelength anomalous diffraction (MAD) techni- que is presented. This approach explicitly treats effects of errors correlated among measurements at different wavelengths and between Bijvoet pairs. The resulting method, which is called Bayesian correlated MAD phasing, gives proper statistical consideration to all data and does not give special treatment to data from a particular wavelength. Results obtained using Bayesian correlated MAD phasing and two other strategies on both a model test case and on data obtained in two actual MAD experiments are compared. Although all proce- dures performed well when the completeness of the data was high, it is shown that Bayesian correlated MAD phasing is more robust with respect to incompleteness of data than the other methods are. At 60% complete- ness the improvement over other methods for the examples given was nearly 50% in the correlation coefficients, and made a substantial difference in the interpretability of an electron-density map. 1. MAD data and its analysis In the multiwavelength anomalous diffraction (MAD) method, crystallographic phases are estimated from the wavelength dependence of diffracted intensities when an X-ray beam from a tunable source is stepped over an absorption edge of some heavy atoms present in small numbers in the asymmetric unit of a crystal (Karle, 1980). MAD experiments measure amplitudes of Bijvoet pairs F + and F- for a crystal at two or more wavelengths chosen so that the f' values for the heavy atoms vary as much as possible among wavelengths and so that the f" values are as large as possible (Hendrickson, 1985). As in a multiple isomorphous replacement (MIR) experiment, the locations of heavy atoms within the asymmetric unit are generally obtained using a Patterson or a difference Patterson synthesis, and parameters describing these heavy atoms are refined. Estimates of heavy-atom structure factors calculated from the heavy-atom model at each wave-