Application of the complex multivariate normal distribution to crystallographic methods with insights into multiple isomorphous replacement phasing

Probabilistic methods involving maximum-likelihood parameter estimation have become a powerful tool in computational crystallography. At the centre of these methods are the relevant probability distributions. Here, equations are developed based on the complex multivariate normal distribution that generalize the distributions currently used in maximum-likelihood model and heavy-atom refinement. In this treatment, the effects of various sources of error in the experiment are considered separately and allowance is made for correlations among sources of error. The multivariate distributions presented are closely related to the distributions previously derived in ab initio phasing and can be applied to many different aspects of a crystallographic structure-determination process including model refinement, density modification, heavy-atom phasing and refinement or combinations of them. The underlying probability distributions for multiple isomorphous replacement are re-examined using these techiques. The re-analysis requires the underlying assumptions to be made explicitly and results in a variance term that, unlike those previously used for maximum-likelihood multiple isomorphous replacement phasing, is expressed explicitly in terms of structure-factor covariances. Test cases presented show that the newly derived multiple isomorphous replacement likelihood functions perform satisfactorily compared with currently used programs.