Multidimensional Histograms for Density Modification

Abstract

Publisher Summary Density modification improves the quality of an approximate electron-density map by imposing physical constraints based on some conserved features of the correct electron density map. Density modification methods generally require an initial map with substantial phase information. In most cases, these phases are obtained from multiple isomorphous replacement (MIR) or multiwavelength anomalous dispersion (MAD), and these phases are of pivotal importance when the experimental source of phase information is bimodal as it is in single-wavelength anomalous scattering (SAD) and single isomorphous replacement methods (SIR). The most commonly used density modification method is solvent flattening that exploits the observation that the solvent region of an electron density map is featureless at medium resolution because of the high thermal motion and disorder of the solvent molecules. Xiang and Carter proposed extending the density histogram to include relationships between neighboring density values via multidimensional histograms defined as joint probabilities of the density values and their higher-order derivatives. In the conventional one-dimensional (1D) electron density histogram matching method, a one-to-one mapping is made on the original electron density to the new electron density so that the density histogram of the modified map matches that of the ideal histogram. The two-dimensional (2D) histogram specifies not only the probability of the electron density value for a given grid point in the map but also the local environment around that grid point as reflected by the density gradients that arise from chemical bonding. The strategy of using alternating 1D histogram matching to achieve 2D histogram matching can be generalized to the matching of higher dimension histograms.