Representing stereochemical information in macromolecular electron-density distributions by multi-dimensional histograms

Previous studies have demonstrated the value of ideal electron-density histograms as targets for the corresponding histograms of experimental electron-density maps. The electron-density histogram makes use of density values as independent objects, and no relationship between them is taken into account. Extension to include the relationships between neighboring density values leads naturally to a multi-dimensional histogram defined as the joint frequency of the density values and their higher order derivatives. We show here that the multi-dimensional histogram including additional dimensions composed of the gradient magnitude and Laplacian of the density is minimally dependent on molecular folding and packing, and captures substantially more stereochemical information than the conventional electron-density histogram. The gradient histogram appears to be much more sensitive to phase errors than the conventional electron-density histogram. Potential uses of the multi-dimensional histogram include improved targets for density modification and more reliable figures of merit for evaluating correct phases.