Overcrossing spectra of protein backbones: characterization of three-dimensional molecular shape and global structural homologies.

A procedure is developed and applied to characterize the global shape and folding features of the backbone of a chain molecule. The methodology is based on the following concept: the probability of observing a rigid placement of a backbone in 3-space as a projected curve with N overcrossings. The numerical computation of these probabilities allows one to construct the overcrossing spectrum of a macromolecule at a given configuration. Although the spectrum is built from the knowledge of the nuclear geometry of the main-chain atoms, the shape descriptor overlooks local geometrical features (such as distances and contacts) and provides a characterization of essential (topological) features of the overall fold, such as its compactness and degree of entanglement. In contrast with other shape descriptors, the present approach gives an absolute characterization of the configuration considered, and not one that is relative to an arbitrarily chosen reference structure. Moreover, it is possible to discriminate between folding features that otherwise may not be distinguished when using other geometrical or topological global descriptors. The overcrossing spectrum is proposed as a tool that complements current structural analyses of macromolecules, especially when monitoring structural homologies within a group of related or unrelated polymers. In this work, we apply the methodology to the analysis of proteins having the globin fold. The results are compared with those of other proteins exhibiting similar size and number of residues. Some basic properties of the spectra as a function of the chain length are also discussed.