Steiner minimal trees, twist angles, and the protein folding problem

J. MacGregor Smith, Yunho Jang, Moon K. Kim
2006 Proteins: Structure, Function, and Bioinformatics  
The Steiner Minimal Tree (SMT) problem determines the minimal length network for connecting a given set of vertices in threedimensional space. SMTs have been shown to be useful in the geometric modeling and characterization of proteins. Even though the SMT problem is an NP-Hard Optimization problem, one can define planes within the amino acids that have a surprising regularity property for the twist angles of the planes. This angular property is quantified for all amino acids through the
more » ... tree topology structure. The twist angle properties and other associated geometric properties unique for the remaining amino acids are documented in this paper. We also examine the relationship between the Steiner ratio q and the torsion energy in amino acids with respect to the side chain torsion angle v 1 . The q value is shown to be inversely proportional to the torsion energy. Hence, it should be a useful approximation to the potential energy function. Finally, the Steiner ratio is used to evaluate folded and misfolded protein structures. We examine all the native proteins and their decoys at http://dd.stanford.edu. and compare their Steiner ratio values. Because these decoy structures have been delicately misfolded, they look even more favorable than the native proteins from the potential energy viewpoint. However, the q value of a decoy folded protein is shown to be much closer to the average value of an empirical Steiner ratio for each residue involved than that of the corresponding native one, so that we recognize the native folded structure more easily. The inverse relationship between the Steiner ratio and the energy level in the protein is shown to be a significant measure to distinguish native and decoy structures. These properties should be ultimately useful in the ab initio protein folding prediction. Proteins 2007;66:889-902. V V C 2006 Wiley-Liss, Inc.
doi:10.1002/prot.21257 pmid:17173288 fatcat:pxg6wmarqjbphiumxmwnrxbn3i