101 optimal PDB structure alignments: a branch-and-cut algorithm for the maximum contact map overlap problem

Structure comparison is a fundamental problem for structural genomics. A variety of structure comparison methods were proposed and several protein structure classification servers e.g., SCOP, DALI, CATH, were designed based on them, and are extensively used in practice. This area of research continues to be very active, being energized bi-annually by the CASP folding competitions, but despite the extraordinary international research effort devoted to it, progress is slow. A fundamental dimension of this bottleneck is the absence of rigorous algorithmic methods. A recent excellent survey on structure comparison by Taylor et.al. [23] records the state of the art of the area: In structure comparison, we do not even have an algorithm that guarantees an optimal answer for pairs of structures …

In this paper we provide the first rigorous algorithm for structure comparison. Our method is based on developing an effective integer linear programming (IP) formulation of protein structure contact maps overlap (CMO), and a branch-and-cut strategy that employs lower-bounding heuristics at the branch nodes. Our algorithms identified a gallery of optimal and near-optimal structure alignments for pairs of proteins from the Protein Data Bank with up to 80 amino acids and about 150 contacts each — problems of instance size of about 300. Although these sizes also reflect our current limitations, these are the first provable optimal and near-optimal algorithms in the literature for a measure of structure similarity which sees extensive practical use. At the heart of our success in finding optimal alignments is a reduction of the CMO optimization to the maximum independent set (MIS) problem on special graphs. For CMO instances of size 300, the corresponding MIS graph instance contains about 10,000 nodes. While our algorithms are able to solve to optimality MIS problem of these sizes, the known optimal algorithms for the MIS on general graphs can at present only solve instances with up to a few hundred nodes. This is the first effective use of IP methods in protein structure comparison; the biomolecular structure literature contains only one other effective IP method devoted to RNA comparison, due to Lenhof et.al. [18].

The hybrid heuristic approach that worked well for providing lower bounds in the branch and cut algorithm was tried on large proteins in a test set suggested by Jeffrey Skolnick. It involved 33 proteins classified into four families: Flavodoxin-like fold CheY-related, Plastocyanin, TIM Barrel, and Ferratin. Out of the set of all 528 pairwise structure alignments, we have validated the clustering with a 98.7% accuracy (1.3% false negatives and 0% false positives).