BACKGROUND: We study the problem of mapping proteins between two protein families in the presence of paralogs. This problem occurs as a difficult subproblem in coevolution-based computational approaches for protein-protein interaction prediction. RESULTS: Similar to prior approaches, our method is based on the idea that coevolution implies equal rates of sequence evolution among the interacting proteins, and we provide a first attempt to quantify this notion in a formal statistical manner. We call the units that are central to this quantification scheme the units of coevolution. A unit consists of two mapped protein pairs and its score quantifies the coevolution of the pairs. This quantification allows us to provide a maximum likelihood formulation of the paralog mapping problem and to cast it into a binary quadratic programming formulation. CONCLUSION: CUPID, our software tool based on a Lagrangian relaxation of this formulation, makes it, for the first time, possible to compute state-of-the-art quality pairings in a few minutes of runtime. In summary, we suggest a novel alternative to the earlier available approaches, which is statistically sound and computationally feasible.

We present a mathematical model and exact algorithm for optimally aligning protein structures using the DALI scoring model. This scoring model is based on comparing the interresidue distance matrices of proteins and is used in the popular DALI software tool, a heuristic method for protein structure alignment. Our model and algorithm extend an integer linear programming approach that has been previously applied for the related, but simpler, contact map overlap problem. To this end, we introduce a novel type of constraint that handles negative score values and relax it in a Lagrangian fashion. The new algorithm, which we call DALIX, is applicable to any distance matrix-based scoring scheme. We also review options that allow to consider fewer pairs of interresidue distances explicitly because their large number hinders the optimization process. Using four known data sets of varying structural similarity, we compute many provably score-optimal DALI alignments. This allowed, for the first time, to evaluate the DALI heuristic in sound mathematical terms. The results indicate that DALI usually computes optimal or close to optimal alignments. However, we detect a subset of small proteins for which DALI fails to generate any significant alignment, although such alignments do exist.

MOTIVATION: Structural alignments of proteins are important for identification of structural similarities, homology detection and functional annotation. The structural alignment problem is well studied and computationally difficult. Many different scoring schemes for structural similarity as well as many algorithms for finding high-scoring alignments have been proposed. Algorithms using contact map overlap (CMO) as scoring function are currently the only practical algorithms able to compute provably optimal alignments. RESULTS: We propose a new mathematical model for the alignment of inter-residue distance matrices, building upon previous work on maximum CMO. Our model includes all elements needed to emulate various scoring schemes for the alignment of protein distance matrices. The algorithm that we use to compute alignments is practical only for sparse distance matrices. Therefore, we propose a more effective scoring function, which uses a distance threshold and only positive structural scores. We show that even under these restrictions our approach is in terms of alignment accuracy competitive with state-of-the-art structural alignment algorithms, whereas it additionally either proves the optimality of an alignment or returns bounds on the optimal score. Our novel method is freely available and constitutes an important promising step towards truly provably optimal structural alignments of proteins. AVAILABILITY: An executable of our program PAUL is available at http://planet-lisa.net/.