On protein structure alignment under distance constraint

Abstract

a b s t r a c tIn this paper we study the protein structure comparison problem where each protein is modeled as a sequence of 3D points, and a contact edge is placed between every two of these points that are sufficiently close. Given two proteins represented this way, our problem is to find a subset of points from each protein, and a bijective matching of points between these two subsets, with the objective of maximizing either (A) the size of the subsets (the LCP problem), or (B) the number of edges that exist simultaneously in both subsets (the CMO problem), under the requirement that only points within a specified proximity can be matched. It is known that the general CMO problem (without the proximity requirement) is hard to approximate. However, with the proximity requirement, it is known that if a minimum inter-residue distance is imposed on the input, approximate solutions can be efficiently obtained. In this paper we mainly show that the CMO problem under these conditions: (1) is NP-hard, but (2) allows a PTAS. The rest of this paper shows algorithms for the LCP problem which improve on known results.

Improved Algorithms for Matching r-Separated Sets with Applications to Protein Structure Alignment

“…Moreover, with the asymptotic time complexity ofÕðn 6:75 Þ, the resulting algorithm even compares favorably to the algorithm of Li and Ng [11] for order-dependent alignments of r-separated sets. For nonsequential alignments, the algorithm of Li and Ng is still faster by a factor of n 0:25 .…”

“…Improvements over the method in [7] have been published, including the method of Li and Ng, which finds an approximate solution to order-independent alignment problem in time Oðn 8 Þ for general point sets and Oðn 7 Þ for protein structures [11]. This algorithm, which we will call AP P ROX ALIGN, is based on the concept of a radial pair.…”

“…It is shown in [11] that, if ðb 1 ; b 2 Þ is a radial pair of Q , then the errors of at most introduced by a transformation T to ðb 1 Þ and ðb 2 Þ will allow for a rotation R around the axis through the points T ðb 1 Þ and T ðb 2 Þ that results in errors of at most 3 from ðbÞ, for all b 2 Q . Furthermore, as explained in [11], if ðb 1 ; b 2 Þ is a radial pair of Q and if g ðb 1 Þ ¼ a 1 and g ðb 2 Þ ¼ a 2 , to find the transformation T that results in at most ð À 1Þ difference from ðbÞ, for each b 2 Q , it is enough to examine Oð1=ð À 1Þ 5 Þ transformations in the discretized space. The algorithm in [11] inspects all pairs of points from B (candidate radial pairs of Q ) and their matched pairs of points from A.…”

“…Furthermore, as explained in [11], if ðb 1 ; b 2 Þ is a radial pair of Q and if g ðb 1 Þ ¼ a 1 and g ðb 2 Þ ¼ a 2 , to find the transformation T that results in at most ð À 1Þ difference from ðbÞ, for each b 2 Q , it is enough to examine Oð1=ð À 1Þ 5 Þ transformations in the discretized space. The algorithm in [11] inspects all pairs of points from B (candidate radial pairs of Q ) and their matched pairs of points from A. In total, there are n 2 Â n 2 ¼ n 4 possibilities.…”

“…More specifically, in Section 3.1, we describe an asymptotically faster approximation algorithm for threshold-LCP for a pair of r-separated sets in IR 3 , which builds upon the method in [7]. In Sections 3.2 and 3.3, the OP T IMAL algorithm [10] and the method of Li and Ng [11] are combined to develop improved running time algorithms for exact threshold-LCP and threshold-PM.…”