Approximation Algorithms for 3-D Common Substructure Identification in Drug and Protein Molecules

Abstract

Identifying the common 3-D substructure between two drug or protein molecules is an important problem in synthetic drug design and molecular biology. This problem can be represented as the following geometric pattern matching problem: given two point sets A and B in three-dimensions, and a real number∈ > 0, find the maximum cardinality subset S ⊆ S for which there is an isometry I, such that each point of I(S) is within (ie253-1) distance of a distinct point of B. Since it is difficult to solve this problem exactly, in this paper we have proposed several approximation algorithms with guaranteed approximation ratio. Our algorithms can be classifed into two groups. In the first we extend the notion of partial decision algorithms for ∈-congruence of point sets in 2-D in order to approximate the size of S. All the algorithms in this class exactly satisfy the constraint imposed by ∈. In the second class of algorithms this constraint is satisfied only approximately. In the latter case, we improve the known approximation ratio for this class of algorithms, while keeping the time complexity unchanged. For the existing approximation ratio, we propose algorithms with substantially better running times. We also suggest several improvements of our basic algorithms, all of which have a running time of O(n ^8.5). These improvements consist of using randomization, and/or an approximate maximum matching scheme for bipartite graphs.

This work was carried out when the author was at IIT Kanpur.

Author to whom all correspondence should be directed. Currently visiting University of Nebraska-Lincoln.