Combinatorial Problems on Strings with Applications to Protein Folding

Abstract

We consider the problem of protein folding in the HP model on the 3D square lattice. This problem is combinatorially equivalent to folding a string of 0’s and 1’s so that the string forms a self-avoiding walk on the lattice and the number of adjacent pairs of 1’s is maximized. The previously best-known approximation algorithm for this problem has a guarantee of $\frac{3}{8}=.375$ [HI95]. In this paper, we first present a new $\frac{3}{8}$-approximation algorithm for the 3D folding problem that improves on the absolute approximation guarantee of the previous algorithm. We then show a connection between the 3D folding problem and a basic combinatorial problem on binary strings, which may be of independent interest. Given a binary string in { a,b }^*, we want to find a long subsequence of the string in which every sequence of consecutive a’s is followed by at least as many consecutive b’s. We show a non-trivial lower-bound on the existence of such subsequences. Using this result, we obtain an algorithm with a slightly improved approximation ratio of at least .37501 for the 3D folding problem. All of our algorithms run in linear time.