Algorithms for Loop Matchings

Abstract

A simplified (two-base) version of the problem of planar folding of long chains (e.g., RNA and DNA biomolecules) is formulated as a matching problem. The chain is prescribed as a loop or circular sequence of letters A and B, n units long. A matching here means a set of A-B base pairings or matches obeying a planarity condition: no two matches may cross each other if drawn on the interior of the loop. Also, no two adjacent letters may be matched. We present a dynamic programming algorithm requiring $O( {n^3 } )$ steps and $O( {n^2 } )$ storage which computes the size of the maximum for the given A-B base sequence and which also allows reconstructing a particular folded form of the original string which realizes the maximum matching size. The algorithm can be adapted to deal with sequences with larger alphabets and with weighted matchings. An algorithm is also presented for a modified problem closer to the biochemical problem of interest: We demand that every match must be adjacent to another match, forcing groups of two or more parallel matches. Some results on the expected maximum matching size are presented. As $n \to \infty $, at least 80% of the vertices can be matched on the average on an A-B string of size n. We briefly discuss the practical application of the algorithm by using contracted versions of very long molecules with a preliminary block construction. A maximum matching is presented for the J-gene of the $\phi $X174 DNA virus. We conclude by stating some problems requiring further study.