Exact and approximation algorithms for DNA sequence reconstruction.

Abstract

The DNA sequence in every human being is a text of three billion characters from a four letter alphabet; determining this sequence is a major project in molecular biology. The fundamental task biologists face is to reconstruct a long sequence given short fragments from unknown locations. These fragments contain errors, and may represent the sequence on one strand of the double-helix, or the reverse complement sequence on the other strand. The Sequence Reconstruction Problem is, given a collection F of fragment sequences and an error rate 0 ≤ ε < 1, find a shortest sequence S such that every fragment F ∈ F, or its reverse complement, matches a substring of S with at most ε|F| errors. Sequence Reconstruction is NP-complete. We decompose the problem into (1) constructing a graph of approximate overlaps between pairs of fragments, (2) selecting a set of overlaps of maximum total weight that induce a consistent layout of the fragments, (3) merging the overlaps into a multiple sequence alignment and voting on a consensus. A solution to (1) through (3) yields a reconstructed sequence feasible at error rate 2ε/(1-ε) and at most a factor 1/1-ε longer than the shortest reconstruction, given some assumptions on fragment error. We define a measure of the overlap in a reconstruction, show that maximizing the overlap minimizes the length, and that approximating (2) within a factor of α approximates Sequence Reconstruction within a factor of (1- ε)α under the overlap measure. We construct the overlap graph for (1) in O(εN²) time given fragments of total length N at error rate ε. We develop two exact and two approximation algorithms for (2). Our best exact algorithm computes an optimal layout for a graph of E overlaps and V fragments in O(K(E + V log V)) time, where K ≤ 2ᴱ is the size of the branch-and-bound search tree. Our best approximation algorithm computes a layout with overlap at least 1/2 the maximum in O(V(E + V log V)log V) time. This is the first treatment of Sequence Reconstruction with inexact data and unknown complementarity.