Picking Alignments from (Steiner) Trees

Abstract

The application of Needleman-Wunsch alignment techniques to biological sequences is complicated by two serious problems when the sequences are long: the running time, which scales as the product of the lengths of sequences, and the difficulty in obtaining suitable parameters that produce meaningful alignments. The running time problem is often corrected by reducing the search space, using techniques such as banding, or chaining of high-scoring pairs. The parameter problem is more difficult to fix, partly because the probabilistic model, which Needleman-Wunsch is equivalent to, does not capture a key feature of biological sequence alignments, namely the alternation of conserved blocks and seemingly unrelated nonconserved segments. We present a solution to the problem of designing efficient search spaces for pair hidden Markov models that align biological sequences by taking advantage of their associated features. Our approach leads to an optimization problem, for which we obtain a 2-approximation algorithm, and that is based on the construction of Manhattan networks, which are close relatives of Steiner trees. We describe the underlying theory and show how our methods can be applied to alignment of DNA sequences in practice, successfully reducing the Viterbi algorithm search space of alignment PHMMs by three orders of magnitude.

The Transitive Minimum Manhattan Subnetwork Problem in 3 dimensions

“…al. [8] use an MMN approach as a preprocessing step to accelerate the Viterbi algorithm for an alignment problem.…”

“…Up to now, the complexity status of the MMN problem is still unknown, but mostly suspected to be NP -hard as well. Hence, the previous work on the MMN problem solely features approximation algorithms: Gudmundson et al [5] presented an 8-approximation in time O(n log n) and a 4-approximation in time O(n 3 ) (which is used in [8]). Benkert et al [1] introduced a 3-approximation in time O(n log n).…”

A rounding algorithm for approximating minimum Manhattan networks

“…Sparse geometric spanners have applications in VLSI circuit design, network design, distributed algorithms and other areas, see for example the survey [6] and the book [9]. Finally, Lam, Alexandersson, and Pachter [8] suggested the use of minimum Manhattan networks to design efficient search spaces for pair hidden Markov model (PHMM) alignment algorithms.…”

The minimum Manhattan network problem: Approximations and exact solutions

“…In this paper we investigate how the extra degree of freedom in routing edges can be used to construct Manhattan networks of minimum total length, socalled minimum Manhattan networks (MMN). The MMN problem may have applications in city planning or VLSI layout, but Lam et al [LAP03] also describe an application in computational biology. For aligning gene sequences they propose a three-step approach.…”