A fundamental problem in computational biology is the construction of physical maps of chromosomes from hybridization experiments between unique probes and clones of chromosome fragments in the presence of error. Alizadeh, Karp, Weisser and Zweig (Algorithmica 13:1/2, 52-76, 1995) first considered a maximum-likelihood model of the problem that is equivalent to finding an ordering of the probes that minimizes a weighted sum of errors and developed several effective heuristics. We show that by exploiting information about the end-probes of clones, this model can be formulated as a Weighted Betweenness Problem. This affords the significant advantage of allowing the well-developed tools of integer linear-programming and branch-and-cut algorithms to be brought to bear on physical mapping, enabling us for the first time to solve small mapping instances to optimality even in the presence of high error. We also show that by combining the optimal solution of many small overlapping Betweenness Problems, one can effectively screen errors from larger instances and solve the edited instance to optimality as a Hamming-Distance Traveling Salesman Problem. This suggests a new approach, a Betweenness-Traveling Salesman hybrid, for constructing physical maps.
One of the key open problems in large-scale DNA sequence assembly is the correct reconstruction of sequences that contain repeats. A long repeat can confound a sequence assembler into falsely overlaying fragments that sample its copies, effectively compressing out the repeat in the reconstructed sequence. We call the task of correcting this compression by separating the overlaid fragments into the distinct copies they sample, the repeat separation problem. We present a rigorous formulation of repeat separation in the general setting without prior knowledge of consensus sequences of repeats or their number of copies. Our formulation decomposes the task into a series of four subproblems, and we design probabilistic tests or combinatorial algorithms that solve each subproblem. The core subproblem separates repeats using the so-called k-median problem in combinatorial optimization, which we solve using integer linear-programming. Experiments with an implementation show we can separate fragments that are over laid at 10 times the coverage with very few mistakes in a few seconds of computation, even when the sequencing error rate and the error rate between copies are identical. To our knowledge this is the first rigorous and fully general approach to separating repeats that directly addresses the problem.
For as long as biologists have been computing alignments of sequences, the question of what values to use for scoring substitutions and gaps has persisted. While some choices for substitution scores are now common, largely due to convention, there is no standard for choosing gap penalties. An objective way to resolve this question is to learn the appropriate values by solving the Inverse String Alignment Problem: given examples of correct alignments, find parameter values that make the examples be optimal-scoring alignments of their strings. We present a new polynomial-time algorithm for Inverse String Alignment that is simple to implement, fast in practice, and for the first time can learn hundreds of parameters simultaneously. The approach is also flexible: minor modifications allow us to solve inverse unique alignment (find parameter values that make the examples be the unique optimal alignments of their strings), and inverse near-optimal alignment (find parameter values that make the example alignments be as close to optimal as possible). Computational results with an implementation for global alignment show that, for the first time, we can find best-possible values for all 212 parameters of the standard protein-sequence scoring-model from hundreds of alignments in a few minutes of computation. © Springer-Verlag Berlin Heidelberg 2006.