A note on scoring clones given a probe ordering

Algorithms for computing and integrating physical maps using unique probes

Current physical mapping projects based on STS-probes involve additional clues such as the fact that some probes are anchored to a known map and that others come from the ends of clones. Because of the disparate combinatorial contributions of these varied data items, it is difficult to design a "tailored" algorithm that incorporates them all. Moreover, it is inevitable that new experiments will provide new kinds of data, making obsolete any such algorithm. We show how to convert the physical mapping problem into a 0/1 linear programming (LP) problem. We further show how one can incorporate additional clues as additional constraints in the LP formulation. We give a simple relaxation of the 0/1 LP problem, which solves problems of the same scale as previously reported tailored algorithms, to equal or greater optimization levels. We also present a theorem proving that when the data is 100% accurate, then the relaxed and integer solutions coincide. The LP algorithm suffices to solve problems on the order of 80-100 probes--the typical size of the 2- or 3-connected contigs of Arratia et al. (1991). We give a heuristic algorithm which attempts to order and link the set of LP-solved contigs. Unlike previous work, this algorithm only links and orders contigs when the join is 90% or more likely to be correct. It is our view that there is no value in computing an optimal solution with respect to some criteria over very noisy data as this optimal solution rarely corresponds to the true solution. The paper involves extensive empirical trials over real and simulated data.

A branch-and-cut approach to physical mapping of chromosomes by unique end-probes

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.