A Dense Graph Model for Haplotype Inference

Phasing genotype data to identify the composite haplotype p airs is a widely-studied problem due to its value for genome-wide association studies, population genetics res ea ch, and other significant endeavors. The accuracy of the phasing is crucial as identification of haplotypes is freque ntly the first step of expensive and vitally important studie s. We present a combinatorial approach to this problem, which we c all Splitting Heirs, that is based on a dense graph model. We have tested Splitting Heirs with several popular existing p hasing methods including PHASE, HAP, and Pure Parsimony, on seven sets of real biological haplotype data. Our method y iel s the highest accuracy obtainable by these methods in al l cases. Furthermore, Splitting Heirs is robust and had highe r accuracy than any of the other approaches for the two data sets with high recombination rates. The success of Splittin g Heirs validates the assumptions made by the dense graph model and highlights the benefits of finding globally optimal solutions. Many of the methods previously used for haplotype inference have favored reduction of the cardinality of unique haplotypes. Pure Parsimony [2] is an extreme case in which a s et of haplotypes are found such that the number of unique haplotypes is the least possible. Splitting Heirs favors re duc d cardinality, but simultaneously considers other fav or ble properties and does not always yield a strictly parsimoniou s solution. Some of the previous algorithms (e.g. PHASE [6]) have favore d additional properties, such as pair-wise similarities between haplotypes. That is, if a potential haplotype is sim ilar to one in the current solution, it is favored. In contras , Splitting Heirs favorscluster-wide similarities by favoring solutions in which many haplotype s are similar to a number of other haplotypes. The dense graph model can be used to quan tify the quality of a solution with regard to reduced cardinality and cluster-wide similarities. Let h equal the number of unique haplotypes in a solution. Conside r a graph withh nodes, in which each node represents a haplotype in the solution. The weight on an edge in the graph is set equal to the distance between the two haplotypes that are endpoints of the edge. Distances betwee n haplotypes can be defined in various ways. A simple distance measure is just the number of sites in which they dif fer. If pair-wise similarities were the only concern, a grap h to consider would contain only edges that connect each haplo type with its nearest neighbor. When relying completely on simple pair-wise distances, it is possible to have h/2 disjoint subgraphs with arbitrarily large distances betwe en them. In real populations, we would expect to find clusters of haplotypes that are similar to each other, so it is desirab le to enforce similarities beyond a single nearest neighbor. In a dense graph model, the density of the graph is required to be greate than or equal to a given value, α. The density of a graph can be defined as e/h, wheree is the number of edges in the graph. By considering these addi tional edges, similarities beyond single nearest neighbors are ta ken into consideration. We evaluate the quality of the dense graph solution using: