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Complex karyotype changes are widespread in cancer genomes. A major gap in cancer genome characterization is the resolution of rearranged chromosomes with chromosome-length continuity. Here, we describe a two-tiered approach to determine the segmental composition of rearranged chromosomes with haplotype resolution. First, we present refLinker, a bioinformatic method for robust determination of chromosomal haplotypes using cancer Hi-C data. By contrast with existing methods, refLinker is insensitive to the presence of large-scale DNA deletions, duplications, and high-level amplification in cancer genomes. Second, we demonstrate a computational strategy to determine the segmental structure of rearranged chromosomes using haplotype-specific Hi-C contacts. We apply these methods to breast cancer genomes and provide direct evidence for long-range transcriptional changes associated with rearrangements of the inactive X chromosome. Together, these results highlight refLinker's broad utility for studying the functional consequences of chromosomal rearrangements.