On the Consistency of a Physical Mapping Method to Reconstruct a Chromosome in Vitro

During recent years considerable effort has been invested in creating physical maps for a variety of organisms as part of the Human Genome Project and in creating various methods for physical mapping. The statistical consistency of a physical mapping method to reconstruct a chromosome, however, has not been investigated. In this paper, we first establish that a model of physical mapping by binary fingerprinting of DNA fragments is identifiable using the key assumption--for a large randomly generated recombinant DNA library, there exists a staircase of DNA fragments across the chromosomal region of interest. Then we briefly introduce epi-convergence theory of variational analysis and transform the physical mapping problem into a constrained stochastic optimization problem. By doing so, we prove epi-convergence of the physical mapping model and epi-convergence of the physical mapping method. Combining the identifiability of our physical mapping model and the epi-convergence of a physical mapping method, finally we establish strong consistency of a physical mapping method.