Inverse inbreeding coefficient problems with an application to linkage analysis of recessive diseases in inbred populations

Medical geneticists connect relatives having the same disease into a family structure called a pedigree. Genetic linkage analysis uses pedigrees to nd the approximate chromosomal locations of disease-causing genes. The problem of choosing a pedigree is particularly interesting for diseases inherited in an autosomal recessive pattern in inbred populations because there are many possible paths of inheritance to choose from. A variety of shortcuts are taken to produce plausible pedigrees from inbred populations. We lay the mathematical foundations for a shortcut that was recently used in a pedigree-disease study of an inbred Mennonite population. Recessive disease genes can be localized using the shortcut of homozygosity mapping by nding regions of the genome where aected persons are homozygous. An important quantity in homozygosity mapping is the inbreeding coecient of a person, which is the prior probability that the person inherited the same piece of DNA on both copies of the chromosome from a single ancestor. Software packages are ill-suited to handle large pedigrees with many inbreeding loops. Therefore, we consider the problem of generating small pedigrees that match the inbreeding coecient of one or more aected persons in the larger pedigree. We call such a problem an inverse inbreeding coecient problem. We focus on the case where there is one sibship with one or more aected persons, and consider the problem of constructing a pedigree so that it is \simpler" and gives the sibship a specied inbreeding coecient. First, we give a construction that yields small pedigrees for any inbreeding coecient. Second, we add the constraint that ancestor-descendant matings are not allowed, and we give another more complicated construction to match any inbreeding coecient. Third, we show some examples of how to use the one-sibship construction to do pedigree replacement on real pedigrees with multiple aected sibships. Fourth, we give a dierent