title: A renewal theory approach to IBD sharing.
abstract: A long genomic segment inherited by a pair of individuals from a single, recent common ancestor is said to be identical-by-descent (IBD). Shared IBD segments have numerous applications in genetics, from demographic inference to phasing, imputation, pedigree reconstruction, and disease mapping. Here, we provide a theoretical analysis of IBD sharing under Markovian approximations of the coalescent with recombination. We describe a general framework for the IBD process along the chromosome under the Markovian models (SMC/SMC'), as well as introduce and justify a new model, which we term the renewal approximation, under which lengths of successive segments are independent. Then, considering the infinite-chromosome limit of the IBD process, we recover previous results (for SMC) and derive new results (for SMC') for the mean number of shared segments longer than a cutoff and the fraction of the chromosome found in such segments. We then use renewal theory to derive an expression (in Laplace space) for the distribution of the number of shared segments and demonstrate implications for demographic inference. We also compute (again, in Laplace space) the distribution of the fraction of the chromosome in shared segments, from which we obtain explicit expressions for the first two moments. Finally, we generalize all results to populations with a variable effective size.
authors.affiliation: Department of Computer Science, Columbia University, New York, NY, 10027, USA. Electronic address: scarmi@cs.columbia.edu.
authors.forename: Shai
authors.initials: S
authors.lastname: Carmi
authors.affiliation: Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, 02138, USA.
authors.forename: Peter R
authors.initials: PR
authors.lastname: Wilton
authors.affiliation: Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA, 02138, USA.
authors.forename: John
authors.initials: J
authors.lastname: Wakeley
authors.affiliation: Department of Computer Science, Columbia University, New York, NY, 10027, USA.
authors.forename: Itsik
authors.initials: I
authors.lastname: Pe'er
mesh_terms: D008040:Genetic Linkage / Q000235:genetics*; D005828:Genetics, Population; D008390:Markov Chains; D008957:Models, Genetic; D008962:Models, Theoretical; D010375:Pedigree
keywords: Coalescent theory; IBD sharing; Recombination; Renewal theory; SMC; SMC’