A renewal theory approach to IBD sharing

Shai Carmi*, Peter R. Wilton, John Wakeley, Itsik Pe'er

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

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.

Original languageAmerican English
Pages (from-to)35-48
Number of pages14
JournalTheoretical Population Biology
Volume97
DOIs
StatePublished - 1 Nov 2014
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2014 Elsevier Inc.

Keywords

  • Coalescent theory
  • IBD sharing
  • Recombination
  • Renewal theory
  • SMC
  • SMC'

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