Abstract
Long genomic segments that are nearly identical between a pair of individuals
and are inherited from a recent common ancestor without recombination are
called identical-by-descent (IBD) segments. IBD sharing has 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 average fraction of the
chromosome found in long shared segments and the average number of such
segments. A number of new results for tree heights in SMC' are proved as
lemmas. 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 use renewal theory to compute
the distribution of the fraction of the chromosome shared. While the expression
is again in Laplace space, we could invert the first two moments and compare a
number of approximations. Finally, we generalized all results to populations
with variable historical effective size.
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