A renewal theory approach to IBD sharing

Area of Science:

• Population Genetics
• Statistical Genetics
• Computational Biology

Background:

• Identical-by-descent (IBD) segments are crucial for understanding genetic relatedness and have applications in various genetic studies.
• Previous models for analyzing IBD sharing, such as Markovian approximations of the coalescent with recombination, have limitations.

Purpose of the Study:

• To provide a theoretical analysis of identical-by-descent (IBD) sharing using Markovian approximations and a novel renewal approximation.
• To derive new results for the number and fraction of shared IBD segments in the infinite-chromosome limit under the SMC' model.
• To explore the implications of IBD sharing distributions for demographic inference in populations with variable effective sizes.

Main Methods:

• Theoretical analysis of the IBD process along the chromosome using Markovian models (SMC/SMC').
• Introduction and justification of a new renewal approximation model where successive segment lengths are independent.
• Derivation of expressions for the distribution of shared segments and the fraction of the chromosome in shared segments using renewal theory, particularly in Laplace space.

Main Results:

• New results derived for the mean number and fraction of shared IBD segments longer than a cutoff under the SMC' model in the infinite-chromosome limit.
• Expressions derived for the distribution of the number of shared segments and the fraction of the chromosome in shared segments.
• Generalization of results to populations with variable effective sizes.

Conclusions:

• The renewal approximation provides a new framework for studying IBD segment lengths.
• The derived theoretical results enhance the ability to perform demographic inference and analyze genetic data.
• The findings are applicable to a broader range of population genetic scenarios, including those with changing population sizes.