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1Department of Ecology and Evolution, The University of Chicago, Illinois 60637-1573, USA.
Journal of Mathematical Biology
|August 1, 2002
Summary
This study explores the structured coalescent for two genes in a one-dimensional model, analyzing genetic drift and migration. It defines a new retrospective stochastic process for digenic samples in the diffusion limit.
Area of Science:
- Population Genetics
- Mathematical Biology
- Stochastic Processes
Background:
- The structured coalescent provides a framework for studying genetic variation within populations.
- Understanding the behavior of gene lineages over time is crucial for inferring population history.
- Previous models often simplified migration patterns and spatial structures.
Purpose of the Study:
- To investigate the structured coalescent for digenic (two-gene) samples in a one-dimensional stepping-stone model.
- To analyze the impact of homogeneous, isotropic migration and random genetic drift.
- To derive the joint probability density of time to the most recent common ancestor (MRCA) and its positional deviation.
Main Methods:
- Utilized the diffusion limit approximation of the stepping-stone model.
- Analyzed single-locus, digenic samples.
- Derived and evaluated the joint probability density function for T (scaled time to MRCA) and Z (scaled positional deviation of MRCA).
Main Results:
- Explicitly evaluated the joint probability density of T and Z.
- Found that both T and Z have infinite expectations in their marginal distributions.
- Established that conditioned on T = tau, Z follows a Gaussian distribution with mean zero and variance 2tau.
- Extended results to anisotropic migration scenarios.
Conclusions:
- Established the existence of a retrospective stochastic process for digenic samples in the diffusion limit.
- Defined this new process for one spatial dimension.
- The findings provide a theoretical foundation for analyzing genetic data in structured populations.
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