Abstract
Erlang renewal models, also called chi-squared models, provide a tractable model for genetic recombination that exhibits positive interference. Closed form expressions for multilocus probabilities are derived for the crossover process when it is a renewal process with the distance between crossovers modeled by a Erlang distribution. These expressions yield explicit formulas for the map functions, coincidence functions and distributions of the identity-by-descent process, giving exact results for a class of models that better model observed biological data.
Highlights
During the process of meiosis, germ cells are produced from the genetic material an organism has inherited from its parents
We show that the matrix functions they consider can be used to specify the multilocus probabilities for the crossover process as well, and we give closed form expressions for both the crossover process and the four strand chiasma process
We show that the thresholds don’t change when the Erlang renewal processes described above are used instead of the Haldane model
Summary
During the process of meiosis, germ cells are produced from the genetic material an organism has inherited from its parents. The main results of this paper are closed form expressions for multilocus probabilities when the inter-event distribution is Erlang. The formula for multilocus probabilities for a crossover renewal process with an Erlang(m, λm) inter-event distribution is given by:. For a gamete formed by a renewal chiasma process with Erlang(m, 2λm) inter-event distribution, the multilocus probabilities are given by. For a chiasma renewal process with NCI and Erlang(m, 2λm) inter-event distribution, the recombination fraction is, rmNCI (d). Quantifying this difference precisely depends on being able to accurately compute cumulative probabilities for multivariate normal distributions with dependence given by (12)
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