A model for biased fractionation after whole genome duplication

Background: Paralog reduction, the loss of duplicate genes after whole genome duplication (WGD) is a pervasive process. Whether this loss proceeds gene by gene or through deletion of multi-gene DNA segments is controversial, as is the question of fractionation bias, namely whether one homeologous chromosome is more vulnerable to gene deletion than the other. Results: As a null hypothesis, we first assume deletion events, on either homeolog, excise a geometrically distributed number of genes with unknown mean mu, and a number r of these events overlap to produce deleted runs of length l. There is a fractionation bias 0 <= phi <= 1 for deletions to fall on one homeolog rather than the other. The parameter r is a random variable with distribution pi(.). We simulate the distribution of run lengths l, as well as the underlying pi(.), as a function of mu, phi and theta, the proportion of remaining genes in duplicate form. We show how sampling l allows us to estimate mu and phi. The main part of this work is the derivation of a deterministic recurrence to calculate each pi(r) as a function of mu, phi and theta. Conclusions: The recurrence for pi provides a deeper mathematical understanding of fractionation process than simulations. The parameters mu and phi can be estimated based on run lengths of single-copy regions.