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A survey of unequal crossover systems and their mathematical properties

We present a model of gene duplication by means of unequal crossover (UCO) where the probability of any given pairing between homologous sequences scales as a penalty factor pz ≤ 1, with z the number of mismatches due to asymmetric sequence alignment. From this general representation, we derive several limiting case models of UCO, some of which have been treated elsewhere in the literature. One limiting case is random unequal crossover (RUCO), obtained by setting p = 1 (corresponding to equiprobable pairings at each site). Another limiting case scenario (the ‘Krueger-Vogel’ model) proposes an optimal ‘endpoint’ alignment which strongly penalizes both overhang and deviations from endpoint matching positions. For both of these scenarios, we make use of the symmetry properties of the transition operator (together with the more general UCO properties of copy number conservation and equal parent-offspring mean copy number) to derive the stationary distribution of gene copy number generated by UCO. For RUCO, the stationary distribution of genotypes is shown to be a negative binomial, or alternatively, a convolution of geometric distributions on ‘haplotype’ frequencies. A different type of model derived from the general representation only allows recombination without overhang (internal UCO or IntUCO). This process has the special property of converging to a single copy length or a distribution on a pair of copy lengths in the absence of any other evolutionary forces. For UCO systems in general, we also show that selection can readily act on gene copy number in all of the UCO systems we investigate due to the perfect heritability (h2 = 1) imposed by conservation of copy number. Finally, some preliminary work is presented which suggests that the more general models based on misalignment probabilities seem to also converge to stationary distributions, which are most likely functions of parameter value p.