Mathematical models for estimating preferential pairing and recombination in triploid hybrids

Different mathematical models for estimating pairing and recombination parameters for triploid hybrids with two-armed chromosomes are discussed. In most models all information on preferential pairing is contained in the ratio of trivalents and ring bivalents and can be estimated independently of the chiasmate association frequencies in the two chromosome arms. The single degree of freedom available only permits the estimation of ranges of the three possible pairing relations between the three genomes. Alternatively, a single parameter can be estimated when additional assumptions are made. When the ratio r between the frequencies of trivalents and ring bivalents, independent of arm length differences and with a theoretical maximum of 2, is about 1, the ranges of the frequencies of the three pairing combinations are wide. When r becomes smaller than 1, very soon the ranges become limited to values where one of the three is relatively large and positive and the other two negative and varying between equal to very different, depending on slight changes in the first. When the frequency of open bivalents is relatively high and the frequency of univalents low, there most probably is a difference in chiasmate association frequency between the two arms of the average chromosome and this difference can be quantified. When the number of univalents is only slightly higher than expected on the basis of the number of open bivalents, the reason may be that (quantifiably) more chiasmata are formed after bivalent pairing than after trivalent pairing within certain ranges of r, and certain ranges of the average chiasmate association frequency. When the excess of univalents is larger, this is best explained by a failure of entire chromosomes to find each other. This degree of pairing failure can be estimated. All models have been applied to the triploid hybrid between allotetraploid Trifolium repens and diploid T. nigrescens. Assuming that the two genomes of T. repens do not pair, which cannot be demonstrated with certainty, the two genomes pair with the nigrescens genome with frequencies of 0.828 and 0.171, respectively. Introgression then occurs into either genome but not with the same frequency. If the repens genomes pair, this would be caused by either genetic factors disturbing the normal pairing behaviour or the absence of strict homologues. Then the relative pairing frequency between T. nigrescens and one of the T. repens genomes would be 0.838 and on an average 0.081 between the two other combinations, with a possibly considerable but unknown difference. The high average chiasmate arm association frequency (0.650) suggests that affinity between the pairing genomes is not very low. The average two arms do not differ in chiasma frequency.Key words: triploid hybrids, preferential pairing, recombination, mathematical models, Trifolium.