Population genetics of unequal crossing over

Summary

The population genetics of unequal crossing over was examined for an infinite population with random mating. The following cases were considered:

• 1. There is an initial portion of duplicated genes which offer the opportunity for unequal crossing over, but the primary event leading to the duplication does not occur any more (model 1a).

• 2. This primary event occurs with a certain (small) probability (model 1b). For both possibilities the long-term consequences for the distribution of “alleles” (i.e. the single gene, the duplicated gene, the triplicated gene etc.) were considered with the following additional assumptions:

• 1. No selection.

• 2. Selection with maximum fitness for an optimum “allele length” (i.e. number of gene repeats).

• 3. For model 1a, selection with general advantage of longer alleles over shorter ones was also examined.

The results are briefly the following: In model 1a under assumption 1 the distribution of allele length tends with increasing generation number to a stationary state which depends on the initial allele distribution (i.e. on the initial frequency of the duplicated gene) but not on the frequency, P, of unequal crossing over; the stationary frequencies of the alleles decrease with increasing allele length. Under assumption 2 there is likewise a stationary allele distribution, but this depends on P as on the strength of selection and not on the initial allele distribution; it is concentrated more or less tightly around the optimal allele length. Under assumption 3 no stationary state seems to be reached: the mean and the standard deviation of the allele distribution increase steadily with the generation number. In model 1b under assumption 1, with certainty no stationary distribution exists. Under assumption 2 the situation is the same as that in model 1a; the stationary distribution of allele length is identical with that in model 1a for the same P and same selection strength, quite independent of the probability of the primary event.

The results were discussed with respect to empirical examples in which unequal crossing over is expected to be important, for example human haptoglobins, immune globulin determining cistrons, and nucleolus organizer regions. The consequences of selection relaxation were considered.