A mathematical treatment of genetical recombination using a four-strand model

Abstract A mathematical treatment of genetical recombination is given, using a four-strand model. Two distinct kinds of interference are recognized: chiasma position interference (‘chiasma interference’), affecting the distribution of the chiasmata along the chromosome, and chiasma type interference (‘chromatid interference’) affecting the strands which take part in adjacent chiasmata. Their treatment is based on the methods of Owen and Weinstein respectively. Type interference is of two kinds, according as adjacent chiasmata more frequently involve two or four strands. When chiasma pairs involving two strands are much more frequent, the recombination fraction rises with map distance to about 25% and thereafter creeps up slowly towards the limiting value, 50%; with map distances of the length met in most material, the recombination fraction might scarcely rise above 25%. When all chiasma pairs involve only two strands, the limiting recombination fraction is 25%. When four-strand chiasma pairs are more frequent than two-strand pairs, the recombination fraction rises with increasing map distance to a maximum above 50% and thereafter performs damped oscillations round 50%. When all chiasma pairs involve four strands, the maximum recombination fraction, in the absence of position interference, is 53-34%. When position interference is present, it enhances the effect of type interference and higher maxima are possible; 100% recombination would occur between points separated by two localized chiasmata always involving four strands. It is thought possible that ‘second-order linkage’ may be detectable in some material, i. e. linkage of less than 50% between two loci, each of which shows linkage exceeding 50% with an intermediate marker.