Gene Arrangement; Crossover Maps 



137 



appear as double crossovers (Figure 10-5). 

 Since the remaining three quarters ( 1 tig) 

 of the meiotic products are noncrossovers 

 or single crossovers, they are not useful in 

 identifying the occurrence of double cross- 

 ing-over events, because they could have 

 been produced in tetrads of other types, for 

 example, those having single or no crossing 

 over. Accordingly, a frequency of 0.04 

 double crossing over would lead us to ex- 

 pect a frequency of .01 double crossovers; 

 and a frequency of only 0.005 would actually 

 be detected were the coefficient of coinci- 

 dence 0.5. In this way, the coefficient of 

 coincidence can be determined from double 

 crossovers observed divided by the double 

 crossovers expected. 



There is another, perhaps simpler, way to 

 calculate the expected frequency of double 

 crossovers. In our example above, the 

 chance a crossing over will occur is 0.2, and 

 the chance that a given strand will be a 

 crossover, 0.5. The chance that both will 

 occur once is 0.1, and that both will occur 

 twice is 0.1 times 0.1 or 0.01. That is, the 

 expected chance that a given strand will be 

 a double crossover is one percent. Accord- 

 ingly, the frequency of observed single cross- 

 overs in the a-b region multiplied by the 

 frequency of observed single crossovers in 

 adjacent b-c region equals the expected fre- 

 quency of double crossovers (one in each 

 region). In practice, therefore, one may 

 readily determine the coefficient of coinci- 

 dence from double crossovers. 



Generally the coefficient of coincidence is 

 negligible — equal to zero for all practical 

 purposes — for short map distances and be- 

 comes larger with increased distance. This 

 relation suggests that a tetrad in which one 

 crossing over occurs is somehow precluded 

 from having a second one occur close by, 

 with this restriction diminishing as the dis- 

 tance to the second region increases. In 

 Drosophila, for example, the coefficient of 



coincidence is zero for distances up to 10-15 

 map units and, consequently, no double chi- 

 asmata (or no double crossovers) occur 

 within such distances. As the distance in- 

 creases beyond \5 map units, however, the 

 coefficient gradually increases to 1, at which 

 point nothing interferes with the formation 

 of double chiasmata. In two equal-armed 

 chromosomes there does not seem to be 

 chiasma interference across the centromere. 



If each tetrad has only a single chiasma, 

 the maximum frequency with which the end 

 genes recombine relative to each other is 0.5. 

 What happens to the frequency of recom- 

 bination for the end genes when the chro- 

 mosome has double chiasmata? 



If each tetrad has two chiasmata, one 

 might think that the end genes would form 

 new combinations with a frequency greater 

 than 0.5. Examination of Figure 10-5 re- 

 veals, however (each type of double chias- 

 mata being equally probable), that on the 

 average eight products (single crossovers) 

 will carry a new combination with respect to 

 one end gene, and eight products will not. 

 Of the latter, four will be noncrossovers and 

 four, double crossovers in which the middle 

 gene has changed position relative to the 

 end genes. Therefore, even if every tetrad 

 has double chiasmata, the maximum fre- 

 quency of recombination for the end genes 

 is 0.5. 



When four loci are studied and three chi- 

 asmata occur in each tetrad — one in each 

 region — one finds that for every 64 meiotic 

 products, 32 are recombinational for the 

 end genes and 32 are not. For cases where 

 four or more chiasmata lie between end 

 genes, the frequency of meiotic products 

 bearing odd numbers of crossover regions 

 is easily calculated to be 0.5. In each of 

 these cases the gene at one end is shifted 

 relative to that at the other. However, the 

 remaining strands contain either even num- 

 bers of crossover regions (which do not 



