LINKAGE AND CROSSL\G-OVER 443 



greater than the cross-over percentage between A and C, and the dis- 

 crepancy increases with the magnitude of the values involved. This 

 fact has been accounted for in two different ways. First, it may be 

 supposed that the arrangement of the genes is really not linear, that 

 B lies out of line with A and C, so that AC will be less than the sum of 

 AB and BC, and that the more distant genes are no farther apart than 

 indicated by the cross-over percentages between them. This ex{)la- 

 nation has met with more difficulties than it has cleared away. The 

 second explanation is that the map-distances indicate proportionate 

 numbers of breaks in the linkage chain between points, not propor- 

 tionate numbers of changes of relation between genes at particular 

 points. Thus, suppose genes ABCDE of a linkage system meet their 

 allelomorphs, abcde, in a cross and gametes are later formed by the 

 cross-bred as follows, (i) ABcde, (2) ABcdE, and (3) AbcDe. Assum- 

 ing that the arrangement is linear, we must suppose that one break 

 in the linkage chain has occurred in (i), two breaks in (2), and three 

 breaks in (3). But if we did not have genes BCD under observation, 

 and merely noted the relation of A to E, we should infer that in case 

 (i) and in case (3) a single crossover had occurred. We should on that 

 basis underestimate the amount of breaking in the linkage chain. 

 Accordingly the construction of maps on the basis of short distances 

 summated is justifiable, provided the arrangement is linear, as it seems 

 to be. But it must be borne in mind that the map distances do not 

 correspond with cross-over percentages (although they are based on 

 them) except in the case of very short distances. Alap distances often 

 exceed 50, but cross-over percentages cannot do so, as already pointed 

 out. To get a distinctive name for the map units, Haldane has called 

 them units of Morgan or simply ''morgans." Haldane has computed 

 a formula for converting cross-over percentages into ''morgans" and 

 vice versa. He finds that the two correspond only for very low values 

 (5 or less) and diverge more and more as the observed cross-over per- 

 centages approach 50. Haldane's formula may be stated thus. If 

 three genes, A, B, and C, occur in a common hnkage group, and ihc 

 cross-over percentages are known between A and B and between B and 

 C, we may predict with a probable error of not over 2 per cent, what 

 cross-over percentage will be found to occur between A and C. Call- 

 ing the cross-over percentage between A and B, w, and that between 

 B and C, n, the cross-over percentage between A and C will lie between 

 (w+n) and (m+n — inm). It will approach the former for amounts 

 of 5 or less, and the latter for amounts of 45 or over. In a useful table 



