A Mathematical Model for the Distribution of Lengths of Chromosomal Deficiences Involving a Specific Locus.

XPERIMENTAL distributions of breaks and chromosomal recombinations in E Drosophila melanogaster have been studied by several authors in attempts to throw light on the mechanism of the origin of chromosomal rearrangement. STADLER ( 1932), BAUER, DEMEREC, and KAUFMAN (1 938), CATCHESIDE (1 938), FANO ( 1 943), KAUFMANN (1946) and many others have contributed to this literature. While there is evidence (BAUER et al. 1938) that breakage, either spontaneous or X-ray induced, occurs largely at random throughout the length of a chromosome, the accumulated evidence of observed chromosomal alterations indicates that recombination may not be random. In obtaining alterations to test the hypothesis of random breakage and recombination, inversions and deficiencies are particularly useful because they admit fairly standard cytological or morphological techniques of identification. Inversion loops may be detected and measured with reasonable facility in the salivary chromosomes of D. melanogaster, and a series of 49 induced inversions of the X chromosome was obtained by BAUER et al. (1938). This distribution was compared by FEDERER, STEEL, and WALLACE (1961) to several mathematical models. The cytological detection of chromosomal deficiencies, however, is impractical unless the deficiency includes a mutant marker. This technique was used by DEMEREC and FANO (1941) to obtain a series of 37 Notch deficiencies of the X chromosome. These authors observed that the expected frequency of deficiencies involving n bands out of a total number of N b'ands is equal to 2 ( N 1 n) / (N 1 ) ( N 2) and that the a priori probability that any specific band is included in a deficiency of length n is approximately equal to n / ( N 2). These results may be obtained by assuming that any two random and independent breaks may lead with equal probability to a deficiency or to an inversion of the chromosomal segment involved. This assumption probably does not obtain, however, for large chromosomal segments which might be expected to be more viable as inversions than as deficiencies, nor for very short segments which might be lost as deficiencies more often than recombining as inversions. This model also does not take into consideration the shorter deficiencies which may arise as a result of a single break or those deficiencies associated with more complicated