must lie laterally about equidistant from both. Other linkage relationships of bifid show that it lies as far as possible removed from the plane in which most of the genes lie. The other case mentioned is that of abnormal abdomen (A, figs. 1 and 2). Its linkage relations are known with only two other genes, yellow and white. But the relation of these two with each other is one of the best determined and the linkage of abnormal with each of them rests on more than 15,000 observations in each case. The yellow-white linkage is 1.1, as already stated; abnormal-yellow is 2.0, and abnormal-white 1.7. These relations make it impossible for abnormal to lie in line with yellow and white. Until a third linkage relation of abnormal is determined, it may swing freely round the line which joins yellow with white but can never come into line with them. A third linkage relation having once been determined for abnormal, its linkage with any other gene in the sex-chromosome could be readily predicted from direct measurement of the reconstructed figure. The actual test of the utility of this method of portraying linkage relationships could easily be made by first forecasting by measurement what undetermined linkage values are likely to be and then actually determining them by experiment. Such predictions could not fail to come nearer the truth than predictions based on a linear map, if as I have suggested the arrangement is really not linear.

What, it might be asked, does this reconstruction signify? Does it show the actual shape of the chromosome, or at any rate of that part of it in which the observed genetic variations lie? Or is it only a symbolical representation of molecular forces? These questions we can not at present answer. A first step toward answering them will be the construction of a model which will give us reliable information as to undetermined genetic relationships. A model which will answer questions truthfully must be a truthful presentation of actual relationships even though we do not know whether they are spatial or dynamic.

If the arrangement of the genes in the chromosome is not linear, Morgan's theory of linkage must be somewhat modified. (1) The fundamental assumption that the genes lie in the chromosomes and have a definite orderly arrangement there is not disturbed. (2) The further assumption that the respective distances between the genes determine their closeness of linkage one with another may also stand unchallenged. (3) But the assumption that the arrangement of the genes within the chromosome is linear cannot be accepted without proof, which at present is lacking. This assumption has made necessary other secondary assumptions, likewise unproved, which are superfluous if this one is abandoned.

Such an unproved secondary hypothesis is that of double-crossing-over. The experimental data show that double-crossing-over must occur, if the arrangement of the genes is linear. For if three genes, A, B, C, are linear in their arrangement in the order named, and all lie in the same gamete, and if subsequently A and C are found together in one gamete and B in another,

it is evident that this rearrangement can have come about only as a result of two breaks in the linkage chain, viz., one between A and B, and another between B and C. But if the arrangement is not linear, double-crossing-over need not be assumed as an explanation of the observed regroupings. For if A, B, and C are linked in a triangle, not in a straight line, then B may be freed from its connections with A and C without necessarily disturbing the connection of A and C with each other. Freeing of B will involve no greater number of breaks than the freeing of either A or C. It will still be true, however, as indicated by the experimental data, that certain groupings of three particular genes are easier to obtain than others. Thus in the case of the three genes white, bifid and vermilion, it is hardest to obtain the regrouping which involves detaching bifid from the other two. Morgan assumes that this is because bifid lies between the other two in a single linkage chain and so could be detached only by two breaks; it is possible, however, that the reason may be that bifid lies peripherally in the 'linkage system and could be detached only by an oblique longitudinal break, whereas either of the others could be detached by a simple transverse break. Similarly in the trio, white-vermilionsable, it is vermilion which is difficult to detach; and in the group, vermilionsable-bar, it is sable. Always it is the middle one considered with reference to the long axis of the system. This may be because, as Morgan supposes, only transverse breaks occur, of which two taking place simultaneously are required to produce the difficult regrouping, or it may be because transverse breaks are more frequent than oblique longitudinal ones, of which a single one would suffice to accomplish the regrouping, if the genes are not strictly linear in arrangement.

The phenomenon of 'coincidence' as described by Weinstein is this. If crossing-over occurs toward one end of a chromosome, it is less likely to occur simultaneously elsewhere in the same chromosome. Crossing-over in one part of a chromosome is thus supposed to 'interfere with' crossing-over elsewhere in the same chromosome. If we adopt the hypothesis of linear arrangement, interference must be assumed to occur. Observed facts require this. But if we do not adopt this hypothesis but suppose that what have been called 'double cross-overs' are really the result of single oblique or single longitudinal breaks, then the supposed phenomenon of interference may mean only this, that transverse breaks are more likely to occur than longitudinal

ones.

Finally, if the genes are not arranged in a single linear chain, the chiasmatype theory will need to be reëxamined. Such a purely mechanical theory seems inadequate to account for interchange of equivalent parts between twin organic molecules, such as the duplex linkage systems of a germ-cell at the reduction division must be. It seems more probable that preceding the reduction division a period of instability within the chromosome molecule comes on. Twin molecules are now closely approximated and parts of one may leave their former connections and acquire new connections with the

corresponding parts of the other twin. It is evident from the experimental data, notably that of Muller, that new connections are not formed with any torn fragment of chromosome which happens to come into the proper position, but that connections are always formed at exactly corresponding points with homologous systems of genes." It is like the replacement of one chemical radicle with another within a complex organic molecule and it seems highly probable that such is its real nature.

1 The distances shown in Morgan's chromosome map in excess of 50 (admittedly not obtained experimentally but only by summation) are therefore too large. Accordingly, if one clings to the assumption that the arrangement of the genes is linear, it must be, not that the longer distances are too short, as Morgan has assumed, but that the short distances are too long. Therefore, any hypotheses framed to account for an apparent shortening of the long distances are superfluous. The long distances given by direct experiment are long enough; they approach the limit of the possible, viz., 50%. Thus in table 65 of Morgan and Bridges, we find the following high cross-over percentages given by direct experiment:-yellow-bar, 47.9; white forked, 45.7; and white-lethal sc, 46.0. What is needed therefore, if the linear arrangement hypothesis is retained, is a secondary hypothesis to explain why the short distances given by experiment are too long.

But if we abandon the hypothesis of linear arrangement, all secondary hypotheses are unnecessary. The experimentally obtained cross-over percentages may be accepted at their face value, which in every case fall within the limits of the possible, O and 50.

2 Morgan, T. H., and Bridges, C. B., Sex-linked inheritance in Drosophila. Carnegie Inst. Washington, Publ., No. 237, 1916, (88 pp., 2 pl.).

3 Weinstein, A., Genetics, 3, 1918, (135–172).

4 Muller, H. J., Amer. Nat., Lancaster, Pa., 50, 1916.

The case of 'deficiency' studied by Bridges (Genetics, 2, pp. 445-460, Sept. 1917) forms an apparent exception to the rule. Here a certain segment of the linkage system was as regularly wanting as it is commonly present. The regularity of the process, however, shows that the principle of union at particular points still holds. In the deficiency race, a new and simplified linkage system had been established and this persisted.

THE LINKAGE SYSTEM OF EIGHT SEX-LINKED CHARACTERS OF DROSOPHILA VIRILIS (DATA OF METZ1)

BY W. E. CASTLE

BUSSEY INSTITUTION, HARVARD UNIVERSITY

Communicated December 9, 1918

In an earlier paper it has been shown that the arrangement of the genes in the sex-chromosome of Drosophila ampelophila is probably not linear, and a method has been developed for constructing a model of the experimentally determined linkage relationships. From such a model one may by direct measurement ascertain what other undetermined linkage values are likely to be. In order to test the utility of this method, it is desirable that it be tried out as widely as possible and the results for different cases compared with each other. For such use, suitable material is found in a paper by Metz1 dealing

with the linkage relations of eight sex-linked characters in Drosophila virilis, a species distinct from D. ampelophila, which has been so exhaustively studied by Morgan and his pupils. Of the eight characters studied by Metz, two agree morphologically and in their linkage relations with each other, with similar characters of D. ampelophila. The six others have no exact counterpart among the known mutations of D. ampelophila. The two characters in question are yellow body and forked bristles. Yellow body, in both species, lies at the extreme, 'zero' end of the linkage system. Forked, in both species lies at a distance of 40 or over from yellow. In D. virilis the distance is exactly 40, according to the observations of Metz; but in D. ampelophila, according to Morgan and Bridges, the distance is about 6 or 7 units greater. But inspection of figure 1 (p. 29) shows that this estimate is probably too

high, since the wire joining white (W) with forked (F) in the model is too long to harmonize fully with other linkage-values given by Morgan and Bridges, the wire being curved. The distance yellow-forked is not given by Morgan and Bridges but it evidently should be about one unit greater than the distance, white-forked, which is given as 45.7. If this estimate of the distance is too high, as figure 1 indicates, then the distance yellow-forked is probably not very different in the two species of Drosophila and will be found to be not far from 40 in both.

The linkage values found by Metz for the eight sex-linked genes of D. virilis have been gathered from his several tables, averaged and brought together in table 1 herewith. They form the basis of the reconstruction shown in figures 3 and 4.

Metz, adopting Morgan's system of linear grouping, shows the eight genes in a linear chain more than 80 units long, although the greatest distance experimentally found between any two genes is 47.3 (yellow-rugose). This discrepancy, like the similar ones observed in D. ampelophila, shows the inadequacy of the hypothesis of linear arrangement. For the maximum possible

FIG. 3. MODEL SHOWING RELATIVE POSITIONS OF GENES OF 8 SEX-LINKED CHARACTERS OF DROSOPHILA VIRILIS

cross-over percentage is 50 and this is in no case exceeded by data given by direct experiment.

Reconstruction in three dimensions (figs. 3 and 4) shows even more clearly in this case than in that of D. ampelophila, that a linear arrangement is out of the question. The reconstructed figure is roughly in the form of a tetrahedron. Figures 3 and 4 are views taken at right angles to each other corre