The Relation Between Chiasmata and Crossing-Over in Diploid and Triploid

THE I%ELATION BETWEEN CHIASMATA AND CROSSING-OVER IN DIPLOID AND TRIPLOID DROSOPHILA MELAI¥OGA~TER.

BY K. MATHEI%. (John Innes Horticultu~'aZ Institution, Me,rton.)

(With Three Text-figures.)

I. INTI%O])UOTION. T1~E genetical observations on Drosol)hila melanogaste~" have, in the past, demonstrated severat principles of paramount importance to the cytolo- gist, some indeed being fundamental to our present theory of meiosis. Note recently work has been done on triptoid forms of D. melanogaste'r, which for its interpretation needs a fuller knowledge of the meiotic behaviour of chromosomes than it has hitherto been possible to derive from cytological observations on Drosophila itself. At the same time, the study of meiosis in more favourable material, notably in monocotyledonous plants, has led to the phenomenon being placed on a sound theoretical basis. It is the object of this paper to apply cytological findings to the fuller interpretation and lmderstanding of some of the recent genetical work on D. melanogaste'r. For the purpose of exposition Darlington's modification of Janssens' chiasmatype theory has been adopted. On this theory ehiasma formation is conditioned by crossing-over, but similar deduc- tions, in the majority of cases, would follow from some other theories of chiasma formation and its relationship with crossing-over. Where this is not the case, special note is made of it.

II. THE DIPLOID--INTERFERENCE. 3/Iost of the work on crossing-ovor in diploid D. ,melanoyaste'r has boon in connection with linkage maps, and the study of coincidence values. ]Recently, however, several experiments involving a mlmber of points scattered along a chromosome have beo~k performed, mainly in order to provide controls for a study of crossing-over in some relatively abnormal forms. 244 Chia~vmata and Crossing-ove~' in Drosophi]~

For instance, l~edfield (1930) has recorded crossing-over in the third chromosome of -~t917 progeny of the cross 1",, h th st c,~ sr e ~ ca 9 × r,~ h tt, st c~ s,, e ~ % d~, + i.e. involving genes from 0.0 to 100.7 ca the standard map. She finds that 27.88 per cent. of the flies in the progeny show no cross-overs, ~5.28 per cent. show one cross-over, 23.52 per cent. two cross-overs, 3.17 per cent. three cross-overs and 0.16 per cent. four cross-overs. Now it has been proved by genetieal means that crossing-over occurs in the four-strand stage, and by cytological observation that a chiasma invoNes only two out of the four strands, i.e. on. any theory of chiasma fomnation and crossing-over ouly two of the four strands will cross over at any given level. This means, as Belling (1931.) points out, that one chiasma will give equal numbers of single cross-over and non-cross-over strands, two chiasmata will give non-crossover, single cross-over and double cross-over strands in the ra,tio 1:2:1 and similarly for higher numbers of chiasmata ~. These ratios may be altered by certain types of interference, which wifl be discussed later. For the present it is assmned that the strands crossing-over at any one chiasma are at random, i.e. not determined by those crossing-over at any other. Sax (1932) has used Belling's formulae to calculate the chiasma frequency of the first chromosome from scute to forked. ]~y using a similar calculation we can estimate the chiasma frequency of the third chromosome of the diploid from Redfield's da~a. A correction must be made to start with, as there is a certain amount of non-observable double crossing-over. From the correction tables given by Bridges and Morgan (1923) it appears that there would be approximately 2 per cent. of non-observable doubIe crossing-over in the cross under consideration. Therefore as 2 per cent. of the double cross-overs appear in the non-cross-over class, 2 per cent. is subtracted from the non-cross- overs and added on to the double cross-overs. No doubt some triple cross-overs appear in the single cross-over class, but the number of triples is so small that the number which, owing to non-observable double crossing-over, will appear in the single cross-over class, will be very small and may be neglected[. Let a be the percentage of bivaleuts with no chiashaata, b those with one chiasma, c those with two chiasmata, d those with three and e those

1 In formul~ting the cross-over ehiasm~ frequency equations, sister strand crossing- over is neglected tln'oughout. On these simple assumptions it has no significance ill this respect, ~md i~s occurrence has been rendered improb~ble by recen~ work. K. MATI-IEI~ 245

with four. The possibility of :five or more ehiasma~a occurring in one bivalen~ is neglect, ed, as flies showing five cross-overs, while ~hey have been seen, are exceedingly rare and occur under relatively abnormal conditions, t-Ienee the equations are:

~YO CrOSS-OVerS Single cross-overs Double cross-overs }o + ~d + ~e = 25.32 Triple cross-overs Quadruple cross-overs ~re = 0.16 which give the solutions: ~ = 3.12 %, ~ = 6.~,~ %, ~ = 67.ss %, ~ = ~0.20 %, e = 2.~ %. The ehiasma frequency curve from ~hese data is plo~tecl in Fig. 1. l~s mean is 2.13 and its variance 0.43. 7O . / ~oso~e A7

~o 50 /// / 2O

/ 2 3 4 5 Number of chiasmata

Fig. 1. The ehiasma frequency curves calculated for the second and third chromosomes from genetical data. Using ttedfleld's (1932) data for the diploid cross, involving the second chromosome genes

l 2 d~, b T; e p~ s~, -9 x az 2 d~, b p,, e 2~ s~, ~,

Journ. of Genetics xx~I 16 246 Cldasmatc~ c~,J~d Orossi~zg-over i9~ Drosophila by a similar calculation we get the following chiasma frequency for the second chrolnosome :

~ = - 9,.57 %, b = ~7.65 %, c = 57.21%, ~z = 15.13 %, e = 2.57 %.

Tlm chiasma frequency curve is plotted from these numbers in Fig. 1. Its mean is 1.93 and its variance 0.51. Since one c]iiasma is equal to 50 centimorgans (of. Darlington, 1931) the above frequencies equal 106.5 centimorgans from % to % and 96.5 centimorgans from aa to s~,, which are practically tlhe same as l~edfield calculates directly from the genetical data in each case. It will be'noticed tha~ in the case of the third chromosome there are apparently 3.12 per cent. of bivalents with no chiasmata, which lead to cross-overs (i.e. on the chiasmatype theory these will be represented by univalents), and in the case of the second chromosome there is a negative number for this class. Now on analogy with flowering ]?lants there should always be at least one cbiasma (i.e. univalents should never occur), and of course the negative number is absurd, hence it is necessary to find out why the above two numbers occur. First of all they may occur owing to errors inherent in the method of calculation. This can easily be tested. Let ~, be the proportion of gametes with no cross-overs,/9 be the proportion with one, y with two, 8 with three and E with four, then substituting these for the observed nnmbers in the above eqnations and solving we get

c* = c,.-/9 + y- 3 + e, b =2/?-47+68-8e, c = 4>, - 128 ÷ 24:e, d = 88 - 32E, e = 16e.

Therefore ~ = 1 2 (/9 + 3). If a = 0 we should expect to find/9 + 8 - 1 1 and its standard error is ~ (where ~ is the total number of files 1 observed), i.e. tim standard error of c~ is 2x 7--O' i.e. 0.71 per cent. for 1 the third chromosome and 2 >< 6~' i.e. 0.84 per cent. for the second chromosome. Hence the deviation1 of a from 0 is in one case distinctly more and in the other case about equal to three times the standard error. Con- sequently these deviations are probably not due to errors in calculation or sampling and so they must be due to either (1) differential viability of K. MATHEg 247 some of the classes or (2) "chromatid" interference in chiasma formation (see below). They are most probably due to diKerential viabilRy, as the numbers deviate in opposite directions. ]-laldano (1931) has shown that if chiasma formation is at random, the chiasma frequencies fall on a Poisson curve and hence the variance and mean are equal. He examined cytological data from several genera of Angiosperms and found in every case that the mean was about fern' times as large as the variance. The above calculated chiasma frequencies show the same phenomenon, and hence demonstrate the interference of one chiasma in the formation of others in precisely the same way as ttaldane observed in the Angiosperms. This close agreement between the frequency curve of chiasmata giving rise to cross-overs (chiasma fre- qneney curve as calculated directly from crossing-over data) and the chiasma frequency curve cytologically observed at several stages of meiosis indicates a direct correlation betwem~ observable chiasmata and crossing-over. Now on the partial chiasmatype theory, as advocated by Belling (1931) and Darlington (1931), this is expected, as chiasma formation is conditioned by crossing-over. On the "classical" theory as advocated by Sax (1%0, 1982) the amount of crossing-over is directly correlated with the number of chiasmata which disappear between diplotene and metaphase, i.e. crossing-over should not be correlated with the number of observable remaining ehiasmata. Hence the data on crossing-over in DrosoFhila , under consideration, very strongly suppor~ the partial chiasmatype theory. The cytological interference can be demonstrated directly fl'om the genetical data, assuming the chiasmatype theory, since, if chiasma for- marion is at random, i.e. if the chiasma frequencies lie on a Poisson curve, the cross-over freqnencies will also lie on a Poisson curve with its mean, and consequently its variance, equal to half those of the chiasma frequency curve (this is capable of mathematical proof). Using I%edfield's data again (uncorrected for non-observable double crossing-over) a mean and variance of 1.02 and 0.65 respectively are found for the thh'd chromosome and 1.01 and 0.60 for the second chromosome, thus demon- strating a cytological interference directly. There are, a priori, two possible kinds of cytological interference, viz. (i) the actual formation of a chiasma interfering with the formation of others in its vicinity, and (ii) the specific s.~rands taking part in the second ehiasma being partially determined by, i.e. not entfl'ely at random with respect to, those crossing-over at the first chiasma. The first type may be termed "chiasma" interference and is capable of both cytological and

16-2 248 , Cl~iasmata and Crossi~g-over i~ Drosophila

genetical demonstration. It always leads to the chiasma frequency curve having a variance less than its mean. It tends to reduce both the numbers of bivalents with a relatively high number of chiasmata (when positive) and of those with a relatively low number of chiasmata (when negative), e.g. in normal diploids where there are no univalents. The negative interference (i.e. the tendency to increase the nmnber of chiasmata in bivalents of low chiasma formation) is very much in evidence where genetioal control leads to a correlation between chiasma frequency and chromosome length such as has been observed by Darlington and Dark (1932) in ~5'tenobothrus parcdldus and by O'Mara (1931) in Yucca jlaccida. In these organisms a 'chiasma is always formed, even by the very short bivalents, and in these very short bivalents in Stenobothrus the variance/mean ratio is 0.04, thus showing very strong "chiasma" interference. Finally this type of interference will never upset the ratios of the chromatids, with the various number of cross-overs, derived from bivalents with a given number of chiasmata, e.g. bivalents with three chiasmata will always give a ratio of 1 triple cross-over : 3 double cross- over : 3 single cross-over : 1 non-cross-over chromatids, if interference is of this type only. The second type of interference may be termed "chromatid" interference and is incapable of direct cytological demonstration. It is possible to distingui:sh, occasionally and only in very favourable materiM, between non-compensating and compensating ehiasmata, but as the latter type may be the result of reciprocal or complementary cross-Overs, which have diametrically opposite effects on the cross-over frequency curve (of. Sansome and Philp, 1932) this distinction is of little vMue. l=Ience this type of interference can only be detected genetically, and then by no means with accuracy, as provided the numbers of reciprocal and complementary cross-overs are equal (giving with two chiasmata the 1:2:I ratio) the proportion of diagonal cross-overs (which give rise to non-compensating chiasmata) is nndetectable, as diagonal cross-overs always give the 1:2:1 ratio with two chiasmata. It is possible to form some estimate of the ratio of reciprocal to complementary compensating ehiasmata genetically, as the former type leads to an increased number of very high and very low cross-over chromatids, i.e. increases the variance/mean ratio, and the latter decreases the number of very high and very low cross-over types, i.e. decreases the variance/mean ratio. These are "negative" aud" positive" interference respectively. If the chiasma frequency curve were calculated on the basis of random assortment of the chromatids at each ehiasma, positive ehromatid interference would K. l~{aa'I-mg 249 be shown by a negative number in the no-chiasma bivalent class (i.e. a iu the above calcuIation would be negative), as is observed for chromosome II, and negative interference would be shown by a positive uumber in the no-chiasma bivalent class (i.e. ¢~ would be positivc) as is observed for chromosome III. No interference woutd be shown by ct = 0 (i.e. no bivalents in this class). It is, however, very likely that the values obtained for c~ in the above calculations are not an indication of inter- ferences but of differential viability. A more sensitive method ot! calculation will be necessary to show fine degrees of "chromatid" interference. This type of interference is probably less common than the other type. In this respect it may be men- tioned that Darlington and Dark (1932) found evidence of an excess of compensating chiasmata in Ste~oboth'rz~s l)C~'cdlel~s, which possibly indicates a connection between the strong "chiasma" interference and "chromatid" interference. In female diploid Drosop/dl~ ib appears from the above calculations that interference is almost, if not quite, completely of the "chiasma" type. '

III. T~E TI~IPLOID--CHI~O~IOSOt~{EPAII~ING AND DISJUNOTION.

Crossing-over in triploid D. ~elc~ogc~ste~' has been studied by Bridges and Anderson (1925) in the X-ehronlosome and l%edfield (1930, 1932) in the third and second chromosomes. Their experimental methods differed to some extent. Bridges aud Anderson marked all three chromosomes and studied their daughters which had derived two X-chromosomes from their triploid mothers, while l~edfield, though marking all three chromosomes in one experiment, con- centrated mainly on offspring carrying one chromosome derived from the triploid mother in which only one chromosome was marked, the other two being wild type. Bridges and Anderson's data are of more value quali- tatively, as two products of the triploid meiosis are present in the offspring studied, but ]~,edfield's data, particularly those for the third chromosome, are of greater value quantitatively, since it is easier to calculate expectation for one product of meiosis than for two. In every ease some of the offspring which had derived two chromosomes from the triploid mother were homozygous for a recessive gene which was present in the heterozygous simplex condition in the mother, so proving that crossing-over takes place in the six-strand (chromagid) stage. 250 Chiasmc~lc~ and C'rossing-ove~" in Drosophil~

In the triploid there are ~hree homologous chromosomes, between any ~wo of which chiasma formation and crossing-over may occur at any level 1. In considering crossing-over in such an individuM we are first concerned with tile way in which the three chromosomes become associated at zygo~ene-paeh}~ene (syilapsis) before they split into daughter- chromatids, and cross-over. The distinction between the prophase pMring of ch~'o'moso,mes and chiasma formation and crossing-over between daugh- ~er ch~'o,mat.fd,s cannot be over-emphasised. In the griploid this distinction is of primary importance in considering chiasma formation and crossing- over, which can only tak6 place between paired chromosomes and hence are, to this extent, dependent upon the actual chromosome pairing at paehytene. Consequently any consideration of crossing-over in a triploid organism is primarily concerned with the prophase pairing of the el~'omosomes (synapsis) which may happen in three ways, viz. : (i) All three chromosomes are effectively associated along their whole length (i.e. all three are effectively associated ag any given level). (ii) Only two of the three chromosomes are effectively associated at any level, but each pairing unit (chromomere or gone according to the method of inference) ae~s independently of the rest. (iii) Only two of the three chromosomes are effectively associated at any level, and the association of any pairing unit is interfered with or partially determined by that of its neighbours. Considering the first possibility--crossing-over would take place at any level between the two unmarked chromosomes in one-third of the eases, and between one unmarked and the marked chromosome in two- thirds of the cases, if only one of the chromosomes is marked as in the majority of gedfield's crosses. Where crossing-over takes place between ~wo unmarked chromosomes there will be no visible result and so it will yield six apparently non-cross-over s~rands. Where crossing-over takes place between the marked and one of the unmarked el~'omosomes, it will yield two single cross-over and four non-cross-over strands. I-tence the net result for one chiasma, allowing for the relative fi'equeneies of crossing-over between the various pairs of the three chromosomes, will be four single cross-over to fourteen, non-cross-over, i.e. two single cross-over strands to seven non-cross-over strands. Expanding this for two, tlR'ee, four and five chiasmata we get the following cross-over ehiasma equations, similar to those for the diploid: Chiasmata have never been observed to concern more ~han two chromosomes,except terminally, which is ~ derived condition. I{. l~i±vr~Er~ ~ 251 a + 7/9b + ~9/slc + 3~3/7~9d, + ~01/6561e + 16s07/590~9f non-eross-overs~ 2/9b ÷ 28/81e ÷ 29~1/729d + 27~/6561e -I- 2~010/590~9f = single cross-overs, 4/81c _L 84/7293 -I- 1176/6561e + 13720/590~9f = double cross-overs, 8/729d + 224-/6561e -I- 3920/590~9f = triple cross-overs, 16/6561e -I- 560/590~9f = q uadnlple cross-overs, 32/590~9f = quintuple cross-overs, where a = proportion of configm'ations (three univalents) with no chins- main, b = proportion of configurations (bivalent + univalent) with one chiasmata, c = proportion of configurations (trivalent or bivalent + univalent) with two chiasmata, d with three, e with four and f with five ehiasmata. igedfietd (1930) found in her second experiment, involving the third chromosome from % to c~, 771 non-cross-overs, ~97 single cross-overs, 160 double cross-overs, 20 triple cross-overs and[ ~1 quadruple cross-overs. Substituting these numbers in the above equation and solving we ge~ a = 743.25, b -- - 2583.0, c = ~19~1.0, d = - 3280.5, c = 1640.25, which, as it includes negative quantities, is absurd. Even assuming ~hat all the quadruple cross-overs were derived from configurations with five or six chiasmata we still get negative numbers in the solutions. Itence this possible method of prophase chromosome pairing does not agree with the observed genetical results. With respect to the second possible method of chromosome association at pachytene, the marked chromosomes wiI1 be effectively associated with one unmarked chromosome at any level in two-thirds of the eases, and unassocia{ed in one-third of the cases. IIence crossing-over at any point will involve the marked chromosome in two-thirds of the eases, and will be between unmarked chromosomes, i.e. will be unobservable, in one- third of the eases. Consequently the equations connecting ehiasma frequency and crossing-over will be the same as those given above for the case where all three chromosomes are associated along their whole length. The eciua~ions have been shown not to fit the genetical results, so this possibility must be relinqnished. This only leaves the last possibility, viz. that only two chromosomes are associated at any level but that the pairing units do not act independently of one another. To derive formulae from this possible method of chromosome pairing two assumptions will be made, viz. (1) that association or pairing of the chromosomes on an average starts at two points 252 Chiasmata and Crossing-over in Drosophila and runs along the chromosomes, i.e. that the mode of association of the pairing units is determiued by the pairing at the two initial points where association is at random and (if) that if a chromosome is partly associated with each of the other two, then the frequency of association with any length of either of the others is constant no matter what regions of the chromosome are involved. Then the marked chromosome wilt be (1) completely unassociated in ~ of the cases, (2) completely associated with one chromosome in ~ of the cases, (3) partially associated with each of the other chromosomes in ~ of the cases, (4) partially associated and partly unassociated in ~ of the cases. ' I I \ I I ! I \ I, I I I / I' I 1 I 1 2 5 4 Fig. 2. The four ways in which ~he three homologous chromosomes of the triploid can pair at paehytene (synapsis). The broken line represents the marked chromosome, the two whole lines the wild type ones. N.B. the change of partner ill 3 and 4 need not occur of necessity in the centre of the chromosomes as illustrated. (For a fuller description see in the t.ext.) The number of chiasmata in which the marked chromosome takes part will be approximately proportional to the length of that chromosome which is associated with either or both of the othersL The genetical consequences of one or more chiasmata are worked oat as for the diploid, bu~ allowance is made for the presence of ~hree chromosomes and for the association of these chromosomes. (This is done by making the number of chiasmata in which a chromosome takes part proportional to the effectively associated length of that chromosome.) So letting a be the number of configurations with no chiasmata, b with one, c with two, d with three, c with four andf with five as before, the derived cross-over chiasma equations are:

1 This is not strictly correct where oN) short lengths of the chromosome are concerned but is reasonably accurate for tong port;ions. It is of course more seem'ate in the triploid than iu the diploid (of, cytological attd diploid and triploid gencticai chronlosonle illapS, Sansome and Philp, 1932). K. MAT,-lEg 253 a + 2/3b + 69/I08c + 319/576d + 867/1728e ÷ 356/72Qf

= no cross-overs ~ 771, 1/3b + 31/i08c ÷ 151/576d + 380/1728e ÷ I13/72Qf = single cross-overs = ,197, 8/I08c + 85/576d + 302/1728e + 118/720f= double cross-overs = 160, 21/576d ÷ 148/1728e ÷ 88/720f = triple cross-overs = 20, 31/1728e ÷ 38/720f = quadruple cross-overs = ~, 7/720f = quintuple cross-overs = 0, again using data from Redfield's second experiment on the third chromosome (1930). Although no quintuple cross-overs were observed it is exceedingly probable that a certain number of configurations had five ehiasmata, but that as quintuple cross-overs would occur only once per hundred strands derived from such configurations the number of flies was not large enough to show them. For purposes of calculation it is assumed that ~ certain number of quadruple cross-overs were derived from such configurations. If two of the quadruple cross-overs are assumed to come from configurations with five ehiasmata the sohtions to the equations are: a : -- 299, b = 84-, c = 1380, d = 137, e = 112, f= 38; and if three of the quadruple cross-overs are assumed to come from such configurations the solutions are a = - 299, b = 111, c = 1308, d = 219, e = 59, f= 57. Similar results are obtained with t~edfield's (1932) data for the second chromosome. These equations fit the geneticaI observations very well except for the negative result for a (i.e. there were less non-cross-over strands observed than expected with tllese equations). The cause of the shortage of non-cross-over strands ~ probably lies in the metaphase orientation and anaphase separation of the configurations at the heterotype division. Except in a few insects, none of which belong to dm Diptera, (el. Darlington, 1932) the metaphase pairing of chromosomes is conditioned by chiasma formation. Hence, where two of the homologous chromosomes of the triploid form ehiasmata with each other, the dlird taking part, in none, these three chromosomes will be represented at me~a]?hase by a bivalent and a univalent. Now the bivalent will always orientate itself so that its constituent chromosomes pass to the opposite poles at anaphase. Consequently no ma{ter to which pole the univalent

1 ]%edfleld points out that the viability relations of the classes in the progeny of this cross are good, hence the shortage of non-cross-overs eaunot be attributed to this cause. 254 Chiasmata and Crossing-over in [Drosophila ]?asses, it will always accompany one of the chromosomes derived from the bivalent (if the univalent divides at the first division this segregation will occur at the second anaphase), i.e. disjunction is not at random (el. Fig. 3). Redfield in the cross under consideration studied diploid progeny only, i.e. progeny which obtained one chromosome from the mother. This chromosome can never have been derived from the univalent where such non-random disjunction occurred. Iqow on the ]?atrial chiasmatype hypothesis chiasmata are the result of cross-overs, therefore where the above non-randon disjunction of chromosomes occurs there

T A

Fig. 3. A illustrates the non-random disjunction which occurs when the Crivalent is replaced by a bivalent and univalent. The univalent must pass to ~;ho ,same pole as one of the constituents of the bivalent. illustrates the random disjnnction of a trivalen~ with same number of ehiasmMa. The two chromosomes ~aking part in either of the ohiasmata may pass to t)he same, or to opposite poles. must be a shortage of non-cross-over chromosomes in ~he diploid progeny, because the mlivalent is never represented. Expressing this in another way, with random trivalent disjunc{ion one chiasma will lead to 33½ per cent. crossing-over (2 cross-over strands : 4 non-m'oss-overs) while with the above type of non-random disjunction ~he bivalent segregation only will be effective in the diploid progeny and hence one ehiasma will lead to 50 per cen~. crossing-over (2 cross-over strands : 2 non-cross-overs). On I£. 1KATIIE~ 255

the so-called "classical" theory of chiasma formation, the chiasmata are formed by the opening out of reductional and equational loops at early diplo~ene and crossing-over by subsequent breakage of a number of these chiasma~a, and so the above argument would not follow directly. On. the other hand, no case has been recorded cytologically of a chromosome being paired at diplotene by chiasmata and unpaired at megaphase, owing to breakage of the ohiasmata, so even on the classical theory ig seems probable that univalents would be chromosomes which had failed to form a chiasma, rather than chromosomes which had formed chiasmata with subsequent breakage. Therefore the occurrence of univalents would probably lead to the shortage of non-cross-overs as on the chiasmatype hypothesis. If the equations are derived as before except that allowance is made for the non-appearance of the univalents in the scored diploid progeny, we get a + 2/3b + 58/108o + b15/324d, + 345/864e -I- 797/2160/'= 771, ]./35 -'r 36/108o + 99/324d -f- 216/8648 6 4-37/2160/' = 497, 14/108o-F 63/324d-F 186/864e -t- 442/2160./" = 160, 17/3s z + 96/86 + 3ss/s16oj' = so, 9 1/86, + 1aT/S160, = 95/:?,16of = 0. The solnfiions are (where two of the q.nadrul~le cross-overs are assumed to come from configurations with five chiasmata) a = - 82, b = 4-32, c = 870, d = 118, e = 82, f = 32, and (where tl~ree quadruple cross-overs are assumed to arise from such configurations)

a = -- 81, b = z'-t-o8, n c = 850, el = 157, c = 41, f= 47. Thus the negative value of a is reduced to about 5 per eent.--a figure which can clN~e easily be due to the comparatively low number of flies scored. These new equations agree very well with the genetieal results, and consequently considerable support is given to the assumptions from which the equations were formed. The actual numerical solutions are not nearly so important as the fact that these equations do fit the genetical Ijesults quite well, certainly much bet%r than the equations derived on the previous possible modes of chromosome pairing. Thus it appears that only two of ~he chromosomes of a ~riploid can be effectively associated at any point, and that furl)her- 256 Chiasmata and Crossing-over in :Drosophila more the association of any pairing unit is not at random with respect to the rest, but that bhe pMring units tend[ to act in blocks. Bridges and kaderson (1925) and ~edfield (1930) have shown that the frequencies of occurrence of progressive and recurrent double cross- overs are approximately equal in the X and third chromosomes respectively. They have con.eluded from this that the chromosomes behave alike in synapsis and that crossing-over at one point does not affect crossing-over at a second. On first considgration this seems to indicate that the chromosomes are all completely associated along their whole length or that if only two chromosomes are associated at any poiut the pMring units behave indepmldentlyof each other. It has, however, been shown above that the genetieal results do not agree with these possi- bilities. At the moment it is impossible to predict the regional frequencies of occurrence of progressive and recurrent double cross-overs on the possible mode of synapsis which the genetical data apparently fit, since to calculate expectation certain minor assumptions have to be made. It seems, however, very probable that they would be equal for the whole length of the chromosomes, hence the frequencies of occurrence of the two different types of double cross-over are not at the moment a critical means of distinguishing between the possible ways of prophase chromosome pairing. The type of chromosome pairing deduced from genetical data in triploid D. melanopaste~" agrees very closely with that observed i~ triploid 51',d@a by Ne~4on and Darliugton (1929), triploid HyacintN~s by Dar- lington (1929), and trisomie Zea Mays by MeOtintock (1932).: They observed cytologically that at the pachytene stage of meiosis the three homologous chromosomes were associated in pairs, i.e. deuoting the three chromosomes as A, B and 6', that A and B could be paired at one point and C unpaired but that changes of partner occur at various planes, i.e. that in another section of the chromosome A and C could be pah'ed and B unpaired, or B and C paired and A unpMred. More recently Dar- tington and Mather (1932) and Stone and Mather (1932) have shown, from a mathematical analysis of chiasma~a in diakinesis configurations of triploid 5l'ul@c~ and in metaphase configurations in triploid Hyaoi~zthus respectively, that only two chromosomes can be effectively associated at one point and that pairing units (chromomeres or genes) do not act independently but in a small number of "pairing btocks"--crossing-over o1' chiasma formatioll occurring between effectively paired chromosomes only. The genetical results on triploid D. mdc~zogaste~"as analysed above agree entirely with these cytological findings, and hence it appears pro- K. ~{AT]~[EI% 257 bable that, if the chromosomes of triploid monocotyledonous plants and triploid flies behave ill the same way, this behaviour may subsequently prove to bc universal in triploid organisms. This "pah'ing block" theory can be applied to other polyploid organisms, but observational diNculties hamper such an application. Nevertheless wherever more than two homologous chromosomes are present in the same celt at meiosis they will, most probably, be found to behave in accordance with the "pairing block" concept, although the number of "pairing blocks" will probabty vary as they have been shown to do in tlyaci~zthus (Stone and Mather). When discussing crossing-over in any organism, particularly one which is polyploid or polysomic, the cytological behaviour of the chromosomes must be continuously borne in mind. As shown above, in a triploid only two of the three homologous chromosomes are e/~ectively associated at any point and consequently any one of the three chromosomes is on an average only two-thirds associated. Hence crossing-over must be considered from this aspecb. Furthermore chiasmata are either the result or cause of crossing-over (cf. Stern (1931), who finds a direct correlation between genetical and cytological crossing-over) and one chiasma in a triploid will give rise to 33½ per cent. crossing-over, whereas 51 a diploid one chiasma leads to 50 per cent. crossing-over. Where, however, the three homologous chromosomes in a triploid are represented at heterotypic metaphase by a bivalen~ and a univalent, one chiasma will again give 50 per cent. of crossing-over when recorded on the diploid progeny. Hence a direct comparison of diploid and triploid crossing-over, at least from certain points of view, is of doubtful validity. Probably the regional comparison of crossing-over in diploid and triploid D. ~nelano- 9aster leads to substantially correct conclusions, but the above facts must be considered when dealing with such data on diploid and polyploid crossing-over, particularly with respect to the total amount and the map distance. In short all treatment of potyploid genetical data, and par- tieularly comparative data concerning diploids and polyploids, must be based on essentially cytological considerations, and must not be wholly genetical (of. Sansome (1933), on the cytological aspect of crossing-over in diploid and tetraploid Nola~zum Lyco~)c~'sicum). In this respect it may be noted that conditions in a tetraploid more nearly correspond[ to those in a diploid, as each chromosome is completely paired, hence interference will be more or less the same as in a diploid. The essential difference between the diploid and tetraploid on the one hand, and the triploid on the other, lies in the presence of an unpaired chromosome at any point in the triploid, and this gives the triploid many of its peculiar features. 258 Chiasmata and Crossing-over in Drosophila

IV. Su~ka¥.

Various genetieal data on diploid and triploid D'rosophila melanogaster are analysed fl'om a cytological point of view. I~ is shown that the frequencies of crossing-over in the diploid agree with the eygologically observed frequencies of chiasma formation and persistence in other organisms, so lending suppor~ to the partial ehiasma- ~ype hypothesis. The various ~ypes o~ interference are discussed. [['he behaviour of the chromosomes at paehytene (synapsis) iu the triploid is deduced and it is shown to agree with the cytologically observed behaviour of chromosomes in certain plants. It is also shown tha~ non-random disj uncgion of the chromosomes may occur in a ~riploid and its genetical consequences are pointed out.

I~EFEgENOES. ]3ELLIOt, J. (1931). "Chiasma.s in flowering plants." Univ. Uab/f. Pub. Bot. 16, 311- 38. ]3RIDaES, C. ]3. and ANDERSON, E. G. (1925). "Crossing-over in the X-clu'omosomes of ~riploid females of Droso2hila, ,melanogaster." Genetics, t0, 418-41. ]3RIDGES, C. ]3. and MORG~-W, T. l~[. (1923). "The thh'd group of mutant characters of Droso~fl~ilc~ melc~nogast~r." Pub. Ua.rneg. Inst. Wash. 327. D~LI~GTO-~, C. D. (1929). "h'~eiosis in po]yploids. IL" Journ. Gen. 2i, 17-56. -- (1931). "h'Ieiosis in diplokl and tetraplokl Primula sinensis." Ibid. 26, 65-96.

-- (1932). l~ecent advances in cytology. London (Churchill). I)~RLI:~GTO~, C. D. and D~K, S. O. S. (1932). "The origin and behaviom' of chiasmata. II. Stenobothrus parallelus." Cytologia, 3, 169-85. DAI~LINGTON, C. D. a.nd ~L~m~, I~. (1932). "The origin and behaviom' of chi~s- mata. IlL Triploid Tulipa." Ibid. 4, 1-15. tLu~DANE, J. ]3. S. (1931). "The cytological basis of genetical hlter~erence." Ibid. 3, 54-65. I~cCLI~TOO]~,]3. (1932). "CySological observNSons in Zea on the intimate association of non-homologous p~l'~s of chromosomes in the nlid-prophase of meiosis and i~s rel~tion to di~kincsis configm'~tions." Proc. Sixth Int. Cong. Goner. pp. 126-8. NEWTON, W. C. F. and ])~!tLIN£~TOlV,C. ]). (1929). "h'Ieiosis in 1)olyplokis. I." Journ. Gen. 2i, 1-16. 0'h~L~Ra, J. (1931). "Chromosome p~iring in Yuccaflaccida." Uytologia, 3, 66-76. I~EmrIELD, H. (1930). "Crossing-over in the thh'd ciu'omosomes of triploids of Drosophila melanogctster." Genetics, i5, 205-52.

-- (1932). "A comparison of triploid and diploid crossing-over for chromosome II of Drosophilc~ melanogastcr." Ibid. 17, 137-52. S±~so~E, F. ~V. (1933). "Chromatid segregation in Solca~um Lycoj)crsicwm." dourly. Gcn. 27, 105-26. K. h{ATtIEI~ 259 S~NSO~E, F. W. and Pi;~v, J. (]932). Recent advances in plant genetics. London (Ch~u.chill). S~x, K. (1930). "Chromosome sbructure and the mechanism of crossing-over." Jou'r~. Arnold Arbor. li, ]93-220. (1932). "The cytological mechanism of crossing-over." 1bid. 13, 180-212. STm~, C. (1931). "Zy~ologisch-genetische Untersuchungen als Beweise fiir die ~{org~nsch~ Thcorie des Faktor~u~ust~usches." Biol. Zbl. 5i, 81-121. STo~, L. H. A. and h~'~'~En, K. (1932). "The origin and behavionr of chia smat~. IV. Diploid and triploid tIyacinthus." Cylologia, 4, 16-25.