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

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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 be- haviour 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 chromo- some 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 approxi- mately 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 60 5O ~o 50 /// / 2O /0 / 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 propor- tion 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 forma- tion 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.
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