A Polymerization Model of Chiasma Interference and Corresponding Computer Simulation

Jeff S. King* and Robert K. Mortimert?* *Graduate Group in Biophysics, University of California, Berkeley, California 94720, tDepartment of Molecular and Cell Biology, Division of Genetics, University of California, Berkeley, Calfornia 94720, and$Division of Cellular and Molecular Biology, Lawrence Berkeley Laboratory, Berkeley, California 94720 Manuscript received April 16, 1990 Accepted for publication August 29, 1990

ABSTRACT A model of chiasma interference is proposed and simulated ona computer. The model uses random events and a polymerization reaction to regulate meiotic recombination between and along chromosomes. A computer simulation of the model generates distributions of crossovers per chromosome arm, position of events along the chromosome arm, distance between crossoversin two-event tetrads, and coincidence as a function of distance. Outputs from the simulation are compared to data from Saccharomyces cerevisiae and the X chromosome of Drosophila melanogaster. The simulation demon- strates that the proposed model can produce the regulationof recombination observedin both genetic andcytological experiments. While the model was quantitativelycompared to data from only Drosophila and Saccharomyces,the regulation observedin these species is qualitatively similarto the regulation of recombination observedin other organisms.

N 1916 MULLERreported that a crossover in one and Saccharomyces gene conversions with a genetic I region of a Drosophila chromosome changes the crossover show chiasma interference but gene conver- probability of a crossover in an adjacentregion sions without exchange of flanking markers do not (MULLER19 16). Subsequent work in other eukaryotes interfere with other gene conversions, either with or revealed in general a distance-dependent reductionin without associated crossovers (STADLER1959; MOR- the probability of a second crossover, known as posi- TIMER and FOCEL1974). Chromatid interference, a tive chiasma interference. Interference is expressed deviation from the 1 :2:1 distribution of 2-, 3-, and 4- in terms of coincidence, which is the ratioof observed strand double crossovers, is not detected via tetrad to expected coincident crossovers in two linked re- analysisin Saccharomyces (MORTIMER and FOGEL gions. Interference has beenobserved atboth the 1974) or in crosses of Drosophila with attached X cytological and genetic levels. Forexample, coinci- chromosomes (EMERSONand BEADLE1933). dence as a function of distance has been measured Electron microscopic studies of pachytene synapto- genetically in Drosophilamelanogaster (WEINSTEIN nemal complexes resulted in the discovery of recom- 19 18) and in Saccharomyces cerevisiae (MORTIMERand bination nodules (SCHRANTZ1970; GILLIES1972) (for FOCEL 1974). Inboth organisms, interference is reviews, see VON WETTSTEIN, RASSMUSSENand HOLM strong for closely linked regions and falls off with 1984; CARPENTER1989). Two types of recombination increasing separation. Cytologically, interference has nodules have been observed.Based on their temporal been examined in terms of coincidence of chiasmata, appearance in meiotic nuclei they have been termed the cross-shaped associations betweenhomologous early and laterecombination nodules (CARPENTER chromosomes seen in diakinesis and considered to be 1989); however, it has not been established that late an outcomeof genetic exchanges (forreview SeeJONES recombination nodules arise from early recombina- 1984). Forexample, on thelong arm of the L3 tion nodules. The demonstration that late recombi- bivalent of Chorthzppusbrunneus complete interfer- nation nodules parallel chiasmata in frequency and ence is observedover distances of 25-30% of the distribution has led to the proposal that late recom- length of the bivalent arm (LAURIE 1980), and no bination nodules are directly involved in meiotic re- interference is observed for regions longer than 60% combination (CARPENTER1975). The distributions of of the length of the bivalent arm. late recombination nodules,chiasmata, and crossovers Geneticexperiments have provided clues tothe are comparable in terms of number per chromosome relationship between recombination and interference. and positions on chromosome arms. Late recombina- In S. cerevisiae and in D. malanogaster, roughly one- tion nodules have been associated with sites of local- half of all gene conversions are associated with recip- ized DNA synthesis (CARPENTER1981), which is be- rocal recombination (HURST, FOCELand MORTIMER lieved to occur during recombination. (For models of 1972; HILLIKER andCHOVNICK 198 1). In Neurospora recombination see MESSELSON and RADDINC1975;

(knrtics 126: 1127-1 138 (December, 1990) 1128 J. S. King and R. K. Mortimer TABLE 1 Comparison of genome sizes and recombination levels for several organisms

Organism DNAco/Mb MbMb/chr chi/biv N co/biv cM/gen SC wn/biv

Schizosaccharomyces pombe 14 3 1,895 12.6 n.d. 4.7 2.7 n.d. Saccharomyces cerevisiae 14 16 4,300 5.4 n.d. 0.88 6.14 1.56 Neurospora crassaNeurospora 7 47 1,000 2.9 n.d. 6.71 0.43 8.29 Drosophila melanogasterDrosophila 1.9 285180 3 n.d. 6015.33 0.032 Caenorhabditiselegans 100 6 300 1.o n.d. 16.7 0.06 5.8 Bombyx mori 500 28 2,900 2.1 n.d.0.12 17.8 9.21 Homo sapiens Homo 3,500 23 3,000 3.0 n.d. 130 0.023 10.26 Mus musculus Mus 3,000 20 1,630 1.6 n.d. 150 0.1 1 n.d. Zea mays 8,000 10 1,300 2.632.5 1.70.0033 800 Lilium long9orumLilium 12180,000 n.d. n.d. 2.4 15,000 0.0000 16 308.3 Amount of DNA in base pairs, number of chromosomes, size of genetic map, and size of synaptonemal complexes. While the organisms listed have radically different amounts of DNA, they are similar in terms of cM per genome andcrossovers per bivalent. Data are from: VON WETTSTEIN,RASSMUSSEN and HOLM(1984), FASMAN(1976), O’BRIEN(1987) and WHITE (1977). Abbreviations: Mb = megabase-pairs, cM = centimorgans, gen = genome, co = crossovers, biv = bivalent, chi = chiasmata, chr = chromosome, SC = synaptonemal complex, N = the haploid number of chromosomes, n.d. = no data. and ORR-WEAVER,SZOSTAK and ROTHSTEIN198 1 .) thy because a growing body of evidence suggests that For thesereasons it is believed that late recombination some DNA sites havemuch higher recombination nodules, chiasmata and genetic crossovers are all man- rates than other sites. Stochastic models can also be ifestations of the same event: reciprocal meiotic re- based on a limited supply of a necessary component. combination. Criticisms of stochastic models stem from their inabil- Regulation of recombination events is apparent at ity to account for interference without additional as- several levels. The number of crossovers per chro- sumptions. Stochastic models based on a small number mosome arm was shown to be nonrandom (HALDANE of sites are not supportedby the fact that in Drosoph- 1931; WEINSTEIN1936) since, compared to a Poisson ila some chromosomal sequences thatare moved distribution with the same average, the number of adopt the exchange distribution characteristic of the Drosophila X chromosomes with no crossovers was new location (BAKERand CARPENTER 1972). Further- underrepresented and the number with one or two more, in Saccharomyces the allelic recombination rate crossovers was overrepresented. The distributions of of a DNA sequence varies significantly, depending on late nodules, chiasmata, and crossovers are nonuni- its location in the genome (LICHTEN,BORTS and HA- form in terms of position along chromosomes and BER 1987).It is also clear from fine-scale genetic nonrandom in terms of distances between them. The analysis that there are many sites at which recombi- positions of crossovers along the telocentric X chro- nation may occur along a chromosome. However, in mosome of Drosophila tend to be centrally located in any one meiosis, recombination occurs at only a few single crossover tetrads and tend to have one cross- of these sites. over near the centromere and the other crossover Other models are based on pairing, in whichcertain near the distal telomere in double crossover tetrads chromosomal regions are assumed to pair first and (CHARLES 1938). are therefore more likely to undergo recombination. Several models have been proposed to account for However,without additional assumptions this does either interference or the distributions of chiasmata not account foreither interferenceor for theobserved alongchromosomes (for review see JONES 1984). distribution of crossovers between chromosomes. Still Some of the models address interference but fail to other models are steric, i.e. the molecules that catalyze account for the observed distributions of chiasmata recombination eventsphysically blockadjacent events. between chromosomes. Others models only address It is difficult to account for the nonrandom distribu- the distributions of chiasmata and donot account for tion of events betweenchromosomes with these interference. A model of particular interest was pro- models, and based on electron micrographs, it is dif- posed by EGELin 1978. EGEL’Smodel is based on ficult to explain how a few nodulesthat are small possibilities of exchanges that are established before relative to the length of the synaptonemal complex synapsis and serve as initiation centers ofsynapsis. could have such a strongeffect on each other. Another Synapsis is then followed by formation of synaptone- criticism is that the source of the steric interference mal complex, which prevents the establishment of would need to be modified significantly between or- further possibilities of exchange. This results in posi- ganisms with different amounts of DNA and synap- tive interference. tonemal complex lengths. Table 1 is a compilation of Stochastic models, based onrecombination sites genomic datafrom several organisms. Theseorga- with individual crossover probabilities, are notewor- nisms have DNA contents that vary over four orders of Chiasma Model of Interference 1129 the model doesnot depend on these speculations. Growing polymers block the binding of additional early structures tothe synaptonemal complex. As these polymers grow, bound structures that have not yet initiated such areaction continue to have the opportunity to do so until they are ejected by the advance of a polymer initiated at a nodule located elsewhere onthe same chromosome. ’I’he ejected structures move into the surrounding medium,where they are either degraded,reattach to an available site on the same synaptonemal complex, or reattach to a site on a different complex. A chromosome that received only a single structure would retain the structure since it could not be ejected,and wouldbe guaranteed to have a single late nodule and thus a single crossover. The number of chromosome arms with zero crossovers is initially determined by a Pois- son distribution based on the average numberof early structures. In this model there are more early structures than late nodules, thus there are fewer chromosome arms with zero crossovers than there would be if the number expected was based solely on the average numberof late nodules. The number of chromosome arms with zero crossovers may be further reduced by the relocation of ejected early structures onto chromosomearms that were initiallyvoid of structures. These features of our model insure that virtually all bivalent arms will eventually obtain at least one late nodule, provided a moderate excess of early structures is synthesized. This model was partly inspired by the proposal of RASMUSSENand HOLM FIGUREI .-A speculative drawing of the model. h-ly structures (1 978) for a redistribution from random recombina- (circular) bind randomly to the synaptonemal complex. Some initi- tion nodules associated with the synaptonemal com- ate polymerization reactions thus becoming late nodules (oval). The plex at zygotene to the nonrandom nodules observed growing polymers eject early structures. The ejected early struc- in pachytene. tures are either degraded or bind to synaptonemal complex that is free of polymer. MATERIALS AND METHODS of magnitude, but (with the exception of Schizosac- A Modula-2 program was written and run on an Apple charomyces pombe) the number of crossovers per biva- Macintosh IIcx computer to simulate the model. The com- lent varies by only a factorof about five. It is apparent puter simulation generated “late nodule” bivalent arms from that a model of interference that is to apply to more “early structure” bivalent arms. Earlystructures were placed than a few organisms must be adaptable such that it at random on a bivalent arm with one hundred binding sites.Random numbers were generated using the Pascal can act over awide range of distances. Polymerization version of the uniform deviate random number generator is a means with biological precedence which may be RAN1 from PRW et al. (1986). The number of early utilized to regulate recombination over such a range. structures placed on the arm followed a Poisson distribution A polymer-based interference model: We propose with an average of roughly twice the number of crossovers a model to account for chiasma interference in which observed in either Drosophila or Saccharomyces. The positions of the early structures were determined by generating early structures randomly attach among and along thea random number between zero and ninety-nine for each synaptonemal complexes of meiotic nuclei (see Figure structure and then designating the corresponding binding 1). Once attached,each structure has an equal chance site as the location of a structure. Each “turn,” a random per unit time of initiating a bidirectional polymeriza- number between zero and one would be generated for each attached early structure. If this number wasless than or tion reaction. It is speculated that the structures ini- equal to an assigned probability of initiating a polymeriza- tiating a polymerization reaction would give rise to tion reaction, that structure would be transformed into a the late recombination nodules, seen in pachytene, late nodule and a polymer would begin to grow outward and would initiate reciprocal exchange. It is further from the nodule. The polymers grewat a constant rate of one site per turn speculated that the early structures are the early re- in both directions and would stop if they encountered an- combination nodules observed in zygotene, however other polymer, a centromere, or an end of a chromosome 1130 J. S. King and R. K. Mortimer

arm. Any early structures encountered by a polymer would RESULTS then be removed and a new round would ensue, checking to see if any remaining early structures had initiated a Figure 2 is a graphic comparison of the observed polymerization reaction. In principle the model allows for number of crossovers along the X chromosome of the reattachment of displaced nodules; however, for sim- plicity this was not done in the computer simulations. When Drosophila to the number predicted by our model. all the early structures had either been ejected or become The probability of initiating apolymerization reaction late nodules, the cyclewould stop andthe results were was set at 0.009.The sample size was 150,000 tetrads. recorded in terms of nodules remaining, distance between The number of bivalent arms with zero nodules in the nodules, and distribution of nodules along the bivalent arm. computer simulation is initially 6.1% based on a Pois- Chiasma interference was analyzed in terms of coincidence son distribution with an average of 2.8 nodules per as a function of distance. For the simulation that was compared to the yeast data, bivalent. The model output and the observed distri- each turn the growing polymers were givena probability of butions are quantitatively similar and both have an independently terminating growth in either direction. The average of 1.4 crossovers per tetrad anddiffer signif- values of the polymerization start and polymerization ter- icantly from the Poisson distribution with the same mination probabilities used in the simulations were deter- average. The same output was then compared to the mined by trial and error, acceptable values being those that Drosophila data in terms of distance between cross- reduced the initial distribution of early structures to a distribution of late nodules that matched the observed distri- overs, coincidence, and distribution of crossovers in bution of number of crossovers. rank one and rank two tetrads. Comparison of our model to genetic observations of the Figure 3 is a comparison of the distributions of number of crossovers and to their distribution along a crossovers in rank one and rank two tetrads plotted chromosome arm requires crosses with markers spanning by CHARLES(1938) and a similar distribution gener- the length of a chromosome arm. Fortunately, relevant ated by the computer simulation. The distributions of published data from Drosophila (BRIDGES1935) andunpub- lished data from Saccharomyces (R. K. MORTIMERand S. crossovers for single- and double-event arms from the FOCEL)were available. In Drosophila thetetrads are in- computer simulation have the characteristics observed ferred from the genotypes of the diploid progeny. The in Drosophila. Single events tend to be located near number of crossovers along the X chromosome in Drosoph- the center of the chromosome. The computer-gener- ilawere determined by WEINSTEIN(1936). Distancesbe- ated distribution of single events does not have the tween crossoversand thedistribution of crossovers in single- sharp peaks (atapproximately 25% and 70% the and double-event tetrads were analyzedby CHARLES(1938). length of the chromosome) and the drop at approxi- The interference values from the computer simulations were compared with the data in STEVENS(1936). In the mately 55% the length of the chromosome, observed comparisons of the model and the experimental observa- in Drosophila. However, CHARLES(1938) concluded tions made in Drosophila, the distances between genetic that due to errorsprimarily from estimating positions markers are based on salivary map data compiled by of markers it can not be concluded that the peaks CHARLES(1938) and are thus physical distances. observed represent a true bimodal distribution. He For the comparisonof the number ofcrossovers per did concludethat in single crossover tetrads, thecross- chromosome arm in the computer simulations with Saccha- romyces, 780 tetrads with markers spanning the right arm over is less likely to be near one of the ends than it is of chromosome III were analyzed. The number of cross- to be near the center of the chromosome. Double- overs or recombinogenic events per tetrad is the tetrad event tetrads tend to have one event near the cen- rank. The distributions of crossovers in rank one and rank tromere and the other towards the telomere, and in two tetrads were calculated. Coincidence values from the this case the model output and the experimental data computer simulation were compared to data from several from Drosophila agree very well. Peaks at approxi- other Saccharomyces crosses, with a total of 8927 tetrads, mately 25% and the length of the chromosome all with at least five closely spaced markers on chromosome 70% VIII. In the comparisons of the model and the experimental are also observed in rank two tetrads. This may indi- observations made in Saccharomyces,the distances between catethat while the overall distribution is predomi- genetic markers are based on genetic data. The two sets of nantly determined by rank there may be a tendency yeast data were necessary sincethe crosses that provided the for events to occur in specific regions regardless of chromosome IIZ data did not have markers close enough rank. The lack of events near the centromere in the together to carry out interference analysis and the crosses experimental distribution of crossovers in two-event used for interference analysis on chromosome VIII did not have markers spanning the length of a chromosome arm. tetrads may be caused by exclusion of events near the Markers spanning the chromosome arm are necessary in centromere. This exclusion may account for the ob- order to record all the recombination events, and this is served distribution of single crossovers near the cen- needed to determine the overall rank of the arm and the tromere being significantly lower than the model out- distribution of crossovers along thearm. Since there is put. To generate distributions of crossovers with ex- typically no interference across centromeres, the two arms cluded regions, amore refined simulation could begin of a chromosome should be independent in terms of the with a nonuniform distributionof early structures. distributions of crossovers along the arms. Because of this we compare the model to chromosome arms. Since the In Figure 4 we compare the distance between nod- Drosophila X chromosome is telocentric this is essentially ules in rank two tetradsfrom Drosophila data the entire chromosome. (CHARLES1938) to data from the computer simula- Model of Chiasma Interference 1131

FIGURE2.-Number of events per tetrad. The computersimulation was based on 150,000 tetrads with an initial distribution of average 2.8 early structures per tetrad and a polymerization start probability of 0.009. The simulation is compared to Drosophila data (WEINSTEIN 1936). and to a Poisson distribution. The results of the computer simulation arelabeled "MODEL" and the data from Dro- sophila are labeled "OBSERVED" The three distributions graphed have an averagevalue of 1.4 crossovers per tetrad.

tions. The average distance between crossovers in Computer simulations were also used to compare both the model output and in Drosophila is approxi- the model to observations made in the yeast S. cermis- mately 50% the length of the chromosome arm. This iae. A probability for a growing polymer to stop was is significantly greater than the average of 33% ex- added to the simulation to obtain coincidence values pected for a random distribution. That the distance that would approach one over long distances. The between crossovers is greater than it would be if the simulations compared to Saccharomyces initially had, events were randomrestates that there is positive on average, three early structures, a polymerization chiasma interference. start probability of 0.02, and a probability of polym- A more informative way of analyzing interference erization termination of 0.0 1. Thirty-thousand model is to graph coincidence vs. distance since it illustrates tetrads were analyzed. The experimental data for the how interference decreases with increasing separation comparison of crossovers per tetrad and thedistribu- of genetic intervals. This was done in Figure 5 for tion of events along the right armof chromosome IZI both the computer generated data and for data ob- come from 780 tetrads from hybrid 4579 (S. FOGEL, tained along theX chromosome in Drosophila. Exper- unpublished data). The genotype of hybrid 4579 is imental coincidence values and uncertainties are from + + leu2-27 MATa + ma12 STEVENS(1936) and distances are from CHARLES a (1 938). This coincidence distance graph illustrates 'us. his4 leu2-I MATa thr4MAL2 that interference data from the model qualitatively + follow experimental results in Drosophila, in that in- The gene leu2 was used as a centromere marker(it is terference is strong over short intervals and falls off about 5 centimorgans to theleft of the actual centrom- with increasing separation. The failure of the com- ere), and recombination events between leu2-27 and putergenerated coincidence values to reach unity MAL2 (MAL2 is within 10 kilobase-pairs of the right over thelength of the chromosome armmay be caused telomere) were recorded. Since leu2 is to the left of by the oversimplification of polymer growth made in the centromere a fractionof the crossovers that were the computer simulation. A real polymer might not recorded ashaving occurred on the right armpresum- have a constant, length-independent growth rate.The ably occurred on the left arm, but since the distance growth rate in long polymers might be limited by between MAT and the centromere (about 30 centi- breakage and a decreasing concentrationof subunits. morgans) is sixtimes longer thanthe distance between At small distances the model does not generate as leu2 and the centromerethis fraction is assumed to be much interference as observed. This may be due to small. our model recombinat.ion structure being only one Table 2 is a comparison of the observed number of potential recombination site in size. A structure that crossovers per tetrad arm (data from chromosome IZI occupied several sites would generate absolute inter- of Saccharomyces), tothe number predicted by a ference over short intervals. computer simulation. Since occasionally four-strand J. S. King and R. K. Mortimer

1.o -

0.8

0.6

FIGURE3.- Distributions of crossovers in tetrads from a com- 0.4 puter simulation and from Drosoph- ila. The experimental data is from CHARLES(1938) inwhich the Dro- sophila tetrads are inferred from the 0.2 genotypes of the diploid progeny. Po- sition 0 corresponds to the centromere and position 100 corresponds to the distal telomere. Frequency is the 0.0 percentage of tetrads with an ex- 0 20 40 60 80 100 change in the specified region per 1/ POSITION 100 the length of the chromosome arm. (A) Distribution of crossovers in rank one tetrads. Model output has been adjusted to account for differ- 1.6 ences in the percentage of single event tetradsreported in CHARLES (1938) and the computer simulation. 1.4 * MODELLEFT + OBSERVEDLEFT (B) Distribution of crossovers in rank 4 MODELRIGKT two tetrads. The left-most and right- 1.2 + OBSEfWEDRIGHT most crossovers are plotted sepa- rately. Model output is adjusted to take into account differences in per- 1.o centage of tetrads withtwo exchanges reported in CHARLES(1 938) 0.8 compared to the computer simulation and the different number of intervals (8 vs. IO). 0.6

0.4

0.2

0.0 0 20 40 60 80 100 POSITION double crossovers occurred betweentwo adjacent that occurred in otherwise rank zero, rank one or markers, resulting in a non-parental ditype tetrad, it rank two tetrads. For example, there were 12 four- can be assumedthat otherdouble crossovers occurred strand double crossovers in tetrads with no other that wereobserved as either parental or tetratype events, so 12 of the rank zero tetrads and 24 of the tetrads. The data are adjusted to take into account rank one tetrads are scored as rank two tetrads. It is these “silent”double exchange events in the following apparent from the data in Table2 that a Poisson way. A four-strand double exchange results in a non- distribution does not approximate the experimental parental ditype tetrad (NPD); assuming a two-strand observationswhile the distribution of late nodules double is as likely (no chromatid interference), there generated by the computer simulation of the model can be expected to be an equal number of two-strand does approximate the observed tetrad ranks. doubles that were scored as parental type tetrads (P). The observed distribution of crossovers in rank one Similarly, for every four-strand double we expect tetrads from hybrid 4579 did not deviate significantly there to be two three-strand doubles that were ob- from a uniform distribution. However, the distribu- served as tetratype tetrads (T). These adjustments are tion of crossovers in rank two tetrads did deviate then weighted in proportion to the numberof NPDs significantly from a uniform distribution and showed Model of Chiasma Interference 1133

FIGURE4.-Distance between crossovers in two-event tetrads vs. frequency of occurrence.Distance is based on physical dataand is in percentage of the chromosome arm length and frequency is thepercentage of rank two tetrads with thecorresponding distancebe- tween exchanges. Experimental data is from Drosophila, and analyzed by CHARLES(1938). In both the computer simulation and in Drosophila the distance betweencrossovers in double- event tetrads are approximately50% the length of the chromosome arm, rather than 33% as would bethe case for a random distribution.

0 60 20 40 80 100 DISTANCE

1.2

1 .o

FIGURE5.-Coincidence vs. distance. 0.8 Experimental data are from theX chro- W mosome ofDrosophila, calculated by Y STEVENS(1936). In both the model and W in Drosophila, interference is strong 0.6 over short distances and falls off with increasingdistance. Inthe computer P simulation coincidence does not reach a 8 value of one even for lor'g distances be- 0.4 cause no limitations were ,,laced on polymer growth. In Drosophi'a coincidence is approximatelyone ior distances 0.2 greater than 50% the length of the X chromosome.

0.0 0 20 40 10060 80

DISTANCE the characteristic of having one of the two events near overs in rank two tetrads tend to be spaced far apart. the centromere and the other near the telomere, as One crossover tends to occur near the centromere was found to occur in Drosophila. The number of (region 1) and the other near thetelomere (region 3), double crossovers in the regions genetically defined thus regions 1 and 3 have a high number of coincident in hybrid 4579 are listed in Table 3. Region 1 is crossovers in rank two tetrads. between leu2 and MAT (no crossovers occurred be- Figure 6 shows thedistribution of crossovers in tween the two leu2 alleles). Region 2 is between MAT model tetrads and tetrads from hybrid 4579 of rank and thr4 and region 3 is between thr4 and MAL2.The one and rank two. The events in rank one tetrads numbers of crossovers expected in the regions listed from the computer simulation have the characteristic if there were no interferenceare in the column labeled of being located nearthe center of the arm;however, "no interference." The numbers from computer sim- it is a moderately flat distribution which is consistent ulations are in the column labeled "model." In Sac- with not detecting a nonuniform distributionof events charomyces and in the computer simulations, cross- in the 253 rank one tetrads examined from hybrid 1134 J. S. King and R. K. Mortimer TABLE 2 defined genetic intervals in hybrid 4579. Crossovers per chromosome arm from Saccharomyces and from In Figure 7, coincidence vs. distance is graphed for the model the model and for the six hybrids listed in Table 4. Allsix hybrids are genetically marked with at least Percentage of tetrads five genes spaced close enough together along the Model right arm of chromosome VZZZ to calculate interfer- Observed No. of Poisson Early Late ence as a function of distance. Positions are in per- crossoversdistribution structures nodules Saccharomyces centage of the total genetic length of the chromosome 0 16.5 5.0 5.0 6.8 arm, beginning atthe centromere with zero, and 1 29.8 14.9 33.4 32.4 moving 40% toward the telomere. The length of the 2 26.8 22.4 38.5 37.1 region from CEN8 to CUPl compared to the length 3 16.1 3 22.4 17.7 19.9 of the chromosome arm was calculated from data in 4 7.2 4 16.8 4.6 3.8 the recent genetic mapof Saccharomyces (MORTIMER >4 3.6 18.5 0.8 0.0 et al. 1989),and the distances between the genes Average 1.80 3.00 1.86 1.80 within the region were calculated from the relative Number of crossovers per tetrad observed on the right arm of frequencies of crossovers in the crosses examined in chromosome III of Saccharomyces compared to the number of early structures and late nodules from a computer simulation. The this study. Coincidence values are calculated for the experimental data are from 780 tetrads of hybrid 4579. The combinations of regions between cen8 and petl, petl simulation was based on 30,000 tetrads with an initial distribution and arg4,arg4 and thrl,thrl and CUPl. Regions of three early structures per tetrad, a polymerization start proba- between arg4 alleles were not used since not enough bility of 0.02, and polymerization termination probability of 0.01. Both the model distribution of late nodules and observed tetrad coincidental events occurred in these intervals to ac- ranks in Saccharomyces differ significantly from a Poisson distri- curately calculate interference. The computer simu- bution. The predicted rank of tetrads from computer simulations lation and experimental observation are basically in based on the model approximate the ranks of tetrads from hybrid 4579. agreement, with the observedinterference being slightly stronger overall. TABLE 3 Positions of crossovers in rank two tetrads DISCUSSION

Regions ObservedRegions No interference Model Since our model proposes that the same cellular 1-2 60 40 54 machinery is used to regulate meiotic recombination 1-3 124133 59 between and along chromosomes, it can explain the 2-3 58 44 65 complex and diverse phenotypes of some meiotic mu- 1-1 16 8 53 tants. For instance Drosophila mutants thatrelax chro- 2-2 8 30 6 mosomal regulation in the form of interference also 3-3 24 65 26 have a more random numberof events between chro- The positions of crossovers along a chromosome arm in rank mosomes (BAKERand HALL1976). In our model, this two tetrads in Saccharomyces are nonuniform and are similar to Drosophila in that one exchange tends toward the centromere and phenotype could be caused by either defective poly- the other toward the telomere. The numbers in the column labeled mer subunits, or by recombination nodules unable to “observed” are from chromosome 111 of hybrid 4579. The number initiate the polymerization reactions. Drosophila mu- of rank two tetrads expected with a crossover in each of the two regions listed in the first column for a uniform distribution and tants, heterozygous for large inversions on chromo- without interference is listed in the column labeled “no interfer- some ZII oron the X chromosome, have reduced ence.” The expected numbers were calculated using the overall recombinationrates on the chromosomes with the recombination rates for each region in hybrid 4579. The numbers in the colunm labeled “model”correspond to thenumber generated inversions and increased recombination rateson their in the computer simulations. For the computer simulations the normal chromosomes (SHULTZand REDFIELD1933). regions were set to the proportions of the regions in hybrid 4579. Thisphenomenon is known as the Shultz-Redfield In rank two tetrads one crossover tends to be in region one andthe other tends to be in region three. effect and has been shown to occur in other organisms. In our model it can be explained as resulting from 4579.Figure 6, parts B and C, are graphs of the early structures being less likely to bind to the mutant distributions of crossovers from simulated rank two chromosomal regions (which may pair more slowly or tetrads and the similar events recorded from hybrid less frequently),resulting in more early structures 4579for the left-most (Figure 6B) and right-most available to bind to the normal chromosomes. (Figure 6C) crossovers. The experimental distances In Drosophila,humans, Neurospora, and several are based on genetic data. The comparison of the other organisms,structures in zygotene associated distribution predicted by the model to the observed with the synaptonemal complex have been observed positions of the crossovers in Saccharomyces illustrates (CARPENTER1979; HOLM and RASMUSSEN1983; that themodel positions crossovers according to tetrad BOJKO 1989). Termed early recombination nodules rank in a pattern similar to that seen experimentally. in Drosophila, these structures outnumber late recom- Analysis of the data is limited by the small number of bination nodules and arerandomly distributed. Struc- Model of Chiasma Interference 1135 A 0.13 1 * Y 0.11 - Lu 3 8 h 0.09 -

0.07 : I I I I 0 20 40 60 80 1 POSITION FIGURE6,”Distribution of crossovers in rank one and rank two tetrads from a computer simulation and from hybrid 4579. The computer simulation

Q DOUBLES, LEFT was based on 30,000 tetrads, a polymerization start probability of 0.02 and a A OBSERVED, LEFT polymerization termination probability of 0.01. (A) Crossovers in rank one tetrads. Crossovers in rankone tetrads tend to belocated near the centerof the chromosome arm, however the localiza- tion is not dramatic. The experimental distribution ofsingle crossovers in hybrid 4579 does not significantly deviate from a uniformdistribution. (B) The positionsof the leftmostcrossover in rank two tetrads. Crossovers tend to be near the centromere, which is near the 0 20 40 60 80 100 left-hand end of the monitored region. POSITION (C) The positions of the rightmost crossover in rank two tetrads. Crossovers tend to be near the telomere. T

Q DOUBLES, RIGHT A OBSERVED, RIGHT

” I I I I 0 20 40 60 80 100 1. POSITION tures interpreted as intermediates between early and currently exist; however this model provides a means late nodules have beenobserved andsupport the for therandomly distributed early nodules to give rise theory that late nodules arise from a subset of the tothe nonrandomlydistributed late nodules using early nodules, but the intermediates observedmay be only random events. partially assembled late nodules (BOJKO1989). In this Several investigators have made proposals for the model these possible morphological changes may possible function of early nodules. CARPENTER(1979) mark theinitiation of a polymerization reaction. Proof proposed that early nodulesmediate nonreciprocal that late nodules arise from early nodules does not recombination and late nodules mediate reciprocal 1136 J. S. King and R. K. Mortimer . .-

0.8 - FIGURE7.-Coincidence us. distance for Saccharomyces and the model. Dis- tance is based on genetic data and is in 0.6 - percentage of the chromosome arm beginning with zero at the centromere. Experimental data is pooled from the six crosses listed in Table 4 (R.K. MORTI- MER and S. FOGEL,unpublished data). 0.4 - The computer simulation is based on 30,000 tetrads, a polymerization start probability of 0.02and a polymerization termination probability of0.01. The 0.2 - computer simulation of the model ap- proximates the experimental observations made in Saccharomyces.

0.0 : -' I I I I 0 10 20 30 40 DISTANCE

TABLE 4 version events are randomly distributed and do not Saccharomycesstrains used to collect data for interference exhibit interference. Late nodules and reciprocal ex- analysis change events are nonrandom and do exhibit interference. Based on our model of interference and the No. of Hybrid Genotype tetrads above observations, and given that the late nodules ~~ ~ arise from the early nodules, there are two possible X2961 + petl + arg4-16thrl - __ orders of events. Both involve the initial random trpl + arg4-3 + + CUP1 2123 placement of early recombination nodules capable of 5571 + pet1 + thrl - initiating gene conversion. In the first, if an early nodule initiates a polymeri- trpl + arg4-3 + CUPI 1074 zation reaction it becomes anchored to the synapto- 5574 + pet1 + + thrl - nemal complex and the necessary steps for reciprocal trpl + a7g4-3arg4-36 + CUPl 1073 recombination occur. In the second possible order of events, cleavage of the DNA resulting in a reciprocal 5577 +arg4-16 petl + thrl - ~~ ~~ ~ ~ exchange precedes and results in the initiation of a trpl + + arg4-36 + CUPI 908 polymerization reaction and cleavage of the DNA not E559 + Pet1 + arg4-16thrl - resulting in crossover does not lead to the initiation of a polymerization reaction. These proposals are trpl + arg4-3 + + CUPl 1140 similar to the proposal, based on the analysis of mu- 5497 trpl + thrlarg4 - tants defective in reciprocal exchange, made by CAR- + petl + + CUP1 2609 PENTER (1982) that isomerization is under genetic Genotypes of strains used in interference analysis. Strains were control. In both cases all early nodules would be at heterozygous for additional markers not used in this analysis. [Hy- sites of initiation of recombination and there would brids from S. FOCELand R. MORTIMER.Data from hybrid X2961 be reciprocal recombination at sites where an early are published in MORTIMERand FOCEL(1 974), the rest are unpublished]. nodule becomes a late nodule. Furthermore the approximately 50% crossover to gene conversion with- recombination. RASMUSSENand HOLM(1978) pro- out crossover ratio would be a reflection of the redis- posed that early nodules mediate both reciprocal and tribution of nodules, not a random resolutionbetween nonreciprocal recombination, but only those that are equivalent forms. involved in reciprocal recombination remain associ- Positive interference is not found in all regions of ated with the synaptonemal complex and thus become the Drosophila and Saccharomyces genomes. For ex- late nodules. There aretwo observations that must be ample MULLER(1916) found no interference across considered in order to fit our model with the genetic the centromere of chromosome ZZ and negative inter- and cytological results. Early nodules and gene con- ference has been reported for some regions (GREEN Model of Chiasma Interference 1137 1975). The lack of interference across the centromere The model would be supported by the discovery of is explained in our model by structures associated with a polymerizing substance associated with the synap- the centromere blocking the advance of the polymer. tonemalcomplex and recombination nodules. Indi- Our model does not provide a means of generating rect evidence for the existence of a polymer may lie negative interference; however, thenegative interfer- in the observation that apparent components of the ence observed may be due to a variation in recombi- core of the synaptonemal complex self-assemblein the nation frequencies (SALLand BENGTSSON1989). Ap- cytoplasm of spermatocytes and oocytes of Ascaris parent negative interference can also be due to gene mum, therebyexhibiting the ability to polymerize conversion of a middle gene, which under some cir- (BOGDANOV1977; FIIL,GOLDSTEIN and MOENS1977). cumstances might be mis-scored as coincident recip- The straightening of the synaptonemal complex be- rocal recombinationevents, straddling the middle tween zygotene and pachytene may also be evidence gene. for polymerization that could provide a force used in The computer simulation used 100 equivalent sites; untangling the chromosomes. however both genetic and molecular studies indicate If this model does represent actual cellular events, that not all DNA sites are equivalent in terms of gene there should also be a randomly distributed precursor conversion and reciprocal recombination frequencies. to the late recombination nodules. The prime candi- Forinstance, genetic and molecular studies of the datefor such aprecursor is, of course,the early ARC4 locus of yeast clearly show a polarity of gene recombination nodules, but whether early recombi- conversion indicatinga preferred site of initiation nation nodules become laterecombination nodules (FOGELet al. 1979; NICHOLASet al. 1989) and the data has yet to be proven. To answer these questions, it plotted in Figure 3 may illustratea tendency for may be necessary to isolate and characterize the com- crossovers to be more frequent in some regions of ponents of the synaptonemal complex and recombi- Drosophila chromosomes than in others, regardless of nation nodules. Analysis of meiotic mutants from a tetrad rank. This model does not address whether variety of organisms has providedimportant clues events are initiated at specific DNA sites, and is in fact regarding the regulationof recombination and its role flexible in terms of the number of sites used. Recom- in meiosis, and future studies will undoubtedly prove bination nodules may bind more readily to certain informative. sites or regions. For example there may be a nonuni- We would like to thank SEYMOREFOCEL for the unpublished form initial distribution of early recombination struc- data andJoHN GAME andDANIEL H. MALONEYfor critical readings tures. The probability of a recombination nodule ini- of the manuscript. This work was supported by a grant from the tiating a polymerization reaction may also be higher office of Health and Environmental Research of the U.S. Depart- at certain sites. ment of Energy under contract DE-AC03-76SF00098 and by U.S. Public Health Service grants GM30990, 5 P40 RR04231-02 and The model presented in this paper is simple and ES07075. biologically reasonable. It regulates thenumber of events per chromosome and thedistribution of events LITERATURE CITED along chromosomes, using a single mechanism. This BAKER,B. S., and A. T. C. CARPENTER, 1972 Geneticanalysis of model also explains how it may be possible for orga- sex chromosome meiotic mutants in Drosophilamelanogaster. nisms with significantly different chromosome lengths Genetics 71: 225-286. to use the same regulatory mechanisms with only BAKER,B. S., and J. C. HALL,1976 Meiotic mutants:genetic minor changes in properties, such as changes in rates control of meiotic recombination and chromosome segrega- of polymerization, or changes in the amount of time tion, pp. 351-434 in Genetics and Biology of Drosophila, Vol. lA, edited by M. A. ASHBURNERand E. NOVITSKI.Academic spent undergoing meiosis. Press, New York. Several possible additions to the model have been BOGDANOV,Y. F., 1977 Formation of cytoplasmic synaptonemal- considered, i.e., nonuniform initial distributions, vari- like polycomplexes at leptotene and normalsynaptonemal com- able initiation probabilities, and variable polymeriza- plexes at zygotene in Ascaris suum male meiosis. Chromosoma tion rates. These variations are considered to be com- 61: 1-21. BOJKO,M., 1989 Two kinds of “recombination nodules” in Neus- plications in simulating the genetic data of Drosophila pora crassa. Genome 32: 309-37 1. and Saccharomyces since even in its simplest form the BRIDGES,C. B., 1935 Constitution of thegerminal material in relation predictions of the model agree well with the genetic to heredity. Yearbook Carnegie Inst. 34 284-291. data examined. Additions to the simulation generate CARPENTER,A. T. C., 1975 Electron microscopy of meiosis in Drosophilamelanogaster females. 11. The recombination nod- a additional variables and without priori knowledge, ule-a recombination-associated structure at pachytene? Proc. it is difficult to assign biologically relevant values to Natl. Acad. Sci. USA 72: 3 186-3 189. these variables. The regulation observed in Drosoph- CARPENTER,A. T. 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