A Polymerization Model of Chiasma Interference and ... · A model of chiasma interference is proposed and simulated on a computer. The model uses random events and a polymerization

Copyright 0 1990 by the Genetics Society of America

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 on a 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 crossovers in 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 regulation of recombination observed in both genetic and cytological experiments. While the model was quantitatively compared to data from only Drosophila and Saccharomyces, the regulation observed in these species is qualitatively similar to the regulation of recombination observed in other organisms.

I N 1916 MULLER reported that a crossover in one region of a Drosophila chromosome changes the

probability of a crossover in an adjacent region (MULLER 19 16). Subsequent work in other eukaryotes revealed in general a distance-dependent reduction in the probability of a second crossover, known as positive chiasma interference. Interference is expressed in terms of coincidence, which is the ratio of observed to expected coincident crossovers in two linked regions. Interference has been observed at both the cytological and genetic levels. For example, coincidence as a function of distance has been measured genetically in Drosophila melanogaster (WEINSTEIN 19 18) and in Saccharomyces cerevisiae (MORTIMER and FOCEL 1974). In both organisms, interference is strong for closely linked regions and falls off with increasing separation. Cytologically, interference has been examined in terms of coincidence of chiasmata, the cross-shaped associations between homologous chromosomes seen in diakinesis and considered to be an outcome of genetic exchanges (for review SeeJONES 1984). For example, on the long arm of the L3 bivalent of Chorthzppus brunneus complete interference is observed over distances of 25-30% of the length of the bivalent arm (LAURIE 1980), and no interference is observed for regions longer than 60% of the length of the bivalent arm.

Genetic experiments have provided clues to the relationship between recombination and interference. In S. cerevisiae and in D. malanogaster, roughly one- half of all gene conversions are associated with reciprocal recombination (HURST, FOCEL and MORTIMER 1972; HILLIKER and CHOVNICK 198 1) . In Neurospora

(knrtics 126: 1127-1 138 (December, 1990)

and Saccharomyces gene conversions with a genetic crossover show chiasma interference but gene conversions without exchange of flanking markers do not interfere with other gene conversions, either with or without associated crossovers (STADLER 1959; MOR- TIMER and FOCEL 1974). Chromatid interference, a deviation from the 1 :2: 1 distribution of 2-, 3-, and 4- strand double crossovers, is not detected via tetrad analysis in Saccharomyces (MORTIMER and FOGEL 1974) or in crosses of Drosophila with attached X chromosomes (EMERSON and BEADLE 1933).

Electron microscopic studies of pachytene synaptonemal complexes resulted in the discovery of recombination nodules (SCHRANTZ 1970; GILLIES 1972) (for reviews, see VON WETTSTEIN, RASSMUSSEN and HOLM 1984; CARPENTER 1989). Two types of recombination nodules have been observed. Based on their temporal appearance in meiotic nuclei they have been termed early and late recombination nodules (CARPENTER 1989); however, it has not been established that late recombination nodules arise from early recombination nodules. The demonstration that late recombination nodules parallel chiasmata in frequency and distribution has led to the proposal that late recombination nodules are directly involved in meiotic recombination (CARPENTER 1975). The distributions of late recombination nodules, chiasmata, and crossovers are comparable in terms of number per chromosome and positions on chromosome arms. Late recombination nodules have been associated with sites of local- ized DNA synthesis (CARPENTER 1981), which is believed to occur during recombination. (For models of recombination see MESSELSON and RADDINC 1975;

1128 J. S. King and R. K. Mortimer

TABLE 1

Comparison of genome sizes and recombination levels for several organisms

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

Schizosaccharomyces pombe 14 3 1,895 12.6 n.d. 4.7 2.7 14 16 4,300 Saccharomyces cerevisiae 5.4 n.d. 0.88 6.14 1.56

Neurospora crassa 47 7 1,000 2.9 n.d. 6.71 0.43 8.29 Drosophila melanogaster 180 3 285 1.9 n.d. 60 0.032 15.33 Caenorhabditis elegans 100 6 300 1 .o n.d. 16.7 0.06 5.8 Bombyx mori 500 28 2,900 2.1 n.d. 17.8 0.12 9.21 Homo sapiens 3,000 23 3,500 3.0 n.d. 130 0.023 10.26 Mus musculus 3,000 20 1,630 1.6 n.d. 150 0.1 1 n.d. Zea mays 8,000 10 1,300 2.6 1.7 800 0.0033 32.5 Lilium long9orum 180,000 12 n.d. n.d. 2.4 15,000 0.0000 16 308.3

n.d.

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 and crossovers 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 ROTHSTEIN 198 1 .) For these reasons it is believed that late recombination nodules, chiasmata and genetic crossovers are all man- ifestations of the same event: reciprocal meiotic recombination.

Regulation of recombination events is apparent at several levels. The number of crossovers per chromosome arm was shown to be nonrandom (HALDANE 1931; WEINSTEIN 1936) since, compared to a Poisson distribution with the same average, the number of Drosophila X chromosomes with no crossovers was underrepresented and the number with one or two crossovers was overrepresented. The distributions of late nodules, chiasmata, and crossovers are nonuniform in terms of position along chromosomes and nonrandom in terms of distances between them. The positions of crossovers along the telocentric X chromosome of Drosophila tend to be centrally located in single crossover tetrads and tend to have one crossover near the centromere and the other crossover near the distal telomere in double crossover tetrads (CHARLES 1938).

Several models have been proposed to account for either interference or the distributions of chiasmata along chromosomes (for review see JONES 1984). Some of the models address interference but fail to account for the observed distributions of chiasmata between chromosomes. Others models only address the distributions of chiasmata and do not account for interference. A model of particular interest was proposed by EGEL in 1978. EGEL’S model is based on possibilities of exchanges that are established before synapsis and serve as initiation centers of synapsis. Synapsis is then followed by formation of synaptonemal complex, which prevents the establishment of further possibilities of exchange. This results in positive interference.

Stochastic models, based on recombination sites with individual crossover probabilities, are notewor-

thy because a growing body of evidence suggests that some DNA sites have much higher recombination rates than other sites. Stochastic models can also be based on a limited supply of a necessary component. Criticisms of stochastic models stem from their inabil- ity to account for interference without additional assumptions. Stochastic models based on a small number of sites are not supported by the fact that in Drosoph- ila some chromosomal sequences that are moved adopt the exchange distribution characteristic of the new location (BAKER and CARPENTER 1972). Further- more, in Saccharomyces the allelic recombination rate of a DNA sequence varies significantly, depending on its location in the genome (LICHTEN, BORTS and HA- BER 1987). It is also clear from fine-scale genetic analysis that there are many sites at which recombination may occur along a chromosome. However, in any one meiosis, recombination occurs at only a few of these sites.

Other models are based on pairing, in which certain chromosomal regions are assumed to pair first and are therefore more likely to undergo recombination. However, without additional assumptions this does not account for either interference or for the observed distribution of crossovers between chromosomes. Still other models are steric, i.e. the molecules that catalyze recombination events physically block adjacent events. It is difficult to account for the nonrandom distribution of events between chromosomes with these models, and based on electron micrographs, it is difficult to explain how a few nodules that are small relative to the length of the synaptonemal complex could have such a strong effect on each other. Another criticism is that the source of the steric interference would need to be modified significantly between organisms with different amounts of DNA and synaptonemal complex lengths. Table 1 is a compilation of genomic data from several organisms. These organisms have DNA contents that vary over four orders

Model of Chiasma Interference 1129

FIGURE I .-A speculative drawing of the model. h - l y structures (circular) bind randomly to the synaptonemal complex. Some initiate polymerization reactions thus becoming late nodules (oval). The growing polymers eject early structures. The ejected early structures are either degraded or bind to synaptonemal complex that is free of polymer.

of magnitude, but (with the exception of Schizosac- charomyces pombe) the number of crossovers per bivalent varies by only a factor of about five. It is apparent that a model of interference that is to apply to more than a few organisms must be adaptable such that it can act over a wide range of distances. Polymerization is a means with biological precedence which may be utilized to regulate recombination over such a range.

A polymer-based interference model: We propose a model to account for chiasma interference in which early structures randomly attach among and along the synaptonemal complexes of meiotic nuclei (see Figure 1). Once attached, each structure has an equal chance per unit time of initiating a bidirectional polymerization reaction. I t is speculated that the structures initiating a polymerization reaction would give rise to the late recombination nodules, seen in pachytene, and would initiate reciprocal exchange. I t is further speculated that the early structures are the early recombination nodules observed in zygotene, however

the model does not depend on these speculations. Growing polymers block the binding of additional early structures to the synaptonemal complex. As these polymers grow, bound structures that have not yet initiated such a reaction continue to have the opportunity to do so until they are ejected by the advance of a polymer initiated at a nodule located elsewhere on the 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 would be 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 number of 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 number of late nodules. The number of chromosome arms with zero crossovers may be further reduced by the relocation of ejected early structures onto chromosome arms that were initially void 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 RASMUSSEN and HOLM (1 978) for a redistribution from random recombination nodules associated with the synaptonemal complex at zygotene to the nonrandom nodules observed in pachytene.

MATERIALS AND METHODS

A Modula-2 program was written and run on an Apple Macintosh IIcx computer to simulate the model. The computer simulation generated “late nodule” bivalent arms from “early structure” bivalent arms. Early structures were placed at random on a bivalent arm with one hundred binding sites. Random numbers were generated using the Pascal version of the uniform deviate random number generator RAN1 from PRW et al. (1986). The number of early structures placed on the arm followed a Poisson distribution with an average of roughly twice the number of crossovers observed in either Drosophila or Saccharomyces. The positions of the early structures were determined by generating a random number between zero and ninety-nine for each structure and then designating the corresponding binding site as the location of a structure. Each “turn,” a random number between zero and one would be generated for each attached early structure. If this number was less than or equal to an assigned probability of initiating a polymerization reaction, that structure would be transformed into a late nodule and a polymer would begin to grow outward from the nodule.

The polymers grew at a constant rate of one site per turn in both directions and would stop if they encountered another polymer, a centromere, or an end of a chromosome


arm. Any early structures encountered by a polymer would then be removed and a new round would ensue, checking to see if any remaining early structures had initiated a polymerization reaction. In principle the model allows for the reattachment of displaced nodules; however, for sim- plicity this was not done in the computer simulations. When all the early structures had either been ejected or become late nodules, the cycle would stop and the results were recorded in terms of nodules remaining, distance between nodules, and distribution of nodules along the bivalent arm. Chiasma interference was analyzed in terms of coincidence as a function of distance.

For the simulation that was compared to the yeast data, each turn the growing polymers were given a probability of independently terminating growth in either direction. The values of the polymerization start and polymerization termination probabilities used in the simulations were determined by trial and error, acceptable values being those that reduced the initial distribution of early structures to a distribution of late nodules that matched the observed distribution of number of crossovers.

Comparison of our model to genetic observations of the number of crossovers and to their distribution along a chromosome arm requires crosses with markers spanning the length of a chromosome arm. Fortunately, relevant published data from Drosophila (BRIDGES 1935) and unpublished data from Saccharomyces (R. K. MORTIMER and S. FOCEL) were available. In Drosophila the tetrads are inferred from the genotypes of the diploid progeny. The number of crossovers along the X chromosome in Drosoph- ila were determined by WEINSTEIN (1936). Distances between crossovers and the distribution of crossovers in single- and double-event tetrads were analyzed by CHARLES (1938). The interference values from the computer simulations were compared with the data in STEVENS (1936). In the comparisons of the model and the experimental observations made in Drosophila, the distances between genetic markers are based on salivary map data compiled by CHARLES (1 938) and are thus physical distances.

For the comparison of the number of crossovers per chromosome arm in the computer simulations with Saccha- romyces, 780 tetrads with markers spanning the right arm of chromosome III were analyzed. The number of crossovers or recombinogenic events per tetrad is the tetrad rank. The distributions of crossovers in rank one and rank two tetrads were calculated. Coincidence values from the computer simulation were compared to data from several other Saccharomyces crosses, with a total of 8927 tetrads, all with at least five closely spaced markers on chromosome VIII. In the comparisons of the model and the experimental observations made in Saccharomyces, the distances between genetic markers are based on genetic data. The two sets of yeast data were necessary since the crosses that provided the chromosome IIZ data did not have markers close enough together to carry out interference analysis and the crosses used for interference analysis on chromosome VIII did not have markers spanning the length of a chromosome arm. Markers spanning the chromosome arm are necessary in order to record all the recombination events, and this is needed to determine the overall rank of the arm and the distribution of crossovers along the arm. Since there is typically no interference across centromeres, the two arms of a chromosome should be independent in terms of the distributions of crossovers along the arms. Because of this we compare the model to chromosome arms. Since the Drosophila X chromosome is telocentric this is essentially the entire chromosome.

RESULTS

Figure 2 is a graphic comparison of the observed number of crossovers along the X chromosome of Drosophila to the number predicted by our model. The probability of initiating a polymerization reaction was set at 0.009. The sample size was 150,000 tetrads. The number of bivalent arms with zero nodules in the computer simulation is initially 6.1 % based on a Pois- son distribution with an average of 2.8 nodules per bivalent. The model output and the observed distributions are quantitatively similar and both have an average of 1.4 crossovers per tetrad and differ significantly from the Poisson distribution with the same average. The same output was then compared to the Drosophila data in terms of distance between crossovers, coincidence, and distribution of crossovers in rank one and rank two tetrads.

Figure 3 is a comparison of the distributions of crossovers in rank one and rank two tetrads plotted by CHARLES (1938) and a similar distribution generated by the computer simulation. The distributions of crossovers for single- and double-event arms from the computer simulation have the characteristics observed in Drosophila. Single events tend to be located near the center of the chromosome. The computer-generated distribution of single events does not have the sharp peaks (at approximately 25% and 70% the length of the chromosome) and the drop at approximately 55% the length of the chromosome, observed in Drosophila. However, CHARLES (1938) concluded that due to errors primarily from estimating positions of markers it can not be concluded that the peaks observed represent a true bimodal distribution. He did conclude that in single crossover tetrads, the crossover is less likely to be near one of the ends than it is to be near the center of the chromosome. Double- event tetrads tend to have one event near the centromere and the other towards the telomere, and in this case the model output and the experimental data from Drosophila agree very well. Peaks at approximately 25% and 70% the length of the chromosome are also observed in rank two tetrads. This may indicate that while the overall distribution is predomi- nantly determined by rank there may be a tendency for events to occur in specific regions regardless of rank. The lack of events near the centromere in the experimental distribution of crossovers in two-event tetrads may be caused by exclusion of events near the centromere. This exclusion may account for the observed distribution of single crossovers near the centromere being significantly lower than the model output. T o generate distributions of crossovers with ex- cluded regions, a more refined simulation could begin with a nonuniform distribution of early structures.

In Figure 4 we compare the distance between nodules in rank two tetrads from Drosophila data (CHARLES 1938) to data from the computer simula-


FIGURE 2.-Number of events per tetrad. T h e computer simulation 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. T h e results o f the computer simulation are labeled "MODEL" and the data from Dro- sophila are labeled "OBSERVED" T h e three distributions graphed have an average value of 1.4 crossovers per tetrad.

tions. The average distance between crossovers in both the model output and in Drosophila is approximately 50% the length of the chromosome arm. This is significantly greater than the average of 33% expected for a random distribution. That the distance between crossovers is greater than it would be if the events were random restates that there is positive chiasma interference.

A more informative way of analyzing interference is to graph coincidence vs. distance since it illustrates how interference decreases with increasing separation of genetic intervals. This was done in Figure 5 for both the computer generated data and for data ob- tained along the X chromosome in Drosophila. Exper- imental coincidence values and uncertainties are from STEVENS (1936) and distances are from CHARLES (1 938). This coincidence 'us. distance graph illustrates that interference data from the model qualitatively follow experimental results in Drosophila, in that interference is strong over short intervals and falls off with increasing separation. The failure of the computer generated coincidence values to reach unity over the length of the chromosome arm may be caused by the oversimplification of polymer growth made in the computer simulation. A real polymer might not have a constant, length-independent growth rate. The growth rate in long polymers might be limited by breakage and a decreasing concentration of subunits. At small distances the model does not generate as much interference as observed. This may be due to our model recombinat.ion structure being only one potential recombination site in size. A structure that occupied several sites would generate absolute interference over short intervals.

Computer simulations were also used to compare the model to observations made in the yeast S. cermis- iae. A probability for a growing polymer to stop was added to the simulation to obtain coincidence values that would approach one over long distances. The simulations compared to Saccharomyces initially had, on average, three early structures, a polymerization start probability of 0.02, and a probability of polymerization termination of 0.0 1. Thirty-thousand model tetrads were analyzed. The experimental data for the comparison of crossovers per tetrad and the distribution of events along the right arm of chromosome IZI come from 780 tetrads from hybrid 4579 (S. FOGEL, unpublished data). The genotype of hybrid 4579 is

+ + leu2-27 MATa + ma12

his4 leu2-I + MATa thr4MAL2 The gene leu2 was used as a centromere marker (it is about 5 centimorgans to the left of the actual centromere), and recombination events between leu2-27 and MAL2 (MAL2 is within 10 kilobase-pairs of the right telomere) were recorded. Since leu2 is to the left of the centromere a fraction of the crossovers that were recorded as having occurred on the right arm presum- ably occurred on the left arm, but since the distance between MAT and the centromere (about 30 centimorgans) is six times longer than the distance between leu2 and the centromere this fraction is assumed to be small.

Table 2 is a comparison of the observed number of crossovers per tetrad arm (data from chromosome IZI of Saccharomyces), to the number predicted by a computer simulation. Since occasionally four-strand

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FIGURE 3.- Dis t r ibu t ions of crossovers in tetrads from a computer simulation and from Drosoph- ila. The experimental data is from CHARLES (1938) in which the Dro- sophila tetrads are inferred from the genotypes of the diploid progeny. Po- sition 0 corresponds to the centromere and position 100 corresponds to the distal telomere. Frequency is the percentage of tetrads with an exchange in the specified region per 1/ 100 the length of the chromosome arm. (A) Distribution of crossovers in rank one tetrads. Model output has been adjusted to account for differences in the percentage of single event tetrads reported in CHARLES

(1938) and the computer simulation. (B) Distribution of crossovers in rank two tetrads. The left-most and rightmost crossovers are plotted sepa- rately. Model output is adjusted to take into account differences in percentage of tetrads with two exchanges reported in CHARLES (1 938) compared to the computer simulation and the different number of intervals (8 vs. IO) .

double crossovers occurred between two adjacent markers, resulting in a non-parental ditype tetrad, it can be assumed that other double crossovers occurred that were observed as either parental or tetratype tetrads. The data are adjusted to take into account these “silent” double exchange events in the following way. A four-strand double exchange results in a non- parental ditype tetrad (NPD); assuming a two-strand double is as likely (no chromatid interference), there can be expected to be an equal number of two-strand doubles that were scored as parental type tetrads (P). Similarly, for every four-strand double we expect there to be two three-strand doubles that were observed as tetratype tetrads (T). These adjustments are then weighted in proportion to the number of NPDs

that occurred in otherwise rank zero, rank one or rank two tetrads. For example, there were 12 four- strand double crossovers in tetrads with no other events, so 12 of the rank zero tetrads and 24 of the rank one tetrads are scored as rank two tetrads. It is apparent from the data in Table 2 that a Poisson distribution does not approximate the experimental observations while the distribution of late nodules generated by the computer simulation of the model does approximate the observed tetrad ranks.

The observed distribution of crossovers in rank one tetrads from hybrid 4579 did not deviate significantly from a uniform distribution. However, the distribution of crossovers in rank two tetrads did deviate significantly from a uniform distribution and showed


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the characteristic of having one of the two events near the centromere and the other near the telomere, as was found to occur in Drosophila. The number of double crossovers in the regions genetically defined in hybrid 4579 are listed in Table 3. Region 1 is between leu2 and MAT (no crossovers occurred between the two leu2 alleles). Region 2 is between MAT and thr4 and region 3 is between thr4 and MAL2. The numbers of crossovers expected in the regions listed if there were no interference are in the column labeled "no interference." The numbers from computer simulations are in the column labeled "model." In Sac- charomyces and in the computer simulations, cross-

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FIGURE 4.-Distance between crossovers in two-event tetrads vs. frequency of occurrence. Distance is based on physical data and is in percentage of the chromosome arm length and frequency is the percentage of rank two tetrads with the corresponding distance between exchanges. Experimental data is from Drosophila, and analyzed by CHARLES (1938). In both the computer simulation and in Drosophila the distance between crossovers in double- event tetrads are approximately 50% the length of the chromosome arm, rather than 33% as would be the case for a random distribution.

FIGURE 5.-Coincidence vs. distance. Experimental data are from the X chromosome of Drosophila, calculated by STEVENS (1936). In both the model and in Drosophila, interference is strong over short distances and falls off with increasing distance. In the computer simulation coincidence does not reach a value of one even for lor'g distances because no limitations were ,,laced on polymer growth. In Drosophi'a coincidence is approximately one ior distances greater than 50% the length of the X chromosome.

overs in rank two tetrads tend to be spaced far apart. One crossover tends to occur near the centromere (region 1) and the other near the telomere (region 3), thus regions 1 and 3 have a high number of coincident crossovers in rank two tetrads.

Figure 6 shows the distribution of crossovers in model tetrads and tetrads from hybrid 4579 of rank one and rank two. The events in rank one tetrads from the computer simulation have the characteristic of being located near the center of the arm; however, it is a moderately flat distribution which is consistent with not detecting a nonuniform distribution of events in the 253 rank one tetrads examined from hybrid


TABLE 2

Crossovers per chromosome arm from Saccharomyces and from the model

Percentage of tetrads

Model Observed

No. of Poisson Early Late crossovers distribution structures nodules Saccharomyces

0 16.5 5.0 5.0 6.8 1 29.8 14.9 33.4 32.4 2 26.8 22.4 38.5 37.1 3 16.1 22.4 17.7 19.9 4 7.2 16.8 4.6 3.8

>4 3.6 18.5 0.8 0.0

Average 1.80 3.00 1.86 1.80

Number of crossovers per tetrad observed on the right arm of chromosome III of Saccharomyces compared to the number of early structures and late nodules from a computer simulation. The experimental data are from 780 tetrads of hybrid 4579. The simulation was based on 30,000 tetrads with an initial distribution of three early structures per tetrad, a polymerization start probability of 0.02, and polymerization termination probability of 0.01. Both the model distribution of late nodules and observed tetrad ranks in Saccharomyces differ significantly from a Poisson distribution. The predicted rank of tetrads from computer simulations based on the model approximate the ranks of tetrads from hybrid 4579.

TABLE 3

Positions of crossovers in rank two tetrads

Regions Observed No interference Model

1-2 60 40 54 1-3 133 59 124 2-3 58 44 65 1-1 8 53 16 2-2 8 30 6 3-3 24 65 26

The positions of crossovers along a chromosome arm in rank two tetrads in Saccharomyces are nonuniform and are similar to Drosophila in that one exchange tends toward the centromere and the other toward the telomere. The numbers in the column labeled “observed” are from chromosome 111 of hybrid 4579. The number of rank two tetrads expected with a crossover in each of the two regions listed i n the first column for a uniform distribution and without interference is listed in the column labeled “no interference.” The expected numbers were calculated using the overall recombination rates for each region in hybrid 4579. The numbers in the colunm labeled “model” correspond to the number generated in the computer simulations. For the computer simulations the regions were set to the proportions of the regions in hybrid 4579. In rank two tetrads one crossover tends to be in region one and the other tends to be in region three.

4579. Figure 6, parts B and C, are graphs of the distributions of crossovers from simulated rank two tetrads and the similar events recorded from hybrid 4579 for the left-most (Figure 6B) and right-most (Figure 6C) crossovers. The experimental distances are based on genetic data. The comparison of the distribution predicted by the model to the observed positions of the crossovers in Saccharomyces illustrates that the model positions crossovers according to tetrad rank in a pattern similar to that seen experimentally. Analysis of the data is limited by the small number of

defined genetic intervals in hybrid 4579. In Figure 7, coincidence vs. distance is graphed for

the model and for the six hybrids listed in Table 4. All six hybrids are genetically marked with at least five genes spaced close enough together along the right arm of chromosome VZZZ to calculate interference as a function of distance. Positions are in percentage of the total genetic length of the chromosome arm, beginning at the centromere with zero, and moving 40% toward the telomere. The length of the region from CEN8 to CUPl compared to the length of the chromosome arm was calculated from data in the recent genetic map of Saccharomyces (MORTIMER et al. 1989), and the distances between the genes within the region were calculated from the relative frequencies of crossovers in the crosses examined in this study. Coincidence values are calculated for the combinations of regions between cen8 and pet l , pe t l and arg4, arg4 and thrl , thrl and CUPl . Regions between arg4 alleles were not used since not enough coincidental events occurred in these intervals to ac- curately calculate interference. The computer simulation and experimental observation are basically in agreement, with the observed interference being slightly stronger overall.

DISCUSSION

Since our model proposes that the same cellular machinery is used to regulate meiotic recombination between and along chromosomes, it can explain the complex and diverse phenotypes of some meiotic mutants. For instance Drosophila mutants that relax chromosomal regulation in the form of interference also have a more random number of events between chromosomes (BAKER and HALL 1976). In our model, this phenotype could be caused by either defective polymer subunits, or by recombination nodules unable to initiate the polymerization reactions. Drosophila mutants, heterozygous for large inversions on chromosome Z I I or on the X chromosome, have reduced recombination rates on the chromosomes with the inversions and increased recombination rates on their normal chromosomes (SHULTZ and REDFIELD 1933). This phenomenon is known as the Shultz-Redfield effect and has been shown to occur in other organisms. In our model it can be explained as resulting from early structures being less likely to bind to the mutant chromosomal regions (which may pair more slowly or less frequently), resulting in more early structures available to bind to the normal chromosomes.

In Drosophila, humans, Neurospora, and several other organisms, structures in zygotene associated with the synaptonemal complex have been observed (CARPENTER 1979; HOLM and RASMUSSEN 1983; BOJKO 1989). Termed early recombination nodules in Drosophila, these structures outnumber late recombination nodules and are randomly distributed. Struc-


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FIGURE 6,”Distribution of crossovers in rank one and rank two tetrads from a computer simulation and from hybrid 4579. The computer simulation was based on 30,000 tetrads, a polymerization start probability of 0.02 and a polymerization termination probability of 0.01. (A) Crossovers in rank one tetrads. Crossovers in rank one tetrads tend to be located near the center of the chromosome arm, however the localiza- tion is not dramatic. The experimental distribution of single crossovers in hybrid 4579 does not significantly deviate from a uniform distribution. (B) The positions of the leftmost crossover in rank two tetrads. Crossovers tend to be near the centromere, which is near the left-hand end of the monitored region. (C) The positions of the rightmost crossover in rank two tetrads. Crossovers tend to be near the telomere.

0 20 4 0 6 0 80 100 POSITION

tures interpreted as intermediates between early and late nodules have been observed and support the theory that late nodules arise from a subset of the early nodules, but the intermediates observed may be partially assembled late nodules (BOJKO 1989). In this model these possible morphological changes may mark the initiation of a polymerization reaction. Proof that late nodules arise from early nodules does not

currently exist; however this model provides a means for the randomly distributed early nodules to give rise to the nonrandomly distributed late nodules using only random events.

Several investigators have made proposals for the possible function of early nodules. CARPENTER (1979) proposed that early nodules mediate nonreciprocal recombination and late nodules mediate reciprocal


. .-

0.8 -

0.6 -

0.4 -

0.2 -

0.0 : -' I I I I 0 10 20 30 4 0

DISTANCE

TABLE 4

Saccharomyces strains used to collect data for interference analysis

Hybrid Genotype tetrads No. of

X2961 + petl __

trpl + + arg4-16 thrl -

~~ ~

arg4-3 + + CUP1 2123

5571 + pet1 + thrl -

trpl + arg4-3 + CUPI 1074

5574 + p e t 1 + + thrl -

trpl + a7g4-3 arg4-36 + CUPl 1073

5577 + petl arg4-16 + thrl - ~

trpl + + arg4-36 + CUPI 908 ~~ ~~ ~

E 5 5 9 + Pet1 + arg4-16 thrl - trpl + arg4-3 + + CUPl 1140

5497 trpl + arg4 thrl -

+ petl + + CUP1 2609

Genotypes of strains used in interference analysis. Strains were heterozygous for additional markers not used in this analysis. [Hy- brids from S. FOCEL and R. MORTIMER. Data from hybrid X2961 are published in MORTIMER and FOCEL ( 1 974), the rest are unpublished].

recombination. RASMUSSEN and HOLM (1978) proposed that early nodules mediate both reciprocal and nonreciprocal recombination, but only those that are involved in reciprocal recombination remain associated with the synaptonemal complex and thus become late nodules. There are two observations that must be considered in order to fit our model with the genetic and cytological results. Early nodules and gene con-

FIGURE 7.-Coincidence us. distance for Saccharomyces and the model. Dis- tance is based on genetic data and is in 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). The computer simulation is based on 30,000 tetrads, a polymerization start probability of 0.02 and a polymerization termination probability of 0.01. The computer simulation of the model ap- proximates the experimental observations made in Saccharomyces.

version events are randomly distributed and do not exhibit interference. Late nodules and reciprocal exchange events are nonrandom and do exhibit interference. Based on our model of interference and the above observations, and given that the late nodules arise from the early nodules, there are two possible orders of events. Both involve the initial random placement of early recombination nodules capable of initiating gene conversion.

In the first, if an early nodule initiates a polymerization reaction it becomes anchored to the synaptonemal complex and the necessary steps for reciprocal recombination occur. In the second possible order of events, cleavage of the DNA resulting in a reciprocal exchange precedes and results in the initiation of a polymerization reaction and cleavage of the DNA not resulting in crossover does not lead to the initiation of a polymerization reaction. These proposals are similar to the proposal, based on the analysis of mutants defective in reciprocal exchange, made by CAR- PENTER (1982) that isomerization is under genetic control. In both cases all early nodules would be at sites of initiation of recombination and there would be reciprocal recombination at sites where an early nodule becomes a late nodule. Furthermore the approximately 50% crossover to gene conversion without crossover ratio would be a reflection of the redistribution of nodules, not a random resolution between equivalent forms.

Positive interference is not found in all regions of the Drosophila and Saccharomyces genomes. For example MULLER (1916) found no interference across the centromere of chromosome ZZ and negative interference has been reported for some regions (GREEN


1975). The lack of interference across the centromere is explained in our model by structures associated with the centromere blocking the advance of the polymer. Our model does not provide a means of generating negative interference; however, the negative interference observed may be due to a variation in recombination frequencies (SALL and BENGTSSON 1989). Ap- parent negative interference can also be due to gene conversion of a middle gene, which under some cir- cumstances might be mis-scored as coincident reciprocal recombination events, straddling the middle gene.

The computer simulation used 100 equivalent sites; however both genetic and molecular studies indicate that not all DNA sites are equivalent in terms of gene conversion and reciprocal recombination frequencies. For instance, genetic and molecular studies of the ARC4 locus of yeast clearly show a polarity of gene conversion indicating a preferred site of initiation (FOGEL et al. 1979; NICHOLAS et al. 1989) and the data plotted in Figure 3 may illustrate a tendency for crossovers to be more frequent in some regions of Drosophila chromosomes than in others, regardless of tetrad rank. This model does not address whether events are initiated at specific DNA sites, and is in fact flexible in terms of the number of sites used. Recom- bination nodules may bind more readily to certain sites or regions. For example there may be a nonuniform initial distribution of early recombination structures. The probability of a recombination nodule initiating a polymerization reaction may also be higher at certain sites.

The model presented in this paper is simple and biologically reasonable. It regulates the number of events per chromosome and the distribution of events along chromosomes, using a single mechanism. This model also explains how it may be possible for organisms with significantly different chromosome lengths to use the same regulatory mechanisms with only minor changes in properties, such as changes in rates of polymerization, or changes in the amount of time spent undergoing meiosis.

Several possible additions to the model have been considered, i . e . , nonuniform initial distributions, variable initiation probabilities, and variable polymerization rates. These variations are considered to be com- plications in simulating the genetic data of Drosophila and Saccharomyces since even in its simplest form the predictions of the model agree well with the genetic data examined. Additions to the simulation generate additional variables and without a priori knowledge, it is difficult to assign biologically relevant values to these variables. The regulation observed in Drosoph- ila and Saccharomyces is qualitatively similar to the regulation observed in other organisms. In organisms with less uniform distributions of events additions to the model may be needed.

The model would be supported by the discovery of a polymerizing substance associated with the synaptonemal complex and recombination nodules. Indi- rect evidence for the existence of a polymer may lie in the observation that apparent components of the core of the synaptonemal complex self-assemble in the cytoplasm of spermatocytes and oocytes of Ascaris mum, thereby exhibiting the ability to polymerize (BOGDANOV 1977; FIIL, GOLDSTEIN and MOENS 1977). The straightening of the synaptonemal complex between zygotene and pachytene may also be evidence for polymerization that could provide a force used in untangling the chromosomes.

If this model does represent actual cellular events, there should also be a randomly distributed precursor to the late recombination nodules. The prime candi- date for such a precursor is, of course, the early recombination nodules, but whether early recombination nodules become late recombination nodules has yet to be proven. To answer these questions, it may be necessary to isolate and characterize the components of the synaptonemal complex and recombination nodules. Analysis of meiotic mutants from a variety of organisms has provided important clues regarding the regulation of recombination and its role in meiosis, and future studies will undoubtedly prove informative.

We would like to thank SEYMORE FOCEL for the unpublished data andJoHN GAME and DANIEL H. MALONEY for critical readings of the manuscript. This work was supported by a grant from the office of Health and Environmental Research of the U.S. Depart- ment of Energy under contract DE-AC03-76SF00098 and by U.S. Public Health Service grants GM30990, 5 P40 RR04231-02 and ES07075.

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Documents

A Polymerization Model of Chiasma Interference and ... · A model of chiasma interference is proposed and simulated on a computer. The model uses random events and a polymerization