A polymerization model of chiasma interference and corresponding computer simulation.

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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 chromo-

somes. 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 posi-

tive chiasma interference. Interference is expressed

in terms of coincidence, which is the ratio of observed

to expected coincident crossovers in two linked re-

gions. Interference has been observed at both the

cytological and genetic levels. For example, coinci-

dence 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 interfer-

ence 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 recom-

bination 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 recombina-

tion nodules. T h e demonstration that late recombi-

nation nodules parallel chiasmata in frequency and

distribution has led to the proposal that late recombination nodules are directly involved in meiotic re-

combination (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 be-

lieved to occur during recombination. (For models of

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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. = n o 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. T h e 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. T h e distributions of late nodules, chiasmata, and crossovers are nonuniform in terms of position along chromosomes and nonrandom in terms of distances between them. T h e

positions of crossovers along the telocentric X chro-

mosome of Drosophila tend to be centrally located in single crossover tetrads and tend to have one cross-

over 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 distribu-

tion of events between chromosomes with these

models, and based on electron micrographs, it is dif-

ficult 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 orga-

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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 re-

ceived only a single structure would retain the struc-

ture since it could not be ejected, and would be

guaranteed to have a single late nodule and thus a single crossover. T h e 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. T h e 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 A N D 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 R A N 1 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.

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1130 J. S. King and R. K. Mortimer

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.

T h e probability of initiating a polymerization reaction

was set at 0.009. T h e sample size was 150,000 tetrads. T h e 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. T h e 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. T h e same output was then compared to the Drosophila data in terms of distance between cross-

overs, 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 gener-

ated 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. T h e 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 approxi-

mately 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 cross-

over 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 cen-

tromere and the other towards the telomere, and in this case the model output and the experimental data from Drosophila agree very well. Peaks at approxi-

mately 25% and 70% the length of the chromosome

are also observed in rank two tetrads. This may indi-

cate 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. T h e 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 ob-

served 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 nod-

ules in rank two tetrads from Drosophila data

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FIGURE 2.-Number o f 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 o f 0.009. T h e simulation is compared to Drosophila data

(WEINSTEIN 1936). and to a Poisson distribu- tion. 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 dis- tributions graphed have an average value o f 1.4 crossovers per tetrad.

tions. T h e average distance between crossovers in both the model output and in Drosophila is approxi-

mately 50% the length of the chromosome arm. This

is significantly greater than the average of 33% ex-

pected 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. T h e failure of the com-

puter 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. T h e

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 polym-

erization termination of 0.0 1. Thirty-thousand model

tetrads were analyzed. T h e experimental data for the

comparison of crossovers per tetrad and the distribu-

tion of events along the right arm of chromosome

IZI

come from 780 tetrads from hybrid 4579 (S. FOGEL,

unpublished data). T h e genotype of hybrid 4579 is

+

leu2-27 MATa

+

ma12

his4 leu2-I

+

MATa thr4MAL2

T h e gene leu2 was used as a centromere marker (it is

about 5 centimorgans to the left of the actual centrom-

ere), 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 centi-

morgans) 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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J. S. King and R. K. Mortimer

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POSITION

6 0 80

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1 0 0

1.6

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0.8

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0.2

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MODELLEFT

+ OBSERVEDLEFT

4 MODELRIGKT

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OBSEfWEDRIGHT

0 20 4 0 6 0 80 100

POSITION

FIGURE 3.- D i s t r i b u t i o n s 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

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Model of Chiasma Interference

1.2

1 .o

0.8

W

Y

P

8

W 0.6

0.4

0.2

0.0

0 2 0 6 0 4 0 80 100

DISTANCE

0 20 4 0 6 0

DISTANCE

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. T h e 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 be-

tween the two leu2 alleles). Region 2 is between

MAT

and thr4 and region 3 is between thr4 and

MAL2.

T h e numbers of crossovers expected in the regions listed

if there were no interference are in the column labeled

"no interference." T h e numbers from computer simulations are in the column labeled "model." In Sac- charomyces and in the computer simulations, cross-

80 100

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. T h e 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

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1134 J. S. King and R. K. Mortimer

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 o f 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 o f tetrads from hybrid 4579.

TABLE 3

Positions of crossovers in rank two tetrads

Regions Observed N o 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. T h e experimental distances are based on genetic data. T h e 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 interfer-

ence 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. T h e 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 p e t l , p e t l and arg4, arg4 and t h r l , t h r l and C U P l . Regions

between arg4 alleles were not used since not enough

coincidental events occurred in these intervals to ac- curately calculate interference. T h e 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 mu-

tants. For instance Drosophila mutants that relax chro-

mosomal regulation in the form of interference also

have a more random number of events between chro-

mosomes (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 chromo-

some 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;

(9)

Model of Chiasma Interference

A

0.13

1 *

Lu

3

Y

0.11

-

8 h

0.09

-

0.07

:

I I I I

0 20 4 0 6 0 80 1

POSITION

Q DOUBLES, LEFT

A OBSERVED, LEFT

0 20 4 0 6 0

POSITION 8 0 1 0 0

Q DOUBLES, RIGHT

A OBSERVED, RIGHT

1.

T

”

I I I I

FIGURE 6,”Distribution of crossovers in rank one and rank two tetrads from a computer simulation and from hybrid 4579. T h e 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. T h e experimental distribution of single crossovers in hybrid 4579 does not significantly deviate from a uniform distribution. (B) T h e 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) T h e positions of the rightmost crossover in rank two tetrads. Crossovers tend to be near the telomere.

0 20 4 0 6 0 8 0 1 0 0

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)

(10)

1136 J. S. King and R. K. Mortimer

.

.-

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 -

~

t r p l

+

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 unpub- lished].

recombination. RASMUSSEN and HOLM (1978) pro-

posed that early nodules mediate both reciprocal and nonreciprocal recombination, but only those that are

involved in reciprocal recombination remain associ-

ated 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 be- ginning 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 observa- tions 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 mu-

tants 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 ap-

proximately 50% crossover to gene conversion with-

out crossover ratio would be a reflection of the redis-

tribution of nodules, not a random resolution between

equivalent forms.

Positive interference is not found in all regions of the Drosophila and Saccharomyces genomes. For ex-

ample MULLER (1916) found no interference across

the centromere of chromosome ZZ and negative inter-

(11)

Model of Chiasma Interference 1137

1975). T h e 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 recombi-

nation 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 recip-

rocal recombination events, straddling the middle

gene.

T h e 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 nonuni-

form initial distribution of early recombination structures. T h e 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, vari-

able 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. T h e 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.

T h e model would be supported by the discovery of a polymerizing substance associated with the synap-

tonemal 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

m u m , thereby exhibiting the ability to polymerize (BOGDANOV 1977; FIIL, GOLDSTEIN and MOENS 1977). T h e 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. T h e prime candi-

date for such a precursor is, of course, the early

recombination nodules, but whether early recombi-

nation 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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