## Abstract

In most organisms that have been studied, crossovers formed during meiosis exhibit interference: nearby crossovers are rare. Here we provide an in-depth study of crossover interference in *Arabidopsis thaliana*, examining crossovers genome-wide in >1500 backcrosses for both male and female meiosis. This unique data set allows us to take a two-pathway modeling approach based on superposing a fraction *p* of noninterfering crossovers and a fraction (1 *− p*) of interfering crossovers generated using the gamma model characterized by its interference strength *nu*. Within this framework, we fit the two-pathway model to the data and compare crossover interference strength between chromosomes and then along chromosomes. We find that the interfering pathway has markedly higher interference strength *nu* in female than in male meiosis and also that male meiosis has a higher proportion *p* of noninterfering crossovers. Furthermore, we test for possible intrachromosomal variations of *nu* and *p*. Our conclusion is that there are clear differences between left and right arms as well as between central and peripheral regions. Finally, statistical tests unveil a genome-wide picture of small-scale heterogeneities, pointing to the existence of hot regions in the genome where crossovers form preferentially without interference.

SEXUALLY reproducing organisms undergo meiosis, thereby producing gametes having a level of ploidy equal to half that of the parental cells. This reduction in ploidy emerges from a complex and tightly controlled sequence of events. In most organisms, prophase I of meiosis begins by the active formation of double-strand breaks (DSBs) mediated by Spo11, a topoisomerase-like transesterase (Keeney *et al.* 1997). Then homologous chromosomes align and pair as the DSBs are repaired, typically using a homolog as template. Such DSB repairs can lead to either a crossover (CO), a reciprocal exchange of large chromosomal fragments between homologs), or to a noncrossover (NCO), a nonreciprocal exchange of small chromosomal segments between homologs, detected through associated gene conversions (Bishop and Zickler 2004)). COs mediate intrachromosomal rearrangement of parental alleles, giving rise to novel haplotypes in the gametes and thus driving genetic diversity. They also provide a physical connection between the homologs, holding them in a stable pair (bivalent) and allowing their correct segregation during anaphase I (Page and Hawley 2004; Jones and Franklin 2006).

Studies in several plants and animals have shown that the average number of COs in meiosis may vary between male and female meiosis (see review by Lenormand and Dutheil 2005), but there is no general rule that governs the direction or degree of CO number variation. Similarly, the distribution of COs along chromosomes can also differ when comparing male and female meiosis (Drouaud *et al.* 2007). Such distributions are generally nonuniform: some portions of the physical chromosome seem more likely to recombine while others hardly ever do (*e.g.*, close to the centromere: Jones 1984; Anderson and Stack 2002). Furthermore, COs do not arise as independent events—there is some “interference” between them. Generally, CO interference reduces the probability of occurrence of nearby COs (Sturtevant 1915; Muller 1916), leading to a lower variability in inter-CO distances than in the absence of interference. It has been suggested that CO interference plays a role in controlling the number of COs formed for each pair of homologs. Indeed, most organisms obey the obligate CO rule whereby each bivalent must have at least one CO to ensure proper segregation of the homologs; one way to ensure such an obligate CO is to have many DSBs and let them develop into COs. But at the same time having many COs may be deleterious, inducing genomic errors or breaking advantageous allelic associations. Interference might then be the signature of a mechanism allowing for the appearance of an obligate CO followed by suppression of CO formation in favor of NCOs. Such an interpretation is supported by the fact that most organisms produce far more DSBs than COs (Baudat and De Massy 2007), so that high interference strengths probably reflect selection pressures to control the number of COs. Interference is thus integral to a mechanistic understanding of meiosis, and it comes as no surprise that most organisms that have been studied do exhibit CO interference.

In the past decade, much progress has been made in identifying key genes and the associated pathways for CO formation. A first pathway (P1) is interfering and depends on proteins of the ZMM family such as MLH1 and MLH3. A second pathway (P2) seems to be noninterfering and depends on MUS81 with other associated proteins. The two pathways have been found to coexist in *Saccharomyces cerevisiae* (Hollingsworth and Brill 2004; Stahl *et al.* 2004), *Arabidopsis thaliana* (Higgins *et al.* 2004; Mercier *et al.* 2005), tomato (*Solanum lycopersicum*, Lhuissier *et al.* 2007), and mouse (*Mus musculus*, Guillon *et al.* 2005). These studies indicate that the proportion of P2 COs varies from species to species. For example, in tomato it hovers at ∼30% (Lhuissier *et al.* 2007) while in mouse it is ∼10% (estimated by putting together results from Broman *et al.* 2002; Froenicke *et al.* 2002; Falque *et al.* 2007). Outliers are *Caenorhabditis elegans* with only interfering COs and *Schizosaccharomyces pombe*, which shows only noninterfering COs. Variations in the proportion of P2 COs also seem to arise within a given species when comparing different chromosomes (Copenhaver *et al.* 2002; Falque *et al.* 2009).

Early ways to detect CO interference were based on the coefficient of coincidence (Ott 1999). More recently, various models of CO formation have been introduced; the fitting of these mathematical models to the experimental data allows one to (1) extract a quantitative estimate of interference strength and (2) dissect interference effects that are entangled in the mixture of interfering and noninterfering pathways, following present biological knowledge summarized in the previous paragraph. Multiple models have been proposed to incorporate interference in CO formation modeling. The most widely used ones take the COs to be generated by a stationary renewal process, which assumes that the genetic distances between successive COs are independently and identically distributed. This is the case of the gamma model (McPeek and Speed 1995), called so because the inter-CO distances follow a gamma distribution. When the shape parameter of this gamma distribution is restricted to be an integer, the model reduces to the chi square (Bailey 1961; Foss *et al.* 1993) or the counting model. All such models of interference have been designed to describe the formation of COs within the P1 pathway. In the case of modeling COs from two pathways (P1 being interfering and P2 not), one first simulates COs from P1 and then one uniformly “sprinkles” additional P2 COs without interference (Copenhaver *et al.* 2002). Using the gamma model for the P1 pathway accompanied by P2 sprinkling has led to estimates of *p* (the proportion of P2 COs) varying from 19 to 20% in *Arabidopsis* chromosomes 1, 3, and 5 (Copenhaver *et al.* 2002), 3 and 5% for *Arabidopsis* chromosomes 2 and 4 (Lam *et al.* 2005), ∼12% for the 10 maize chromosomes (Falque *et al.* 2009), 0–21% for humans depending on the chromosome (Housworth and Stahl 2003), and ∼10% for baker’s yeast chromosome 7 (Malkova *et al.* 2004). Such modeling studies have also provided estimates of interference strengths via the fitted value of the shape parameter of the gamma model distribution. Note that the availability of confidence intervals on these parameters is often an issue, and systematic testing of differences between chromosomes has not yet been attempted. Are the currently estimated differences in interference strength or the P2 proportion *p* significant? Would it be possible instead to assign an overall (chromosome and sex-independent) interference and/or P2 proportion value to each species? Finally, is there any evidence of variation in interference at the intrachromosomal level?

As a first answer to these questions, previous work on *Arabidopsis* male chromosome 4 (Drouaud *et al.* 2007) found that both the local recombination rates and the synaptonemal complex lengths were significantly different when comparing male and female meiosis. Furthermore, those authors concluded that interference strength varied along the length of chromosome 4 through tests using the classical coefficient of coincidence (Ott 1999). But these tests had two limitations. First, large intervals were used to avoid statistical noise, erasing any small-scale variations in interference. Second, different interval sizes were pooled together to gain statistical power, introducing biases. It is thus worthwhile to see whether the modeling approach (based on fitting parameters of CO formation models rather than on measuring coefficients of coincidence) gives results similar to those of Drouaud *et al.* (2007) while adding more insight. A first step in this direction was provided by Giraut *et al.* (2011) using the single pathway gamma model. Although these researchers found higher interference strengths in female than male meiosis for all five chromosomes of *Arabidopsis*, they provided no tests and thus made no claims of statistically significant differences. As we shall see, the single-pathway approach has serious shortcomings, making it essential to use two-pathway modeling, which allows one to resolve interference into properties of the two pathways P1 and P2.

In this work, we exploit the two large reciprocal backcross populations produced by Giraut *et al.* (2011) to study CO interference in *A. thaliana*. The gamma single and two-pathway models are used to fit the data of male and female meiosis for all five chromosomes. The variability in interference at several levels is then analyzed: between different chromosomes (separately for male and female meiosis), between male and female meiosis for each chromosome pair, and finally, between different regions of the same chromosome. Significant differences are found at all levels. We also obtain a genome-wide picture of candidate intervals that are anomalously hot for the proportion of the noninterfering pathway. Finally, the discrepancies unveiled in this work between the *Arabidopsis* data and the fits demonstrate the need for more sophisticated models than the ones available today.

## Materials and Methods

### Experimental data

#### Plant material:

The two *A. thaliana* accessions, Columbia-0 (Col) (186AV) and Landsberg *erecta* (Ler) (213AV), were obtained from the Centre de Ressources Biologiques at the Institut Jean Pierre Bourgin, Versailles, France (http://dbsgap.versailles.inra.fr/vnat/). The Col accession was crossed with Ler to obtain F_{1} hybrids. These hybrids were backcrossed with Col: their pollen was used for male meiosis, while Col pollen was used for female meiosis. Further details of the crossings are given in Giraut *et al.* (2011).

#### DNA extraction:

The plant material from the Colx(ColxLer) and (ColxLer)xCol populations was lyophilized (specifics in Giraut *et al.* 2011). Then DNA was extracted as explained in Giraut *et al.* (2011).

#### Selection of single-nucleotide polymorphism markers and genotyping:

For the two populations associated with male and female meiosis, a set of 384 SNP markers (Supporting Table S1 of Giraut *et al.* 2011) were chosen from the Monsanto and Salk Institute databases based on even physical spacing along the chromosomes (details in Giraut *et al.* 2011). Markers and plants with too many undetermined genotypes were removed from the final data set. The resulting populations consisted of 1505 and 1507 plants having genotype data from 380 and 386 markers for the male and the female populations, respectively (380 markers in common). Totals of 222 and 163 singletons were verified in the male and female populations, respectively, using PCR and DNA sequencing.

### Single-pathway interference modeling

#### Model:

We have worked within the standard hypothesis that the two CO formation pathways produce COs independently (Copenhaver *et al.* 2002; Argueso *et al.* 2004) and that P2 has no interference at all (Copenhaver *et al.* 2002). P2 participates only in the two-pathway modeling, which will be discussed below; here we need to consider only the first pathway that is interfering. To specify the P1 pathway framework, we used the gamma model (McPeek and Speed 1995) at the level of the bivalent (two homologs, four chromatids). To completely define the model, one has to provide the genetic length *L _{G}* of the chromosome considered and the interference parameter

*nu*, which can take any value in [1, ∞]; these two quantities are independent.

*L*is simply set to the experimentally observed genetic length. The parameter

_{G}*nu*quantifying the pathway’s interference strength corresponds to the “shape parameter” of the gamma distribution used in the process of generating the genetic positions of successive COs. In addition, 2*(

*nu*) is the “rate” of that gamma distribution on the bivalent, ensuring that the density of COs is two per Morgan as it should be by definition of genetic distances. Note that the backcross data lead to information on only one of the four gametes produced during each meiosis. Properties of CO patterns at the gamete level were deduced using the assumption of no “chromatid interference” (Zhao

*et al.*1995; Copenhaver 1998) (details in Supporting Information).

#### Estimation:

Given a value *nu* of interference strength, the likelihood for each backcross genotype was computed. Since each backcross is associated with a different meiosis, the likelihood *L* for the whole data set is the product of the likelihoods of each meiosis. Then we obtained the “best” value of the interference strength *nu* by maximizing *L* (this is the classical maximum-likelihood method), adjusting the model parameter *nu* using a “hill-climbing” procedure (Gauthier *et al.* 2011) (details in Supporting Information). Although the computation of likelihoods has been provided previously for whole chromosomes, in the present work we were also able to compute the likelihoods when chromosome portions under consideration did not form a continuous stretch (details given in Supporting Information). Such calculations allowed us to perform comparisons of interference strength between the central regions and the extremities of chromosomes. Not surprisingly, our whole-chromosome single-pathway estimates agree with those reported by Giraut *et al.* (2011) as these were based on the same data and same maximum-likelihood approach. Confidence intervals were computed using the Fisher information matrix.

### Two-pathway modeling via sprinkling

The P2 (noninterfering) COs were put down randomly with uniform density in genetic position (that is, along the genetic map) and then superimposed or “sprinkled” (Copenhaver *et al.* 2002) onto the P1 COs. On the bivalent, the density of P2 COs (defined as their mean number per Morgan) was 2 times the proportion (chromosome-wide) of the noninterfering pathway COs, *p*, where *p* lies in [0,1]. Similarly, the density of P1 COs was 2 times (*1 − p*), leading to a value of the rate parameter 2*(*nu)**(*1 − p*). This two-pathway gamma model is then specified by the genetic length *L _{G}*,

*nu*, and

*p*; these three quantities are independent. The first is set to its experimentally observed value while the other two are adjustable. The adjustment was obtained by maximizing the likelihood

*L*for a given chromosome as described above except that here

*L*had two parameters (

*nu*,

*p*) spanning a two-dimensional parameter space. Again, the hill-climbing algorithm (Gauthier

*et al.*2011) was used for maximization. After the adjustment, confidence intervals were obtained from the Fisher information matrix.

### Statistical procedures and comparison tests

#### Comparing two data sets (separate chromosomes or different regions of one chromosome):

We examined differences of interference strength at three levels. We compared the effective interference (using the single-pathway model) as well as the P1 interference and the proportion of non-interfering COs (using the two-pathway model) between (1) male and female meiosis for the same chromosomes, (2) between the different chromosomes but for a given sex, and (3) between different regions or the two arms of the same chromosomes. We tested the null hypothesis (H_{0}) that the two data sets being compared have equal means, using the Welch *t*-test rather than a *t*-test. Indeed, the classical two-sample *t*-test assumes the sample variances to be equal, which is not valid for the comparisons here. The Welch *t*-test generalizes the standard *t*-test to allow for unequal variances for the two samples. When the null hypothesis is rejected, it indicates that there is a statistically significant difference between the means of the two samples (details given in Supporting Information).

#### Detecting intervals hot for P2 COs:

We compared simulated and experimental data sets to detect hot intervals specific for P2. For each interval between adjacent markers, we first selected the plants having a CO in that interval. The frequency of COs in each of the other intervals was then computed, separating the cases of gametes with a total of two and three COs. The same was done for simulated data, generated with the *simdata* option in CODA (Gauthier *et al.* 2011) using the *nu* and *p* values obtained by fitting the experimental data. Expected (simulated or “theoretical”) and observed (experimental) distributions of COs for each intermarker interval were contrasted by Pearson’s chi-square test (Lindsey 1995) within the R statistical software, *chisq.test()*. This allowed us to test for each interval the null hypothesis that the two distributions (experimental and simulated) are similar. Furthermore, to better exploit the data, we merged the values from the two-CO and three-CO cases by taking the sum of the corresponding chi-square values (for intervals having data for two COs and three COs). The corresponding *P*-value was then computed by the R function, *pchisq()*. For intervals with data for one case only, the previous chi square and *P*-values were retained (details in Supporting Information). Since Pearson’s test is performed for all the intervals, we applied the Bonferroni correction at a family-wise error rate (FWER) of 5% for male and female meioses and each chromosome.

## Results

### Whole-chromosome analyses

#### Single-pathway analyses:

For each of the five chromosomes in *A. thaliana*, we estimated the values of the effective interference strength (given by the parameter *nu*) in male and female meiosis with the corresponding 95% confidence intervals, using the gamma model (see *Materials and Methods*). The fitted values of *nu* fall in a rather small range—from 2.4 to 4.1. Interestingly, female meiosis consistently exhibits higher values of *nu* than male meiosis (Figure S1). The highest female/male (F/M) interference ratios are seen for chromosomes 5 (1.3) and 4 (1.2) although these differences are not significant statistically (diagonal entries of Table S1). Comparing now different chromosomes for male meiosis, chromosome 4 has the largest *nu*, which is statistically different from the *nu* of chromosomes 1, 2, and 3 (top triangular part of Table S1). Furthermore, chromosome 5 has the second highest effective interference, and when compared to the two chromosomes with the lowest values (2 and 3), the differences in *nu* are statistically significant. Finally, for female meiosis, chromosome 4 has the highest effective interference while chromosome 3 has the lowest and the difference between them is significant (bottom triangular part of Table S1).

#### Two-pathway analyses:

For the gamma-sprinkling two-pathway model, we found values of *nu* between 8 and 37, with the range for male meiosis being 8–15, and for female meiosis, 13–37 (Figure 1). These values are systematically higher than the ones obtained in the single-pathway modeling. Such a trend is expected since the single-pathway modeling provides only an effective interference strength that mixes contributions from the two pathways; whenever *p* is appreciable, effective interference strength will necessarily be low. Considering now the estimates of *p*, the values found lay within the range 0.06–0.19, with 0.12–0.19 for male and 0.06–0.12 for female meiosis (Figure 2).

Comparing the male and female meioses, just as in the single-pathway analyses, we find that *nu* is consistently higher in female meiosis than in male meiosis for all chromosomes; in particular, the female-to-male ratios for *nu* are highest for chromosomes 2 and 4 (3.9 and 2.2, respectively). These differences are statistically significant for three chromosome pairs (diagonal entries of Table 1 associated with *nu*). Furthermore, the F/M ratios for *nu* are much higher than those obtained within the single-pathway analyses that do not dissect the interference signal into two pathways.

We obtain confidence intervals on *p* that do not contain the point *p* = 0. We therefore exclude at the 5% significance level the possibility of having only P1 COs: it indeed is necessary to use the two-pathway framework for all of the chromosomes for a sensible modeling. Furthermore, just as *nu* is larger for female meiosis than for male meiosis, we find that female meiosis has lower values of *p* than male meiosis; the highest male-to-female ratio (1.9) occurs for chromosome 3. This difference is significant for two among the five chromosome pairs (diagonal entries of Table 1 associated with *p*).

Compare now the different chromosomes for their level of P1 interference strength *nu* and proportion *p* of P2 COs for male and female meiosis separately. Beginning with male meiosis, chromosome 4 has the highest *nu* value (14.6) that is statistically different from that for chromosomes 1 and 5 (male-male comparisons, top triangular part of Table 1, entries associated with *nu*). Considering the values of *p* in male meiosis, chromosome 3 has a significantly larger proportion of P2 COs than chromosomes 1, 4, and 5 (top triangular part of Table 1, entries associated with *p*). For female meiosis, chromosomes 2 and 4 have higher values of *nu* as compared to chromosomes 1, 3, and 5, and many of the associated comparisons are statistically significant (female-female comparisons, bottom triangular part of Table 1, entries associated with *nu*). We also find that chromosomes 1 and 2 have greater values of *p* than the others, while chromosome 4 has the lowest; most of the statistically significant comparisons arise when including chromosome 4 (bottom triangular part of Table 1, entries associated with *p*).

### Intrachromosomal variation of interference

Uniformity of interference along chromosomes was tested via the difference in interference strength *nu* or parameter *p* (1) between the two arms of the chromosome (denoted “left” and “right” and separated by the centromere) or (2) between the central region (corresponding to half of the genetic length, taken between the fractions 0.25 and 0.75 of the whole chromosome) and the rest of the chromosome (extremities). These analyses were performed on individual chromosomes and when pooling the acrocentric chromosomes 2 and 4 on the one hand and the metacentric chromosomes 1, 3, and 5 on the other.

#### Single-pathway analyses:

A few of the comparisons suggest interference strength heterogeneities. For example, chromosome 4F shows a significant difference between the *nu* values of the left and right arms (the suffix M or F denotes male or female meiosis, respectively); the right arm that is longer shows a higher interference strength (first column of Table S2). But when merging data sets into two groups — metacentric chromosomes 1, 3, and 5 and acrocentric chromosomes 2 and 4 — no significant differences are found between left and right arms in either groups, be it for male or female meiosis.

When comparing the central region to the extremities, there is no overall trend for *nu*: five chromosomes show higher interference in the central region while the remaining exhibit the opposite behavior. In spite of that, the difference is significant for certain chromosomes (see the second column of Table S2). Here again, merged data sets for chromosomes 1, 3, 5 and for chromosomes 2 and 4 do not yield significant differences.

Being based on a single pathway, all these results should be considered in a qualitative spirit only since the two-pathway analyses exclude the possibility that *p* = 0.

#### Two-pathway analyses:

First consider the P1 interference strength parameter, *nu*. Among the comparisons between left and right arms, only chromosome 2M shows a significant difference, with a higher value for the right arm (third column of Table S2). Merged data sets for chromosomes 1, 3, 5 and for chromosomes 2, 4 give no significant differences. However, for comparisons between the central region and extremities, *nu* tends to be higher in the extremities for the majority of the chromosomes (fifth column of Table S2; see also Figure 3). For the metacentric merged data consisting of chromosomes 1M, 3M, and 5M, we find a significant difference between these regions, with higher *nu* for the extremities. The other merged data show no trend.

Second, for the parameter *p*, the difference between left and right arms is significant only for chromosomes 2M and 3M (fourth column of Table S2). Considering merged data sets for acrocentric chromosomes 2 and 4, the right long arm shows significantly higher *p* than the left short arm in male as well as female meiosis. For comparisons between the central region and extremities, *p* is observed to be higher in the extremities for most chromosomes with several significant differences (sixth column of Table S2). For the merged data sets, metacentric chromosomes (1, 3, and 5) exhibit (significantly) higher values for *p* in the extremities for both male and female meiosis.

The passage from single to two-pathway modeling leads to fewer statistically significant differences because there is an additional parameter to fit and thus loss of power. Nevertheless, the merged data analysis provides an unambiguous trend of intrachromosomal interference heterogeneity, namely higher interference as well as a higher proportion of noninterfering COs in the extremities.

### Hot intervals for the noninterfering (P2) pathway

Scatter plots of positions of CO pairs (see Figure S2 and Figure S3) reveal the presence of pairs close to the diagonal, indicating an anomalously low effective interference and thus presumably a high contribution of P2 in the corresponding regions. Furthermore, if an interval I* is hot for P2, that is, if the fraction of P2 COs arising in that interval is much higher than the value *p* inferred from the standard two-pathway modeling, then there will be an enrichment phenomenon whereby gametes with two or three (or more) COs will have a particularly high probability of having I* be recombinant. Such an enrichment leads to an excess of points on the horizontal or vertical line associated with that interval in the scatter plot of pairs of COs; this is indeed what is observed for a number of intervals (*cf*. Figure 4 and Figure S2 and Figure S3). Note that some of the events displayed correspond to CO position pairs that arise in several gametes; that is, there are points with multiplicities going up to 4 [Figure 4, made visible by introducing random noise in both axes using the R function *jitter()*].

To have an objective criterion for considering an interval to be hot for P2, we apply Pearson’s chi-square test, comparing the theoretical and observed distribution of genetic distances between successive COs. The Bonferroni correction is then applied to take into account that there are as many *P*-values calculated for each chromosome as there are intervals (*cf*. *Materials and Methods* and Supporting Information). These tests reveal highly significant *P*-values for several intervals along most of the chromosomes, showing that the current two-pathway modeling does not adequately describe all of the statistical features in the experimental CO patterns. From the *P*-values derived for each interval, we obtain a putative genome-wide profile of hot P2 intervals (Figure 5). We see that the intervals for which *P*-values are highly significant suggest the presence of hot regions for the noninterfering (P2) pathway. In addition to the heterogeneity within these P2 hot regions, the pattern varies between chromosomes and between male and female meiosis. Some chromosomes show several average and major peaks while others show only one high peak. The positions of the peaks also vary, sometimes occurring next to the centromere in particular for male meiosis, sometimes farther down each arm as seems to be typical in female meiosis (Figure 5).

The profiles of Figure 5 suggest hot regions for P2 COs, but our test would also generate small *P*-values if there were many gene conversion events (due to NCOs) affecting our data. Noting that these events would give rise to double recombinants in adjacent intervals, we have reanalyzed the data after removing all such cases. For this modified data set, 29 intervals lead to significant *P*-values (Figure S4). Thus we reject the hypothesis that current two-pathway modeling (where P2 COs are uniformly sprinkled along chromosomes) describes the statistical features of the experimental CO patterns. This result was also reached before removing double recombinants in adjacent intervals, so gene conversions on their own do not explain the heterogeneities we find in either Figure 5 or Figure S4.

## Discussion

### Female meiosis exhibits higher “effective” interference than male meiosis

Fitting data to the single-pathway gamma model provides a value for the associated interference parameter. The estimated values here are in agreement with those reported earlier (Giraut *et al.* 2011). We further find that the five chromosomes show higher effective interference in female meiosis than in male meiosis (Figure S1 and Table S1). However, we also find that the single-pathway model gives rise to poor adjustments to the experimental data. Such behavior is not surprising since, when using the two-pathway model, fitting leads to estimates of *p* (the P2 parameter) that are always incompatible with zero. To be on the safe side, single-pathway approaches should be considered to provide qualitative information only.

### Female meiosis exhibits higher P1 interference and lower P2 proportion than male meiosis

We find that two-pathway modeling points to higher P1 interference strength in female than in male meiosis (Figure 1). Furthermore, the values for *p* are significantly higher in male than in female meiosis (Figure 2). These systematic differences, arising in all chromosomes, suggest that the action of interference is affected by the cellular environment; *i.e.*, the male and female meiocytes provide environments where the interference strength and presumably the proportions for each pathway are modulated at a systemic level. Clearly, the cellular environment effects recombination rates, and thus its effecting interference strength does not come as a surprise.

In the light of these results for P1 and P2, we can look back at the results of the single-pathway analysis. Because P1 is more interfering in females and *p* is higher in males, one expects the effective interference inferred by the single-pathway modeling to be stronger in females. As shown above, this is indeed what the single pathway finds; in fact, it does so for all chromosomes.

In previous studies (Vizir and Korol 1990; Giraut *et al.* 2011), it was observed that the M/F overall recombination ratio in *A. thaliana* is ∼1.93. Could the extra genetic length of the male genetic maps be due to just an increase in COs from P2, keeping P1 unchanged both for the number of COs and their level of interference? The answer is “no”: we know that P1 COs see both their numbers increased and their interference level reduced when going from female to male meiosis because the male–female differences in *nu* are often statistically significant.

### Comparison to previous two-pathway studies

Genome-wide CO interference in *Arabidopsis* has been studied previously by other authors (Copenhaver *et al.* 2002; Lam *et al.* 2005) but only for male meiosis. Using the single-pathway gamma model, it was concluded that the effective interference parameter *nu* lies in the range 4–10 (Copenhaver *et al.* 2002) while in our analyses we have *nu* going from 2.4 to 3.5 (Figure S1); it is not possible to make a more quantitative comparison because those authors do not provide confidence intervals. They also performed two-pathway analyses, estimating *nu* to be between 10 and 21 and *p* between 0 and 0.2. Our values range from 8.8 to 14.6 for *nu* (Figure 1) and from 0.12 to 0.19 for *p* (Figure 2), so the results are qualitatively similar, but again a more detailed comparison cannot be given in the absence of confidence intervals. Also, our confidence intervals for *p* do not include zero, precluding the presence of only the interfering CO-formation pathway in agreement with Copenhaver *et al.* (2002). Another two-pathway analysis was performed (Lam *et al.* 2005) with a larger data set for chromosomes 2M and 4M, each of which bears a nucleolus-organizing region (NORs). Those authors find *p* to be 0.029 [confidence interval (0.003, 0.059)] for chromosome 2M and 0.054 [confidence interval (0.023, 0.097)] for chromosome 4M. These estimates are lower than what we find here, namely 0.14 [confidence interval (0.106, 0.176)] for 2M and 0.12 [confidence interval (0.087, 0.151)] for 4M (Figure 2). The difference might be attributed to several factors: (i) we have 71 SNP markers (they had 17) on chromosome 2 and 44 markers (they had 21) on chromosome 4; (ii) our data set contains >1500 gametes while theirs contains 143 tetrads (tetrad data bring roughly four times more power to the analysis as compared to the same number of gametes, so 1500 gametes here should be compared to the equivalent of ∼572 tetrads); and (iii) the plants were not subject to exactly the same growth conditions.

### Chromosome-specific effects

Analysis using the two-pathway framework leads to markedly higher P1 interference parameter values in female meiosis for chromosomes 2 and 4. It may be a coincidence, but these are the two short, acrocentric, NOR-bearing chromosomes. Within the NOR regions, one does not have exploitable markers, so no COs are detected there. Such missing data can very well lead to fitting biases, especially if the remaining COs are few as in the female meioses. This effect may thus explain why chromosomes 2 and 4 are *nu* outliers for female meiosis (Figure 1 and bottom triangular part of Table 1); note furthermore that chromosome 4 is a *nu* outlier for male meiosis also (top triangular part of Table 1), giving further credence to the hypothesis that chromosomal architecture and in particular NOR regions are responsible for high P1 interference parameters.

A similar analysis for the parameter *p* reveals that chromosome 4 has a markedly lower proportion of P2 COs than the other chromosomes in male meiosis (Figure 2 and top triangular part of Table 1). The same reasons as above are plausible causes.

### Heterogeneity of interference within chromosomes

In a previous study (Drouaud *et al.* 2007) of chromosome 4M of *A. thaliana*, it had been observed by analyzing coefficients of coincidence that the left side of that chromosome had higher effective interference than the right side. Even though our data set comes from a plant panel different from the one of Drouaud *et al.*, the two have similar interference characteristics (see Supporting Information and Figure S5 and Figure S6). In particular, using the coefficient of variation of inter-CO distances, which provides a qualitative measure of effective interference, we find very good agreement between the two data sets (see Figure S6) and that effective interference is stronger on the left side than on the right side of chromosome 4M.

The present work extends intrachromosomal interference comparison to all five chromosomes while also basing such comparisons on model fitting. We compare left *vs.* right arms and central region *vs.* extremities (Table S2). When doing so, we consider only a portion of each chromosome, and so the number of CO events is reduced, thus diminishing statistical power to a large extent. This difficulty explains why our comparison tests are often inconclusive. One striking result of the two-pathway modeling is the generally higher value of the interference strength parameter *nu* as well as *p* in the extremities, compared to the central region, for both male and female meiosis (see Figure 3). In addition to several significant comparisons yielded by analyzing one chromosome at a time, the merged data sets also provided conclusive results. The metacentric chromosomes 1, 3, and 5 together give significant differences between the central region and the extremities for *nu* and *p*. Both parameters are higher in the extremities. This may indicate that, while interference strength (or *nu*) increases toward the extremities, reducing the number of interfering COs, the fraction of noninterfering COs (or *p*) rises in compensation. Whereas toward the central region, the opposite behavior presents itself, with the proportion of noninterfering COs (or *p*) decreasing and interfering COs populating the region, which is more in keeping with the decreased interference (or *nu*). Perhaps this effect is governed by some architectural properties of the chromosomes. These could involve, for example, the level of compaction of the chromatin or mechanical stiffness that certainly plays a role in a number of other phenomena (Kleckner *et al.* 2004). Or the centromere itself could play a role, given that the merged data for the three metacentric chromosomes (1, 3, and 5) gives lower *nu* in the central part than in their extremities in both male and female meiosis.

The heterogeneities hereby demonstrated have never been considered in any interference model. Models to date consider interference to be constant along the chromosome with a single representative parameter (*nu* in the gamma model). Our results suggest that modifications of the gamma model should be considered, for example, by replacing the single value of *nu* for the entire chromosome by a vector of local *nu* values. Unfortunately, this increases greatly the number of parameters to be estimated so that a much larger data set would be required to perform reliable parameter estimation.

There were some instances when it was difficult to obtain estimates for the parameters *nu* and/or *p* when comparing subparts of the chromosomes, especially for chromosomes 2 and 4. In addition to being the smaller chromosomes, for female meiosis in particular, where interference is higher, the number of COs plummet rapidly in general and further when we look at smaller regions rather than the whole chromosome. Also, the left arm in general is very small, which leads the maximum-likelihood algorithm to allow for large values for interference.

### P2-associated hot regions within chromosomes

In light of the evidence for intrachromosomal variations of (i) interference strength and (ii) proportion *p* of noninterfering COs, we tested within our modeling framework for intervals that may be anomalously hot for P2. This possibility is not considered by any of the currently available interference models, but may be of strong biological relevance. Indeed, one already knows that double-strand breaks mature into crossovers or into noncrossovers in proportions depending on the locus (Mancera *et al.* 2008); such a propensity may extend to the choice of using one CO pathway rather than the other. Differences in the treatment of double-strand breaks may in fact tie in with the different mechanisms that are used for mis-match repair in the two pathways of CO formation (Getz *et al.* 2008).

Performing our tests for male and female meiosis in each interval, we found a number of very strong candidate intervals where P2 COs likely arise at significantly higher frequencies than expected (Figure 5 and Figure S4). This result suggests that not only does interference strength vary along chromosomes, but so does *p*, the relative contribution of P2 COs to recombination rates. Our genome-wide exploration revealed a heterogeneous pattern; on average some large-scale regions are likely to be hotter than others, but otherwise there do not seem to be any global trends. Clearly, current models, in particular the two-pathway gamma model in which P2 COs are sprinkled uniformly, are simply too crude. A new class of models has to be formulated to incorporate this knowledge. The molecular mechanisms specifying the relative proportions of P1 and P2 COs are only beginning to be unveiled (Crismani *et al.* 2012); one may also speculate that the chromosomal architectural properties can play an important role in determining these proportions. With high-quality data, some of these speculations could provide useful guidance for the modeling.

In summary, our use of crossover formation models to analyze meiotic recombination in *A. thaliana* has led to a genome-wide view of interference. Although some trends are obtainable from the coefficients of coincidence as developed for a single chromosome in (Drouaud *et al.* 2007), the use of the two-pathway modeling provides numerous new insights. For example, there are marked differences in the inferred model parameters when comparing male and female meiosis as well as when comparing different chromosomes. A number of trends emerge such as higher P1 interference strength in female meiosis and higher proportions of P2 COs in male meiosis. Furthermore, we find that the model parameters have clear intrachromosomal variations. For example, the interference strength as well as proportion of noninterfering COs is higher in the extremities compared to the central region for most chromosomes. And, when merging data sets, we find this trend to be significant for male and female meiosis for the metacentric chromosomes 1, 3, and 5. In fact, we reveal genome-wide intrachromosomal heterogeneities arising at scales going from centimorgan distances to the size of a whole chromosome. In particular, the large data set used in this study (taken from Giraut *et al.* 2011) allowed us to present the first genome-wide picture of candidate hot regions for the (noninterfering) P2 pathway. It remains to be seen whether this phenomenon is specific to *Arabidopsis* or more general. Finally, given these strong heterogeneities, it will be necessary to introduce more sophisticated models of crossover formation that allow for such behavior. Just as when going from single- to two-pathway modeling, these improvements will bring deeper biological insights, but because of the increase in their number of parameters, the use of such models will require still larger data sets.

## Acknowledgments

We thank Denise Zickler, Raphaël Mercier, and Mathilde Grelon for fruitful discussions of this work and Eric Jenczewski for a critical reading of the manuscript.

## Footnotes

*Communicating editor: A. Houben*

- Received July 21, 2013.
- Accepted September 2, 2013.

- Copyright © 2013 by the Genetics Society of America