A Two-Pathway Analysis of Meiotic Crossing Over and Gene Conversion in Saccharomyces cerevisiae

Franklin W Stahl; Henriette M Foss

doi:10.1534/genetics.110.121194

. 2010 Oct;186(2):515–536. doi: 10.1534/genetics.110.121194

A Two-Pathway Analysis of Meiotic Crossing Over and Gene Conversion in Saccharomyces cerevisiae

Franklin W Stahl ^1,¹, Henriette M Foss ¹

PMCID: PMC2954474 PMID: 20679514

Abstract

Several apparently paradoxical observations regarding meiotic crossing over and gene conversion are readily resolved in a framework that recognizes the existence of two recombination pathways that differ in mismatch repair, structures of intermediates, crossover interference, and the generation of noncrossovers. One manifestation of these differences is that simultaneous gene conversion on both sides of a recombination-initiating DNA double-strand break (“two-sidedness”) characterizes only one of the two pathways and is promoted by mismatch repair. Data from previous work are analyzed quantitatively within this framework, and a molecular model for meiotic double-strand break repair based on the concept of sliding D-loops is offered as an efficient scheme for visualizing the salient results from studies of crossing over and gene conversion, the molecular structures of recombination intermediates, and the biochemical competencies of the proteins involved.

EUKARYOTES transit from the diplophase to the haplophase via meiosis, which is associated with a number of interrelated processes, including crossing over and gene conversion. These processes involve meiosis-specific, programmed DNA double-strand breaks (DSBs) and their repair (DSBr). DSBr, in turn, is associated with mismatched base pairs and their rectification, referred to as “mismatch repair” or MMR (Bishop et al. 1987). Current efforts to accommodate both the genetic and molecular phenomena associated with meiotic DSBr in yeast (Saccharomyces cerevisiae) have been thoroughly reviewed (e.g., Hollingsworth and Brill 2004; Hoffmann and Borts 2004; Surtees et al. 2004; Hunter 2007; Berchowitz and Copenhaver 2010), but none of the reviews commits to an overall picture with quantitative predictions. This work aims to remedy that lack. Specifically, we have made use of salient published studies to develop, step-by-step, a comprehensive model of meiotic DSBr and MMR. The main features of this model are summarized in Table 1.

TABLE 1.

Proposed properties of two DSBr pathways

Features	Pairing pathway	Disjunction pathway
Products	Crossovers and noncrossovers	Crossovers only
Crossover Interference	No positive interference	Positive interference
Msh4–Msh5 dependence	None	Total
Bimolecular intermediate	Long with junctions not fully ligated	Short with fully ligated Holliday junctions
Invasion heteroduplex	Partly ephemeral	Ephemeral
MMR at invasion and annealing	Dependent on Msh2 and Mlh1	None
MMR near the DSB site	Directed by 3′ invading and annealing ends	Mlh1 dependent; directed by junction resolution
Role of Msh2 in MMR	Recognizes mismatches and attracts Mlh1	None
Role of Msh4–Msh5 in MMR	None	Attracts Mlh1

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RESULTS

For readers who are unfamiliar with yeast genetics and/or the known details of MMR, we begin by reviewing (1) the basic principles and vocabulary of tetrad analysis in yeast, which expose the products of individual acts of meiosis, (2) the DSBr model of Szostak et al. (1983) as modified by Sun et al. (1991), which has provided a basic molecular interpretation of meiotic recombination, and (3) the known roles of mismatch-repair proteins such as Msh2 and Mlh1.

Relative frequencies of tetrad types provide measures of linkage distance and crossover interference:

Consider a population of diploid yeast cells heterozygous for two linked sites, A/a and D/d. When meiosis proceeds without a hitch, the resulting tetrads each contain four viable haploid spores. Because the genotypes of the spores are identifiable by the phenotypes of the colonies they give rise to, each spore in the tetrad can be characterized as a crossover or a noncrossover with respect to sites A/a and D/d. When the A/a and D/d sites are closely linked, the most frequent tetrads contain only the two genotypes that characterized the parents; i.e., they have two AD and two ad spores and are therefore referred to as “parental ditype” tetrads (PD). Two other types of tetrads may be found in various frequencies: “tetratype” tetrads (T), i.e., those in which the spores are all different (AD, Ad, aD, ad), and “nonparental ditype” tetrads (NPD: Ad, Ad, aD, aD) representing the two recombinant genotypes. This type of tetrad analysis—assessing the relative frequencies of PDs, T's, and NPDs—allows a quantitative measure of crossing over (i.e., linkage distances and crossover interference).

Gene conversion as evidence of DSBr:

The same diploid may be marked at one or two other sites, B/b (and/or C/c), closely bracketing a “DSB hotspot,” i.e., a site that receives a high frequency (e.g., 20%) of programmed, meiosis-induced double-strand breaks. The B/b (and C/c) sites serve to identify tetrads that have undergone a DSBr event at the hotspot (Gilbertson and Stahl 1996). Such tetrads are recognized by their failure to exhibit Mendelian (i.e., 2:2) segregation of the markers among the spores. Instead, a “conversion tetrad” may contain three spores with a marker derived from one parent, and one spore with the allele from the other parent (e.g., 3 B:1 b, or 3 b:1 B). This non-2:2 distribution of genotypes is also termed “aberrant” or “non-Mendelian.” Furthermore, because the chromosome, a double-stranded DNA molecule, carries genetic information on each of its complementary strands, a haploid spore may give rise to a “mixed colony,” consisting of both B and b (or C and c) cells. Tetrads containing such a spore will be referred to as “half-conversions” (HCs) or 5:3. They can be diagrammed as BB, BB, Bb, bb when B is in excess over b, or as bb, bb, bB, BB when b is in excess. In this article, we deal with the two most common types of conversions: the HCs or 5:3's as described above, and the “full conversions” (FCs) or 3:1's (aka 6:2's), which may be diagrammed as BB, BB, BB, bb or bb, bb, bb, BB). In general, tetrads with an excess of B or of b are found at similar frequencies. A significant deviation from this expectation is referred to as “disparity.”

Successful application of tetrad analysis requires careful placement of the markers ABCD/abcd. Sites B/b and C/c must be located within a few hundred base pairs from a hotspot to efficiently register DSBr events as gene conversions. In contrast, markers A/a and D/d, designed to monitor crossing over and crossover interference, must be far enough from the hotspot to rarely suffer gene conversion, but close enough to ensure that most crossovers between A/a and D/d are products of the DSBr events monitored by B/b and/or C/c.

Because of limitations on the availability of markers, only one of the two sites closely bracketing a hotspot has typically been in a gene that determines a conveniently scored phenotype. Screening for conversions at that site has been used to select for analysis those tetrads that have undergone a DSBr event, involving nonsister chromatids, at the marked hotspot. Within that population, the markers at the second, less conveniently scored site, have then been determined by DNA analysis. The use of conversion at one marked site, guaranteeing that homologous DSBr has occurred at the hotspot, allows more meaningful scoring of the other site for 2:2 segregation (as well as for FC or HC) of the marker. By this procedure, tetrads are identified as “two-sided” (conversions for both B/b and C/c) and “one-sided” (conversions for only the more conveniently scored site).

The Szostak/Sun model for meiotic DSBr:

We can use the Szostak/Sun DSBr model (Figure 1) to illustrate how DSBr could generate a variety of the observed tetrads. For example, Figure 1A illustrates how mismatched base pairs created by the loss of genetic information from the chromatid undergoing DSBr, and its replacement with information from the unbroken homolog, can create a HC. Should a 3′-end be degraded past the site of a marker near the DSB, the repair process is called “gap repair” and results, invariably, in FC for the marker, as shown in Figure 1B. Such gap repair presumably occurs independently of MMR proteins. MMR, too, can generate FCs, but it operates only in the presence of known MMR proteins. MMR occurring at invasion or annealing will be directed by the invading or annealing termini and results only in FCs (Figure 1C). Should a mismatch escape MMR at invasion or annealing, it becomes subject to MMR directed by the termini created by Holliday junction resolution. Because, in this model, both the Watson and the Crick chains are cut to effect resolution, MMR can, with presumed equal probability, result in an FC or in restoration to 2:2 segregation (Figure 1C and 1D). Insofar as 2:2 segregation for a marker close to a DSB hotspot (in a tetrad with conversion for a close marker on the other side of the DSB site) can be attributed to restoration, MMR directed by junction resolution can be inferred. Note that Figure 1 does not show all four chromatids, but illustrates the fate of only the two chromatids directly involved in the DSBr process.

Many basic features of the Szostak/Sun model have survived more than two decades' worth of tests. These features include steps leading up to the formation of the joint-molecule, ligated DSBr intermediate (Figure 1A, step 4), and the existence of the intermediate is not in doubt (Schwacha and Kleckner 1995). The ligated intermediate, however, is now understood to give rise (by an, as-yet, unknown mechanism) to crossovers only (Allers and Lichten 2001a; reviewed in Bishop and Zickler 2004). In this work we present further departures from the Szostak/Sun model inspired (1) by the demonstration (Getz et al. 2008) that, in yeast, meiotic DSBr occurs via either a “pairing pathway,” first proposed by Zalevsky et al. (1999), or a “disjunction pathway” (see Stahl et al. 2004), and (2) by the extensive data of Hoffmann et al. (2005), which allow us to propose and quantify a set of DSBr-pathway-specific properties.

Since this analysis rests, to a large extent, on the known functions of MMR proteins in meiosis, and since much of our understanding of these proteins is based on studies of the bacterial anti-mutation proteins MutS and MutL, we summarize here the current understanding of MutS, MutL and their eukaryotic homologs.

Conventional skinny on MMR in meiosis:

The bacterial MutS and MutL proteins have a demonstrated role in correcting mismatches that arise during DNA replication (Modrich 1991). Several MutS and MutL homologs have been identified in S. cerevisiae and other eukaryotes (reviewed in Nakagawa et al. 1999; Culligan et al. 2000; Argueso et al. 2002). In both bacterial and eukaryotic MMR, rectification of the mismatch is directed by a discontinuity, on one strand or the other, in the DNA duplex. MMR sacrifices the mismatched marker that is on the strand with the nearby end. The functions of most eukaryotic homologs appear to be similar to those of their bacterial cousins. For example, MutS homologs recognize mismatches and attract MutL homologs to the region, and MutL homologs attract the remaining components of the mismatch-repair machinery. The expected manifestations of such MMR in meiosis are dependent on the DSBr structures in which it is presumed to operate as well as on the intrinsic repairability of the particular mismatch.

Among MutS homologs, the Msh4–Msh5 heterodimer (Burns et al. 1994; Hollingsworth et al. 1995; Pochart et al. 1997) stands out in several respects. First, the Msh4–Msh5 heterodimer alone is meiosis specific (Ross-MacDonald and Roeder 1994) and required for wild-type levels of crossing over and interference (Novak et al. 2001) as well as for the formation of an intermediate (Figure 1A, step 1) leading to (detectable) ligated joint-molecule DSBr intermediates (Figure 1A, step 4) (Börner et al. 2004). The meiosis specificity is consistent with the failure of Msh4–Msh5 to affect the mitotic mutation rate. However, the absence of Msh4–Msh5 also fails to manifest an increase in the frequency of HCs, the expected consequence of defective meiotic MMR (Ross-MacDonald and Roeder 1994; Hollingsworth et al. 1995; Wang et al. 1999). This, plus the observation that Msh4–Msh5 protein has no identified mismatch recognition sequence like that of MutS and its yeast homolog Msh2 (Culligan et al. 2000), has been taken as evidence that Msh4–Msh5 lacks the ability to effect MMR. (We revisit this view below.) The msh4–msh5-induced loss of crossing over, without a detected loss of either DSBr or MMR, suggests that the DSBs that were fated to have been interhomolog crossovers have, instead, been repaired using the sister chromatid as jig and template. Armed with this background, we first review the observations that force revision of the Szostak/Sun model of DSBr and, second, develop the alternative model (Table 1) for DSBr and MMR, along with arguments demonstrating the adequacy and economy of the model.

Two pathways for meiotic DSBr:

In an effort to reveal the relationship between DSBr and MMR, Getz et al. (2008) used tetrad analysis to monitor, at each of two hotspots, conversion of a palindrome marker (Nag et al. 1989). DSBr-induced mismatches involving such palindrome markers often fail to undergo MMR in the presence of normal MMR proteins (Figure 1A). We refer to such mismatches as “poorly repairable mismatches” or PRMs. In addition, Getz et al. (2008) monitored crossing over between markers bracketing the palindrome and the DSB site. They conducted these crosses in the presence or absence of Msh4, a protein required for wild-type levels of crossing over and interference as described above. Their data show that the frequencies of only those crossover tetrads with FC or with nonconversion for the palindrome marker (in many of which a PRM had presumably undergone restorational repair) were dependent on the presence of Msh4 protein. In contrast, the frequencies of both crossover and noncrossover tetrads with HC for the marker (i.e., those in which MMR of the PRM had failed) were, to a good approximation, Msh4 independent. Similarly, in wild-type yeast, the crossover tetrads with FC or nonconversion for the palindrome marker showed positive interference, while the crossover tetrads with HC for the palindrome lacked such interference.

Pathway-specific rules for MMR:

The observations by Getz et al. (2008) provide good evidence that wild-type yeast, unlike S. pombe (Cromie and Smith 2008), has two DSBr pathways with different MMR properties, both of which yield crossovers. Specifically, when DSBr creates a PRM via the Msh4-dependent, interference-generating disjunction pathway, the PRM undergoes efficient repair, yielding a crossover with either FC or nonconversion for the palindrome marker. In contrast, when the PRM is created by DSBr in the Msh4-independent, noninterference, “pairing pathway,” the PRM is refractory to repair, yielding either a HC crossover or a HC noncrossover. (Note the implication that if any FCs for the palindrome marker arise in the absence of Msh4, they must have resulted from gap repair.)

Msh2 deletion and PRMs are equivalent ways of eliminating Mmr from the pairing pathway:

The results of Getz et al. (2008) provide an incentive to revisit, in the framework of two pathways for DSBr, the powerful study by Hoffmann et al. (2005), which was also designed to explore relationships between DSBr and MMR. The methods used by Hoffmann et al. (2005) differ from those used by Getz et al. (2008). Specifically, Hoffmann et al. (2005) monitored DSBr as conversion for markers that make rather “well-repairable mismatches” (WRMs) in a wild-type background, and they created MMR deficiencies by deletion of the MutS homolog Msh2 or the MutL homolog Mlh1. Getz et al. (2008), instead, monitored DSBr events as conversions for a marker that makes PRMs near a DSB site. They noted that failed vs. successful MMR for the marker was associated with Msh4 independence or dependence, respectively, which enabled them to assign DSBr events to one or the other DSBr pathway. In particular, they noted that MMR was effective only in the disjunction pathway (Getz et al. 2008). Lest the reader protest that comparing Hoffmann et al. (2005) with Getz et al. (2008) is comparing apples and oranges, we argue, on the basis of the following observations, that failure to undergo repair of PRMs on the one hand and deletion of Msh2 on the other are equivalent manifestations of failed MMR in the pairing pathway, and that pathway only.

Stone and Petes (2006) demonstrated that, for a WRM near a DSB site, deletion of MSH2 causes a shift from FCs to HCs independently of Msh4. This implies that msh2-deletion mutants lack MMR in the Msh4-independent, pairing pathway, but have no conspicuous MMR in the Msh4-dependent, disjunction pathway. Getz et al. (2008) demonstrated that, in a MSH2 background, failure of the PRM to be repaired is also independent of Msh4. This implies that the observed HCs for the PRM, too, represent failure of MMR in the pairing pathway. Together, these observations suggest that a PRM involving a palindrome in a MSH2 background may be defined as a mismatch that is refractory to Msh2-dependent MMR. The work of Wang et al. (2003), showing that Msh2 interacts aberrantly with a palindrome mismatch, further supports our thesis that the methods of Hoffmann et al. (2005) and those of Getz et al. (2008) support each other.

Deletion of Mlh1 causes greater loss of MMR than does deletion of Msh2:

In their exploration of DSBr and MMR, Hoffmann et al. (2005) had analyzed tetrads of the ABCD/abcd-type illustrated above, with markers his (which makes rather well repairable mismatches) and BIK (which makes WRMs) representing the generic B/b and C/c sites bracketing the hotspot. Accordingly, the authors registered DSBr events as conversions for his and/or BIK. Scoring, initially, conversion for his, they showed that the elimination of either Mlh1 or Msh2 resulted in a shift from full- to half-conversion, as predicted by the Szostak/Sun model. However, the absence of Mlh1 caused a greater MMR deficiency, i.e., a greater increase in HC/(FC+HC), than did the absence of Msh2 (Appendix A). In a one-pathway model for DSBr, this might imply that Mlh1 simply affects MMR more efficiently than does Msh2. In a two-pathway context, on the other hand, the quantitative difference in phenotypes suggests a requirement for Mlh1 to affect MMR not only in the pairing pathway in conjunction with Msh2, but also, without Msh2, in the disjunction pathway. Further analysis of their data supports the hypothesis of Mlh1-dependent MMR in the disjunction pathway, as described below.

Evidence for disjunction-pathway-specific, Mlh1-dependent MMR:

Like msh4–msh5 mutants, but unlike msh2 mutants, mlh1 mutants show reduced meiotic crossing over (Hunter and Borts 1997) and reduced interference (Abdullah et al. 2004; Appendix B). Moreover, these mlh1 phenotypes are observed in a MSH4–MSH5 background only (Wang et al. 1999; Argueso et al. 2004). Thus, at least with respect to crossing over and interference, MLH1 functions in the disjunction pathway. The data of Hoffmann et al. (2005) indicate that the observed mlh1-induced gain in HCs mentioned above represents twice the mlh1-induced loss in FCs (Appendix A). Data in the same article suggest (1) that the disjunction-pathway-specific, Mlh1-dependent FC tetrads were crossovers, while the mlh1-induced HCs were noncrossovers and (2) that these noncrossovers represented twice the number of Mlh1-dependent crossovers (Appendix C). These observations imply that Mlh1 does, indeed, play a role in MMR in the disjunction pathway and that ∼50% of the time, Mlh1-dependent MMR in the disjunction pathway restores Mendelian (2:2) segregation of the marker.

In the disjunction pathway, Mlh1-dependent MMR occurs only in response to junction resolution:

The evidence that Mlh1-dependent, disjunction-pathway-specific MMR yields restoration and FC tetrads at equal frequencies implies that such MMR was directed by resolution of the Holliday junctions of the ligated DSBr intermediate (which is the molecular hallmark of the disjunction pathway) (Figure 1D). A corollary of this view is that mismatches created at the invasion and/or annealing phases of disjunction-pathway DSBr fail to undergo MMR prior to being incorporated in the ligated intermediate. Work by Allers and Lichten (2001b) supports this interpretation. These authors used gel electrophoresis of DNA from a MSH2 MLH1 strain to characterize DSBr intermediates with respect to a palindrome marker that makes PRMs near a DSB site. As expected, they found that the intermediate frequently contained the marker in mismatched, heteroduplex DNA, indicating a paucity of MMR prior to ligation of the intermediate. Moreover, in none of the intermediates were all four of the marked DNA strands derived from only one parent or the other, which would have been an indication of MMR.

The results of Allers and Lichten (2001b) indicate that PRMs in the disjunction pathway generally escape MMR at the invasion and annealing phases of DSBr. As discussed above, the data of Hoffmann et al. (2005), who used markers that make rather well-repairable mismatches, show that disjunction-pathway-specific MMR of WRMs, too, is directed by junction resolution. Thus, in wild-type yeast, the lack of MMR at the invasion and annealing phases appears to be a regular feature of disjunction-pathway DSBr. Yet, as demonstrated by Getz et al. (2008), mismatches induced by DSBr in the disjunction pathway are invariably repaired. Thus, MMR in this pathway occurs always, and only, in response to Holliday junction resolution.

msh2-induced lack of Mmr reduces two sidedness:

As shown above, lack of MMR in the disjunction pathway revealed DSBr events that would not have been detected in the presence of MMR. The data of Hoffmann et al. (2005) allow us to ask if the absence of Msh2-dependent (i.e., pairing-pathway specific) MMR would produce the same result. These authors screened for tetrads with conversion at his. Within this his-conversion population, they compared conversion frequencies for their second marker, BIK, in msh2 vs. MSH2 strains. Their results showed that deletion of Msh2 caused an increase in HCs among conversion tetrads, as expected. At the same time, however, two sidedness (the conversion frequency of BIK among his conversions) was decreased, rather than increased in response to loss of Msh2. Getz et al. (2008) reported equivalent results from crosses in a MMR-proficient background. To select for DSBr events at the hotspot, they used a “B/b” site (close to a DSB hotspot) that made WRMs. The crosses contained a “C/c”site (on the other side of the DSB hotspot) to assess two sidedness. When the C/c site made WRMs, most of the C/c conversions tetrads were FCs, as expected. When they replaced the WRM C/c site with a C/c site that made PRMs, most of the conversions at the C/c site were now HC, also as expected, but two sidedness was significantly reduced.

Migrating D-loops and transient heteroduplex:

How could loss of MMR cause these apparent reductions in conversion? One way is illustrated in Figure 2. The figure focuses on the heteroduplex created at the invasion stage of a DSBr event, and on the D-loop resulting from the invasion. While, in the Szostak/Sun model (Figure 1), extension of the 3′-invading end simply enlarges the D-loop, in Figure 2 extension of the invading end causes the lagging as well as the leading end of the D-loop to move toward the DSB site (Ferguson and Holloman 1996; Hoffmann and Borts 2005). As a result, a mismatch formed at invasion is undone as the invading strand is extruded to reunite with its original partner. If the mismatch undergoes MMR before it is “undone” by the migrating D-loop, full conversion for the marker will indicate that a mismatch had been created. If the mismatch fails to be repaired promptly, evidence that a mismatch had been created at invasion may be erased by migration of the D-loop.

A role for Msh4–Msh5 in disjunction-pathway MMR:

The concept of transient heteroduplex at invasion, together with “use-it-or-lose-it” conversion opportunities, satisfactorily accounts for MMR-dependent two sidedness in the pairing pathway. [Unwinding of the invasion heteroduplex (SDSA, Paques and Haber 1999) could provide a second “use-it-or-lose-it” route to MMR-dependent two sidedness. As such, one sidedness due to lack of MMR would be enriched among noncrossovers relative to crossovers (Merker et al. 2003; Getz et al. 2008)]. A migrating D-loop, causing transient heteroduplex, may well characterize disjunction pathway DSBr also. Indeed, data from Allers and Lichten (2001b) and Schwacha and Kleckner (1995) demonstrated that a conspicuous fraction of ligated (i.e., disjunction-pathway) DSBr intermediates had both Holliday junctions on the same side of the DSB site. If the disjunction pathway does, in fact, have transient invasion heteroduplex, mismatches created at invasion will be lost without a trace, because the disjunction pathway appears to routinely forego MMR at invasion and annealing, even in wild-type crosses.

What could prevent MMR in wild-type strains from acting at invasion and annealing in the disjunction pathway? By way of answer, we suggest that the lack of MMR prior to completion of the ligated intermediate in this Msh4–Msh5-dependent pathway is due to the absence of MutS function required for recognizing mismatches in duplex DNA. Msh2 and Msh4–Msh5, the only known candidates for this role, are both disqualified, although for different reasons—Msh2, which does recognize mismatches in duplex DNA, does not operate in the disjunction pathway (see Appendix A), while Msh4–Msh5, which, by definition, does operate in the disjunction pathway, fails to recognize mismatches in duplex DNA. It remains to be considered how, in the absence of mismatch recognition, PRMs, and WRMs too, in the disjunction pathway are nevertheless invariably repaired to yield either FC or nonconversion tetrads (Getz et al. 2008). Work by Snowden et al. (2004) suggests how the unique properties of Msh4–Msh5 might allow the dimer to promote MMR in a joint-molecule double-Holliday-junction DSBr intermediate.

Snowden et al. (2004), working with human MutS and MutL homologs, concluded that the behavior of Msh4–Msh5 protein at a Holliday junction is like that of a MutS protein at a mismatch in duplex DNA—Msh4–Msh5 binds to a Holliday junction and then slides away (Acharya et al. 2003). The high concentration of Msh4–Msh5 could then attract the MutL homolog. In the case of the double-Holliday-junction intermediate, reiteration of such behavior, with sliding in either direction, could lead to a traffic jam of Msh4–Msh5 in the region between the junctions, attracting Mlh1 to the entire region. Now, when nicks are introduced to resolve a junction, every mismatch between the junctions is rectified. Whether the DNA removal and replacement required for rectification is an inevitable consequence of junction cutting or is dependent on the mismatch is not answerable at this time.

The dog that didn't bark:

If, as we propose, there is such a thing as Msh4–Msh5/Mlh1-dependent MMR in the disjunction pathway, why then would msh4–msh5 deletion mutants not have a MMR-deficiency phenotype? This lack of msh4–msh5-induced increase in HCs is economically explained by the requirement of Msh4–Msh5 for the establishment of the disjunction pathway (Börner et al. 2004). Without the disjunction pathway and its products to register the presence or absence of MMR, there can be no msh4-induced increase in HCs to signal that MMR had failed. Hence, the apparent paradox—any MMR-deficiency phenotype of msh4–msh5 mutants should be detectable only in the presence of Msh4–Msh5 protein. Msh4–Msh5's vital contribution to the establishment of the disjunction pathway makes it impossible to challenge directly the proposal that, within the disjunction pathway, MMR is Msh4–Msh5 dependent. A critical test of Msh4–Msh5's involvement in MMR may require an (as-yet hypothetical) msh4–msh5 mutant that has retained the ability to form/stabilize the double-Holliday-junction DSBr intermediate, but has lost the ability to recruit Mlh1. Perhaps such a separation-of-function mutant will be found and will exhibit the typical meiotic MMR-deficiency phenotype, viz., an increase in HCs.

Where the rubber meets the road:

The DSBr and MMR data from the extensive study by Hoffmann et al. (2005) were reported in terms of HCs and FCs for his and BIK in wild-type, msh2 and mlh1 crosses. In addition, as noted above, these authors screened tetrads for two sidedness. These data can be used to test whether the model is capable of generating those observed values. Such a test requires that we first identify and evaluate (see Appendix A) the probabilities for each of the steps that can lead to a specified outcome. They are of two kinds—those whose values are preset by the model (nonadjustable parameters) and those that are specific to the HIS4 DSB hotspot and the markers in Hoffmann's strains (adjustable parameters).

Our model implies the following values for the four nonadjustable parameters:

The probability that his is on the annealing side of any DSBr intermediate = ½ (When his is not so situated, BIK is)
The probability that a mismatch on the annealing side of the DSBr intermediate in the disjunction pathway is repaired = 1
The probability that repair in the disjunction pathway leads to FC = ½
The probability that a mismatch on the invading side of DSBr in the disjunction pathway remains within the migratory D-loop = 0.

The adjustable parameters are:

g, the probability, specific for each marker, that a mismatch in the pairing pathway becomes FC by gap repair (Szostak et al. 1983) or FC-biased “short-patch repair” (Coïc et al. 2000)
R, the probability that heteroduplex rejection in the pairing pathway (Chambers et al. 1996; Goldfarb and Alani 2005) does not occur (see Appendix A)
m, the probability, specific for each marker, that mismatches in the pairing pathway are repaired (always to FC rather than 2:2)
E, the probability that, on the invasion side of a pairing pathway DSBr event, the mismatch remains covered by the traveling D-loop so that it appears as an HC in MMR-deficient crosses
P, the probability (expressed as number of DSBs per thousand tetrads) that a DSB is repaired by way of the pairing pathway
D, the probability (expressed as number of DSBs per thousand tetrads) that a DSB is repaired by way of the disjunction pathway.

This list of parameters intentionally excludes the possibility of restoration of 2:2 segregation by MMR acting on a mismatch that is close to an initiating DSB. In so doing we minimize the number of parameters.

Sudoku:

To estimate values for the adjustable parameters, we adopted the conventional strategy of starting with the parts of the puzzle that look easiest. For example, to obtain a value for D, we made use of the model's feature that DSBr products contributed by the pairing pathway should be strictly the same for mlh1 as for msh2 crosses. Hence, any differences between those two crosses should lead directly to estimates of D, the only adjustable parameter in the disjunction pathway. To determine the values of g_his, P, and E, we first estimated E on the basis of the frequency of tetrads, in the MMR mutants, that were simultaneously conversions for BIK and his (two-sided tetrads). We assume a single value for E, rather than assuming his- and BIK-specific values, on the grounds tha, if we can fit the data with a single value, we could surely fit them with separate values. The remaining two parameters were then chosen to give satisfactory fits to the FC and HC data for the MMR mutants (Appendix A).

To obtain a his-specific value for m and a value of R, we turned to the HC and FC frequencies in the wild-type cross. We held D, g_his, E, and P at the values deduced from the MMR-mutant crosses, and, for simplicity, assumed m to apply equally to mismatches created at invasion or annealing (Appendix A). The strategy for obtaining values for the only remaining parameters, m_bik and g_bik, is described in Appendix A.

With a value for each of the parameters, it was then possible to compare the expected values for FC, HC, and two sidedness with the observed values. The summary (Table 2) demonstrates that a single set of plausible parameter values satisfies both the HC and FC data, as well as the two sidedness, for each of the three genotypes, msh2, mlh1, and WT. It is gratifying that, with eight adjustable parameters, the model can account for 12 observations.

TABLE 2.

Summary of Sudoku

Parameter		Value^a
A. Values for adjustable parameters at his
D, disjunction pathway events per 1000 tetrads		154
P, pairing pathway events per 1000 tetrads		171
E, fraction of P retaining heteroduplex on invasion side		0.3
R, probability that heteroduplex rejection does not occur		0.643
g_his, probability of gap repair at his		0.127
g_BIK, probability of gap repair at BIK		0.04
m_his, probability of mismatch rectification to FC at his in pairing pathway		0.726
m_BIK, probability of mismatch rectification to FC at BIK in pairing pathway		0.96
Class	Obs.^b	Calc.^c
B. Conversion at his
mlh1 HC	171	174
mlh1 FC	21	21.8
msh2 HC	97	97
msh2 FC	61	60.3
WT HC	17	17.1
WT FC	120	122
Cross	Obs.^d	Calc.^e
C. Fraction of two-sided his conversions
mlh1	0.26–0.47	0.31
msh2	0.22–0.46	0.39
WT	0.58–0.77	0.71
Cross	Obs.^f	Calc.^e
D. FC/(FC+HC) for BIK among his conversions
mlh1	4/41	0.10
msh2	3/51	0.10
WT	89/92	0.97

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From Tables A2, A3, and A4.

From Table A1.

From Tables A2 and A3.

95% confidence interval on data from Table 5 of Hoffmann et al. (2005, Table 5).

From Table A4.

From Hoffmann et al. (2005, Table 5).

DISCUSSION

The data of Hoffmann et al. (2005) and Getz et al. (2008) call for an updated view of DSBr and MMR. Our interpretation of these data, above, has allowed us to assign specific attributes to each of two pathways for DSBr in wild-type yeast (Table 1), leading to a molecular model that illustrates how repair of the programmed, meiotic double-strand breaks might occur in each of the two pathways (Figures 3 and 4).

Figure 3.— — Proposed features of the disjunction pathway of DSBr in *MLH1* meiosis. (Step 1) The 3′-strand invades the homolog (blue) creating a D-loop (Hunter and Kleckner 2004). (Steps 2 and 3) Strand extension, shown as bricks of blue, often moves the trailing edge of the D-loop across the DSB site, undoing any mismatch that might have been created by invasion. D-loop migration is then blocked by annealing. The absence of MMR at invasion and annealing is characteristic of wild-type DSBr in the disjunction pathway. (Steps 4 and 5) Formation of the right-hand junction, promoted by Msh4–Msh5 binding, limits the length of the four-stranded structure, often to a region shorter than the resection. (Steps 6 and 6′) Junction resolution by cutting gives crossover products that are 4:4 on the invasion side and either FC or 4:4 on the annealing side, with the stretch of DNA that has lost information by MMR shown speckled. The arrows identify the junctions that are cut “vertically” and “horizontally.” “Vert. first” and “Hori. first” indicate whether the vertical or the horizontal cut was made first, respectively. When resolution is by 6′, the DNA synthesis is on different product duplexes, as reported by Terasawa *et al*. (2007). (In an *mlh1* mutant, the resulting products would be noncrossovers that are 4:4 on the invasion side and HC on the annealing side.)

Figure 4.— — Proposed features of the pairing pathway of DSBr in *msh2* and *mlh1* meioses. (A) Noncrossovers: (Step 1) A D-loop is initiated by strand invasion on one side of the DSB. (Steps 2, 3, 3′) Under the impetus of strand extension, the D-loop migrates, causing the region of heteroduplex arising from invasion to be reduced in extent (Step 3) or even eliminated (Step 3′) before annealing brings the D-loop migration to a stop. (Step 4) DNA synthesis proceeds leftward as far as the junction. (Steps 4′, 5) Withdrawal and reannealing (SDSA of Paques and Haber 1999) reconstitute the duplexes. (Steps 5, 5′) In the absence of gap repair, the tetrads are HC on both sides or 4:4 on the invasion side and HC on the annealing side. In the event of gap repair (or of MMR in wild-type cells), the invading side and/or the annealing side would be FC. (B) Crossovers: Through steps 3 and 3′, events are as for noncrossovers. (Steps 4 and 4′) Strand extension continues on the annealing side, which expands the D-loop. (Steps 5 and 5′) Strand extension is followed by trimming as necessary. (Steps 6, 6′; 7, 7′) Following a proposal by Cromie and Smith (2008), the intermediates are drawn without a pair of Holliday junctions and are presumed to be resolved by Mus81–Mms4 as pictured in of Hollingsworth and Brill (2004, Figure 3, A and B). (Steps 7 and 7′) In the absence of gapping, the resulting crossovers, like the noncrossovers, are HC on both sides or 4:4 on the invasion side and HC on the annealing side of the DSB. In the event of gap repair (or of MMr in wild-type cells), the invading side and/or the annealing side would be FC. The role of extended synthesis in the production of crossovers derives from Maloisel *et al*. (2004), who proposed “ …a model in which DNA synthesis determines the length of strand exchange intermediates and influences their resolution toward crossing over.” Unbridled creation and extension of structures 4 and 4′, accompanied by branch migration that creates invasive 3′-ends could lead to the accumulation of multimolecular structures in meioses lacking the endonuclease Mus81–Mms4 and the helicase Sgs1 (Oh *et al*. 2008; Jessop and Lichten 2008).

The model invites us to revisit several meiotic phenomena previously published and interpreted. These include the wide variation in the estimated lengths of regions of heteroduplex resulting from DSBr, MMR-dependent two sidedness, the apparent mutual exclusivity of interference and gene conversion in Sordaria fimicola, and an unexpected DSBr intermediate. Below, we review these phenomena and offer interpretations within the framework of our two-pathway model with traveling D-loops.

Heteroduplex lengths:

Several studies of DSBr in yeast have yielded data that create impressions of the length of the regions of heteroduplex created at the invasion and/or the annealing stages of DSBr. Some estimates were based on physical analyses (e.g., microscopy or gel electrophoresis), others on genetic analyses. The problem is that the impressions appear to contradict each other. For example, genetic analyses suggest that conversion tracts (which depend on regions of heteroduplex) may often be “long” (Detloff et al. 1992; Foss et al. 1999). Microscopy by Bell and Byers (1983), on the other hand, indicates that double-Holliday-junction intermediates tend to be “short.” This estimate by microscopy is consistent with gel electrophoresis data reported by Schwacha and Kleckner (1995), which imply that the region of DNA between two Holliday junctions in observed intermediates is usually short. Our model suggests that these apparent discrepancies reflect the differences between the pairing and disjunction pathways. Both the gel electrophoresis and the microscopy focus on double-Holliday-junction intermediates, i.e. on disjunction-pathway intermediates, which, in our model, are short (Figure 3, and see below). In contrast, the genetic studies indicating long regions of heteroduplex (Figure 4; Detloff et al. 1992; Foss et al. 1999; Hillers and Stahl 1999) were based on HC tetrads, which are manifestly products of the pairing pathway (Getz et al. 2008; and see above). This interpretation is also consistent with the msh4-induced increase in the average length of conversion tracts, as deduced from genetic data of Mancera et al. (2008)—according to our model, the absence of Msh4 would eliminate the disjunction pathway with its short conversion tracts, thereby increasing the average conversion tract length.

It should be noted that electrophoresis studies have yielded little evidence on the structure of pairing-pathway intermediates. While this could reflect an ephemeral nature, we suggest that it reflects (instead or also) the variable and “unfinished” nature of these intermediates (Figure 4) and/or a length that frequently exceeds the distance between the restriction sites used by the investigator to liberate the intermediates from the chromosome. Furthermore, frequent erosion of the 3′-single-strand ends at the DSB in the pairing pathway (g > 0) could confound the detection of DSBs specific to that pathway.

MMR-dependent two sidedness:

Hoffmann et al. (2005) explained the phenomenon of MMR-dependent two sidedness with the suggestion “... that in wild-type cells the initial DSB repair event is two sided. The absence of MMR, by either mutation or use of poorly repaired palindromes, allows a second, unbiased mispair removal pathway to restore a proportion of heteroduplexes, leading to apparent one-sided events.” This proposal suffers from several problems, one of which is that the short-patch system hypothesized by Hoffmann et al. (2005) to be responsible for this unbiased MMR is claimed by its discoverers in yeast (Coïc et al. 2000) to be biased against the marker on the invading strand, thus favoring FCs over restorations (and see Appendix A, Disparity between the two classes of HCs). Our model suggests, instead, that a traveling D-loop in the pairing pathway allows a mismatch created at invasion only a transient opportunity to enjoy MMR (Figure 4). Thus, the one sidedness (fraction of conversions for one of the bracketing markers that are 2:2 for the other) reflects the failure of MMR to turn such a mismatch into an FC before the heteroduplex containing the mismatch is undone.

Gene conversion in Sordaria:

Our model for DSBr obliges us to revisit the observation by Kitani (1978) that, unlike yeast crossovers, Sordaria crossovers that exhibit gene conversion have no crossover interference, and vice versa. Stahl and Foss (2008) had suggested that, in both organisms, MMR in the disjunction pathway is directed by junction resolution. In yeast such MMR would result in interfering crossovers with FC or 2:2 segregation for the marker with equal probability while, in Sordaria, junction-directed MMR would yield interfering crossovers with only 2:2 segregation. This proposal rested on the assumption that the disjunction pathways for the two organisms differ from each other in the same manner as do the pairing pathways; viz. yeast has predominantly asymmetric heteroduplexes, revealed by 5:3 segregation, while Sordaria has a high frequency of symmetric heteroduplex, revealed as aberrant 4:4 segregation (Kitani and Olive 1967; Stahl and Hillers 2000; Stahl and Foss 2008). We now suggest a second possibility as to why interference and conversion appear mutually exclusive in Sordaria. This possibility is based on our understanding that Kitani's monitored markers were not selected for proximity to a DSB hotspot. If, as for yeast, Sordaria's regions of heteroduplex in disjunction-pathway intermediates are relatively short, Kitani's markers might only rarely generate disjunction-pathway mismatches, in which case the interfering crossovers would usually lack conversion regardless of whether the intermediate had symmetric or asymmetric heteroduplex. In contrast, crossovers in the pairing pathway, with its longer conversion tracts, would be relatively more likely to involve a marker far from the DSB site in heteroduplex.

This explanation (for the frequent lack of detectable DSBr participation in the disjunction pathway for marked sites relatively remote from the DSB site) may apply also to the apparent lack of BIK's participation in disjunction pathway DSBr (Appendix A, The BIK data).

An unexpected DSBr intermediate:

Allers and Lichten (2001b) reported a DSBr intermediate that they labeled JM2. JM2 had two Holliday junctions, identifying it as a disjunction-pathway intermediate. The junctions, however, were on the same side of the DSB site, beyond a palindrome marker located near the DSB site (on the right in Figure 5). Moreover, the palindrome marker was in parental configuration, which the authors had some difficulty explaining. Within the framework of our model featuring a traveling D-loop and junction-directed repair in the disjunction pathway, we offer the scheme illustrated in Figure 5. A most attractive feature of the scheme is that, in harmony with the observations of Allers and Lichten (2001a), it yields restoration intermediates but never FC ones. We note that if JM2s had been frequent in the data of Hoffmann et al. (2005), our model would demand that the excess of HCs seen in the mlh1 cross would be more than twice the excess of FCs seen in the MLH1 msh2 cross. The data do not lean that way.

Figure 5.— — The JM2 DSBr intermediate of Allers and Lichten (2001a,b). Steps 1 and 2 are as in Figure 3. (Steps 3 and 4) If the traveling D-loop overshoots the marker to the right of the DSB site before being halted by annealing, the marker is in heteroduplex composed of an old red strand and a new blue strand. (Steps 5 and 6) Prior to the completion of the Holliday junction on the right, the unligated junction, posing as a nicked junction, directs Mlh1-dependent MMR of the mismatch, restoring the parental state (speckled) of this disjunction pathway intermediate. Schwacha and Kleckner (1995) reported a high frequency of JM2s (their type IIA). Prediction: double-Holliday-junction intermediates isolated from a strain lacking Mlh1 would lack JM2s.

Predictions:

Our view of the roles of the various Mut homologs in meiotic MMR makes a number of predictions:

Crosses carried out with a poorly repairable mismatch close to a DSB site in a MSH2 background should have the same HC/(FC + HC) ratios as do crosses in a msh2 background with a marker, at the same site, that makes well-repairable mismatches.
The proposal that deletion of MSH4 in yeast eliminates a crossover pathway that yields only nonconversions and one-sided FCs implies that msh4 mutants should show a modest increase not only in the HC/(FC + HC) ratio for a palindrome marker at HIS4 (by decreasing FC crossovers as shown by Getz et al. 2008), but also in the fraction of DSBr events that are two sided (by decreasing the one-sided tetrads).
Getz et al. (2008) noted that, in msh4 mutants, the loss of conversion crossovers appeared to result conspicuously in a gain in nonconversion noncrossovers (presumably by sister repair). The studies of Hoffmann et al. (2005) suggest that, in contrast, deletion of MLH1 appears to turn FC interfering crossovers into HC noncrossovers. The possibility that these HC noncrossovers would interfere with each other is a heady one, but difficult to test.
Another test of the model concerns the effect of Msh4 on HC/(FC + HC) ratios in msh2 and mlh1 mutants. Although these two single mutants differ appreciably with respect to HC/(FC + HC) ratios at his, when the disjunction pathway is removed by deletion of MSH4 the msh2 msh4 and mlh1 msh4 double mutants should be seen to have identical values for that ratio.
According to our model, only the pairing pathway produces two-sided tetrads, implying that two-sided crossovers should not manifest positive interference. Expected and observed phenotypes for a variety of relevant genotypes are summarized in Table 3.

A prudent investigator aiming to challenge these predictions would probably choose to work at HIS4 and with the strains of Hoffmann et al. (2005).

TABLE 3.

Mutant phenotypes of MutS and MutL homologs

A. Effects on crossing over and interference
Genotype	Meiotic phenotype		Explanation
msh4	Reduced crossing over, interference eliminated^a		Disjunction pathway eliminated^b
msh2	None		NA
mlh1	Reduced crossing over and interference^c		Turns disjunction pathway COs into NCOs
msh2 msh4	Like msh4^d		Msh2 does not affect crossing over or interference.
mlh1 msh4	Like msh4^e		Mlh1 does not affect crossing over or interference in the absence of the disjunction pathway
B. Effects on MMR for well repairable mismatch near a DSB site
	Conventional wisdom		Model
Genotype	Phenotype	Explanation	Predicted/explained phenotype	Explanation
msh4	None obvious^b ^d ^f ^g	Msh4 not a MMR protein	Increased HC/(HC+FC)	Loss of disjunction pathway, in which all conversions are FCs
msh2	Increased HC/(HC+FC)^h	Reduces meiotic MMR	Increased HC/(HC+FC) for NCOs and noninterfering COs	Eliminates MMR in the pairing pathway only, resulting in both more HCs and fewer conversions
mlh1	Increase in HC/(HC+FC) greater than for msh2^h	Removal of Mlh1 reduces MMR more efficiently than does removal of Msh2	Increase in HC/(HC+FC) greater than for msh2^h	Eliminates MMR in both pathways
msh2 msh4	Like msh2^d	Msh4 not a MMR protein	Sum of msh2 and msh4 phenotypes	The mutants affect distinct pathways
mlh1 msh4	Not known	NA	Like msh2 msh4	In the absence of the disjunction pathway, mlh1 has a phenotype only in the pairing pathway
C. Effects on MMR for poorly repairable mismatch near a DSB site
	Wisdom according to Getz et al. (2008)		Model
Genotype	Phenotype	Explanation	Predicted/explained phenotype	Explanation
msh4	Increased HC/(HC+FC)	Eliminates disjunction pathway	Increased HC/(HC+FC)	Loss of disjunction pathway, in which all conversions are FCs
msh2	Not known	NA	No effectⁱ	PRMs not repaired by Msh2^j
mlh1	Not known	NA	Increased HC/(HC+FC) for NCOs	Turns disjunction pathway FC and 4:4 COs into HC NCOs
msh2 msh4	Not known	NA	Like msh4	Eliminates disjunction pathway
mlh1 msh4	Not known	NA	Like msh2 msh4 = like msh4	PRMs not repaired in pairing pathway; Disjunction pathway is gone

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Abbreviations: DSBr, double-strand break repair; MMR, mismatch repair; COs, crossovers; NCOs, noncrossovers; HC, half conversion; FC, full conversion; NA, not applicable.

Novak et al. (2001)

Getz et al. (2008)

Abdullah et al. (2004)

Stone and Petes (2006)

Hunter and Borts (1997)

Ross-MacDonald and Roeder (1994)

Hollingsworth et al. (1995)

Hoffmann et al. (2005)

ⁱ

If PRMs escape heteroduplex rejection in wild type.

Wang et al. (2003)

Acknowledgments

We thank Eva Hoffmann, John Fowler, Rhona Borts, Nancy Hollingsworth, Elizabeth Housworth, Michael Lichten and anonymous reviewers for helpful comments. Jette Foss provided invaluable editorial assistance on Appendix A. Unless otherwise specified, statistical calculations were conducted with the aid of VassarStats (http://faculty.vassar.edu/lowry/VassarStats.html).

APPENDIX A: ASSIGNING PARAMETER VALUES (F.W.S.)

Hoffmann et al. (2005) reported the values for the relative frequencies of two- vs. one-sidedness as well as those of HCs and FCs for his and BIK in wild-type, msh2, and mlh1 crosses. To test whether the DSBr model proposed above (Table 1 and Figures 3 and 4) can accommodate these values it was necessary to evaluate the parameters that were identified as defining the model (Table 2). This appendix describes how such evaluation was achieved.

Parameter values were assigned on the basis of data in Hoffmann et al. (2005, Table 5). The key observation is that the fraction of FCs among conversions in the mlh1 mutant strain (12/112) is significantly less than that in the msh2 strain (33/86; P < 0.0001). These data provide the statistical support for the view that, whereas Mlh1 is required for MMR in both the pairing and disjunction pathways, Msh2 has no role in the disjunction pathway.

To allow comparison between the observed and expected his conversion frequencies (Table 2) in msh2, mlh1, and wild-type strains, we expressed the entries in Hoffmann et al. (2005 Table 5) as events per 1000 tetrads, rounding to whole numbers (Table A1; a convenience that exaggerates the significance of the msh2 and mlh1 data, while reducing that of the wild-type data). In addition, we combined the four conversion classes (6:2, 2:6, 5:3, 3:5) of Hoffmann et al. (2005) into two classes, FC and HC, ignoring for now the implications of some conversion disparities (Appendix D). Finally, we assumed that the BIK and his ends of the broken chromosome are equally likely to initiate invasion.

TABLE A1.

Conversions at his

	Conversion type per thousand tetrads^a
	6:2	2:6	FC^b	5:3	3:5	HC^c	Tetrads	(FC + HC)%	HC/(FC + HC)%
Wild type	56	64	120	8	9	17	1731	13.7	12.4
msh2	28	33	61	31	66	97	545	15.8	61.4
mlh1	9	12	21	60	111	171	585	19.2	89.1

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Conversions of the marker his4–ATC (his), close to the DSB site. Adapted from Hoffmann et al. (2005, Table 5).

Ignores a few rare tetrad classes. Conversion types are rounded to the nearer whole number.

6:2 + 2:6.

5:3 + 3:5.

We now have the tools to evaluate the adjustable parameter D. Following the model, which assumes that msh2 and mlh1 have identical phenotypes in the pairing pathway, the excess of HCs in the mlh1 strain over HCs in the MLH1 msh2 strain (171 − 97 = 74: Table A1) corresponds to D/2, the number of tetrads in which his is involved on the annealing side of a disjunction-pathway event. Also by the model, the excess of FCs in MLH1 msh2 over FCs in mlh1 should be D/4. That excess is 61 − 21 = 40 (Table A1). We arrive at a value for D by comparing the two mutants with respect to HCs (D/2 = 74; D = 148) and, independently, by comparing them with respect to FCs (D/4 = 40; D = 160). We average these values to get a working estimate of D = 154 (Table A2).

TABLE A2.

Worksheet for confirming parameter values for his in MMR mutants

	Pairing pathway: P = 171^a				Disjunction pathway: D = 154^b
	Invasion: 1/2		Annealing: 1/2		Invasion: 1/2		Annealing: 1/2
	FC	HC	FC	HC	FC	HC	FC	HC
msh2	g 0.127	E(1 − g) 0.262	g 0.127	1 − g 0.873	0	0	1/2	0
N	10.9	22.4	10.9	74.6	0	0	38.5	0
Summed conversions:			Calculated:			Observed:
FC			10.9 + 10.9 + 38.5 = 60.3			61
HC			22.4 + 74.6 = 97			97
mlh1	g 0.127	E(1 − g) 0.262	g 0.127	1 − g 0.873	0	0	0	1
N	10.9	22.4	10.9	74.6	0	0	0	77
Summed conversions:			Calculated:			Observed:
FC			10.9 + 10.9 = 21.8			21
HC			22.4 + 74.6 + 77 = 174			171

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Estimated number of tetrads per thousand that enjoyed a DSB at HIS4 and were repaired on a homolog via the pairing pathway.

Estimated number of tetrads per thousand that enjoyed a DSB at HIS4 and were repaired on a homolog via the disjunction pathway. g = 0.127 is the probability of gap repair (or short-patch repair to FC in the pairing pathway). E = 0.3 (see Table A4) is the probability that the marker (BIK or his) on the invasion side of the DSB remains within the heteroduplex. N is the number, per thousand tetrads, expected by the model evaluated with these parameter values. The observed numbers are from Table A1. Assigning the same probability, ½, to the invasion and annealing sides of the DSB implies an assumed lack of left vs. right sequence preference in the invasion events.

To obtain values for P and g_his for each of the mutant strains, msh2 and mlh1, we write expressions for the number of tetrads demanded by the model to result in FC or HC for his:

Based on the approximately 30% of two sidedness (measured as the fraction of his conversion tetrads that are also conversions for BIK) and the rarity of BIK FCs among these two-sided events (Hoffmann et al. 2005, Table 5; Table A4A), we set E, the fraction of pairing pathway events in which the sliding D-loop comes to rest over the BIK marker on the invasion side of the DSB at E = 0.3. Absent an estimate of E for his, we assume the same 0.3 value. With D = 154 and E = 0.3, values for P can be extracted from the equations. Solving the FC and HC equations for msh2, we get P = 172, while solving the equations for mlh1 gives P = 166. We can then get values for g_his from both the msh2 and the mlh1 data. These are 0.131 and 0.127, respectively. The P and g_his values for the two strains are, as expected, similar, and we settled on P = 171 and g_his = 0.127 for personal reasons. (Recall that our modest goal is to determine whether there is a set of parameters that allows the model to fit the data, rather than to determine best estimates of those parameters.) Calculating the values for the FCs and HCs in the wild-type crosses (Table A3) required the evaluation of two additional parameters—one for the probability (m_his) that Msh2–Mlh1-dependent MMR leads to full conversion in the pairing pathway and one for the probability (R) that heteroduplex rejection does not occur. The need for R in the wild-type cross is signaled by the otherwise puzzling observation (Table A1) that total conversions at his are lower in wild type than in the MMR mutants despite the greater two sidedness in wild type. In limiting the model to eight adjustable parameters, we ignore a variety of possible additional factors with the hope that in so doing we are better exposing the skeletal features of the model. For instance, we are assuming that, for our markers, MMR in the pairing pathway is directed by the invading and annealing ends created by the DSB, with the result that all MMR in that pathway results in FC. Justification for this simplification is found in the adequacy of our simple model (Table 2).

TABLE A4.

Work sheet for testing parameter values for BIK and his with regard to two-sidedness and conversion types for BIK

A. MMR-defective pathways	Pairing pathway: P = 171				Disjunction pathway: D =154
	Invasion: 1/2		Annealing: 1/2		Invasion: 1/2		Annealing: 1/2
	FC	HC	FC	HC	FC	HC	FC	HC
msh2	g BIK: 0.04 (3.4) his: 0.127 (10.9)	E(1 − g) BIK: 0.288 (24.6) his: 0.262 (22.4)	g BIK: 0.04 (3.4) his: 0.127 (10.9)	1 − gBIK: 0.96 (82.1) his: 0.873 (74.6)	0	0	1/2 BIK: 0.5 (38.5) his: 0.5 (38.5)	0
	FC + HC BIK: 0.328 (28.0) his: 0.389 (33.3)		FC + HC BIK: 1 (85.5) his: 1 (85.5)				FC + HC BIK: 0.5 (38.5) his: 0.5 (38.5)
Properties of BIK conversions			Calculated				Observed
Freq. two-sided tetrads among his conversions			(28.0 + 33.3)/(33.3 + 85.5 + 38.5) = 0.39				20/61 = 0.22 − 0.46^a
FC/(FC + HC)_BIK among his conversions			(3.4/28)(1/1.389) + (3.4/85.5)(0.389/1.389) = 0.10				4/41^b
FC/(FC + HC}_BIK in unselected tetrads			(6.8 + 38.5)/(113.5 + 38.5) = 0.30				0/9^b
(FC + HC)_BIK /(FC + HC)_his			(113.5 + 38.5)/(118.8 + 38.5) = 0.97				(9/101)/(96/545) = 0.51 ± 0.33^b
mlh1	g BIK: 0.04 (3.4) his: 0.127 (10.9)	E(1 − g) BIK: 0.288 (24.6) his: 0.262 (22.4)	g BIK: 0.04 (3.4) his: 0.127 (10.9)	1 − gBIK: 0.96 (82.1) his: 0.873 (74.6)	0	0	0	1 BIK: 0.5 (77) his: 0.5 (77)
	FC + HC BIK: 0.328 (28.0) his: 0.389 (33.3)		FC + HC BIK: 1 (85.5) his: 1 (85.5)		FC + HC BIK: 0 his: 0		FC + HC BIK: 0.5 (77) his: 0.5 (77)
Properties of BIK conversions			Calculated				Observed
Freq. two-sided tetrads among his conversions			(28.0 + 33.3)/(33.3 + 85.5 + 77) = 0.31				29/81 = 0.26 − 0.47^a
FC/(FC + HC)_BIK among his conversions			As in msh2 = 0.10				3/51^b
FC/(FC + HC}_BIK in unselected tetrads			(6.8)/(113.5 + 77) = 0.04				1/11^b
(FC + HC)_BIK /(FC + HC)_his			(113.5 + 77)/(118.8 + 77) = 0.97				(14/106)/(116/585) = 0.67 ± 0.34^b
B. WT	Pairing pathway: RP = 0.643 × 171				Disjunction pathway: D = 154
	Invasion: 1/2		Annealing: 1/2		Invasion: 1/2		Annealing: 1/2
	FC	HC	FC	HC	FC	HC	FC	HC
	g + m(1 − g) BIK: 0.962 (52.9) his: 0.761 (41.8)	E(1 − g)(1 − m) BIK: 0.0115 (0.6) his: 0.0718 (3.9)	g + m(1 − g) BIK: 0.962 (52.9) his: 0.761 (41.8)	(1 − g)(1 − m) BIK: 0.0384 (2.1) his: 0.239 (13.2)	0	0	1/2 BIK: 0.5 (38.5) his: 0.5 (38.5)	0
	FC + HC BIK: 0.973 (53.5) his: 0.833 (45.8)		FC + HC BIK: 1.0 (55.0) his: 1.0 (55.0)				FC + HC BIK: 0.5 (38.5) his: 0.5 (38.5)
Properties of BIK conversions			Calculated				Observed
Freq. two-sided tetrads among his conversions			(53.5 + 45.7)/(45.7 + 55.0 + 38.5) = 0.71				61/90 = 0.57 − 0.77^a
FC/(FC + HC)_BIK among his conversions			1 − (0.6/53.5)(1/1.831) − (2.1/55.0)(0.831/1.831) = 0.97				89/92 = 0.97^b
HC/(FC + HC}_BIK in unselected tetrads			(2.7)/(108.5 + 38.5) = 0.02				0/9^b
(FC + HC)_BIK /(FC + HC)_his			(108.5 + 38.5)/(100.7 + 38.5) = 1.06				(10/107)/(243/1731) = 0.67 ± 0.40 ^b

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Values for P, D, R, and E, as well as for m_his and g_his, are as in Tables A2 and A3. E = 0.3 and g_BIK = 0.04 were picked to fit the (FC + HC) and the FC/(FC + HC) data for BIK among his conversions. m_BIK = 0.96 was then selected to fit the WT sidedness data. Expected numbers (N) are in parentheses after each expected conversion frequency.

Hoffmann et al. (2005, Table 6).

Hoffmann et al. (2005, Table 5).

TABLE A3.

Worksheet for confirming parameter values for his in wild type

	Pairing Pathway: RP = 0.643 × 171				Disjunction Pathway: D = 154
	Invasion: 1/2		Annealing: 1/2		Invasion: 1/2		Annealing: 1/2
	FC	HC	FC	HC	FC	HC	FC	HC
	g+m(1−g) 0.760	E(1−g)(1−m) 0.0718	g+m(1−g) 0.760	(1−g)(1−m) 0.239	0	0	1/2	0
N	41.8	3.9	41.8	13.2	0	0	38.5	0
Summed conversions:			Calculated:		Observed:
FC			41.8 + 41.8 + 38.5 = 122		120
HC			3.9 + 13.2 = 17.1		17

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P, D, g and E values are as in Table A2. R = 0.643 and m = 0.726 were determined by fitting the model to the observed HC and FC numbers (per thousand tetrads, Table A1) for this wild-type cross.

We next address the features of the model that link two sidedness directly to MMR (Table A4). Since BIK and his are simultaneously present in all the crosses, their conversions must depend on common values of P, D, and R. Because we have assumed that E, also, is the same for his and BIK, we need pick only m and g values for BIK. If there were a large body of BIK data analogous to the his data, we might have estimated m_BIK and g_BIK as we did m_his and g_his. However, the best data come from the BIK conversions among tetrads selected for being his conversions, forcing a change in strategy. Thus, for both the MLH1 msh2 and mlh1 crosses, we sought and found a g_BIK value (0.04, by trial and error) that gave satisfactory fits to these two-sidedness data as well as to the FC/(FC + HC) value for BIK among his conversions in the two MMR-defective strains (Table A4A). For the wild-type cross, the only remaining parameter to be estimated is m_BIK, which we chose to fit the two-sidedness data exactly (Table A4B). This m_BIK value proved to give a good fit to the wild-type HC/(FC + HC) ratio for BIK among his conversions, supporting the view that the Mlh1- and Msh2-dependent two sidedness is a reflection of MMR per se.

Disparity between the two classes of HCs:

Data of Hoffmann et al. (2005) showed disparities in the rates of conversion to his and HIS. In our Sudoku, we ignored the disparity, raising the possibility that in doing so we have concealed important information. The default hypothesis for disparity is differential rates of DSBs on the two homologs, and disparity so caused would be without consequence for our analysis. However, Hoffmann et al. (2005), noting that the disparity was statistically significant only for the HCs, attributed it to different rates of restoration, by short-patch repair (Coïc et al. 2000), for the two different mismatches. This interpretation appeared to strengthen the authors' proposal that short-patch repair, operating primarily in the absence of Msh2 and Mlh1, was responsible for the one sidedness seen in the msh2 and mlh1 crosses. Shortcomings of this proposal, along with support for the differential DSB hypothesis, are detailed in Appendix D. The significance of a well-supported proposal for differential DSBs is that it undermines restoration by short-patch repair as an explanation for the one sidedness.

The BIK data:

Hoffmann et al. (2005) noted that the rate of conversion at BIK, especially in the MMR mutants, is less than that at his. Other aspects of the BIK data combine with this observation to suggest that our Sudoku is not quite finished. Most of the BIK data, in Hoffmann et al. (2005, Table 5), were collected from tetrads that were preselected as his conversions. As such, they were two sided and, according to our model, must be from the pairing pathway. Consequently, those BIK data were expected to be the same for the mlh1 and the msh2 crosses, which they seem to be (mlh1—3 FC, 48 HC; msh2—4 FC, 37 HC; P = 0.7). That's cool, but BIK, in unselected tetrads, fails to show the HC and FC differences that characterize the his data; i.e., for BIK, there is no evidence of mlh1-specific HCs or of MLH1 msh2-specfic FCs. (mlh1—1 FC, 10 HC; msh2—0 FC, 9 HC). The numbers are small and could be ignored for that reason. However, the 0:9 ratio for BIK in the msh2 cross is significantly different (P = 0.02) from the corresponding ratio, 33:53, for his. An economical interpretation for both the relatively low conversion rate of BIK (Hoffmann et al. 2005) and its failure to show properties characteristic of his in the disjunction pathway is that the disjunction pathway conversion tracts usually fail to include BIK. This may be simply because conversion tracts in the disjunction pathway are short and BIK is farther (maximum ∼600 bp) from the HIS4 hotspot than is his (maximum ∼266 bp) (E. R. Hoffmann, personal communication). This possibility, which is of no consequence for our basic Sudoku, predicts that most or all BIK conversion crossovers come from the pairing pathway and, consequently, will have weaker interference than do his conversion crossovers.

Multiple events?

In all studies of conversion at recombination hotspots in yeast there are tetrads that would be interpreted on the basis of any model as due to multiple DSBr events, and these tetrads are usually exempted from interpretation. However, as pointed out by Merker et al. (2003), the identification of the less obvious multiple events is unavoidably model dependent. Furthermore, the estimation of expected frequencies of multiple events is confounded by the possibility of negative interference between DSBs at the same level (Lamb and Wickramaratne 1973). Among other candidates for complex events we note two-sided HCs that are crossovers with heteroduplex on the same chromatid in the trans configuration (Hoffmann and Borts 2005). Such tetrads might result from a pairing-pathway noncrossover being accompanied by a not-so-incidental exchange (Ray et al. 1989) provoked by the 3′-end of a protruding SS-DNA whisker (Hotchkiss 1971) (Figure A1).

Figure A1.— — Crossovers that are *trans*-HC on the same chromatid. Steps 1–4 are as steps 1–4 in Figure 4. Steps 4 and 5: An intermediate in the pairing pathway has enjoyed D-loop expansion but unwinds and reanneals. Step 6: The 3′-whisker arising by branch migration on the resulting noncrossover product may promote crossing over with the blue chromatid or a third chromatid.

APPENDIX B: CROSSOVER INTERFERENCE IN mlh1 CROSSES (F.W.S.)

Deletions of Mlh1 cause a reduction in crossing over, generally to a lesser degree than do deletions of Msh4 or Msh5 (Wang et al. 1999; Abdullah et al. 2004; Argueso et al. 2004). Deletions of Mlh1 are hypostatic to deletions of Msh4 (Wang et al. 1999; Argueso et al. 2004), clearly indicating that Mlh1 promotes crossing over in the disjunction pathway. Consequently, it is a strong expectation of our simple two-pathway model that mlh1Δ mutants should have reduced interference, commensurate with the degree to which they have reduced crossing over. Confirmation of this expectation is befogged by claims in the literature regarding the effect, if any, of Mlh1 on crossover interference.

Abdullah et al. (2004) wrote, “ … as might be predicted for genes of the MSH4 pathway, interference was abolished by deletion of MLH1 and MLH3.” In the same year, Argueso et al. (2004) wrote, “In wild type, interference was significant at all intervals analyzed in chromosome XV…/ These values did not significantly change in mlh1 … strains, which were shown previously to maintain interference.” These two authors have reached extreme, opposite conclusions, neither of which is in concordance with the two-pathway model (Getz et al. 2008).

Shortcomings in the analyses by Abdullah et al. (2004):

Abdullah et al. (2004) presented a three-factor test for interference on chromosome III. The data revealed interference in wild type and fit well with the null hypothesis of no interference in the mlh1 deletion mutant. The conclusion that mlh1Δ lacked interference was embraced by the authors as being expected. However, the authors appear to have similar data for chromosome VII, but they present no analysis of those data for interference. The authors make no tests of their two-factor data for either chromosome.

Shortcomings in the analysis by Argueso et al. (2004):

Argueso et al. (2004) conducted both three-factor and two-factor tests for interference and reported significant levels of interference in mlh1Δ crosses, leading them to conclude that deletion of MLH1 is without effect on interference. However, their analyses, too, have several problems:

The authors represent their tests as supporting the conclusion that “[interference] did not significantly change in mlh1 … strains… .” However such a conclusion requires statistical tests of mlh1 vs. wild-type data, while the only tests presented are of mlh1 and wild type vs. the null hypothesis of no interference.
To test their two-factor data for interference, the authors used an inefficient method (Papazian 1952) for calculating the expected frequency of NPDs. In 2004, shortcomings of statistical tests based on that expectation were not generally appreciated. More recently (Stahl 2008) such tests were shown to give false positives—i.e., to give chi-square P-values that are too small. Two of Argueso's conclusions of significance at the 5% level seem to have fallen victim to that shortcoming of the test (LYS–HIS and ADE–HIS, Table B1). Further undermining the usefulness of these data for concluding that interference did not significantly change in mlh1 strains is that the ADE–HIS interval comprises a major fraction of the LYS–HIS interval, so those two observations are not independent.
As a test for interference revealed by three-factor crosses, Argueso et al. (2004) calculated coefficients of coincidence). These “COC tests” look convincing for the presence of interference in the mlh1 strain (Shinohara et al. 2003, Table 4) for the URA–LEU–LYS intervals, but not for the LEU–LYS–ADE intervals, where the tetrad data do not indicate interference at the 5% level of significance. Their “Spore” data for the LEU–LYS–ADE intervals, on the other hand, are reported as manifesting interference significant at the 5% level. However, the description provided by Argueso et al. (2004) of their COC test for “Spore” data suggests that it has incorrectly indicated significance. Since these data are simply their disaggregated tetrad data, supplemented by spore data from tetrads with fewer than four viable spores, the conclusion of significance is justified only if the analysis recognizes that many of the recombinants in the data set arose in pairs. Since the “RANA” software employed assumes that all spores arise from independent events (S. E. Zanders and E. Alani, personal communication), P-values based on those data are underestimated. Furthermore, as a test for an mlh1-induced change in interference, COCs suffer from being a function both of the map distances involved and the distribution of crossovers with respect to each other (Stahl and Housworth 2009).

TABLE B1.

Map distances and interference

Chromosome XV^a	ADE–HIS			URA–LYS			LYS–HIS			URA–LEU
	wt	mlh1	msh5	wt	mlh1	msh5	wt	mlh1	msh5	wt	mlh1	msh5
PD:	343	400	496	264	351	513	278	344	465	607	486	643
T:	709	211	215	759	261	300	744	261	242	456	128	76
NPD:	16	5	9	45	4	7	48	11	13	5	2	1
cM:	37.7	19.6	18.7	48.2	23.1	16.8	47.8	26.5	22.2	22.8	11.4	5.7
m:	4.5	0.8	0.0	3.1	1.9	0.9	2.7	0.7	0.0	2.1	0.5	0.0
P:	<0.0001	0.06	0.73	<0.0001	0.0006	0.01	<0.0001	0.06	0.91	<0.0001	0.35	0.94

Chromosome III^b	LEU–MAT							HIS–LEU
	wt	mlh1	msh4	msh5	wt	mlh1	msh4	msh5
PD:	595	570	213	153	744	583	193	121
T:	722	370	62	46	496	191	39	16
NPD:	51	14	3	1	20	5	1	1
cM:	37.6	22.0	14.4	13.0	24.4	14.2	9.7	8.0
m:	0.6	0.6	<0	0.3	0.6	0.2	0.0	<0
P:	0.0044	0.042	0.55	0.66	0.025	0.46	0.94	0.19
Chromosome VII^b	TRP–CYH							CYH–MET
	wt	mlh1	msh4	msh5	wt	mlh1	msh4	msh5
PD:	316	487	413	141	1039	781	514	190
T:	1023	451	128	61	395	140	25	9
NPD:	113	25	4	2	6	2	1	0
cM:	58.6	31.2	13.9	17.9	15.0	8.2	2.9	2.3
m:	2.0	0.6	0	0.3	0.9	0.1	<0	—
P:	<0.0001	0.028	0.82	0.62	0.011	0.59	0.042	0.82

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Map lengths (cM) for the intervals are calculated on the assumption that the number of exchanges per bivalent in any interval is not more than two (Perkins 1949). m is an index of the strength of interference (Stahl and Lande 1995; Stahl and Housworth 2009); m = 0 implies no interference. P is the probability that such an observed deviation of the data from the hypothesis of no interference would occur by chance alone, calculated according to Stahl (2008). The map for chromosome XV is URA–LEU–LYS–ADE–HIS. The intervals on chromosomes III and VII are all nonoverlapping.

Argueso et al. (2004).

Abdullah et al. (2004).

Reanalyzing the data:

We test our view that deletion of MLH1 reduces, but does not eliminate, interference using the two-factor data of Abdullah et al. (2004) and Argueso et al. (2004). For a set of tetrads that contains at least one NPD, we assess the magnitude of the deviation of the data set from the hypothesis of no interference by determining the counting number (m; Foss et al. 1993; Stahl and Lande 1995). Although the counting model was written for integer values of m, the calculator at Stahl Lab Online Tools (http://www.molbio.uoregon.edu/∼fstahl/tetrad.html) allows estimates of noninteger values of m by visual interpolation. If we do this for the wild-type, mlh1, and msh4/5 data of Abdullah and of Argueso, we expect to see that m goes down as crossing over goes down, reaching zero for msh4/5. This analysis is tabulated in Table B1 along with P-values for the null hypothesis of no interference.

The m values for mlh1 are between the wild-type and msh4 values everywhere but once (Table B1), where interference was very low in the wild type. We conclude that mlh1 has interference that is intermediate between that of wild type and msh4, as expected for a strain that is missing a fraction of its disjunction pathway crossovers (and for which the linkage-map distances between the remaining disjunction pathway exchanges may have become more variable). Data by Wang et al. (1999) are fully compatible with this conclusion (Table B2). For two intervals and the inclusive interval, the m-values for both mlh1 and mlh3 deletion strains are less then those for wild type, indicating reduced interference. The three P-values for wild type are all less than 0.05, while the mlh1 and mlh3 P-values for one interval (in which wild-type interference is weak) are greater than 0.05, while for the other interval and the inclusive interval they both are less than 0.05. Thus, interference, although weakened, is abolished in neither the mlh1 nor mlh3 mutants.

TABLE B2.

Map distances and interference: Chromosome III (Wang et al. 1999)

	URA–HISLEU			HISLEU–MAT			URA–MAT
	wt	mlh1	msh3	wt	mlh1	msh3	wt	mlh1	msh3
PD:	483	564	697	506	603	689	258	391	450
T:	483	345	370	451	301	388	646	502	585
NPD:	7	8	6	24	12	17	72	37	47
cM:	27.0	21.4	18.9	30.3	20.4	21.9	55.2	38.9	40.1
m:	2.7	0.9	1.0	0.6	0.3	0.3	1.1	0.6	0.5
P:	<0.0001	0.004	0.0015	0.027	0.34	0.24	0.0004	0.016	0.027

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See Table B1 for explanations.

This analysis implicates Mlh3 as the partner to Mlh1 in promoting crossing over in the disjunction pathway and raises the question of which MutL homolog cooperates with Mlh1 in disjunction-pathway MMR. Mlh3 would surely be the prime suspect were it not for data that failed to demonstrate an MMR phenotype for mlh3 mutants (Wang et al. 1999). However, for a test marker to manifest an Mlh1-dependent MMR phenotype, the marker must participate in conversion in the disjunction pathway. As pointed out in Appendix A, BIK appears to be a marker that, while participating in conversion in the pairing pathway, fails to do so in the disjunction pathway, perhaps because it is rarely included in the short intermediate. Reported failures (e.g., Wang et al. 1999) to see a meiotic MMR phenotype for mlh3 may simply mean that the few markers monitored for meiotic MMR in those crosses happened to be of that sort.

APPENDIX C: DELETION OF mlh1 CHANGES SOME ONE-SIDED CROSSOVERS INTO ONE-SIDED NONCROSSOVERS (F.W.S.)

Hoffmann et al. (2005) scored tetrads from wild-type, msh2, and mlh1 crosses for one vs. two sidedness for conversion at sites B/b and C/c and for being crossed over (or not) with respect to linked sites A/a and D/d (Hoffmann et al. 2005, Table 7). They then compared each of the mutants to wild type and to each other with respect to the distribution of events among the four classes scored. They wrote, “Both MMR mutant strains showed a difference in the distribution of events into those four classes compared to the wild-type strain (P < 0.05; G-test of homogeneity), reflecting that the mlh1Δ and msh2Δ strains contain more one-sided events. When we compared the distribution of mlh1Δ to that of the msh2Δ strain, we did not observe a significant difference” (p. 1299).

Had Hoffmann et al. (2005) thought to compare the mutants with respect to crossover vs. noncrossover frequencies for the one-sided tetrads only, they would have found a significant difference in crossover/noncrossover ratios between the mlh1 and the MLH1 msh2 strains (Table C1). Among the one-sided tetrads, the loss of Mlh1 resulted in an mlh1-specific loss of one-sided crossovers accompanied by a gain of twice as many noncrossovers. Since the loss of Mlh1 results in a major failure of MMR, these data imply that, among disjunction-pathway one-sided tetrads, the mlh1-induced increase in HCs represents twice the number of FCs lost. The larger data set in Appendix A supports this suggestion. We propose that the twofold mutation-induced excess of HCs gained over FCs lost reflects the existence of disjunction-pathway-DSBr intermediates whose mismatches, in the presence of Mlh1, would have been rectified equally to either 2:2 (restoration) or FC of the his marker. Furthermore, MMR resulting in restoration must have been directed by the junction because, as posited in Appendix A, the invasion and annealing opportunities for MMR would have yielded only FCs for a marker so close to the DSB. The further implication of this interpretation, that disjunction-pathway mismatches usually persist throughout the formation of the ligated double Holliday junction intermediate, is supported by the observations of Allers and Lichten (2001a,b) and is consistent with our hypothesis that disjunction-pathway mismatches are repaired only and inevitably via Mlh1-dependent, resolution-directed MMr (and see Getz et al. 2008; Stahl and Foss 2008).

TABLE C1.

mlh1 phenotype in two- and one-sided tetrads

	Two-sided tetrads			One-sided tetrads
	CO	NCO	p^a	CO	NCO	p^a	p^b	Total tetrads
Wild type	35	19	1	9	8	0.067	0.39	1731
mlh1	15	9		10	29		0.007	585
MLH1 msh2	9	6	0.86	17	15	0.027	0.76	545

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Among one-sided tetrads, the mlh1 mutation causes the mlh1-specific loss of approximately 17 − 10 = 7 crossovers (COs) coupled with the appearance of 29 − 15 = 14 noncrossovers (NCOs). The similarity in total population size of the mlh1 and msh2 data sets justifies this direct comparison of observed numbers. Data from Table 7 of Hoffmann et al. (2005).

Probability by two-tailed Fisher exact probability test that these data and those for mlh1 could have been drawn randomly from the same universe.

Probability by two-tailed Fisher exact probability test that the one-sided and two-sided data could have been drawn randomly from the same universe.

APPENDIX D: DISPARITY AND ONE SIDEDNESS (F.W.S.)

In the total data of Hoffmann et al. (2005) there are 182 conversions to HIS and 252 to his, a clear indication of disparity (P = 0.0009). In the MMR-defective crosses, 3:5 (HIS/his) tetrads were about twice as frequent as 5:3 (HIS/his) tetrads, while the 2:6 and 6:2 tetrads (at HIS) showed a statistically insignificant disparity in the same direction (Table D1). Hoffmann et al. (2005) reported that the wild-type cross showed no significant disparity among HCs, FCs, or total conversions. To account for their observations, Hoffmann et al. (2005) proposed that the G:G HIS/his mismatches that would give 5:3 segregation in the absence of Msh2 or Mlh1 be subject to unbiased short-patch repair rather than terminus-directed MMR (Radford et al. 2007; but see Coïc et al. 2000, who reported that short-patch repair in yeast favored FCs). We note, however, that the wild-type HC data of Hoffmann et al. (2005) are not significantly different from either of the two mutant data sets or from the sum of those two sets. Similarly, for none of the crosses are the HC data significantly more disparate than the FC data. More troubling for the unbiased short-patch-repair explanation for the disparity is the observation that the bias that is shown by the FCs, albeit insignificant, is in the direction opposite to that predicted. In their model, selective removals of 5:3 HIS/his mismatches, while generating disparity in the HCs, would result in a disparity in the FCs which was half as great, but in the opposite direction, i.e., favoring 6:2 HIS/his. Since the disparities in the FCs and HCs are in the same direction, a differential frequency of DSBs on the two homologs has to be the favored explanation.

TABLE D1.

Disparity at his

	FC		HC
	6:2	2:6	5:3	3:5	P^a	P^b
Wild type	96	111	14	15	1.0	–
mlh1	5	7	35	65	0.75	0.28
msh2	15	18	17	36	0.31	0.23
mlh1+msh2	20	25	52	101	0.27	0.21
Σ three crosses	116	136	66	116	0.053	–
P^c	0.23		0.0003

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Data from Hoffman et al. (2005, Table 5).

Chi-square probability that a difference in disparity between FCs and HCs that is this great or greater could arise by chance alone.

Chi-square probability that a difference in disparity between the mutant strain and WT that is this great or greater could arise by chance alone.

Chi-square probability for the data summed over the three crosses that an observed disparity this great or greater could arise by chance alone.

The simple explanation of differential DSBs is further supported by the disparity observed at BIK. If the disparity in HCs at his were due to mismatch-specific restoration of incipient 5:3 tetrads, there would be no expectation that unselected BIK conversions would show a related disparity. However, the unselected “subset” BIK conversions do manifest disparity [20 (2:6 + 3:5) and 9 (6:2 + 5:3)], which is significant (P = 0.03) and in the direction expected if the disparities at the two loci both resulted from the homologs being unequally subject to DSBs.

What might account for a differential frequency of DSBs on the two homologs? Since sporulations were conducted with little or no growth following mating, it is plausible that the HIS4 hotspots on the two chromosomes were in different states with respect to Spo11 sensitivity (Abdullah and Borts 2001; Cotton et al. 2009). Hoffmann et al. (2005) physically measured DSBs at HIS4 and found no differences between the homologs. However, the physical studies were conducted on an established HIS/his diploid culture in which the homologs could not be expected to manifest a physiology-dependent difference.

We conclude that the work of Hoffmann et al. (2005) provides no support for their thesis that one sidedness reflects restoration by short-patch repair in MMR-defective crosses. At the same time, their work contradicts a proposal by Getz et al. (2008) that one sidedness is caused by resolution-directed MMR in the disjunction pathway, affected by an unidentified MMR enzyme (as imagined in Figure 1D).

This article is dedicated to the Institute of Molecular Biology, established January 1, 1959.

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PERMALINK

A Two-Pathway Analysis of Meiotic Crossing Over and Gene Conversion in Saccharomyces cerevisiae

Franklin W Stahl

Henriette M Foss

Abstract

TABLE 1.

RESULTS

Relative frequencies of tetrad types provide measures of linkage distance and crossover interference:

Gene conversion as evidence of DSBr:

The Szostak/Sun model for meiotic DSBr:

Figure 1.—

Conventional skinny on MMR in meiosis:

Two pathways for meiotic DSBr:

Pathway-specific rules for MMR:

Msh2 deletion and PRMs are equivalent ways of eliminating Mmr from the pairing pathway:

Deletion of Mlh1 causes greater loss of MMR than does deletion of Msh2:

Evidence for disjunction-pathway-specific, Mlh1-dependent MMR:

In the disjunction pathway, Mlh1-dependent MMR occurs only in response to junction resolution:

msh2-induced lack of Mmr reduces two sidedness:

Migrating D-loops and transient heteroduplex:

Figure 2.—

A role for Msh4–Msh5 in disjunction-pathway MMR:

The dog that didn't bark:

Where the rubber meets the road:

Sudoku:

TABLE 2.

DISCUSSION

Figure 3.—

Figure 4.—

Heteroduplex lengths:

MMR-dependent two sidedness:

Gene conversion in Sordaria:

An unexpected DSBr intermediate:

Figure 5.—

Predictions:

TABLE 3.

Acknowledgments

APPENDIX A: ASSIGNING PARAMETER VALUES (F.W.S.)

TABLE A1.

TABLE A2.

TABLE A4.

TABLE A3.

Disparity between the two classes of HCs:

The BIK data:

Multiple events?

Figure A1.—

APPENDIX B: CROSSOVER INTERFERENCE IN mlh1 CROSSES (F.W.S.)

Shortcomings in the analyses by Abdullah et al. (2004):

Shortcomings in the analysis by Argueso et al. (2004):

TABLE B1.

Reanalyzing the data:

TABLE B2.

APPENDIX C: DELETION OF mlh1 CHANGES SOME ONE-SIDED CROSSOVERS INTO ONE-SIDED NONCROSSOVERS (F.W.S.)

TABLE C1.

APPENDIX D: DISPARITY AND ONE SIDEDNESS (F.W.S.)

TABLE D1.

References

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