Abstract
Genetic interference means that the occurrence of one crossover affects the occurrence and/or location of other crossovers in its neighborhood. Of the three components of genetic interference, two are well modeled: the distribution of the number and the locations of chiasmata. For the third component, chromatid interference, there exists only one model. Its application to real data has not yet been published. A further, new model for chromatid interference is presented here. In contrast to the existing model, it is assumed that chromatid interference acts only in the neighborhood of a chiasma. The appropriateness of this model is demonstrated by its application to three sets of recombination data. Both models for chromatid interference increased fit significantly compared to assuming no chromatid interference, at least for parts of the chromosomes. Interference does not necessarily act homogeneously. After extending both models to allow for heterogeneity of chromatid interference, a further improvement in fit was achieved.
DURING meiotic prophase 1 in diploid individuals, each chromosome is paired with its homologue. Each homologue is duplicated, producing two identical chromatids, the sister strands. A crossover represents an event where two nonsister chromatids form chiasmata, break, and reunite, enforced by the tight contact and the twisting between the chromatids and the subsequent repair mechanism. After meiosis, one of the four resulting gametes is randomly chosen for further inheritance. For clarity we use the term chiasma at the fourstrand stage, while the term crossover is used with single strands or gametes. Hence, from one nonsister strand chiasma, two crossovers result. An example of a meiosis at the fourstrand stage is given in Figure 1. It has often been proven that chiasma or crossover events are not independent. The notion of genetic interference describes the effect on crossovers of neighboring crossovers. The components of interference are as follows:

Non(complete)randomness in the number of crossovers: The nointerference model applies if the crossover numbers are Poisson distributed. All other count distributions yield deviations from no interference.

Non(complete)randomness in crossover locations: The suppression of nearby crossovers has been modeled by nonuniformly distributed locations and by nonexponentially distributed intercrossover distances with renewal point processes.

Chromatid interference (CI): The strands actually involved in a crossover depend in some way on those strands involved in neighboring crossovers.
Substantial progress has been made in investigating and modeling the first two components. For these cases we use the term suppression interference (SI). For SI models, no chromatid interference (NCI) is assumed. For recent reviews see Karlin and Liberman (1994) and McPeek and Speed (1995). The χ^{2}model of recombination (Fosset al. 1993; Zhaoet al. 1995a) is accepted as a satisfying model for positive SI. Negative SI, i.e., one chiasma enforcing the occurrence of another one, can be described, for example, by a negative binomial count distribution for the number of chiasmata.
The investigation of CI has not reached the same level yet. It started in the 1930s when recombination fractions >0.5 had been observed. This phenomenon was termed pseudolinkage. Models have been developed for data exhibiting pseudolinkage (Winge 1935; Mather 1938). Particularly, Mather found that this phenomenon could result only from CI. However, little evidence was found for it in diploid organisms. For a review and a test procedure, see Zhao et al. (1995b). Recently, Zhao and Speed (1998, 1999) developed a model for CI. To our knowledge, an application has not been published so far. Summarizing the literature, the general view is that CI is not evident. This is reflected by widely distributed mapping software (e.g., CRIMAP) that reduce recombination fractions >0.5 to 0.5. Yet, recombination fractions θ > 0.5 are accepted in tetraploids. Recombination fractions of up to 0.8 have been found in tetraploid fish (Wrightet al. 1983). Since θ ≤ 0.5 is valid even for tetraploid species under NCI [see formula (6) with Q = 0.25], there is evidence for an action of CI.
Hence, CI cannot be excluded, either from a theoretical point of view, or from empirical evidence. It therefore appears to be helpful to derive alternative models to the existing model of Zhao and Speed (1998), either to increase the evidence for CI in diploids or to strengthen the conviction that there is none. The objective of the present study was to develop a model allowing parallel nonsister strand chiasmata, which could not arise under high SI if one assumes suppression on all four strands, and sister strand chiasmata, which also have not been modeled so far but could play a role at high SI. In analyzing a number of recombination data sets we found evidence for SI and CI. Until now heterogeneity of interference has not been investigated. We incorporated CI heterogeneity into both CI models and obtained further improvement in fit.
METHODS
The CI(η) model of Zhao and Speed (1998): Zhao and Speed (1998) treated Weinstein's (1936) and Mather's (1938) approach of modeling CI by introducing a parameter η that defines the probability that a strand involved in a chiasma is also involved in the following chiasma. From this, the probabilities for complementary, diagonal, and reciprocal chiasma pairs (cf. Figure 1) are (1 – η)^{2}, 2η(1 – η), and η^{2}, respectively. Under NCI these are ¼, ½, and ¼. Thus η > 0.5 indicates an increased and η < 0.5 a reduced amount of reciprocal chiasma pairs compared to complementary pairs. The force of CI is assumed to be independent of the distance between the neighbored chiasmata. For the underlying SI process the χ^{2}model of recombination was chosen; i.e., suppression of nearby chiasmata is working on all four strands. This model here is called CI(η).
The CI(Q) model allowing sister strand and parallel nonsister strand chiasmata: Jarrell et al. (1995) found an increased occurrence of fourstrand double crossovers at the centromere of bovine chromosome 23. For all models of SI derived for the fourstrand bundle, such a finding can occur only if suppression is absent, since otherwise SI would act on all four strands.
To develop a CI model that takes this into account and gives additional information compared to the CI(η) model, we assume here that CI acts only in the neighborhood of a chiasma. Then a complementary chiasma pair is modeled by a parallel nonsister strand pair; i.e., two complementary chiasmata occur close to one site. As mentioned above, this would lead to the phenomenon of double crossovers. On the other hand, nearby reciprocal chiasmata would look like sister strand chiasmata since they would seldom lead to observable recombinations. Thus we also allow sister strand chiasmata.
Let us again consider the question whether suppression of nearby chiasmata acts on all four strands or only on those involved in the chiasma. So far an answer to this question is not known. If we assume the highest amount of suppression in the first case, then the maximum average suppression distance is 0.5 M since within this distance the next chiasma must appear. In the latter case, the maximum average suppression distance is 1 M, since a chiasma on the complementary strands could restore the needed expected number of chiasmata. In this way, complementary pairs are enforced, which can be considered to be parallel from a model point of view. Additionally, sister strand chiasmata have been observed repeatedly. Although they are invisible to the observer of recombinations, they influence the location of neighbored nonsister strand chiasmata if suppression is a property of nonsister as well as sister strand chiasmata. Therefore, they must be taken into account.
Assume the fourstrand stage of meiosis of diploids. We define a chiasma site to be a location on a chromosome, where strands break and reunite. Let P be the probability that exactly two nonsister strand chiasmata occur at a chiasma site and S that one or two sister strand chiasmata occur. Consequently, 1 – P – S is the probability that exactly one nonsister chiasma appears at the chiasma site. The chance of a chiasma site producing a crossover on a gamete is then Q = (1 + P – S)/2. We assumed that there are no dependencies between different chiasma sites; i.e., for this, NCI is assumed.
Consider two loci on a random gamete with map distance x; i.e., the expected number of crossovers is x and an appropriate recombination fraction θ. The relationship between x and θ is given by the map function θ(x). Let
Before we can apply the model we have to define the chiasma site distribution {c_{i}(x)}. As done by Zhao and Speed (1998), we use the χ^{2}model of recombination. The combined model is denoted with CI(Q).
Note that (5) can be viewed as a general formula for an arbitrary strand stage of meiosis under NCI. A chiasma is involved in two of, say, a strands. A crossover will result from a chiasma on a random gamete with probability Q = 2/a. Therefore, for Q = 1, (5) coincides with the map function of the twostrand stage,
The map function for CI(Q): Recall the χ^{2}model of recombination. The chiasma formation process is described by a stationary renewal process, where the intercrossover distances follow a χ^{2}distribution with 2(m + 1) d.f. For the theory derived by Zhao et al. (1995a) the model of Foss et al. (1993) of the same process is helpful. In this model the locations C of socalled gene conversions are assumed to be uniformly and independently distributed on a scale y. Their values follow a Poisson distribution. By this, not every gene conversion C leads to a nonsister strand chiasma Cx. One realized Cx is followed by m gene conversions Co, from which no crossovers result. Afterward the next Cx is produced. Parameter m is the interference parameter, m = 0 indicates no interference. The probability of no crossovers was determined to be
The application of the CI(Q) model is based on the assumption that an event Cx represents a chiasma site now, which leads to a crossover on a gamete with probability Q. The relation between the model scale y and the genetic scale x then changes to y = (m + 1)x/Q. The probability of the crossover number i = 0 is given by (10) and those for i = 1, 2,... are found to be
Note that for m = 0 (Haldane's model) the CI parameter Q does not influence the map function if 0 < Q ≤ 1 holds. This can be proved by solving (3) and (5) with {c_{i}(x)} following the Poisson distribution.
The distribution of recombination patterns for CI(Q): To find the theoretical distribution
{γ(i)} of the multilocus recombination pattern i = (i_{1}, i_{2}, · · ·, i_{r}) for the gametes, where i_{k} = 1 indicates a recombination and i_{k} = 0 indicates absence of recombination between markers k and k + 1 on a randomly chosen gamete, we can follow the Appendix of Zhao et al. (1995a). Besides y = (m + 1)x/Q, a difference is the chance of s chiasmata in an interval to produce a recombination, which is (1 – (1 – 2Q)^{s})/2 now, derived from (5). Thus we obtain
Definition of models CI(Q_{i}) and CI(η_{i}) for the investigation of heterogeneity of interference: Assume the frequent case that a model of a crossover formation process does not fit recombination data. Two reasons might explain this. The data set may be too small or unreliable, or the model might not be appropriate. For the latter case we have to check the assumptions. In all models created so far the real process is assumed to be homogeneous over the whole chromosome. This is not necessarily true. We therefore investigate heterogeneous processes. Our hypothesis is that when finding the same degree of interference for two adjacent or overlapping regions of a chromosome, the model is applicable if it fits both parts together. On the other hand, we cannot expect a model to fit data for a certain region if it already failed for a part of it, or if different parts of it show different characteristics of interference.
From a preliminary analysis we concluded that interference is likely to act heterogeneously. A generalization of the CI models to regard heterogeneous SI appears to be difficult. We propose the incorporation of CI heterogeneity. We therefore introduce CI parameters into the models that may vary between different parts of the chromosome. For the CI(Q) model, like the CI(η) model of Zhao and Speed (1998), we assign CI parameters Q_{i} and η_{i} to intervals i, which does not change the theory in any significant way. These models are denoted by CI(Q_{i}) and CI(η_{i}), respectively. However, the twolocus recombination fractions then are not only a function of the distance between the loci but of their locations. To ensure 0 ≤ Q_{i}, η_{i} ≤ 1 during numerical calculations, we fit auxiliary variables z_{i} and put Q_{i}, η_{i} = sin^{2}(z_{i}).
Given a recombination data set we apply the following procedure. First we fit Haldane's nointerference model, then a NCI model like the χ^{2}model of recombination for positive SI or alternatively the negative binomial map function for negative SI. To test CI vs. no interference, we use the model CI(η) with m = 0. Then the homogeneous CI models are applied. If they do not fit the data, the models for heterogeneous CI are used. If these models are significantly better than the homogeneous CI models, one source of the heterogeneity is proved. If the data still do not fit satisfactorily, heterogeneous SI appears to act and is subject to further investigations: We divide the chromosome and start with the analysis of all pairs of adjacent intervals. For each pair there are four possible recombination patterns, (0, 0), (1, 0), (0, 1), and (1, 1), to which we have to fit the observations. Since the sample size is known, three equations have to be solved. Practice shows that this can be realized by the two genetic distances to be estimated and by the introduction of one parameter to estimate SI. Following this, we analyze all triples of adjacent intervals, then the quadruples, etc.
The test criterion for the fit of the models to real data: Commonly, the multilocus concept of Weeks et al. (1993) is used to analyze recombination data. For a directly observable multilocus recombination pattern, the criterion of fit is the loglikelihood
Haldane's nointerference model is nested in the x^{2}model of recombination, representing the NCI model. The NCI model is nested in both models for homogeneous CI; i.e., a likelihoodratio test can be applied to test whether CI is acting or not. Also, the CI(Q) and CI(η) models are nested in the CI(Q_{i}) and CI(η_{i}) models, respectively, and we can test whether CI heterogeneity is significant. To compare two nested models, a likelihoodratio test is applied. Following the asymptotic theory, twice the difference of the two loglikelihoods of the models is χ^{2}distributed with the difference of the number of parameters being the degrees of freedom.
In the application of the models for heterogeneous CI, the intervalwise CI parameters Q̂_{i} and
We evaluate the quality of a fitted model by comparing its loglikelihood value with the ideal loglikelihood value theoretically achievable by an unknown model that fits the observed gamete distribution exactly; i.e., for this model, γ(i) = n(i)/N with N = R_{i} n(i) is valid. If twice the difference of the two loglikelihoods is smaller than the 1 – αquantile of the χ^{2}distribution with 1 d.f., which is 3.841 for α = 0.05, a model of a significantly better fit than that of the model analyzed does not exist. For this case we use the phrase that the model fits the data.
RESULTS
Application to the data of Morgan et al. (1935): The ninelocus Drosophila data of Morgan et al. (1935) have often been used to compare crossover formation models (see McPeek and Speed 1995). The data have not been fitted well by any model. Compared with the ideal loglikelihood of –36,899.5, we obtained ln L = –37,956.6 for Haldane's nointerference model, ln L = –36,987.1 for the NCI model (χ^{2}model of recombination with m̂ = 4; similar to the gamma model of McPeek and Speed 1995), and ln L = –37,122.1 for the CI(η) model under absence of SI (m = 0,
Then we analyzed the quadruples of intervals. The results are shown in the appendix, Table A2. For the first quadruple, termed the “fivelocus data of Morgan et al. (1935)” in the literature, and the second and fourth quadruple a highly significant improvement over the NCI model is evident. Deviating from the former conclusions, both CI models indicate a significant effect of dominating reciprocal chiasmata at quadruple (4, 5, 6, 7). However, from quadruples onward, no model fits the data. The analysis of the quintuples (Table A3) underlines the tendency of the telomeric part to carry an increased number of reciprocal chiasma pairs. The dominance of complementary chiasma pairs at the centromere region is still visible. The analysis of six neighboring intervals with the CI(Q) model indicates the same result. There, model CI(η) showed significance only at the telomeric part. With seven or eight intervals, no improvements over the NCI model are found.
We applied the CI(Q_{i}) and CI(η_{i}) models to the fourinterval case. The results compared with the eightinterval analysis are displayed in the appendix, Tables A4 and A5. A significant improvement over the models assuming homogeneous CI is observed.
Application to the data of Weinstein (1936): The sevenlocus Drosophila data of Weinstein (1936) have also often been analyzed (e.g., Zhaoet al. 1995a). All models applied here yield a significant gain in fit when compared to nested models. The loglikelihood for Haldane's model is –56,394.3 and that of the CI(η) model under absence of interference (m = 0,
Then we analyzed the quadruples of adjacent intervals. The results are shown in the appendix, Table A7. For the first and second quadruples we again found a highly significant improvement over the NCI model.
We applied the CI(Q_{i}) and CI(η_{i}) models to the fourinterval case and compared the results with the sixinterval analysis (Table A8). Some significant improvements over the models of homogeneous CI were found.
Application to the data of Blank et al. (1988): The recombination data of Blank et al. (1988) on chromosome 12 of mice have been used repeatedly for investigations on the phenomenon of interference (Weekset al. 1994; Lin and Speed 1996). The analysis of the whole data set led to results shown in Table 2. Furthermore, we obtained ln L = –564.67 for fitting Haldane's model and ln L = –547.63 for fitting CI(η) under absence of SI (m = 0,
For intervals 2–7 the NCI model already fits the data (m̂ = 29). We evaluated ln L = –430.03, which was highly significantly better than Haldane's model (ln L = –452.48). For comparison, the ideal loglikelihood was –428.96.
DISCUSSION
A longknown reason for the occurrence of chiasmata is to help organize meioses by fixing strands. On the other hand, it turned out that molecular strategies of inheritance play an important role in the biological process of evolution. It is not clear if or how chiasmata are involved in this process. Only trivial statements can be made, such as negative or absence of genetic interference increases genetic variability on the gametes more than high positive interference. The investigation of meiotic processes with particular sets of recombination data is only one but an important step in contributing knowledge in this area.
The NCI models routinely used in mapping today reflect only SI. We have proved that, as well as SI, CI and even interference heterogeneity may play an important role in meiosis. We found evidence for heterogeneous SI and CI. The SI heterogeneity differed among the three data sets considered. While we found positive, slightly varying intermediate SI for the data of Morgan et al. (1935) and Weinstein (1936), we found slightly negative or absent SI at the centromere region and high SI at the telomeric part of the chromosome for the data of Blank et al. (1988).
The results on CI also differ between data sets. We therefore conclude that no general rules for interference heterogeneity exist. Using the data of Morgan et al. (1935) and Weinstein (1936) we found an increased amount of complementary chiasmata at the centromere region. For the data of Morgan et al. (1935) we additionally found an increased amount of reciprocal chiasma pairs at the telomeric part of the chromosome. This is particularly evident from the results for the CI(Q) model and could result from sister strand chiasmata. Using the data of Blank et al. (1988) we proved an intensification of complementary chiasma pairs that are concentrated mainly in the telomeric part. The findings for the centromere weakly indicate either an increased amount of reciprocal chiasmata or simply negative SI.
One should note that Morgan et al. (1935) and Weinstein (1936) examined the same chromosome of the same species. It is of particular interest that the first five loci and the seventh locus were identical. Comparing Tables A1 and A6, A2 and A7, A4 and A8, and A5 and A8, we indeed observe a congruence of parameter estimates and interference behavior. Both models for CI behaved similarly. Therefore it is not clear whether CI works better at small or large distances. We note that numerically the CI(Q) model is about five times faster than the CI(η) model.
We have shown that chromatid and suppression interference are not completely separable. Especially, positive SI may partially be compensated by a greater amount of complementary chiasma pairs and negative SI by an enlarged amount of reciprocal chiasma pairs. For the data of Morgan et al. (1935) and Weinstein (1936), SI dominates CI, while for the data of Blank et al. (1988) both kinds of interference have the same strength. However, separation would be easier if we were dealing with data exhibiting obvious deviations from NCI, such as recombination fractions >0.5 or decreasing recombination fractions for increasing distances. Let us consider the data of Blank et al. (1988). The data do not cover the whole chromosome. The maximum recombination fraction is 0.498. The CI models fitted suggest that the recombination fractions would exceed 0.5 if the whole chromosome had been investigated. If this is not true, the gain in fit explained with CI is an artifact. Then we would have proved that the χ^{2}model of recombination is not appropriate to the data, but that another chiasma formation process is adequate, and may be a nonstationary renewal one. However, our experience with mouse markers (Brockmannet al. 1998) shows that recombination fractions can exceed 0.5.
From our results and from the indications found for CI acting in tetraploids, we conclude that chromatid interference is likely to act. We encourage geneticists to be aware of recombination fractions exceeding onehalf and that the phenomenon of interference heterogeneity should not be ruled out.
Acknowledgments
The authors are indebted to Dr. T. P. Speed, Phillip Huff, and two anonymous reviewers for numerous suggestions on an earlier version of the manuscript.
APPENDIX
Footnotes

Communicating editor: G. A. Churchill
 Received April 12, 1999.
 Accepted July 25, 2000.
 Copyright © 2000 by the Genetics Society of America