Two-Parameter Characterization of Chromosome-Scale Recombination Rate
The genome-wide recombination rate () of a species is often described by one parameter, the ratio between total genetic map length () and physical map length (), measured in centiMorgans per Megabase (cM/Mb). The value of this parameter varies greatly between species, but the cause for these differences is not entirely clear. A constraining factor of overall in a species, which may cause increased for smaller chromosomes, is the requirement of at least one chiasma per chromosome (or chromosome-arm) per meiosis. In the present study, we quantify the relative excess of recombination events on smaller chromosomes by a linear regression model, which relates the genetic length of chromosomes to their physical length. We find for several species that the two-parameter regression, provides a better characterization of the relationship between genetic and physical map length than the one-parameter regression that runs through the origin. A non-zero intercept () indicates a relative excess of recombination on smaller chromosomes in a genome. Given , the parameter predicts the increase of genetic map length over the increase of physical map length. The observed values of have a similar magnitude for diverse species, whereas varies by two orders of magnitude. The implications of this strategy for the genetic maps of human, mouse, rat, chicken, honeybee, worm and yeast are discussed.
The rate of meiotic recombination rate (), defined as the ratio between genetic and physical map length and measured in centiMorgan per Megabase (cM/Mb), is known to vary widely between the genomes of different species. As a rule of thumb for the human genome, 1cM genetic map length equals 1Mb physical map length, see e.g. (Collins and Morton 1998; Ulgen and Li 2005). This rate is about twice as large as the genome-wide observed in the mouse genome (Jense-Seaman et al. 2004), but far less than the of 340cM/Mb that is observed in the yeast genome (Mortimer et al. 1992; Baudat and Nicolas 1997). Understanding of these differences in between different species is of fundamental importance for evolutionary and medical genetics (Nachman 2002). In addition to these differences between species, it was also noted that differs between chromosomes within a species, with smaller chromosomes showing higher (Nachman and Churchill 1996; Broman et al. 1998; Lander et al. 2001; Venter et al. 2001; Kong et al. 2002; Matise et al. 2007). Therefore, species differences in genome-wide may be best studied under a model that also considers the intragenomic differences between chromosomes.
From a population genetic perspective, the main role of recombination is the production of new combinations of alleles by shuffling of parental haplotypes, which increases the efficiency of natural selection in theoretical and empirical model systems (Maynard-Smith 1978; Barton and Charlesworth 1998; Rice 2002; Otto and Lenormand 2002). Many recent empirical studies have addressed the question at which sites in a genome recombination is most likely to occur (Petes 2001; McVean et al. 2004; Hey 2004; Myers et al. 2005; Coop 2005; Mancera et al. 2008). In this context it was also found that evolves extremely fast on a kb-scale (Ptak et al. 2005; Winckler et al. 2005) and that historical recombination hotspots are associated with specific gene functions in human, which was hypothesized to indicate an influence of natural selection on hotspot locations (Freudenberg et al. 2007; The International HapMap Consortium 2007). When is examined at a megabase scale instead of a kilobase scale, the evolution of local is more constrained (Myers et al. 2005), and differs much less between closely related species, such as human and chimpanzees (Winckler et al. 2005; Ptak et al. 2005). However, the mechanism behind this conservation of on the larger scale is unclear. One contributing explanation could be the requirement of a minimal or fixed number of chiasmata per chromosome during meiosis to stabilize homologous chromosome pairs (Mather 1938).
The question how many chiasmata are exactly required per chromosome or per chromosome arm has not been resolved yet and might not have a generally valid answer (Lynn et al. 2004; Laurie and Hultén 1985). Nevertheless, these meiotic constraints can explain the excess of recombination on shorter chromosomes. Consistent with an influence of karyotype on overall recombination rate, a correlation was found between the number of chromosome arms in a genome and the genetic map length (De Villena and Sapienza 2001a). Altered recombination may lead to aneuploidy (Hassold and Hunt 2001; Lynn et al. 2004), which may impose strong selective constraints and explain the tight relationship between karyotype structure and recombination rate (Dumas and Britton-Davidian 2002; De Villena and Sapienza 2001b).
On the other hand, domesticated plants and animals show evidence for increased chiasma formation (Burt and Bell 1987), which suggests that there exist additional determinants of genome-wide than karyotype. For instance, the level of interference in chiasma formation could differ between species (Broman et al. 2002). Therefore, it would be useful to apply a formal method that separates the contribution of karyotype structure from the relationship between physical and genetic map length. This is not accomplished by the genome-wide cM/Mb ratio: although the cM/Mb ratio is a convenient single parameter measurement, it does not model the higher contribution of smaller chromosomes to the genome-wide of a species.
To address this problem and better understand the overall of a genome, we propose a novel strategy that explicitly models, if and to what extend, the overall in a genome is influenced by the relative excess of recombination on smaller chromosomes. This proposed two-parameter strategy takes into account that a certain minimal amount of recombination is required to maintain genome integrity during meiosis (Mather 1938) and that a genome therefore has minimal genetic map length. This idea becomes more clear, if we use a statistical regression framework to compare the proposed strategy with the one parameter strategy that is typically applied to shorter scales than the chromosome-scale. Since the one-parameter characterization of implies that genetic length is proportional to the physical length and recombination events occur independently on different chromosomes, the cM/Mb ratio is the slope of the linear regression of genetic lengths of chromosomes over their sequence lengths, with the requirement that the regression line goes through the origin. In our new approach, we drop the requirement that the regression line must go through origin by using two parameters to fit the genome-wide genetic map information at the chromosomal scale.
From a biological perspective, the one-parameter model considers the length of the genetic map of a genome to be determined by the length of the underlying physical map and the species-specific . Building on this, the two-parameter model also includes a separate effect of karyotype structure that may produce a disportional distribution of recombination events over chromosomes of different length. Under the two-parameter model, the value of the -intercept quantifies the relative excess of recombination events on a hypothetical chromosome with length zero, whereas the slope of the regression measures the increase of genetic with physical map length in the same way as the one-parameter model. Our results show that in human, as well as other species, the two-parameter regression provides a much better fit for describing the genetic map length of chromosomes.
A two-parameter regression model fits the genetic map length of human chromosomes better than the one-parameter model
To look for systematic differences in recombination rate between human chromosomes, we started by reproducing the Marey map (Chakravarti 1991; Rezvoy et al. 2007), a cumulative plot similar to those used in DNA sequence representation or analysis (Li 1997; Grigoriev 1998), for 22 human autosomes and 34 arms of metacentric chromosomes (Appendix Figure A1). The chromosome-scale or chromosome-arm scale recombination rate may be defined as the slope of a straight line that links the first and the last marker. For smaller chromosomes or chromosome arms, the end points in the Marey map tend to lie above the line with a slope equal to 1 (cM=Mb), i.e., smaller chromosomes(-arms) have larger cM/Mb ratios (see also Figure 16 of (Lander et al. 2001) and Table 12 of (Venter et al. 2001)).
We next regressed the genetic map length of chromosomes over their physical map length (Figure 1(A), similar plot can be found in (Housworth and Stahl 2003)). When sex-averaged, female and male genetic lengths are fitted separately, the three regression lines are described by:
The normality assumption of regression residuals was tested graphically by a QQ-plot (Appendix Figure A2), and the normality condition does not seem to be violated.
Equation (A two-parameter regression model fits the genetic map length of human chromosomes better than the one-parameter model) shows that the -intercept for female data is 29% larger than for male data, whereas the slope is 92% larger. Thus, the different length of the male and female map mainly manifests as a different slope and less so as a different -intercept. As can be seen from Figure 1(A), all human chromosomes exceed the minimal length of 50cM both for the male and the female genetic map.
To test the robustness of the -intercept value, we added random noise to the genetic map length and repeated the regression analysis. The histogram of 50000 -intercepts from this procedure is shown in Appendix Figure A3. Although values of range from 35 to 60, they are all far from zero.
We next repeated the analysis using chromosome arms instead of full chromosome as separate data points (Figure 1(B)). This leads to the regression equations:
The intercepts at chromosome arm-scale regression is now reduced to somewhat more than half of the intercept at the full chromosome scale. This reduction shows that cytogenetic constraints exert a smaller influence on the chromosome-arm scale than on the full chromosome scale.
Several methods can be used to show that the two-parameter regression model fits the data better than the one-parameter regressions. To this end, we first compared the coefficient of determination which is the proportion of variability explained by the regression model. The observed values of the one-parameter regression range between 0.48 and 0.87, whereas the values of the two-parameter regressions range between 0.86 and 0.98 (Table 1), indicating that the two-parameter regression explains more of the variability in the data.
We further cast the comparison between the one- and two-parameter regression as a model selection problem. Two such model comparison strategies are provided by the Akaike information criterion (AIC) (Akaike 1974) and Bayesian information criterion (BIC) (Schwarz 1978). Both AIC and BIC values for the two-parameter regression model are smaller than those for the one-parameter model, indicating a better statistical model (Appendix Table A1).
Finally, we tested the null hypothesis that is zero. The -values in this test range between and (Table 1). Because the null hypothesis is that all chromosomes have the same RR, a simulated distribution of can be obtained from the regression over data that are obtained by the permuting of chromosome-specific cM/Mb ratios, while leaving the physical chromosome length unchanged. Out of 50000 such permutations, only two showed a value that is larger than the observed value of 48.1 (for sex-averaged full chromosome data), corresponding to a -value of . To summarize, all evaluation methods support the conclusion that the two-parameter regression model is better than the one-parameter model.
|human chromosome, sex-averaged||0.976||0.817||3.1|
|human chromosome arm, sex-averaged||0.955||0.775||9.3|
As the deCode data were published more than six years ago, we further tested the chromosome-scale regression strategy on a more recent dataset, the Rutgers Map v.2 (Matise et al. 2007). The regression lines are (sex-average), (female), and (male), respectively. These results are consistent with the parameter estimations in Eq.(A two-parameter regression model fits the genetic map length of human chromosomes better than the one-parameter model), again showing that male and female data differ more in the slope than in the -intercept.
Two other quantities can be derived from that help to interpret the -intercept parameter. The first is the physical length on the regression line that corresponds to a specified minimum genetic length such that . If we set 50cM, then we obtain 2.45Mb for sex-averaged chromosome data. One may assume that for any hypothetical chromosome with , its genetic length remains constant at 50cM and does not decrease for shorter chromosome length. As the second quantity of interest, we define the percentage of genetic length that is explained by the inclusion of into the model as: . For the sex-averaged data, we find that of variability is explained by the -intercept. This value can also be obtained from the decomposition of : 1.13= 0.35 + 0.78, because 0.35/1.13 = 31%. The relatively large percentage value once again highlights the importance of the -intercept for modeling chromosome-scale recombination rate in human.
Different intercept but similar slope in the two-parameter regression models for rat and mouse chromosomes
Both rat (Rattus norvegicus) and mouse (Mus musculus) genome are known to have lower recombination rates than human (Jense-Seaman et al. 2004), with rat having a higher overall than mouse. The rat genome has a roughly equal physical map length, but contains one more chromosome (n=20) than the mouse genome (n=19). Furthermore, rat chromosomes show a greater heterogeneity in their physical length and one may hypothesize that these karyotype differences contribute to the somewhat higher in rat (0.62 cM/Mb vs. 0.57 cM/Mb in mouse). The regression models of the sex-averaged genetic length of rat and mouse chromosomes over their sequence lengths (Figure 2(A)) are:
These models display a similar slope and the different overall of rat and mouse mainly manifests as a different intercept value .
Testing for the rat genome is significant (-value= 0.0012), whereas testing for mouse genome (the fitted value for mouse is 69% of that for rat) is not significant (-value = 0.11). AIC/BIC calculation confirms that the two-parameter regression is a convincingly better model for rat than the one-parameter regression, whereas this barely holds for the mouse data (Appendix Table A1). Thus, the mouse genome displays a non-significant excess of recombination on smaller chromosomes, which is consistent with the smaller variation of chromosome size in the mouse genome. The greater -intercept for the rat genome supports the hypothesis that cytogenetic factors contribute more to the genetic map length of rat than mouse.
Because the rat karyotype consists of both meta- and acrocentric chromosomes, we repeated the analysis after splitting all metacentric rat chromosome into two parts, based on the location of the centromere. Different from the human genome, for the rat genome this mainly affects the smaller chromosomes, which are often metacentric. The regression line is now described by and testing the intercept is still significant (-value=0.024), though at a less stringent level. Thus, at the scale of chromosome-arms, the likelihood of crossovers in an interval is more determined by its physical length and less by influenced by any obligate recombination requirements.
Recombination rate of small and large chromosomes in the chicken genome
The chicken (Gallus gallus) genome consists of both large (macro-) and small (micro-) chromosomes (Hillier et al. 2004; Smith et al. 2000), with length ranging from a few Mb to close to 200Mb. The two-parameter regression model for the chicken genetic data in Figure 2(B) leads to:
In this regression model, the non-zero intercept is significant with a a -value of and there is a considerable difference of AIC/BIC for the one- and two-parameter regression favoring the two-parameter model (Appendix Table A1). Both coefficients of determination for the 2- and 1-parameter regressions attain a high value: 0.98 and 0.93 respectively. A reason that the 1-parameter regression only marginally reduces the value is given by the fact that larger chromosomes contribute much more to the total variance, which is equally well captured by the 1-parameter model. Thus, two of the three methods confirm a relative excess of recombination on short chromosomes.
However, the orders of magnitude difference between the size of chicken chromosomes raises the question of the robustness of the regression. From (Hillier et al. 2004) and Appendix Figure A4, it is clear that the genetic length reaches a plateau at the level of 50cM for microchromosomes smaller than 8Mb. When chromosomes below a certain length threshold are discarded from the regression analysis, the -intercept value changes slightly, but not dramatically. For example, if the length thresholds for removal are 8Mb and 25Mb, for Eq.(4) decreases to 32.95 and 31.88. When the regression model is only fitted to the five largest chromosomes (longer than 50Mb), the model parameters are . On the other hand, if we remove the largest five chromosomes, the regression line is .
To see how the quality of the map distance measurements may influence these results, we next looked at the recently updated chicken map (Groenen et al. 2009), which contains more genetic markers and higher marker density. Applying the two-parameter regression leads to
As can be seen, the overall reduction of as compared to the older map (Hillier et al. 2004) mainly manifests as reduced estimate of , whereas the estimate of remains almost unchanged.
Exceptionally high recombination on the largest honey bee chromosome leads to a better fit of the one-parameter than the two-parameter model
Notably, the two-parameter regression does not provide a better fit for the genetic map data from honey-bee(Apis mellifera) (Beye et al. 2006) than the one-parameter model. When plotting the genetic length over physical length (Figure 2(C)), the -intercept of the regression line does not significantly differ from zero (-value ):
The coefficient of determination for both the two- and one-parameter regression is around 0.95. In contrast to other genomes, AIC/BIC analysis favors the one-parameter regression model (Appendix Table A1).
As can be seen from Figure 2(C), the longest chromosome (chromosome 1) is four times the length of the shortest chromosome, and the regression result may depend on the presence of this “outlier”. To check this possibility, we repeated the analysis after chromosome 1 was removed, which led to the regression equation: . However, also in this model, testing is not significant (-value=0.37), both the two- and one-parameter regressions exhibit similar coefficient of determination ( 0.80, 0.79), and the zero-intercept regression is still the better model according to AIC/BIC analysis (Appendix Table A1). Therefore, different from other species, the honey-bee genome does not display any significant excess of recombination on smaller chromosomes.
Two-parameter regression at much shorter length scales: the example of budding yeast
Yeast (S. cerevisiae) has been extensively used to study the molecular machinery of recombination and it has a much smaller ( 12Mb) and more compact genome (Cherry et al. 1997). Although the physical length of yeast chromosomes only ranges from 200kb to 1.5Mb, their genetic length is between 100 and 500cM, even longer than the genetic length of human chromosomes. The best fitting regression line for the yeast genetic map is (Figure 2(D)):
The non-zero -intercept is significant (-value = 0.009). The two-parameter regression is superior to the one-parameter model as judged by AIC/BIC (Appendix Table A1). The value of -intercept, 49.12 cM is very close to 50cM which corresponds to almost one crossing over event for a hypothetical chromosome of physical length of zero.
The extremely high recombination rate in the yeast genome is surprising. From the molecular perspective, one can speculate about various hypotheses, such as a different meiotic regulatory system which makes a denser spatial distribution of chiasmata possible, a lack of secondary chromatin structure as compared to higher organisms so that the actual physical distance between two locations on chromosome is more or less equal to the linear sequence distance, or the lack of other supporting mechanism to hold chromatids together so that more chiasmata per chromosome arm are required for proper chromosome segregation. On the other hand, the -intercept of the regression has a similar magnitude as that observed for higher organisms, indicating a similar relative excess of recombination on smaller chromosomes.
The difference between the central gene cluster and telomeric regions in worm genome is due to a difference in
Finally, we used genetic map data from the worm C. elegans to show that the two-parameter regression strategy can also be useful to compare different regions within a genome. The chromosomes of C. elegans are unusual, because discrete centromeres are missing and the chromosomes are holocentric, i.e. microtubules attach at many sites for chromatid segregation (Tyler-Smith and Floridia 2000). Accordingly, the Marey map analysis of the worm genome indicates that each worm chromosome can be partitioned in three regions: the central gene-rich region with a low recombination rate and two distal telomeric regions with high recombination rates (Barnes et al. 1995). Therefore, we separately performed the regression analysis of genetic length over physical length for these two types of regions (Figure 3). The fitted regression coefficients are:
Within the single parameter framework without the intercept term , the two types of regions would have a very different cM/Mb ratio: 4.57 for telomeric regions, 0.68 for central regions. However, when allowing non-zero value, the two regions display similar slope values, 1.01 and 0.94. This indicates a constant excess of recombination in the distal region as compared to the central region in C. elegans, which is combined with a similar incremental cM/Mb ratio. Thus, after accounting for a fixed amount of recombination in a distal chromosome region, the likelihood of any additional recombination depends in similar strength on physical length in distal regions and central regions.
As a note of caution, one may point out that the regression coefficient in Eq.(The difference between the central gene cluster and telomeric regions in worm genome is due to a difference in ) is obtained from only a few data points. Nevertheless, further regression diagnostics supports our conclusion. For example, testing is significant for the distal regions (-value= 0.006), but not significant for central regions (-value=0.57). AIC/BIC analyses lead to the same conclusion (Appendix Table A1).
Discussion and Conclusions
Our results show that instead of the simpler genetic-to-physical length ratio, the relationship between the physical and genetic map length at chromosome scale is better described by a statistical model that contains a second parameter , which is the -intercept of the regression of genetic map length over the physical chromosome length. A conceptually similar approach was used earlier in measuring the genome-wide recombination rate of a species by counting the chiasmata on each chromosome in excess of one (Burt and Bell 1987).
The consideration of this intercept parameter is important, because karyotype structure has been established as an important determinant of genome-wide (De Villena and Sapienza 2001a; Coop 2005) and smaller chromosomes display higher (Lander et al. 2001; Hillier et al. 2004). Our proposed two-parameter model provides a formal expression of this size dependency of : , i.e., a constant term plus a second term that increases for smaller chromosome sizes (if is positive). This is what we observe for human, mouse, rat, chicken and yeast genomes. When writing as , the -intercept measures the amount of recombination after the physical map length has been accounted for. Therefore, one would expect that the total map length of a chromosome increases by after splitting it up into two separate parts. In fact, this has already been quantitatively observed for the experimental alteration of yeast chromosome I (Kaback et al. 1992).
When comparing between species, the usage of instead of the genome-wide cM/Mb ratio will reduce the influence of karyotype differences on the result. This was also the intention behind the counting of chiasmata per chromosome in excess of one (Burt and Bell 1987). In our study, the order of species remains unchanged, whether ranked by or by cM/Mb ratio. However, due to the different values of , we cannot use a single regression line to model the genetic-physical length relationship across species. Thus, a molecular mechanism must exist that drives, within a particular species, the proportional increase of genetic over physical map length. This mechanism might typically act with weaker strength in larger genomes, which could contribute to the inverse correlation between genome size and (Lynch 2006).
Among mammals it was furthermore found that is more similar for more closely related species (Dumont and Payseur 2008), which could be partly due to their similar karyotype. It might be interesting to test where in the phylogenetic tree the signal might be altered, when using instead of the genome-wide cM/Mb ratio. In this context, it is also important that genome-wide typically differs between genders and individuals (Broman et al. 1998; Kong et al. 2002, 2004; Cheung et al. 2007; Petkov et al. 2007). The biological factors that were invoked as possible explanations, such as differences in synaptonemal complex formation or crossover interference, may be more plastic than karyotype structure. The respective strength of these factors could also contribute to species differences and may be better measured by using than by using the genome-wide cM/Mb ratio.
If would equal to zero with the obligate chiasma requirement holding true, then were required to be 50cM. This pattern can be observed for female Opossum (Monodelphis domestica), where each chromosome acquires exactly one crossover near one of its telomeres (see (Mikkelsen et al. 2007) and Appendix Figure A5). Similarly, very small chromosomes may always acquire exactly one crossover, despite reduced chromosome size, as seen for the microchromosomes in the chicken genome. In order to predict the transition from this plateau to the linear regression, we derived the minimum physical length parameter from a given and the estimated regression parameters. Note that if both physical and genetic length are measured as those in excess of and , their ratio is exactly equal to : .
Because reduced recombination may result in aneuploidy of smaller chromosomes (Warren et al. 1987; Brown et al. 2000), it is conceivable that the length of smaller chromosomes could influence genome-wide by introducing a lower bound for the propensity for chiasma formation in a species. Our analysis supports the size of the smaller chromosomes as a strong determinant of genome-wide for the six genomes studied in this paper (Appendix Figure A6). In log-log scale, the correlation coefficient between and the shortest chromosome length is (-value= 0.008). If the recombination rate is measured by , in log-log scale the correlation coefficient is (-value=0.01). This correlation is nearly as strong as the reported correlation between and the total physical length for over 100 genomes (cc= , -value= 0.0003 on log-log scale) as reported in (Lynch 2006). Obviously, data on more species are needed for a more conclusive analysis. Nevertheless, it may be interesting to point out that the genome with the lowest known recombination rate, Opossum, lacks any short chromosome (Mikkelsen et al. 2007; Samollow et al. 2007).
Obviously, any genome-wide analysis relies on the availability of high quality data. We are convinced that the data used in this study are of sufficient quality to study recombination on the chromosomal scale. However, some error might be introduced by the fact that the used genetic maps are not perfect and, in particular for telomeres, missing some data. This can be seen for the chicken genomes, where the two chromosomes fall below the minimum genetic length of 50cM in the older map and climb to about 50cM in the newer map (Appendix Figure A4). Data selectively missing crossovers at the telomeres might lead to an underestimation of in the regression model.
We note that we restricted our analysis to chromosomes or chromosome-arms. If the genetic length is regressed over the length of much smaller regions, the coefficient of determination is expected to be much lower due to a mixture of recombination hot- and cold- spots. From a biological perspective, we also would not expect a positive value in such a regression, because there is no requirement for a Mb-sized region to have at least one chiasma to maintain meiotic integrity.
In summary, we find that the introduction of the parameter helps to understand the recombination rate differences between species, because it separates the effect of the requirement for at least one chiasma formation on smaller chromosomes from the factors that determine the amount of recombination on larger chromosomes. More specifically, the partitioning of chromosome-scale recombination rate leads to the following list of conclusions: i) human male-female differences disproportionally affect larger chromosomes; ii) the higher recombination rate in the rat genome as compared to the mouse genome is likely to be caused by the higher number of smaller chromosomes that constitute the rat karyotype; iii) both chicken micro- and macro-chromosomes display a high and the extraordinarily high of some micro-chromosomes does not lead to an extraordinary excess of recombination on smaller chromosomes; iv) the honey-bee genome does not display any significant excess of recombination on smaller chromosomes; v) yeast displays a relative excess of recombination on smaller chromosomes that is similar to higher organisms, despite its outstandingly high overall recombination rate; vi) recombination of the worm genome mainly occurs in telomeric regions and given one recombination per chromosome arm, the likelihood of a second recombination is determined by physical map length. These examples demonstrate that the proposed statistical framework allows to pinpoint differences of genomic recombination rate, which should be useful for the further study of genome-wide recombination rate as a quantitative trait of fundamental importance.
Materials and methods
Genetic map data
The human genetic map was obtained from (Kong et al. 2002) that uses
5136 microsatellite markers with 1257 meiotic events,
and is estimated from pedigree data
(Supplementary Table E:
http://www.nature.com/ng/journal/v31/n3/suppinfo/ng917_S1.html). The rat (Rattus norvegicus) and mouse (Mus musculus) genetic map data were obtained from Table 1 of (Jense-Seaman et al. 2004), based on 2305 markers in rat and 4880 markers in mouse. The two chicken (Gallus gallus) genetic maps were obtained from Supplementary Table S2 of (Hillier et al. 2004), which is built from 1471 markers, and from Table 1 of (Groenen et al. 2009) built from 9258 markers. The honey bee (Apis mellifera) genetic map was obtained from Table 2 of (Beye et al. 2006) based on 1500 markers. The budding yeast (Saccharomyces cerevisiae) genetic map was downloaded from
http://downloads.yeastgenome.org/chromosomal_feature/SGD_features.tab. The worm (Caenorhabditis elegans physical and genetic lengths of central “gene clusters” and distal “arms” were obtained from Table 1 of (Barnes et al. 1995) based on 168 markers.
Measuring how good a linear regression is by coefficient of determination
Regression analyses were carried out by the subroutine in statistical package. For genetic lengths, (, e.g., for the chromosome-scale regression and for the chromosome-arm-scale regression), one can regress them over sequence lengths () allowing -intercept (non-zero when approaches 0):
or, without the -intercept ( approach 0 as approaches 0):
How good a linear regression model fits the data can be measured by the coefficient of determination , which is the proportion of variability that is explained by the model. More specifically, if is the total sum of squares of the genetic lengths of chromosomes, the term for allowing non-zero -intercept, or the term, for not allowing -intercept, is the residual sum of squares (RSS), then
Model selection by Akaike information criterion
Akaike information criterion (AIC)(Akaike 1974) of a statistical model is defined as where is the number of parameters in the model, and is the maximum likelihood estimated from the data. Similarly, Bayesian information criterion (BIC)(Schwarz 1978) is defined as , where is the number of samples used to calculate the likelihood. For linear regression, AIC/BIC is related to the residual sum of squares (RSS) according to (Venables and Ripley 1999) by:
where is the number of sample points for the regression analysis. Between two statistical models that are fitted to the same dataset, the model with a smaller AIC/BIC value is considered to be better than the model with a larger AIC/BIC value.
For the comparison between the two- and one-parameter regressions, we have:
If the second term, , is larger than two (for AIC, or for BIC), then the two-parameter regression can be seen as the better model than the single-parameter regression.
Quantities derived from
The linear relationship between and cannot extend to the physical length of zero, if the -intercept is greater than zero and the obligate chiasma requirement holds. Therefore, a point must exist below which genetic map length remains constant at , independent from the actual physical map length of a chromosome. We can define this transition point as follows: is the physical length for which the regression line crosses the horizontal line defined by the minimum genetic length , thus .
Another derived quantity is the genome-wide percentage of genetic length that is explained by . For a single chromosome (), this percentage is . For the whole genome, it is , where is the number of chromosomes. This definition of is valid only when the -intercept is positive ( ).
We thank Alejandro Morales for participating in the initial stage of this work, Tara Matise, Zhiliang Hu, Hong Ma, Oliver Clay for discussions and the anonymous reviewers for their valuable comments and suggestions. J.F. was supported by a NARSAD Young Investigator award.
1 Marey Map for human chromosomes (Fig.A1)
2 Checking the normality assumption of the regression (Fig.A2)
3 Testing the robustness of regression result by adding random noise to the genetic map (Fig.A3)
4 AIC and BIC of two-parameter regression models vs. one-parameter models (Table A1)
|source of genetic length data||AIC= AICAIC||BIC= BICBIC|
|human chromosome, sex-averaged|
|human ch. arm, sex-averaged|
|rat, chromosome, sex-averaged|
|mouse, chromosome, sex-averaged|
|chicken, chromosome, sex-averaged|
|honeybee, chromosome, sex-averaged||1.9||2.7|
|remove chromosome 1||1.0||1.7|
|yeast, chromosome, sex-averaged|
5 Chicken genetic length vs. physical length in log-log scale (Fig.A4)
6 Two-parameter regression model of Opossum chromosome genetic length (Fig.A5)
7 Recombination rates of six genomes as a function of the smallest chromosome size (Fig.A6)
- Akaike (1974) Akaike H. 1974. A new look at statistical model identification. IEEE Trans Automatic Control 19:716-722.
- Barnes et al. (1995) Barnes TM, Kohara Y, Coulson A, Hekimi S. 1995. Meiotic recombination, noncoding DNA and genomic organization in Caenorhabditis elegans. Genetics, 141:159-179.
- Barton and Charlesworth (1998) Barton NH and Charlesworth B. 1998. Why sex and recombination? Science 281:1986-1990.
- Baudat and Nicolas (1997) Baudat F and Nicolas A. 1997. Clustering of meiotic double-strand breaks on yeast chromosome III. Proc Natl Acad Sci 94:5213-5218.
- Beye et al. (2006) Beye M, Gattermeier I, Hasselmann M, Gempe T, Schioett M, Baines JF, Schlipalius D, Mougel F, Emore C, Rueppell O, et al. 2006. Exceptionally high levels of recombination across the honey bee genome. Genome Res 16:1339-1344.
- Broman et al. (1998) Broman KW, Murray JC, Sheffield VC, White RL, Weber JL. 1998. Comprehensive human genetic maps: individual and sex-specific variation in recombination. Am J Hum Genet 63:861-869.
- Broman et al. (2002) Broman KW, Rowe LB, Churchill GA, Paigen K. 2002. Crossover interference in the mouse. Genetics 160:1123-1131
- Brown et al. (2000) Brown AS, Feingold E, Broman KW, Sherman SL. 2000. Genome-wide variation in recombination in female meiosis: a risk factor for non-disjunction of chromosome 21. Hum Mol Genet 9:515-523.
- Burt and Bell (1987) Burt A and Bell G. 1987. Mammalian chiasma frequencies as a test of two theories of recombinations. Nature 326:803-805.
- Chakravarti (1991) Chakravarti A. 1991. A graphical representation of genetic and physical maps: the Marey map. Genomics 11:219-222.
- Cherry et al. (1997) Cherry JM, Ball C, Weng S, Juvik G, Schmit R, Adler C, Dunn B, Dwight S, Riles L, Mortimer RK. 1997. Genetic and physical maps of Saccharomyces cerevisiae. Nature 387:67-74.
- Cheung et al. (2007) Cheung VG, Burdick JT, Hirschmann D, Morley M. 2007. Polymorphic variation in human meiotic recombination. Am J Hum Genet 80:526-30.
- Collins and Morton (1998) Collins A and Morton NE. 1998. Mapping a disease locus by allelic-association. Proc Natl Acad Sci 95:1741-1745.
- Coop (2005) Coop G. 2005. Can a genome change its(hot)spots? Trends Ecol Evol 20:643-645.
- De Villena and Sapienza (2001a) De Villena FPM and Sapienza C. 2001a. Recombination is proportional to the number of chromosome arms in mammals. Mamm Genome 12:318-322.
- De Villena and Sapienza (2001b) De Villena FPM and Sapienza C. 2001b. Female meiosis drives karyotypic evolution in mammals. Genetics 159:1179-1189.
- Dumas and Britton-Davidian (2002) Dumas D and Britton-Davidian J. 2002. Chromosomal rearrangements and evolution of recombination: comparison of chiasma distribution patterns in standard and robertsonian populations of the house mouse. Genetics 162:1355-1366.
- Dumont and Payseur (2008) Dumont BL and Payseur BA. 2008. Evolution of the genomic rate of recombination in mammals. Evolution 62:276-94.
- Freudenberg et al. (2007) Freudenberg J, Fu YH, Ptácek LJ. 2007. Enrichment of HapMap recombination hotspot predictions around human nervous system genes: evidence for positive selection? Eur J Hum Genet 15:1071-1078.
- Grigoriev (1998) Grigoriev A. 1998. Analyzing genomes with cumulative skew diagrams. Nucl Acids Res 26:2286-2290.
- Groenen et al. (2009) Groenen MAM, Wahlberg P, Foglio M, Cheng HH, Megens HJ, Crooijmans R, Besnier F, Lathrop M, Muir WM, Wong GKS, et al. 2009. A high density SNP based linkage map of the chicken genome reveals sequence features correlated with recombination rate. Genome Res 19:510-519.
- Hassold and Hunt (2001) Hassold T and Hunt P. 2001. To err(meiotically) is human: the genesis of human aneuploidy. Nature Rev Genet 2:280-291.
- Hey (2004) Hey J. 2004. What’s so hot about recombination hotspots? PLoS Biol 2:e190.
- Hillier et al. (2004) Hillier LW, Miller W, Birney E, Warren W, Hardison RC, Ponting CP, Bork P, Burt DW, Groenen MAM, Delany ME, et al., International Chicken Genome Sequencing Consortium. 2004. Sequence and comparative analysis of the chicken genome provide unique perspectives on vertebrate evolution. Nature 432:695-716.
- Housworth and Stahl (2003) Housworth E and Stahl FW. 2003. Crossover interference in humans. Am J Hum Genet 73:188-197.
- The International HapMap Consortium (2007) The International HapMap Consortium. 2007. A second generation human haplotype map of over 3.1 million SNPs. Nature 449:851-861.
- Jense-Seaman et al. (2004) Jensen-Seaman MI, Furey TS, Payseur BA, Lu Y, Roskin KM, Chen CF, Thomas MA, Haussler D, Jacob HJ. 2004. Comparative recombination rates in the rat, mouse, and human genomes. Genome Res 14:528-538.
- Kaback et al. (1992) Kaback DB, Guacci V, Barber D, Mahon JW. 1992. Chromosome size-dependent control of meiotic recombination. Science 256:228-232.
- Kong et al. (2004) Kong A, Barnard J, Gudbjartsson DF, Thorleifsson G, Jonsdottir G, Sigurdardottir S, Richardsson B, Jonsdottir J, Thorgeirsson T, Frigge ML, et al. 2004. Recombination rate and reproductive success in humans. Nat Genet 36:1203-1206
- Kong et al. (2002) Kong A, Gudbjartsson DF, Sainz J, Jonsdottir GM, Gudjonsson SA, Richardsson B, Sigurdardottir S, Barnard J, Hallbeck B, Masson G, et al. 2002. A high-resolution recombination map of the human genome. Nature Genet 31:241-247.
- Lander et al. (2001) Lander ES, Linton LM, Birren B, Nusbaum C, Zody MC, Baldwin J, Devon K, Dewar K, Doyle M, FitzHugh W, et al., International Human Genome Sequencing Consortium. 2001. Initial sequencing and analysis of the human genome. Nature 409:860-921.
- Laurie and Hultén (1985) Laurie DA and Hultén MA. 1985. Further studies on chiasma distribution and interference in the human male. Ann Hum Genet 49:203-214.
- Li (1997) Li W. 1997. The study of correlation structures of DNA sequences: a critical review. Comp Chem 21:257-271.
- Lynch (2006) Lynch M. 2006. The origins of eukaryotic gene structure. Mol Biol Evol 23:450-468.
- Lynn et al. (2004) Lynn A, Ashley T, Hassold T. 2004. Variation in human meiotic recombination. Ann Rev Genomics Hum Genet 5:317-349.
- McVean et al. (2004) McVean GA, Myers SR, Hunt S, Deloukas P, Bentley DR, Donnelly P. 2004. The fine-scale structure of recombination rate variation in the human genome. Science 304:581-584
- Mancera et al. (2008) Mancera E, Bourgon R, Brozzi A, Huber W, Steinmetz LM. 2008. High-resolution mapping of meiotic crossovers and non-crossovers in yeast. Nature 454:479-485.
- Mather (1938) Mather K. 1938. Crossing-over. Biol Rev 13:258-292.
- Matise et al. (2007) Matise TC, Chen F, Chen W, De La Vega FM, Hansen M, He C, Hyland FCL, Kennedy GC, Kong X, Murray SS, et al. 2007. A second-generation combined linkage – physical map of the human genome. Genome Res 17:1783-1786.
- Maynard-Smith (1978) Maynard-Smith J. 1978. The Evolution of Sex, Cambridge University Press, Cambridge, UK.
- Mikkelsen et al. (2007) Mikkelsen TS, Wakefield MK, Aken B, Amemiya CT, Chang JL, Duke S, Garber M, Gentles AJ, Goodstadt L, Heger A, et al. 2007. Genome of the marsupial Monodelphis domestica reveals innovation in non-coding sequences. Nature 447:167-177.
- Mortimer et al. (1992) Mortimer RK, Contopoulou CR, King JS. 1992. Genetic and physical maps of Saccharomyces cerevisiae, edition 11. Yeast 8:817-902
- Myers et al. (2005) Myers S, Bottolo L, Freeman C, McVean G, Donnelly P. 2005. A fine-scale map of recombination rates and hotspots across the human genome. Science 310:321-324.
- Nachman (2002) Nachman MW. 2002. Variation in recombination rate across the genome: evidence and implications. Curr Opin Genet Devel 12:657-663.
- Nachman and Churchill (1996) Nachman MW and Churchill GA. 1996. Heterogeneity in rates of recombination across the mouse genome. Genetics 142:537-548.
- Otto and Lenormand (2002) Otto SP and Lenormand T. 2002. Resolving the paradox of sex and recombination. Nature Rev Genet 3:252-261.
- Petes (2001) Petes TD. 2001. Meiotic recombination hot spots and cold spots. Nature Rev Genet 2:360-309.
- Petkov et al. (2007) Petkov PM, Broman KW, Szatkiewicz JP, Paigen K. 2007. Crossover interference underlies sex differences in recombination rates. Trends Genet 23:539-542.
- Ptak et al. (2005) Ptak SE, Hinds DA, Koehler K, Nickel B, Patil N, Ballinger DG, Przeworski M, Frazer KA, Pääbo S. 2005. Fine-scale recombination patterns differ between chimpanzees and humans. Nature Genet 37:429-434.
- Rezvoy et al. (2007) Rezvoy C, Charif D, Guéguen L, Marais GAB. 2007. MareyMap: an R-based tool with graphical interface for estimating recombination rates. Bioinformatics 23:2188-2189.
- Rice (2002) Rice WR. 2002. Experimental tests of the adaptive significance of sexual recombination. Nature Rev Genet 3:241-251.
- Samollow et al. (2007) Samollow PB, Gouin N, Miethke P, Mahaney SM, Kenney M, VandeBerg JL, Graves JAM, Kammerer CM. 2007. A microsatellite-based, physically anchored linkage map for the gray, short-tailed Opossum (Monodelphis domestica). Chrom Res 15:269-282.
- Schwarz (1978) Schwarz GE. 1978. Estimating the dimension of a model. Ann Stat 6:461-464.
- Smith et al. (2000) Smith J, Bruley CK, Paton IR, Dunn I, Jones CT, Windsor D, Morrice DR, Law AS, Masabanda J, Sazanov A, et al. 2000. Differences in gene density on chicken macrochromosomes and microchromosomes. Anim Genet 31:96-103.
- Tyler-Smith and Floridia (2000) Tyler-Smith C and Floridia G. 2000. Many paths to the top of the mountain: diverse evolutionary solutions to centromere structure. Cell 102:5-8.
- Ulgen and Li (2005) Ulgen A and Li W. 2005. Comparing single-nucleotide-polymorphism marker-based and microsatellite marker-based linkage analyses. BMC Genet 6:S13.
- Venables and Ripley (1999) Venables WN and Ripley BD. 1999. Modern Applied Statistics with S-PLUS, 3rd edition, Springer-Verlag, New York.
- Venter et al. (2001) Venter JC, Adams MD, Myers EW, Li PW, Mural RJ, Sutton GG, Smith HO, Yandell M, Evans CA, Holt, RA, et al. 2001. The sequence of the human genome. Science 291:1304-1351.
- Warren et al. (1987) Warren AC, Chakravarti A, Wong C, Slaugenhaupt SA, Halloran SL, Watkins PC, Metaxotou C, Antonarakis SE. 1987. Evidence for reduced recombination on the nondisjoined chromosomes 21 in Down syndrome. Science 237:652-654.
- Winckler et al. (2005) Winckler W, Myers SR, Richter DJ, Onofrio RC, Gabriel SB, Reich D, Donnelly P, Altschuler D. 2005. Comparison of fine-scale recombination rates in humans and chimpanzees. Science 308:107-111.