Predicting recombination frequency from map distance

Map distance is one of the key measures in genetics and indicates the expected number of crossovers between two loci. Map distance is estimated from the observed recombination frequency using mapping functions, the most widely used of those, Haldane and Kosambi, being developed at the time when the number of markers was low and unobserved crossovers had a substantial effect on the recombination fractions. In contemporary high-density marker data, the probability of multiple crossovers between adjacent loci is negligible and different mapping functions yield the same result, that is, the recombination frequency between adjacent loci is equal to the map distance in Morgans. However, high-density linkage maps contain an interpretation problem: the map distance over a long interval is additive and its association with recombination frequency is not defined. Here, we demonstrate with high-density linkage maps from humans and stickleback fishes that the inverses of Haldane’s and Kosambi’s mapping functions systematically underpredict recombination frequencies from map distance. To remedy this, we formulate a piecewise function that yields more accurate predictions of recombination frequency from map distance. Our results demonstrate that the association between map distance and recombination frequency is context-dependent and without a universal solution.Subject terms: Genetic linkage study, Evolutionary genetics     

A mathematically tractable family of genetic mapping functions with different amounts of interference

By extension of the argument of Kosambi (1944), a family of mapping functions can be derived, which has a parameter regulating the intensity of interference. Different values of this parameter yield the Haldane (1919) and Kosambi mapping functions as special cases. The parameter is the coincidence coefficient for nearby small intervals. The family includes mapping functions for negative interference. A simple rule for combining recombination fractions in adjacent intervals is also obtained.     

A Mathematical Model of Interference for Use in Constructing Linkage Maps from Tetrad Data

In determining genetic map distances it is necessary to infer crossover frequencies from the ratios of recombinant and parental progeny. To do this accurately, in intervals where multiple crossovers may occur, a mathematical model of chiasma interference must be assumed when mapping in organisms displaying such interference. In Saccharomyces cerevisiae the model most frequently used is that of R.W. Barratt. An alternative to this model is presented. This new model is implemented using a microcomputer and standard numerical methods. It is demonstrated to fit ranked tetrad data from Saccharomyces more closely than the Barratt model and thus generates more accurate estimates of map distances when used with two-point data. A computer program implementing the model has been developed for use in calculating map distances from tetrad data in Saccharomyces.     

On the bias of recombination fractions, Kosambi's and Haldane's distances based on frequencies of gametes

The estimation of recombination frequencies is a crucial step in genetic mapping. For the construction of linkage maps, nonadditive recombination fractions must be transformed into additive map distances. Two of the most commonly used transformations are Kosambi's and Haldane's mapping functions. This paper reports on the calculation of the bias associated with estimation of recombination fractions, Kosambi's distances, and Haldane's distances. I calculated absolute and relative biases numerically for a wide range of recombination fractions and sample sizes. I assumed that the ratio of recombinant gametes to the total number of gametes can be adequately represented by a binomial function. I found that the bias in recombination fraction estimates is negative, i.e., the estimator is an underestimate. However, significant values were only obtained when recombination fractions were large and sample sizes were small. The relevant estimates of recombination fractions were, therefore, nearly unbiased. Haldane's and Kosambi's distances were found to be strongly biased, with positive bias for the most interesting values of recombination fractions and sample sizes. The bias of Kosambi's distance was considerably smaller than the bias of Haldane's distance.     

Linkage mapping and genome length in eastern white pine (Pinus strobus L.)

 Haploid linkage analysis of eastern white pine, Pinus strobus L., was carried out using mainly RAPD markers and microsatellite, or simple-sequence-repeat, markers. Ninety one loci mapped to 12 linkage groups of three or more markers. The resulting framework genome map, the first for a soft pine species, contained 69 markers. The map covered 58% of the estimated genome length of 2071 cM(K), with a 95% confidence interval of 1828–2242 cM(K). A systematic comparison of linkage data from eastern white pine, longleaf pine (P. palustris Mill.) and maritime pine (P. pinaster Ait.), gave genome-length estimates for all three species very close to either 2000 cM(K) or 2600 cM(H), depending on whether the Kosambi(K) or Haldane(H) map functions, respectively, were employed. Differences among previous pine genome-length estimates were attributed to the divergent criteria used in the methods of estimation, and indicate the need for the adoption of uniform criteria when performing genome-length estimates. Current data suggest that members of the two pine subgenera, which diverged during the late Mesozoic era, have highly conserved rates of recombination. Received: 5 January 1997/Accepted: 24 January 1997     

A corrected Haldane’s map function to calculate genetic distances from recombination data

The estimation of genetic distances from recombination data has no direct relationship due to the fact that multiple crossovers do not generate recombinant gametes that can be recognized in the progeny. The Haldane’s map function is the most widely used mathematical formulation able to relate the observed recombination frequency with the actual number of crossovers. Here I show that the model in which the Haldane’s correction is based on is not correct, and I present a modified map function that takes into consideration the actual number of recombinant gametes produced in cells in which different number of crossovers have occurred. My correction generates shorter genetic distances than the Haldane’s one.     

Negative crossover interference in maize translocation heterozygotes.

Negative interference describes a situation where two genetic regions have more double crossovers than would be expected considering the crossover rate of each region. We detected negative crossover interference while attempting to genetically map translocation breakpoints in maize. In an attempt to find precedent examples we determined there was negative interference among previously published translocation breakpoint mapping data in maize. It appears that negative interference was greater when the combined map length of the adjacent regions was smaller. Even positive interference appears to have been reduced when the combined lengths of adjacent regions were below 40 cM. Both phenomena can be explained by a reduction in crossovers near the breakpoints or, more specifically, by a failure of regions near breakpoints to become competent for crossovers. A mathematical explanation is provided.     

Alignment of molecular and classical linkage maps of rice,Oryza sativa

Application of genetic linkage maps in plant genetics and breeding can be greatly facilitated by integrating the available classical and molecular genetic linkage maps. In rice, Oryza sativa L., the classical linkage map includes about 300 genes which correspond to various important morphological, physiological, biochemical and agronomic characteristics. The molecular maps consist of more than 500 DNA markers which cover most of the genome within relatively short intervals. Little effort has been made to integrate these two genetic maps. In this paper we report preliminary results of an ongoing research project aimed at the complete integration and alignment of the two linkage maps of rice. Six different F₂ populations segregating for various phenotypic and RFLP markers were used and a total of 12 morphological and physiological markers (Table 1) were mapped onto our recently constructed molecular map. Six linkage groups (i.e., chr. 1, 3, 7, 9, 11 and 12) on our RFLP map were aligned with the corresponding linkage groups on the classical map, and the previous alignment for chromosome 6 was further confirmed by RFLP mapping of an additional physiological marker on this chromosome. Results from this study, combined with our previous results, indicate that, for most chromosomes in rice, the RFLP map encompasses the classical map. The usefulness of an integrated genetic linkage map for rice genetics and breeding is discussed.Abbreviations RFLP restriction fragment length polymorphism - chr chromosome - cM centiMorgan     

Linkage Group Xix of Chlamydomonas Reinhardtii Has a Linear Map

Linkage group XIX (or the UNI linkage group) of Chlamydomonas reinhardtii has been reported to show a circular meiotic recombination map. A circular map predicts the existence of strong chiasma and chromatid interference, which would lead to an excess number of two-strand double crossovers during meiosis. We have tested this prediction in multipoint crosses. Our results are consistent with a linear linkage group that shows positive chiasma interference and no chromatid interference. Chiasma interference occurs both within arms and across the centromere. Of the original loci that contributed to the circular map, we find that two map to other linkage groups and a third cannot be retested because the mutant strain that defined it has been lost. A second reported unusual property for linkage group XIX was the increase in meiotic recombination with increases in temperature during a period that precedes the onset of meiosis. Although we observed changes in recombination frequencies in some intervals on linkage group XIX in crosses to CC-1952, and in strains heterozygous for the mutation ger1 at 16°, we also show that our strains do not exhibit the previously observed patterns of temperature-sensitive recombination for two different pairs of loci on linkage group XIX. We conclude that linkage group XIX has a linear genetic map that is not significantly different from other Chlamydomonas linkage groups.     

Mapping the d1 and d2 dwarfing genes and the purple foliage color locus P in pearl millet

The d(1) and d(2) dwarfing genes and the P purple foliage color gene were placed on the restriction fragment length polymorphism (RFLP)-based molecular marker linkage map of pearl millet [Pennisetum glaucum (L.) R. Br.] using a mapping population based on a cross of inbred lines IP 18293 (D(1)/D(1), d(2)/d(2), P/P) and Tift 238D1 (d(1)/d(1) D(2)/D(2) p/p). A skeleton genetic linkage map of 562 cM (Haldane function) was constructed using 33 RFLP markers and these three morphological markers. The D(1)/d(1) plant height locus mapped to pearl millet linkage group 1, while the D(2)/d(2) plant height locus and the P/p foliage color locus mapped to pearl millet linkage group 4. Loose genetic linkage was observed between the D(2)/d(2) and P/p loci, with 42% repulsion-phase recombination corresponding to 92 cM (Haldane). This loose linkage of morphological marker loci detected on pearl millet LG4 can likely find use in applied pearl millet breeding programs, as host plant resistances to both downy mildew and rust have previously been identified in this genomic region. Such exploitation of these morphological markers in an applied disease resistance breeding program would require development of appropriate genetic stocks, but the relatively loose genetic linkage between d(2) and P suggests that this should not be difficult.     

Evidence for negative interference: clustering of crossovers close to the am locus in Neurospora crassa among am recombinants.

In response to a conflict between two mapping studies in the predicted orientation of the allele map with respect to the centromere, Fincham proposed that recombination events at the Neurospora am locus rarely have an associated crossover. Fincham considered that the elevated levels of crossing over between flanking markers in am recombinants resulted from negative interference, an increased probability of a nearby second event, and on this basis predicted a clustering of crossing over near am in these recombinants. In this article we reevaluate the data from three mapping studies of the am locus and report molecular evidence that shows crossovers to be clustered immediately proximal to am in am recombinants.     

A Composite Linkage Map from Three Crosses Between Commercial Clones of Cacao, Theobroma cacao L.

In this paper, we report the construction of the first composite map of cacao from linkage data of one F₂ and two F₁ mapping populations with a high number of codominant markers in common. The combination of linkage information from all three maps results in the currently most precise estimates of marker locations and distances between markers, especially in densely marked areas. JoinMap®V4 software was used for all marker quality assessment and mapping. Individual (sub-composite) maps and the composite map contained 10 major linkage groups, corresponding to the number of cacao chromosomes. Homogeneity of marker placement was very high among sub-composite maps, the composite map, and the designated “reference” map. Care was exercised in the re-creation of sub-composite maps and the composite map to include only markers with acceptable mapping quality parameters. The composite map places more markers with higher precision than any individual map. This research clearly demonstrates for the first time a very high level of marker homogeneity among commercial cacao clones compared to other species. The observed homogeneity between different maps, including the composite one, is probably due to a narrow genetic base of commercial cacao clones. Markers linked to identified quantitative trait loci (QTLs) are more likely to retain linkage in other commercial clones, rendering the QTLs in cacao potentially more stable than in other species.     

Recombination patterns reveal information about centromere location on linkage maps

Linkage mapping is often used to identify genes associated with phenotypic traits and for aiding genome assemblies. Still, many emerging maps do not locate centromeres – an essential component of the genomic landscape. Here, we demonstrate that for genomes with strong chiasma interference, approximate centromere placement is possible by phasing the same data used to generate linkage maps. Assuming one obligate crossover per chromosome arm, information about centromere location can be revealed by tracking the accumulated recombination frequency along linkage groups, similar to half‐tetrad analyses. We validate the method on a linkage map for sockeye salmon (Oncorhynchus nerka) with known centromeric regions. Further tests suggest that the method will work well in other salmonids and other eukaryotes. However, the method performed weakly when applied to a male linkage map (rainbow trout; O. mykiss) characterized by low and unevenly distributed recombination – a general feature of male meiosis in many species. Further, a high frequency of double crossovers along chromosome arms in barley reduced resolution for locating centromeric regions on most linkage groups. Despite these limitations, our method should work well for high‐density maps in species with strong recombination interference and will enrich many existing and future mapping resources.     

Investigation of crossover interference in barley ( Hordeum vulgare L.) using the coefficient of coincidence

Interference, the interaction between recombination events, was analysed in seven mapping populations of barley (Hordeum vulgare L.). The coefficient of coincidence was applied to investigate the type and position of interference within the genome. Interference was analysed with respect to dependence on the recombination fraction, and simulations were used to obtain test statistics which consider the small sample size of 71-150 double haploid lines. In addition to positive interference in intermediate intervals, strong negative interference, i.e. encouraged double recombination, was found in short intervals. The relationship between recombination fraction and interference could not be described with a uniform function, neither for the entire genome nor for individual chromosomes. The analysis of the position of interference within the genome revealed that interference does not act in the same way in the whole genome. Intervals spanning the centromere exhibited significantly higher means for the coefficient of coincidence than intervals within the chromosome arms, especially with regard to small intervals. In general, positive interference was found in the chromosome arms and no or negative interference in the genetically small but physically large centromeric region.     

A comparative map of wild rice (Zizania palustris L. 2n=2x=30)

We present the first genetic map of wild rice (Zizania palustris L., 2n=2x=30), a native aquatic grain of northern North America. This map is composed principally of previously mapped RFLP (restriction fragment length polymorphism) genetic markers from rice (Oryza sativa 2n=2x=24). The map is important as a foundation for genetic and crop improvement studies as well as a reference for genome organization comparisons among species of Gramineae. A comparative mapping approach with rice is especially useful because wild rice is grouped in the same subfamily, Oryzoideae, and no other mapping comparison has yet been made within the subfamily. As rice is the reference point for mapping and gene cloning in cereals, establishing a consensus map within the subfamily identifies conserved and unique regions. The genomes of wild rice and rice differ in total DNA content (wild rice has twice that of rice) and the number of chromosome pairs (wild rice=15 versus rice=12). The wild rice linkage map reported herein consists of 121 RFLP markers on 16 linkage groups spanning 1805 cM. Two linkage groups consist of only two markers. Colinear markers were found representing all rice linkage groups except #12. The majority of rice loci mapped to colinearly arranged arrays in wild rice (92 of 118). Features of the map include duplication of portions of three rice linkage groups and three possible translocations. The map gives basic information on the composition of the wild rice genome and provides tools to assist in the domestication of this important food source. Received: 25 August 1998 / Accepted: 20 February 1999     

Models of multilocus recombination: nonrandomness in chiasma number and crossover positions.

Cytological evidence indicates that genetic interference can be partitioned into two empirical components: nonrandomness in the number of chiasmata that occur and nonrandomness in the locations of multiple chiasmata. Previous studies have incorporated the first effect into genetic models for analyzing multipoint data. An extension to this approach is presented which allows for the second component of interference by modeling the probability density function of the locations of multiple crossovers. Results of reanalyses of multilocus data for the Drosophila X chromosome show that models that incorporate only the first effect give a better fit to these data than do standard mapping functions and that the extended model significantly improved the fit by decreasing the predicted frequency of multiple crossovers in nearby regions. Our results demonstrate that chiasma-based models of multilocus recombination, which are unique in incorporating direct estimates of the frequency of multiple crossovers for a chromosome region, can provide a powerful and realistic means of accounting for genetic interference when applied to the problems of gene localization, locus ordering, and exclusion mapping.     

A comparative map of wild rice (Zizania palustris L. 2n=2x=30)

We present the first genetic map of wild rice (Zizania palustris L., 2n=2x=30), a native aquatic grain of northern North America. The map is composed principally of previously mapped RFLP (restriction fragment length polymorphism) genetic markers from rice (Oryza sativa 2n=2x=24). The map is important as a foundation for genetic and crop improvement studies, as well as a reference for genome organization comparisons among Gramineae species. A comparative mapping approach with rice is especially useful because wild rice is grouped in the same subfamily, Oryzoideae, and no other mapping comparison has yet been made within the subfamily. As rice is the reference point for mapping and gene cloning in cereals, establishing a consensus map within the subfamily identifies conserved and unique regions. The genomes of wild rice and rice differ in total DNA content (wild rice has twice that of rice) and chromosome pairs (wild rice=15 versus rice=12). The wild rice linkage map reported herein consists of 121 RFLP markers on 16 linkage groups spanning 1805 cM. Two linkage groups consist of only two markers. Colinear markers were found representing all rice linkage groups except #12. The majority of rice loci mapped to colinearly arranged arrays in wild rice (92 of 118). Features of the map include duplication of portions of three rice linkage groups and three possible translocations. The map gives basic information on the composition of the wild rice genome and provides tools to assist in the domestication of this important food source. Received: 25 August 1998 / Accepted: 20 February 1999 (Corrected version. Originally published in TAG 99:793–799)     

Random variability of map distances based on Kosambi’s and Haldane’s mapping functions

The majority of published genetic maps are based on Kosambi distances or on Haldane distances. For a comparison of both map distance measures, their random variability is of particular interest. For the statistic ‘variance’, this paper presents a relationship between Kosambi distances and Haldane distances. The results suggest that Kosambi distances exhibit a smaller random variability. The theoretical results are applied to an experimental data set for molecular AFLP markers linked to the bolting gene of sugar beet (Beta vulgaris L.).     

A gene‐based SNP linkage map for pacific white shrimp,Litopenaeus vannamei

Pacific white shrimp (Litopenaeus vannamei) are of particular economic importance to the global shrimp aquaculture industry. However, limited genomics information is available for the penaeid species. We utilized the limited public information available, mainly single nucleotide polymorphisms (SNPs) and expressed sequence tags, to discover markers for the construction of the first SNP genetic map for Pacific white shrimp. In total, 1344 putative SNPs were discovered, and out of 825 SNPs genotyped, 418 SNP markers from 347 contigs were mapped onto 45 sex‐averaged linkage groups, with approximate coverage of 2071 and 2130 cm for the female and male maps, respectively. The average‐squared correlation coefficient (r²), a measure of linkage disequilibrium, for markers located more than 50 cm apart on the same linkage group, was 0.15. Levels of r² increased with decreasing inter‐marker distance from ～80 cm , and increased more rapidly from ～30 cm . A QTL for shrimp gender was mapped on linkage group 13. Comparative mapping to model organisms, Daphnia pulex and Drosophila melanogaster, revealed extensive rearrangement of genome architecture for L. vannamei, and that L. vannamei was more related to Daphnia pulex. This SNP genetic map lays the foundation for future shrimp genomics studies, especially the identification of genetic markers or regions for economically important traits.     

Microsatellite marker polymorphism and mapping in pea (Pisum sativum L.)

This paper aims at providing reliable and cost effective genotyping conditions, level of polymorphism in a range of genotypes and map position of newly developed microsatellite markers in order to promote broad application of these markers as a common set for genetic studies in pea. Optimal PCR conditions were determined for 340 microsatellite markers based on amplification in eight genotypes. Levels of polymorphism were determined for 309 of these markers. Compared to data obtained for other species, levels of polymorphism detected in a panel of eight genotypes were high with a mean number of 3.8 alleles per polymorphic locus and an average PIC value of 0.62, indicating that pea represents a rather polymorphic autogamous species. One of our main objectives was to locate a maximum number of microsatellite markers on the pea genetic map. Data obtained from three different crosses were used to build a composite genetic map of 1,430 cM (Haldane) comprising 239 microsatellite markers. These include 216 anonymous SSRs developed from enriched genomic libraries and 13 SSRs located in genes. The markers are quite evenly distributed throughout the seven linkage groups of the map, with 85% of intervals between the adjacent SSR markers being smaller than 10 cM. There was a good conservation of marker order and linkage group assignment across the three populations. In conclusion, we hope this report will promote wide application of these markers and will allow information obtained by different laboratories worldwide in diverse fields of pea genetics, such as QTL mapping studies and genetic resource surveys, to be easily aligned.Electronic Supplementary Material Supplementary material is available for this article at