Mapping functions

Accurate estimation of map distance between markers is important for the construction of large-scale linkage maps because it provides reliable and useful linkage information of markers on chromosomes. How to improve accuracy of estimating map distances depends on an appropriate mapping function. We used the coefficient of coincidence to integrate the Haldane function, in which crossovers are assumed to be independent and the Morgan function, in which crossovers are assumed to be interfered, and produce a new mapping function. The mapping function based on positive interference is referred to as the positive function and that on negative interference as the negative function. In these two mapping functions, map distances between loci are determined by both recombination frequencies and the coefficient of coincidence. We applied our mapping functions to four examples and show that our map estimates have much higher goodness-of-fit to the observed mapping data than the Haldane and Kosambi functions. Therefore, they can provide much more precise estimates of map distances than the two conventional mapping functions. Furthermore, our mapping functions produced almost linear (additive) map distances.     

RAPD-based genetic linkage maps of yellow passion fruit (Passiflora edulis Sims. f. flavicarpa Deg.).

A single cross between two clones of passion fruit (Passiflora edulis Sims. f. flavicarpa Deg., 2n = 18) was selected for genetic mapping. The mapping population was composed of 90 F1 plants derived from a cross between 'IAPAR 123' (female parent) and 'IAPAR 06' (male parent). A total of 380 RAPD primers were analyzed according to two-way pseudo-testcross mapping design. The linkage analysis was performed using Mapmaker version 3.0 with LOD 4.0 and a maximum recombination fraction (theta) of 0.30. Map distances were estimated using the Kosambi mapping function. Linkage maps were constructed with 269 loci (2.38 markers/primer), of which 255 segregated 1:1, corresponding to a heterozygous state in one parent and null in the other. The linkage map for 'IAPAR123' consisted of 135 markers. A total of nine linkage groups were assembled covering 727.7 cM, with an average distance of 11.20 cM between framework loci. The sizes of the linkage groups ranged from 56 to 144.6 cM. The linkage map for 'IAPAR 06' consisted of 96 markers, covering 783.5 cM. The average distance between framework loci was 12.2 cM. The length of the nine linkage groups ranged from 20.6 to 144.2 cM. On average, both maps provided 61% genome coverage. Twenty-four loci (8.9%) remained unlinked. Among their many applications, these maps are a starting point for the identification of quantitative trait loci for resistance to the main bacterial disease affecting passion fruit orchards in Brazil, caused by Xanthomonas campestris pv. passiflorae, because parental genotypes exhibit diverse responses to bacterial inoculation.     

An integrated AFLP and RFLP Brassica oleracea linkage map from two morphologically distinct doubled-haploid mapping populations

Genetical maps of molecular markers in two very different F₁-derived doubled-haploid populations of Brassica oleracea are compared and the first integrated map described. The F₁ crosses were: Chinese kale×calabrese (var. alboglabra×var. italica) and cauliflower×Brussels sprout (var. botrytis×var. gemmifera). Integration of the two component maps using Joinmap v.2.0 was based on 105 common loci including RFLPs, AFLPs and microsatellites. This provided an effective method of producing a high-density consensus linkage map of the B. oleracea genome. Based on 547 markers mapping to nine linkage groups, the integrated map covers a total map length of 893 cM, with an average locus interval of 2.6 cM. Comparisons back to the component linkage maps revealed similar sequences of common markers, although significant differences in recombination frequency were observed between some pairs of homologous markers. Map integration resulted in an increased locus density and effective population size, providing a stronger framework for subsequent physical mapping and for precision mapping of QTLs using substitution lines. Received: 5 February 1999 / Accepted: 16 June 1999     

Construction of dense linkage maps “on the fly” using early generation wheat breeding populations

In plant species, construction of framework linkage maps to facilitate quantitative trait loci mapping and molecular breeding has been confined to experimental mapping populations. However, development and evaluation of these populations is detached from breeding efforts for cultivar development. In this study, we demonstrate that dense and reliable linkage maps can be constructed using extant breeding populations derived from a large number of crosses, thus eliminating the need for extraneous population development. Using 565 segregating F₁ progeny from 28 four-way cross breeding populations, a linkage map of the hexaploid wheat genome consisting of 3,785 single nucleotide polymorphism (SNP) loci and 22 simple sequence repeat loci was developed. Map estimation was facilitated by application of mapping algorithms for general pedigrees implemented in the software package CRI-MAP. The developed linkage maps showed high rank-order concordance with a SNP consensus map developed from seven mapping studies. Therefore, the linkage mapping methodology presented here represents a resource efficient approach for plant breeding programs that enables development of dense linkage maps “on the fly” to support molecular breeding efforts.     

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.     

Multilocus Analysis for Gene-Centromere Mapping Using First Polar Bodies and Secondary Oocytes

Polar body and oocyte typing is a new technique for gene-centromere mapping and for generating female linkage maps. A maximum likelihood approach is presented for ordering multiple markers relative to the centromere and for estimating recombination frequencies between markers and between the centromere and marker loci. Three marker-centromere orders are possible for each pair of markers: two orders when the centromere flanks the two markers and one order when the centromere is flanked by the two markers. For each possible order, the likelihood was expressed as a function of recombination frequencies for two adjacent intervals. LOD score for recombination frequency between markers or between the centromere and a marker locus was derived based on the likelihood for each gene-centromere order. The methods developed herein provide a general solution to the problem of multilocus genecentromere mapping that involves all theoretical crossover possibilities, including four-strand double crossovers.     

Genome-wide analysis of the frequency and distribution of crossovers at male and female meiosis in Sinapis alba L. (white mustard)

We present the first genetic linkage maps of Sinapis alba (white mustard) and a rigorous analysis of sex effects on the frequency and distribution of crossovers at meiosis in this species. Sex-averaged maps representing recombination in two highly heterozygous parents were aligned to give a consensus map consisting of 382 loci defined by restriction fragment length polymorphisms and arranged in 12 linkage groups with no unlinked markers. The loci were distributed in a near-random manner across the genome, and there was little evidence of segregation distortion. From these dense maps, a subset of spaced informative markers was used to establish recombination frequencies assayed separately in male and female gametes and derived from two distinct genetic backgrounds. Analyses of 746 gametes indicated that recombination frequencies were greater in male gametes, with the greatest differences near the ends of linkage groups. Genetic background had a lesser effect on recombination frequencies, with no discernible pattern in the distribution of such differences. The possible causes of sex differences in recombination frequency and the implications for plant breeding are discussed.     

Casares’ map function: no need for a ‘corrected’ Haldane’s map function

A genetic map function M(d) = RF provides a mapping from the additive genetic distance d to the non-additive recombination fraction RF between a given pair of loci, where the recombination fraction is the proportion of gametes that are recombinant between the two loci. Genetic map functions are needed because in most experiments all we can directly observe are the recombination events. However, since a recombination event is only observed if there are an odd number of crossovers between the two loci, recombination fractions are not additive. One of the most widely used map functions is Haldane’s map function, which is derived under the assumptions of no chiasma and no chromatid interference, and has been in widespread use since 1919. However, Casares recently proposed a ‘corrected’ Haldane’s map function – we show here that this ‘corrected’ map function is not correct due to faulty assumptions and mistakes in its derivation.     

Recombination disequilibrium in ideal and natural populations

Following Hardy-Weinberg disequilibrium (HWD) occurring at a single locus and linkage disequilibrium (LD) between two loci in generations, we here proposed the third genetic disequilibrium in a population: recombination disequilibrium (RD). RD is a measurement of crossover interference among multiple loci in a random mating population. In natural populations besides recombination interference, RD may also be due to selection, mutation, gene conversion, drift and/or migration. Therefore, similarly to LD, RD will also reflect the history of natural selection and mutation. In breeding populations, RD purely results from recombination interference and hence can be used to build or evaluate and correct a linkage map. Practical examples from F₂, testcross and human populations indeed demonstrate that RD is useful for measuring recombination interference between two short intervals and evaluating linkage maps. As with LD, RD will be important for studying genetic mapping, association of haplotypes with disease, plant breading and population history.     

Construction of High-Density Linkage Maps of Populus deltoides × P. simonii Using Restriction-Site Associated DNA Sequencing

Although numerous linkage maps have been constructed in the genus Populus, they are typically sparse and thus have limited applications due to low throughput of traditional molecular markers. Restriction-site associated DNA sequencing (RADSeq) technology allows us to identify a large number of single nucleotide polymorphisms (SNP) across genomes of many individuals in a fast and cost-effective way, and makes it possible to construct high-density genetic linkage maps. We performed RADSeq for 299 progeny and their two parents in an F₁ hybrid population generated by crossing the female Populus deltoides ‘I-69’ and male Populus simonii ‘L3’. A total of 2,545 high quality SNP markers were obtained and two parent-specific linkage maps were constructed. The female genetic map contained 1601 SNPs and 20 linkage groups, spanning 4,249.12 cM of the genome with an average distance of 2.69 cM between adjacent markers, while the male map consisted of 940 SNPs and also 20 linkage groups with a total length of 3,816.24 cM and an average marker interval distance of 4.15 cM. Finally, our analysis revealed that synteny and collinearity are highly conserved between the parental linkage maps and the reference genome of P. trichocarpa. We demonstrated that RAD sequencing is a powerful technique capable of rapidly generating a large number of SNPs for constructing genetic maps in outbred forest trees. The high-quality linkage maps constructed here provided reliable genetic resources to facilitate locating quantitative trait loci (QTLs) that control growth and wood quality traits in the hybrid population.     

Comparison of linkage maps from F2 and three times intermated generations in two populations of European flint maize (Zea mays L.)

Intermated mapping populations are expected to result in high mapping resolution for tightly linked loci. The objectives of our study were to (1) investigate the consequences of constructing linkage maps from intermated populations using mapping methods developed for F₂ populations, (2) compare linkage maps constructed from intermated populations (F₂Syn3) with maps generated from corresponding F₂ and F₃ base populations, and (3) investigate the advantages of intermated mapping populations for applications in plant breeding programs. We constructed linkage maps for two European flint maize populations (A × B, C × D) by mapping 105 SSR markers in generations F₂ and F₂Syn3 of population A × B, and 102 SSR markers in generations F₃ and F₂Syn3 of population C × D. Maps for F₂Syn3 were constructed with mapping methods for F₂ populations (Map A) as well as with those specifically developed for intermated populations (Map B). Both methods relate map distances to recombination frequencies in a single meiosis and, therefore, did not show a map expansion in F₂Syn3 compared with maps constructed from the respective F₂ or F₃ base populations. Map A and B differed considerably, presumably because of theoretical shortcomings of Map A. Since loosely linked markers could not unambiguously be mapped in the F₂Syn3 populations, they may hamper the construction of linkage maps from intermated populations.     

The Theory and Applications of Measuring Broad-Range and Chromosome-Wide Recombination Rate from Allele Frequency Decay around a Selected Locus

Recombination is the exchange of genetic material between homologous chromosomes via physical crossovers. High-throughput sequencing approaches detect crossovers genome wide to produce recombination rate maps but are difficult to scale as they require large numbers of recombinants individually sequenced. We present a simple and scalable pooled-sequencing approach to experimentally infer near chromosome-wide recombination rates by taking advantage of non-Mendelian allele frequency generated from a fitness differential at a locus under selection. As more crossovers decouple the selected locus from distal loci, the distorted allele frequency attenuates distally toward Mendelian and can be used to estimate the genetic distance. Here, we use marker selection to generate distorted allele frequency and theoretically derive the mathematical relationships between allele frequency attenuation, genetic distance, and recombination rate in marker-selected pools. We implemented nonlinear curve-fitting methods that robustly estimate the allele frequency decay from batch sequencing of pooled individuals and derive chromosome-wide genetic distance and recombination rates. Empirically, we show that marker-selected pools closely recapitulate genetic distances inferred from scoring recombinants. Using this method, we generated novel recombination rate maps of three wild-derived strains of Drosophila melanogaster, which strongly correlate with previous measurements. Moreover, we show that this approach can be extended to estimate chromosome-wide crossover interference with reciprocal marker selection and discuss how it can be applied in the absence of visible markers. Altogether, we find that our method is a simple and cost-effective approach to generate chromosome-wide recombination rate maps requiring only one or two libraries.     

Construction of a sequence-tagged high-density genetic map of papaya for comparative structural and evolutionary genomics in brassicales

A high-density genetic map of papaya (Carica papaya L.) was constructed using microsatellite markers derived from BAC end sequences and whole-genome shot gun sequences. Fifty-four F(2) plants derived from varieties AU9 and SunUp were used for linkage mapping. A total of 707 markers, including 706 microsatellite loci and the morphological marker fruit flesh color, were mapped into nine major and three minor linkage groups. The resulting map spanned 1069.9 cM with an average distance of 1.5 cM between adjacent markers. This sequence-based microsatellite map resolved the very large linkage group 2 (LG 2) of the previous high-density map using amplified fragment length polymorphism markers. The nine major LGs of our map represent papaya's haploid nine chromosomes with LG 1 of the sex chromosome being the largest. This map validates the suppression of recombination at the male-specific region of the Y chromosome (MSY) mapped on LG 1 and at potential centromeric regions of other LGs. Segregation distortion was detected in a large region on LG 1 surrounding the MSY region due to the abortion of the YY genotype and in a region of LG6 due to an unknown cause. This high-density sequence-tagged genetic map is being used to integrate genetic and physical maps and to assign genome sequence scaffolds to papaya chromosomes. It provides a framework for comparative structural and evolutional genomic research in the order Brassicales.     

Continuous linkage map of human chromosome 14 short tandem repeat polymorphisms.

Nine moderately to highly informative short tandem repeat polymorphisms were assigned to chromosome 14 using somatic cell hybrids and were mapped using linkage analysis. The nine markers formed a continuous linkage map covering almost the entire long arm from 14q11.2 to q32. The markers filled a large gap within previously reported linkage maps for this chromosome. Best order of the new loci from q11.2 to q32 was D14S50, D14S54, D14S49, D14S47, D14S52, D14S53, D14S55, D14S48, and D14S51. The order shown for all adjacent pairs of loci was very strongly favored with the exception of loci pair D14S55 and D14S48, for which the order was moderately favored. Map lengths for the nine loci were 142 cM in females and 72 cM in males. Female recombination frequencies exceeded male recombination frequencies in the middle and distal portions of the map.     

The Double-Reduction Landscape in Tetraploid Potato as Revealed by a High-Density Linkage Map

The creation of genetic linkage maps in polyploid species has been a long-standing problem for which various approaches have been proposed. In the case of autopolyploids, a commonly used simplification is that random bivalents form during meiosis. This leads to relatively straightforward estimation of recombination frequencies using maximum likelihood, from which a genetic map can be derived. However, autopolyploids such as tetraploid potato (Solanum tuberosum L.) may exhibit additional features, such as double reduction, not normally encountered in diploid or allopolyploid species. In this study, we produced a high-density linkage map of tetraploid potato and used it to identify regions of double reduction in a biparental mapping population. The frequency of multivalents required to produce this degree of double reduction was determined through simulation. We also determined the effect that multivalents or preferential pairing between homologous chromosomes has on linkage mapping. Low levels of multivalents or preferential pairing do not adversely affect map construction when highly informative marker types and phases are used. We reveal the double-reduction landscape in tetraploid potato, clearly showing that this phenomenon increases with distance from the centromeres.     

Combined RAPD and RFLP molecular linkage map of asparagus.

Two linkage maps of asparagus (Asparagus officinalis L.) were constructed using a double pseudotestcross mapping strategy with restriction fragment length polymorphisms (RFLPs), random amplified polymorphic DNAs (RAPDs), and allozymes as markers in a population generated from crossing MW25 x A19, two heterozygous parents. All data were inverted and combined with the natural data to detect linkages in repulsion phase. Two sets of data, one for each parent, were formed according to the inheritance patterns of the markers. The maternal MW25 map has a total of 163 marker loci placed in 13 linkage groups covering 1281 cM, with an average and a maximum distance between adjacent loci of 7.9 and 29 cM, respectively. The paternal A19 map has 183 marker loci covering 1324 cM in 9 linkage groups, with an average and a maximum distance between two adjacent loci of 7.7 and 29 cM, respectively. Six multiallelic RFLPs segregating in the pattern a/c x b/c and eight heterozygous loci (four RAPDs, and four RFLPs segregating in the pattern a/b x a/b (HZ loci)) were common to both maps. These 14 loci were used as bridges to align homologous groups between the two maps. In this case, RFLPs were more frequent and informative than RAPDs. Nine linkage groups in the MW25 map were homologous to six groups in the A19 map. In two cases, two or more bridge loci were common to a group; thus, the orientation of homologous linkage groups was also determined. In four other cases, only one locus was common to the two homologous groups and the orientation was unknown. Mdh, four RFLPs, and 14 RAPDs were assigned to chromosome L5, which also has the sex locus M.     

A high-density consensus map of A and B wheat genomes

A durum wheat consensus linkage map was developed by combining segregation data from six mapping populations. All of the crosses were derived from durum wheat cultivars, except for one accession of T. ssp. dicoccoides. The consensus map was composed of 1,898 loci arranged into 27 linkage groups covering all 14 chromosomes. The length of the integrated map and the average marker distance were 3,058.6 and 1.6?cM, respectively. The order of the loci was generally in agreement with respect to the individual maps and with previously published maps. When the consensus map was aligned to the deletion bin map, 493 markers were assigned to specific bins. Segregation distortion was found across many durum wheat chromosomes, with a higher frequency for the B genome. This high-density consensus map allowed the scanning of the genome for chromosomal rearrangements occurring during the wheat evolution. Translocations and inversions that were already known in literature were confirmed, and new putative rearrangements are proposed. The consensus map herein described provides a more complete coverage of the durum wheat genome compared with previously developed maps. It also represents a step forward in durum wheat genomics and an essential tool for further research and studies on evolution of the wheat genome.     

Expanding the genetic map of maize with the intermated B73 × Mo17 (IBM) population

The effects of intermating on recombination and the development of linkage maps were assessed in maize. Progeny derived from a common population (B73 × Mo17) before and after five generations of intermating were genotyped at the same set of 190 RFLP loci. Intermating resulted in nearly a four-fold increase in the genetic map distance and increased the potential for improved genetic resolution in 91% of the intervals evaluated. This mapping population and related information should connect research involving dense genetic maps, physical mapping, gene isolation, comparative genomics, analysis of quantitative trait loci and investigations of heterosis.     

A High-Density Genetic Map for Soybean Based on Specific Length Amplified Fragment Sequencing

Soybean is an important oil seed crop, but very few high-density genetic maps have been published for this species. Specific length amplified fragment sequencing (SLAF-seq) is a recently developed high-resolution strategy for large scale de novo discovery and genotyping of single nucleotide polymorphisms. SLAF-seq was employed in this study to obtain sufficient markers to construct a high-density genetic map for soybean. In total, 33.10 Gb of data containing 171,001,333 paired-end reads were obtained after preprocessing. The average sequencing depth was 42.29 in the Dongnong594, 56.63 in the Charleston, and 3.92 in each progeny. In total, 164,197 high-quality SLAFs were detected, of which 12,577 SLAFs were polymorphic, and 5,308 of the polymorphic markers met the requirements for use in constructing a genetic map. The final map included 5,308 markers on 20 linkage groups and was 2,655.68 cM in length, with an average distance of 0.5 cM between adjacent markers. To our knowledge, this map has the shortest average distance of adjacent markers for soybean. We report here a high-density genetic map for soybean. The map was constructed using a recombinant inbred line population and the SLAF-seq approach, which allowed the efficient development of a large number of polymorphic markers in a short time. Results of this study will not only provide a platform for gene/quantitative trait loci fine mapping, but will also serve as a reference for molecular breeding of soybean.     

Estimation of Recombination Frequencies and Construction of RFLP Linkage Maps in Plants from Crosses between Heterozygous Parents

The construction of a restriction fragment length polymorphism (RFLP) linkage map is based on the estimation of recombination frequencies between genetic loci and on the determination of the linear order of loci in linkage groups. RFLP loci can be identified as segregations of singular or allelic DNA-restriction fragments. From crosses between heterozygous individuals several allele (fragment) configurations are possible, and this leads to a set of formulas for the evaluation of p, the recombination frequency between two loci. Tables and figures are presented illustrating a general outline of gene mapping using heterozygous populations. The method encompasses as special cases the mapping of loci from segregating populations of pure lines. Formulas for deriving the recombination frequencies and information functions are given for different fragment configurations. Information functions derived for relevant configurations are also compared. A procedure for map construction is presented, as it has been applied to RFLP mapping in an allogamous crop.