Notes hinb hinb hina hina hina hina gsp gsp gsp Marker 3_0163 1_0745 2_0894 2_0553 3_0543 3_0983 3_0984 2_0226 3_0978 3_0979 3_0980 3_0975 3_0976 3_0977 3_0001 3_1022 3_1023 3_0591 2_0206 1_1381 2_1202 2_1221 1H 2H 3H 4H 5H 6H 7H Total SM 0 0 0 0 0 0 0 0 0 0 0 0 0 1.09 2.17 2.17 341 485 475 335 535 352 417 2940 Abbreviations are SM = Steptoe × Morex, OWB = Oregon Wolfe Barley, MB = Morex × Barke, HO = Haruno Nijo × OUH602. Blank entries indicate the marker was not mapped in that population. package for the R computing environment called DAG- GER. The package offers both linear and quadratic pro- gramming options for linearizing the consensus graph. When these two methods were compared using simu- lated data, the consensus maps generated by quadratic programming had less compression and higher accuracy. When applied to four barley linkage maps genotyped at nearly 3000 SNP markers, in less than one minute DAGGER produced a consensus map with improved fine structure compared to the existing barley consensus SNP map. The RMS error between the linkage maps Table 3 Barley consensus map statistics Chromosome # Markers The DAGGER map was generated with the QP method. Endelman BMC Genomics 2011, 12:407 http://www.biomedcentral.com/1471-2164/12/407 Table 2 Barley linkage maps in the 5HS telomere region OWB 0 2.15 4.30 5.38 5.38 5.38 5.38 5.38 5.38 10.78 MB 0 0 1.07 1.07 2.15 2.15 2.15 3.23 3.23 4.30 141 161 174 125 198 133 167 1099 HO 0 Page 7 of 9 and the DAGGER map was 0.82 cM per marker interval compared to 2.28 cM for the existing consensus map. Examination of the barley hardness locus at the 5HS tel- omere confirmed that the DAGGER output was more accurate for fine structure analysis. DAGGER is an effective, freely available resource for integrating the information from a set of consistent linkage maps. Methods Linearizing the consensus graph DAGGER uses the R packages quadprog [27] and Rglpk [28] for the QP (Equation 4) and LP (Equation 5) options, respectively. The package quadprog is only for strictly con- vex quadratic programs, i.e., those in which the quadratic form is positive definite. Because the adjacency matrix A is for an acyclic graph, the addition of row vector [1 0 0 ... 0] makes a full column rank matrix Ã (the corresponding entry in d is an arbitrary constant, say zero, which fixes the origin of the consensus map). With this addition, the combined adjacency matrix C∼ (formed by stacking Ã on B) is also full column rank, and the quadratic form is posi- 2 tive definite x C˜ QC˜ x = Q1/2 C˜x = 0 ⇔ x = 0 . 2 Validation with simulated data To simulate g linkage maps, 2g parents with a fixed average level of sequence identity were randomly gener- ated and paired to create g doubled haploid populations. For each population, gamete formation was simulated for 200 individuals assuming no crossover interference. Linkage maps were constructed using Haldane’s map- ping function and a LOD-score weighted-least-squares approach [2,29]. The additional constraint of fixed mar- ker order (equal to the simulated physical order) was used to create linkage maps with no ordering conflicts. The results in this manuscript are based on four-locus linkage maps; changing to three or five loci had little effect. Mean Linkage Map Length (cM) Consensus Map Length (% of Mean) DAGGER MergeMap # Bins DAGGER MergeMap 88% 86% 85% 92% 87% 91% 86% 87% 144% 148% 140% 143% 152% 154% 138% 146% 242 310 320 207 315 229 263 1886 117 153 149 111 174 109 129 942