As for most phenotypes, the amount of variance in educational achievement explained by SNPs is lower than the amount of additive genetic variance estimated in twin studies. Twin-based estimates may however be biased because of self-selection and differences in cognitive ability between twins and the rest of the population. Here we compare twin registry based estimates with a census-based heritability estimate, sampling from the same Dutch birth cohort population and using the same standardized measure for educational achievement. Including important covariates , we analyzed 893,127 scores from primary school children from the years 2008–2014. For genetic inference, we used pedigree information to construct an additive genetic relationship matrix. Corrected for the covariates, this resulted in an estimate of 85%, which is even higher than based on twin studies using the same cohort and same measure. We therefore conclude that the genetic variance not tagged by SNPs is not an artifact of the twin method itself. There are evident individual differences in children's educational achievement. While some learn at such a fast pace that they are bored at school, others are struggling in each subject. This inevitably leads to the nature vs. nurture debate: To what extent can we explain these differences in school performance by genetic and environmental influences? One of the most-often used methods to quantify the relative influence of nature and nurture is the twin study design. Twin pairs are either identical  and share the same genetic material, or non-identical  and share on average only half of the segregating genes. When identical twins are more similar  than non-identical twins, this implies that genetic influences are at play. We then conclude that to some extent, individuals are different from each other at least in part because of differences in genotypes. Twin researchers often use the so-called ACE model that decomposes variance in observed measures into additive genetic variance (A), common-environmental variance (C) and unique-environmental variance (E). Common-environmental variance represents variance due to the environmental influences that make siblings similar but cannot be attributed to their genetic resemblance. Unique-environmental variance refers to variance explained by non-genetic influences that render siblings different. Heritability is then defined as the proportion of the total observed variance that can be explained by genetic differences. Twin studies are usually based on large twin registries where study participation is voluntary. Results of twin studies suggest that educational achievement is a highly heritable trait. Including 61 studies from 11 cohorts and 7 continents/countries , de Zeeuw et al.  conductepolygenic inheritance), studies often use polygenic scores rather than assessing the pure presence or absence of a single gene mutation. A polygenic score aggregates the effect of all significant single nucleotide polymorphisms (SNPs) into one genotypic score for a particular trait  GWAs on educational achievement was performed on years of education, a proxy measure for the trait. They found that the DNA variant with the largest effect size accounted for 1% of the variance in years of education, but the polygenic score accounted for only 2% in a replication sample , a revised polygenic score based on a new GWAS was used, which almost doubled the effect size: 3.9% of the variance in years of education could be explained in an independent sample  than Dutch twins with a normal birth weight. Furthermore, singletons received higher rating from their teachers for, among others, arithmetic skills. These differences could however be largely accounted by the birth order within a family: Twins who were first in birth order had the same (or even higher) ratings as their non-twin sibling. Contrary, twins who were not first in birth order had lower rating than their non-twin sibling(s). Second, participation in registries is voluntary and might therefore introduce self-selection bias  that shows no self-selection bias, (2) of children that are more representative regarding suboptimal pre- and perinatal circumstances, (3) with which we can draw inference regarding the influence of genetics, and (4) where test scores are available and can be compared meaningfully. Here we investigated whether earlier Dutch twin study results on educational achievement can be validated and generalized to the general Dutch population by making use of a sample of Eindtoets Basisonderwijs test, which is an achievement tests that is administered yearly in the final year of primary school (around age 12). The Eindtoets Basisonderwijs consists of 290 multiple choice items in four different subjects  and assesses what a child has learned during primary school. Together, the performance scales result in a standardized total score between 500 and 550 that is used to give an advice on the choice for secondary education. From these scores we selected only the scores between 500 and 550, since other values cannot be valid. Thus, we obtained a total of 902,884 test scores with 41,242 test scores from 2008, 93,642 from 2009, 146,363 from 2010, 152,971 from 2011, 156,356 from 2012, 154,821 from 2013, and 157,489 from 2014. For some children, two scores were found in different years . In those cases, the earliest observed score was selected and used for further analysis, resulting in a total of 902,884–1,204 = 901,680 scores. For every pupil in our dataset, the sex, migrant background and adoption status were determined, using information obtained from the Netherlands Bureau of Statistics. Parentage information from the Netherlands Bureau of Statistics  was available for 899,437 pupils, but children that were known to be adopted were left out of the analysis, reducing the number of phenotyped pupils to a total of 894,750. On the basis of the census data, it was not possible to determine zygosity  of twins included in the dataset such that there are some pairs of individuals in our analysis that were assumed fraternal while they were actually identical.The phenotypic data consisted of achievement scores from all pupils registered in the Dutch primary education system during the years 2008 through 2014 using the N = 16,334). The pedigree was simplified by omitting individuals that were involved in a loop (N = 593): for instance persons being their own grandparent. The pedigree was further reduced by omitting branches without phenotypic information . In this way, all relations in the phenotypic data that were either full siblings, half siblings, half cousins, or full cousins were identified, but all other relations were assumed to be nonexistent . Data on great-grandparents of phenotyped children was available but deemed too unreliable by the Netherlands Bureau of Statistics and therefore ignored .Pedigree data  was available for 14,875,512 individuals. Individuals that were known to be adopted themselves were given unknown parents , School denomination , Migrant status , Sex and SES for which we used a variable that indicated the weight that is, depending on the educational level of the parents, given to a pupil by the Dutch government as a proxy measure. In the following, these covariates are described in more detail.As covariates, we used N = 1,623) did not have a covariate value for the weight covariate . These children were excluded from the analysis with covariates, which led to a total sample size of 893,127 children.Information on all covariates was retrieved from the Netherlands Bureau of Statistics. All phenotyped children had complete data on covariates, except that children from special education . Public schools provide secular education, but can also teach around specific pedagogical principles . Private schools are inspired by a religion . Special schools are schools that are distinct from both public and private education. They are generally administered by an independent board, as opposed to government authority, while still receiving government funding. At the Netherlands Bureau of Statistics, these three types are further divided into 37 subcategories. In our data, 18 different school denominations occurred. Following the same coding as at the Netherlands Bureau for Statistics, in our data, the four largest school denominations in our data were the following: ABZ (4.49%), OPB (30.20%), PC (23.83%), and RK (36.35%). The category ABZ captures general special schools , which is neutral education in the sense that education is not bound to ideologies or social movements. OPB  refers to public schools, PC are Protestant-Christian schools  and RK are Roman-Catholic oriented schools . For a complete list of all denominations that occurred in the data, the reader is referred to the official documentation at the website of the Netherlands Bureau of Statistics. The three-level covariate Migrant status indicates whether a pupil is either a first generation migrant (1), a second generation migrant (2), or neither (0). A first generation migrant was defined as a pupil born abroad with at least one parent born abroad. A second generation migrant was defined as a pupil born in the Netherlands with at least one parent born abroad. All other pupils got a score of 0. This variable was treated as a categorical variable with the 0 category serving as the reference category, as was the Sex covariate  with Female serving as reference category.The covariate SES of the pupils.In the Dutch educational system, depending on the educational level of the parents, pupils are assigned to one of the following weights: 0.25, 0.30, 0.40, 0.70, 0.90, and 1.20. A weight of 0.25 is given to pupils with poorly educated indigenous parents who both have a schooling to our through pre-vocational exam. Pupils whose parents did not exceed the level of practical training or pre-vocational education from the basic vocational program or the management vocational program get a weight of 0.30. A weight of 0.40 is assigned to pupils who are staying in a boarding school or foster home and whose father or mother is or has been engaged in a skipper company. When the parents work in the circus or fair business and live or have lived in a caravan, a weight of 0.70 is assigned and pupils with poorly educated immigrant parents with an education up to graduation from preparatory vocational education get a weight of 0.90. A weight of 1.20 is assigned to pupils with one parent that only finished primary school and whose other parent finished only the highest pre-vocational education of the basic vocational program or the management vocational program. All other children get a weight of 0.00. This weight variable was used as a proxy for the Table h2, is then defined as the proportion of phenotypic variance that can be explained by additive genetic . For the data preparation, the software package R  could be explained by the covariates.The results for the empty model without any covariates suggested that the largest part of the variance could be explained by genetic differences, resulting in a posterior mean for JOO schools , ABZ schools, HIND schools  and ISL  showed the highest average scores, whereas the lowest averages were observed in IC , SCA , EVB and EVA  and SOP  schools. Note that these effects are corrected for immigrant status and SES. First and second generation immigrant pupils clearly showed scores of around 1.5–2.0 points lower than non-immigrant pupils. Group size was clearly but weakly related to performance: for every extra pupil in the classroom, the expected test score increased by 0.03 points. Note that in this analysis of the full model, special primary education pupils were left out. Boys scored on average 0.35 points higher than girls. In sum: the majority of the variance in test scores was explained by family relationships, and only a small part (around 6%) by the covariates. After correction for the covariate effects, 85% of the remaining variance could be explained by genetic differences.When these covariate effects were taken into account, the heritability estimate decreased to a posterior mean of 0.85 for the full model, suggesting that part of the variance shared by family members that is captured in the Eindtoets basisonderwijs, partly determine the level of secondary education suitable for a child. Several twin studies have looked at the heritability of individual indifferences on this test, but due to self-selection bias and possible differences in singletons and twins, these results might not generalize to the general population of Dutch pupils. Here we determined the heritability of test scores using population-wide census data. For the estimation, pedigree-based mixed models were used, a method borrowed from the field of animal genetics. We found a heritability of 0.94. When corrected for several school-related covariates, this estimate dropped to 0.85. How does this fairly high heritability estimate compare to that based on twin studies?In the Netherlands, the results of a national educational achievement test, the Eindtoets basisonderwijs). They found that genetic influences explained 57% of the variance in test scores and environmental influences 43%. Twenty-seven percent of the environmental variance could be explained by common-environmental influences and 16% by unique-environmental influences. Schwabe et al.  the biological parents. But if there are non-biological relationships in our analyses, this can only have led to an underestimation of heritability.Second, since our analysis included all pupils with available data, there are a number of monozygotic twin pairs (or multiples) in our data. As on the basis of census data, zygosity cannot be determined, there are some pairs of individuals in our analysis that were assumed fraternal while they were actually identical. This may have led to overestimation of the heritability. The seriousness of this bias is however proportional to the number of MZ twins, which is only a small part of the Dutch population (roughly one in every 200 births).Third, as we have used only biological parentage data in our analysis (parentage on paper) we have not taken into account whether sibling pairs and half-sibling pairs actually share the same home environment, nor whether cousins share the same household (could be true for some families), nor whether nonrelated children share the same household. We were therefore not able to estimate non-genetic variance that is shared due to being brought up in the same home environment. If ignored, this variance, as far as it is not related to the covariates that we used in the analysis, ended up in the heritability estimate. This may have led to an overestimation of heritability. Note however, that the shared environmental component was rather small  in both the de Zeeuw et al.  and the Put differently, if we acknowledge that we can only estimate A + C together in our analysis , and when we compare that to the A + C estimates based on the twin studies using the exact same birth cohorts, the results are rather similar: 94% here, 74 + 8 = 82% in the de Zeeuw et al.  and 66 +In future research, a method should be developed that makes it possible to estimate also variance due to common-environmental influences. This might for example be possible by adding a random effect for every household in the data set, functioning as a proxy measure for the shared environment. Note that this is not an easy feat in an age where many parents split up, children have two mothers or two fathers (or even three parents), and children grow up in multiple households in different combinations of siblings, half-siblings and unrelated children. Here instead we have tried to capture the most important environmental factors available in the census data that might explain correlation in family members: the cultural and educational background of the parents, the denomination of the school, and the number of children in the same classroom.So irrespective of study design, twins or census, we find a high heritability for educational achievement in the Netherlands. The high heritability might be explained by the homogeneity in educational opportunities: most schools are tied to a standard curriculum and funded by the government (Sturm et al., SvdB developed the study concept and performed the statistical analyses. IS drafted the manuscript and SvdB and LJ provided critical revisions and feedback. All authors approved the final version of the manuscript for submission.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


In both cases, the genetic association was stronger with the composite facial factor phenotype than with any of the individual linear distances that comprise those factors. While the biological role of PARK2 in the craniofacial complex is currently unclear, there is evidence from both mouse models and Mendelian syndromes that FREM1 may influence facial variation. These results highlight the potential value of data-driven multivariate phenotyping for genetic studies of human facial morphology.Several studies have now shown evidence of association between common genetic variants and quantitative facial traits in humans. The reported associations generally involve simple univariate measures and likely represent only a small fraction of the genetic loci influencing facial morphology. In this study, we applied factor analysis to a set of 276 facial linear distances derived from 3D facial surface images of 2187 unrelated individuals of European ancestry. We retained 23 facial factors, which we then tested for genetic associations using a genome-wide panel of 10,677,593 single nucleotide polymorphisms (SNPs). In total, we identified genome-wide significant (p < 5 × 10 PAX3 and nasal root morphology in two independent studies [DCHS2, RUNX2, GLI3, PAX1 and EDAR [ALX3). Of interest, this study observed the same association between soft-tissue nasal width and PAX1 reported by Adhikari et al. [A number of studies have reported associations between genetic variants and normal-range variation in facial morphology. These include candidate gene studies focusing on a small number of genetic loci chosen based on their known roles in craniofacial development or in genetic syndromes –3 and ge studies ,5. More and EDAR . Anotherand EDAR  identifii et al. .Prior association studies have used several diverse approaches to generate and test facial shape phenotypes. There is currently no agreement on the optimal phenotyping strategy. The variety of different measures and approaches used in prior studies makes it difficult to compare results and may partly explain the lack of replication across studies. To date, univariate tests involving simple linear distances or qualitatively graded facial features have generally shown the greatest success in GWAS designs. Such measures are often correlated, however, as the human craniofacial complex shows strong evidence of morphological integration . The patTo overcome some of these limitations, we used factor analysis  to derive composite measures of facial morphology based on a large number of traits in a well-characterized cohort of US individuals of European ancestry. Specifically, we applied factor analysis to a set of 276 facial linear distances derived from 3D facial surface images and then tested the resulting composite phenotypes for genetic associations using a genome-wide panel of single nucleotide polymorphisms (SNPs).Our study sample was comprised of 2187 unrelated self-described White individuals of European ancestry from the United States . Participants were recruited at research centers in Pittsburgh, Seattle, Houston and Iowa City as part of the FaceBase Consortium’s 3D Facial Norms dataset . Partici3D facial surfaces were captured via digital stereophotogrammetry  using published protocols ,13. Twen2, height, and weight using linear regression. This resulted in 276 adjusted phenotypes , which were entered into the factor analysis. Visual inspection of the scree plot and parallel analysis were used to determine the number of factors to retain. Specifically, factors were retained if the observed eigenvalues from the correlation matrix were greater than the mean obtained from random uncorrelated data [A set of 276 facial measurements was generated for each individual in the dataset. These 276 measurements represent every possible unique Euclidean distance calculated between the 3D coordinates of our set of 24 facial landmarks. These distances were calculated using the program WinEDMA v1.0.1 . Factor ted data . To aid Genotyping and data cleaning was performed as previously described . In brieImputation was performed to capture information on unobserved SNPs as well as sporadically missing genotypes among genotyped SNPs, using all haplotypes from the 1000 Genomes Project  Phase 3 2 < 0.1 in a sliding window of 10 Mb). Linear regression, testing the association between each principal component (PC) and each SNP in the genome, confirmed that none of the first 20 PCs of ancestry were due to local variation in specific genomic regions. Based on the scree plot and joint distributions, we determined that four PCs were sufficient for capturing the population structure. Joint distributions of the four PCs, and their joint distributions with the 23 factors, are provided in the Population structure was assessed with principal component analysis using 96,700 autosomal SNPs pruned from the total panel based on call rate (> 95%), MAF (> 0.05), and LD  the threshold for genome-wide statistical significance. Because this threshold is conservative and the overall approach here can be considered hypothesis-generating, we also reported “suggestive” evidence of association at p < 5 × 10−6 in the Supplemental Material. In order to account for multiple GWAS scans corresponding to the 23 factors, we consider p < 2.17 × 10−9  the strict threshold for study-wide significance.As customary in the field, we accounted for the issue of multiple testing by considering p < 5 × 10A total of 276 factors were extracted, 23 of which were retained. These 23 factors captured approximately 94% of the variation. The first four factors alone captured over 54% of the variation. As expected, moving from the first to the last factor revealed a general shift away from more global aspects of facial variation to more localized regional effects. Factor 1, for example, described 36% of the variation and involved multiple measures capturing the overall horizontal breadth of the face. In contrast, factor 21explained 0.6% of the variation and involved measurements comprised of only two landmarks on the nasal alae. −8) was located within a narrow LD block within an intron of PARK2, a gene that spans 1.3Mb (FREM1 (−8). Factor 14, which captured the horizontal spacing of the inner canthi of the orbits (−8). The four remaining association signals at 8q12 , 12q24.2 , 16p12.1 , and Xp11.3  involved isolated imputed SNPs , which are detailed in For three of the seven genome-wide significant signals, we observed that genetic associations with the factors were stronger than with any of the constituent linear distances that comprised the factors. For example, the association between rs9456748 and factor 9 was 4.99 × 10−8).In this study, we performed a GWAS of composite facial traits in a sample of 2187 unrelated healthy individuals. To derive these traits, we applied factor analysis to 276 facial linear distances calculated between the 3D coordinates of 24 facial surface landmarks. Analysis of 23 distinct factors, accounting for 94% of the variation, revealed seven genetic associations exceeding the strict threshold for genome-wide statistical significance  syndrome [Frem1 mutant mice have similar phenotypic features including reduced snout projection and a shorter philtrum [FREM1 influencing normal variation in philtrum and central upper lip morphology in humans.We observed a novel association between SNPs in adhesion . Frem1 i midface –24. Of p midface  reportedlion lip . These asyndrome , Manitobsyndrome , and triphiltrum . These fPARK2 gene and a factor capturing aspects of midfacial height  do not exhibit strong craniofacial expression. Finally, this interval is largely devoid of chromatin signatures for regulatory elements and bioinformatic analysis of the top SNPs did not reveal compelling annotations. Despite a strong statistical signal at this locus, there is currently little biological evidence pointing toward a possible mechanism by which these SNPs influence midfacial height.We observed a novel association between SNPs in the l height . PARK2 ehis gene . Howeverneration , cholestneration , leprosyneration , and, fr embryos . SimilarHDAC8, which is associated with Cornelia de Lange syndrome–a condition characterized by hypertelorism. As is evident on the LocusZoom plot  based on physical proximity and known biology, the tasks of determining exactly which causal variant accounts for an association signal, which gene the variant impacts, and through what mechanism the variant acts, are difficult. Typically observational evidence, alone, is insufficient in identifying a putative causal allele, and additional experimental work is usually necessary to understand the mechanism. Therefore, more work is needed, beginning with replication of these genetic associations in independent cohorts, in order to truly understand the contributions of associated variants to human facial morphology.A major challenge in gene-mapping studies is making the jump from associated locus to causal variant. Though we describe biologically plausible candidates Click here for additional data file.S2 Table(DOCX)Click here for additional data file.S3 Table(XLSX)Click here for additional data file.S4 Table(XLSX)Click here for additional data file.S1 FigLandmarks are labeled as follows: n = nasion; prn = pronasale; sn = subnasale; ls = labiale superius; sto = stomion; li = labiale inferius; sl = sublabiale; gn = gnathion; en = endocanthion; ex = exocanthion; al = alare; ac = alar curvature point; sbal = subalare; cph = crista philtri; ch = chelion; and t = tragion. For bilateral landmarks, left and right indicated by _l and _r after the landmark abbreviation.(PDF)Click here for additional data file.S2 FigAll pairwise combinations of Eigenvectors  and factors are depicted via scatterplots. Pearson correlation coefficient (r) and significance of the correlation (p) are indicated for each pair.(PDF)Click here for additional data file.S3 FigLocusZoom plots showing genome-wide significant associations observed for Factor 3 (A), Factor 21 (B), and Factor 22 (C). LocusZoom plots show the association  with each factor. Genotyped SNPs are depicted by stars and imputed SNPs are depicted by circles. Shading of the points represents the linkage disequilibrium  between each SNP and the top SNP, indicated by purple shading. The blue overlay shows the recombination rate (right y-axis). Positions of genes are shown below the plot.(PDF)Click here for additional data file.S4 Fig−8 for genome-wide significance and 5 x 10−6 for suggestive significance.Chromosomes are arranged in order along the x-axis. The y-axis shows the log base 10 p-value. Lines for p-value thresholds set at 5 x 10(PDF)Click here for additional data file.


Mus musculus musculus and M. m. domesticus. We performed the first genome-wide association study for molar shape and used 3D surface morphometrics to quantify subtle variation between individuals. We show that many loci of small effect underlie phenotypic variation, and identify five genomic regions associated with tooth shape; one region contained the gene microphthalmia-associated transcription factor Mitf that has previously been associated with tooth malformations. Using a panel of five mutant laboratory strains, we show the effect of the Mitf gene on tooth shape. This is the first report of a gene causing subtle but consistent variation in tooth shape resembling variation in nature.Numerous loci of large effect have been shown to underlie phenotypic variation between species. However, loci with subtle effects are presumably more frequently involved in microevolutionary processes but have rarely been discovered. We explore the genetic basis of shape variation in the first upper molar of hybrid mice between  Understanding the genetic basis of evolution requires the identification of genes and mutations responsible for phenotypic variation between individuals and populations. A complete catalog of such variants will allow the identification of the genetic paths favored by evolution . RegardiThe mouse tooth is one of the traits extensively studied regarding shape variation. In paleontology, it is a key character for phylogenetic and dietary inferences . It has Salazar-Ciudad and Jernvall, 2002Surprisingly, only two studies have tried to map loci responsible for such morphological variation. Using F2 crosses between LG/J and SM/J inbred lines of mice, Mus musculus musculus and Mus musculus domesticus collected in the Bavarian hybrid zone  . Three-dWe focused on the first upper molar given that it is the first molar to develop, and therefore it is not directly constrained by the development of other molars, as is the case for the second and third molars . In addiTo quantify the 3D shape of the molar tooth, a template was designed based on a randomly chosen tooth. The template corresponds to the surface describing the erupted part of the tooth, with the roots and UM1/UM2 junction manually removed. On this surface, 1500 equally spaced 3D semi-landmarks were requested, resulting in 1588 automatically sampled points. Ten landmarks were placed manually on the tooth surface using the Landmark Editor software  and were used to anchor the template on the original surface. For each specimen, the template was then deformed to match the tooth surface using the R package Morpho .The semi-landmarks were allowed to slide along tangent planes to the surface using the minimum bending energy criterion, that is minimizing the bending energy necessary to produce the changes in each specimen relative to the Procrustes consensus configuration . SpecimeIn addition, we used an approach that directly addresses the effect of wear on tooth shape. It is known that the degree of wear is a major cause of non-heritable shape variation. Following For each data set (complete and wear-free template), a principal component analysis (PCA) was performed on the covariance matrix of shape variables. Because of the large number of variables (more than 1000 points in three dimensions), a reduction of dimensionality  was performed. We used PCs representing >1% of variance as phenotypes for mapping; this dataset comprises 18 PCs representing 81.4% and 86.6% of total shape variation in the complete and wear-free template, respectively.The relationship between shape and age was investigated using a multivariate regression of the 18 PCs on mouse age.Centroid size (CS), estimated as the square root of the summed squared distance between each semi-landmark and the centroid, was used as indicator of tooth size.SNP genotypes for the 183 mice used in the mapping were obtained from 183 mice, 145 378 SNPs, and 18 PCs, each representing more than 1% of shape variation were used to map loci associated with shape variation of the first upper molar. Each PC was analyzed separately, and therefore, the approach used here corresponds to a univariate mapping of shape variables (PCs). This approach was used in order to implement a linear mixed-model (LMM) to control for family structure (see below); however, it should be noted that mapping PCs might not identify genetic associations that are not strongly aligned with single PCs but that are spread across multiple PCs. CS was used to find associations with size variation. The PC scores and CS data used for the mapping are available as We performed mapping using the LMM implemented in GEMMA . A cente2 >0.8 was used. For significant SNPs without neighboring SNPs in tight linkage, we report regions extending 250 Kb to each side of the best  SNP resulting in regions of 500 Kb. This value is based on results from a previous study mapping cranial morphology in this sample  and weakly linked  to significant SNPs. LD calculations were done in PLINK 1.07 (The linkage disequilibrium (LD) between the significant SNPs and neighboring SNPs  was used to delimit the significant regions associated with molar shape variation. A threshold of rs sample ; intermeINK 1.07 .2) between the 18 PCs and the genotype of the best SNP per region. In this way, the effect size of an SNP is relative to total phenotypic variation, and not to the individual PC it was associated with in the mapping. It should be kept in mind that given that siblings and half-siblings were used for mapping, the coefficient of determination could result in an overestimation of the effect size.The effect of the significant QTLs on the phenotype was calculated based on Procrustes distances as the coefficient of determination  analysis implemented in GCTA . Due to microphthalmia (Mitf) in molar shape variation, we took advantage of four existing laboratory mouse strains carrying mutant alleles, including mi-vga9Mitf, mi-enu22(398)Mitf, Mi-whMitf, and miMitf. Details about each mutant allele and associated phenotypes are reported in To further explore the role of the candidate gene mi-vga9Mitf/+, five mi-vga9Mitf/mi-vga9,Mitf 10 mi-enu22(398)Mitf/mi-enu22(398)Mitf, two Mi-whMitf/+, four Mi-whMitf/Mi-whMitf, and five compound heterozygotes Mi-whMitf/miMitf. All these mutations are on C57Bl/6J (B6) background, and therefore five B6 mice were used as the wild-type control. Heterozygous and homozygous mice for the Mi-whMitf allele were siblings, as well as mice with the mi-vga9Mitf allele. The mutant mice were male and female ranging from 5 to 10 weeks. Due to the small sample size per group, it was not possible to test for sexual dimorphism in molar shape; however, it has been shown that sex has very small effect, if at all, on tooth shape in mice . Since there are only two Mi-wh/+Mitf mice, they were not included in this analysis. p-Values were corrected for multiple testing using the Holm-Bonferroni method in R.A PCA including all The comparison between the morphological changes associated with each mutant and the effects associated with the SNP identified in the mapping encountered the problem that each analysis was done with independent Procrustes superimpositions. This means that the PC axes from both analyses are not comparable. We therefore developed an approach to be able to compare the morphological signature of each effect on the tooth.Shape changes were visualized using reconstructed surfaces . Then, for two surfaces, the distance between each vertex was calculated. The correlation between the effect of an SNP and (i) a PC or (ii) a mutation was assessed as follows: (1) Each effect was characterized by a pair of reconstructed surfaces: a surface for each homozygous genotype; the mean surface of the wild-type vs the mean surface of a mutant; the shapes reconstructed from the scores at the two extremes of each PC used in the mapping. (2) For each pair of surfaces, the distances for each of their vertices were calculated. This provided, for each effect, a range of 8150 between-vertices distances. (3) The between-vertices distances were compared between two effects: if they are similar, a large difference in a part of the tooth for one effect should also correspond to a large difference for the other effect. This provided a quantitative indicator of the degree of resemblance between two factors (e.g. an SNP and mutation or a PC). (4) To generate a proxy for the correlation to be expected between orthogonal directions of change, the shape changes represented by the PCs used in the mapping were compared to each other. However, it should be kept in mind that although PC axes are statistically independent, this is not necessarily true regarding the underlying genetics.M. m. musculus and M. m. domesticus and represent a hybridization continuum between the two subspecies , and this is reflected in the significant correlation between age and some PCs (r2(PC1) = 2.1%, p-value=0.027; r2(PC4) = 2.2%, p-value=0.025; r2(PC5) = 3.3%, p-value=0.008; r2(PC14) = 1.8%, p-value=0.04; r2(PC18) = 4.9%, p-value=0.002). More importantly, a qualitative assessment indicates that wear-like patterns are present already in the first axis of variation (PC1)  -a.2(PC3) = 4.6%, p-value=0.002; r2(PC18) = 3.5, p-value=0.007).Although age correlates with wear patterns, there are other factors, such as diet and behavior that also influence tooth morphology through wear . The weaThe heritability values estimated in this study correspond to the effect that all SNPs used in the mapping have on the phenotype; this value is also known as SNP or chip heritability, and it serves as a proxy of the additive genetic variance underlying phenotypic variation. Substantial genetic variance was found in all PCs . A weighWhen exploring heritability from a chromosomal point of view , insteadWe decomposed variation in molar tooth shape in PCs. The mapping was performed using 183 mice and ~145,000 SNPs. Centroid size was used as proxy for tooth size, however, no genomic regions were significantly associated with size variation.When shape data derived from the complete template was used in the mapping, no significant associations were found. In contrast, the data derived from the wear-free template resulted in nine SNPs significantly associated with molar shape variation, clustered in five genomic regions. Hereafter, we will thus focus on the results of the wear-free approach.These five loci are found in chromosomes 1, 5, 6, and X, and were associated with PC7, PC11, PC16, and PC18 . The namRps6ka3 , namely Rps6ka3  and the a3 , and deafness Mitf is closest to wild type (0.0237), and mi-vga9Mitf/mi-vga9Mitfmi is most distant (0.0466) (As shown in an shape . In term(0.0466) .Mi-whMitf/miMitf compound heterozygotes have a similar molar shape as Mi-whMitf/+ heterozygotes, suggesting that the miMitf allele alone has no additional effect on molar shape  PCs. Together these loci explain around 10% of the total phenotypic variation, hence there are additional contributing loci yet to be identified. This is expected for a highly polygenic trait; increasing sample size and implementing multivariate mapping of shape traits will likely identify additional genes contributing to natural variation in molar shape.Mitf, a candidate gene identified by association mapping, affects the shape of the upper first molar in mice. All Mitf mutant alleles that were studied here had a significant effect on molar shape, regardless of severity of the mutation. Interestingly, even mice heterozygous for the mutant alleles showed consistent shape changes. The consistency in direction and magnitude of the phenotypic effect of each mutation  has a strong additive genetic basis. Such variation in molar shape is due to many small effect loci, but identification of loci and validation of causative genes is feasible. We report that the candidate gene,  In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.eLife. Your article has been favorably evaluated by Patricia Wittkopp (Senior Editor) and three reviewers, one of whom, Craig T Miller (Reviewer #1), is a member of our Board of Reviewing Editors. The following individual involved in review of your submission has agreed to reveal their identity: Alistair R. Evans (Reviewer #3).Thank you for submitting your article "Genomic regions controlling shape variation in the first upper molar of the house mouse" for consideration by The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.Summary:Mitf, and the authors show in lab mice with different mutant alleles of Mitf a role for Mitf in regulating molar shape. Overall this study presents the first GWAS for molar shape in natural populations, identifies the first genomic regions found to underlie quantitative variation in molar shape in nature, and identifies a specific gene (Mitf) regulating quantitative aspects of molar shape in the lab that appears to also regulate natural variation in tooth shape.Pallares and colleagues study mouse molar shape variation in two subspecies of house mice. Using a GWAS and 3D surface morphometric approach in hybrid mice, the authors identify five genomic regions associated with molar shape. One genomic region contains the candidate gene Essential revisions:Mitf mutant comparisons are done between mutant mice and a panel of non-sibling B6 mice. It seems possible that differences in genetic background in these mouse lines contribute to the differences in tooth shape. One nice argument against this is the authors observe differences in tooth shape between mice heterozygous and homozygous for the mi-vga9 allele. It would strengthen the authors' conclusions if in Mitf, not genetic background differences are responsible.1) One potential concern is many of the Mitf-associated QTL is beyond the scope of this study, can the authors at least comment on whether coding changes are found between the domesticus and musculus mice used in their study, or whether Mitf coding changes are present in the reference genome assemblies available for domesticus and musculus derived strains or populations?2) Although finding actual mutations underlying the 3) Subsection “Association mapping”, third paragraph: Can the authors further justify the decision to pick arbitrary 500 kb intervals for most QTL? It seems they have data that would speak to patterns of LD in this sample, including genome-wide averages and within these QTL, so not sure whether 500 kb is an overly conservative estimate or not.4) Subsection “Association mapping”, second paragraph: I was surprised that no genomic regions were significantly associated with size variation, given how polygenic and strong the signal appears for tooth shape. Was mouse size corrected for in mapping centroid size?Mitf on molar shape comes from mutations in a mouse laboratory strain, and it is therefore not equivalent to comparing the effect of naturally occurring alleles”.5) "The other mutant alleles do not exhibit osteopetrosis in homozygotes." Can the authors clarify for this statement, whether they mean osteopetrosis has not been reported in homozygotes for these alleles, or whether they (or other groups) looked for osteopetrosis in homozygotes? If the latter, the authors should provide references or data to back up this claim. If the former, the sentence should be edited. Same comment for “The evidence presented here for effect of 6) Materials and methods: Only the first upper molar is analyzed. Why not, but the rational for this choice is never explained through the manuscript.7) Geometric morphometrics: You used 10 landmarks to anchor the template. They are very important features to know because they impose some constrains, but there is no way for the readers to get this information.For the full template, 1588 semi-landmarks are used and for the "wear-free" 1532. This is much denser in the second case than in the first given that all tips of cusps are removed. I don't understand why you increase the sampling.8) Subsection “Tooth shape, mouse age and wear”. Your idea is that age is related to wear, as once erupted tooth shape doesn't change except by the effect of wear. In my opinion you should explain it to readers.9) Wear is expected to be on PC1. People have used Burnaby-like procedure to get rid of such artifacts (many examples with fish) based on PC1 or expected shape changes. You decide to use a very different alternative approach (quite a disappointing one) by cutting cusp tips. Why not, but why? Because age impacts a lot of PCs and thus you consider that the effect of wear is spread all over the shape space? If so, I think you should explain your rational. Interestingly, age explain the same amount of variance with the 'wear-free' template but is less significant.10) Finally you analyzed something very close to what is captured with 2D outline. The amount of 3D information you have in your sampling depend on the cusp height kept in the template. You didn't say anything about that. How did you decide the height at which you cut the tips?Mitf”, second paragraph). It adds a very complex way of comparing shape changes. I can understand that you were afraid that weird mutant twists your Procrustean space for GWAS but in my opinion, you could have added the mutant as supplementary observations into the hybrid GPA (i.e. superimposing each mutant to the hybrid mean shape). It will have allowed the comparison of shape changes using more classical tools of geometric morphometrics . In your approach, I don't get how the two can be centered and aligned to ensure that trivial variation are removed.11) I don't understand why you run separate Procrustes superimpositions . The math ensures that they are orthogonal but nothing ensure that they are independent given the underlying generative processes (for ex genetics).12) One additional remark on this method of comparing shape changes is that in order to get a threshold you consider the values obtained between PCs. PCs are orthogonal but not independent  GWAS: The family structure in your sample is very strong and you correctly use linear mixed model to handle it but, as in your previous papers, running multiple univariate LMM on PCs is not the most powerful approach, but I understand that running multivariate LMM is challenging, but the cost is evident here as you don't have a very powerful design with 183 mice."To facilitate association mapping" but it is at the price of some power lost because PCs are not aligned to the underlying genetics.Nonetheless, you correctly want to assess the effect size of best SNPs simultaneously on the full shape space , but why using only the first 18 PCs? Why estimating these effect sizes only on 86% variance? Did you consider that 14% are just analytical errors?14) About that  you say that you used the coefficient of determination but with multivariate data I guess you used an analogue based on distances (a ratio of sums of sum-of-squares) but there are other analogue based on classical multivariate statistics (Pillai etc.). It may be better to precise that it is procrustes distances based R2.15) It is unclear that you used leave-one chromosome out approach to handle the pop structure in your LMM .16) SNP heritability: subsection “SNP heritability”, first paragraph: In my opinion, the way you present your computation of the "total heritability" is right in calculus but is little weird in term of genetics. In my opinion, you assess the sum of the additive genetic variances and do the ratio with the total variance to get a quite poor proxy of h2, and indeed means not much in term of heritability (see discussions of Monteiro and Klingenberg between 2002 and 2010).Mitf”: I don't agree with this statement about sexual dimorphism. There are plenty of phenotypes with SD in mammals and more specifically in the house mouse . The fact that SD is low on bone shape doesn't insure that it is the case in tooth.17) Functional evaluation: subsection “Samples used to functionally evaluate this character and are not distinguishable from within-species loci. I don't think we need to ask for transgressive variation or hybrid instability.18) Results, first paragraph: The most parsimonious hypothesis is simply it is within species variation and that the inter-subspecific variance is smaller than the within, species-specific loci have small effects on 19) Discussion: subsection “Genetic architecture”, first paragraph: Actually this conclusion is based on the huge amount of missing heritability . However, as your GWAS doesn't properly handle the multivariate nature of the shape space, you don't know anything about that because for instance a locus, orthogonal to all PCs, and explaining 1% of variance on each, will finally explained a lot of variance but will never be captured with your approach.2. Will you do a t-test with N = 5 samples? Same conditions apply to Hotelling. Moreover you have twice more additional parameters plus correlation between the two variables to estimate. Maybe doing something non-parametric based on distances will be more reliable.20) You have a very small sample size to run Hotelling T21) Some p-values for statistical tests were not reported in the manuscript. Essential revisions:1) One potential concern is many of the Mitf mutant comparisons are done between mutant mice and a panel of non-sibling B6 mice. It seems possible that differences in genetic background in these mouse lines contribute to the differences in tooth shape. One nice argument against this is the authors observe differences in tooth shape between mice heterozygous and homozygous for the mi-vga9 allele. It would strengthen the authors' conclusions if in Mitf mutations are on a B6 background , and the B6 line used to generate the Mitf mutations is the same one used as control; this line is maintained in the same facilities and under the same conditions than the Mitf mutant lines derived from it.All the Mitf”. We have also added the comparison between mi-vga9/mi-vga9 and mi-vga9/+ in The mi-vga9/mi-vga9 and mi-vga9/+ were indeed full siblings. This information was added in the first paragraph of the subsection “Samples used to functionally evaluate 2) Although finding actual mutations underlying the Mitf-associated QTL is beyond the scope of this study, can the authors at least comment on whether coding changes are found between the domesticus and musculus mice used in their study, or whether Mitf coding changes are present in the reference genome assemblies available for domesticus and musculus derived strains or populations?Mitf sequences in samples from 6 wild populations of mice, including M. m. musculus and M. m. domesticus, M. m. castaneus and related M. spretus  doi:10.1038/sdata.2016.75), and eight hybrid mice from the same population used in this study .We compared Mus spretus and Mus castaneus. One variant is present in a population of M.m. musculus from Kazakhstan, and one in a population of M.m. domesticus in Germany. Another variant is present in mice from the hybrid zone used in this study.We found seven predicted missense variants in wild populations of mice. Five are found in We have added this information in the Discussion (last paragraph), and added a table of the predicted variants as The SNP data can be accessed in the UCSC genome browser (genome.ucsc.edu) -> My Data -> Public Sessions -> wildmouse3) Subsection “Association mapping”, third paragraph: Can the authors further justify the decision to pick arbitrary 500 kb intervals for most QTL? It seems they have data that would speak to patterns of LD in this sample, including genome-wide averages and within these QTL, so not sure whether 500 kb is an overly conservative estimate or not.In a previous study where we used the same SNP data  we showed that the median size of the associated regions with LD > 0.8 was 150kb, and the regions with LD > 0.2 (weakly linked) was ~ 1.8 Mb. In the present study, some significant SNPs fell in regions with poor SNP density resulting in zero neighboring SNPs with LD> 0.8. Therefore, we decided to use the intermediate value 500 Kb here, similar to what we observed in the previous study, but not as stringent. We added a clarification of this choice in the third paragraph of the subsection “Association mapping”.4) Subsection “Association mapping”, second paragraph: I was surprised that no genomic regions were significantly associated with size variation, given how polygenic and strong the signal appears for tooth shape. Was mouse size corrected for in mapping centroid size?No, we didn’t account for the effect of body size on tooth centroid size. The first upper molar is mineralized around birth in mice, and although it might be correlated with body size at birth (we don’t have this data), it is not correlated with adult body size at the population level . Using body size as a covariate might introduce variation that is actually not relevant for molar size. Besides, the lack of associations with size is not rare, usually such loci are of very small effect, and the studies that have been able to identify them are usually performed with F2 crosses of mouse inbred lines that were previously selected for body size differences.5) "The other mutant alleles do not exhibit osteopetrosis in homozygotes." Can the authors clarify for this statement, whether they mean osteopetrosis has not been reported in homozygotes for these alleles, or whether they (or other groups) looked for osteopetrosis in homozygotes? If the latter, the authors should provide references or data to back up this claim. If the former, the sentence should be edited. Same comment for “The evidence presented here for effect of Mitf on molar shape comes from mutations in a mouse laboratory strain, and it is therefore not equivalent to comparing the effect of naturally occurring alleles”.Mitf mutations, the references for individual studies are found in the following review that we have now added as reference in the pertinent sections : Steingrimsson et al., 2004, annual review genetics. And, we have also added, in the same sections, the following reference where the alleles mi, ew, Wh, and vga9 used in this study were directly tested for osteopetrosis: Steingrimsson et al. 2002, PNAS.Osteopetrosis has been characterized for many of the 6) Materials and methods: Only the first upper molar is analyzed. Why not, but the rational for this choice is never explained through the manuscript.We decided to focus on the first molar because it is the first one to develop in the molar row and therefore, it is relatively independent from the development of the second and third molars given that they develop following the Inhibitory-Cascade model . The second and third molar, in contrast, are not only a simplified version  of the first molar, but are also directly affected/constrained by the development of the first one. Along these lines, the effect of genetic variation relative to environmental variation (in this case the development of other molars) on molar shape is enhanced on the first molar.Regarding the lower molars, these are less morphologically complex compared to the upper molars. And therefore, exploring the first upper molar maximizes the amount of phenotypic variation available for mapping.Given the results found here for the first upper molar we think it will be very interesting to explore the genetic basis of molar shape in the second and third molars independently of each other, but also in terms of the correlations that are created between them given the Inhibition-Cascade model. This however, will need to be explored in a different manuscript.We added a line in the Introduction specifying that we focused on the first upper molar (fourth paragraph) and expanded on the reason for this in the second paragraph of the Materials and methods subsection “3D surface morphometrics”7) Geometric morphometrics: You used 10 landmarks to anchor the template. They are very important features to know because they impose some constrains, but there is no way for the readers to get this information.For the full template, 1588 semi-landmarks are used and for the "wear-free" 1532. This is much denser in the second case than in the first given that all tips of cusps are removed. I don't understand why you increase the sampling.The 10 landmarks were only used to pre-position the template to the original template before this was deformed to match the tooth surface. These landmarks were not used in any analysis and therefore they should not affect any of the morphometric results and interpretations. We have, however, added the landmarks to the surfaces on The semi-landmarks are automatically sampled following a Poisson distribution with the aim of obtaining equally-spaced semi-landmarks. The number requested was 1500 for both templates with the aim of obtaining a dense sampling and therefore capture variation at a very small scale. To fulfill the equally-spaced requirement, the actual number selected by the algorithm is approximately the requested number, therefore the specific numbers used in each template are not identical. Since the number of semi-landmarks is very high, going from, e.g. 1500 in the full template to e.g. 1000 in the wear-free template won’t affect the observed results, and therefore we opted for using the same number for both.We have now added in the Materials and methods, a statement regarding the number of requested landmarks requested and the number finally sampled .8) Subsection “Tooth shape, mouse age and wear”. Your idea is that age is related to wear, as once erupted tooth shape doesn't change except by the effect of wear. In my opinion you should explain it to readers.We have expanded the Results section “Tooth shape, mouse age and wear”to include the explanation for this. In short, yes, tooth shape doesn’t change once the teeth have erupted. Only wear affects tooth morphology after eruption, and this is why wear and age are correlated. However, other non-age-related factors like diet and behavior also affect the degree of wear, and this is one of the reasons why we used a wear-free template approach, to be able to address not only age-related wear patterns, but also other unknown factors.9) Wear is expected to be on PC1. People have used Burnaby-like procedure to get rid of such artifacts (many examples with fish) based on PC1 or expected shape changes. You decide to use a very different alternative approach (quite a disappointing one) by cutting cusp tips. Why not, but why? Because age impacts a lot of PCs and thus you consider that the effect of wear is spread all over the shape space? If so, I think you should explain your rational. Interestingly, age explain the same amount of variance with the 'wear-free' template but is less significant.Wear is a very difficult aspect to address especially because, as correctly suggested by the reviewers, it impacts several PCs. Age, although correlated with wear is an incomplete proxy for wear since there are other factors affecting wear patterns as described in the response to the previous comment and explored in Renaud and Ledevin 2017. Before choosing the wear-free method, we tried several approaches such as multivariate regressions, or using the dentine-enamel interface as the surface of interest. At the end, removing the cusp tips appeared to be the most convincing protocol. We also considered the fact that it should be possible to apply this phenotyping method to wild animals for which age is unknown .10) Finally you analyzed something very close to what is captured with 2D outline. The amount of 3D information you have in your sampling depend on the cusp height kept in the template. You didn't say anything about that. How did you decide the height at which you cut the tips?It is true that traditional 2D and our 3D phenotyping seem similar as they both focus on the crown, however, our 3D approach detects shape details that a 2D outline will not be able to detect. It captures information from the crown outline but also from the central parts of the tooth such as the inter-cusp space and the morphology of the cusp basal area. The slope at the front of the cusps, described by our 3D method, is not accessible with a 2D outline.The height at which we cut the tip was decided empirically by comparing teeth of different wear stages. The cut section was designed so that the most worn teeth of our dataset can still be described by our template. We have now added this in the Materials and methods section .11) I don't understand why you run separate Procrustes superimpositions . It adds a very complex way of comparing shape changes. I can understand that you were afraid that weird mutant twists your Procrustean space for GWAS but in my opinion, you could have added the mutant as supplementary observations into the hybrid GPA (i.e. superimposing each mutant to the hybrid mean shape). It will have allowed the comparison of shape changes using more classical tools of geometric morphometrics . In your approach, I don't get how the two can be centered and aligned to ensure that trivial variation are removed.Adding mutants as supplementary specimens is an option that we considered, however the probability for mutants would have been outside of the range of variation of the wild-derived mice. The risk of incorrect projection of the mutants into the shape space was therefore not negligible, and we preferred to run two separate analyses. Now, since there is not a Procrustes superimposition, there is not centering and alignment, and therefore our method doesn’t compare effects in the shape space, but directly on the tooth surface. The question we tried to answer in this way is whether two effects  modify the tooth in the same way.12) One additional remark on this method of comparing shape changes is that in order to get a threshold you consider the values obtained between PCs. PCs are orthogonal but not independent . The math ensures that they are orthogonal but nothing ensure that they are independent given the underlying generative processes (for ex genetics).Mitf”, last paragraph and legend We have now re-written this part to make clear that the phenotypic changes associated with PCs are not necessarily independent given the underlying genetics. However, since it is not feasible to get totally independent shape changes to be used as a null in 13) GWAS: The family structure in your sample is very strong and you correctly use linear mixed model to handle it but, as in your previous papers, running multiple univariate LMM on PCs is not the most powerful approach, but I understand that running multivariate LMM is challenging, but the cost is evident here as you don't have a very powerful design with 183 mice."To facilitate association mapping" but it is at the price of some power lost because PCs are not aligned to the underlying genetics.Nonetheless, you correctly want to assess the effect size of best SNPs simultaneously on the full shape space , but why using only the first 18 PCs? Why estimating these effect sizes only on 86% variance? Did you consider that 14% are just analytical errors?We agree with this assessment. The mapping of univariate PCs is limited, and it is not the optimal way of mapping shape data. Whenever possible, shape data should be mapped in a multivariate manner. However, this was the way to circumvent the strong family structure of the data and therefore to be able to implement the LMM. We have now mentioned the limitations of this approach  and removed the line “to facilitate association mapping”.The choice of the first 18 PCs was totally arbitrary, as it usually is when it comes to decide on the “relevant” PCs. Here we just chose as a cutoff, the PCs that explained at least 1% of the variance.14) About that  you say that you used the coefficient of determination but with multivariate data I guess you used an analogue based on distances (a ratio of sums of sum-of-squares) but there are other analogue based on classical multivariate statistics (Pillai etc.). It may be better to precise that it is procrustes distances based R2.Yes, we calculated r2 as the explained variance from sums of squares summed over all responses. We have clarified this in the third paragraph of the subsection “Association mapping”.15) It is unclear that you used leave-one chromosome out approach to handle the pop structure in your LMM .We didn’t use the leave-one-chromosome-out approach. We used a unique kinship matrix including all chromosomes to evaluate associations in each chromosome. We however acknowledge that excluding the focal chromosome might have increased our power a bit.16) SNP heritability: subsection “SNP heritability”, first paragraph: In my opinion, the way you present your computation of the "total heritability" is right in calculus but is little weird in term of genetics. In my opinion, you assess the sum of the additive genetic variances and do the ratio with the total variance to get a quite poor proxy of h2, and indeed means not much in term of heritability (see discussions of Monteiro and Klingenberg between 2002 and 2010).The specific way in which heritability was estimated in this study is again, as a comment above adequately pointed out, a limitation associated with performing univariate mapping with a multivariate trait, and we acknowledge that. However, we don’t completely agree with the idea that estimating heritability for each shape dimension is the right approach for the question we want to answer in this study (referring to the Monteiro and Klingenberg discussion). We think that depending on the objective of the study (as we think it was the conclusion of Monteiro and Klingenberg discussion), a univariate estimate of variance is adequate. If the objective is, for example, to address response to selection, a scalar heritability value is of little use. But, if, as in this study, the intention is to get an impression on how much additive genetic variance underlies a trait, a scalar value is the only way of making it comparable with other studies or traits.We have now clarified the intention behind calculating a scalar value for heritability .17) Functional evaluation: subsection “Samples used to functionally evaluate Mitf”: I don't agree with this statement about sexual dimorphism. There are plenty of phenotypes with SD in mammals and more specifically in the house mouse . The fact that SD is low on bone shape doesn't insure that it is the case in tooth.bone shape”, however we meant tooth shape, and not bone shape. There’s evidence  that sexual dimorphism is very small, and most of the time is absent in molar tooth shape. Therefore, although our sample size is small and doesn’t allow us to test for sexual dimorphism, we are still confident that such factor doesn’t affect our results. We have now corrected the text .We totally agree with this comment, and with the results shown in the paper suggested by the reviewers. By no means we wanted to imply that sexual dimorphism, in general, is not important in mice. Thanks to the reviewer comment, we noticed that we had stated “sex has very small effect in 18) Results, first paragraph: The most parsimonious hypothesis is simply it is within species variation and that the inter-subspecific variance is smaller than the within, species-specific loci have small effects on this character and are not distinguishable from within-species loci. I don't think we need to ask for transgressive variation or hybrid instability.We agree with this interpretation. We have now added it to the text , including reference where such pattern was described for bone shape. We have, however, kept transgressive phenotypes and hybrid instability as other sources of within-pop variation given the hybrid nature of the sample used in this study.19) Discussion: subsection “Genetic architecture”, first paragraph: Actually this conclusion is based on the huge amount of missing heritability . However, as your GWAS doesn't properly handle the multivariate nature of the shape space, you don't know anything about that because for instance a locus, orthogonal to all PCs, and explaining 1% of variance on each, will finally explained a lot of variance but will never be captured with your approach.Our conclusion has two lines of evidence. First, as the reviewers mention, the large amount of missing heritability. And second, the results derived from the chromosomal partitioning of the variance shown in We agree with the fact that our univariate mapping approach misses genetic associations that are spread across many PCs, and this will affect the missing heritability estimates. However, the percentage of variance explained by a locus increases, but not in a dramatic way, when it is calculated based on the PC it is associated with, vs. on the combined effect of all PCs. We therefore, don’t expect single loci to explain a lot of variance in this phenotype, even if multivariate approach were used.Yet, we offer another line of evidence for the polygenicity of tooth shape that is totally independent of the associations found in GWAS: the results of the chromosomal partitioning of the variance. This test has been used, mostly in human studies, to determine “how polygenic a polygenic trait is”. In this test, no individual associations between SNPs and phenotype are calculated, but the joint effect of all SNPs found in the different chromosomes. The rationale behind this is: if the effect of individual SNPs is small and many SNPs underlie phenotypic variation, there will be a linear correlation between chromosomal length and phenotypic effect. This is exactly the pattern that we see for tooth shape, and therefore, although we acknowledge and share the reviewers concerns regarding the first line of evidence, we are confident on the interpretation of our results.We have added lines in the Results section to make clear the importance of the chromosomal-partitioning-of-the-variance analysis.2. Will you do a t-test with N = 5 samples? Same conditions apply to Hotelling. Moreover you have twice more additional parameters plus correlation between the two variables to estimate. Maybe doing something non-parametric based on distances will be more reliable.20) You have a very small sample size to run Hotelling T2) is one of the most performant tests given a small sample size . More important than the significance of this test, is the distribution of the different mutant shapes that show a clear effect of each of the mutant alleles on molar tooth shape.It is always tricky to test for differences when the sample size is small. However, given the circumstances (it was not possible to obtain more mutant mice), a t-test (here a T21) Some p-values for statistical tests were not reported in the manuscript.We have now added the missing p-values in the last paragraph of the subsection “Tooth shape, mouse age and wear”, and subsection “Heritability”.


Finding epistatic interactions in large association studies like genome-wide association studies (GWAS) with the nowadays-available large volume of genomic data is a challenging and largely unsolved issue. Few previous studies could handle genome-wide data due to the intractable difficulties met in searching a combinatorial explosive search space and statistically evaluating epistatic interactions given a limited number of samples. Our work is a contribution to this field. We propose a novel approach combining K-Nearest Neighbors (KNN) and Multi Dimensional Reduction (MDR) methods for detecting gene-gene interactions as a possible alternative to existing algorithms, e especially in situations where the number of involved determinants is high. After describing the approach, a comparison of our method (KNN-MDR) to a set of the other most performing methods  is carried on to detect interactions using simulated data as well as real genome-wide data.Experimental results on both simulated data and real genome-wide data show that KNN-MDR has interesting properties in terms of accuracy and power, and that, in many cases, it significantly outperforms its recent competitors.The presented methodology (KNN-MDR) is valuable in the context of loci and interactions mapping and can be seen as an interesting addition to the arsenal used in complex traits analyses.The online version of this article (doi:10.1186/s12859-017-1599-7) contains supplementary material, which is available to authorized users. These last years have seen the emergence of a wealth of biological information. Technical improvements in genotyping and sequencing technologies have facilitated the access to the genome sequence and to massive data on genes expression and on proteins. This large availability of molecular information has revolutionized the research in many fields of biology. In parallel to these technical developments, methodological advances are needed to address the various questions of scientific interest that have been targeted when developing these new molecular tools. For example, the identification of up to several millions genomic variations in many species and the development of chips allowing for an effective genotyping of SNPs panels in large cohorts have triggered the need for statistical models able to associate genotypes from individuals and interacting SNPs to phenotypic traits such as diseases, physiological and productions traits . Our papThe systematic exploration of the universe of variants spanning the entire genome through genome-wide association studies (GWAS) has already allowed the identification of hundreds of genetic variants associated to complex diseases and traits, and provided valuable information into their genetic architecture  while alConsequently, an important question still remains about the exact relationship between the genomic configuration, including the interactions between the involved genes, and the phenotypic expression. The major idea in this respect is to try to associate observed variations at the macroscopic level (phenotype) to identified variations and their interactions at the molecular level.This view introduces at least two challenges. First, the genetic mechanisms underlying most traits of interest are complex and probably involve most of the time many genes and many interactions between these genes, leading to a complex relationship between genomic variants and phenotypes. So, modeling and identification of every, and even of any, interaction is a potentially very challenging task . Second,In the literature, various statistical methods have been used to detect gene-gene or gene-environment interactions , 11. ManOne of the most successfully used family of methods in the gene-interactions problems is multifactor dimensionality reduction (MDR) . The MDRRefinements of the method have been proposed to deal with potential limitations. Cattaert et al.  has propAlthough applied to numerous genetic studies , 34, MDRIn this paper, we propose a novel MDR approach using K-Nearest Neighbors (KNN) methodology (KNN-MDR) for detecting gene-gene interaction as a possible alternative to current MDR methods in situations where the number of involved determinants is potentially high and the number of tested markers is large. After explaining the rationale of our method, we will provide results on the comparison of KNN-MDR to a set of competitor methods on both simulated and real datasets.KNN stands for “K Nearest Neighbors” and is one of the most popular algorithms for pattern recognition and classification. Roughly, classification of an observation can be made using a majority vote within the K nearest neighbors of the observation , where thigh risk’ or ‘low risk’). To illustrate, if a set of N SNP markers is used in a case-control study to define the multi-locus genotype, 3N genotypes are possible. Each of these genotypes can be mapped to a status with only 2 values (case or control) using a majority vote on the statuses of the training set individuals falling into that genotype. The classification performances of any set of markers used to define the genotypes can then be assessed, typically using a cross-validation procedure, where the performance is estimated on a test set for each partition trough a measure involving sensitivity and/or sensibility of the classifier, and averaged over all partitions. For all computations reported in this paper, we have used a 10-fold cross-validation procedure and assessed the performances using ‘balanced accuracy’, which is a simple average of the sensibility and the sensitivity of the classifier. Repeating this procedure over all possible markers sets allows obtaining the best model, which is defined as the set of markers providing the best allocation performances. In practical situations, the potential number of tested markers sets might be huge: if an exhaustive search is to be performed on all P-markers interactions in a GWAS with M markers, about M!/[P!*(M-P)!] ~ MP/P! combinations would need to be checked, a huge number with nowadays available markers panels. Significance for the optimal model can be obtained through a permutations test, in which the potential links between the individuals’ genotypes and the phenotypes are disrupted by randomly shuffling the phenotypes. The p-values obtained using this test have then to be corrected for multiple testing, where multiple tests are due to the number of models that are successively tested.The method will be described for dichotomous traits for the sake of simplicity, but could be extended to other situations using the approach described for GMDR [32]. The Multi-Dimensional Reduction (MDR) method is designed to replace large dimension problems with reduced dimension ones, allowing to make inferences based on a smaller set of variables. In the context of genomic studies, the idea in  is to reAlthough a widely used and well-established technique, MDR faces several problems, as detailed above. The computational load described in the previous section remains a major issue. Although recent publications  have proOur proposal is therefore to slightly modify MDR to allow facing some of the shortcomings of the method. The only modification is in the status allocation procedure: while MDR uses a majority vote among the  set of individuals sharing the same multilocus genotype as the tested individual, we propose to use a majority vote within a set of the K nearest neighbors of the tested individual. This procedure has the obvious advantage to eliminate the problem of potentially scarce or empty genotypic configurations mentioned above. On the other hand, this strategy introduces the need to define the neighborhood: a “distance” between individuals based on the genotypic configurations at the selected markers will be needed, and the size K of the neighborhood will have to be provided. These parameters of the method - the chosen distance, K - are further discussed in the Note that, in KNN-MDR, the computer burden scales quadratically with the number of individuals since the distances between pairs of individuals are needed, but is less sensitive to the number of markers since markers are pooled into windows. So, the important parameter from a computing point of view is the number of windows W, which does not necessarily increase when the number of markers increases.AntEpiSeeker  and BOOSMDR  is one oMegaSNPHunter  is targeAll these methods have been applied successfully to real datasets,These methods have different search strategies: exhaustive search , stochastic search (MegaSNPHunter) and heuristic search (AntEpiSeeker),Software implementing the methods is available.After designing our method, we needed to compare the performances of our approach to some of the other proposed algorithms. Since many methods are available , we deciIn order to assess the performances of the proposed method, we have simulated various situations and ran MDR, BOOST, MegaSNPHunter, AntEpiSeeker and KNN-MDR on the same datasets to compare the performances in terms of detection power and accuracy. The generation of the simulation datasets will be described in the following lines.To obtain a linkage disequilibrium (LD) pattern similar to patterns that can be observed in humans, SNPs spanning the human chromosome 9 (HSA9) have been obtained from a study on Crohn disease in Caucasians  for 197 200 potentially different individuals while conserving some LD.Since many different individuals are needed in the simulations, we used a trick similar to  to generG SNP were then randomly chosen as having an effect on the simulated phenotype, where G = 2, 3, 4 or 5. Since SNP selection is random, SNP might be linked or not.A penetrance is computed for each multi-locus (G SNP) genotype in such a way that each of the G SNP shows no marginal effect:i denotes the genotype for locus i , 0, 1, 2 are the number of instances of the minor allele in the SNP genotype, A means Affected, P(A | Gi) is the penetrance for genotype Gi, and P is the prevalence of the disease in the sample .where G1 = k, G2 = m, …) where k, m, … = 0, 1 or 2 are obtained to meet the requirement of no marginal effect (see previous step). An algorithm to compute these penetrances is provided in an appendix  are then obtained by randomly sampling a uniform distribution between 0 and 1 and comparing the obtained deviate d to the multi-locus penetrance MP: if d < (>) MP, the individual is (not) affected.Selected SNP genotypes were then used to generate the binary phenotypes. More details of the algorithm are given in an appendix  and those potentially significant . When a higher precision was needed for the p-values (see below), an adaptative permutations scheme was used, in which windows not reaching a pre-determined p-value threshold are progressively abandoned in the permutations scheme since these windows are very unlikely to finally reach a significant result p-value bWhen working on large sets of markers, such as for example those commonly met in GWAS analyses, splitting the complete set into a reasonable set of windows could necessitate including large numbers of markers in each window, which would eventually swamp the signals of interest, as explained in the p < 0.05), and the “corrected power” as the proportion of simulations where the association was detected and involved the causal SNP . The comparison of the five tested methods is presented for situations where G = 2 in Table Since performing classical MDR analyses on a large number of markers is not an obvious task, especially when the number of putative involved SNPs (noted G) is 3 or more, we restricted our analyses to G = 2 and G = 3 to make comparisons to other methods feasible. We have defined the “power” as the proportion of simulations where an association signal was detected (p = 0.0143 when comparing KNN-MDR to its closest competitor for G = 2 and p = 7.23e-7 for G = 3).As can be seen from Tables A short literature survey , 48–50 lWe ran another set of simulations to assess the respective effects of the sizes of the windows and of the number K of neighbors on the (corrected) detection power. Results of these simulations are reported in Table 20 k SNP were first extracted from the data. Although several selection procedures could be applied, we simply selected 1 SNP every 15 SNP.We divided the data into 200 windows of 100 SNP each.We then tested each of the 19900 possible pairs of windows (sets of 200 SNP) using KNN-MDR.p-values were lower than 2.5e-6 (a threshold obtained after Bonferroni correction at level 0.05). To reach that significance level using a permutations procedure, we used the following adaptative scheme: after 100 permutations performed on the 19900 possible pairs of windows, only those reaching the 0.05 level were considered for the next round of permutations, assuming that those not reaching that level of significance were very unlikely to reach the desired significance at the end of the process. This left us with 2319 combinations. In the next round, 900 more permutations were performed, and only the combinations reaching the 0.005 level were kept . Repeating this procedure for 1.0e4, 1.0e5, 1.0e6 and 2.0e6 permutations, and respective thresholds equal to 5.0e-4, 5.0e-5, 5.0e-6 and 2.5e-6, we ended up with the 83 sets cited above.We extracted the 83 sets for which the The SNP hidden in step 1 were then recovered, leading to 83 sets of 3000 SNP .KNN-MDR was applied on every set from step 5: the sets were divided into 30 windows of 100 SNP and all 435 combinations of windows pairs in each set were considered by KNN-MDR.p-value < 1.15e-4 (Bonferroni correction at level 0.05).We kept the 241 sets of 200 SNP with a MDR was then used for the sets from the previous step, leading to examine 19900 SNP-SNP interactions for each set.p-value < 2.51e-6 (Bonferroni correction at level 0.05) were then considered as significant.The interactions with a Since working on such a large dataset (>300 k SNP) is very demanding in terms of computing time, we proceeded as follows:Results from this analysis are presented in Table This paper has introduced a new MDR approach to find markers interactions in genomic scans. It could also be used for other attributes than markers, such as environmental factors, leading to a gene-environment interaction search method. Due to the proposed strategy relying on the MDR approach, and in parallel with a recent study  using . This might allow to highlight interactions between markers far apart on the genomic map (trans-interactions), while some strategies proposed to restrict the scans to close-by markers (cis-interactions) to reduce the amount of computations.We now discuss some of the features of the method:In this paper, although the algorithm given in the appendix can be used for G larger than 3, only 3 markers have been used to generate the phenotypes. Nevertheless, in practical applications, it is not unlikely that situations involving more than 3 loci might exist. These situations might increase the interest of using methods such as KNN-MDR. Indeed, when more regions are involved in the phenotype, this could decrease the distance measure between individuals sharing some or all of these regions and better cluster individuals sharing the same status. Conversely, in MDR, discovering such complex patterns would likely necessitate to increase the number of loci scanned simultaneously, which would make computations even more difficult. Also, increasing the number of loci increases the number of cells with no (or very few) observations, making status allocation potentially inaccurate or even impossible.We mentioned earlier that parameters setting in KNN-MDR mainly involves defining the sizes, positions and the number of windows, the number K of neighbors and the distance measure. All parameters are problem dependent, making it difficult to devise general rules. Nevertheless, some guidelines might be given.In all the analyses performed in this study, we have only used Mahalanobis distances, as already mentioned. The reason was that this distance allows to take into account potential correlations between attributes  and because it makes it possible to weight the attributes in the sum . In our studies, only SNPs have been used, for which the distance proposed in the Mahalanobis measure makes sense, with D = D = 0.5* D, where AA, AB and BB are the three possible SNP genotypes. This might be different and might need more investigations if other types of genetic variants are used. Note also that, in most computations, to reduce the computational burden, the correlation between neighboring markers has not been estimated but set to 0 , which might potentially affect the power. Although we did not explicitly test this, we expect that including the correlations would lead to better take into account the linkage disequilibrium, which should have a positive effect on the detection power. So, using this information might be favorable in terms of power, but at the cost of an increase in the computation time. Note also that using this kind of distance makes less sense when working with markers with more than 2 alleles, unless it can be postulated that the distance between, for example, alleles 1 and 3 is roughly twice the distance between alleles 1 and 2. An easy to compute and similar distance measure would then be to square the number of differing alleles  between two compared genotypes, to normalize as for the Mahalanobis distance, to sum over all markers in the window and to take the square root of the product. This “binary” distance is implemented in our KNN-MDR software.For the windows dimensions, our idea is to use the assumption that individuals sharing mutations responsible for the trait should look more similar in the surroundings of these mutations than those not sharing these mutations. The resemblance should thus extend to neighboring markers, where the neighborhood size is a function of the linkage disequilibrium (LD) in the region. In situations where LD increases (due to the studied population and/or the markers density), distance between individuals sharing genomic regions  should decrease and detection power should increase. Note that this genomic feature is ignored in the other tested methods. Accordingly, the windows sizes W should ideally be defined to capture the local linkage disequilibrium. Since the measurable LD is dependent on the population history and on the markers density, assessment of this measure should first be made in order to have reference dimensions for the various windows to be used in KNN. Note that the extent of LD need not be the same across the whole genome: accordingly, the size of the windows might be varied along the genome to better reflect the underlying structure and better capture the relevant information.To illustrate that expected behavior, we have performed the simulations leading to Table Next, the number K of neighbors should somehow reflect the number of animals sharing regions harboring causal mutations. This number is of course unknown and difficult to evaluate a priori because it is dependent on various population and trait parameters such as the history of the population or the genetic heterogeneity of the trait. Furthermore, it might vary from region to region, making it difficult to devise general rules allowing to infer relevant values of K. Possible “brute force” approaches would be to rerun the algorithm with varying number of neighbors (grid search) or to use bootstrap methods . This stOur simulations have shown that, as reported in other studies, results are often penalized by high false positive rates Table . One obvThe reason for the drop in the power of the alternative methods when considering the accuracy is not completely clear, but we can suggest a tentative explanation.As can be seen from Table A potential criticism on our “accuracy measure” is that using windows sets makes it more likely to cover the culprit regions, and so this “accuracy” measure is biased in favor of KNN-MDR. For that reason, and to make the comparison fair between the methods, we have chosen the parameters to end up with similar number of markers in the finally selected markers sets in each approach. Note nevertheless that the resolution of KNN-MDR could eventually be increased in these analyses, for example using the strategy described for large datasets in the Figure So, in our study as in the other ones, statistically significant SNP interactions have been identified using KNN-MDR and MDR in a genome-wide association study. Their biological relevance is obviously not clear at this stage and needs more investigations in the future. We can nevertheless say that some of our results are consistent with other results in the domain of Rheumatoid Arthritis , 46, 54), 5446, 5In our results, the comparisons between  and KNN-MDR in terms of computer resources has not been fully addressed. Nevertheless, it has been shown how and why KNN-MDR decreases the computer load with respect to MDR, making it a potential candidate to analyze large datasets, as shown for the RA data. To be fair, it should be mentioned that computing nearest neighbors is more computer intensive than a majority vote in the subset sharing the same multi-locus genotype. Nevertheless, as shown in the simulations, and as can be understood from the previous discussion, computations remain more affordable in KNN-MDR than in MDR and the other methods for similar scans. Furthermore, strategies could also be devised to make KNN-MDR efficient, such as pre-computing distances for windows and using distance additivity properties to compute distance over several windows.Another point that might be worth adding is that, although KNN is natively a classification method, we have used it here in a detection context. KNN-MDR could nevertheless as well be used as a classification tool: to that end, the best model  could be used to compute the neighborhood of a new individual and classify the latter in one or the other category.In summary, KNN-MDR is an alternative to existing methods for detecting epistatic interactions, with interesting features. Among these, we have demonstrated that KNN-MDR is more computationally efficient than other exhaustive strategies, facilitating the analysis of large-scale data sets with potentially genome-wide SNPs. The method is also capable to detect high-order interactions and to take into account linkage disequilibrium (LD). Another advantage is that it is able to detect interactions between SNPs even in the absence of marginal effects. Also, the method is non-parametric: no prior distribution is assumed, unlike many parametric-statistical methods. Nevertheless, parameters  are available to allow some flexibility in the search strategies, which could help to render the method useful in other classification contexts.Although KNN-MDR is potentially beneficial for epistasis detection, several aspects would nevertheless deserve more investigations. For example, the burden associated to the computation of the K nearest neighbors could become an issue when the dataset is very large. Since the load increases quadratically with the number of individuals, and linearly with the number of markers, improving the computational performances of the method could necessitate some code optimization to make the program more efficient. Another point necessitating more work is the tuning of the parameters allowing an optimal detection power. This includes the optimal sizes of the windows - which should be dependent on the studied population, the markers density, the LD pattern, the optimal size of the neighborhoods to be considered, the pre-selection of markers in the early phase of large dataset analyses, the distance measure or the adaptative selection scheme for the selection of markers in large studies, among others.Additional file 1:Computing multi-locus penetrances. (DOCX 19 kb)Additional file 2:The data set(s) supporting the results of Table Additional file 3:The data set(s) supporting the results of Table Additional file 4:Competitor methods. (DOCX 14 kb)Additional file 5:The data set(s) supporting the results of Table Additional file 6:The data set(s) supporting the results of Table Additional file 7:Table Additional file 8:KNN MDR user’s guide. (PDF 116 kb)