The results of our investigation into SNPs associated with mathematical ability are compatible with those of similar studies of SNPs associated with reading  and g , in providing strong molecular genetic support for the Generalist Genes Hypothesis which has emerged from quantitative genetic research. We have demonstrated that SNP sets identified on the basis of their association with a composite measure of mathematics at age 10 correlate significantly with each of the three diverse components of mathematics assessed both by web-based tests and teacher ratings. We have shown that the influence of these SNP sets on mathematics can be seen as early as age 7 and continues to age 12, and that these associations, which become weaker further away from the original target age, follow the simplex pattern expected from quantitative genetic longitudinal results [9, 23]. The strongest support for the Generalist Genes Hypothesis comes from the mathematics SNP-set associations with reading and general cognitive ability at age 10.
Although we demonstrate overlap in the influence of the mathematics SNP sets with other cognitive abilities, our results also provide some evidence for genetic specificity. For example, when we regressed out variance due to both web-tested and teacher-rated reading scores at age 10 as well as variance due to age-10 general cognitive ability, the SNP sets still correlated with age-10 mathematical ability: 0.046 (p = 0.060; N = 1139) and 0.072 (p = 0.007; N = 1141) with the 10- and 43-SNP sets, respectively. Although the 10-SNP-set correlation is no longer significant, the 43-SNP set still accounts for 0.52% of the variation in mathematical ability at age 10 in our sample.
It would be interesting to know if this specificity was due to the specific action of particular SNPs. We individually analysed the three SNPs most strongly associated with mathematics at age 10 - rs11225308, rs363449 and rs17278234. Our analyses indicate that all three of these SNPs are associated with the other cognitive traits, though admittedly to varying degrees. Such a general influence may be expected, as the most proximal genes to these three SNPs - matrix metalloproteinase 7 (MMP7), glutamate receptor ionotropic kainate 1 (GRIK1) and dynein axonemal heavy chain 5 (DNAH5) - all play important yet quite general roles in development. Although MMP7 is involved in the breakdown of extracellular matrix during normal physiological processes such as embryonic development, growth and tissue repair , GRIK1 and DNAH5 have more direct links to the brain. GRIK1 encodes a glutamate receptor which mediates neurotransmission and synaptic plasticity [25, 26], and dysfunction of this gene is implicated in a number of psychiatric phenotypes [27, 28]. DNAH5 encodes the dynein axonemal heavy chain 5 protein, which is the force-generating component of cilia, the correct functioning of which is essential in all areas of embryonic growth ; DNAH5 in particular has been demonstrated as vital for normal brain development . If MMP7, GRIK1 and DNAH5 underlie the SNP associations that we have reported, they may be indicative of the variety of 'generalist' genes involved in development - especially of the brain - which one might expect to exert small effects over a range of cognitive abilities.
In the present study, rs11225308, rs363449 and rs17278234 yielded correlations of 0.074, 0.069 and 0.067, respectively, with the 10-year mathematics composite (Table 1). With Ns of 2182 to 2188, power is marginal to detect such small correlations. However, the 10- and 43-SNP sets yielded correlations of 0.167 and 0.164 with the 10-year mathematics composite, and were significantly associated with many of the traits assessed here, even when the individual SNPs tested were not, which speaks to the value of using SNP sets. Moreover, despite containing 33 SNPs which failed to replicate their association with mathematics in the original study, the 43-SNP set performed just as well, and in some cases far better, than the 10-SNP set. Due to the nature of their selection - in a genome-wide association study bound by chance to generate some false positive results - it is unlikely that all of these 33 SNPs represent true associations. However, the superior performance of the 43-SNP set suggests that many of them may exhibit true effects on mathematics and other cognitive abilities that are too small to be detected when analysed alone. It also indicates that adding some false positive results into a composite SNP-set score containing a number of true positives is not detrimental to its performance in association analyses.
One of the main limitations of this study is its reliance on the same sample used to identify the 10-year mathematics SNPs. The TEDS sample, in which subjects have been assessed throughout development across a wide variety of traits, offers a prime opportunity for empirically testing the Generalist Genes Hypothesis at the molecular genetic level. However, if SNPs are associated with mathematics in this sample, it may not be surprising that they show associations with other traits with which mathematics is phenotypically correlated. Analyses conducted on two behavioural traits (data not shown) partially address this argument. Although measures of academic motivation and mathematics liking at age 12 correlate with mathematics at age 10 (0.356 and 0.448 respectively), they do not yield significant associations with the mathematics SNPs or SNP sets. Therefore, rather than simply reflecting phenotypic correlations between mathematics and other traits, our results seem to be more in line with the genetic correlations estimated in quantitative genetic research, supporting the Generalist Genes Hypothesis. Nevertheless, the fact that the cognitive measures assessed are correlated limits our ability to draw firm conclusions from this study. As all work on these SNPs has thus far been conducted within the TEDS sample, the replication of our findings in independent samples is essential before they can be generalised to the wider population.
A second limitation is the inclusion of individuals with missing genotype data in our 43-SNP-set analyses. As none of the 43 SNPs were in linkage disequilibrium, we were unable to impute the likely value of a missing genotype based upon available genotype information. Our substitution of missing genotypes with mean sample-scores did not affect our sample means, however it did artificially reduce variation. Though the use of a multiple imputation method  in forming a 43-SNP-set would have been complicated, the uncertainty associated with using missing values would have been more properly accounted for. Nonetheless, as the 43-SNP-set used here performed in much the same way as the 10-SNP-set, which did not include missing data, we would not expect multiple imputation to produce vastly different results. Our cross-sectional and univariate approach may also be seen as a limitation. As we assessed genetic associations in mathematical ability across several ages and abilities, it would have been possible to use longitudinal and multivariate models . However, we suggest that the present focus on the Generalist Genes Hypothesis is better served by comparing associations across ages as well as across abilities, especially as the sample sizes often differ considerably.
Another possible limitation of this study lies with the measures used. Although quantitative genetic studies report genetic overlap between different cognitive abilities, and although we report an overlap in the association of specific genetic markers, this may be due to the complex nature of the tests employed. It is possible that the tests conventionally used to assess different learning abilities tap such a wide variety of cognitive processes that they will inevitably share many of the same genetic underpinnings. For this reason, it might be interesting to examine the association between the mathematics SNP sets and more specific cognitive processes. However, as there is evidence to suggest that even very specific tasks, such as different aspects of executive function  or information-processing measures , are highly genetically correlated, it is unlikely that we would see a significant deviation from the Generalist Genes perspective.