We previously noted that levels of genetic complexity underlying heritable variation in growth differed among chemical conditions in a single cross 
. Here, we sought to determine the generality of our previous finding by examining additional crosses. We first generated the strains and microarrays to conduct X-QTL in all 6 pairwise crosses of the BY, RM, YJM, and YPS strains (Materials and Methods
). Because the statistical power of X-QTL is dependent on effective enrichment of highly resistant cross progeny in a segregating pool, and the crosses vary in their genetic compositions, leading to different distributions of resistance among the progeny of each cross, we used dose-response experiments to determine cross-specific, highly selective drug concentrations for each of 13 diverse chemicals that resulted in similar selection intensities for all crosses (Materials and Methods
; File S1
). Once the selective doses were determined, we conducted one X-QTL experiment for each chemical and cross combination.
We observed substantial variation in the number of loci detected in different conditions and crosses (). Across all 78 X-QTL experiments, we identified 837 total peaks at a False Discovery Rate (FDR) of 1%, or an average of 10.7 peaks per trait per cross (; Figure S1A
). Both the chemical and the cross had significant effects on the number of peaks detected in an X-QTL experiment (ANOVA, chemical effect F
; cross effect F
0.014), with the effect of the chemical (partial R2
0.46) being much larger than the effect of the cross (partial R2
0.11). An ANOVA testing the effects of chemical and strain resulted in a similar effect of chemical on the number of detected peaks (partial R2
), but no strain had a significant effect on its own (partial R2
<0.02; F<2.5, d.f.
1, p>0.12; Materials and Methods
). Consistent with a comparatively small effect of strain background on genetic complexity, only one trait showed a significant excess of peaks in crosses involving any one strain: crosses in which RM was one of the parents had an excess of peaks in diamide (χ2
1, Bonferroni-corrected p
; ). These results suggest that genetic complexity in yeast is mainly a property of the trait being examined rather than of the strain background.
Numbers of detected peaks and distinct loci.
For each trait, we expected to detect loci at the same genomic positions in different crosses sharing a parent. To identify only the distinct loci affecting each trait, we performed a grouping procedure on the peaks identified in all crosses for a given chemical condition. We found 411 distinct loci (an average of 32 loci per condition), with a minimum of 8 loci for growth in cycloheximide and a maximum of 57 loci for growth in zeocin ( and ). We then examined the extent to which these loci showed effects on growth in multiple conditions. For a range of genomic window sizes, we considered peaks detected for multiple chemicals within a window to correspond to the same underlying locus, and counted the number of conditions in which the locus showed an effect. With 50-kilobase (kb) windows, we found that 40% of the distinct loci had effects in only one condition, 29% had effects in two conditions, 11% had effects in three conditions, and only 20% had effects in four or more conditions (; Materials and Methods
). Although the numbers differed across window sizes, the general observation that most of the detected loci had effects in a relatively small number of the tested conditions, and only a small number of loci showed effects across a large number of conditions, held over the entire range of plausible sizes. With 50 kb windows, three loci exhibited effects in more conditions than expected by chance (Materials and Methods
). These loci were located on Chromosome V near the X-QTL control marker CAN1
, Chromosome X near ENT3
, and VPS70
, and Chromosome XIV near the pleiotropic gene MKT1
Genome-wide plots of detected loci.
We next examined the patterns of detection of loci for each trait across the six crosses. With four strains, two simple patterns are possible at bi-allelic loci: one strain can carry an allele that confers susceptibility or resistance relative to the allele carried by the other three strains, or two strains can carry the more susceptible allele and two strains the more resistant allele. We refer to these cases as “allelic singletons” and “allelic doubletons,” respectively. These two cases should give rise to different patterns of peaks: peaks with a consistent direction of effect in all three crosses involving one strain for allelic singletons, and peaks with specific effect directions in four specific crosses for allelic doubletons (Table S1
; ). Allowing for false-negative peaks, 135 of the 411 distinct loci showed patterns consistent with allelic singletons, and 28 showed patterns of peaks consistent with allelic doubletons (Table S1
Patterns used to identify allelic singletons and allelic doubletons in the X-QTL data, and the number of loci detected with these patterns.
We attempted to narrow the number of candidate genes for each of the bi-allelic loci by scanning the parental genome sequences for SNP alleles that are found in the four strains in a pattern consistent with the peaks. Using this approach, we found an average of 10 candidate genes per locus, with a range of 1 to 18 genes. Further restricting the list of candidate genes to those that carry nonsynonymous polymorphisms with appropriate allelic patterns reduced the average number to 6 per locus. We attempted to validate the genes underlying some of these loci by constructing allele replacement strains, and found reproducible evidence that HXT6
harbor functional polymorphisms that confer growth differences in rich medium and tunicamycin, respectively (Figure S2
; Materials and Methods
is a high affinity glucose transporter 
, suggesting that variability in glucose uptake may contribute to growth differences among the strains. The effect of RED1
on tunicamycin resistance is less clear, as this gene is thought to be involved in chromosome segregation 
, and tunicamycin affects the unfolded protein response. We also constructed allele replacement strains for two other genes: NUP157
, which lies within a copper sulfate resistance locus with the resistance allele coming from BY, and PTK1
, which lies within a paraquat resistance locus with the resistance allele coming from YPS. However, we obtained inconsistent results for NUP157
: the allele replacements produced effects on resistance that were in the opposite direction from those seen in the X-QTL selections, and also caused growth defects on standard rich medium, suggesting that we did not identify the right candidate genes for these loci.
In addition to the simple bi-allelic patterns, we observed other more complex patterns of peaks (). Some of these are consistent with the presence of allelic series, in which either three or four alleles with different phenotypic effects are present among the four strains; we observed 29 examples involving at least 3 alleles and 9 examples that can only be explained by the presence of 4 different alleles (Table S2
). The other 210 loci (51% of all loci) showed patterns of peaks that were not easily interpretable in terms of specific allelic classes. This probably reflects a mixture of false negatives in which a peak was present but not detected in a given cross, and cross-specific effects due to non-additive interactions and linkage between loci.
The allele frequency spectrum of causal loci is critical for the design of genetic mapping studies and for understanding sources of missing heritability in natural populations, including humans. As discussed above, we were able to distinguish and enumerate two simple allelic classes—singletons and doubletons. We used a maximum likelihood approach that accounted for false negatives to estimate the ratio of allelic singletons to doubletons. We estimated the peak detection rate to be 51%, with a 95% confidence interval of 39%–62%, and the ratio of allelic singletons to doubletons to be 3.03, with a 95% confidence interval of 1.7–5.3 (; Figure S3
). This result suggests that despite the high statistical power of X-QTL, a substantial fraction of loci with weaker effects likely still go undetected in any one cross. Interestingly, the estimate of the ratio of allelic singletons to doubletons is similar to that observed for nonsynonymous polymorphisms in the genomes of the parent strains (2.97), and is shifted toward singletons relative to both the neutral expectation of 2.67 and the observed ratio of 2.57 for 109,585 SNPs genome-wide (). Thus, the frequency spectrum of variants that contribute to complex trait variation in yeast appears to be mildly shifted toward lower frequencies by purifying selection, but, given the wide confidence interval for the estimated ratio of allelic singletons to doubletons, we cannot rule out that the variant frequencies follow the neutral spectrum.
Population genetics of identified loci.
Several lines of evidence suggest that lineage-specific selection or demography has shaped variation among the four strains. We observed an excess of allelic singletons at detected loci for BY and RM, and a deficit for YJM and YPS, relative to the numbers of singleton SNPs in the parent genomes (χ2
3, p<0.0001; ). The laboratory strain BY also exhibits other signatures of selection for both general and chemical-specific resistance. For instance, BY carries a marginally significant excess of allelic singletons that confer resistance relative to the other three strains (Fisher's exact test, Bonferroni-corrected p
0.06; ; ). In addition, trait-specific sign tests 
identified one significant result: an excess of copper sulfate resistance alleles contributed by BY in the BYxRM cross (18 loci with BY carrying the resistance allele and 2 loci with RM carrying the resistance allele; binomial test, Bonferroni-corrected p
0.031; ). Interestingly, BY is among the most copper-resistant S. cerevisiae
, and our data suggest that this resistance in BY may be the result of selection, possibly due to the use of high levels of copper or another chemical with similar effects in standard growth media. However, the BYxYJM and BYxYPS crosses do not show significant excess of BY alleles, and RM is also among the more highly copper-resistant strains 
, making the excess of BY resistance alleles in the BYxRM cross difficult to explain. Overall, our results are consistent with previous analyses that have shown lab strains isogenic to BY exhibit high evolutionary rates relative to other yeast isolates 
, probably due to both relaxed purifying selection 
and adaptation 
We have shown that variation in chemical resistance among yeast strains is typically due to a large number of underlying loci. The level of genetic complexity, as measured by the number of loci detected, is largely a property of each resistance trait, although it is also affected to a lesser extent by the choice of parent strains. The total number of distinct loci detected for a trait in these crosses among four strains ranged from 8 to 57, and these numbers substantially exceeded those seen in any one cross. These observations suggest that the total number of loci affecting certain resistance traits in S. cerevisiae can be very large, since many of them will have escaped detection because they don't vary among the four parent strains examined here, have effect sizes that are too small, or are too closely linked to be resolved as separate loci by our mapping technique. Our results suggest that the functional variants underlying complex traits are broadly distributed across the frequency spectrum from rare to common alleles, and that many loci harbor more than two allelic variants. These findings provide multiple non-exclusive explanations for the sources of the “missing heritability” of complex traits, and illustrate the power of a simple model system for probing genetic complexity.