Interactions Among Flower-Size QTL of Mimulus guttatus Are Abundant but Highly Variable in Nature

Abstract

The frequency and character of interactions among genes influencing complex traits remain unknown. Our ignorance is most acute for segregating variation within natural populations, the epistasis most relevant for quantitative trait evolution. Here, we report a comprehensive survey of interactions among a defined set of flower-size QTL: loci polymorphic within a single natural population of yellow monkeyflower (Mimulus guttatus). We find that epistasis is typical. Observed phenotypes routinely differ from those predicted on the basis of direct allelic affects in the isogenic background, although the direction of deviations is highly variable. Across QTL pairs, there are significantly positive and negative interactions for every trait. Across traits, specific locus pairs routinely exhibit both positive and negative interactions. There was a tendency for negative epistasis to accompany positive direct effects and vice versa for the trait of corolla width, which may be due, at least in part, to the fact that QTL were identified from their direct effects on this trait.

EPISTASIS contributes significantly to intrapopulation variation in floral morphology, development time, and male fitness components of Mimulus guttatus (Kelly 2005). The aggregate effect of interactions among QTL substantially alters the resemblance of relatives and phenotypic response to inbreeding. However, previous experiments did not identify the specific character of interactions between QTL. For example, it is not clear whether interactions change the rank order of QTL genotypes. If the direction of allelic effect changes with genetic background, so-called “sign epistasis” (Weinreich et al. 2005), the same selection pressure may favor different alleles in different genomic contexts, e.g., different subpopulations of a species (De Brito et al. 2005). When the trait is fitness, these kinds of interactions naturally generate peaks and valleys in genotypic fitness landscapes (Wright 1932; Burch and Chao 2000).

Sign epistasis can involve a reversal of allelic effect at one or both loci of an interacting pair. Poelwijk et al. (2007) define the double reversal as “reciprocal sign epistasis” and contrast sign epistasis generally to “magnitude epistasis” where the magnitude but not the direction of allelic effects changes with genetic background. Magnitude epistasis includes synergistic interactions (alleles have greater effect in combination than individually) and less-than-additive or diminishing returns interactions (alleles have lesser effect in combination) (see Crow and Kimura 1970). Alternatively, one can classify interactions as positive or negative (Phillips et al. 2000)—positive if the observed phenotype of an allelic combination exceeds that predicted from direct effects at each locus, and negative if the phenotype of the combination is less than the additive prediction. Unfortunately, there is no simple logical mapping from the magnitude/sign epistasis classification to the positive/negative classification, nor to epistasis in the classical sense (Bateson 1909), wherein one locus masks the effect of another. The taxonomy of epistasis is further complicated by dominance (Routman and Cheverud 1997), higher-order interactions (Templeton 2000), and environmental dependencies (Brock et al. 2010).

Molecular genetic studies provide clear examples of sign epistasis. Here, the interacting polymorphisms are often within the same gene. For example, the stability of RNA secondary structures requires matching of nucleotides at different positions. Whether a particular nucleotide change increases or reduces stability depends entirely on the identity of the nucleotide at its paired site (Chen et al. 1999). With “compensatory evolution” (Moore et al. 2000), a mutation that is neutral or detrimental in the original genetic background becomes advantageous by compensating for some other mutation that has recently fixed or at least become prevalent within the population. Mutations conferring antibiotic resistance often have deleterious side effects that reduce bacterial fitness in the absence of the drug. These side effects are attenuated by secondary mutations that are often detrimental in the original genotype (Levin et al. 1997; Schrag et al. 1997).

Despite the progress in research on microbes (Weinreich et al. 2005; Elena et al. 2010), we currently know little about the prevalence or nature of epistasis for quantitative traits (Carlborg and Haley 2004) and particularly its impact on standing (segregating) variation within natural populations. Eshed and Zamir (1996) found extensive epistasis among QTL in Lycopersicon (tomato), but interactions primarily influenced the magnitude of single-locus effects and not their direction. In contrast, Kroymann and Mitchell-Olds (2005) documented a case of QTL effect reversal in Arabidopsis thaliana. The high allele for biomass accumulation in the Ler-0 accession becomes the low allele when introgressed into another line (the Col-0 accession). Patterns of gene sequence variation suggest that this polymorphism is maintained by balancing selection. In Avena barbata, two loci with negligible average effects exhibit sign epistasis for fitness in a cross between mesic and xeric genotypes (Latta et al. 2010).

Even with these examples, it is difficult to evaluate the quantitative frequency of epistasis or regularities in its nature from the current literature. In part, this is because discovery of QTL×QTL interactions is typically idiosyncratic. In a segregating mapping population such as F₂’s or recombinant inbred lines, there is limited replication of particular multi-locus genotypes. As a consequence, it is difficult to accurately estimate the mean phenotype of any particular multi-locus genotype. Also, there are an enormous number of pairwise tests in a full simultaneous scan, which leads to very stringent significance levels (see box 2 of Carlborg and Haley 2004). Thus, while genomic scans have successfully identified interactions (e.g., Li et al. 1997; Cheverud 2000; Montooth et al. 2003), it is hard to know if the many nonsignificant interactions are due to absence of effect or absence of power.

Our intention in this study was to conduct a comprehensive survey of interactions among a specific set of monkeyflower QTL. These loci are polymorphic within a single contiguous natural population. We first mapped flower-size QTL within nearly isogenic lines (NILs). Measurements on the NILs isolate the “direct effect” of each QTL, i.e., how the polymorphism affects phenotype in a single uniform genetic background. We then intercrossed the various single-QTL NILs in all combinations to generate the four double homozygotes for each pair of QTL: aabb, AAbb, aaBB, and AABB. This design allows high replication of each multi-locus genotype and hence reasonable power to detect even moderate epistasis (e.g., Moyle and Nakazato 2009). Applying this methodology, we find that epistasis is more the rule than the exception, although the form of interaction is highly variable among QTL pairs.

Materials and Methods

Mapping populations

Three panels of nearly isogenic lines (Molly-2, Molly-3, and Molly-4) were synthesized by repeatedly backcrossing a donor genotype into a fixed genetic background (the recipient line). A distinct donor genome was used for each panel. Each donor was an F₁ from a distinct cross of large- and small-flowered parents produced by artificial selection. The initial source population for the selection experiment was a collection of genotypes derived from a single natural population on Iron Mountain in central Oregon (Kelly 2008). As a consequence, our donor genotypes (the F₁’s) were highly heterozygous for flower-size QTL specific to that natural population. The same recipient genome, IM767, was used for each panel. It is one of several hundred randomly extracted lines from Iron Mountain (Willis 1999). It was chosen for this experiment because it has high pollen fertility relative to most inbred lines, yet is intermediate in flower size.

The first generation of backcrossing to IM767 yielded ∼200 lineages per panel. After five subsequent generations, Molly-2 retained 198 lines, Molly-3 retained 156 lines, and Molly-4 retained 139 lines. The Mendelian expectation is that each NIL would be (1/2)⁶ donor (1.5% on average) and the remaining genome IM767. We self-fertilized each sixth generation NIL and collected leaf tissue for genotyping (see below). We grew 12 progeny from each selfed family under standard greenhouse conditions (see Arathi and Kelly 2004) and measured the widest width of the corolla on the first two flowers. Each panel was assayed in a distinct grow-up. Including 154 IM767 individuals, a total of 5708 plants were measured across the three panels. There was significant variation among NILs within each panel. A summary of NIL means is given in supporting information, File S1.

QTL identification

We used a selective genotyping method to identify genomic regions affecting corolla width. NILs were ranked by mean corolla width within panels, and we selected NILs from the high and low tails of each distribution for genotyping. Twenty-four NILs were selected from Molly-2, 16 from Molly-3, and 10 from Molly-4. These 50 NILs (the parents of the plants measured for phenotype) were genotyped at 236 length-polymorphic markers. Except for five microsatellite loci, all markers were selected from the MgSTS marker set (http://www.mimulusevolution.org). The marker list, with genomic locations on the current Genome build, is reported in File S2. The F₂ map length of M. guttatus is ∼1500–1700 cM (Fishman et al. 2008; Lee 2009), which implies one marker every 6–7 cM. Markers were chosen on the basis of map position to effectively span the 14 autosomal chromosomes of M. guttatus.

As expected, the great majority of NILs were homozygous for the IM767 allele at all markers. Heterozygous loci were largely idiosyncratic to each NIL, given that different portions of the donor genome were randomly retained in each lineage. However, some donor regions, defined by two to five contiguous markers, were over-represented in high or low selected NILs or in both sets. We genotyped the 24 markers in these regions in the selfed progeny of the 50 selected NILs and then estimated genotype–phenotype association, i.e., cosegregation of putative QTL with flower size. This procedure identified seven QTL for subsequent study: x1, x5a, x5b, x8, x9, x10a, and x10b where the number refers to the linkage group according the convention of Fishman et al. (2002). The donor allele at three of these QTL—x1, x5b, and x8—reduced corolla width relative to IM767. The donor allele increased corolla width for QTL x5a, x9, and x10b, while x10a exhibited overdominance in the NIL progeny data set.

The next three generations of the experiment consisted of “cleaning” the background of selected NILs. From the genotype data, we identified a single NIL containing each of the target QTL (File S1). The choice of this NIL among numerous NILs harboring a particular QTL was based on the relative scarcity of donor markers across the genome. We then used a combination of backcrossing to IM767 and selfing to eliminate remaining donor segments from other parts of the genome. The QTL diagnostic markers were MgSTS_198 and MgSTS_757 for x1; MgSTS_40 for x5a; MgSTS_641 and MgSTS_385 for x5b; MgSTS_432, MgSTS_736, and MgSTS_31 for x8; MgSTS_523 for x9; MgSTS_70 and MgSTS_82 for x10a; and MgSTS_308 for x10b. The resulting single-QTL NILs were then intercrossed in all possible combinations to produce 21 doubly heterozygous genotypes. These double heterozygotes were the grandparents of the plants measured for phenotype and used to estimate epistasis.

Each double heterozygote was self-fertilized. We then grew 100–200 progeny from each progeny set and genotyped these plants at the two segregating QTL (the loci heterozygous in that particular parent). Within each progeny set, we selected multiple individuals of each double-homozygote category—AABB, AAbb, aaBB, and aabb—for subsequent self-fertilization and intercrossing of identical genotypes. These four genotypes are true breeding and one of them (aabb) reiterates the recipient line IM767. Their progeny can be assigned to genotype without directly genotyping individuals. The double homozygote, AABB, is unique to each set, but there is considerable redundancy of the other genotypes across sets. A complete array from seven QTL would involve 29 distinct two-locus homozygous genotypes. However, due to failures in crossing or seed germination, the current data set lacks four of the double-donor homozygotes: x1 with x5a, x1 with x8, x5a with x5b, and x5a with x8.

Estimating genotypic effects

Plants were grown in the University of Kansas greenhouses in seven distinct cohorts over a period of 9 months in 2009–2010. The genotypes measured varied across cohorts, but there was extensive overlap. The effects of cohort and genotype are easily distinguished. Each plant was measured for corolla width on the first two flowers, days to first flower, pistil length, anther length, and corolla length of the first flower. We collected pollen from the first two flowers. Total pollen per flower and pollen viability were measured using an updated version of our grain-size method (Kelly et al. 2002): Pollen viability was estimated as the proportion of grains in the larger size class divided by 0.9.

During the course of measurements, we noted a flower color polymorphism within the genotype set for x5b with x8. This is likely caused by a heterozygous locus in the double-heterozygote x5b/x8 grandparent, although we do not know if it represents a de novo mutation or residual variation from the original donor/recipient cross. Regardless, we recorded the “pale corolla” on all affected plants, and it is included as an additional factor in the statistical analysis. In each cohort, we genotyped a subset of plants to confirm QTL identity. A small number of contaminants were recognized and excluded from the data set.

The phenotypic data were analyzed using general linear models with cohort as a random factor and genotypes as fixed effects. Calculations were conducted using the REML option in JMP v8. We applied two variants of the same statistical model. In the first variant, QTL are characterized in a factorial way with a direct effect of each locus and an interaction term for each pair of loci (Table S2 of File S3). Epistasis is implied by a significantly non-zero coefficient on the cross-product for a pair of loci. In the second variant model, the 25 distinct two-locus genotypes are treated as different levels of the same factor (Table S3 in File S3). This parameterization does not provide tests for epistasis but does yield a standard error for the mean phenotype of each genotype (Figure 1; Table 1). While the coefficients estimated from each variant of the model are different, they yield the same predicted phenotypic values for each genotype.

Figure 1 .

QTL interactions for pistil length (y-axis in mm): For each QTL pair, the deviation of AABB (yellow), AAbb (blue), and aaBB (red) from the mean phenotype of IM767 (aabb) is reported. The locus A QTL defines rows while the locus B QTL defines columns. A double asterisk (**) implies that the interaction remains significant after Holm–Bonferroni correction. A single asterisk (*) denotes interactions that are significant in marginal tests (P < 0.05) but not after multiple test correction. Error bars are ±1 standard error (taken from model fits in Table S3 in File S3). The vertical scale is different for the panel with locus A = x5b.

Table 1 . Genotypic effect estimates reported for corolla width, days to flower, and pistil length.

	Corolla width (mm)			Days to flower			Pistil length (mm)
	Coefficient	SE	P-value	Coefficient	SE	P-value	Coefficient	SE	P-value
IM767 (aabb)	17.56	0.42		27.60	1.12		13.77	0.21
x10a	0.31	0.10	0.0022	−0.25	0.16	0.1203	−0.13	0.05	0.0120
x9	0.60	0.13	0.0000	0.92	0.20	0.0000	0.53	0.07	0.0000
x1	−0.01	0.12	0.9406	0.38	0.20	0.0535	−0.05	0.07	0.4876
x5a	0.70	0.16	0.0000	−1.53	0.25	0.0000	−0.13	0.08	0.1120
x5b	−1.08	0.12	0.0000	0.39	0.20	0.0474	0.09	0.07	0.1887
x10b	0.95	0.10	0.0000	0.53	0.16	0.0009	0.61	0.05	0.0000
x8	−0.64	0.12	0.0000	0.16	0.19	0.3963	−0.28	0.06	0.0000
Pale corolla	−0.34	0.15	0.0213	−1.76	0.24	0.0000	0.00	0.08	0.9928
x10a/x9	0.02	0.42	0.0432	−1.37	0.68	0.0039	−0.21	0.22	0.0082
x10a/x1	−0.38	0.21	0.0073	−0.87	0.34	0.0127	−0.09	0.11	0.4812
x10a/x5a	0.40	0.16	0.0073	−2.70	0.25	0.0103	−0.62	0.08	0.0035
x10a/x5b	0.62	0.30	0.0000	−1.48	0.48	0.0020	0.63	0.16	0.0001
x10a/x10b	1.54	0.17	0.1601	1.04	0.26	0.0173	0.03	0.09	0.0000
x10a/x8	−0.55	0.13	0.2541	−0.13	0.22	0.9006	−0.83	0.08	0.0000
x9/x1	0.86	0.18	0.2502	1.47	0.28	0.6381	0.88	0.09	0.0015
x9/x5a	0.74	0.17	0.0235	−0.27	0.27	0.3810	0.11	0.10	0.0363
x9/x5b	−0.45	0.17	0.9209	0.36	0.26	0.0071	0.62	0.09	1.0000
x9/x10b	1.12	0.25	0.1241	1.19	0.41	0.5771	0.82	0.14	0.0409
x9/x8	0.57	0.23	0.0242	−0.30	0.37	0.0014	0.76	0.12	0.0004
x1/x5b	−0.61	0.18	0.0445	2.24	0.29	0.0001	0.02	0.10	0.8837
x1/x10b	0.67	0.16	0.2011	−0.56	0.26	0.0000	0.39	0.09	0.1323
x5a/x10b	0.53	0.21	0.0000	−0.23	0.34	0.0754	−0.28	0.11	0.0000
x5b/x10b	1.03	0.32	0.0010	4.28	0.50	0.0000	1.55	0.17	0.0000
x5b/x8	−0.52	0.19	0.0000	−0.67	0.29	0.0006	0.15	0.10	0.0059
x10b/x8	−0.35	0.15	0.0011	0.59	0.25	0.7557	−0.69	0.08	0.0000

P-value for locus pairs refers to the test for a significant interaction among loci (from Table S2 in File S3). SE is standard error for the coefficient, and P-values reported as 0 are <0.00005 (boldface type indicates an interaction test with P < 0.05).

For each locus pair and each trait, the statistical model yields an observed phenotype for AABB and a predicted value in the absence of epistasis (from the estimated direct effect of alleles in the isogenic background). We regressed observed AABB phenotypes onto predicted values using least squares to estimate the slope and test whether it was different from 1 (Figure 2). For this, we applied a bias correction to account for estimation error in the predicted phenotype (Fuller 1987). The estimation error variance for predicted values of loci A and B was calculated as $s_A^2+s_B^2$, where $s_A$ and $s_B$ are the standard errors for estimated direct effects at loci A and B, respectively (Table 1). The overall error variance, $v_e$, was taken as the average of this quantity across locus pairs. The bias corrected slope, b′, was then calculated as $b^{\text{'}}=b(\frac{v_o}{v_o-v_e})$, where $v_o$ is the observed variance in predicted values (the x-axis values). The standard error of b′ is calculated as $(\frac{v_o}{v_o-v_e})$ times the uncorrected standard error. These calculations treat $(\frac{v_o}{v_o-v_e})$ as known (Fuller 1987, section 1.1) and assume that correlations between estimation errors of direct effects are negligible.

Figure 2 .

The relationship between the observed phenotype of double-donor homozygotes (AABB) and that predicted from the direct effects of each QTL is given for (A) corolla width, (B) days to flower, (C) pistil length, and (D) log[pollen per flower]. The IM767 phenotype (aabb) is the origin (0,0) in these graphs, and significance of the interaction is denoted by symbol. The solid line is the 1:1 relationship (no epistasis), and the broken line is the bias-corrected linear regression of observed onto predicted.

Results

Across cohorts, 6523 plants were measured and included in the analysis, although we could not measure all traits on all plants. The mean, standard deviation, standard error, and sample size for each trait are reported in Table S1, File S3. The model fits to each trait including the interaction term are reported in Table S2 in File S3. Table 1 reports mean values for the 25 distinct two-locus genotypes for three different traits (the full reporting of estimates for this variant of the model are given in Table S3 in File S3). The first row of Table 1 reports the mean phenotype for IM767, and subsequent rows are deviations from this value for genotypes that have the donor allele at one or two loci. The coefficients for each individual QTL (rows 2–8 of Table 1) are the estimated effect of each QTL in an isogenic background. Pale corolla is the effect of the visible polymorphism. The remaining rows detail deviations of double-donor homozygotes (AABB) from the recipient genome (aabb). Unlike other rows, the test for each locus pair is not for whether the coefficient is different from zero. Instead, the P-value refers to the test for a significant interaction among loci (see Table S2 in File S3).

All QTL had highly significant effects on corolla width except x1. At the other QTL, the direction of effects was always as expected from phenotypes measured in the initial NIL mapping. The average magnitude of direct effects was 0.31 environmental standard deviations. Pleiotropy was extensive (Table S3 in File S3). For example, x1 had significant direct effects on days to flower, stigma–anther separation, anther length, pollen viability, and pollen number per flower, while x10a had significant direct effects on anther length, pistil length, corolla length, and pollen viability. Epistasis is evident for the majority of gene combinations (Tables 1 and 2; Tables S2 and S3 in File S3). For corolla width, 11 of 17 locus pairs exhibit a significant interaction (Table 1). Epistasis was equally frequent for other traits: 11 significant interactions for days to flower, 7 for anther length, 13 for pistil length, 8 for corolla length, 8 for stigma–anther separation, 6 for log[pollen viability], and 8 for log[total pollen per flower]. The number of significant interactions per trait was directly related to the within-experiment “heritability,” i.e., the magnitude of the genotypic variance relative to the environmental variance of a trait (Table S1 in File S3).

Table 2 . Estimated epistatic deviation reported for each QTL pair for each trait.

	Days to flower	Corolla width	Anther length	Pistil length	Corolla length	Stigma–anther separation	Log(pollen viability)	Log(pollen per flower)	Log(viable pollen per flower)
x10a/x9	−0.884	−2.040	−0.595	−0.606	−1.370	−0.006	0.004	0.033	0.036
x10a/x1	−0.680	−1.006	0.237	0.095	−0.310	−0.139	0.081	0.185	0.267
x10a/x5a	−0.607	−0.923	−0.196	−0.353	−0.625	−0.125	−0.043	−0.062	−0.105
x10a/x5b	1.391	−1.626	0.060	0.679	0.954	0.616	0.041	0.120	0.163
x10a/x10b	0.281	0.757	−0.021	−0.445	0.233	−0.419	−0.050	−0.034	-0.084
x10a/x8	−0.215	−0.038	−0.272	−0.425	−0.295	−0.146	0.022	−0.017	0.005
x9/x1	0.268	0.172	0.117	0.395	0.219	0.286	0.018	0.129	0.147
x9/x5a	−0.559	0.341	0.155	−0.282	0.011	−0.411	−0.022	−0.058	−0.080
x9/x5b	0.022	−0.952	0.212	0.000	0.356	−0.214	0.000	0.017	0.018
x9/x10b	−0.436	−0.254	0.139	−0.319	0.019	−0.452	0.020	−0.022	−0.002
x9/x8	0.615	−1.375	0.576	0.510	0.764	−0.041	0.043	−0.009	0.035
x1/x5b	0.479	1.470	0.052	−0.019	−0.426	−0.093	−0.003	0.015	0.014
x1/x10b	−0.272	−1.472	0.196	−0.180	0.250	−0.348	0.052	0.083	0.135
x5a/x10b	−1.115	0.765	−0.020	−0.758	−1.002	−0.707	−0.005	−0.080	−0.086
x5b/x10b	1.151	3.354	0.667	0.850	0.976	0.192	0.084	0.119	0.206
x5b/x8	1.201	−1.219	0.197	0.336	0.915	0.197	0.018	0.083	0.088
x10b/x8	−0.658	−0.100	−0.904	−1.029	−0.961	−0.047	−0.050	−0.027	−0.077

The epistatic deviation is the difference between the observed AABB phenotype and that predicted without epistasis. Boldface type indicates a significant interaction test.

The significant interaction counts given above are based on a 0.05 critical value per test. If one applies a Holm–Bonferroni (Holm 1979) adjustment to “correct” for the fact that there are 17 simultaneous interaction tests (critical value for smallest P is 0.05/17; critical value for next most significant P is 0.05/16, etc.), then fewer interactions are significant: 5 for corolla width, 7 for days to flower, 5 for anther length, 10 for pistil length, 4 for corolla length, 6 for stigma–anther separation, 1 for log[pollen viability], and 3 for log[total pollen per flower]. All QTL combinations produced at least two significant interactions across traits. For certain QTL pairs, interactions were observed for multiple traits even when neither QTL had significant direct effects. Table 2 reports epistatic deviations—the observed AABB phenotypevminus the value expected without epistasis—for each two-locus combination.

Figure 1 illustrates the spectrum of gene interactions for pistil length. Given that all QTL combinations share the aabb genotype (mean pistil length = 13.77 mm), the remaining genotypes are expressed as deviations from this common value (as in Muir and Moyle 2009). The combination of x8 with x10b exhibits sign epistasis: The donor allele at x10b increases pistil length in the isogenic background (AAbb > aabb) but decreases this trait in the x8 donor background (AABB < aaBB). Locus pairs x10a/x5b, x5a/x10b, and x9/x8 are other possible cases of sign epistasis in this trait (Figure 1). Classical (masking) epistasis (Bateson 1909) is apparent for x9/x1: The x1 donor allele changes from neutral in the isogenic background (aabb ≈ aaBB) to positive in the x9 donor homozygote (AAbb < AABB). The “less-than-additive” interaction is illustrated by x9/x10b where the direct effect of each donor allele is to increase pistil length. The pistil length of the double-donor homozygote is also elevated, but less than expected given the direct effect of each QTL. Synergistic interactions, where alleles from different QTL have magnified effects when combined, are illustrated by x5a/x10a and x5b/x10b. The estimates for other traits also reveal high variability of interactions (Table 1 and Table S3 in File S3).

An informative summary of interactions for each trait is the plot of observed AABB phenotypes against the values predicted in the absence of epistasis (Figure 2). Without epistasis, the relationship should be isometric and locus pairs with nonsignificant interactions are usually close to the 1:1 line. Positive/negative epistasis is estimated by the residual from the isometric line (these are the values reported in Table 2). Statistical tendencies in epistatic interactions are revealed by the regression of observed onto predicted phenotypic values (dashed lines in Figure 2; Table 3). The data from most traits are consistent with a slope of 1, although the high variability of interactions produces broad confidence bands. The regression slope is significantly <1 for corolla width and pollen per flower (Figure 2). For both traits, epistasis is always positive if the predicted direct effect is negative. Epistasis is negative if the direct effect is positive.

Table 3 . Summary of estimates for the bias-corrected slope and SE obtained from the linear regression of observed AABB phenotype onto that predicted without epistasis.

Trait	Slope	SE	Lower bound	Upper bound
Days to flower	1.12	0.44	0.19	2.04
Corolla width	0.45	0.15	0.14	0.75
Anther length	1.21	0.27	0.65	1.78
Pistil length	0.96	0.35	0.23	1.68
Corolla length	0.97	0.28	0.37	1.56
Stigma–anther separation	0.87	0.33	0.18	1.56
Log(pollen viability)	0.65	0.35	−0.08	1.39
Log(pollen per flower)	0.23	0.24	−0.27	0.74
Log(viable pollen per flower)	0.33	0.25	−0.20	0.86

Boldface type indicates that the slope is significantly <1.

Table 4 summarizes correlations among traits. In the full data set (aggregating measurements across cohorts and genotypes), we find strong positive correlations of flower-size dimensions (corolla length and width, pistil length, anther length) and moderate positive correlation of days to flower with flower-size dimensions. Pollen viability and pollen per flower exhibit weak but positive correlations with flower-size measurements and moderate but negative correlation with days to flower. Statistical significance is much less frequent for correlations on the basis of genotypic effects across traits (lower diagonal of Table 4) due to smaller sample sizes (genotypic effect correlations are based on 25 values). However, there is generally excellent agreement between corresponding numerical values in the upper and lower diagonal of Table 4. All of the raw data (genotypes and phenotypes) are presented in File S4.

Table 4 . Raw phenotypic correlations and correlations of genotypic effects (deviations from IM767).

	Corolla width	Days to flower	Anther length	Pistil length	Corolla length	Stigma–anther separation	Log(pollen viability)	Log(pollen per flower)	Log(viable pollen per flower)
Corolla width		0.26	0.63	0.64	0.77	0.08	0.14	0.09	0.13
Days to flower	0.19		0.27	0.23	0.21	−0.02	−0.19	−0.30	−0.30
Anther length	0.50	0.56		0.69	0.71	−0.30	0.17	0.12	0.16
Pistil length	0.49	0.55	0.81		0.73	0.49	0.17	0.11	0.16
Corolla length	0.65	0.49	0.91	0.81		0.11	0.11	0.05	0.09
Stigma–anther separation	0.05	0.08	−0.15	0.45	−0.01		0.01	−0.01	0.00
Log(pollen viability)	0.34	−0.15	0.36	0.30	0.23	−0.05		0.42	0.77
Log(pollen per flower)	0.26	−0.40	0.08	0.14	0.09	0.11	0.71		0.90
Log(viable pollen per flower)	0.32	−0.32	0.20	0.22	0.15	0.05	0.89	0.95

Raw phenotypic correlations are reported above the diagonal, and correlations of genotypic effects (deviations from IM767) are reported below the diagonal. Boldface type indicates a correlation significantly different from zero (P < 0.05).

Discussion

Epistasis clearly contributes to the genetic differences among species and among populations within a species (e.g., Hard et al. 1992; Palopoli and Wu 1994; Burton et al. 1999; Fenster and Galloway 2000; Demuth and Wade 2007), and such interactions have been mapped to particular QTL in many systems (Carlborg and Haley 2004; Elnaccash and Tonsor 2010). In the great majority of cases, however, it is not clear that interacting loci were ever simultaneously polymorphic within the same evolving population. The present study of yellow monkeyflower considers genetic variability in a single natural population at Iron Mountain in Oregon. The estimates thus speak directly to the consequences of epistasis for quantitative trait evolution.

A principal finding is that QTL for flower size routinely interact, and these interactions are highly variable in form. Variable interactions among novel or visible mutations have been demonstrated in several systems (Clark and Wang 1997; de Visser et al. 1997; Elena and Lenski 1997). The present data extend this generalization to intrapopulation variation in quantitative characters. This is a nontrivial extension, given that standing variation has been sieved by selection and is likely to have different properties from new mutations. Notably, we find sign epistasis, i.e., the reversal of allelic effect with changes in genetic background, for segregating variation. However, other kinds of interactions (less-than-additive, synergism, effect masking) are also routinely observed.

Variability of interactions is multifaceted. Across QTL pairs, there are significantly positive and negative interactions for every trait (Table 2). Across traits, specific locus pairs routinely exhibit both positive and negative interactions. With a couple of notable exceptions (e.g., Figure 2A), the frequency of positive relative to negative interactions is largely unrelated to the direct effect of a locus in the isogenic background. There was some consistency of effects by a locus pair across genetically correlated traits. For example, x10a and x9 interact in a consistently negative way for flower-size dimensions, as does x10a with x5a and x10b with x8 (Table 2). x5b and x10b interact in a consistently positive way for the same collection of measurements. The finding of variation in epistatic effects was clearly anticipated by previous quantitative genetic experiments (Kelly 2005). Kelly (2005) found that the statistical signal of epistasis could not be removed by any scale transformation. Scale transformation can eliminate epistasis if it is simple and directional, but not if interactions are variable in nature.

Directional epistasis is routinely invoked in evolutionary models. For example, synergism between deleterious mutations is a key component of the mutational deterministic model for the maintenance of sex (Kondrashov 1988). However, a number of novel evolutionary consequences emerge from variation in epistatic effects (Phillips et al. 2000). With recurrent deleterious mutation, Otto and Feldman (1997) show that epistatic variation reduces the likelihood that selection will favor sexual reproduction. Variable epistasis also alters the progression of Muller’s ratchet (Butcher 1995) and the rate that adaptive evolution within populations generates reproductive isolation among them (Gavrilets and Gravner 1997; Wade 2000).

Epistasis and the genetic variance in flower size

Flower size and shape have diversified extensively in the genus Mimulus associated with changes in mating system and pollination syndrome (Vickery 1978; Bradshaw and Schemske 2003). In the Iron Mountain population of M. guttatus, flower-size variation is highly polygenic and under strong natural selection (Willis 1996; Mojica and Kelly 2010). Narrow sense heritabilities for floral dimensions are in the range of 0.3–0.5 (Kelly and Arathi 2003; Kelly 2008; Scoville et al. 2009), and >40 intrapopulation flower-size QTL have been mapped (Lee 2009). The additive genetic variance in flower size depends on allelic effects (including dominance) at QTL, on the interactions among QTL, and on the frequency of alternative alleles in the natural population.

At present, we have no direct estimates for allele frequency for any of these loci. However, we have estimated the additive genetic variance contribution (within Iron Mountain) of nine flower-size QTL, including x5a, x5b, x9, and x10a of the present study (Scoville et al. 2011). V_Q, the additive genetic variance contributed by a locus (as a percentage of the total phenotypic variance), was estimated to be 1.8, 4.8, 1.8, and 1.9 for x5a, x5b, x9, and x10a, respectively. In the absence of epistasis, the additive genetic variance of a trait is a sum of V_Q across all QTL in the genome [plus an increment for any loci exhibiting linkage disequilibria (Bulmer 1980)]. With epistasis, the additive value of an allele is an average across genetic backgrounds. As a consequence, V_Q for a locus will depend on allele frequencies at other QTL.

Epistasis may explain an intriguing pattern from the V_Q study. We found that QTL with moderate effects on corolla width, as measured in NILs, had the largest V_Q estimates. The QTL with largest effect in the NILs had the lowest V_Q estimates (see figure 1 of Scoville et al. 2011). This could be due to the fact that mutations with large effects on corolla width are detrimental for fitness and thus exist at low frequency. Epistasis provides an alternative explanation in that allelic effects inferred from a particular NIL population might be quite different than the average effect obtained by averaging over a diversity of genetic backgrounds. The V_Q estimates of Scoville et al. (2011) were obtained by integrating marker-trait associations over 138 distinct “line-cross families” representing genetic variation within the natural population. Relevant here is that our mapping method preferentially selected QTL with the most pronounced effects on corolla width within a single genetic background. We created NILs by repeatedly backcrossing a donor genotype into a fixed genetic background (the IM767 recipient line) and focused on NILs with the most extreme mean corolla widths (File S1). If this procedure preferentially identified QTL with magnified effects in the IM767 background, then we should expect reduced signal as a QTL is assayed in more diverse backgrounds. All nine QTL with V_Q estimates were initially identified by selective genotyping within NILs (Scoville et al. 2011).

The net effect of epistasis on the amount of genetic variation in quantitative characters of M. guttatus cannot be evaluated from the current data. However, it is informative to consider the variance of observed AABB phenotypes relative to the variance of predicted values in the absence of epistasis. In other words, we compare the variance along the y-axis of Figure 2 to the variance along the x-axis. For most of the traits, the y-axis variance is greater than the x-axis variance, suggesting that epistasis inflates the genetic variation (ignoring estimation issues). The exceptions are corolla width, log[pollen per flower], and log[viable pollen per flower]. Each of these traits exhibits a slope significantly <1 in the regression of observed AABB phenotypes onto predicted values (Figure 2; Table 3). The isolation of QTL on the basis of corolla width effects might explain why this trait differs from other floral dimensions. The pollen traits exhibit lower genetic variance (relative to environmental effects) than other characters (Table S1 in File S3).

Evolutionary consequences

An important question is how variable epistasis affects response to directional selection on a trait (Hallander and Waldmann 2007; Crow 2010). In a large, randomly mating population, response should be proportional to the additive genetic variance. The principal effect of epistasis is in determining the additive value of an allele. Recombination ensures that an allele is expressed in a variety of genetic backgrounds and that its additive value is an average over backgrounds. With sign epistasis, whether an allele increases or decreases trait values will depend on the relative frequencies of different genetic backgrounds. The same selection pressure imposed on different populations, with different initial allele frequencies, might cause different alleles at the same locus to be favored in different populations.

An interesting property of variable epistasis among QTL is that it may be essentially “invisible” at the phenotypic level in a population undergoing sustained directional selection. As allele frequencies change at QTL for the selected trait, V_Q at these loci will also change. With directional epistasis, for example, when alleles increasing trait values interact synergistically, changes in V_Q may be directionally consistent across QTL. This can produce substantial changes in the overall additive variance (see figure 3 of Hansen 2006), causing substantial accelerations or decelerations in the rate of phenotypic change. With variable epistasis, however, changes in V_Q are unlikely to be directionally consistent across QTL. Positive changes at some QTL cancel negative changes at others, yielding approximate constancy of the overall genetic variance in the short term. In this way, the effects of variable epistasis may not be evident at the phenotypic scale, particularly given other factors influencing the genetic variance such as mutation and genetic drift. However, variable epistasis will definitely complicate prediction of allele frequency change at individual QTL, particularly if allelic effects are estimated in a single uniform genetic background and assumed to be constant.

The epistasis for stigma–anther separation (Figure 3) is interesting to consider in relation to the contrasting views of Ronald Fisher and Sewall Wright on the role of epistasis in evolution (Provine 1971). Fisher argued that recombination effectively averages interactions across backgrounds and selection acts on average allelic effects, while Wright felt that selection of specific gene combinations is essential for adaptation. The recombination essential to Fisher’s view depends on random mating. Self-fertilization reduces the efficacy of recombination and may allow selection on gene combinations. In fact, selfing is a more potent mechanism to preserve gene combinations than the population structure/drift mechanism of Wright’s shifting balance model.

Figure 3 .

The estimated stigma–anther separation (y-axis in mm) is reported for each of the homozygous genotypes for three loci: x10a (A/a), x5a (B/b), and x10b (C/c). Genotypes differ from IM767 at one locus (shaded bars), two loci (stippled bars), or three loci (open bars). The error bars are ±1 standard error for the estimated deviation from aabbcc. No error bars are reported for AABBCC because this genotype was not present in the data set. Its value is predicted from the estimated direct and pairwise interactions (Table S2 in File S3), assuming no higher-order interaction.

Stigma–anther separation is a determinant of selfing rate in many flowering plants (e.g., Vallejo-Marin and Barrett 2009), and several M. guttatus experiments have shown correlation of reduced stigma–anther separation with increased selfing (Carr and Fenster 1994; Fishman and Willis 2008b). The M. guttatus species complex includes a number of highly selfing lineages that have evolved reduced stigma–anther separation (Ritland and Ritland 1989; Fenster and Ritland 1994; Wu et al. 2008). Figure 3 illustrates an interesting feature of interactions among QTL for this trait: x5a, x10a, and x10b each have minimal direct effects, but combinations of alleles can produce striking reductions in stigma–anther separation. These QTL are potential mating system modifiers, but only in combination. Our pairwise interaction model (Table S2 in File S3) predicts a 1-mm reduction in stigma–anther separation for the x5a/x10a/x10b donor homozygote. A change of this magnitude could substantially elevate selfing rate in Iron Mountain plants (see Fishman and Willis 2008; figure 4 of Bodbyl Roels and Kelly 2011). The stigma–anther results are intriguing in that the phenotypic effect of epistasis might itself facilitate the persistence of gene combinations across generations.