The aim of this App is to show how the combination of gene action and allele frequencies at causal loci translate to genetic variance and genetic variance components for a complex trait. Although the theory underlying the App is more than a century old, it is highly relevant in the current era of genome-wide association studies (GWAS). The App can be used to demonstrate the relationship between a SNP effect size estimated from GWAS and the variation the SNP generates in the population, i.e., how locus-specific effects lead to individual differences. In addition, it can also be used to demonstrate how within and between locus interactions (dominance and epistasis, respectively) usually do not lead to a large amount of non-additive variance relative to additive variance, and therefore that these interactions usually do not explain individual differences in a population.
The three models described below mainly illustrate the Chapters 7 and 8 of Falconer and Mackay (1996) and the Chapter 5 of Lynch and Walsh (1998).
In this single-locus model, we consider a biallelic locus with allele A1 and A2 in frequencies p and 1-p. Under panmixia (i.e., random mating) and Hardy-Weinberg equilibrium, the expected genotype frequencies are (1-p)2,2p (1-p) and p2, for A2A2, A1A2 and A1A1 respectively.
We arbitrarily assign genotypic values (the mean trait values for the different genotype class) -a, d and a to the three genotypes, d representing the dominance effect (within locus interaction, no interaction when d = 0) and 2a the difference between the two homozygotes. Under this model, the population mean is:
M = (2p-1)a + 2p(1-p)d
Average effect of gene (allele) substitution (also called additive effect in the literature)
The transmission of value from parents to offspring occurs through their genes (alleles) and not their genotypes. The average effect of gene substitution (𝛼) is defined as the average effect on the trait when substituting alleles at this locus in the population. It can also be defined as the mean value of genotypes produced by different gametes:
𝛼 = a + (1-2p)d
Importantly, 𝛼 is also the slope of the linear regression of the genotype means, weighted by their frequency, on the A1 allele dosage (0, 1 or 2).
When performing a standard GWAS, individual phenotypes y are regressed on the number x (x = 0, 1, 2) of reference alleles at a given locus, , i.e., the allelic “dosage”, where the reference allele for this dosage count is arbitrarily the major or minor allele (but this arbitrary choice is reflected in the sign of the regression coefficient β:
y = μ + βx + e
Where the residuals e include both the non-additive genetic effects at the locus, the genetic effects (additive and non-additive) at other loci and an environmental and/or chance effect (non-genetic). The quantity of interest is the slope β of the model (the effect size of the locus), which is the average effect of allele substitution, hence β = 𝛼.
Additive (breeding) values and dominance deviations
The breeding values are the expected genotypic values under additivity (the predictions from the linear model). Expressed as deviations from the population mean M, the breeding values of the 3 genotypes A2A2, A1A2 and A1A1 are
-2p𝛼, (1-2p)𝛼 and (2-2p)𝛼. The residuals of the linear regression are the deviations due to the within locus interaction (dominance)
Genetic variance
The total genotypic variance (VG) is partitioned into Additive (VA) and Dominance (VD) variance.
VG = VA + VD
Additive variance
The Additive variance (VA) is the variance of additive (breeding) values. When values are expressed in terms of deviation from the population mean, the variance simply become the mean of the squared values. Hence, VA is obtained by squaring the additive (breeding) values described above, multiplying by the corresponding frequencies and summing over the 3 genotypes, leading to:
VA = 2p(1-p)𝛼2 = 2p(1-p)[a+d(1-2p)]2 = H𝛼2,
with H the heterozygosity at the locus. Note that VA is the variance explained by a SNP in GWAS (2p(1-p)𝛼2 = 2p(1-p)β2) and contain both a term due to additivity (a) and dominance (d) through the average effect 𝛼.
Dominance variance
Similarly, the dominance variance (VD) is the variance of dominance deviations:
VD = (2p (1-p)d)2 = H2d2
Therefore, the dominance variance disproportionally depends on the locus heterozygosity compared to the additive variance (H2 versus H).
We extend the one-locus to a two-locus model with additive and additive-by-additive epistatic interaction only, assuming no within loci dominance effects (d = 0 at both loci). We introduce a second (unlinked) locus with alleles B1 and B2 in frequencies q and 1-q respectively. The genotypic values and allele frequencies of the 9 genotypes are:
| A2A2 | A1A2 | A1A1 | |
|---|---|---|---|
| B2B2 | -aA-aB+aAB (1-p)2(1-q)2 |
-aB 2p(1-p)(1-q)2 |
aA-aB-aAB p2(1-q)2 |
| B1B2 | -aA (1-p)22q(1-q) |
0 4p(1-p)q(1-q) |
aA 2p2q(1-q) |
| B1B1 | -aA+aB-aAB (1-p)q2 |
aB 2p(1-p)q2 |
aA+aB+aAB p2q2 |
Population mean
In our model, the mean of the genotypic values is:
M = aA(2p-1) + aB(2q-1) + aAB(1-2(p+q)+4pq)
Note that the expression of M depends on the arbitrarily assigned genotypic values.
Average effect of gene (allele) substitution
In this model, the locus specific average effects are:
𝛼A = aA + 2qaAB
𝛼B = aB + 2paAB
Genetic variance
The total genotypic variance (VG) of the model is partitioned into Additive (VA) and Additive-by-Additive (VAA) variance.
VG = VA + VAA
Additive variance
The additive variance of the model is:
VA = ∑iHi𝛼i2, with Hi the heterozygosity at locus i (i = A, B) and 𝛼i the average effect of locus i. Hence:
VA = 2p(1-p)[aA+2qaAB] + 2q(1-q)[aB+2paAB]
Note that VA contains a term due pairwise additive-by-additive interaction between locus A and B (aAB).
Additive-by-Additive variance
The additive-by-additive variance of the model is:
VAA = ∑i∑j>iHiHjaij2, with Hi the heterozygosity at locus i (i = A, B) and aij the additive-by-additive interaction effect between locus i and j. Hence:
VAA = 4p(1-p)q(1-q)aAB
Therefore, the additive-by-additive variance disproportionally depends on the locus heterozygosity as compared to the additive variance.
Lastly, we use a generalized two-locus model where the user can provide all the genotypic values in an interactive table and choose the allele frequencies at the two loci (p and q). The genotypic values as well as the linear regressions are plotted as a function of the A1 allelic dosage for the different genotypes at locus B, as well as the linear regression of the genotypic values weighted by their frequency on the A1 allele dosage. The total genotypic variance (VG) of this model is then partitioned in five components:
VG = VA + VD + VAA + VAD + VDD
Where VAD is the additive-by-dominance variance and VDD the dominance-by-dominance variance. We use the least square approach described in Lynch and Walsh (1998) Chapter 5 to derive the different variance components and display their values.