7: Phenotypic Variation and the Resemblance Between Relatives

The covariance between identical twins

Let’s first consider the case of a pair of identical twins, monzygotic (MZ) twins, from two unrelated parents. Our pair of twins share their maternal and paternal allele identical by descent (X1M = X2M and X1P = X2P ). As their maternal and paternal alleles are not correlated draws from the population, i.e. have no probability of being IBD as we’ve said the parents are unrelated, the covariance between their effects on the phenotype is zero (i.e. Cov(X1P , X2M ) = Cov(X1M , X2P ) = 0). In that case, eqn. 7.12 is

\[Cov(X1, X2) = Cov(X1M , X2M )+Cov(X1P , X2P ) = V ar(X1M )+V ar(X1P ) = VA\]

where the middle step follows from the fact the maternal (or sim- ilarly the paternal) allele in a pair of twins is the same allele so Cov(X1M , X2M ) = Cov(X1M , X1M ) = V ar(X1M ), as the covariance of random variable with itself is just its variance, and then the addi- tive variance of the maternal allele contribution is V ar(X1M ) = VA/2 following from eqn(7.9).

To calculate the narrow sense heritability we could then in principal divide the covariance of our pairs of MZ twins (MZ1 and MZ2) by the trait variance to give

\[\]

where ρMZ is the correlation of pairs of MZ twins (see Appendix eqn (A.43) for more on correlations). For example, we could estimate the heritability of a measure of body from the MZ correlation in Figure \(\PageIndex{9}\). In general, this simple estimator isn’t great as the correlation of identical twins includes the effects of the shared family environment of the twins (i.e. Cov(X1E,X2E)).

Moreover, it can be inflated by non-additive effects as identical twins don’t just share alleles, they share their entire genotypes, and thus resemble each other in phenotype also because of shared dominance effects (we’ll discuss non-additive effects in Section 7.1.1). Better twin-based estimates of heritability are commonly used based on the comparison of MZ vs twins that bypass some of these issues.

The covariance in phenotype between parent and child Children re- semble their biological parents because children inherit their genome from their parents (putting aside shared environments for the moment). If a mother and father are unrelated individuals, i.e. they are two random draws from the population, then this mother and her child share one allele IBD at each locus (i.e. r1 = 1 and r0 = r2 = 0). Let’s assume that our mother (ind 1) transmits her paternal allele to the child (ind 2), in which case XP1 = XM2, and so Cov(XP1,XM2) = V ar(XP 1) = 12 VA, and all the other covariances in eqn. 7.12 are zero. We’d also arrive at this result if instead we had thought of the mother transmitting her own maternal allele. Thus Cov(X1, X2) = 12 VA, we can leverage this form of the covariance to directly estimate h2 by regression.

We can estimate the narrow sense heritability through the regression of child’s phenotype on the parental mid-point phenotype. The parental mid-point phenotype is simply the average of the mum and dad’s phenotype. See Figure \(\PageIndex{10}\) for an example from song sparrows.

We denote the child’s phenotype by Xkid and mid-point phenotype by Xmid, so that if we take the regression Xkid ∼ Xmid this regres- sion has slope β = Cov(Xkid, Xmid)/V ar(Xmid). The covariance of Cov(Xkid, Xmid) = 21 VA, and V ar(Xmid) = 12 VP , as by taking the aver- age of the parents we have halved the variance, such that the slope of the regression is

\[\]

i.e. the regression of the child’s phenotype on the parental midpoint phenotype is an estimate of the narrow sense heritability. If much of the phenotypic variation is due to the (additive) differences in genotypes among individuals (h2 ≈ 1), then children will closely resemble their parents. Conversely if much of the variation is environmental (h2 ≈ 0), and there is no shared environment between parent and child, children will not resemble their parents.

Applying this heritability estimate to the Song sparrow sample we find h2 = 0.43 and h2 = 0.3 for beak depth and tarsus length respectively from Figure \(\PageIndex{10}\). So in Smith and Zach (1979) analysis, for example, 30% of the variance in tarsus length is atttributal to the additive effect of genetic differences among individuals. Smith and Zach (1979) also regressed the average offspring phenotype agains their fathers or mothers against their offspring, giving
a slope of βdad,avg.kid and βmum,kid. For tarsus length, for example, they found βdad,avg.kid = 0.19 and βmum,avg.kid = 0.17. Following a similar argument to that in eqn (7.15) we find that these slopes are βdad,kid = VA/2/VP = h2/2, and the same for mums. Thus the regression of offspring’s phenotype on a particular parent is an estimate of half the narrow-sense heritability, in line with the reduced slopes found by Smith and Zach (1979), this halfing of the slope is due to the fact that a single parent’s phenotype is a noisier estimate of the parental mid-point and so less informative about the child s phenotype. These parent specific estimates of heritability are particularly useful as they allow us to investigate sex-specific inheritance and sexual dimorphism (we’ll explore this in a later section).

Estimating heritability by these various parent-offspring regression have the issue of not controlling for environmental correlations between parent and offspring, which can inflate our estimates of hertability (as we will mistake environmentally mediated resemblance for genetics). Raising the organisms in the lab could remove much of the potential for shared environment between parent and offspring, but it also removes much of the environmental variation and we (as evolutionary geneticists) are usually not primarily interested in knowing the heritability in the lab bur rather in the field. In some organisms, notably plants, we can begin to sidestep these issues by raising offspring in a common set of randomized field conditions (a so called “common garden”). Another option is cross-foster animals, for example Smith and Dhondt (1980) returned to the song sparrow population and swapped eggs between parents nests. They found that the covariance between biological parents and children was still high despite these children being raised in a different nest, but that there was no significant covariance between foster parents and their non-biological children (see Figure \(\PageIndex{13}\) for beak depth). This suggests that family environment is not confounding the estimate of heritability in this song sparrow sample. However, such manipulations are often impossible in many systems, and issues of shared environmental covariance due to maternal resources from egg (or seed) are still present.

Despite its issues, this measure of heritability provides useful intuition and is directly relevant to our discussion of the response to selection in the next chapter. That’s because our regression allows us to attempt to predict the phenotype of the child given the phenotypes of the parents; how well we can do this depends on the slope. See Figure \(\PageIndex{12}\) for examples. If the slope is close to zero then the parental phenotypes hold little information about the phenotype of the child, while if the slope is close to one then the parental mid-point is a good predictor of the child’s phenotype. As we will see, natural selection will only efficiently drive evolution if children resemble their parents.

Thinking abour our prediction of child’s phenotpye more formally, the expected phenotype of the child given the parental phenotypes is

\[E(Xkid|Xmum, Xdad) = μ + βmid,kid(Xmid − μ) = μ + h2(Xmid − μ)\]

which follows from the definition of linear regression. So to find the child’s predicted phenotype, we simply take the mean phenotype and add on the difference between our parental mid-point and the population mean, multiplied by our narrow sense heritability.

The covariance between general pairs of relatives under an additive model The above examples above make clear that to understand
the covariance between phenotypes of relatives, we simply need to think about the alleles they share IBD. Consider a pair of relatives (1 and 2) with a probability r0, r1, and r2 of sharing zero, one, or two alleles IBD respectively. When they share zero alleles Cov((X1M + X1P ), (X2M + X2P )) = 0, when they share one allele Cov((X1M + X1P),(X2M +X2P)) = Var(X1M) = 12VA, and when they share two alleles Cov((X1M + X1P ), (X2M + X2P )) = VA. Therefore, the general covariance between two relatives is

\[\]

where F1,2 is our coefficient of kinship, i.e. the probability that two alleles sampled at random from our pair individuals 1 and 2 are IBD (see eqn (2.5)). So under a simple additive model of the genetic basis of a phenotype, to measure the narrow sense heritability we need to measure the covariance between pairs of relatives (assuming that we can remove the effect of shared environmental noise). From the covariance between relatives we can calculate VA, and we can then divide this by the total phenotypic variance to get h2.

Estimating additive genetic variance across a variety of different relationships (The animal model). In many natural populations we may have access to individuals with a range of different relationships to each other (e.g. through monitoring of the paternity of individuals), but relatively few pairs of individuals for a specific relationship (e.g. sibs). We can try and use this information on various relatives as fully as possible in a mixed model framework. Building from equation 7.3, we can write an individual’s phenotype Xi as

\[Xi = μ + XA,i + XE,i\]

where XE,i ∼ N(0,VE) and XA,i is normally distributed across individuals with covariance matrix VAA, where the the entries for a pair of individuals i and j are Aij = 2Fi,j and Aii = 1. Given the matrix A we can estimate VA. We can also add fixed effects into this model to account for generation effects, additional mixed effects could also be included to account for shared environments between particular individuals (e.g. a shared nest). This approach is sometimes called the “animal model”, and is widely used to in modern quantitative gentics to estimate genetic variances and heritabilities.

Non-additive variation.

Up to now we’ve assumed that our alleles contribute to our pheno- type in an additive fashion. However, that does not have to be the case as there may be non-additivity among the alleles present at a locus (dominance) or among alleles at different loci (epistasis). We can accommodate these complications into our models. We do this
by partitioning our total genetic variance into independent variance components. We’ll see that dominant and epistatically interacting loci can contribute to the additive genetic variance. In constructing these variance components models we’ll assume that we know the alleles contributing to variation in our trait and their effects, but in reality we rarely know these. However, as we’ll see we don’t need to know these details and we can partition our variance and estimate additive variance and other forms of non-additive variation using the resemblance between various types of relatives.

Dominance. To understand the effect of dominance, let’s consider how the allele that a parent transmits influences their offspring’s phenotype. A parent transmits one of their two alleles at a locus to their offspring. Assuming that individuals mate at random, this allele is paired with another allele drawn at random from the population. For example, assume your mother transmitted an allele 1 to you: with probability p it would be paired with another allele 1, and you would be a homozygote; and with probability q it’s paired with a 2 allele and you’re a heterozygote.

Now consider an autosomal biallelic locus l, with frequency p for allele 1, and genotypes 0, 1, and 2 corresponding to how many copies of allele 1 individuals carry. We’ll denote the mean phenotype of an individual with genotype 0, 1, and 2 as Xl,0, Xl,1, Xl,2 respectively. This mean is taking an average phenotype over all the environments and genetic backgrounds the alleles are present on. We’ll mean center (MC) these phenotypic values, setting X′l,0 = Xl,0 − μ, and likewise for the other genotypes.

We can think about the average (marginal) MC phenotype of an individual who received an allele 1 from their parent as the average of the MC phenotype for heterozgotes and 11 homozygotes, weighted by the probability that the individual has these genotypes, i.e. the probability they receive an additional allele 1 or an allele 2 from their other parent:

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Similarly, if your parent transmitted an 2 allele to you, your average MC phenotype would be

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Let’s now consider the average phenotype of an offspring by how many copies of the allele 1 they carry

i.e. the mean phenotype of each genotypes’ offspring averaged over all possible matings to other individuals in the population (assuming individuals mate at random). These are the additive MC genetic values (breeding values) of our genotypes. Here we are simply adding up the additive contributions of the alleles present in each genotype and ignoring any non-additive effects of genotype.

To illustrate this, in Figure \(\PageIndex{20}\) we plot two different cases of dominance relationships; in the top row an additive polymorphism and in the second row a fully dominant allele. The additive genetic values of the genotypes are shown as red dots. Note that the additive values of the genotypes line up with the observed MC phenotypic means in the top row, when our alleles interact in a completely additive manner. Our additive genetic values always fall along a linear line (the red line in our figure). The additive values are falling along the best fitting line of linear regression for our population, when phenotype is regressed against the additive genotype (0, 1, 2 copies of allele 1) across all individuals in our population. Note in the dominant case the additive genetic values differ from the observed phenotypic means, and are closer to the observed values for the genotypes that are most common in the population.

The difference in the additive effect of the two alleles al,2 − al,1 can be interpreted as an average effect of swapping an allele 1 for an allele 2; we’ll call this difference αl = al,2 − al,1. Our αl is also the slope of the regression of phenotype against genotype (the red line in Figure \(\PageIndex{20}\). Note that the slope of our regression of phenotype on genotype (αl) does not depend on the population allele frequency for our completely additive locus (top row of 7.20). In contrast, when there is dominance, the slope between genotype and phenotype (αl) is a function of allele frequency (bottom row of 7.20). When a dominant allele (1) is rare there is a strong slope of phenotype on genotype, bottom left Figure 7.20. This strong slope is because replacing a single copy of the 2 allele with a 1 allele in an individual has a big effect on average phenotype, as it will most likely move an individual from be- ing a 22 homozygote to being a 12 heterozygote. In contrast, when the dominant allele (1) is common in the population, replacing a 2 allele by a 1 allele in an individual on average has little phenotypic effect, leading to a weak slope (bottom right Figure \(\PageIndex{20}\)). This small effect is because as we are mainly turning heterozygotes into homozygotes (11), who have the same mean phenotype as each other.

As as an example of how dominance and population allele frequencies can change the additive effect of an allele, let’s consider the genetics of the age of sexual maturity in Atlantic Salmon. A single allele of large effect segregates in Atlantic Salmon that influences the sexual maturation rate in salmon (Ayllon et al., 2015; Barson et al., 2015), and hence the timing of their return from the sea to spawn (sea age). The allele falls close to the autosomal gene VGLL3 (Cousminer et al., 2013, variation at this gene in humans also influences the timing of puberty). The left side of Figure \(\PageIndex{22}\) shows the age at sexual maturity in males. The L allele associated with slower sexual maturity is recessive in males. While the LL homozygotes mature on average a whole year later, the additive effect of the allele is weak while the L allele is rare in the population. The right panel shows the effect of the L allele in females. Note how the allele is much more dominant in females, and has a much more pronounced additive effect. The dominance of an allele is not a fixed property of the allele but rather a statement of the relationship of genotype to phenotype, such that the dominance relationship between alleles may vary across phenotypes and contexts (e.g. sexes).

The variance in the population phenotype due to these additive breeding values at locus l, assuming HW proportions, is

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The total additive variance for the whole genotype can be found by summing the individual additive genetic variances over loci

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Having assigned the additive genetic variance to be the variance ex- plained by the additive contribution of the alleles at a locus, we define the dominance variance as the population variance among genotypes at a locus due to their deviation from additivity. We can calculate how much each genotypic mean deviates away from its additive prediction at locus l (the length of the arrows in Figure \(\PageIndex{20}\)). For example, the heterozygote deviates

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away from its additive genetic value, with similar expressions for each of the homozygotes (dl,0 and dl,2). We can then write the dominance variance at our locus as the genotype-frequency weighted sum of our squared dominance deviations

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Writing our total dominance variance as the sum across loci

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Having now partitioned all of the genetic variance into additive and dominant terms, we can write our total genetic variance as

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We can do this because by construction the covariance between our additive and dominant deviations for the genotypes is zero. We can define the narrow sense heritability as before h2 = VA/VP = VA/(VG + VE ), which is the proportion of phenotypic variance due to additive genetic variance. We can also define the total proportion of the phenotypic variance due to genetic differences among individuals, as the broad-sense heritability H2 = VG/(VG + VE).

The additive and dominance variance can be estimated by the re- semblance among relatives. When dominance is present in the loci influencing our trait (VD > 0), we need to modify our phenotype covariance among relatives to account for this non-additivity. Specifi- cally, our equation for the covariance among a general pair of relatives (eqn. 7.17 for additive variation) becomes

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where r2 is the probability that the pair of individuals share 2 alleles identical by descent, making the same assumptions (other than additivity) that we made in deriving eqn. 7.17. In table 7.1 we show the phenotypic covariance for some common pairs of relatives. Importantly, in the presence of dominance variance, the regression of offspring phenotype on parental midpoint still has a slope VA/VP , as a parent and offspring share precisely one of their autosomal alleles IBD but never their genotype IBD (assuming no inbreeding).

Full sibs and parent-offspring have the same covariance if there is no dominance variance (as they have the same kinship coefficient F1,2). However, when dominance effects are present (VD > 0), full-sibs resemble each other more than parent-offspring pairs. That’s because full-sibs can share both alleles (i.e. the full genotype at a locus) identical by descent. We can attempt to estimate VD by comparing different sets of relationships. For example, non-identical twins (full sibs born at same time) should have 1/2 the phenotypic covariance of identical twins if VD = 0. Therefore, we can attempt to estimate VD by look- ing at whether identical twins have more than twice the phenotypic covariance than non-identical twins.

The most important aspect of this discussion for thinking about evolutionary genetics is that the parent-offspring covariance is still only a function of VA. This is because our parent (e.g. the mother) transmits only a single allele, at each locus, to its offspring. The other allele the offspring receives is random (assuming random mating), as
it comes from the other unrelated parent (the father). Therefore, the average effect on the child’s phenotype of an allele the child receives from their mother is averaged over all possible random alleles the child could receive from their father (weighted by their frequency in the population). Thus we only care about the additive effect of the allele, as parents transmit only alleles (not genotypes) to their offspring. This means that the short-term response to selection, as described by the breeder’s equation, depends only on VA and the additive effect of alleles. Therefore, if we can estimate the narrow-sense heritability we can predict the short-term response.

While our VA predicts the short term response to selection, if alleles display dominance, our value of VA will change as alleles at our loci change in frequency. For, example as dominant alleles become common in the population their contribution to VA decreases, we can see this in Figure 7.20 our rare dominant allele (bottom left) contributes to the additive variance far more than when it is at high frequency (bottom right). So if selection favours higher values of our trait, the response to selection will push the dominant allele to higher frequency decreasing VA. Therefore, if there is dominance our value of VA will not be constant across generations.

Up to this point we have only considered dominance and not epistasis. However, we can include epistasis in a similar manner (for ex- ample among pairs of loci). This gets a little tricky to think about, so we will only briefly explain it. We can first estimate the additive effect of the alleles by considering the effect of the alleles averaging over their possible genetic backgrounds (including the other interact- ing alleles they are possibly paired with), just as before. We can then calculate the additive genetic variance from this. We can estimate the dominance variance, by calculating the residual variance among genotypes at a locus unexplained by the additive effect of the loci. We can then estimate the epistatic variance by estimating the residual vari- ance left unexplained among the two locus genotypes after accounting for the additive and dominant deviations calculated from each locus separately. In practice these high variance components are hard to estimate, and usually small as much of our variance is assigned to the additive effect. Again we would find that we mostly care about VA for predicting short-term evolution, but that the contribution of loci to the additive genetic variance will depend on the epistatic relationships among loci.