Why we study resemblance?

Resemblance (=covariance) between related individuals.

Heritability is the central concept in quantitative genetics

• Proportion of variation due to additive genetic values (Breeding values)

$$\begin{align*} h^2 = \frac{\sigma^2_A}{\sigma^2_P} \end{align*}$$

Phenotypes (and $\sigma^2_P$) can be directly measured.

How about $\sigma^2_A$ ?

• Estimates of $\sigma^2_A$ require known collections of relatives.

Types of Relatives

Ancestral relatives

• Parent and offspring

• Grandparent and offspring

Collateral relatives

• Half sibs: have one parent in common

• Full sibs: have both parents in common

• Twins

Relatives (full sib families)

Full sibs

• have both parents in common

Relatives (half sib families)

Half sibs

• have one parent in common

Relatives (half sib families)

A thought experiment

Body weight of newborn piglets

• In the 1st half family $O_{11}$ ... $O_{3k}$, mean $=3 \pm 0.1$ lb
• In the 2nd half sib family $O_{n1}$ ... $O_{nk}$, mean $=4 \pm 0.1$ lb

Two key observations

A thought experiment

Body weight of newborn piglets

• In the 1st half family $O_{11}$ ... $O_{3k}$, mean $=3 \pm 0.1$ lb
• In the 2nd half sib family $O_{n1}$ ... $O_{nk}$, mean $=4 \pm 0.1$ lb

Obs. One:

If trait variation has a significant genetic basis, the closer the relatives, the more similar their phenotypic values.

Obs. Two:

The amount of phenotypic resemblance among relatives for the trait provides an indication of the amount of genetic variation for the trait.

Resemblance and genetic covariance

Genetic covariance

• Resemblance between relatives has the genetics basis
• Genetic covariances arise because two related individuals are more likely to share alleles than are two unrelated individuals

Identical by descent (IBD)

• Sharing alleles means having alleles that are identical by descent (IBD).
• If two alleles are IBD, then both copies of alleles can be traced back to a single copy in a recent common ancestor.

Genetic covariance between relatives

Identify types of IBD?

• No allele IBD
• one allele IBD
• two alleles IBD

Identical by descent (IBD)

If allele $x_i$ carried by individual X is IBD to allele $y_k$ in Y, then the covariance due to this allele is:

$$\begin{align*} Cov(\alpha_i, \alpha_k) & = E[(\alpha_i - \mu_{\alpha})(\alpha_k - \mu_{\alpha})] \\ & = E[(\alpha_i - \mu_{\alpha})^2] \\ & = \sigma_{\alpha}^2 \end{align*}$$

Because $\alpha_i = \alpha_k$ if alleles $x_i$ and $y_k$ are IBD.

Additive genetic covariance

The additive genetic covariance between relatives is generated through individuals sharing average effects of alleles.

Alleles in individuals $X (x_ix_j)$ and $Y (y_ky_l)$ can be IBD through four possible events:

$$\begin{align*} & x_i \equiv y_k \\ & x_i \equiv y_l \\ & x_j \equiv y_k \\ & x_j \equiv y_l \\ \end{align*}$$

$$\begin{align*} Cov_\alpha(X, Y) = & P(x_i \equiv y_k)Cov(\alpha_i, \alpha_k) + P(x_i \equiv y_l)Cov(\alpha_i, \alpha_l) \\ & + P(x_j \equiv y_k)Cov(\alpha_j, \alpha_k) + P(x_j \equiv y_l)Cov(\alpha_j, \alpha_l) \\ = & 4f_{XY}\sigma_\alpha^2 \\ = & 2f_{XY}\sigma_A^2 \\ \end{align*}$$

Because $\sigma_A^2 = \sigma_{\alpha_i}^2 + \sigma_{\alpha_j}^2 = 2\sigma_\alpha^2$ and $\alpha_i = \alpha_j$ when alleles $i$ and $j$ are IBD.

Parent-offspring

Recall that the coefficient of co-ancestry ( $f_{XY}$ from Ch.5) between a non-inbred parent and non-inbred offspring is 1/4.

• Thus, the coefficient for $\sigma_A^2$ is 1/2.

Consider the convariance between parent (P) and offspring (O),

• If we assume the two alleles of the parent are unrelated,

• Then all offspring can't share a common dominance deviation ( $\sigma^2_D =0$ ).

Therefore,

$$\begin{align*} Cov(P, O) = \frac{1}{2}\sigma_A^2 \end{align*}$$

Parent-offspring

From Breeding value

• Parent genotypic value: $G = A + D$.

• Offspring (half the breeding value of the parents from Ch.7): $G=1/2A$

$$\begin{align*} Cov(P, O) & = Cov(A + D, \frac{1}{2}A) \\ & = \frac{1}{2}Cov(A, A) + \frac{1}{2}Cov(A, D) \\ & = \frac{1}{2}\sigma_A^2 \\ \end{align*}$$

Because $Cov(A, D) = 0$ from Ch.8.

Parent-offspring

Parent-offspring

$$\begin{aligned} Cov(X, Y) & = E(XY) - E(X)E(Y) \\ \end{aligned}$$

where,

$$\begin{aligned} E(XY) = \sum_i \sum_j x_i y_j Pr(X = x_i, Y = y_j) \end{aligned}$$ $$\begin{aligned} E(PO) = & p^2 \times 2q(\alpha-qd) \times q\alpha + 2pq \times ((q-p)\alpha+2pqd) \times 1/2(q-p)\alpha \\ & + q^2 \times (-2p(\alpha+pd)) \times (-p\alpha) \\ = & [2p^2q^2\alpha^2 - 2p^2q^3d\alpha] + [pq\alpha^2(q^2-2pq+p^2) + 2p^2q^2d\alpha(q-p)] \\ & + [2p^2q^2\alpha^2 + 2p^3q^2d\alpha] \\ = & pq\alpha^2(2pq + q^2 -2pq +p^2 + 2pq) + 2p^2q^2d\alpha(-q+q-p+p) \\ = & pq\alpha^2 \end{aligned}$$

Parent-offspring

$$\begin{aligned} Cov(P, O) & = E(PO) - E(P)E(O) \\ \end{aligned}$$ $$\begin{aligned} E(PO) = & pq\alpha^2 \\ E(O) = & 0 \end{aligned}$$ Therefore,

$$\begin{aligned} Cov(P, O) & = E(PO) - E(P)E(O) \\ & = pq\alpha^2 \\ & = 1/2 V_A \end{aligned}$$ Because $V_A = 2pq\alpha^2$.

Offspring and mid-parent

The covariance of the mean of the offspring and the mean of both parents (commonly called the mid-parent)

Let $P$ and $P'$ be the values of the two parents, therefore $\bar{P}=1/2(P + P')$

$$\begin{aligned} Cov(\bar{P}, O) & = Cov(1/2(P + P'), O) \\ & = 1/2(Cov(P,O) + Cov(P', O))\\ & = 1/2 V_A \end{aligned}$$ If $P$ and $P'$ have the same variance.

See P149 of F&M for the algebraic reduction.

The genetic covariance for half sibs?

Genetic covariance for half sibs?

$$\begin{align*} Cov(1/2A, 1/2A) & = 1/4V_A\\ \end{align*}$$

$$\begin{align*} Cov_{HS} = & p^2(q\alpha)^2 + 2pq \times 1/4(q-p)^2\alpha^2 + q^2 \times p^2\alpha^2 \\ =& pq\alpha^2[pq + 1/2(q-p)^2 + pq] \\ =& pq\alpha^2[1/2(p+q)^2] \\ =& 1/2pq\alpha^2 \\ = & 1/4V_A \end{align*}$$

Additive genetic covariance for full sibs?

• Additive value for full sibs: $$\begin{align*} G_{o1} & = \frac{1}{2}A + \frac{1}{2}A'\\ G_{o2} & = \frac{1}{2}A + \frac{1}{2}A'\\ \end{align*}$$

• The additive genetic covariance for full sibs: $$\begin{align*} Cov(G_{o1}, G_{o2}) = & Cov(\frac{1}{2}A + \frac{1}{2}A', \frac{1}{2}A + \frac{1}{2}A') \\ = & Var(\frac{1}{2}(A + A')) \\ = & \frac{1}{4}(\sigma_A^2 + \sigma^2_{A'}) \\ = & \frac{1}{2}\sigma_A^2 \\ \end{align*}$$

Dominance genetic covariance for full sibs?

• Among the progeny, only four possible genotypes, each with a frequency of 1/4.

  • $A_1A_3$, $A_1A_4$, $A_2A_3$, $A_2A_4$

• Let the first sib has any of these genotypes. The 2nd has the same genotype is 1/4.

  • Thus one-quarter of all sib-pairs have the same genotype and consequently the same dominance deviation, $D$.

• Thus, the cross-product of the dominance deviation is $\sigma_D^2$, times frequency 1/4 = $1/4\sigma_D^2$

Genetic covariance for full sibs?

The genetic covariance of full sibs is therefore,

$$\begin{align*} Cov_{FS} = \frac{1}{2}\sigma_A^2 + \frac{1}{4}\sigma_D^2 \end{align*}$$

Covariance of half sibs

$$\begin{align*} Cov_{HS} = \frac{1}{4}\sigma_A^2 \end{align*}$$

In principle, dominance variance can be calculated using full sibs and half sibs:

$$\begin{align*} Cov_{FS} - 2Cov_{HS} = & \frac{1}{2}\sigma_A^2 + \frac{1}{4}\sigma_D^2 - 2 \times \frac{1}{4}\sigma_A^2 \\ = & \frac{1}{4}\sigma_D^2 \end{align*}$$

Twins

Monozygotic (identical) twins

$$Cov_{MZ} = V_G$$

Dizygotic (fraternal) twins

• Not identical twins
• = Full sibs

Summary

General framework for calculating resemblance

Alleles shared between relatives that are identical by descent (IBD) contribute to the covariance between relatives.

For example, if $\sigma_A^2 = 0$, meaning no variation in breeding values exists in the population for the trait of interest, then shared average allelic effects will not contribute to resemblance.

$u \neq 0$ only if the relatives having the same genotype through two alleles IBD.