Sources of Genetic Variation

Epistasis

Epistasis describes the nonadditivity of effects between loci.

Consider an individual with alleles \(A_i\) and \(A_j\) at one locus and \(B_k\) and \(B_l\) at another.

\[G_{ijkl}=\mu_G+(\alpha_i+\alpha_j+\delta_{ij})+(\alpha_k+\alpha_l+\delta_{kl})+\epsilon_{ijkl}\]

With only two loci, there are actually three ways in which interactions can arise between loci: \(additive \times additive\)(\(\alpha\alpha\)), \(additive \times dominance\)(\(\alpha\delta\)), \(dominance \times dominance\)(\(\delta\delta\)).

A general least-squares model for genetic effects

Additive effects:

\[\alpha_i=G_{i...}-\mu_G\]

Dominance effects:

\[\delta_{ij}=G_{ij..}-\mu_G-\alpha_i-\alpha_j\] \[\delta_{kl}=G_{..kl}-\mu_G-\alpha_k-\alpha_l\]

\[(\alpha\alpha)_{ik}=G_{i.k.}-\mu_G-\alpha_i-\alpha_k\]

\[(\alpha\delta)_{ikl}=G_{i.kl}-\mu_G-\alpha_i-\alpha_k-\alpha_l-\delta_{kl}-(\alpha\alpha)_{ik}-(\alpha\alpha)_{il}\]

\[(\delta\delta)_{ijkl}=G_{ijkl}-\mu_G-\alpha_i-\alpha_j-\alpha_k-\alpha_l-\delta_{ij}-\delta{kl}-(\alpha\alpha)_{ik}-(\alpha\alpha)_{il}-(\alpha\alpha)_{jk}-(\alpha\alpha)_{jl}-(\alpha\delta)_{ikl}-(\alpha\delta)_{jkl}-(\alpha\delta)_{ijk}-(\alpha\delta)_{ijl}\]

complete description of a genotypic value:

\[G_{ijkl....}=\mu_G+[\alpha_i+\alpha_j+\alpha_k+\alpha_l]+[\delta_{ij}+\delta_{kl}]+[(\alpha\alpha)_{ik}+(\alpha\alpha)_{il}+(\alpha\alpha)_{jk}+(\alpha\alpha)_{jl}]+[(\alpha\delta)_{ikl}+(\alpha\delta)_{jkl}+(\alpha\delta)_{ijk}+(\alpha\delta)_{ijl}]+(\delta\delta)_{ijkl}+...\]

The mean value of each type of effect is always equal to zero.

Total genetic variance:

\[\sigma^2_G=\sigma^2_A+\sigma^2_D+\sigma^2_{AA}+\sigma^2_{AD}+\sigma^2_{DD}+...\]

Linkage

Genes that lie on the same chromosome tend to be inherited as a group, a tendency that declines with increasing distance between the loci. Crossing-over is responsible for this decline.

Coefficient of linkage disequilibrium or Coefficient of gametic phase disequilibrium

\[D_{A_iB_j}=P_{A_iB_j}-p_{A_i}p_{B_j}\]

The expected dynamics of gametic phase disequilibrium depend on the recombination fraction between loci, c (0~0.5).

\[P_{A_iB_j}(t+1)=(1-c)P_{A_iB_j}(t)+cp_{A_i}p_{B_j}\]

\[D_{A_iB_j}(t+1)=(1-c)D_{A_iB_j}(t)\]

\[D_{A_iB_j}(t)=(1-c)^tD_{A_iB_j}(0)\]

f <- function(c, t) {
    value <- (1-c)^t
}
plot(x = 0:100, y = f(0.5, 0:100), type = 'l', col = 'red', lwd = 1, xlab = 'Generation(t)', ylab = 'D(t)/D(0)')
text(9, 0.13, 'c=0.5')
lines(0:100, f(0.1, 0:100), col = 'blue', lwd = 1.5)
text(22, 0.19, 'c=0.1')
lines(0:100, f(0.01, 0:100), col = 'orange', lwd = 1.8)
text(50, 0.68, 'c=0.01')
lines(0:100, f(0.001, 0:100), col = 'yellow', lwd = 2)
text(82, 0.95, 'c=0.001')

Effect of disequibrium on the genetic variance

• repulsion disequilibrium
• coupling disequilibrium

\[\sigma^2(x+y)=\sigma^2(x)+\sigma^2(y)+2\sigma(x,y)\] \[\sigma^2(\sum_i^nx_i)=\sum_i^n\sum_j^n\sigma(x_i,x_j)=\sum_i^n\sigma^2(x_i)+2\sum_{i<j}^n\sigma(x_i,x_j)\]

\[\sigma^2_A=2\sum_{i=1}^{n}\alpha(i)^2p_iq_i+2\sum_{i=1}^n\sum_{j\neq i}^n\alpha(i)\alpha(j)D_{ij}\] \[\sigma^2_D=4\sum_{i=1}^n(a_ik_ip_iq_i)^2+4\sum_{i=1}^n\sum_{j\neq i}^na_ia_jk_ik_jD_{ij}^2\]

When the disequilibrium covariance is negative, we refer to it as hidden genetic variance.