Consequences of small population

In the absence of migration, mutation, or selection, what is the allele freq over time?

Random drift

• Leads to allele fixation

• Leads to genetic differentiation and local group (or geographic isolation)

• Reduces diversity and becomes more alike in genotype in local groups

Increased homozygosity

• Reduces heterozygotes and results in inbreeding

Fixation

• Over time, each sub-population fluctuates in allele freq and they become more spread apart

• Eventually, each line will become fixed

After fixation

Mean

• The mean allele freq of the lines is still $p_0$ and $q_0$

• $p_0$ is the fraction of lines expected to be fixed for $A_1$ and $q_0$ is the fraction fixed for $A_2$

Variance

• $V(p) = V(q) = p_0q_0(1-(1-\frac{1}{2N})^t) = p_0q_0$

Genotype frequencies

• Genotype frequencies can be deduced from the variance in allele frequencies

• Homozygotes are gained (equally to $p$ and $q$) at the expense of heterozygotes

Inbreeding

The mating together of individuals that are related to each other by ancestry.

• For an unrelated ancestry, one individual will have two parents, four grand-parents, eight great-grandparents, ...

• $t$ generations back it has $2^t$ ancestors ( $2^{20}$ > 1 million)

• In small populations, individuals are related to each other through common ancestors in the more or less remote past.

Consequences of Inbreeding

Inbred individuals (offspring produced by inbreeding)

• May carry two alleles at a locus that are replicates of one and the same allele in a previous generation.

Identical by descent (IBD)

• Individuals carry two alleles that have originated from the replication of one single allele in a previous generation.

• IBD provides a basis for the measurement of the relationship between the mating pairs.

Inbreeding coefficient ( $F$ )

Is the probability that the two alleles at any locus in an individual are IBD.

The range of $F$

$F$ measure the degree of relationship between the individual's parents

• If parents at any generations have mated at random, then $F=0$

• The higher the level of inbreeding the closer the $F$ approaches 1

Inbreeding in the idealized population

Considering hermaphrodite marine organism, capable of self-fertilization, shedding eggs and sperm into the sea.

Inbreeding after t generations

$$\begin{align*} F_t = \frac{1}{2N} + (1-\frac{1}{2N})F_{t-1} \end{align*}$$

Part1: Increment

Attributable to the new inbreeding

Part2: Reminder

Attributable to the previous inbreeding and having the inbreeding coefficients of the previous generation.

Rate of inbreeding

• Let "new inbreeding"
  • $\frac{1}{2N} = \Delta F$
  • (Remember this when we get to variance)

$F_t$ expresses as a function of $\Delta F$

$$\begin{align*} F_t & = \frac{1}{2N} + (1-\frac{1}{2N})F_{t-1} \\ & = \Delta F + (1- \Delta F)F_{t-1} \\ \end{align*}$$

Rewritten the equation

$$\begin{align*} \Delta F = \frac{F_t - F_{t-1}}{1 - F_{t-1}} \end{align*}$$

Panmictic (random mating) index

Using a symbol $P$ for the complement of the inbreeding coefficient $1-F$

• Random mating index or panmictic index
• $P = 1 -F$

$$\begin{align*} \Delta F & = \frac{F_t - F_{t-1}}{1 - F_{t-1}} \\ & = \frac{(F_t -1) - (F_{t-1} - 1)}{1-F_{t-1}} \\ &= \frac{-P_t + P_{t-1}} {P_{t-1}} \end{align*}$$

$$\begin{align*} & \frac{P_t}{P_{t-1}} = 1 - \Delta F \\ \end{align*}$$

Thus, the $P$ index is reduced by a constant proportion each generation.

Panmictic (random mating) index

The $P$ index is reduced by a constant proportion each generation.
$$\begin{align*} & \frac{P_t}{P_{t-1}} = 1 - \Delta F \\ \end{align*}$$

• At generation 2:

$$\begin{align*} & \frac{P_t}{P_{t-2}} = (1 - \Delta F)^2 \\ \end{align*}$$

• From gen $0$ to gen $t$:

$$\begin{align*} & \frac{P_t}{P_0} = (1 - \Delta F)^t \\ & P_t = (1 - \Delta F)^t {P_0}\\ \end{align*}$$

• In base population, $F=0$, so $P=1$
• Then, inbreeding in any generation $t$, relative to the base population is

$$\begin{align*} & F_t = 1- (1 - \Delta F)^t \\ \end{align*}$$

Variance of allele frequency

• Previously, $V(p) = V(q) = \frac{p_0q_0}{2N}$ under random sampling

• New inbreeding: $\Delta F = \frac{1}{2N}$

So, in terms of inbreeding

$$\begin{align*} V(p) = V(q) = p_0q_0 \Delta F \end{align*}$$

Following the relationship after $t$ generations

$$\begin{align*} V(p_t) = V(q_t) = p_0q_0 F_t \end{align*}$$

Back to our previous definition

$$\begin{align*} V(p_t) = V(q_t) = p_0q_0 (1 - (1 - \frac{1}{2N})^t) \end{align*}$$

• Remember that $F_t = 1- (1 - \Delta F)^t$
• Then $V(p_t) = V(q_t) = p_0q_0 F_t$

• $\Delta F$ is the rate of dispersion
• $F_t$ is the cumulative effect of drift

Genotype Frequencies

• Change in allele freq due to drift (sampling process)

• Now, same thing in terms of inbreeding:

Genotype Frequencies

• But, now we know more, if we can distinguish between IBD and IBS (identity by state)

• A deficiency of heterozygotes may be the indication that it is a subdivided population.