class: center, middle, inverse, title-slide .title[ # Inbreeding ] .author[ ### Jinliang Yang ] .date[ ### Feb. 5, 2024 ] --- # 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 <div align="center"> <img src="f_var.png" height=300> </div> - 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 | Genotype | Frequency in whole population | | :-------: | : ------ : | | `\(A_1A_1\)` | `\(p_0^2 + V(q)\)` | | `\(A_1A_2\)` | `\(2p_0q_0 - 2V(q)\)` | | `\(A_2A_2\)` | `\(q_0^2 + V(q)\)` | -- - 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. .pull-left[ <div align="center"> <img src="drift.png" height=300> </div> ] -- .pull-right[ - In based population, alleles at a locus are __non-identical__ - `\(N\)` individual, each shedding equal numbers of gametes - `\(2N\)` different sorts of alleles - Any gamete has a `\(1/2N\)` chance to unit with another of the same sort - IBD zygotes: `\(1/2N\)` - Remaining proportion: `\(1 - 1/2N\)` ] --- # 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) | Genotype | Frequency in whole population | | :-------: | : ------ : | | `\(A_1A_1\)` | `\(p_0^2 + V(q)\)` | | `\(A_1A_2\)` | `\(2p_0q_0 - 2V(q)\)` | | `\(A_2A_2\)` | `\(q_0^2 + V(q)\)` | -- - Now, same thing in terms of inbreeding: | Genotype | Base pop. | Change due to F | | :-------: | : ------ : | : ------ : | | `\(A_1A_1\)` | `\(p_0^2\)` | `\(+p_0q_0 F\)` | | `\(A_1A_2\)` | `\(2p_0q_0\)` | `\(-2p_0q_0 F\)` | | `\(A_2A_2\)` | `\(q_0^2\)` | `\(+p_0q_0 F\)` | --- # Genotype Frequencies - But, now we know more, if we can distinguish between IBD and IBS (identity by state) | Genotype | Base pop. | Change due to F | Independent (HWE) | Identical | :-------: | : ------ : | : ------ : | : ------ : | : ------ : | | `\(A_1A_1\)` | `\(p_0^2\)` | `\(+p_0q_0 F\)` | `\(= p_0^2 (1-F)\)` | `\(+p_0 F\)` | | `\(A_1A_2\)` | `\(2p_0q_0\)` | `\(-2p_0q_0 F\)` | `\(= 2p_0q_0(1-F)\)` | | | `\(A_2A_2\)` | `\(q_0^2\)` | `\(+p_0q_0 F\)` | `\(= q_0^2 (1-F)\)` | `\(+q_0 F\)` | - A deficiency of heterozygotes may be the indication that it is a subdivided population.