class: center, middle, inverse, title-slide .title[ # Coancestry ] .author[ ### Jinliang Yang ] .date[ ### Feb. 28, 2024 ] --- # Individual Inbreeding: `\(F_X\)` `\begin{align*} F_{X} = (\frac{1}{2})^n(1 + F_{A}) \end{align*}` - Where `\(n\)` is the number of individuals in the path from the individual’s sire (dad), through the common ancestor, to the dam (mom). - If multiple common ancestors, must sum the individual estimates. -- ------------ The inbreeding coefficient of an individual X depends on the amount of common ancestry in its two parents. # Coancestry (Kinship): `\(f_{XY}\)` - Coancestry of two parents = inbreeding of their offspring - Probability of IBD of two haplotypes, one drawn from each parent X and Y, symbolized by `\(f_{XY}\)` --- # Coancestry Coancestry ( `\(f_{XY}\)` ) is the probability of two gametes, one from each parent (X and Y), will contain haplotypes that are IBD ### Example: `\(X \times Y\)` mating .pull-left[ <div align="center"> <img src="dog1.png" width=280> </div> ] -- .pull-right[ ### Coancestry `\(f_{XY}\)` `\begin{align*} f_{XY} = & \frac{1}{4}Pr(x_1 \equiv y_1) \\ & + \frac{1}{4}Pr(x_1 \equiv y_2) \\ & + \frac{1}{4}Pr(x_2 \equiv y_1) \\ & + \frac{1}{4}Pr(x_2 \equiv y_2) \\ \end{align*}` ] --- # Case study 1: selfing ### Coefficient of coancestry with one’s self. <div align="center"> <img src="corn1.png" width=250> </div> -- - If __ `\(x1x2\)`__ are sampled, the probability they are IBD depends on whether or not the individual could have obtained these gametes from a common ancestor - This is reflected by the inbreeding coefficient of the individual __ `\(F_X\)`__ - If inbreeding = 0, this combination of gametes does NOT contribute to the possibility of IBD. --- # Case study 1: selfing ### Coefficient of coancestry with one’s self. <div align="center"> <img src="corn1.png" width=280> </div> `\begin{align*} f_{XX} = & \frac{1}{4}Pr(x_1 \equiv x_1) + \frac{1}{4}Pr(x_1 \equiv x_2) + \frac{1}{4}Pr(x_2 \equiv x_1) + \frac{1}{4}Pr(x_2 \equiv x_2) \\ = & \frac{1}{2}(1 + F_X) \\ \end{align*}` -- This result suggests that `\(f_{XX} = \frac{1}{2}\)` if individual X is a non-inbred with `\(F_X=0\)`. --- # Case study 2: Parent-Offspring Draw gametes at random (one from each individual X and its progeny A), what is probability they are IBD (or `\(f_{XA}\)`)? <div align="center"> <img src="dog2.png" width=280> </div> --- # Case study 2: Parent-Offspring Draw gametes at random (one from each individual X and its progeny A), what is probability they are IBD (or `\(f_{XA}\)`)? <div align="center"> <img src="dog3.png" width=450> </div> -- #### Eight possible combinations - 1) `\(x_1x_1\)` 2) `\(x_1x_2\)` 3) `\(x_2x_1\)` 4) `\(x_2x_2\)` - 5) `\(x_1y_1\)` 6) `\(x_1y_2\)` 7) `\(x_2y_1\)` 8) `\(x_2y_2\)` --- # Case study 2: Parent-Offspring .pull-left[ <div align="center"> <img src="dog2.png" width=200> </div> ] .pull-right[ #### Eight possible combinations - 1) `\(x_1x_1\)` 2) `\(x_1x_2\)` 3) `\(x_2x_1\)` 4) `\(x_2x_2\)` - 5) `\(x_1y_1\)` 6) `\(x_1y_2\)` 7) `\(x_2y_1\)` 8) `\(x_2y_2\)` ] ------ - If gamete from X is `\(x_1\)`, and gamete from A is `\(y_1\)`, the probability of IBD depends on if there is a relationship between X and Y (the parents of A) -- - This is quantified by the coefficient of coancestry between those parents `\(f_{XY}\)` - In total, of the 8 possible combinations, 4 include a gamete from parent Y where this relationship holds --- # Case study 2: Parent-Offspring .pull-left[ <div align="center"> <img src="dog2.png" width=200> </div> ] .pull-right[ #### Eight possible combinations - 1) `\(x_1x_1\)` 2) `\(x_1x_2\)` 3) `\(x_2x_1\)` 4) `\(x_2x_2\)` - 5) `\(x_1y_1\)` 6) `\(x_1y_2\)` 7) `\(x_2y_1\)` 8) `\(x_2y_2\)` ] ------ `\begin{align*} f_{XA} = & \frac{1}{8}[Pr(x_1 \equiv x_1) + Pr(x_1 \equiv x_2) + Pr(x_2 \equiv x_1) + Pr(x_2 \equiv x_2) )\\ & + Pr(x_1 \equiv y_1) + Pr(x_1 \equiv y_2) + Pr(x_2 \equiv y_1) + Pr(x_2 \equiv y_2)] \\ = & \frac{1}{8}(1 + F_X+ F_X+1 + 4f_{XY})\\ = & \frac{1}{4}(1 + F_X + 2f_{XY})\\ \end{align*}` -- Assuming no inbreeding, i.e., `\(F_X = 0\)` and `\(f_{XY}=0\)` Coefficient of coancestry of parent-offspring, `\(f_{XA}=\frac{1}{4}\)` --- # Relationship vs Coancestry ### Coefficient of coancestry: Pick an allele from X. The probability that you will __pick__ that allele in Y is the coefficient of coancestry. -- ### Coefficient of relationship: Pick an allele from X. The probability that Y __has__ that allele is the coefficient of relationship -- ---------------- Therefore: __coancestry = 1⁄2 relationship__ --- # Relationship vs Coancestry | | Coancestry | Relationship | | :-------: | : ------ : | :-------: | | Parent-offspring | 0.25 | 0.5 | | Full-siblings | 0.25 | 0.5 | | Half-siblings | 0.125 | 0.25 | - Full-sibs have two parents in common - Half-sibs have one parent in common --- # The basic rule relating coancestries <div align="center"> <img src="cow_ped.png" width=350> </div> #### What is coancestry of X and Y (or `\(f_{XY}=Pr(x \equiv y)\)`)? -- Coancestry between X and Y depends on relationship among parents `\begin{align*} f_{XY} & = Pr(x \equiv y) \\ & = Pr(a \equiv c) + Pr(a \equiv d) + Pr(b \equiv c) + Pr(b \equiv d ) \end{align*}` -- Consider one part of this equation: `\(Pr(a \equiv c)\)` - This is the probability that a random allele from X is derived from individual A - And a randomly chosen allele from Y is from individual C - And these are IBD --- # The basic rule relating coancestries <div align="center"> <img src="cow_ped.png" width=350> </div> `\begin{align*} f_{XY} & = Pr(x \equiv y) \\ & = Pr(a \equiv c) + Pr(a \equiv d) + Pr(b \equiv c) + Pr(b \equiv d ) \end{align*}` ### Consider one part of this equation: `\(Pr(a \equiv c)\)` `\begin{align*} Pr(a \equiv c) & = \frac{1}{4}\frac{1}{4}Pr(a_1 \equiv c_1) + \frac{1}{4}\frac{1}{4}Pr(a_1 \equiv c_2) + \frac{1}{4}\frac{1}{4}Pr(a_2 \equiv c_1) + \frac{1}{4}\frac{1}{4}Pr(a_2 \equiv c_2) \\ & = \frac{1}{16}(Pr(a_1 \equiv c_1) + Pr(a_1 \equiv c_2) + Pr(a_2 \equiv c_1) + Pr(a_2 \equiv c_2)) \\ & = \frac{1}{16}(4f_{AC}) = \frac{1}{4}(f_{AC}) \end{align*}` --- # The basic rule relating coancestries ### Repeat for the 3 other terms <div align="center"> <img src="cow_ped.png" width=350> </div> `\begin{align*} f_{XY} & = Pr(x \equiv y) \\ & = Pr(a \equiv c) + Pr(a \equiv d) + Pr(b \equiv c) + Pr(b \equiv d ) \\ & = \frac{1}{4}(f_{AC} + f_{AD} + f_{BC}+ f_{BD}) \end{align*}` -- In summary, `\(f_{XY}\)` is equal to the average coefficient of coancestry between the parents of X and Y - Avg coancestry between parents: AC, AD, BC, and BD --- # Relationship vs Coancestry | | Coancestry | Relationship | | :-------: | : ------ : | :-------: | | Parent-offspring | 0.25 | 0.5 | | Full-siblings | 0.25 | 0.5 | | Half-siblings | 0.125 | 0.25 | - Full-sibs have two parents in common <div align="center"> <img src="cow2.png" width=350> </div> -- `\begin{align*} f_{XY} & = Pr(x \equiv y) \\ & = \frac{1}{4}(f_{AA} + f_{AB} + f_{BA}+ f_{BB}) \end{align*}` With no previous inbreeding or relationship, and `\(f_{AA}=f_{BB}\)`=1/2, `\(f_{XY}=1/4\)`