class: center, middle, inverse, title-slide # Heterozygosity ### Jinliang Yang ### Sept. 14, 2022 --- # Systematic and dispersive processes ### Systematic process: mutation, migration, and selection - Bring the allele frequencies to stable equilibria at particular value -- ### Dispersive process - drift - Scatter the allele frequencies away from these equilibrium - Eventually lead to all alleles being either fixed or lost --- # Non-recurrent neutral mutation ### Infinite allele model (IAM) of mutation - Probability of the same mutation occurs more than once is very, very, very small -- ### Neutral model (new allele has no effect on fitness) - Probability new allele becomes fixed in the absence of selection = `\(\frac{1}{2N}\)` - If `\(\mu\)` is mutation rate at the locus, total number of new mutants = `\(2N\mu\)` - Probability of a new mutant being fixed `\(= 2N \mu \times \frac{1}{2N} = \mu\)` > Kimura, 1983 --- # Effective neutral mutation #### Probability of fixation of new mutation is influenced by selection - Favorable (beneficial) or unfavorable (deleterious) -- `\begin{align*} Pr(fixation) \approx \frac{1}{2N_e} \end{align*}` > Kimura, 1983 - __Effective neutral mutation__ is one a coefficient of selection `\(s\)` against it - Range from `\(s=0\)` (neutral) to `\(s=\frac{1}{2N_e}\)` or -- - `\(N_es < 1/2\)`: effectively neutral - `\(N_es > 1/2\)`: under selection --- # Effective neutral mutation ### `\(N_es < 1/2\)`: effectively neutral For example, inbred lines of mice resulting from sib matings have `\(N_e \approx 2.5\)` - To keep `\(N_es < 1/2\)`, `\(s\)` can be as high as 0.2 and this allele could become fixed due to drift and have a negative effect on fitness -- - The indication: favorable mutants are far too rare to conter balance the loss of fitness in small population! --- # Selection and drift ### Selection counteracts drift - Drift shifts allele frequency away from equilibrium - This reduces the average fitness of the population -- ### Probability of fixation under selection `\begin{align*} Pr(fixation) = \frac{1 - e^{-2s}}{1- e^{-4N_es}} \end{align*}` > Moran, 1959 -- - If `\(2s\)` and `\(-4N_es\)` are small, numerator becomes `\(\approx 2s\)` and denominator `\(\approx 4N_es\)` `\begin{align*} Pr(fixation) = \frac{1 - e^{-2s}}{1- e^{-4N_es}} \approx \frac{2s}{2N_es} = \frac{1}{2N_e} \end{align*}` --- # Probability of fixation under selection `\begin{align*} Pr(fixation) = \frac{1 - e^{-2s}}{1- e^{-4N_es}} \end{align*}` > Moran, 1959 - If `\(2N_es\)` is small, an advantageous mutation has about the same Pr(fixation) as a neutral mutation -- ### R function for the fixation equation ```r pfix <- function(ne, s){ # exp computes the exponential function num <- 1 - exp(-2*s) # numerator den <- 1 - exp(-4*ne*s) # denominator return(num/den) } ``` --- # Probability of fixation under selection - In this case, `\(s\)` is selective advantage ```r ne <- seq(1, 100, by=1) plot(ne, 1/(2*ne), type="l", lwd=3, col="red", xlab="Effective population size (Ne)", ylab="Pr(fixation)") lines(ne, pfix(ne, s=0.01), lty=2, lwd=3, col="blue") lines(ne, pfix(ne, s=0.05), lty=2, lwd=3, col="black") lines(ne, pfix(ne, s=0.1), lty=2, lwd=3, col="orange") ``` <img src="week4_c2_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" /> --- # Probability of fixation under selection <div align="center"> <img src="nes.png" width=450> </div> - Favorable mutations likely be lost - Mutation with a 1% advantage (heterozygous, `\(s=0.01\)`), has 2% probability of being fixed - If 10% advantage `\(s=0.1\)`, probability becomes 20%