Wright-Fisher model

• The population size stays constant
• In a model with infinite population size, the allele frequency doesn't change if there is no selection.
• But in a finite population, allele frequency fluctuates due to sampling.
• Transition probability
  • Imagine filling the empty circles in gen. t + 1 randomly.
  • What is the probability that the 1st circle in gen. t+1 is red?
  • What is the probability that all circles become red?
  • Familiar?? Let's generalize this process:
    Let $X_t$ be the number of red alleles, $Prob(X_{t+1} = k \vert Xt)$?
    
    
    
    
    
    
  • $E[X_{t+1}] = E[X_{t}] = E[X_{t-1}] = E[X_{t-2}] = \cdots = X_0$.
    This means that drift is not expected to change allele frequencies on average.
  • $Var[X] =$
    The amount of drift is inversely proportional to the population size. Small populations are prone to random changes.
• Absorbing states
  • Let p be the initial frequency of an allele in N diploids.
  • What is the fate of this allele?

  • Calculate the probability of each fate.