http://www.wsu.edu/~mmorgan/gencb511f98/drift/node1.html#SECTION00012000000000000000

Back: Genetic drift (pdf)

Population Genetics Martin Morgan Fall, 1998

Real populations are finite in size, so that stochastic forces influence gene frequency

Some alleles leave zero descendants

• Allele representation following reproduction is stochastic, given by Poisson distribution.
• Expected number of copies produced per allele must be, in the long term, near 1.
• Constant sized population implies $\lambda=1$ so $\Pr{0\mathrm{\,descendants}} = \lambda^0e^{-\lambda}/0! = e^{-1}\approx 0.37$.
• New mutation lost 37% of time after 1 generation, 53% after two,...

Binomial sampling describe stochastic change at intermediate allele frequency

• Random change following binomial distribution:
  
  $$\Pr{j\,\mathrm{copies}}=\binom{2N}{j}p^jq^{2N-j}$$
  
  

• Mean, variance of binomial distribution.
• Iterative application.
• Hard-core: diffusion approximation for change in probability distribution, $\phi(x,t)$ (Hartl and Clark, pp. 277-283.

Genetic drift results in differentiation between populations, ultimately leading to fixation or loss of alleles

• Average change? Mean of binomial distribution.
• Differentiation between populations? Binomial variance.
• Probability of fixation? Initial frequency. Of loss? 1-initial frequency.
• New mutation: initial frequency? Time to loss $\approx 2\log {2N}$ (relatively insensitive to population size). Time to fixation: $\approx 4N$ (large when population size large).

Identity by descent increases in small populations because of genetic drift

• $F$: Expected homozygosity, rather than inbreeding coefficient. $F_1=(1-1/2N)F_0+1/2N$.
• Expected heterozygosity: $H=1-F$ so $H_t =1-F_t =(1-1/2N)(1-F_{t-1}) =(1-1/2N)H_{t-1}= \cdots =(1-1/2N)^tH_0$. Corollary: $F_t = 1-(1-1/2N)^t(1-F_0)$
• Does greater $F$ imply deviation from Hardy Weinberg proportions? What is $F$ measuring?

Martin Morgan Department of Genetics & Cell Biology Department of Botany

Back: Genetic drift (pdf)