Selection

Deterministic models to describe selection:  (diploid organisms, two alleles A₁ and A₂)

    codominance (kind of logistic equation) q=frequency of allele A2, 

Genotype:                                     A₁A₁        A₁A₂        A₂A₂

Relative number of offspring           1             1+s           1+2s

Fitness                                           w11          w12          w22

genotype frequency                        p^2             2pq          q^2

(pq: allele frequencies,=> genotype frequencies in Hardy Weinberg equilibrium)

Change in frequency (approximately):  
dq/dt= s* q*(1-q) and

q(t)=1/(1+((1-q₀)/q₀)*e^-st)

    over dominance

Genotype:                                                A₁A₁    A₁A₂    A₂A₂

          s₁>s₂  :   balancing selection (try it)

Go to Kent Holsinger's collection of JAVA applets here and explore some of the time courses with different values of s₁ and s₂.  

Under which conditions of w11, w12, and w22 can one maintain both alleles over long periods of time?

Stochastic approaches -- random drift - neutral evolution:

Law of the gutter (see also Steven J Gould's interpretation on the trend to increasing complexity)

Explore some simulations: 
     Drift only (vary the population size N),

How does the survival of multiple alleles in a population depend on the population size.

     Drift and Selection (interesting setting: P=0.01, N=50)

Note: Even though the allele conveys a strong selective advantage of 10%, the allele has a rather large chance to go extinct quickly.

Mutation rate versus Substitution rate

The following assumes co-dominance or no selection:

s=0:  Probability of fixation, P, is equal to frequency of allele in population, q

mutation rate (per gene/per unit of time) = u ;  

frequency with which new alleles are generated in a diploid population size N equals to u*2N

Probability of fixation for each new allele = 1/(2N)

Substitution rate = frequency with which allele is generated * Probability of fixation= u*2N *1/(2N) = u

Therefore:
The substitution rate is independent of population size if s=0 and equal to the mutation rate!!!!
This is the reason that there is hope that the molecular clock might sometimes work.

For advantageous mutations: 
      Probability of fixation, P, is approximately equal to 2s;
      e.g., if selective advantage s = 1% then P = 2%

      Does this correspond to the simulations you performed above?

Fixation time

Neutral mutations:  t_av=4*N_e generations 
(N_e=effective population size; For n discrete generations N_e= n/(1/N₁+1/N₂+?..1/N_n)

S unequal to 0:  t_av= (2/s) ln (2N) generations  (also true for mutations with negative s --  How can this be??)

E.g.:  N=10⁶, s=0:  average time to fixation: 4*10⁶ generations

N=10⁶, s=0.01:  average time to fixation: 2900 generations