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Selection versus genetic drift.

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

 Relative number of offspring                1         1+s1     1+s2

          s1>s:   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.

     This simulation follows many populations (with the selected parameters) over time. It plots a histogram that shows how many of the populations have the allele frequency indicated on the y-axis. If you set the mutation rate to 0, this provides a nice illustration of the law of the gutter. (In the presence of the alleles converting back and forth, fixation does not occur.)

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:  tav=4*Ne generations 
(Ne=effective population size; For n discrete generations Ne= n/(1/N1+1/N2+?..1/Nn)

S unequal to 0:  tav= (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

Neutral theory: 

The vast majority of observed sequence differences between members of a population are neutral (or close to neutral). These differences can be fixed in the population through random genetic drift. Some mutations are strongly counter selected (this is why there are patterns of conserved residues). Only very seldom is a mutation under positive selection. 

The neutral theory does not say that all evolution is neutral and everything is only due to to genetic drift.

(Nearly neutral theory:  Even synonymous mutations do not lead to random composition but to codon bias.  Small negative selection might be sufficient to produce this bias. )

Note: the larger the population the better selection works, and the closer to neutral a mutation needs to be in order to be fixed by genetic drift. (If N*s<<1 the mutation behaves as neutral, and the fixation probability is 1/N; if N*s~1 then fixation probability is only about 2s, which is small, but seems to work.)

Is Evolution in humans only neutral? Does selection still play a role? E.g., here , the distribution of alleles that encode a protein presumably involved in brain development (here for the article, in case you are interested to read more, a similar case reported here), here for a comment that argues the haplotype frequencies might be due to drift and small founder populations and not reflect selection.