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
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.
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
