Natural Selection
Let's talk about natural selection from the population genetics point of view.
One of the easiest
ways to start is to ignore all the stages at which selection can occur and to just lump them all together
into a term that we will call
Fitness
.
Fitness is defined as the (probability of survival) X (the probability
of successful mating) X (fecundity).
This is just the success of different genotypes at producing new
individuals.
Now we will assign a fitness value to genotypes that have been produced by random mating.
At generation 0,
before selection
, we will have
allele frequencies of p and q and
genotype frequencies
of p
2
, 2pq, and q
2
.
Now let's introduce differential fitnesses of w
BB
, w
Bb
, w
bb
, which represent the average
fitnesses of the individuals with each genotype
It is convenient but not necessary that the w's are
expressed as
relative fitnesses
, with one of the fitnesses (usually the highest) set equal to 1 and the others
expressed as a proportion of that fitness.
After selection
, the genotypes will be represented in the
following relative numbers:
Genotype
BB
Bb
bb
Numbers
p
2
w
BB
2pqw
Bb
q
2
w
bb
These are just (the probability of getting the genotype with random mating) times (the
genotypes' average viability, mating success, and fecundity).
It is easier to think about fitnesses by
considering only viability selection, but the math works with selection at any stage in the life cycle.
Note that these are not frequencies anymore.
[A good way to visualize this is to let p = .5 , q
=.5, w
BB
= 1 and w
Bb
= w
bb
= 0.
Now all of our individuals are BB, but p
2
w
BB
= p
2
, which not 1.]
In order
to make them frequencies, we need to divide these numbers by the total number.
The total number is just
the sum of the three genotype numbers:
p
2
w
BB
+ 2pqw
Bb
+ q
2
w
bb
= W
Note that
W = the average fitness of the population
(since the sum of frequencies of a class times the
value of the class equals the average value.
See, I told you that you Problem Set 1 would be useful!
Note also that W is the average of all the individuals in the population, not just the unweighted average
of the 3 genotype fitnesses.)
So now our
genotype frequencies after selection
are:
BB
Bb
bb
p
2
w
BB
/ W
2pqw
Bb
/ W
q
2
w
bb
/ W
We can calculate our
allele frequencies after selection
as the sum of homozygote frequency and 1/2 the
heterozygote frequency, just as we did before:
p
after selection
= [p
2
w
BB
/ W] + [pqw
Bb
/ W] = (p
2
w
BB
+ pqw
Bb
) / W
if all w are the same (= no selection) then p
after selection
= p but if they differ then p
after selection
will not
necessarily equal p.
That is,
if there is differential fitness among genotypes, allele frequencies may
change due to selection
.
Since we will now mate randomly among this post-selection population, it also follows that