1. Genetic Terms
Gene - a unit of inheritance that usually is directly responsible for one trait or character.
Allele - an alternate form of a gene. Usually there are two alleles for every gene, sometimes as many a three or four.
Homozygous - when the two alleles are the same.
Heterozygous - when the two alleles are different, in such cases the dominant allele is expressed.

2. Genetic Terms
Dominant - a term applied to the trait (allele) that is expressed irregardless of the second allele.
Recessive - a term applied to a trait that is only expressed when the second allele is the same (e.g. short plants are homozygous for the recessive allele).
Phenotype - the physical expression of the allelic composition for the trait under study.
Genotype - the allelic composition of an organism.
Punnett squares - probability diagram illustrating the possible offspring of a mating.

3. Genetic Code

4. Example of Pedigree
A2/A2
A1/A2
Female
A1/A1
Male

5. Two alleles model
Let us assume a gene containing two mutually exclusive alleles:A1,A2.
There are three possible combinations:A1/A1,A1/A2 and A2/A2, with initial frequencies: u,v and w.
u+v+w=1
Let us compute their density after one division

6. First generation frequency Offsprings Mating type 2uv
0.5A1/A1+ 0.5A1/A2
A1/A1xA1/A2
2uw
A1/A2
A1/A1xA2/A2 U2
A1/A1
A1/A1xA1/A1 V2
0.25A1/A1+0.5A1/A A2/A2
A1/A2xA1/A2
2vw
0.5A1/A2+0.5A2/A2
A1/A2xA2/A2 w2
A2/A2
A2/A2xA2/A2

7. In the next generation Hardy-Weinberg Equilibrium
A1/A1=u2+uw+0.25v2=(u+0.5v)2=p12
A2/A1=uv+2uw+0.5v2+vw=2(u+0.5v)(w+0.5v)=2p1p2
A2/A2=0.25v2+vw+w2=(w+0.5v)2=p22

8. Third generation frequency Offsprings Mating type 2p13p2
0.5A1/A1+ 0.5A1/A2
A1/A1xA1/A2
p12p22
A1/A2
A1/A1xA2/A2
p14
A1/A1
A1/A1xA1/A1
4p12p22
0.25A1/A1+0.5A1/A A2/A2
A1/A2xA1/A2
2p1p23
0.5A1/A2+0.5A2/A2
A1/A2xA2/A2
p24
A2/A2
A2/A2xA2/A2

9. X linked loci
If an allele is present in the X chromosome, the situation is more complicated.
If qn is the allele frequency in women in the generation n, and rn the allele frequency in mens.
Rn=qn-1
Qn=0.5qn-1+0.5rn-1
2/3qn+1/3rn=2/3(0.5qn-1+0.5rn-1)+1/3qn-1
=2/3qn-1+1/3rn-1=p
Qn-p=-0.5(qn-1-p)

10. Dynamics to equilibrium

11. When the Hardy-Weinberg Law Fails to Apply
To see what forces lead to evolutionary change, we must examine the circumstances in which the Hardy-Weinberg law may fail to apply. There are five:
mutation
gene migration
genetic drift
nonrandom mating
natural selection

12. Mutation
The frequency of gene B and its allele b will not remain in Hardy-Weinberg equilibrium if the rate of mutation of B -> b (or vice versa) changes.
By itself, mutation probably plays only a minor role in evolution; the rates are simply too low. But evolution absolutely depends on mutations because this is the only way that new alleles are created. After being shuffled in various combinations with the rest of the gene pool, these provide the raw material on which natural selection can act.

13. Gene Migration
Many species are made up of local populations whose members tend to breed within the group. Each local population can develop a gene pool distinct from that of other local populations.
However, members of one population may breed with occasional immigrants from an adjacent population of the same species. This can introduce new genes or alter existing gene frequencies in the residents. In many plants and some animals, gene migration can occur not only between subpopulations of the same species but also between different (but still related) species. This is called hybridization. If the hybrids later breed with one of the parental types, new genes are passed into the gene pool of that parent population. This process, is called introgression.

14. Genetic Drift
As we have seen, interbreeding often is limited to the members of local populations. If the population is small, hardy-Weinberg may be violated. Chance alone may eliminate certain members out of proportion to their numbers in the population. In such cases, the frequency of an allele may begin to drift toward higher or lower values. Ultimately, the allele may represent 100% of the gene pool or, just as likely, disappear from it. Drift produces evolutionary change, but there is no guarantee that the new population will be more fit than the original one. Evolution by drift is aimless, not adaptive.

15. Rate of Genetic Drift

16. Nonrandom Mating
One of the cornerstones of the Hardy-Weinberg equilibrium is that mating in the population must be random. If individuals (usually females) are choosy in their selection of mates the gene frequencies may become altered. Darwin called this sexual selection.
Assortative mating -Humans seldom mate at random preferring phenotypes like themselves (e.g., size, age, ethnicity). This is called assortative mating.

17. Natural Selection
If individuals having certain genes are better able to produce mature offspring than those without them, the frequency of those genes will increase. This is simple expressing Darwin's natural selection in terms of alterations in the gene pool. (Darwin knew nothing of genes.) Natural selection results from
differential mortality and/or
differential fecundity.

18. Mortality Selection
Certain genotypes are less successful than others in surviving through to the end of their reproductive period.
The evolutionary impact of mortality selection can be felt anytime from the formation of a new zygote to the end (if there is one) of the organism's period of fertility. Mortality selection is simply another way of describing Darwin's criteria of fitness: survival.

19. Fecundity Selection
Certain phenotypes (thus genotypes) may make a disproportionate contribution to the gene pool of the next generation by producing a disproportionate number of young. Such fecundity selection is another way of describing another criterion of fitness described by Darwin: family size. In each of these examples of natural selection certain phenotypes are better able than others to contribute their genes to the next generation. Thus, by Darwin's standards, they are more fit. The outcome is a gradual change in the gene frequencies in that population.

20. Effect of Natural Selection on Gene Frequencies.
Let us define the frequency of each genotype in the population as: w, and the initial allele distribution as p and q for A1 and A2.
wA1/A1=1-r
wA1/A2=1
wA2/A2=1-s

21. Fitness The average fitness is: W=(1-r)p2+2pq+(1-s)q2=1-rp2-sq2
DP=([(1-r)p2+pq]/W)-p=pq[s-(r+s)p]/W

22. Hetrozygote Advantage
Pn+1-(s/(r+s))=[(1-r)pn2+pnq]/W -(s/(r+s))
=[(1-r)pn2+pnq]-(s/(r+s))W]/W=
=[(1-r)pn2+pnq]-(s/(r+s)) (1-rpn2-sqn2)]/W
= (1-rpn-sqn)/W[pn-(s/(r+s)]
The difference decreases to zero only for positive r and s. Thus the scenario in which both alleles can survive is Hetrozygote Advantage

23. Recessive diseases
If r>0, and s=0, the disadvantage appears only homozygotic A1.
In this case: pn+1=pn(1-rpn)/(1-rpn2)
1/pn+1-1/pn=1/pn[(1-rpn2)/(1-rpn)-1]=
[r(1-pn2)/(1-rpn)]
1/pn-1/p0=nr

24. Fitness Summary
Third fix point is in the range [0,1] only if r and s have the same sign.
It is stable only of both r and s are positive
In all other cases one allele is extinct.
If r>0 and s=0 then the steady state is still p=0, but is is obtained with a rate pn=1/(nr+1/p0)

25. Balance between Mutation and selection.
Mutations can provide a balancing force to selection.
Let us assume a mutation rate of m from A2 to A1. The dynamics equation is:
DP=[(1-r)p2+pq]/W-p+(1-m)*q
An equilibrium is obtained when
q=[pq+(1-s)q2]/W(1-m)1+m(1-s)/s

26. Summary
In the absence of selection an allele concentration equilibrium is obtained after one generation.
In the presence of selection, usually a single allele survives.
There are many mechanisms which can lead to the failure of the hardy weinberg equilibrium.