Populations and genetic dynamics

A population is a group of organisms of the same species occupying a particular space at a particular time that can potentially interbreed.

A population can contain one or more gene pools, i.e. a number of alleles (of genes) within organisms with a restricted (geographical) range, but which can be passed from individual organisms to their offspring by the normal process of reproduction.

Depending on the environment, different alleles may give advantages to organisms holding them, and their frequency within the gene pool may consequently increase. This is the basis of evolution, and it is responsible for the formation of new and different species.

As such, the term population has an ecological and a genetic component.

This topic is centred on the Hardy-Weinberg principle and its application within a population, but I extended the title above to draw attention to changes in the abundance of different genetic characteristics and their equilibrium in the world.

It follows on from inheritance and genetics, and it can be seen to have relevance within the ecosystem and the evolution of species which have specialisations that allow them to be successful in this context.

Frequencies within a population

Often the abundance of different types within a population are counted and expressed as a proportion or a percentage, but it is normal to convert these into a value between 0 and 1.

Allele frequency

Although we think of alleles circulating within a gene pool within a population, not all are immediately noticeable.

Genotype frequency

Dominant alleles have a higher profile than recessive alleles, but in diploid organisms it is the combinations of (pairs of) alleles that affect the individuals within the population at a cellular level.

Phenotype frequency

This is a relatively simple concept, involving the counting of of organisms showing or not showing the characteristics that are observable as a result of the expression of the genes. But often homozygous individuals and heterozygous individuals carrying one or two dominant genes show the same phenotype. Only homozygous recessive types are available to be counted.

In a stable population, the frequencies above will remain fairly constant. Positive changes result in the formation of new species (speciation) whereas decreases can lead to the loss of species (extinction).

Organisms reproducing sexually give rise to offspring which show a spectrum of variation, rather than aexually-reproducing organisms which produce identical offspring.

As a result a population can be considered to be a mosaic of different genotypes, only some of which can be directly seen in the phenotypes, but interactions will occur between recessive alleles hidden in heterozygotes, and there is a dynamic equilibrium between the different forms.

The Hardy-Weinberg principle

This provides a mathematical model, based on phenotype, genotype and allele frequencies. This can confirm when allele frequencies are stable, not changing from generation to generation.

Underlying this are a number of assumptions

(1) random mating - in proportion to the frequencies of each genotype
(2) the absence of natural selection - no differential success for one phenotype or another
(3) a very large population size - minimising the effect of genetic drift
(4) no gene flow or migration - no input of individual organisms, with different genetic makeup
(5) no mutation (in recent times) - although different alleles are expected to have arisen by mutation in the past
(6) the locus is autosomal - with sex linkage alleles are only represented on single versions of a sex chromosome in one sex (so no heterozygotes), whereas the other sex has two sex chromosomes. It is possible to make allowance for this by different application of the mathematics below.

Before the emergence of this principle, it was fairly widely assumed that, within a population, dominant alleles would tend to take precedence.

It is now seen that the heterogeneity of a population depends on a number of underlying factors, including interactions between carriers of 'hidden' recessive genes.

Applying the Hardy-Weinberg equation

If a gene in a population has a dominant allele A and a recessive allele a, assume that
A has the allele frequency p and
a has the allele frequency q.
Since we are covering the whole population; p+q=1

The population will contain the genotypes AA, Aa and aa.

The frequencies of these genotypes will be given by the expression:
p² + 2pq + q² = 1
AA will have the genotype frequency = p²
Aa will have the genotype frequency = 2pq
aa will have the genotype frequency = q².


It is likely that the dominant homozygotes AA and heterozygotes Aa have the same phenotype, so they are indistinguishable.
But the recessive homozygotes aa should be obviously different and countable.

By taking the aa phenotype frequency, which is numerically the same as the aa genotype frequency - q² - it is possible to work out the recessive allele a frequency - q - by taking the square root of this value.

The dominant allele frequency p is 1-q. From this the genotype frequencies for AA and Aa can be calculated by multiplication.

Multiple alleles

If more than 2 alleles are involved, the Hardy-Weinberg equation will be extended with the addition of more terms. For 3 alleles: p² + 2pq + q² + 2pr + r² + 2qr = 1

For example, for the ABO blood group, there are 4 phenotypic classes:

Phenotype (blood group)	Genotypes	Expected frequency	%
A	IAIA, IAIo	p2 + 2pr	38
B	IBIB, IBIo	q2 + 2qr	10
AB	IAIB	2pq	3
O	IoIo	r2	48

In the event of polyploidy, the equation expands into a more complicated form.

Examples

Heathland flowers

Heather or ling (Calluna vulgaris) usually has purple flowers, but if a large area of heathland is examined, plants with white flowers can be found. This is caused by a recessive gene w so white-flowered plants have the genotype ww rather than the normal colouration caused by gentotypes WW and Ww.

The frequency of plants with white flowers on an area of bogland has been estimated at 1 in 1,000,000.

Calculate the frequency of the w allele in this sample. Show your working.

White phenotype frequency = ww genotype frequency = q² = 1/1,000,000 = 0.000001

So w allele frequency (q) = 1/1000 = 0.001

What is the proportion of the population that is heterozygous for the white colour allele (Ww)?

p=1-q = 1 - 0.001 = 0.999

So 2pq = 0.001998 ( 1 in 500.5)

This means that for every single white-flowered plant there are 500 carrier plants (genotype Ww) - amongst 999,499 pure breeding normal purple flowered plants (with genotype WW).

Metabolic condition

Phenylketonuria (PKU) is a genetic disorder in humans in which the amino acid phenylalanine in food is not converted to other amino acids like tyrosine and so it can build up to toxic levels. Newborn children are tested to check that phenylalanine levels in blood are within acceptable levels, and if not they must have a low phenylalanine diet for life.

The condition results from a recessive allele usually known as PAH. Phenylalanine hydroxylase is the enzyme which normally converts phenylanine into tyrosine and mutant forms are inactive.

The frequency of phenylketonuria in the general population is about 1 in 10,000.

We can use this to answer a number of questions

- What is the frequency of heterozygotes (carriers) for this condition?

q² is 1/10000 , i.e. 0.0001

so q is 0.01, and p is 0.99.

and the frequency of heterozygotes - 2pq - is 0.0198 - which is close to 1 in 50.

- What is the chance of passing it on to the next generation ?

So the chance of 2 heterozygotes pairing up would be 0.000392

And only 1/4 of their children would be homozygous for the PAH allele.

This is 0.000098, which is close to 0.0001. ← the starting point for this

- What is the frequency of homozygotes completely unaffected by this condition?

Homozygous dominants (non-carriers) - p² - are 0.9801

Check total: AA 0.9801 + Aa 0.0198 + aa 0.0001 = 1.0000

Heterozygote Advantage

In some cases a genetic condition may be a very bad thing in its homozygous form, but an advantage if only one copy of the allele is present.

Sickle cell anaemia is an example.

Sickle cell disease is caused by a mutant form of the allele coding for β-haemoglobin - the red oxygen-carrying pigment in red blood cells. It is a single base substitution - (DNA triplet GAG coding for glutamic acid being replaced by GTG which codes for valine). This causes the molecule to pack differently into the red blood cells, causing sickling. These mis-shapen cells are likely to block minor blood vessels, causing painful sickle cell crises. Sickle cell anaemia is actually caused by a double dose of the HbS allele and homozygotes suffer in a number of ways.

It is thought that the modified red blood cells are less likely to be infected by the Plasmodium falciparum parasite which causes malaria. Carriers for sickle cell anaemia - often described as having sickle cell trait - have only a single dose of the HbS allele and in areas where malaria is common, these heterozygotes appear to have an advantage over people with ordinary haemoglobin in their blood and those with a double dose of the HbS allele.

Sickle cell disease occurs more often among people from parts of the world where malaria is or was common.

This is reflected in the increased frequency of sickle cell disease (SCD) in people of Black (African / Caribbean) or Asian ethnic origin, as well as those from other tropical parts of the world.
In the US, the CDC reports that
SCD affects approximately 100,000 Americans.
SCD occurs among about 1 out of every 16,300 Hispanic-American births.
SCD occurs among about 1 out of every 365 Black or African-American births.
About 1 in 13 Black or African-American babies is born with sickle cell trait (SCT).

Working on from the Black or African-American SCD figures above I calculate that the SCT carriers should have a frequency of 0.0991 which is 1 in 10, so it may be that this mismatch reflects the mortality caused by the double dose of the HbS allele which causes full blown sickle cell anaemia.

Other genetic advantages and disadvantages

Numerous surveys have established a connection between various blood groups and susceptibility to diseases, especially cancers. Individuals with blood group O show a decreased risk for several thromboembolic diseases, so blood clotting is another factor. .

Sex-linked genes

Red-green colour blindness

This is caused by a recessive b allele on the X chromosome, and it is said to affect around 1 in 11 men and 1 in 120 women.

Because males have only one X-chromosome, sufferers have the genotype X^bY, and the b allele frequency is q - not q².
So q=1/11 = 0.091
[and p (the frequency of the normal colour vision allele) = 0.909]

Females have 2 X-chromosomes, and the homozygous b allele frequency (genotype X^bX^b) is q².
This is 0.0083 - 1 in 120.

It is possibly reasonable to state that colour blindness is a condition does not affect viability - it is not life-threatening - so there is no reason for it to become eliminated from the population.