The following tasks deal with the law of Hardy and Weinberg, a fundamental law of theoretical population genetics. For those not familiar with it, I'll explain it shortly.

Basically, this law describes the distribution of genetic variants among an "ideal" population (that is a large population without mutations, without migration and with totally random selection of partners). A version of a gene is called "allele". Human beings are diploid organisms; this means that every one of us basically has two copies of each gene in his genome: one inherited by our mothers, the other by our fathers. (This was a bit simplified; in reality this does not apply to all genes. But it applies to all of the following tasks.) These copies may be identical, but they may also be different alleles.

Most genes have only two widespread alleles which do not have a lethal effect on the individual. In the general formulae of Hardy and Weinberg the frequencies of these alleles are called p and q. If there are only two alleles of one gene, a simple law applies to them: p + q = 1. (Or, if you are using percents: p + q = 100%.)

This primitive equation describes the distribution of the alleles but it doesn't describe the distribution of the genotypes. In a diploid organism there are three different genotypes: pp (homozygotous for p), qq (homozygotous for q) and pq (heterozygotous). In order to compute their frequencies, both sides of our basic equation must be multiplied with themselves. We thus get: (p + q)^2 = 1. This further leads to: p^2 + 2 * p * q + q^2 = 1.

So in a two-allele system in a diploid organism, the frequency of individuals homozygotous for p is p^2, the frequency of individuals homozygotous for q is q^2, and the frequency of heterozygotous individuals is 2 * p * q.

This is the most widely used version of the Hardy-Weinberg law. Similar laws can easily be derived for genes with more than two different alleles available and with organisms whose cells have more than two copies of each gene (e.g. triploid ones have three copies of each gene).

There are some allels which "dominate" over the others. That means: Heterozygotous individuals have about the same outward appearance ("phenotype") as those homozygotous for this specific, "dominant" allele. The other allele is called "recessive". Only individuals homozygotous for the recessive allele will have the outward appearance defined by this allele. The heterozygotous ones don't have the symptoms of the recessive allele, but they will pass the allele on to 50% of their offspring; such people are called "conductors". The most important feature of individuals homozygotous for a dominant allele is that all their offspring will have the same phenotype as they have.

This is all knowledge of biology you need in order to solve the following tasks.

1. 9% of a human population are suffering from a genetic disease which is triggered by a recessive allele. 42% are healthy, but they are conductors. Might this be an ideal population? Why (not)?

2. Assume that a tiny human population of only 5 persons has the same distributions of genes as an ideal population. There are two allels of the gene in question, with the frequencies p and q. 48% of the population are heterozygotous. Calculate p and q, where p is the frequency of the more widely spread allele.

3. 36% of an ideal, human population are suffering from a genetic disease which is triggered by a dominant allele. How many percent of the sick ones will definitely pass the illness on to the next generation?

4. There are three alleles of a gene, with the frequencies of p, q and r. In an ideal, human population a fifth of the people carry the allel p. The genotype qr has a frequency of 12%. Calculate q and r, where r is the frequency of the more widely spread allele.

5. Derive the Hardy-Weinberg law for a gene with two alleles and an ideal, triploid population.