# Biology as Poetry: Evolutionary Biology

## Hardy-Weinberg Theorem

Sorry about the math, but the italics are exam questions… ☺

*For a one locus, two allele, diploid system, of allele frequencies **p* and *q*, the following given __random mating__ will hold: *p*^{2} + 2*pq* + *q*^{2} = (*p* + *q*) × (*p* + *q*) = 1.

Other assumptions are that mutation, genetic drift, migration, and natural selection are not occurring. Given these assumptions then *p*^{2} and 2*pq* and *q*^{2} are the expected frequencies of three possible genotypes, i.e., *PP*, *PQ*, and *QQ*, here using the upper case *P* and *Q* as allele names. For further explanation, see Hardy-Weinberg equilibrium.

For an equivalent three-allele system, the equation instead would consist of:

*p*^{2} + *q*^{2} + *r*^{2} + 2*pq* + 2*qr* + 2*pr* = (*p* + *q* + *r*) × (*p* + *q* + *r*) = 1

To reiterate for the two-allele system:

*p*^{2} + 2*pq* + *q*^{2} = 1.

(*p* + *q*) × (*p* + *q*) = 1.

And, mathematically, this occurs because *p* + *q* = 1. That is, since there are only two alleles in the system, in this example, the sum their frequencies will account for all of the alleles in the system, i.e., __100%__, which is another way of saying 1.

Furthermore, the odds of generating a specific homozygote – e.g., *PP* or *QQ* – are either *p*^{2} or *q*^{2} (i.e., the odds of picking one allele twice in a row or, instead, the other allele twice in a row). The odds of generating the heterozygote, i.e., *PQ*, are 2*pq*, and this actually is *pq* + *qp*, i.e., the odds of picking one allele type and then the other plus the odds of picking the same combination of alleles but in the opposite order.

Similarly, for the three-allele system also illustrated above, *p* + *q* + *r* = 1, and so on.

For more on this topic, see Wikipedia
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