respectively p + r = 0.4 and r + q = 0.6. So the distribution of next generation
of genotypes is 16%, 48% and 36%. Successive proportions of the genotypes
bb, bB, and BB after random matings are and the proportions settles down after
one generation, as before, and the transmission probabilities of 0.4 and 0.6
remain constant. With these figures 16% of the population will have blue eyes
and 48% + 36% = 84% will have brown eyes because B is dominant in the
genotype bB.
10%, 60%, 30% → 16%, 48%, 36% →... → 16%, 48%, 36%
So the Hardy–Weinberg law implies that these proportions of genotypes bb,
bB and BB will remain constant from generation to generation whatever the initial
distribution of factors in the population. The dominant B gene does not take over
and the proportions of genotypes are intrinsically stable.
Hardy stressed that his model was only approximate. Its simplicity and
elegance depended on many assumptions which do not hold in real life. In the
model the probability of gene mutation or changes in the genes themselves has
been discounted, and the consequence of the transmission proportions being
constant means it has nothing to say about evolution. In real life there is ‘genetic
drift’ and the transmission probabilities of the factors do not stay constant. This
will cause variations in the overall proportions and new species will evolve.
The Hardy–Weinberg law drew together Mendel’s theory – the ‘quantum
theory’ of genetics – and Darwinism and natural selection in an intrinsic way. It
awaited the genius of R.A. Fisher to reconcile the Mendelian theory of inheritance
with the continuous theory where characteristics evolve.
What was missing in the science of genetics until the 1950s was a physical
understanding of the genetic material itself. Then there was a dramatic advance
contributed by Francis Crick, James Watson, Maurice Wilkins and Rosalind
Franklin. The medium was deoxyribonucleic acid or DNA. Mathematics is needed
to model the famous double helix (or a pair of spirals wrapped around a
cylinder). The genes are located on segments of this double helix.
Mathematics is indispensable in studying genetics. From the basic geometry of
the spirals of DNA and the potentially sophisticated Hardy–Weinberg law,
mathematical models dealing with many characteristics (not just eye-colour)
including male–female differences and also non-random mating have been
developed. The science of genetics has also repaid the compliment to
mathematics by suggesting new branches of abstract algebra of interest for their
marcin
(Marcin)
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