Evolution is Math is Fun!

Evolution is ultimately mathematical.  This is in part why I have taken so long to continue with Dobzhansky’s book, Genetics and the Origin of Species – we are now getting to the math.  But the math cannot be ignored – in the early 1900s, some brilliant mathematicians developed formulae that predicted what biologists would spend the next seventy-plus years confirming experimentally.  Darwinian evolution is a case in which number (largely) preceded biology.

Case in point: check out what is now known as the Hardy-Weinburg equilibrium (what Dobzhansky refers to as Hardy’s formula of 1909 – I guess Weinburg’s contribution had not yet been discovered by the 1930s).  In the simplest case, imagine we have one dominant and one recessive allele for a gene.  These could be, say, two different eye colours, which we will denote as A for dominant and a for recessive.  In a population, some individuals have two copies of A (AA = homozygous dominant), some have aa (= homozygous recessive), and some are Aa (= heterozygous dominant).  If you count up all of the A alleles and all of the a alleles, and divide each by the total number of alleles, you get a frequency or proportion.  We will call p the frequency of A, and q the frequency of a.

To see how this works so far: imagine that we have a population with 10 AA, 10 aa and 20 Aa. 

p = ((10 * 2) + 20)/((10 + 10 + 20) * 2) = 40/80 = 0.5

q = ((10 * 2) + 20)/((10 + 10 + 20) * 2) = 40/80 = 0.5

As a check, since we are dealing with proportions, p + q = 1, and indeed this is the case.

Hardy-Weinburg equilibrium says that, given some proportion p and some proportion q, and given random mating, no selection, large population size, no mutations and no migration, then the alleles will be partitioned according to the formula p² + 2pq + q² = 1, where:

p² = Proportion of AA

q² = Proportion of aa

2pq = Proportion of Aa

In our above example, p = 0.5, q = 0.5, and there are 40 individuals randomly breeding with 80 alleles in total.  In the next generation, according to the Hardy-Weinburg equilibrium:

p² = 0.5² = 0.25

q² = 0.5² = 0.25

2pq = 2(0.5)(0.5) = 0.5

Therefore in the next generation for every 0.25 AA there should be 0.25 aa and 0.5 Aa.  In a population of 40 individuals, we would have 10 AA, 10 aa and 20 Aa, just as before.

Hardy-Weinburg equilibrium shows us that, so long as its expectations are met, the ratios of homozygous dominant to homozygous recessive to heterozygous individuals within a population will not change over time.  In this particular example, the parental generation began in Hardy-Weinburg equilibrium, but this need not always be the case.

Imagine that we began with 100 individuals, 40 AA, 20 aa and 40 Aa.

p = ((40 * 2) + 40)/(100*2) = 120/200 = 0.6

q = ((20*2) + 40)/200 = 80/200 = 0.4   (check: 0.6 + 0.4 = 1)

p² = 0.36

q² = 0.16

2pq = 2(0.6)(0.4) = 0.48

If this second generation were to have 100 individuals, we would now have 36 AA, 16 aa and 48 Aa, which is different from the parental generation.  The parents were therefore not in equilibrium.

If we were to do it again,

p = ((36 * 2) + 48)/200 = 120/200 = 0.6

q = ((16 * 2) + 48)/200 = 80/200 = 0.4

Notice that these are the same p and q as before.  This is an important lesson from Hardy-Weinburg: allele frequencies remain constant, even if there are changes to genotype frequencies (ie. AA, aa and Aa).  Since p and q are the same as before, the remaining calculations will be the same, leaving us once again with 36 AA, 16 aa and 48 Aa in a third generation of 100 individuals.  Equilibrium will continue to be maintained, even though the original parents were not in equilibrium.

This little mathematical model may not look like much, but it is remarkably powerful.  For example, it can be used to figure out the frequency of a dominant allele.  Ordinarily, we cannot tell AA individuals from Aa individuals, but we can always see and count aa individuals.  Since aa = q², by taking the square root of q² we get q, and since p + q = 1, 1 – q = p.  We can therefore easily calculate p, and can then calculate the expected frequencies of AA and Aa.

But this model is even more important when a population with known p and q does not give us the expected AA, Aa and aa numbers.  Such populations are in violation of Hardy-Weinburg equilibrium. 

Dobzhansky argues that there are forces that try to shift the genetic equilibrium, to cause violations of Hardy-Weinburg equilibrium.  ‘A living population is constantly under the stress of the opposing forces; evolution results when one group of them is temporarily gaining the upper hand over the other group.’

Dobzhansky provides three examples of what he means:

1.  Mutation and genetic equilibrium

Let us start with a population of individuals all homozygous for allele A.  But let’s suppose that there is a mutation rate, such that allele A mutates to allele a numerous times, albeit slowly.  And let us suppose that this is not reversible – that is, a, once formed, cannot mutate back to A.  Eventually, given enough time, there will be no A alleles left – the population will be homozygous for a.  We would say that this population has evolved solely due to mutation pressure; selection played no role.  In this scenario, the change in the frequency of A can be mathematically represented by  –up, where u is the mutation rate and p is the frequency of A. 

This scenario is rather unlikely.  It is more likely that, as A mutates to a, there will be an opposite force that mutates a back to A.  Let us suppose that this mutation rate occurs at a different speed, which we will call v.  The change in the frequency of A can now be described by –up + v(1-p). 

These two opposing mutation rates will alter p and q, but not indefinitely – eventually they will reach a point of equilibrium.  Equilibrium is when the frequency of A is no longer changing; therefore, the change in the frequency of A equals 0.  Therefore, -up + v(1-p) = 0 at equilibrium.  Solving for p, we get p = v/(u+v).  Whatever p is, this is the frequency of A that will persist under these mutation pressures.

Mutations, therefore, will cause changes in p and q that Hardy-Weinburg cannot account for.

2. Population size

Hardy-Weinburg equilibrium assumes that the population is infinitely large.  Obviously this is never true, but reasonably large populations do not appreciably deviate from expectation.  Small populations, however, do.  Why?

Imagine that you had a large population with 100 different alleles for a single gene.  And imagine that that population were to suddenly get fragmented, divided into smaller populations.  By chance, it is likely that each population will start off with different frequencies for each of the 100 alleles. Some populations may be lacking some alleles entirely – the small population size alone has already reduced variability.  Now let’s imagine that parents only have two offspring.  If an AA individual were to breed with an Aa individual, statistics would dictate that there should be one AA for every one Aa individual produced.  But, by chance, 25% of the time such a mating will produce two AA individuals instead of one AA and one Aa, removing the a allele from that generation.  And sometimes there will be two Aa individuals, suddenly doubling the a allele.  The vagaries of reproduction in small populations can thus entirely remove alleles or make them more frequent than expected, without there being any selection.  This is known as genetic drift. 

In one drift experiment, Dobzhansky noted that, after 127 generations, only 153 out of an initial 10 000 neutral alleles managed to avoid extinction.  Since drift is random, repeated experiments would have different results, with different alleles avoiding extinction.  Indeed, this is seen in nature.  When a population gets fragmented into many small populations, and the environment in each population is identical, the populations can still evolve to become highly differentiated from one another.  Genetic drift seems to be the reason for this – alleles survive in one population that go extinct in the other, causing the populations to be quite different from one another.

There is a formula to track the loss of variation due to drift, but I won’t bore you with it.

The important thing to note is that a deviation in Hardy-Weinburg could be due to genetic drift.

The third example that Dobzhansky discusses is selection.  Since he devotes an entire chapter to it, we will take it up next time.

What is important to note for now is that evolution is a highly mathematical science.  I don’t think that the average Creationist really appreciates this point.  Mathematical models were developed in the early 1900s to predict what would happen to alleles in a population if Darwinian evolution were true.  These models made predictions; we field biologists have been verifying and refining these predictions for the last several decades.  Numbers speak volumes – a made-up and fanciful theory should not produce numbers that correspond so well with reality.

Random quotes from chapter V of Genetics and the Origin of Species:

‘Evolutionary plasticity can be purchased only at the ruthlessly dear price of continuously sacrificing some individuals to death from unfavourable mutations.  Bemoaning this imperfection of nature has, however, no place in a scientific treatment of this subject.’

‘The eugenical Jeremiahs keep constantly before our eyes the nightmare of human populations accumulating recessive genes that produce pathological effects when homozygous.  These prophets of doom seem to be unaware of the fact that wild species in the state of nature fare in this respect no better than man does with all the artificiality of his surroundings, and yet life has not come to an end on this planet.’