Re: Are there mathmatical equations for evolution or any biological science.

Thanks for the great question Stephan,

In fact, we have quite a lot of equations that we use to describe the processes of evolution in particular and biology in general. As you might be aware, the various fields of science are similar in that they all deal with phenomena that are observable and reproducible. If we can collect enough data on a particular natural phenomenon, we can often describe it in terms of mathematical equations. I’ll go through a few examples, so you can see what I mean.

The Golden Ratio

One of the oldest equations that scientists have applied to biological phenomena was first recognized by the ancient Greeks as something called "the Golden Ratio." The Greek philosopher/mathematicians loved ratios, and they thought that the most perfect ratio was one where the ratio of a smaller number to a larger number was the same as the ratio of the larger number to the sum of the two numbers, or X/Y = Y/(X+Y). Try drawing this out so you can see it for yourself. The value of the ratio is approximately 1.618, but like the value of Pi, the digits of the Golden Ratio repeat forever without ending.

Somewhere way back when, these Greek philosophers noticed that the structures of quite a lot of plants (and some animals) follow this Golden Ratio. I’ll get to some examples of this in a bit, but it turns out that the best way to approximate the value of the Golden Ratio is by using what we call the Fibonacci Sequence. This is a sequence of numbers in which you generate the next number in the sequence by adding the previous two numbers. The first two numbers are 0 and 1; so 0+1 = 1, and the sequence then becomes 0, 1, 1. Then you add 1 + 1 to get 2, and the sequence becomes 0, 1, 1, 2. Then you add 2 + 1 to get 3, and the sequence is 0, 1, 1, 2, 3. Eventually, you will have a string of numbers beginning with 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.

Now, you might ask, "how does this relate to the Golden Ratio?" Well, the further out you go in the Fibonacci sequence, the closer you get to the actual value of the Golden Ratio when you divide successive numbers. So, 1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.6666, 8/5 = 1.6, etc, and eventually we find that 144/89 = 1.617977. Try this on your own, and you will see how it works.

Now, if you look at the way that certain plants and animals grow, you will see that they follow the Fibonacci sequence. If you look at a pine cone, or a pineapple, or the petals of flowers, you will notice that the scales on the pine cone run in a spiral around the outside of the cone. Count the number of scales in each spiral as they proceed upward along the cone; you will find that the number of scales is a number in the Fibonacci sequence, and that the ratio of scales in successive spirals is an approximation of the Golden Ratio. Similarly, if you look at the chambers of a Nautilus shell, you will find that the angles relating the various sized chambers all approximate the Golden Ratio.

Of course, the real question here is, "what does it all mean?" I’m afraid that we don’t have an answer for that yet.

Evolution

Now Stephan, what you really want to know about is Evolution. There are a number of different processes of evolution, and we have a variety of equations that we use to express them. But first lets look at the actual definition of evolution, because it is actually quite different from what you expect. I know that the general public thinks of evolution as "survival of the fittest", but that is really difficult to quantify. Besides, sometimes the fittest don’t survive; sometimes "the fittest" happened to be too close to the volcano when it erupted, and what you really have is "survival of the furthest from the volcano." Because of things like this, when evolutionary biologists are talking about evolution, we are referring to changes in gene frequencies in a population over time. As evolutionary biologists, we examine and deduce the forces that cause those changes.

Fortunately for us, there is a whole field of biology called "Population Genetics" which makes use of mathematics to investigate these changes. When we look at a set of gene frequencies for a population, one of the first things that we check are the so-called Hardy-Weinberg Equilibrium ratios for the genes in the population. If you recall, a given gene can have multiple, variant DNA sequences called "alleles." By looking at the ratios of the alleles for a given gene, we can tell if the population in which those alleles are observed is experiencing changes in its allele frequencies (evolution).

Here’s how it works. We take the percentages of each allele for a given gene in a population and put them into a simple equation. If there are only two alleles for a gene (allele P and allele Q), then one allele has the value q and the other has the value p. Since percentages must total to 1, p + q = 1. If there are 3 alleles, then p + q + r = 1.

Now, because everyone has two copies of each gene, we know that they have 1 alleles for each gene, so a single person can only have two alleles, no matter how many alleles there are in the population. If there are 2 alleles (P and Q), then some people will have 2 P alleles (PP), some will have 2 Q alleles (QQ) and some will have one of each (PQ). This knowledge allows us to apply a second equation to the same population, and learn what percentage of people in the population should have which pairs of alleles. The second equation tells us that, p^2 ( p squared) + 2pq + q^2 = 1.

With these two equations in mind, we can look at the number of alleles in the population, as well as the number of individuals in the population with each possible pair of alleles, and determine if p + q = p^2 + 2pq + q^2 . If this is not true, then we know that evolutionary forces are at work on the population, because the frequencies of the alleles are changing between generations.

Evolutionary Forces

In general, when a population is experiencing a change in gene frequency, we attribute it to one of three forces. The first is "selection", where environmental factors result in certain alleles being passed on to the next generation in preference to other alleles. The second is "mutation", where a new allele is generated (through one of a variety of mechanisms) from an old allele, and the third is "genetic drift", where random factors working on the population result in changes to gene frequencies. Examples of genetic drift: not every organism which can reproduce in a given generation does so (in our species, some people choose to be celibate) and their genes won’t be passed on to the next generation. Or perhaps a population splits into a larger and a smaller group; these two populations might not have the exact same gene frequencies. Or perhaps the volcano erupts and the only members of the population who were far enough away happen to be related; the new population will have different gene frequencies from the old one. These last extreme examples of genetic drift are much more pronounced for small populations than for large populations.

We have a few simple equations that we use to determine if gene frequencies are changing due to selection or if they are proceeding under something like genetic drift. We use these equations to calculate the Ewens-Watterson Homozygosity statistic (which we just call F). As with Hardy-Weinberg Equilibrium proportions, we look at the frequencies of the alleles for a given gene. F is the sum of all the squares of the frequencies of the various alleles for a given gene.

F = Sum (f^2) For all values of f, where f = the frequency of a given allele.

Once we have calculated F, we can compare the actual value for the population to the values that we expect to get for a population of the same size and with the same number of alleles, that is experiencing only random drift (Fexpected). To do this, we use a second equation to calculate a Z value:

Z = (F - Fexpected) / (Standard Deviation of F expected).

If the value of F is very close to Fexpected, the value of Z will be close to zero, which tells us that the population is evolving under a genetic drift model. If the value of Z is negative, it tells us that one type of selection is operating on the population, while if the value of Z is positive, it tells us that a different type of selection is operating on the population. However, I think that a discussion of the different types of selection should be the subject of another question.

So Stephan, I hope this helps to give you a basic idea of the sort of equations that can be used to decribe biological processes, and evolutionary processes in particular.

If you are really interested, I suggest you look up a couple of evolution and population genetics textbooks in your school library. I recommend, "Principles of Population Genetics", by Hartl and Clark, and also "Genetic Data Analysis", by Weir. "Principles of Population Genetics" also contains all of the references for the Hardy-Weinberg equations and the Ewens- Watterson equations.