Originally, as I was mulling over a new topic for EvC, I was going to offer a rebuttal to Faith’s argument that creationism can indeed work as an explanatory model for horizontal gene transfer and phylogenetic incongruences.
However, I’ve decided to instead tackle Faith’s argument in the The Science in Creationism thread that evolution always reduces genetic variability, which is the opposite of what evolution needs. This argument seems to have been floated by Faith way back in 2005 (and earlier), in the thread titled Natural Limitation to Evolutionary Processes.
I wasn’t around EvC at that time, but the argument presented by Faith seems come back again and again, most lately in The Science in Creationism thread. I believe that thread has mostly outlived its usefulness, so I will just propose a new topic. It also seems that there’s been some difficulty coming up with a viable mathematical approach to understanding the subject.
Thus, my refutation of Faith’s argument here will take a molecular population genetics approach. Most of the material discussed below is pretty standard introductory content in evolutionary biology courses, and can be found online, as well. Only high school algebra, high school probability theory, and a smattering of Mendelian genetics should be needed to grasp the basis of my argument.
I will begin by noting that
genetic drift eliminates diversity in a population. Genetic drift, of course, is basically a sampling error -- there are a number of cool images on the web that can help you visualize why genetic drift, by itself, removes genetic diversity in a population.
So both genetic drift and selection eliminate diversity in a population; both of these processes weed out alleles at a given locus. As both of these processes remove genetic diversity in a population of organisms, I will focus on genetic drift. Note that my argument holds perfectly well for selection; however, the mathematics for selection are slightly more intricate, so I will go there only if necessary.
Now comes the math. Why does genetic drift weed out diversity? For starters, let G = the probability that 2 alleles (from the same locus) randomly chosen from the population are identical.
Thus, G = how much genetic variation there is in the population. If G = 1, then there is absolutely no variation at that locus; all the alleles are the same. When G = 0, every single allele (from that locus) in the population is different. Perhaps they differ by a few nucleotides here, and a few there -- but they are all different if G = 0.
So far so good.
A quick side note: from now on, whenever I say 2 alleles or 2 identical alleles or the 2 alleles are different, or whatever, I mean 2 alleles of the same loci. And by locus, of course, I mean the chromosomal location of the allele under consideration, or Where in the genome you can find this allele.
Now, then, let G’ = the probability of randomly picking 2 identical alleles from the population
after one generation, or one round of more-or-less random mating.
The value of G’ = 1/2N + (1-1/2N)*G.
You might think Where the actual freak did that come from?
So let me explain.
There are basically only two ways for 2 alleles picked at random to be identical. First, 2 alleles could be identical because they share an immediate ancestor -- a parent -- in the previous generation. The probability of 2 randomly picked alleles being identical because of this is 1/2N.
Here, N = the size of the mating population. But 2N is used because we are dealing with diploid organisms, so there are 2 allele copies per organism. So when you think about it -- yes, if I’m looking at one allele, and I want to randomly find an identical allele that has the same parent, then the probability of doing that successfully is 1/2N.
The other way for 2 alleles to be identical is if they do
not share a parent, but if the
immediate ancestors of each allele have identical alleles. So, for example, let’s say A and B are two identical alleles, with different parents. A and B can be identical if the parent allele of A is identical to the parent allele of B. Think through it and it will make sense.
So what’s the probability of 2 alleles being identical through this way? It’s (1-1/2N)*G. Why 1-1/2N?
Well, say you have 2 identical alleles. The probability that they are identical is 100%, or 1. There are two basic ways for them to be identical: they share the same immediate ancestor in the prior generation (1/2N), or they are identical because both their parents were identical. These must add up to 1, so the probability of the alleles being identical because their parent alleles were identical must be 1-1/2N. But 1-1/2N must be multiplied by G to get the actual probability of the alleles being identical because their parent alleles were identical; this is because G = the probability that 2 alleles randomly chosen from the population are identical.
G is independent from 1-1/2N, so the two are multiplied together to get the actual probability of 2 alleles (picked randomly from the population) being identical because their parent alleles were identical.
Then we add 1/2N to (1-1/2N)*G to get the value for G’, which is the probability of randomly picking 2 identical alleles from the population
after one generation. This brings us to the equation above:
G’ = 1/2N + (1-1/2N)*G
Now, let’s take this equation out for a test drive.
Suppose N = 100, and the initial probability of two alleles being identical = 50% After 1 generation, G’ = 1/200 + (1-1/200)*.5 = 50.25%. In other words, the probability that the two alleles, drawn at random from the population, will be identical has increased. We can carry this on through for several generations.
G2 = 1/200 + (1-1/200)*.5025 = 50.5%. The probability keeps increasing, inching the population towards homozygosity. After 10 generations, we’re at 52.6%. Eventually, it will reach 100% in a real population. In other words, all the alleles will be identical -- so there is no genetic diversity in the population.
But now let’s take a look at the role of mutation in this.
Mutation puts variation into a population at the rate 2Nu, where u = the mutation rate for selectively neutral alleles. Why 2Nu? Well, there are 2N gene copies per generation, and u = mutation rate, so these are multiplied together to get the overall rate at which variation enters the population.
More precisely, u = the mutation probability for a given allele. So u = the probability that an allele in a given locus of a gamete will have a mutation. When mutation rate is taken into consideration, then, we must revise the equation for G’. Remember, G’ is the probability of 2 alleles picked at random from the population will be identical after one generation; if G = 1, all alleles are the same; if G = 0, no alleles are the same.
So the new equation, taking mutation into consideration, is this:
G’ = (1-u)^2 * [1/2N + (1-1/2N)*G]
What is 1-u? The factor 1-u is the probability that a mutation did not occur in one allele. But remember these are diploid organisms (2 allele copies), so the probability that a mutation didn’t take place in either allele is (1-u)^2 -- it is multiplied by itself (the two events are independent, so
a la basic probability theory, they are multiplied instead of added). Why 1-u? Well, u = the probability that it a mutation
does happen; u must necessarily be less than 1, so 1-u is the probability of the allele not having a mutation.
Okay, now let’s plug in some numbers. Say the mating population size is 100 and G = 50%. Let’s say the mutation rate is 10^-5 (pretty standard mutation rate for a number of diploid organisms). That means there’s a 1 in 100,000 chance that a given allele will mutate. After one generation, G’, the probability of 2 alleles being identical (picked randomly from given loci) is: 50.248995%, which is extremely close to the 50.25% reported above, where mutation was NOT taken into consideration. However, what happens when the population size is increased?
When the population size is 100,000 (instead of a mere 100), the probability of randomly picking 2 identical alleles begins decreasing. After one generation, it’s 49.99925%. Each generation, the probability inches closer to 0. In other words, because of mutation, the probability of randomly picking out 2 identical alleles increasingly becomes 0. This, in turn, means that there’s an enormous amount of genetic diversity in the population. In short, mutation has an effect that can and does counter both genetic drift and the forces of selection.
The challenge is for Faith to show that:
(1) The mathematics undergirding these processes become irrelevant in isolated founding populations, which represent a sampling of the allele frequency of the ancestor population. Clearly, if founding populations are quite small and geographically isolated, then genetic drift and selection will work to eliminate genetic diversity. Often, the result will be extinction. But if the founding population is sufficiently large, then mutation will be enough to continue adding diversity to the gene pool, generation after generation -- despite selection and genetic drift.
(2) Most mutations are too detrimental for any of this to be of real meaning in biology, or beneficial mutations too rare for positive selection to have something to work on.
(3) While mutation can overcome the reduced diversity wrought by genetic drift, selection is
necessarily always strong enough that mutation cannot overcome its reductive effects.
Any questions, just ask!
Note: when I say "parent" in the above piece, I mean "parent allele," of course.
Edited by Genomicus, : No reason given.
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