So, I didn’t take a lot of biology in college, and what I did I can’t remember. Now that @swamidass has introduced me to the Neutral Theory of Evolution I have some questions (pardon my ignorance). From the article quoted below:
Now imagine that, by genetic drift, one allele reproduces enough to replace another in the population. When it does so, any mutations in its neighboring regions will tag along. This is a strategy that allows neutral changes to accumulate very rapidly, but it is one not generally available to beneficial mutations.
How does an allele “reproduce”? Does that mean that once a mutation occurs it will be carried on to future generations (is that subject to passing on of only half the genetic code? it could be missed?)
How exactly does one allele replace another in the population? It seems like if they are random and non-selected then you would always end up with roughly even mixture of alleles.
Related to question 2, how could that happen so rapidly when it’s neutral and random?
I’m sure I’ll have more questions later. I’ll try to add them to this thread.
P.S. I think this the first thread I’ve ever created, yay me!
Sloppy language. What it means is that individuals that have that allele just happen to reproduce (in the ordinary meaning) more than individuals without it.
In the case we’re talking about here, purely by chance. In any finite population, the frequencies of alleles will fluctuate from generation to generation. Sometimes that random fluctuation eventually results in one allele being the only one left in the population. In fact, if there is no natural selection happening, every allele will inevitably become either lost or will replace others.
It generally doesn’t happen rapidly. What that’s saying is that it can happen to several nearby mutations at once. If I recall, the average time from mutation to fixation, for alleles that do become fixed, is 4N generations, where N is the population size. I wouldn’t call it a strategy either, just a thing that happens.
So it seems like it would be a low probability to actually have fixation because in my mind the allele frequency would be very noisy and random and it would mostly be an even shot. Also, are there generally just a two (or a few) alleles for a given gene or are the many? In my basic intro genetics it always seemed like it was “there are two options” but I’m not sure if that was just to keep the math easy (like 2x2 Punnett squares).
So this is totally reasonable intuition, the same intuition most people had, but it turns out to be false. The reason why is because the number of mutations is generated is proportional to the population size. While the chance than any individual mutation is low, there are just so many of them that we can be certain that some are being fixed all the time.
The chances a mutation will be fixed is 1/2N. We are talking about a diploid population with two copies of the genome. That means there are 2N genomes floating around out there. How many new mutations are formed each generation? m*2N with m as the mutation rate times the genome size. So, how many mutations are fixing each generation?
m * 2N / 2N → m
That is the surprising thing that happens. The N cancels, and we find out that the fixation rate equals the mutation rate. That means, for example, if there are 100 mutations per generation per genome in humans, no matter what the population is, we expect there to be about 100 fixed mutations per generation. This is, in essence, the basis for the molecular clock.
The probability that a new mutation will eventually become fixed (in a completely neutral situation) is 1/(2N). But since a whole lot of them happen, a lot become fixed, in fact a number in the population, each generation, equal to the number of mutations per individual in that generation.
There are many, just through random mutation. Given the human population, almost every possible point mutation probably happens at least once in each generation. In a gene of 10,000 bases, that’s at least 10,000 alleles (ignoring multiple hits for the sake of simplicity) at some frequency. Most of them will be lost within one generation, though.
I don’t believe that’s true. What you’re talking about is the probability of eventual fixation. The time to fixation is 4N generations. Of course the variance is huge.
And there are caveats you should always mention: this is a description of perfectly neutral evolution. Any selection, positive or negative, changes the expectation. Also, it assumes a constant population size. In a rapidly increasing population, there will be way fewer fixations, if any. In a rapidly decreasing one, there will be more.
OK, I did it and it is very non-intuitive, especially the Random Drift simulation. Here’s where I was getting hung up I think. I was envisioning that the mutation rate would be high enough that it would essentially scramble everything up often enough that fixation would be unlikely to occur. But it’s really a combination of mutation being relatively low probability and passing on of the allele, whatever it is, being high probability. It’s not like flipping a coin 100 times. Each generation is not an independent random sample.
Mutation rate is usually exceedingly low. To first approximation you can assume once a spot is mutated, it won’t mutate again. Of course rarely that is not the case, but there are so many mutations in a large population it is bound to happen somewhere. To understand they basic process, however, you can neglect this because most of the time it doesn’t happen.
OK, new insight. Initially the simulator defaults to allele frequency of 0.4 and so it goes to fixation most of the time. So, based on your comment I set the allele frequency low (0.05 since the simulation starts out with 20 individuals) then it starts looking better. I can run the simulation many times (I did 500 replicates) and almost always it dies out quickly, but in my case 1 out of 500 replicates reached fixation within 100 generations.
So mutation itself is a random, very low probability event, and fixation is also a random, low probability event, but it’s not 0 probability and it didn’t need any selection to do it. Is that right?
That’s right. Turns out the number or mutations in a populations scales with N and the probability of fixation scales with 1/N. So the per individual mutation rate equals the population wide fixation rate. That is the basic finding of the molecular clock.
If an allele is lost by mutation much more often than it is gained by mutation, then mutation, as well as drift, may influence the time to loss.
How would an allele be lost by mutation much more often than it is gained by mutation? That would mean it’s not random, right? Is that saying something about the chemistry of the DNA strands? Not all bases pairs are equally likely? (are we talking about mutation = switching out a base pair?)
As a general case, the probability of an allele being fixed is equal to its initial frequency. If you start out with a frequency of 0.4, the probability of it becoming fixed is 0.4, and the probability of the other allele (assuming there are only two) becoming fixed is 0.6.
No. “Random” doesn’t require that all probabilities be equal. The probability of a loss could easily be higher than the probability of gain.
Yes. Even if they were, the probability of, for example T → C would be lower than the probability of C → T, G, or A, any of which would remove the mutation.
Yeah, my bad. I meaning more if they were independent events that just undid each other, then the probability of loss should equal the probability of gain. As you showed though, they aren’t simply the reverse of each other. Thanks.
Even if they were, the probabilities wouldn’t have to be equal. Consider: if base frequencies are at an equilibrium in which G is less common than A, then in order to preserve that equilibrium the probability (per site) of A → G must be different from the probability (per site) of G → A.
Nice link. I wonder if anyone knows of a similar type of software that simulates the continued occurence of new alleles rising by mutation, and their subsequent drift in the population?
So instead of just following the population history of two alleles at some set initial frequency, for a set number of generations, instead it also shows how every generation new alleles arise and are either lost or fixed by selection.
I’ve often come across the misconception that because the fixation probability of a newly arising neutral mutation is low, this means one can in effect assume newly arising neutral mutations never reach fixation.
Which is of course very wrong, but instead of just throwing the equations at people that prove this to be wrong, a graphical simulation that actually show the effect continously over time would be very instructive.
I’m thinking something like the “Twenty generations of random drift” example in your link, except it keeps running and we just see generations scroll by and new mutations continously arise in the population and then drift as they do. This would also help illustrate how, at any given moment, the extant population will most often constitute some mix of alleles having arose many generations in the past, and how this is a continous process where new alleles constantly arise in that same fashion.
It would help illustrate how the current standing genetic variation in a population is a product of a vast history of mutations arising in the past, and therefore how despite the initial probability of fixation being low, this continued onslaught of new mutations arising still unavoidably leads to variation and eventual continued fixation.
This keeps running, but unfortunately doesn’t include a spontaneous mutation rate. If there was some small chance of new alleles being generated by mutation every generation (and you could perhaps set the mutation rate) it would be exactly what I had in mind.
Makes sense, except why are they at equilibrium? I think I need a better idea of what a mutation is at a chemical level perhaps. I’m still thinking of them as basically coin flips. Are you saying it could be more like a loaded die?
This thread is really illustrating how going from an environmental science BA to a physical chemistry PhD can lead to significant holes around biochemistry and molecular biology. I did a little bit of conservation biology and introductory genetics as an undergrad, but most of my training is on molecules < 50 atoms in size.
I did a little bit of conservation biology and introductory genetics as an undergrad, but most of my training is on molecules < 50 atoms in size.