I've been procrastinating way too much lately so have been putting off responding in this thread in favor of gaming too much.
leroy wrote:Rumraket wrote:For a population size of 1 individual who reproduces by cloning and replacement. That's not the current situation for any species on the planet.
In my simplistic equations the size of the population is irrelevant because we have that variable both in the numerator and in the denominator, thus the variable can be canceled.
Your ability to cancel out numerators and denominators is irrelevant if your equation doesn't correctly describe the probability of occurrence of a particular change as it scales with population size.
Also, you have apparently managed to confused yourself about what it is you are actually calculating or whether it even makes sense. You are using the equation for probability of fixation of a neutral allele, when you yourself defined the specific case to be beneficial. And you're calculating the probability of fixation for a population of 1 individual.
Let's just review what a fixation probability is. Per wikipedia:
"In the process of substitution, a previously non-existent allele arises by mutation and undergoes fixation by spreading through the population by random genetic drift or positive selection. Once the frequency of the allele is at 100%, i.e. being the only gene variant present in any member, it is said to be "fixed" in the population."Okay, so an allele that rises from being present in 1 individual, to being present in ALL individuals (100% of individuals) is said to have become fixed.
With this we can proceed.
in reality it is "easier" to evolve when the population size is small.
No, in reality
fixation is more likely when the population size is small, and the allele is neutral or very nearly so, and if other values are held constant and if no new mutations are introduced along the way.
But evolution is not identical with the probability of fixation, so if the selection coefficient is low, and the mutation rate is low and the population size is small, it can still be unlikely to fix a beneficial mutation against genetic drift. However, if the population size is ginormous, then that mutation will come to exist in many individuals with every new generation added to the population due to new mutations occurring, rather than the mutation having to "get a foothold" only through drift or selection.
Unless of course the population size is 1 individual, then sure, fixation is not just more likely,
it's unavoidable. Any mutation that happens to a population size of one individual is therefore fixed.
You think A is already present in 100% of the population. So you want to determine the probability that B will happen and then fix in the population. Right?
Well, if the population size is only 1 individual, then the probability of fixation is 100% if and when B occurs. Then all that needs to happen is that the B mutation occurs to that one individual, then it will have become fixed.
So the probability of A+B fixation when A is already present in the 1 individual, must simply be the probability of occurrence of B. Which for 100/3,000,000,000 = 1 in 30 million.
Obviously the probability of a particular mutation happening to a single individual is low. Fixation probability is guaranteed IF the mutation happens, but mutation probability is low. That's why if the population is 1 individual that reproduces by cloning and replacement, it's going to take
on average 30 million generations before it happens.
Okay, but what if the population size is huge? Well, now the mutation has more opportunity to happen in other individuals in the population. And we already determined the probability of the mutation is 1 in 30 million for a single individual.
But if we have more than 1 in individual, we can use the frequentist interpretation to just estimate that events that have a probability of 1 in 30 million, will
on average happen once in 30 million trials. Stated another way, it happens at a frequency of 1 in 30 million trials. And every new individual that was born for the population to become it's present size, was a trial.
There are 7.4 billion people on Earth = so there must have been at least 7.4 billion trials for the population to get to it's present size. (In reality there were many many more than this, because reproducing couples had many many more offspring than survived, but we'll keep this simple for practical reasons (and because I honestly can't do the math if we make it very realistic).
7.4 billion / 30 million = on average ~246,7 will get the B mutation.
But what is the fixation probability of the A+B mutation? Well, those two mutations in combination are selectively beneficial, remember? Not neutral. You are incorrectly using the formula for the probability of fixation of neutral alleles. But A+B is beneficial. That means we need a very different formula. A formula that takes the
selection coefficient into account. Becaues if the A+B combo has a benefical effect on reproductive success, that will increase it's probability of fixation. And if the allele starts out with a frequency of roughly 246 individuals in a population of 7.4 billion, it already exists in a considerable number of individuals which will help it's probability of fixation.
So a population of "1" is your best possible scenario.
For fixation, sure. For evolution in a more realistic scenario? No, it isn't.
You are focusing only on fixation and ignoring probability of independent occurrence of the mutation.
Rumraket wrote:In the human population of 7.4 billion, it would happen about 123 times every generation.
My equation already took in to account that size of the population, you can´t multiply it again.
You are extremely confused about what your own equation even means.
Leroy for fucks sake please just think for a moment.
The probability of fixation in a population size of 1, is 100%. It can't be any other way. In so far as the mutation occurs,
it will have occurred to the only individual that makes up the population, and so will have instantly become fixed. For a population size of 1, a mutation happening is the same as the mutation fixing in the population.
But I see your point, sure with a population size of 7.4B it would be almost certain to have hundreds of "A +B combos" but in such big population fixation becomes very unlikely.
How unlikely? We need the formula for the probability of fixation of a beneficial mutation, where it factors in population size, the initial frequency of the allele in the population, and the selection coefficient of the beneficial allele compared to non carriers. Since we are talking about a hypothetical example, we will have to agree on a realistic selection coefficient. Those vary quite wildly in nature. Alleles are known that affect reproductive success anywhere on a spectrum of fitness effects that span from outright lethal (a cell that has the mutation is flat out dead straight after cell division) to basically doubling or tripling reproductive success (carriers have double or three times the average number of offspring).
To keep things conservative, I suggest we use a number like 2% increase in reproductive success of carriers.
The probability of fixation is 1/2N
... for a
neutral allele and with no new mutations occurring, in a population of constant size.
This is not what we are trying to calculate. The A+B combo is beneficial, remember?
Initially we were trying to determine the probability of occurrence of a double neutral mutation that shows beneficial epistatic interaction. As in they are neutral alone, but beneficial together.
That's what you set out to determine. To begin with I (still, entirely correctly) informed you that you can't calculate this number, because the frequency by which individually neutral mutations
are beneficial when simultaneously present is an empirical question. You have to
observe how often mutations that are individually neutral, have beneficial effects when together.
Only by observing the frequency by which this occurs, can you calculate the probability that it will occur in the future.
In response you've been blathering very incompetently about population genetics. I'm not a population geneticist and neither are you, but I can see already with the little I know that you know even less than me and you can't think straight.