Rumraket wrote:I thought you meant besides the example of chloroquine resistance. You are claiming to have found examples of adaptations that could not have evolved, right? Then showing an example of an adaptation that evolved seems rather self-defeating in my view.

At these point I am simply establishing the fact that at least some “benfitial steps” require more than 1 mutation.

Rumraket wrote:In the argot of chemical kinetics, getting beyond the deleterious mutation is the "rate-limiting step."

That is not quite relevant; any significant number of neutral mutations in path would limit evolution (feel free to do the math by yourself)

Rumraket wrote:
Because YOU are the one claiming to be in possesion of an argument that should cause us to doubt the veracity of the evidence that has already been collected.

I would say that you are the one who has to provide the step by step path that would produce an eye or a flagellum, and justify that the path is achievable with Darwinian evolution. After you do that any objector would have to show that at least some steps are not achievable.

Rumraket wrote:The evidence from the past history of life from comparative genetics, comparative anatomy, chronology of embryological developments, from the fossil record, from observations of molecular change in experiments, to observations of change of wild populations, from change due to domestic breeding and artificial selection and so on and so forth ad infinitum.

Most of the evidence that you mentioned would prove that universal common ancestry is true, as far as I know Behe (and I) grant universal common descend.

The rest bits of evidence show that some diversity is achievable by Darwinian evolution, which is also something that Behe and I would grant. The disagreement is o on whether if Darwinian evolution can account for all the diversity of life.

Rumraket wrote:No, we are affirming that we don't have reason to think there isn't one given all the evidence we already have. This goes back to Darwin himself....,

Well and I am affirming that there are good reasons to think that for example detecting light and reacting when light is detected are independent and codependent systems. This is to say that you can´t get both systems with a single mutation and each system by itself is useless without the other.

There are also good reasons to say that it would have been statistically unlikely to have a neutral mutation that would allow the organism to detect light, survive genetic drift, and get a second mutation that would allow the organism to react positively when light is detected and survive genetic drift.

Just to be clear what is exactly your point of disagreement?

1 that each system would have been useless without the other

2 that you can’t get both systems with a single mutation

3 that it is statistically unlikely to have a path that involves too many neutral mutations

Rumraket wrote:Rather it is YOU who is demanding of US that we prove a negative to you. All YOU have to do is find a SINGLE example of something that genuinely could not have evolved. Instead, you are demanding that WE prove to you that there are NO such examples. You want of us to prove that no barriers exist out there, an impossible task.

The number of possible paths is “potentially infinite” even if I show that a path would have been impossible you could always say that there might be an other path. I cant falsify a potentially infinite number of paths.

I can’t prove ether that you can’t get a system that can detect light and a system that causes a reaction when light is detected could have not been accrued by a single mutation, because the number of possible mutations is also potentially infinite. You are the one that would have to provide an example of such mutation.

Well anyway all I can do is:

Show that a significant number of neutral mutations in a path would we statistically unlikely and therefore constitute a barrier and that there are good reasons to assume that a significant amount of neutral steps would have had to occur if the eye or the flagellum would have had evolved by Darwinian mechanism.

To me this is enough to justify my skepticism towards the claim that all (or most) of the diversity of life is a result of Darwinian evolution.


And this is not even a big deal, many “non creationists” share the same skepticism, the issue is widely being discussed in journals

leroy wrote:

Rumraket wrote:I thought you meant besides the example of chloroquine resistance. You are claiming to have found examples of adaptations that could not have evolved, right? Then showing an example of an adaptation that evolved seems rather self-defeating in my view.

At these point I am simply establishing the fact that at least some “benfitial steps” require more than 1 mutation.

For what purpose? That is a trivial statement.

leroy wrote:

Rumraket wrote:In the argot of chemical kinetics, getting beyond the deleterious mutation is the "rate-limiting step."

That is not quite relevant; any significant number of neutral mutations in path would limit evolution (feel free to do the math by yourself)

I'm not disputing the math, I'm disputing the claim that we know of such barriers to the evolution of any species or biological entity or structure.

Give an actual concrete example of something that had to evolve while requiring "a significant number of neutral mutations" and show that this entity could not reasonably have evolved.

leroy wrote:

Rumraket wrote:Because YOU are the one claiming to be in possesion of an argument that should cause us to doubt the veracity of the evidence that has already been collected.

I would say that you are the one who has to provide the step by step path that would produce an eye or a flagellum, and justify that the path is achievable with Darwinian evolution.

That has already been done. Read Matzke 2003: http://www.talkdesign.org/faqs/flagellum.html

leroy wrote:After you do that any objector would have to show that at least some steps are not achievable.

Then do that.

leroy wrote:

Rumraket wrote:No, we are affirming that we don't have reason to think there isn't one given all the evidence we already have. This goes back to Darwin himself....,

Well and I am affirming that there are good reasons to think that for example detecting light and reacting when light is detected are independent and codependent systems. This is to say that you can´t get both systems with a single mutation and each system by itself is useless without the other.

What are these systems that are useless without the other that you are talking about? Give examples.

leroy wrote:There are also good reasons to say that it would have been statistically unlikely to have a neutral mutation that would allow the organism to detect light, survive genetic drift, and get a second mutation that would allow the organism to react positively when light is detected and survive genetic drift.

That chain of events is literally what happened in the Long Term Evolution Experiment with E coli, when aerobic citrate transport evovled (a neutral potentiating mutation happened, survived genetic drift, and later a second beneficial gene duplication put the citrate transporter under control of an operong active with oxygen present, leading to the ability to transport citrate into the cell and thus be metabolized). So your imagined statistical unlikelihood, isn't.

Just to be clear what is exactly your point of disagreement?

1 that each system would have been useless without the other

2 that you can’t get both systems with a single mutation

What are "each system" here? Give examples.

leroy wrote:3 that it is statistically unlikely to have a path that involves too many neutral mutations

How many is too many? Given that you manifestly thought that something which happened in the LTEE, is "statisticallly unlikely", which I take to mean you think that such an event could not possibly have been expected to happen in the history of life, then I think your intuitions about what can or can't happen aren't worth anything.

Your hunches about what is "too many" needs some clarification. Stop talking in vague generalities and give specifics.

leroy wrote:

Rumraket wrote:Rather it is YOU who is demanding of US that we prove a negative to you. All YOU have to do is find a SINGLE example of something that genuinely could not have evolved. Instead, you are demanding that WE prove to you that there are NO such examples. You want of us to prove that no barriers exist out there, an impossible task.

The number of possible paths is “potentially infinite” even if I show that a path would have been impossible you could always say that there might be an other path. I cant falsify a potentially infinite number of paths.

You have confused yourself. We aren't talking about mere possibility, we are talking about probabilities.

Your argument is that certain combinations of mutations are very unlikely, and if such combinations of mutations are required to evolve X, then it is unlikely that evolution is the explanation for X.

Your job is then to find examples of X where such very unlikely combinations of mutations are required. Should you find such a thing, our job would then be to find either an alternative pathway to X, or show that you are wrong when you say it is unlikely.

If we were to just declare that there are alternative pathways to X, you could reasonably ask for evidence of this. And if we could not provide any evidence that implies that there were such alternative pathways to X, you could then reasonably say that there is no good evidence that X could have evolved.

leroy wrote:I can’t prove ether that you can’t get a system that can detect light and a system that causes a reaction when light is detected could have not been accrued by a single mutation, because the number of possible mutations is also potentially infinite. You are the one that would have to provide an example of such mutation.

I don't have to provide an example of something I don't see any reason to think is how sight evolved.

To make sure you understand, I think both the proteins that react to light (opsins, which evolved from GPCRs, G-protein coupled receptors), and the biochemistry whereby a cell produces a genetically controlled reaction to the light stimulus(or another external stimulus) were both in place before sight evolved. That it is a modification of a series of duplications of an already existing number of systems.

leroy wrote:Well anyway all I can do is:
Show that a significant number of neutral mutations in a path would we statistically unlikely and therefore constitute a barrier and that there are good reasons to assume that a significant amount of neutral steps would have had to occur if the eye or the flagellum would have had evolved by Darwinian mechanism.

Then do that. And please show the actual math this time. How much is a significant number? Because whatever you thought it was up above, it happened in the LTEE.

leroy wrote:And this is not even a big deal, many “non creationists” share the same skepticism, the issue is widely being discussed in journals

Please give examples where evolutionary biologists think that a significant number of neutral steps had to occur for some entity X to have evolved, and show how they are then skeptical that it could have evolved because it would be "statistically unlikely" and thus constitute a barrier.

I think you can bring zero support for that claim.

@leroy
I think with respect to your last point, that you might have confused two different subjects here.

It is certainly true to say that there is a very real debate about the relative contributions of genetic drift, and natural selection, to the diversification of life. How much of this evolution owes to genetic drift at the molecular and morphological level, and how much of it owes to natural selection?

That is still very much debated between people who come down at different degrees of support spanning the full spectrum from "natural selection had a strong influence on everything" to biologists who say genetic drift has been a much bigger factor.

But you seem to be saying that there is "widespread" doubts about whether evolution could even have taken place if genetic drift has been a major component. That's just not true.

Rumraket wrote:Then do that. And please show the actual math this time.

Well do you see anything wrong with the mathematical approach that I suggested earlier? If you think I should do some corrections please let me know, after we agree on the approach I will proceed in presenting the real (approximate) values for each point.

to Calculate, the probability of getting 2 independent and codependent mutatoins occuring at a differen time


Pretend that an organism requires mutation A and Mutation B in order to have a beneficial step towards the evolution of the eye.

Mutation A and mutation B would be useless by themselves. Only the combination of both would have a benefit.


1 the probability of having mutation A in any individual member of a population
2 times the probability of that mutation being fixed in the population
3 times the probability of having mutation B in any individual member of the population that already posse’s mutation A
4 times the probability of fixation

leroy wrote:

Rumraket wrote:Then do that. And please show the actual math this time.

Well do you see anything wrong with the mathematical approach that I suggested earlier? If you think I should do some corrections please let me know, after we agree on the approach I will proceed in presenting the real (approximate) values for each point.

leroy wrote:to Calculate, the probability of getting 2 independent and codependent mutatoins occuring at a differen time

Pretend that an organism requires mutation A and Mutation B in order to have a beneficial step towards the evolution of the eye.

Mutation A and mutation B would be useless by themselves. Only the combination of both would have a benefit.


1 the probability of having mutation A in any individual member of a population
2 times the probability of that mutation being fixed in the population
3 times the probability of having mutation B in any individual member of the population that already posse’s mutation A
4 times the probability of fixation

Leroy how would this calculation tell us anything meaningful? Please follow me in this thought experiment:

Suppose we travel back in time 25 years to right before Richard Lenski starts the long-term evolution experiment with E coli. Then we ask: "what is the probability that 25 years in the future a set of these particular 600 mutations will have fixed, in the particular order they did, in this particular lineage out of the 12 running in parallel?"

We do the calculation (and simplify to just look at substitutions), and find that the very fist mutation has a probability of 1 in 4(the number of substitutions) in 4.5 million(roughly the E coli genome size in basepairs).
So does the next mutation.
So does the next mutation
...
600 times.

The compound probability of those 600 mutations happening over the next 25 years in that particular lineage of E coli, is (4.5x10^6 x 4)^-600= 1 in 1.457*10⁴³⁵³.

One point four five seven times ten to the fourthousand threehundred and fiftythird power. An incomprehensibly unlikely event. Yet 600 particular mutations happened in that lineage. And roughly a similar number in the other eleven lineages running in parallel. And each outcome was different, and they all had roughly the same incredibly low likelihood.

These kinds of ad-hoc probability calculations are meaningless. They don't tell us whether something could have happened in the past. Mutations accumulate over time, and they are subject to drift and selection. But any long historical sequence of such accumulations is going to look unfathomably unlikely in the future when we look back at them.

What were the odds that that particular sequence of mutations would have happened? Incomprehensibly low. But since mutations constantly accumulate, and since they are all subject to drift and selection, many years and generations in the future some sequence of them will be the one that has happened.

Calculating it's probability will be meaningless. We can't falsify history with a probability calculation. Otherwise we would be forced to conclude that the Long term evolution experiment (LTEE) did not take place. Which is obviously ridiculous.

The kind of thing you are trying to achieve with these calculations is a fool's errand. It doesn't actually tell you what caused something to happen, whether it was by design or it was just blind chance mutations filtered by drift and selection.

Rumraket wrote:Leroy how would this calculation tell us anything meaningful?

Well what mathematical approach do you suggest we should use?




Suppose we travel back in time 25 years to right before Richard Lenski starts the long-term evolution experiment with E coli. ..........

What were the odds that that particular sequence of mutations would have happened?
.[/quote]

I understood your point, but I am not committing the mistake that you think I am. I am not pretending to calculate the probability of having a particular set of mutations in a particular order.

In my example “A” and “B” could be any mutation, the only limitation is that A and B must be neutral by themselves but positive when they work together. With “eye” I mean ether an “eye” or something equivalent, something that would be considered irreducible complex by creationists.

So if my mathematical approach is wrong, do you have any suggestion on how to calculate the odds?
I am not ignoring the rest of your post, it is just that I want to clear this point before answering to you points.

leroy wrote:

Rumraket wrote:Leroy how would this calculation tell us anything meaningful?

Well what mathematical approach do you suggest we should use?

Rumraket wrote:Suppose we travel back in time 25 years to right before Richard Lenski starts the long-term evolution experiment with E coli. ..........

What were the odds that that particular sequence of mutations would have happened?

I understood your point, but I am not committing the mistake that you think I am. I am not pretending to calculate the probability of having a particular set of mutations in a particular order.

In my example “A” and “B” could be any mutation, the only limitation is that A and B must be neutral by themselves but positive when they work together.

Then you'd need to know the frequency by which otherwise neutral mutations are beneficial together. That can't be calculated, it has to be measured empirically.

You can't calculate the probability that a mutation will have a particular effect (will it be beneficial, deleterious or netrual?). You can only calculate the probability that particular combinations of mutations will happen, not the probability of what effect they will have.

The effects of particular mutations have to be measured. We can't just sit in our armchairs and calculate whether changing nucleotide 21,073,140 on chromosome 6 is going to be beneficial unless we happen to know a lot about what that nucleotide does. Those are the kinds of things math can't solve, we have to do experiments to determine the answer.

But in order to get the kind of number you want, you would need to establish how frequent two neutral mutations have beneficial epistatic effects. That can only be done by measuring the effects of combinations lots of mutations and even then it's going to be context specific and the number might change over time as the organism moves from one environment to another, and goes from near an adaptive peak, to the foot or somewhere on the slope of a "hill" in the fitness landscape. Even so, my guess is some work has been done on that so you'd have to try and find some literature on the subjec to find estimates of such numbers.

Rumraket wrote:

But in order to get the kind of number you want, you would need to establish how frequent two neutral mutations have beneficial epistatic effects. That can only be done by measuring the effects of combinations lots of mutations and even then it's going to be context specific and the number might change over time as the organism moves from one environment to another, and goes from near an adaptive peak, to the foot or somewhere on the slope of a "hill" in the fitness landscape. Even so, my guess is some work has been done on that so you'd have to try and find some literature on the subjec to find estimates of such numbers.

Is not as hard as you make it seem, specially because we are not interested in the exact values.

1 the probability of having mutation A in any individual member of a population
The probability of getting any neutral mutation at any time to any individual is 100% (1) this is to say that we have almost 100% certainty that a neutral mutation will occur

2 times the probability of that mutation being fixed in the population
Again 100% (1) we know with near certainty that at least one mutation will become fixed

3 times the probability of having mutation B in any individual member of the population that already posse’s mutation A
This is where it becomes interesting, since at this point you do need a specific kind of mutation, the speficic type of mutation that would produce a benefit in the “A” + “B” system.
Well the mutation rate is roughly 100 mutations per generation and genomes on average have 3,000,000,000 locations (or base pairs) so the probability of having a specific mutation in a individual member of the population would be (100/3,000,000,000)*N (N=POPULATION SIZE)


4 times the probability of fixation
This is easy 1/2N*

You can see that we have N both in the numerator and in the denominator, so N can be canceled.

So all you have to do is multiply all those yellow numbers.

The result is 1/60,000,000

This means that the probability is 1/60,000,000 or to put it this way you need 60,000,000 tries (60,000,000 generations) to get something as simple as A+B.

Considering the age of the earth and the number of populations (“species”) that have ever lived we do have enough probabilistic resources to get A+B every once in a while. Specially if you are dealing with unicellular organism living under a controlled environment (Experiment with E coli, for example) but A+B is a limited resource, you can’t say that A+B made a mayor contribution in the evolution of the eye. Otherwise it would be like claiming mount improbable

*Well to be fair 4 is suppose to be positive, so the probability of fixation is a little big greater than 1/2N But even in the best possible (and realistic) scenario you will still get a small number as a result

leroy wrote:

Rumraket wrote:

But in order to get the kind of number you want, you would need to establish how frequent two neutral mutations have beneficial epistatic effects. That can only be done by measuring the effects of combinations lots of mutations and even then it's going to be context specific and the number might change over time as the organism moves from one environment to another, and goes from near an adaptive peak, to the foot or somewhere on the slope of a "hill" in the fitness landscape. Even so, my guess is some work has been done on that so you'd have to try and find some literature on the subjec to find estimates of such numbers.

Is not as hard as you make it seem, specially because we are not interested in the exact values.

You are confused. A lot.

You specifically said that you wanted to calculate the odds of two neutral mutations being beneficial in combination. That means you want to calculate the odds of two mutations having a particular effect.

Your own words: "I understood your point, but I am not committing the mistake that you think I am. I am not pretending to calculate the probability of having a particular set of mutations in a particular order.

In my example “A” and “B” could be any mutation, the only limitation is that A and B must be neutral by themselves but positive when they work together."

Yo can't calculate that. It's an empirical question. How often ARE two mutations beneficial when combined? You can't just calculate a number for that, you can only settle that by observation. It's like asking how to calculate how tall I am. You can't calculate that, you need to measure it.

1 the probability of having mutation A in any individual member of a population

Bzzzt. That's the odds of a particular mutation, which you manifestly said you weren't trying to do.

You need to try to get your story straight.

The probability of getting any neutral mutation at any time to any individual is 100% (1)

Over what period of time? For what species? 1 generation? 500? A single bacterium undergoing binary fission?

That depends on the mutation rate, genome size, the fraction of neutral mutations out of all mutations, and the population size. Leroy you're doing this wrong.

2 times the probability of that mutation being fixed in the population

Technically it doesn't need to be fixed. If it fixes in the population then all the individuals have it, sure. And that increases the odds by a lot that it can combine with the second mutation if it occurs, since if A has already fixed, we are basically just waiting for the 2nd mutation to occur if the 1st mutation has happened and become fixed in the population.

But how big is the population? With a trillion individuals, that mutation is going to happen much faster than with 10,000.

3 times the probability of having mutation B in any individual member of the population that already posse’s mutation A[/color]
This is where it becomes interesting, since at this point you do need a specific kind of mutation, the speficic type of mutation that would produce a benefit in the “A” + “B” system.

Yeah, and that is just the probability of B, since you assumed A became fixed. But we still need a population size. If A has fixed in the population, and if B has a probability of 1 in 3 billion as you go on to assert, with 7.4 billion people on Earth chances are ~2.47 individuals in the population has the A+B combination already. With every new generation, on average 2.47 new individuals will get the B mutation in addition to the A they already have.

Well the mutation rate is roughly 100 mutations per generation and genomes on average have 3,000,000,000 locations (or base pairs)

For humans.

so the probability of having a specific mutation in a individual member of the population would be (100/3,000,000,000)*N (N=POPULATION SIZE)

4 times the probability of fixation
This is easy 1/2N*

Where does fixation enter into this? If A has already fixed, then every member in the population has A. Then B just needs to occur in any individual and then you have the combination A+B. And A+B is beneficial you say, but then the probability of fixation depends on the selection coefficient, (how much does it increase average reproductive success of carriers?) and the population size (because if the population is huge, there will be more instances of A+B cropping up independently, and these in turn will be selected for).

You can see that we have N both in the numerator and in the denominator, so N can be canceled.

So all you have to do is multiply all those yellow numbers.

The result is 1/60,000,000

This means that the probability is 1/60,000,000 or to put it this way you need 60,000,000 tries (60,000,000 generations) to get something as simple as A+B.

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.

About a trillion (10¹²) bacteria live in your gut right now. Their genome size is in the 3-10 megabasepairs range. An event with a probability of 1 in 60 million happens ~16.700 times every new generation for a population of a trillion individuals, assuming they get 1 new mutation each every time they divide.

In the human population of 7.4 billion, it would happen about 123 times every generation. In prehistory during human evolution, assuming we had a population size of about 100.000 individuals for most of our evolution, the A+B combo would happen once every 600 generations on average if A has already fixed and each couple gets 2 children on average. Of course, they probably had more than 2 children on average. Even going back a few centuries it was not uncommon for reproducing couples to raise 10 children or more.

Considering the age of the earth and the number of populations (“species”) that have ever lived we do have enough probabilistic resources to get A+B every once in a while.

There are roughly 10³⁰ organisms on the planet right now. Events with a probability of 1 in 6.0*10⁷ would happen about 1.67*10²² times every generation for a population of that size.

Specially if you are dealing with unicellular organism living under a controlled environment (Experiment with E coli, for example) but A+B is a limited resource, you can’t say that A+B made a mayor contribution in the evolution of the eye. Otherwise it would be like claiming mount improbable

I think I have shown the opposite.

I don't think you've thought this one through.

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.

in reality it is "easier" to evolve when the population size is small. So a population of "1" is your best possible scenario.

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.


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.

The probability of fixation is 1/2N

so with a population( N) of 7.4Billion the probability would be 1/14.8 Billion.

So even if you have 123 "a+b combos" in every generation you still need 60,000,000 generations to get a single combo that becomes fixed.

I am simply trying to prove a trivial point, it shouldn't be controversial, all I am saying is that A+B combos are unlikely and not expected to be a mayor component in the evolution of a complex system.

A+B+C Combos are expected to be very rare and A+B+C+D combos virtually impossible, all I am saying is that any model that tries to explain the evolution of the eye has to take this probabilities in to account.

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.

To calculate the probability of fixation for a beneficial allele in a diploid population of constant size we can use the diffusion-approximation (because it is easy to use in practice) from Felsenstein's freely available Theoretical Evolutionary Genetics, page 315 equation (VII-91):

U(p) ≃ 1-e^-4Nsp / 1-e^-4Ns

The letter e is Euler's number, which has approximately the value 2.71828. N is population size, s is selection coefficient of the beneficial allele, p is the frequency of the allele in the population.

Plugging in the values of the current human population of 7.4 billion, a selection coefficient of 0,02 (2% average increase in number of offspring of carriers), and an initial frequency of 0,000000033338 (246,7 individuals out of 7.4 billion), we get

U(p) = 0,9999999973160780.

That is, a beneficial allele that gives carriers a 2% increase in number of offspring on average, which is present in 246,7 individuals out of 7.4 billion, has a 99,9999997316078% chance of fixation. That's pretty much guaranteed to happen.

Even if it initially only exists in 1 individual it still has about 7,69% chance of fixation.

Greetings,

Nice work, Rumraket.

Just a minor addition...

For every person alive today, it's said that 30 have died, so the 246.7 would be 7401.

Kindest regards,

James

Greetings,

Nice work, Rumraket.

Just a minor addition...

For every person alive today, it's said that 30 have died, so the 246.7 would be 7401.

EDIT: Acksherly, it should be 7647.7 - the 30 already dead plus the 1 currently alive.


Kindest regards,

James