Behe's Ratchet

[Rumraket (Mikkel R.)]

@Roy

Finger trouble, I think - 25*39998 = 999,950 years.

This looks more reasonable.

I read too many nines, you are correct.

[Ashwin_s]

@Rumraket

and I get a time to fixation for a neutral allele initially arising in a single individual in a population of 10.000 of 39998 generations, which is 9.999.950 years if the generation time is 25 years.

I got Something similar. Its 4Ne (Not 4N). Anyway,he simplifies the time in generation to fixation to 4Ne in equation 22. He gives the value of Ne as 0.8XN. So for N=10000; We get 32000 generations and for generation time of 25 years, it become 800K years.
What you missed is that this calculation is where probability for fixation is 100%.

In cases where the mutation is not fixed, He calculates the probability from eqn 16 as below in the same paper (refer eqn 23) - (Quoted below)

Next, let us consider the number of generations until a neutral mutant gene
is lost from the population disregarding the cases in which it is eventually fixed.
This is given by formula (16) by putting p=I/(2N). ,
t(1/2N)= (4NE/(2N-1))* loge (2N) = 2(Ne/N)loge(2N)

Since the ratio N,/N is around 0.8 in man (CROW 1954), a single mutant gene
which appeared in a human population will be lost from the population on the
average in about 1.6 log,2N generations. If N = 10^4, this amounts to about 16
generations. These results show that a great majority (fraction 1 - (1/2N) of neutral or nearly neutral mutant genes which appeared in a finite population are
lost from the population within a few generations, while the remaining minority
(fraction 1/(2N)) spread over the entire population (i.e. reach fixation) taking a
very large number of generations.

So his conclusion is that the mutations that are not fixed are lost comparatively faster. I agree with you that the mutations that are fixed eventually last for a long time.

763.pdf

[Rumraket (Mikkel R.)]

In any case you’re right, and I was wrong about the time to fixation by an order of magnitude. I also misunderstood the point you initially made which I responded to. I’ll take that as a hint I need a break.

[Ashwin_s]

@Rumraket

In any case you’re right, and I was wrong about the time to fixation by an order of magnitude. I also misunderstood the point you initially made which I responded to. I’ll take that as a hint I need a break

Welcome…
All this is totally new stuff for me (to the extent of having to look up what the various terminologies mean)… so I appreciate being questioned/corrected.

Thanks.

[T_aquaticus]

@Ashwin_s

I was just pointing out the time it takes for a set of mutations to get fixed (4Ne generations)… this amounts to 800k years for a generation size of 25years.

Time to fixation depends on the population size, with smaller populations allowing for quicker fixation. Fixation is an ongoing process, so the 800k ticker starts on new mutations every generation. For every generation, there are 50 mutations that reach fixation (on average) that first appeared 800k years ago. Just think of how whiskey distilleries are able to send out 18 year old whiskey every year. They are making new batches every year.

[T_aquaticus]

@Ashwin_s

This is not possible as per Kimura’s paper. For human populations, he estimates the neutral mutations that are lost (and not fixed) in the population disappear within 16 generations (Edit: For a population of 10000). He doesn’t get a result where mutations which are yet to be fixed linger on for millions of years.

That’s only an average. If you think of a bell curve, there are rare neutral mutations on the edges of the curve that reach fixation which is why we see an estimate of 50 neutral mutations reaching fixation out of 500,000 that occur each generation in a population of 10,000.

[T_aquaticus]

@Roy

Whiskey (‘e’) comes from Ireland.

I was wondering if someone would pick up on that. Whiskey also comes from the US, which is where I am sitting. For what it’s worth, a lot of that Scottish whisky is aged in whiskey barrels from the US, so there is a bit of whiskey in the whisky.

[Ashwin_s]

@T_aquaticus

Time to fixation depends on the population size, with smaller populations allowing for quicker fixation.

Agreed. I was calculating based on a population of 10000. which is a fair number for human beings.

@T_aquaticus

Just think of how whiskey distilleries are able to send out 18 year old whiskey every year. They are making new batches every year

I did think of this. But the situation is different. Let me point out a problem-

The population will not be stable for 800k years and the calculation collapses. (Human beings had atleast two bottleneck events in the last 800k years).

It seems to me that saying the mutations will be fixed in 800k years is the same as saying, we cant predict when the mutations will be fixed.

Yet, If I understand correctly, the data matches the idea that the no: of mutations fixed per generation are the same as the mutation rate. The fixation time should be not be fixed… so the no: of mutations that are fixed should not be constant…
It’s a mystery to me. Perhaps bottlenecks are always followed by exponential population growth at similar rates and it balances things out. Or bottlenecks are typically short and fast such that it doesn’t have much of an impact on the process.

[T_aquaticus]

@Ashwin_s

The population will not be stable for 800k years and the calculation collapses. (Human beings had atleast two bottleneck events in the last 800k years).

The calculation doesn’t “collapse”, it just changes. Bottlenecks will have effects, but they will not stop the fixation of neutral mutations.

@Ashwin_s

It seems to me that saying the mutations will be fixed in 800k years is the same as saying, we cant predict when the mutations will be fixed.

You’re an engineer. Haven’t you ever used simulations to get an estimation? Have you ever thought that we should throw out 200 years of discoveries in physics because those models aren’t 100% accurate?

@Ashwin_s

Yet, If I understand correctly, the data matches the idea that the no: of mutations fixed per generation are the same as the mutation rate. The fixation time should be not be fixed… so the no: of mutations that are fixed should not be constant…

Founder effects can be modeled using population genetics as well. In fact, there is an online simulator here:

http://virtualbiologylab.org/NetWebHTML_FilesJan2016/RandomEffectsModel.html

A small population after a bottleneck can fix mutations faster, which is worth remembering.

[colewd (Bill Cole)]

@swamidass

Behe agrees. @Agauger agrees. John Sanford agrees. Everyone who does population genetics, YEC, OEC, and ID together all agree. Why do you disagree with them here?

I don’t think they read Kimura’s paper carefully enough. Ashwin has pointed out the key problem which is the probability of a new mutation appearing in the population and making it to fixation before it gets eliminated.

The simulator that T posted confirms this.

[T_aquaticus]

@colewd

I don’t think they read Kimura’s paper carefully enough. Ashwin has pointed out the key problem which is the probability of a new mutation appearing in the population and making it to fixation before it gets eliminated.

What is that probability, and how is it a problem?

[davecarlson]

Bill, the probability of a neutral mutation being fixed by drift is simply its frequency in the population. Most mutations are lost by drift. But there are lots, and lots of mutations occurring in each generation. It’s inevitable that some will be fixed.

[colewd (Bill Cole)]

@T_aquaticus

What is that probability, and how is it a problem?

The probability of getting fixed before elimination given a population of 80 and a mutation that has reached 10% of the population is about 10%. A fresh mutation in a population of 10K is going to have a much larger problem.

[T_aquaticus]

@colewd

The probability of getting fixed before elimination given a population of 80 and a mutation that has reached 10% of the population is about 10%. A fresh mutation in a population of 10K is going to have a much larger problem.

I fully agree that the vast majority of neutral mutations are going to be lost. What you keep ignoring is the number of neutral mutations that appear in each generation. For a population of 80 and a mutation rate of 50 mutations per individual per generation, that is 4,000 mutations in one generation.

Nearly all lottery tickets are losers, but people still win.

[colewd (Bill Cole)]

@davecarlson

Bill, the probability of a neutral mutation being fixed by drift is simply its frequency in the population. Most mutations are lost by drift. But there are lots, and lots of mutations occurring in each generation. It’s inevitable that some will be fixed.

We agree here.

The issue is how do you reconcile a transition where DNA differences are greater than 30 million mutations and the time is less than 20 million years.

[T_aquaticus]

@colewd

The issue is how do reconcile a transition where DNA differences are greater than 30 million mutations and the time is less than 20 million years.

The fixation rate is the mutation rate. If the mutation rate is 50 mutations per individual per generation and a generation time of 25 years, then we would expect 40 million mutations to reach fixation in that lineage over 20 million years.

[colewd (Bill Cole)]

@T_aquaticus

The fixation rate is the mutation rate.

Can you show mathematically this is true?

[T_aquaticus]

@colewd

Can you show mathematically this is true?

We did this yesterday.

For a diploid population of size N and neutral mutation rate {\displaystyle \mu }, the initial frequency of a novel mutation is simply 1/(2 N ), and the number of new mutations per generation is {\displaystyle 2N\mu }. Since the fixation rate is the rate of novel neutral mutation multiplied by their probability of fixation, the overall fixation rate is {\displaystyle 2N\mu \times {\frac {1}{2N}}=\mu }. Thus, the rate of fixation for a mutation not subject to selection is simply the rate of introduction of such mutations.
Fixation (population genetics) - Wikipedia

The copy pasta messes up some of the symbols, so go to the Wiki to get the full details. The math is all there.

[colewd (Bill Cole)]

Can you reconcile 1/2n with 4n as the time of mutations that are not eliminated becoming fixed?

[Ashwin_s]

@T_aquaticus

The fixation rate is the mutation rate. If the mutation rate is 50 mutations per individual per generation and a generation time of 25 years, then we would expect 40 million mutations to reach fixation in that lineage over 20 million years.

I didn’t understand it that way. The probability for a mutation to get fixed in neutral conditions is (1/2N).
Let m be the mutation rate. Then total no: of mutations in a diploid population is 2Nm.
No: of mutations fixed is = (1/2N) X2Nm
=m…
I.e irrespective of the population, only m mutations will get fixed. This was shared by @swamidass. I hope I understood it correctly.