Research

[Climbing Air]

So I'm writing a piece of fiction based on the introduction of a new allele, and since I'm not a scientist, I have a few questions.

 

The existing allele is a wild gene and the new allele is dominant. In my story, a single individual introduces the new allele to the world and it quickly populates due to natural selection.

 

I'm wondering if 'a' could ever become completely extinct? I know that 'Aa' would dominate but could 'AA' ever take over? Or would 'AA' need a survival advantage over 'Aa' too?

 

Later in my story, I was thinking of introducing a third allele which I will call 'B'. This allele is dominant to 'a' as well and using a bit of creative license, 'A' and 'B' would cancel each other out rendering any offspring 'aa'. However, I was wondering if it was mathematically possible for 'A' and 'B' to wipe each other out or if the population would remain constant?

 

Finally, supposing 'B' doesn't exist, I was wondering if there would be any way for 'a' to make a comeback and possibly annihilate 'A'? Keeping in mind, 'A' had a survival advantage over 'a' which lead to the population growth of 'AA' and 'Aa'.

 

 

Sorry if I haven't been clear, if anything needs clarifying, let me know. And any help will be much appreciated. Thank you.

[CharonY]

You are probably thinking of fixation. However, assuming that there is a positive selection for AA as well as Aa, it will take a long time before a gets eliminated (the probability function is a bit complicated, it depends on the selection coefficient, frequency in the population, dominance parameter, mutation rate etc.

Note that fixation is always probabilistic, if your question is whether one or the other allele may get extinct or fixated, the answer is yes. It is just the probability that varies.

[Climbing Air]

You are probably thinking of fixation. However, assuming that there is a positive selection for AA as well as Aa, it will take a long time before a gets eliminated (the probability function is a bit complicated, it depends on the selection coefficient, frequency in the population, dominance parameter, mutation rate etc.

Note that fixation is always probabilistic, if your question is whether one or the other allele may get extinct or fixated, the answer is yes. It is just the probability that varies.

 

 

So what would be the quickest way to get 'A' fixated and 'a' gone for good? And how long would that take from a single individual among maybe a million? A rough estimate will do...

[CharonY]

Did a lengthy answer and then the browser crashed.

 

In short, your limiting factor is most likely to be your low mutation rate (i.e. a singular event). From memory (i.e. better check it out somewhere else, it may be wrong) the time to fixation for a neutral allele in the diploid case and for very low mutation rates goes towards 4N (i.e. four times the population size in generations).

 

For selected traits it is (2/s)ln(2N) generations, with s being the selection coefficient. Note that these are average times (finally it may or may not happen). Note that even deleterious mutations can become fixed.

 

For more details you may want to check out the papers from Kimura or some text books (e.g. Futuyma).

[Climbing Air]

Hey, I realise this is an old thread now but if anyone could provide any more help, it would be much appreciated.

 

I ditched the whole thing after a bit of reading when I realised what I was suggesting couldn't be done, at least in an adequate time frame. However, a friend prompted me to look into it, I've done a bit more reading and realised it may not have been as bad as I first thought.

 

 

I'd really appreciate it if someone could provide a rough number for me or at least give me a way to figure it out. I'll provide as much information as possible and if any more is needed, ask away.

 

I'd like to determine the time to fixation for a single allele in a population of about 1 million.

 

The original mutant would be 'AA'... and everybody else would be 'aa'. The mutation 'A' would be dominant and it would be highly beneficial... if you need a number, give me a range I'll provide one. The mutation would also encourage killing of the wild type 'a'... So this is another edge for the 'A'. They would also produce a lot more children than 'a' would. So hopefully with those 3 combined edges, even from a single individual, fixation of 'AA' shouldn't take too long? Reproduction would also be more or less random.

 

 

However, am I right in thinking that a dominant allele would not be fixed... and that Aa would become more common? Because if the survival benefit is in 'A' then 'AA' would have no more benefit than 'Aa' which would automatically become more common through random reproduction, right?

 

This explains what I mean: Complete recessiveness of an allele alone prevents the loss of that allele due to selection alone. Because the heterozygote's relative fitness is equal to the homozygote with both beneficial alleles, the deleterious allele is maintained in the population. Loosely put, the deleterious allele "is hidden" from selection in the heterozygous state.

 

If this is correct... then switching 'A' and 'a' and making the mutant allele recessive would solve the problem, correct? Either way, I would like to know how long it would take for a beneficial 'A' or 'a' to become fixated in a population of 1 million. 200,000 and 20,000. dominant or recessive.

 

Or if you could provide a simple formula for me to use, that would be appreciated to!

[CharonY]

Even if it was dominant there would be chance that it becomes fixed. In fact, deleterious alleles can also be fixed, especially in small populations (remember, we are talking about probabilities). There are different models that explore the question you have, but I do not know it precisely off the top of my head. Kimura has developed models in that area, though. You may want to check his papers/books (unfortunately I do not have a copy with me).

[Climbing Air]

Well, even if you could explain that formula in more detail for me...? The (2s) ln(2N)

 

Could you run an example by me? Say... selection coefficient of 0.1 and a population of 1 million? I've got a number somewhere... but I don't know how the number was arrived at and I'm curious t see if your number will match up.

[CharonY]

Under the assumption that it fixes, it will take about 290 generations. This model does not take parameters such as dominance into account.

Note that time to fixation does not imply that it will really fix. It is just the average time that is required in case it really fixes.

[Climbing Air]

Thank you That's pretty much the number I had in mind! I'll call it an even 300 and leave it at that... Thank you for all your help!

[Climbing Air]

Sorry, one more question...

 

I don't know how to phrase it... but is it scalable? I mean... would 10% of 'N' = 10% f the output... So for example... A population of 100,000 would only take 29 generations... Or am I simplifying it?

[CharonY]

It scales with ln(2N) (i.e logarithmic). ln(2E6) = 14.5, however ln(2E5) = 12.2. That would be 244 generations.

[Climbing Air]

LOL Okay... I like this! You're telling me all I need to know. Look, I apologise... I have TRIED to understand ln and your formula etc... I have looked all over the net and done a fair bit of reading but I really can't grasp it.

 

So if you can help me here, I promise I will leave you alone.

 

I am also after answers for 50k, 20k and 10k.

 

And I am wondering if dominance would play a part... Would a dominant allele go to fixation quicker than a recessive one? Would it make much difference? I assume it doesn't factor into the formula anyway...

 

I was also wondering if you could put a rough estimate on the selection co-efficient for me... Obviously, I would have to explain a bit about the allele. Basically, it would predispose people to 'evil'. It would manifest itself in different ways... however, it has 3 distinct survival advantages. 1. It creates tougher individuals, who are more inclined to fight for survival. 2. People with the allele would actively hunt those without it. 3. They are more inclined to proliferate through rape and promiscuity...

 

I don't know if an 's' number of 0.1 is fair or not... What do you reckon? If you can come up with a better number, you might have to adjust the answers for 1 mill, 100k etc...

 

Part of me feels like I am trying too hard here... but I know how annoying it is when a piece of fiction is completely out of touch with the topic it's talking about. I mean, I want it to be realistic. If only there was a piece of software to calculate all of this for me. I've looked!