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Author
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Topic: The Mathematics Of Natural Selection?
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Dr Adequate
Member (Idle past 312 days) Posts: 16113 Joined: 07-20-2006
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Message 1 of 10 (396481)
04-20-2007 10:14 AM
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I've been doing a bit of computer modelling again. A new mutation arises in a (haploid) population, the size of which is taken to be constant. In the long run, it will achieve either fixation or extinction. How does its probability of fixation relate to the relative advantage it confers upon its possessors? (Note that a relative advantage less than 1 is in fact a disadvantage.) Here's the answer from my computer model for effective populations of 10 and 100.
It occurs to me that you ought to be able to use probability theory to calculate the points on this graph directly, rather than by endless simulations. I know, for example, that it can be shown that for neutral mutations (i.e. with relative advantage 1) and haploid organisms the probability of fixation is 1/population (as is in fact shown by my model); however, the reasoning behind this result depends crucially on the fact that the mutation is neutral. In principle, though, there should be some formula where we put in the population size, the ploidy, and the relative advantage, and get out the probability of fixation. Does anyone know of such a formula and how it's derived? Edited by Dr Adequate, : Fixed graph.
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AdminNosy
Administrator Posts: 4754 From: Vancouver, BC, Canada Joined: 11-11-2003
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Message 2 of 10 (396484)
04-20-2007 10:22 AM
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Thread moved here from the Proposed New Topics forum.
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PaulK
Member Posts: 17827 Joined: 01-10-2003 Member Rating: 2.3
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The field of study you're looking for is Population Genetics.
| This message is a reply to: | | | Message 1 by Dr Adequate, posted 04-20-2007 10:14 AM | | Dr Adequate has not replied |
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dwise1
Member Posts: 5952 Joined: 05-02-2006 Member Rating: 5.2
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Some of that math is covered in a textbook by John Maynard Smith, "Evolutionary Genetics" (1989).
| This message is a reply to: | | | Message 1 by Dr Adequate, posted 04-20-2007 10:14 AM | | Dr Adequate has not replied |
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Wounded King
Member Posts: 4149 From: Cincinnati, Ohio, USA Joined: 04-09-2003
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I'm not au courrant with the latest in pop. gen. these days but I found a paper which seems to be relevant to your question.
The Probability of Fixation in Populations of Changing SizeGenetics. 1997 June; 146(2): 723-733. S. P. Otto and M. C. Whitlock The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size. The full text of this article is available online as a PDF. In their introduction they refer to a diffusion approximation by Kimura which I think fulfills your criteria. The formula for the probability of fixation is...
[qs]1 - exp N esp[/i] ------------------- 1 - exp N es[/i]][/qs] Where Ne is the variance effective size for a population size N, s is the additive selective effect and p is the initial frequency of the allele. TTFN, WK Edited by Wounded King, : No reason given.
| This message is a reply to: | | | Message 1 by Dr Adequate, posted 04-20-2007 10:14 AM | | Dr Adequate has replied |
| Replies to this message: | | | Message 9 by Dr Adequate, posted 04-20-2007 2:12 PM | | Wounded King has replied |
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Dr Adequate
Member (Idle past 312 days) Posts: 16113 Joined: 07-20-2006
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Message 6 of 10 (396505)
04-20-2007 11:27 AM
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Heck, it's already been promoted. The graph needs correcting. Thanks for the responses already.
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Allopatrik
Member (Idle past 6215 days) Posts: 59 Joined: 02-07-2007
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Finite Populations and Stochastic Processes
If you are dealing with finite populations, your model needs to incorporate accommodation for stochastic effects. A very nice discussion of the biological contributions to stochasticity and the mathematical underpinning can be found here: Kimura. M. (1957). Some problems of stochastic processes in genetics. The Annals of Mathematical Statistics 28(4): 882-901. A
Natural Selection is not Evolution-- R.A. Fisher
| This message is a reply to: | | | Message 1 by Dr Adequate, posted 04-20-2007 10:14 AM | | Dr Adequate has replied |
| Replies to this message: | | | Message 8 by Dr Adequate, posted 04-20-2007 2:02 PM | | Allopatrik has not replied |
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Dr Adequate
Member (Idle past 312 days) Posts: 16113 Joined: 07-20-2006
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Message 8 of 10 (396532)
04-20-2007 2:02 PM
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Reply to: Message 7 by Allopatrik
04-20-2007 1:20 PM
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Re: Finite Populations and Stochastic Processes
If you are dealing with finite populations, your model needs to incorporate accommodation for stochastic effects. Clearly it does: if it was deterministic, the new mutation would always achieve fixation when relative advantage was > 1 and extinction when it was < 1. --- I'll annotate this later, I'm a bit busy now. This is one run through from mutation to fixation/extinction. The graph is produced by counting the proportion of such simulations that achieve fixation for each value of r. function fixed(r:real; pop:integer) : boolean; var cp, ncp, nc1, m: integer; begin cp:=1; q:=r+1; repeat ncp:=0; nc1:=0; repeat; m:=random(pop); if (m
begin
if q*random > 1 then ncp:=ncp+1
end
else if q*random < 1 then nc1:=nc1+1
until ncp + nc1 = pop;
cp:=ncp;
until (cp = 0) or (cp = pop);
if cp = pop then fixed:=true else fixed:=false;
end; Edited by Dr Adequate, : No reason given.
| This message is a reply to: | | | Message 7 by Allopatrik, posted 04-20-2007 1:20 PM | | Allopatrik has not replied |
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Dr Adequate
Member (Idle past 312 days) Posts: 16113 Joined: 07-20-2006
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That might be it, though I'd have to check (a) that the equation doesn't rely on some sort of approximation which works for small s and large N; and (b) whether s can take negative values (which it can't if you use Haldane's linearization). I'll look up how it's derived. (Or I could just stick it into a program and see what graph it draws.) Thank you. --- ETA: it doesn't matter if s can't take negative values --- if it can't I can just change p from 1/ Ne to ( Ne - 1)/ Ne and take s positive for those cases, can't I? * slaps forehead * Edited by Dr Adequate, : No reason given. Edited by Dr Adequate, : No reason given. Edited by Dr Adequate, : No reason given. Edited by Dr Adequate, : No reason given.
| This message is a reply to: | | | Message 5 by Wounded King, posted 04-20-2007 10:57 AM | | Wounded King has replied |
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Wounded King
Member Posts: 4149 From: Cincinnati, Ohio, USA Joined: 04-09-2003
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I have a further link for an article criticising Kimura's formulation and suggesting an alternative they consider more accurate ( Wang and Rannala, 2004). Unfortunately their alternative is considerably more complex and I can't reproduce it here, so you will just have to read the paper to find out what it is. TTFN, WK
| This message is a reply to: | | | Message 9 by Dr Adequate, posted 04-20-2007 2:12 PM | | Dr Adequate has not replied |
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