Talk:Fisher's fundamental theorem of natural selection

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[edit] Edwards' genetics paper out of scope

In the list of references, Edwards' 2000 paper in Genetics is listed. This paper is a historical review with anecdotes of things that have happened in the population genetics community, and only shortly mentions the fundamental theorem. I think it is out of scope for this wikipedia entry, so it could be deleted because of lack of relevance. Anyone who thinks otherwise? --Anthony Liekens 00:46, 26 Nov 2004 (UTC)

[edit] Quote

Why is the Edward's quote in the introduction? It is not really more modern terminology, and there is confusion about gene frequency (should be allele frequency) and genic variation. What does the Edwards' quote give us? Ted 15:47, 31 May 2006 (UTC)

[edit] What about Gaussian adaptation?

Fisher's theorem is based on a model of evolution selecting genes. Gaussian adaptation and the theorem of Gaussian adaptation is based on a model selecting individuals. Unfortunately, the theorems differ from each other. In Fisher’s theorem a maximal mean fitness means zero variances, while Gaussian adaptation carries out a simultaneous maximization of mean fitness and variances (average information) according to the entropy law. But, the scientific community has not yet accepted Gaussian adaptation. Would it not be possible to temporarily accept Gaussian adaptation as a possible alternative until the discrepancies has been fully understood?--Kjells 09:27, 5 July 2007 (UTC)

Please note that talk pages are for discussions aabout improving the article at hand, not for soapboxing your pet theories.Sjö 19:39, 6 July 2007 (UTC)
Would the following contribution be interpreted as an improvement?
== Some criticism ==
As a random process one would expect evolution to increase the phenotypic variances (average information) according to the second law of thermodynamics (entropy law) dependent on the mutation rate. But unfortunately, Fisher’s fundamental theorem tells us nothing about this. So, a possible impact of the entropy law seems to be missing. If this impact is taken into consideration, it may as well happen that evolution maximizes the average information keeping the mean fitness constant; or something between the extremes.--Kjells 09:27, 10 July 2007 (UTC)


This is not correct. Fisher's theorem is talking about a population undergoing directional selection, which leads to a narrowing of phenotypic space. Natural selection does not lead to a increase in phenotypic divergence under most circumstances (excluding density dependent selection).
Yes, but it is missleading in the sense that a maximum in mean fitness can only be reached when variances are = 0. But in reality and in a state of selective equilibrium mean fitness may be constant with variances greater than 0. In addition evolution is a random process and such processes produce disorder according to the entropy law. As a "fundamental" theorem it should not ignore the impact of the entropy law.
In Gaussian adaptation the gradient of mean fitness is
dP(m)/dm = M-1 P (m* – m), where M is the moment matrix, P is the mean fitness, m is the centre of gravity of phenotyhpes before selection and m* is centre of gravity after selection. The gradient also gives the direction in phenotypic space.--Kjells 06:44, 2 August 2007 (UTC)
This has been discussed on the thread
http://www.iidb.org/vbb/showthread.php?t=163730&page=5&highlight=rogerg
--Kjells 07:06, 2 August 2007 (UTC)
This is the fundamental theorem of Gaussian adaptation taking also the entropy law into consideration. In this case selection takes place on the individual level (not the genetic dito). As can be seen P may be maximal when m = m*, even though the variance is not = 0.--Kjells 08:00, 2 August 2007 (UTC)

I see no conflict of interest. My interest was to find out why there is a paradox here. I now see why the theorem of Fisher differs from the theorem of Gaussian adaptation (GA), in which only one definition of mean fitness is used. But, in Fisher's theorem two different definitions are used: 1 the mean fitness of offspring (before selection) and 2 the mean fitness of the parents to offspring in the next generation (after selection). Thus far I have always used the same definition of a mathematical entity when trying to investigate its increase. Therefore the theorem tells me nothing about the increase in the mean fitness of offspring from one generation to the next (my main concern) or likewise for the parents. The entropy law is ignored and without entropy there will be no evolution. GA will also be unable to work properly without a suitable increase in entropy. I look forward to see a new Fisher-theorem considering also the entropy law.--Kjells 18:09, 6 August 2007 (UTC)

That is ok, but Wikipedia is not the place to develop new ideas or conduct additional research, and then publish instructional data. See - no original research, wikipedia is not a textbook 130.216.191.182 01:41, 8 August 2007 (UTC)
But, in order to avoid misunderstanding for readers who are not biologists by profession I suggest that at least some lines in the article should point to the fact that the Fisher theorem is not about the increase in mean fitness of the offspring from one generation to the offspring in the next (what the layman may think) but only from offspring to parents in the same generation.--Kjells 12:10, 8 August 2007 (UTC)


[edit] A closer look at Fisher's theorem

I have now been able to get a closer look on the proof of Fisher’s fundamental theorem.

Following Maynard Smith (see reference page 117), we may let the frequency of genotypes

  g = (g1, g2, …, gn) before selection be 
  p = (p1, p2, …, pn)

And their fitness

  w = (w1, w2, …, wn).

Then the mean fitness, W, becomes (summation is over the set of indices i)

  W = Σ piwi);         (1)

After selection has operated, the frequency_after_selection of gi becomes

  pi* = piwi/W;               (2)

and hence the mean fitness of of the selected parents W* is

  W* = Σ piwi2/W;

Hence the selection differential on fitness is

  S = W* – W = variance in fitness before selection / mean_fitness

Equation (2) is perfect as long as selection takes place on the level of individuals where w describes the fitness of the individuals. But Maynard Smith defines fitness as a property, not of an individual, but of a class of individuals - - (page 38). Suppose for instance that selection is triangular in a phenotypic 2-dimensional space (x,y) according to the figure below, and that the digits in x and y represent genotypes. The probability of selection for green points may be = 1 and for red points = 0. But this seems impossible if individuals have no fitness.

http://www.evolution-in-a-nutshell.se/select_triangle.gif

In order to secure the correctness, the fitness of genotypes must always - in accordance to equation (2) - be proportional to

  wi = W * frequency_after_selection / pi.       (3)

Philosophically and numerically this seems tricky because W requires pi and wi given beforehand according to (1), while wi requires frequency_after_selection and W given according to (3). I don’t know how to put it in English; Fisher’s theorem is formally correct, but the proof seems to go in a circle. What happens in the phenotypic space is not correctly dealt with.

On the other hand, Gaussian adaptation (GA) – using selection of individuals - gives the gradient (the steepest direction) of W with respect to changes in m (the centre of gravity of Gaussian distributed phenotypes). The variance or variability of interest here is represented by M; the moment matrix of the Gaussian.

 gradient_of_mean_fitness = dW/dm = W M-1(m* – m)

where M is the moment matrix of the Gaussian and m* is the centre of gravity of phenotypes after selection. A possible correspondence to Fisher’s increase in W is

  S = (dW/dx)*Δx + (dW/dy)Δy
    = W(m* - m)’M-1(m* – m)

In this case the increase in W is from the offspring in one generation to the offspring in the next. It is assumed that the mutation rate is adapted such that M is fairly constant from generation to generation.

Simulations gave the results:

   Selection_differential = 0.0116      using the GA-gradient at black m
                          = 0.0112      by moving the Gaussian to dark-green m_aft
                          = 0.1295      According to Fisher.

It seems to me that the GA-model gives about the same result if the gradient is used for estimation, or if the Gaussian is moved to m*. But the small figures are uncertain and mean results of a hundred simulations have been used.

Thus, Fishers model gives a seemingly fast increase in W because the mutation rate and the offspring in the next generation is not considered. GA shows a very slow increase in this case because the triangle is thin. But both evolution and GA may adapt M to become more proportional to M*, which may speed up the process considerably. Therefore, I think that GA may contribute (a little) to the theory of evolution.

When GA is used for optimization the gradient will suffice.

Reference: Maynard Smith, J. Evolutionary Genetics. Oxford University Press, 1998. --Kjells 09:11, 29 August 2007 (UTC) Changed --Kjells 17:34, 20 September 2007 (UTC) Changed --Kjells 15:09, 1 October 2007 (UTC) --Kjells (talk) 15:04, 15 January 2008 (UTC)

[edit] Creationists have reason to doubt the classical theoy of evolution

A discussion about Fisher's fundamental theorem has recently been held at ScienceBlog, where I have been encouraged to publish some new paper about it. Here is my blog:

Submitted by kjellstrom on Sat, 2008-01-12 03:04. bioscience and medicine

Creationists have reason to doubt the theory based on Fisher’s fundamental theorem of natural selection published in 1930. It relies on the assumption that a gene (allele) may have a fitness of its own being a unit of selection. Historically this way of thinking has also influenced our view of egoism as the most important force in evolution; see for instance Hamilton about kin selection, 1963, or Dawkins about the selfish gene, 1976 in http://en.wikipedia.org/wiki/Gaussian_adaptation#References

On the other hand, if the selection of individuals rules the enrichment of genes, then Gaussian adaptation will perhaps give a more reliable view of evolution (see the blog “Gaussian adaptation as a model of evolution”).

In modern terminology (see Wikipedia) Fisher’s theorem has been stated as: “The rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequencies is exactly equal to its genic variance in fitness at that time”. (A.W.F. Edwards, 1994).

A proof as given by Maynard Smith, 1998, shows the theorem to be formally correct. Its formal validity may even be extended to the mean fitness and variance of individual fitness or the fitness of digits in real numbers representing the quantitative traits.

But, if the selection of individuals rules the enrichment of genes, I am afraid there might be a risk that the theory becomes nonsense, and that this is not very well known among biologists.

A drawback is that it does not tell us the increase in mean fitness (see my blog “The definition of fitness of a DNA- or signal message”) from the offspring in one generation to the offspring in the next (which would be expected), but only from offspring to parents in the same generation. Another drawback is that the variance is a genic variance in fitness and not a variance in phenotypes. Therefore, the structure of a phenotypic landscape – which is of considerable importance to a possible increase in mean fitness - can’t be considered. So, it can’t tell us anything about what happens in phenotypic space.

The image shows two different cases (upper and lower) of individual selection, where the green points with fitness = 1 - between the two lines - will be selected, while the red points outside with fitness = 0 will not. The centre of gravity, m, of the offspring is heavy black and ditto of the parents and offspring in the new generation, m* (according to the Hardy-Weinberg law), is heavy red.

http://www.evolution-in-a-nutshell.se/image001.gif

Because the fraction of green feasible points is the same in both cases, Fisher’s theorem states that the increase in mean fitness is equal in both upper and lower case. But the phenotypic variance (not considered by Fisher) in the horizontal direction is larger in the lower case, causing m* to considerably move away from the point of intersection of the lines. Thus, if the lines are pushed towards each other (due to arms races between different species), the risk of getting stuck decreases. This represents a considerable increase in mean fitness (assuming phenotypic variances almost constant). Because this gives room for more phenotypic disorder/entropy/diversity, we may expect diversity to increase according to the entropy law, provided that the mutation is sufficiently high.

So, Fisher’s theorem, the Hardy-Weinberg law or the entropy law does not prove that evolution maximizes mean fitness. On the other hand, Gaussian adaptation obeying the Hardy-Weinberg and entropy laws may perhaps serve as a complement to the classical theory, because it states that evolution may maximize two important collective parameters, namely mean fitness and diversity in parallel (at least with respect to all Gaussian distributed quantitative traits). This may hopefully show that egoism is not the only important force driving evolution, because any trait beneficial to the collective may evolve by natural selection of individuals.

Gkm

http://www.scienceblog.com/cms/creationists-have-reason-doubt-classical-theory-evolution-15214.html

http://www.scienceblog.com/cms/blog/kjellstrom

--Kjells (talk) 13:00, 16 January 2008 (UTC)

Please don't add links to your own texts. As I understand it, talk pages aren't indexed by search engines, so the link here and on Talk:Gaussian adaptation won't increase the rating anyway.Sjö (talk) 17:24, 15 January 2008 (UTC)
Thanks for the tip. --Kjells (talk) 13:00, 16 January 2008 (UTC)