Dispatches from the Seminar: FTNS

Last Monday (29 January), my HPB seminar met to talk about Fisher's "Fundamental Theorem of Natural Selection." (Go here for the reading list.)

It's well known that Fisher's own statement of the FTNS, whether he made it in the 1930 or 1958 edition of The Genetical Theory of Natural Selection or in the 1941 paper, "Average Excess and Average Effect of a Gene Substitution", is virtually impenetrable. And so the immediate question to ask in a seminar is "What does the FTNS say?" The second question, and perhaps the more interesting one, is, "What's fundamental about the FTNS?"

The Price-Ewens interpretation is widely acknowledged to be the one that captures what Fisher wanted to say with the theorem (Ewens 1989, Price 1972). At the same time, probably the clearest statement of the theorem in words is Edwards' (1994). Edwards accepts the Price-Ewens derivation of the theorem, as he should it seems, but he words the theorem in a way that captures what Price and Ewens accomplished in more "Fisherian" terms. He says (Edwards 1994, p. 450):

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.

Compare this to Fisher's 1958 (p. 37) version of the theorem:

The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.

Edwards (1994, pp. 450-451) explains his statement of the FTNS as follows:

This [the statement of the theorem above] is as close as I can get to Fisher's wording whilst following the interpretation of Price and Ewens. "Genic" has replaced "genetic" [to capture that Fisher means 'additive genetic variance'] and 'mean fitness' has replaced 'fitness of any organism'; these are uncontroversial rewordings. ... I have added 'ascribable to natural selection acting through changes in gene frequencies' from Fisher's own words: 'ascribable to natural selection' is from his 1941 explanation of the Theorem and '[due to all] changes in gene frequencies' is from the sentence preceding his statement of the theorem [in 1930 and 1958] modified as suggested by Price. Price and Ewens repeatedly emphasize that Fisher thought  of the immediate effect of natural selection as being only through the changes in gene frequencies. For the sake of minimizing the changes I have kept Fisher's word 'organism', though it would not now be the first choice; in fact Fisher himself replaced it by 'species' in the Summary at the end of Chapter II of The Genetical Theory. 'Population' might now be the preferred word.

So let's say, then, that the FTNS states the following:

The rate of increase in the mean fitness of any population 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.

Let's also be clear about the theorem's assumptions: no mutation, fixed fitness values, no fertility differences, no random effects, no geographical structure, one sex only, and all the other usual assumptions except that there is no restriction on the mating structure (Ewens 1989,p. 167-168).

With all that said, Fisher's FTNS is true and exact (Edwards 1994, Ewens 1989, Price 1972). It seems to me that this is all quite uncontroversial and barely worth mentioning except for background to the more important question of the theorem's biological "fundamentality."

Out of Fisher's Genetical Theory, the papers by Edwards, Ewens, Price and a new philosophical paper by Plutynski (2006), who agrees with the Price-Ewens derivation of the theorem, our seminar listed several candidate answers to the question:

Fisher (1958, pp. 37-39)

For Fisher, the theorem makes a "deep" biological claim about the nature of selection, that is, that the partial change in mean fitness is due to single-locus gene frequency changes. Fisher also thought that the theorem's biological depth and more generally its scope afforded a comparison to the Second Law of Thermodynamics.

Price (1972, pp. 139-140)

Price thinks the main importance of the theorem for biology is its scope (it requires only statistical smoothing through large N and on assumptions of no meiotic and gametic selection). Price also thinks that the theorem is the best anyone had said (at least by 1972) about natural selection. At the same time, Price thinks the theorem isn't as important, biologically, as Fisher thought, due to what he (and others) consider(s) to be the defect of treating non-additive gene effects as "environment." He further thinks that the theorem is defective because it appears to treat mean population fitness as always increasing but generally close to zero.

Ewens (1989, p. 179)

Ewens agrees with Price's assessment of the biological importance of the theorem. He's a bit clearer, however: Ewens doesn't see any justification for singling out the partial change in mean fitness Fisher does as isolating natural selection from the total change in mean fitness. After all, all of the terms of the FTNS, mean fitness, average excess, and average effect all depend on gene frequency and all change with changes in gene frequency.

Edwards (1994, p. 469-470)

Edwards says "Fisher's Fundamental Theorem of Natural Selection is important for three reasons. (p. 469)" First, the theorem directly influenced Wright's construction of the adaptive topography. Second, he agrees, mostly, with Fisher's assessment of the theorem and thinks we can't have expected more out Fisher than he gave (contra Price and Ewens). Third, it's extendable.

Plutynski (2006, p. 75)

According to Plutynski, Fisher regarded his theorem as "so very fundamental" because it was "a culmination of Fisher's lifelong project to vindicate Darwinism and unify the biometrical gradualist model of evolution and Mendelism in a rigorous mathematical theorem analogous to the physical sciences."

My own view about these five candidate answers to the "fundamentality" question is this: Price and Ewens are right. And I think we've all known this for a long time. Their assessment just follows from the math (see especially Ewens 1989, p. 171). If Fisher had captured what he thought he did with the theorem, then I think the FTNS would be fundamental in evolutionary genetics in the same way that the Hardy-Weinberg Equilibrium Principle is --you can't do population genetics without it.

Fisher is incorrect about the "depth" of the theorem for precisely the reasons Price and Ewens give about the lack of justification for singling out the partial change in mean fitness as isolating natural selection from the total change in mean fitness. This means that Edward's third reason for the importance of the theorem is mistaken. I'm certain Edwards' first reason is wrong, that is, that Wright didn't get the idea of the adaptive topography from Fisher. This is easy to see just by reading Provine's (1986) biography. And I'm not terribly confident about the substance of Edwards' third reason, although it's more a question that will be answered by history (but see Frank and Slatkin 1992 and Lessard and Castilloux 1995 for extensions of the theorem to clutch size and to fertility selection respectively).

I admit I'm not sure I understand Plutynski's claim. (Neither did the seminar participants.) Actually, I think there are lots of problems with the paper. But focusing just on Plutynski's claim about the importance of the theorem, one wonders what she means when she asks, "Why did Fisher regard his theorem as so very 'fundamental'?" (2006, p. 75). Actually, I don't know if any answer to that question is much more than historical/philosophical speculation about what was going on in Fisher's head at the time. Anyway, it's not the right "fundamentality" question to ask. But I can't see in the paper where Plutynski clearly answers the question about the theorem's biological "fundamentality." She does agree with the Price-Ewens derivation of the theorem, but it's not clear she agrees with their assessment of its importance.

Ultimately, there's not much new historically or philosophically to say about Fisher's FTNS specifically. The more interesting question concerns the place of the theorem in Fisher's larger argument for neo-Darwinism. And, actually, this is where Plutynski's paper is quite relevant. I don't think focusing on the theorem as she does, though, is the right way to answer it. But what Plutynski does say about the structure of Fisher's argument strikes me as backwards. She gets it right when she says that the theorem culminates Fisher's neo-Darwinian argument, but that's small beer --the theorem came pretty much last in the effort, so of course it does. It's the details of Fisher's argument Plutynski gets wrong. This is a major research topic of mine at the moment, and you'll see some of it on this blog in due time. For now, see here and here; there are glimmers.

References

Edwards, A. W. F. (1994), "The Fundamental Theorem of Natural Selection", Biological Reviews of the Cambridge Philosophical Society 69: 443-474.

Ewens, W. (1989), "An Interpretation and Proof of the Fundamental Theorem of Natural Selection", Theoretical Population Biology 36: 167-180.

Fisher, R. A. (1930, 1958, 1999), The Genetical Theory of Natural Selection. Oxford: Oxford University Press.

Fisher, R. A. (1941), "Average Excess and Average Effect of a Gene Substitution", Annals of Eugenics 11: 53-63.

Frank, S. and M. Slatkin (1992), "Fisher's Fundamental Theorem of Natural Selection", TREE 7: 92-95.

Lessard, S. and A.-M. Castilloux (1995), "The Fundamental Theorem of Natural Selection in Ewens' Sense: Case of Fertility Selection. Genetics 141: 733-742.

Plutynski, A. (2006), "What was Fisher's Fundamental Theorem of Natural Selection and What was it For?", Studies in History and Philosophy of Biological and Biomedical Science 37: 59-82.

Price, G. (1972), "Fisher 'Fundamental Theorem' Made Clear", Annals of Human Genetics 36: 129-140.

Provine, W. (1986), Sewall Wright and Evolutionary Biology. Chicago: University of Chicago Press.

Comments

[Roberta Millstein]

Great post, Rob, thanks. It really put into words some of my inchoate intuitions about Fisher's "fundamental" theorem.

[David B]

I agree that Wright didn't get the idea of the adaptive landscape from Fisher, but didn't he often use Fisher's FTNS to prove that populations (usually) climb up the hills in the landscape?

[Rich Lawler]

In a really wonderful, brief overview of Fisher's work, here's what ALAN GRAFEN says,

"While preparing this paper, I have come to believe that Fisher was right in his beliefs about the importance of the theorem, and that the modern ‘completers’, to coin a term, are barking up the wrong tree or, at least, are barking up a non-Fisherian tree.

Fisher had a grand view of evolution, and we have already seen that his main interest in evolutionary biology, and I dare to say his main interest in genetics also, was providing an underpinning in genetical terms for Darwin’s ideas about natural selection. Indeed, the title of his book was The Genetical Theory of Natural Selection. Darwin’s natural selection informally was that fitter organisms left more offspring; their offspring resembled those parents, and so the population came to resemble the fittest forebears. Fisher was perfectly aware that population genetics was more complicated than that, and that mean fitness could, like the value of modern investments, go up or down. Come to think of it, as Fisher was writing in 1929, the value of investmentswas plummeting throughout thewesternworld. Where, in the complexities of population genetics models, could he find an expression of Darwin’s principle of natural selection? In my view, Fisher thought that his fundamental theorem isolated what we might call the adaptive engine of Darwinian natural selection.

Thus the partial nature of change is not an inability to find a stronger result. Fisher believed that this partial change was the only aspect of the changes in a population’s genetic constitution that was progressive, that could create design. Environmental change, of course, generally messes things up, presenting new challenges, and demanding new responses. Fisher explicitly lumps the segregation of genes into genotypes with environmental change as non-adaptive, as non-design-creating."

This is from a paper called "Fisher the Evolutionary Biologist" Journal of the Royal Statistical Society: Series D (The Statistican), 52, 319-329

[Robert Skipper]

Rich,

I know the Grafen paper; I'm not convinced by it. Ewens' argument in the 1989 paper is quite convincing, and if Grafen is to suggest otherwise, he owes us an argument to that effect.

[Robert Skipper]

David B.,

Phase II of Wright's shifting-balance process involves mass selection, but I don't think Wright would've said that the FTNS accounts for it. In part, his reasoning would be based on a misunderstanding of the FTNS. At the same time, for Wright, mass selection acts on complex genetic interaction systems, and it's not clear that, even using a proper understanding of the FTNS, Wright would think the theorem captured his sort of mass selection.

[Rich Lawler]

I can't speak for Grafen, but I believe his arguments are put forth verbally in the "Fisher" paper and mathematically in his 2002 paper, "First and formal link between Price equation and optimization" (among other places...see all his papers on Darwinian unification). Grafen suggests that Fisher's theorem *is* fundamental because, of all the mechanics of evolutionary change, Fisher found and focused on that component which changes non-randomly and has an implicit optimizing tendency: gene frequencies. Essentially, Fisher separated out "design-creating, non-random gene frequency changes" from "everything else." Wright, Price, Ewens and many others would suggest that "everything else" is important and leads his theorem to be less than fundamental. Fisher was concerned with adaptation (hence his focus on design/optimization--a theme Grafen is currently fleshing-out); Wright was concerned with selection in progress and the attendent complexities of how gene frequencies change, sometimes in non-optimal ways (hence his emphasis on Fst, inbreeding, etc.). It would seem that Fisher's stance is, "given that adaptations are found everywhere in nature, what is the most important thing we should focus on with respect to how selection produced a given adaptation?" Wright's stance is, "in thinking about the mechanics of evolutionary change, what are the important factors involved in the process of adaptive evolutionary change?"