Rejoinder to Basener and Sanford’s reply, part I

William Basener and John Sanford have responded here to my post concerning whether R.A. Fisher’s Fundamental Theorem of Natural Selection is critical to work on the theoretical population genetics of the interaction between mutation and natural selection. (This reply by Basener and Sanford is also reposted here.)

They had published a paper in Journal of Mathematical Biology in which they argued that the FTNS was the foundation of all subsequent work on the theoretical population genetics of natural selection, that the FTNS ignored mutations, and that it needed correcting. They then added terms for the effects of mutation to a limited version of the FTNS, one which was published by Crow and Kimura (1970). They reported numerical simulation results using that model, which showed natural selection to be unable to
prevent most deleterious mutations from fixing in the population.

My reply showed that the basic mathematical theory of the population genetics of deleterious mutations was not dependent on the Fundamental Theorem
of Natural Selection, but had been published before, by William Ernest Castle (1903), H.T.J. Norton (1915), R.A. Fisher (1922, 1929), J.B.S. Haldane (1927), and Sewall Wright (1928, 1929). The FTNS was an interesting result, but was not at all critical to subsequent work on mutations and selection.

In this rejoinder, I will deal with two issues: (1) Is the FTNS the foundation of the mathematical treatment of mutation and natural selection in populations, and (2) did R.A. Fisher draw from the FTNS the conclusion that natural selection led to ever-increasing mean fitnesses. I will show that the answer to both is “no”.

Basener and Sanford’s quotations

In their reply Basener and Sanford present a cornucopia of quotations from population geneticists and evolutionary biologists, myself included.  These quotations are presented to demonstrate the importance of R.A. Fisher to evolutionary biology, and the importance of his 1930 book. They are right about both of those.

But those are not the right questions.

Arlin Stolzfus, in a comment on that thread, summarized the situation admirably succinctly, saying in part:

The issue is not whether Fisher was influential, or Fisher’s 1930 book was influential, but whether FTNS is actually some sort of foundation. That is, are various key conclusions in evolutionary thinking dependent on FTNS, either directly or through other theoretical developments that depend on FTNS?

Rather than addressing that question directly, Basener and Sanford try to establish a positive answer by way of quotations. But the quotations don’t say the right things, and they are not quoting the right people. Nearly all of the quotations are about Fisher or his 1930 book or his 1920s work, without specifying FTNS.

I cited, in my earlier post, the substantial papers that established the theory of how natural selection affects the gene frequencies of deleterious mutant alleles.

Let’s check this by looking at the presentation of the mathematics of mutation-versus-selection in some of the leading theoretical population genetics texts. I have looked at major texts by C. C. Li (1954), Douglas Falconer (1960), J. F. Crow and M. Kimura (1970), L. L. Cavalli-Sforza and W. F. Bodmer (1971), and Warren Ewens (1969, 1981, 2006). Many do not make specific citations as to who did the foundational work on the mathematics of mutation and selection. Those who do cite anyone, cite the papers of Haldane (1927 onward) and C. H. Danforth (1923). Of course, being published before Fisher’s 1930 book, neither of these papers cited the Fundamental Theorem of Natural Selection as the foundation of theoretical work on the genetics of natural selection.

Basener has a response to these citations. In a comment (herehere) in the discussion after their post, he says that

I think Joe F misread our paper when they say we argue that the FTNS “is the basis for all subsequent theory in population genetics”. If anything in our paper seemed to say that, it was unintentional.

Well, OK, he’s right about one point. Saying that it “is the basis for all subsequent theory” in population genetics is not what they did. What Basener and Sanford actually argued in their paper was that the FTNS is the basis for all subsequent theory involving natural selection.

What is the evidence that they said that? Could it be that I am misreading statements, or that they are unintentional? Well, here are some quotes from their Journal of Mathematical Biology paper:

His book, The Genetical Theory of Natural Selection, established for the first time the connection between genetics and natural selection. Within that pioneering book, Fisher presented his famous fundamental theorem of natural selection.

and

At the heart of Fisher’s conception was his famous fundamental theorem of natural selection (Fisher’s Theorem).

and

His fundamental theorem of natural selection was an enormous step forward, in that for the first time he linked natural selection with Mendelian genetics, which paved the way for the development of the field of population genetics.

and that in his 1930 book

Fisher conceptually linked natural selection with Mendelian genetics, which had not been done up to that time.

Was it a misreading for me to interpret those as claims that the theoretical population genetics of natural selection had not been done prior to Fisher’s 1930 book? Can a claim repeated that insistently be “unintentional”?

These statements were also interpreted by
David Coppege, at his Creation Evolution Headlines site where Coppege says that

Fisher was the first to reconcile the apparent conflict between the ideas of Darwin and the experimental observations of Mendel. Fisher accomplished this by showing mathematically how natural selection could improve fitness by selecting for desirable genetic units (beneficial alleles), and simultaneously selecting against undesirable genetic units (deleterious alleles).

Thus if I misread Basener and Sanford’s statements, so did Coppege.

The Role of Fisher’s Fundamental Theorem in Basener and Sanford’s simulations

An interesting contradiction in Basener and Sanford’s paper is they provide computer simulations involving the interaction of mutation and natural selection. For the processes that involve natural selection they present equations of change of haploid genotype frequencies. In deriving them, they do not find it necessary to use the
Fundamental Theorem of Natural Selection. Instead they use equations from Crow and Kimura’s 1970 book for the change of genotype frequencies, with some modifications from a review article by Claus Wilke (2005).  None of those equations are derived
from Fisher’s Fundamental Theorem. If it was the Fundamental Theorem that first
brought Mendelian genetics and natural selection together, one would think that
we could trace the equations for change in genotype frequencies back to
the FTNS.

The mathematics of natural selection in Mendelian populations, even without
taking mutation into account, preceded Fisher’s 1930 book, involving Castle’s 1903 paper, Fisher’s 1922 paper, Danforth’s 1923 paper, and a series of papers by J.B.S. Haldane starting in 1924 that covered many cases.

Fisher’s general conclusion?

The second issue Basener and Sanford raise is whether Fisher drew from his Fundamental Theorem the conclusion that natural selection would always (or almost always) increase mean fitnesses. Basener and Sanford argue that he did.

For example, they said that

Despite the limitations in Fisher’s theorem, Point (A) above (that natural selection can result in an optimization process of allele frequencies) is widely accepted. Thus, while his methods to compute fitness from the genetic level have not become universally accepted, his general conclusion concerning Point (A) has been accepted.

and that

In terms of Fisher’s primary thesis, we cannot overstate the essential role of new mutations and their fitness effects. Fisher’s theorem by itself actually shows that, apart from new mutations, a population can only optimize the frequencies of the pre-existing alleles, followed by stasis. Yet Fisher argued forcefully that his theorem was so fundamental in its nature, that it essentially guaranteed that any population would increase in fitness without limit (essentially constituting a mathematical proof that Darwinian evolution is inevitable).

and they conclude that

Our analysis shows that Fisher’s primary thesis (universal and continuous fitness increase) is not correct.

 

This interpretation has also been taken up by creationist and ID blogs such as the Discovery Institute’s Evolution News and Science Progress site (here) by Denyse O’Leary (“News”) at Uncommon Descent (herehereherehere, here, here, and here), by David Coppege (here), at the World Magazine blog (here), and in a lengthy Youtube video by Paul Giem.  All echo Basener and Sanford’s interpretation of Fisher
as believing that increase of mean fitness was guaranteed.

Looking at Fisher’s Fundamental Theorem more closely

That Fisher did not interpret his theorem in that way can be seen if we look at it carefully. In Fisher’s 1930 book, his Fundamental Theorem is given as this formula

    \[\frac{dM}{dt} + \frac{M}{C} \ = \ W \ - \ D\]

which by moving one term from the left to the right side becomes of course

    \[\frac{dM}{dt} \ = \ W \ - \ D \ - \ \frac{M}{C}\]

This writes the rate of change of the mean fitness M at the instant t in time, dM/dt, as equal to the additive genetic variance of fitnesses, W minus two other terms. Plain-language accounts of FTNS often omit the decrease in fitness due to environmental change, -D, and also the reduction of fitnesses by density-dependent effects when mean fitness increases, -M/C.  The latter happens as a population approaches its carrying capacity, and fitnesses are reduced by competition and lack of resources.

There is no general reason why the second and third terms on the right-hand side could not make the net change in fitness negative. It depends on such factors as how quickly the environment is changing and how strong a negative effect those changes have on fitness. That in turn depends on details of the biological situation. Any assertion that Fisher thought he had proven that mean fitness always increases is in obvious contradiction with the equations he gave.

In fact, Fisher commented on this in the chapter in which the Fundamental Theorem of Natural Selection is discussed, concluding in the Summary of that chapter that we do not expect the Malthusian parameter (the fitness expressed as the growth rate) to exceed zero in the long term:

Any net advantage gained by an organism will be conserved in the form of an increase in population, rather than an increase in the average Malthusian parameter, which is kept by this adjustment always near zero.

Thus Basener and Sanford’s reply to my post misses the mark on both points.

I trust that Basener and Sanford will now go around posting or commenting at the creationist and ID sites, explaining to the these creationists and ID advocates that they too have misread an unintentional statement.

The third point

I have not dealt here with the third major conclusion in Basener and Sanford’s paper, which was that, under realistic conditions, natural selection would not prevent a rapid buildup of deleterious mutations. Michael Lynch and I argued in a second post at TSZ that their simulations showed this result because they ignored genetic recombination and the independence of fitness effects at different loci. They have posted a second reply, arguing that we ignored the finiteness of the population. In a rejoinder to this second reply, I hope (with a co-author or co-authors) to re-emphasize the validity of the reply here by Michael Lynch and I, showing by redoing Basener and Sanford’s simulations that when a more realistic distribution of fitnesses of haploid genomes is used, that natural selection is much more able to resist a rapid decline in fitness when deleterious mutations occur at many loci.

 

17 thoughts on “Rejoinder to Basener and Sanford’s reply, part I

  1. From the OP:

    Was it a misreading for me to interpret those as claims that the theoretical population genetics of natural selection had not been done prior to Fisher’s 1930 book?

    That wasn’t the question.

    Here is the question you were supposed to be answering in your quotes from their paper:

    What is the evidence that they said that? Could it be that I am misreading statements, or that they are unintentional? Well, here are some quotes from their Journal of Mathematical Biology paper:

    The quotes are supposed to establish that they claimed:

    … that the FTNS is the basis for all subsequent theory involving natural selection.

    Not that there was no work in population genetics done prior to Fisher’s book.

    So much ink spilled over nothing. Back to the drawing board Joe.

    Would you like me to proofread your OPs before you post them?

  2. Joe,

    I e-mailed Bill, John and Dave and alerted them to this OP. I think I can say this on their behalf, that they are appreciative of your willingness to write about their paper even though you may have sharp disagreements about what it says. Thank you for the dialogue.

  3. Mung: Not that there was no work in population genetics done prior to Fisher’s book.

    Joe acknowledged that they weren’t making quite that strong a statement, and went on to qualify.

    OP: Well, OK, he’s right about one point. Saying that it “is the basis for all subsequent theory” in population genetics is not what they did.

  4. Mung,

    You do more reading than most folks here. You really should take a look at Chapter 2, “The Fundamental Theorem of Natural Selection,” in Fisher’s (1930) The Genetical Theory of Natural Selection. Pay close attention to pages 42-44 (as numbered in the book), the context of Fisher’s model that Joe summarized quickly in the OP:

    An increase in numbers of any organism will impair its environment in a manner analogous to, and probably more definitely than, an increase in the numbers or efficiency of its competitors. It is a patent oversimplification to assert that the environment determines the numbers of each sort of organism which it will support. The numbers must indeed be determined by the elastic quality of the resistance offered to increase in numbers, so that life is made somewhat harder to each individual when the population is larger, and easier when the population is smaller. The balance left over when from the rate of increase in the mean value of m produced by Natural Selection, is deducted the rate of decrease due to deterioration in environment, results not in an increase in the average value of [fitness, i.e., Malthusian growth rate] m, for this average value cannot greatly exceed zero, but principally in a steady increase in population.

    Fisher has just said that the mean Malthusian growth rate (fitness expressed on a logarithmic scale, with 0 corresponding to a multiplicative factor of 1) cannot increase indefinitely. Instead, the mean fitness maxes out at a value not much greater than 0, and then the population size increases. Continuing…

    The situation is represented by the differential equation [given by Joe in the OP]

    [EQUATION]

    in which M is the mean of the Malthusian parameter, C is a constant expressing the relation between fitness and population increase, and defined as the increase in the natural logarithm of the population, supposed stationary at each stage, produced by unit increase in the value of M, W is the rate of actual increase in fitness determined by natural selection, and D is the rate of loss due to the deterioration of the environment. If C, W and D are constant the equation has the solution

    [EQUATION]

    in which A is an arbitrary constant, dependent upon the initial conditions. C has the physical dimensions of time, and may therefore be reckoned in years or generations, and the equation shows that if C, W, and D remain constant for any length of time much greater than C, the value of M [the average fitness, i.e., the mean value of the Malthusian growth rate] will approach to the constant value given by

    [EQUATION]

    In this steady state the whole of the organism’s advantage or disadvantage will be compensated by change in population, and not at all by change in the value of M [the average fitness, i.e., mean value of the Malthusian growth rate].

    Again, Fisher says that change in the average Malthusian growth rate (fitness) ceases, though change in the population size continues.

    Now tell me that what Basener and Sanford dub “Fisher’s Corollary 1” is something other than a preposterous fabrication:

    Fisher’s Corollary 1

    Fisher’s fundamental theorem, plus a steady supply of new mutations, necessarily results in unbounded fitness increase, as mutations continuously replenish variance, and as selection continuously turns that variance into increased fitness.

    The term “corollary” is justified here because Fisher believed that if Fisher’s fundamental theorem is true, then the corollary is true as a necessary logical consequence. Fisher never derived his corollary mathematically. Moreover, most modern evaluations of Fisher’s theorem focus on the theorem itself and do not address the role of mutations.

    I’d also like to know what you think of Basener and Sanford’s decision to bury under 2500 words of introductory text the revelation that it was not Fisher’s model that they had modified, but instead a simplified model due to Crow and Kimura:

    In order to understand Fisher’s theorem in light of newly arising mutations, we need to reformulate the original theorem to allow for incoming new mutations. Instead of building the model up from the genetic allele level, we consider the resulting fitness to be equal to the Malthusian growth rate of the population in its environment, such that a “special example” (Crow and Kimura 1970, p. 10) of Fisher’s theorem can be proven. This new version of the theorem includes an objective metric of fitness which allows for dynamic modeling of the mutation–selection process over time. In this special case, we are exchanging Fisher’s derivation of the theorem based upon pre-existing Mendelian alleles for a new derivation that has the ability to quantify fitness with an objective metric that can be applied to a changing population. The statement of Fisher’s fundamental theorem becomes “the rate of change of fitness at any instant, measured in Malthusian parameters, is equal to the variance in fitness at that time” (Crow and Kimura 1970, p.10).

    To put it straight, Basener and Sanford pin on Fisher a model that leaves out the negative terms that put the brakes on the change in average fitness. And they assiduously avoided quoting the parts of his book that make it clear how very wrong they are to do that.

    Really, Mung, you don’t want to spend your goodwill defending this patent BS (pun intended). I’ve not commented on it previously, because I thought I wasn’t qualified. Turns out, it is incredibly easy to see that Basener and Sanford are ignoring what Fisher wrote, and making up an “icon of evolution” to bash. All you need to do is to read a few pages of Fisher. Oh, yeah — I hope you’ve actually read the article.

  5. Mung:
    Maybe Joe could edit his most recent OP to include links to the previous posts.

    There’s a ton of links in my post, including to Basener and Sanford’s paper, to my posts on to it, to their replies, to a free online copy of Fisher’s 1930 book, to the Evolution News trumpeting of the Basener and Sanford paper, to Denyse O’Leary’s every post on it and on responses to it, to a number of creationist blog comments on it, and of course to Arlin Stolzfus’s comment which was a concise demolition of Basener and Sanford’s quote-filled reply. It was a lot of work getting all those in the right form for WordPress to be happy with them.

    So no, I won’t. Read my post and look for the parts where the text is highlighted, indicating a link. And do so before commenting with a suggestion to put in links.

  6. Alan Fox:
    Stickied the post. I’m wondering which OPs still need a sticky. Joe?

    Maybe the most recent one from Basener and Sanford, and the most recent reply by our side.

    Having 4-6 stickied posts is too many, it takes people too long to get past them to see the most recent posts of other sorts. I have tried to put in my post lots of links to the previous ones in the debates (and to commentary on this in other blogs).

  7. From the OP:

    What Basener and Sanford actually argued in their paper was that the FTNS is the basis for all subsequent theory involving natural selection.

    None of the quotes Joe provided even address this point.

    Instead, they address something he brings up after the quotes:

    Was it a misreading for me to interpret those as claims that the theoretical population genetics of natural selection had not been done prior to Fisher’s 1930 book?

    He raises a question [was the FTNS the basis for all subsequent theory involving natural selection] but then answers a completely different one [was Fisher’s work the first to connect mendelian genetics and natural selection in a theoretical way].

    All of his quotes address the latter question but not the original question he raised.

    Pardon me for noticing. 🙂

    In all that mess the really interesting questions get glossed over:

    Qouting Coppedge:

    Fisher was the first to reconcile the apparent conflict between the ideas of Darwin and the experimental observations of Mendel.

    Was there a conflict between Mendelism and Darwinism.

    Did Fisher resolve that conflict and was he the first to do so.

    How.

    If the answer is yes then I think the authors have a point about subsequent population genetics theory involving natural selection.

  8. Tom English: You do more reading than most folks here. You really should take a look at Chapter 2, “The Fundamental Theorem of Natural Selection,” in Fisher’s (1930) The Genetical Theory of Natural Selection.

    I will do that.

    But from your quotes he appears to be talking about RATE of increase, not whether there is an increase.

  9. From the OP:

    An interesting contradiction in Basener and Sanford’s paper is they provide computer simulations involving the interaction of mutation and natural selection. For the processes that involve natural selection they present equations of change of haploid genotype frequencies. In deriving them, they do not find it necessary to use the Fundamental Theorem of Natural Selection. Instead they use equations from Crow and Kimura’s 1970 book for the change of genotype frequencies, with some modifications from a review article by Claus Wilke (2005).

    It’s not a contradiction if they actually argued in their paper that the FTNS is the basis for all subsequent theory involving natural selection.

  10. Mung: But from your quotes he appears to be talking about RATE of increase, not whether there is an increase.

    Here’s the clearest passage:

    In this steady state the whole of the organism’s advantage or disadvantage will be compensated by change in population, and not at all by change in the value of M [the average fitness, i.e., mean value of the Malthusian growth rate].

    He’s saying, flat out, that the mean fitness does not change. I have to do stuff in the real world for a while, so I’m handing this over to Joe.

  11. Here’s the quote I like:

    …it appears impossible to conceive that the detailed action of Natural Selection could ever be brought completely within human knowledge…

  12. Mung:

    Was there a conflict between Mendelism and Darwinism.

    No there wasn’t a conflict, as it turns out.

    Did Fisher resolve that conflict and was he the first to do so.

    How.

    If the answer is yes then I think the authors have a point about subsequent population genetics theory involving natural selection.

    The answer is, however, no, the material in Fisher’s 1930 book was not the first to show that there was no conflict. And in particular, the FTNS material in that book did not show that there was no conflict.

    The population geneticists of the 1910s and 1920s (including Fisher) had already developed the relevant theory of how natural selection affected gene frequencies, and what was the balance between deleterious mutation and the natural selection against those mutants.

    This is all discussed in my earlier Reply to Basener and Sanford, part I, to which I have linked in the present OP.

  13. Mung:
    Here’s the quote I like:

    I agree, it’s similar to saying that it’d be impossible to know the exact trajectory of every molecule of water in a river. We can still say the water is flowing downhill though.

    In a similar way, it’s impossible to know how every single relevant physical event actively affects the survival and reproduction of all members of a whole species. But we can still make true and sensible general statements about how natural selection affects that population without knowing literally every possible detail. We can make sense of the becoming white of the ancestors of polar bear species, and the being and staying white of polar bears in the arctic.

    Sensible, logical, unproblematic.

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