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.
At the heart of Fisher’s conception was his famous fundamental theorem of natural selection (Fisher’s Theorem).
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 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.
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 (here, here, here, here, 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
which by moving one term from the left to the right side becomes of course
This writes the rate of change of the mean fitness at the instant in time, , as equal to the additive genetic variance of fitnesses, minus two other terms. Plain-language accounts of FTNS often omit the decrease in fitness due to environmental change, and also the reduction of fitnesses by density-dependent effects when mean fitness increases, . 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.