I [We] shall call the
quantity sq/(1-sq), summed over all the generations involved in a gene
substitution, the cost of a gene substitution. Designating this by C, we
have 
(Crow, 1968 and 1970, italics as in the originals)
Back to discussion on Haldane's Dilemma
My paper, "Cost Theory and the Cost of Substitution – a clarification," was submitted to the journal Theoretical Population Biology, where it was peer-reviewed by James Crow, Warren Ewens, and two additional reviewers. The peer-review process went terribly awry, and warrants a public airing (see here). Dr. Crow has responded by posting some comments.1 (Hereafter, Crow's post.)
Crow's post acknowledges my paper is "essentially correct and made some interesting points", but nonetheless Crow and Ewens rejected it from publication on the grounds that my paper is not sufficiently new or different from what was known by them and some of their colleagues in the 1970s. In particular, Crow cites his 1968 and 1970 papers2 as taking priority over mine. I here show his defense is inadequate.
Crow's post attempts to justify his many decades of negligence concerning the cost of substitution.
(All text within boxes are from Crow's post.)
Since my name has been brought up repeatedly by Walter ReMine and others, mostly in connection with my early articles, I would like to take this chance to restate what I wrote and thought in the late 1960s. My, that was a long time ago. I'll answer as best I can the comments and criticisms from Walter ReMine in his note posted 01-25-2005, some of which I agree with. (Crow's post)
Crow acknowledges my paper is “essentially correct.”4 So the issue at hand is whether my paper is a new, needed, clarification beyond the previous literature. What Crow "thought" in the late 1960s is irrelevant here. If he had something in his head, but did not publish, then it's irrelevant. Likewise he cannot "restate" or re-word what he wrote in the 1960s. Any new assertions in his post are irrelevant. What counts is what he actually published previously, and whether it was correct, clear, and unambiguous, because on that basis he rejected my paper.
I claim there are many confusion factors in the literature that remain to this day, and my paper makes a clean clarification of it, as never before. My cost concept is worthy, not just for what it is (it is clear, practical, general-purpose, and applies to any model of any evolutionary scenario), but also for what it is not (it is not the various confusion factors). No other author did those things.
The confusions remain overwhelming today, even for researchers who have spent years studying the cost literature. The confusions are in the literature. The confusions on this thread at ARN. The confusion is in our inability to rationalize genetic death, particularly in the cases of greatest interest (diploids and multiple concurrent substitutions). The confusion is in the contradictions that remain unresolved, and not even explicitly acknowledged, such as claims by Brues, Feller, Felsenstein, Hoyle, or Moran. The confusion is in the extremely common notion (originated by Wallace) that soft selection makes the cost of substitution “disappear.” My paper shows all those are false. Confusions, contradictions, and errors abound in the literature, on the Internet, and at the highest levels of peer review, as I will show. I took pains to document that point for Ewens and Crow during the review process. (see more examples here.) Their rejection of my paper amounts to a de facto preference for the existing confusion, a move I consider profoundly negligent.
To the present moment, Ewens and Crow scarcely acknowledge the confusion. Yet it is overwhelming, and they, as much as anyone, are responsible for it. The proof of the pudding is in the tasting. If they believe the cost literature was clear in the 1970s, then I invite them to explain the current state of confusion. It is overwhelming.
At a highly respected genetics journal, Heredity, Ranjan Chaudhuri (peer-reviewer, from the Institute of Genetics), and John Brookfield (editor) both responded that my cost concept is "different from its previous usage in population genetics". Brookfield wrote, "I really believe that defining the cost of a substitution [in the way that you do] makes this concept inconsistent with previous uses of the concept of a cost of selection". These reviewers recognize my cost concept is different, which contradicts Ewens and Crow, and demonstrates that the prevailing confusions rise to the highest level of peer-review at major genetics journals. The cost concept that Ewens and Crow claim was widely known in the 1970s, is not even recognized by other peer-reviewers.
In fact, my cost concept is so different that Chaudhuri and Brookfield initially opposed it, both of them indicating "it is not interesting that the spread of an allele requires a reproductive excess". The field needs clarification, and Ewens/Crow are wrong to reject it.
At the journal, Theoretical Population Biology, Ewens and Crow were contradicted by their colleague and co-reviewer, reviewer #4 who wrote:
"I strongly recommend this paper be published. I believe it will revitalize discussion/investigation within an area of population biology which has otherwise become bogged down and neglected. I believe the conceptual framework of 'genetic load' has generally led to an unfruitful morass. ReMine offers a fresh perspective on this old problem. Instead of understanding cost/load in terms of "genetic deaths", ReMine forcefully argues that we should understand this issue entirely in terms of required reproductive excess. He shows this not only clarifies the whole problem conceptually, but allows much cleaner and more generalized computations of cost - in a way that is very clearly connected to the real world. This paper has significantly impacted my own understanding of the problem of substitution cost. Even those who may take exception to ReMine's general approach, should benefit from the resulting stimulation of dialog." (Reviewer #4 at Theoretical Population Biology)
At the Journal of Theoretical Biology (John Tyson, editor), reviewer #2 said my paper is "not appropriate for JTB because it aims at a parochial audience of population geneticists" That contradicts Crow, who rejected my paper from an audience of population geneticists, remarking that "It might well be considered as a semi-popular review, but not in a journal of theoretical population genetics." (Crow, October 2004, from his original peer-review comments) Crow rejected my paper under the notion (the falsehood) that the field does not need clarification, rather, all that's needed is a "semi-popular review" for the unwashed masses. Crow is in denial concerning the depth of confusion today. The confusions are thoroughly engrained in the field, and therefore ought be clarified directly within a population genetics journal, such as Theoretical Population Biology.
Next we examine Crow’s two papers from 1968 and 1970.2 This is tedious, but necessary, since Crow claims it as his defense.
I'll answer as best I can the comments and criticisms from Walter ReMine ..., some of which I agree with. But there are major differences. Most of these have to do with the verbal description of a concept that is essentially mathematical. (Crow's post)
Crow's post says our differences have to do with the "verbal description of a concept that is essentially mathematical." That is an odd defense, and we are forced to take him literally.5 What does he mean, the concept is "essentially mathematical"? In Crow 1968 & 1970, he gives some introductory discussion, and then comes the precise moment where he explicitly and formally defines cost:
I [We] shall call the quantity sq/(1-sq), summed over all the generations involved in a gene substitution, the cost of a gene substitution. Designating this by C, we have
In other words, his cost definition is an equation, in search of a physical meaning.
But his equation is not general-purpose; rather his equation is solely for single substitutions, in haploids, with a constant population size. That requires him to posit a different definition (i.e. a different equation) for every situation in nature. Whereas my cost concept is defined once (physically), and it applies consistently and uniformly to any model of any evolutionary scenario. That makes my definition different from his. QED
Surrounding his cost definition (his equation), Crow's papers spend several pages repeatedly equating cost with genetic death (i.e. elimination of the unfit individuals).
Crow 1968 begins with his account of Haldane’s cost concept:
Concentrating on a particular locus, Haldane showed that the total number of selective deaths (or the equivalent in lowered fertility) required for this process depends mainly on p0, the initial frequency of the gene that subsequently is favored by natural selection. If the cost, for example were ten this would mean that during the process of gene substitution the number of selective deaths is ten times the population number in a single generation. (Crow, 1968, p 167)
Then Crow 1968 continues with his own uses of genetic death:
Relative to the [favored] A genotype, a fraction s of the [unfavored] A' genotype will fail to survive or reproduction will be lower by an equivalent amount. In the whole population the ratio of individuals (or genes) eliminated (i.e., not represented in the next generation because of preadult death or differential fertility) to those not eliminated will be sq0/(1 – sq0). (Crow, 1968, p 168)
In that last sentence Crow explicitly equates his general-purpose cost equation with a concept of genetic death (as a ratio of individuals eliminated to those not eliminated).
In this population, if sq0 = .2 and therefore 1 – sq0 = .8, the ratio of eliminated to non-eliminated is .2/.8 or .25. (Crow, 1968, p 168)
(Haldane simply summed sq, which therefore measures the ratio of eliminations to the number of zygotes rather than to the number of adults.) (Crow, 1968, p 168)
Thus, if Na and Nb are roughly constant, C [the cost of substitution] measures the ratio of the total number of eliminations throughout the process to the number of adults in a single generation. (Crow, 1968, p 169)
For example, if s = .1, the total cost is 4.8; that is to say, that the total number of “genetic deaths” during the gene substitution is about five times the number of adults in a single generation. (Crow, 1968, p 169)
Most of the cost is during the early generations of the gene substitution while the favorable gene is rare and the ratio of eliminations to non-eliminations is high. (Crow, 1968, p 170)
Likewise Crow 1970 repeatedly uses genetic death:
Relative to the A genotype, a fraction s of the A' genotype will fail to survive or their reproduction will be lowered by an equivalent amount. In the whole population the ratio of individuals (or genes) not transmitted to those that are transmitted will be [equation = sq0/(1 – sq0). (Crow, 1970, p 245) [Note: In that sentence he is explicitly connecting the concept of "eliminations" (i.e. genetic death) to his cost equation.]
In this population, if sq = .2 and 1 – sq = .8, the ratio of eliminated to noneliminated is .2 : .8 or .25. (Crow, 1970, p 246)
For example, if s = .1, the total cost is 4.8 nonsurvivors per survivor, or in terms of numbers of individuals the total number of extinctions is 4.8 times the number of survivors in a single generation. (Crow, 1970, p 246)
These numbers are all quite large. This means that, if the typical allele has an initial frequency of 10-4, a population of 1 million individuals will have to have 9 million genetic deaths each generation if it is to substitute an average of one allele per generation. (Crow, 1970, p 248)
Thus, in a diploid system, the cost of substituting a moderately rare mutant (e.g. one previously maintained by a mutation rate of 10-5 to 10-4 is from 10 to 100. That is to say, the total number of eliminated individuals is 10 to 100 times the number of adults in any single generation. Haldane suggests 30 as a representative number. (Crow, 1970, p 249)
Next is an especially relevant quote, for it explicitly gives Crow’s cost definition, and explicitly shows how it does, and does not, differ from Haldane’s cost definition. His definition twice uses Haldane’s concept of genetic death: Once in the ratio’s numerator (sq), and again in the denominator (1-sq).
Our formulation differs from Haldane’s in giving the cost as the ratio of zygotes eliminated by differential viability and fertility to those not eliminated. Haldane gave it as the ratio of those eliminated to all zygotes. When s is small these are practically equivalent, so he has the same equations. However, our formulation gives a more exact representation for larger values; and perhaps the meaning is more transparent. (Crow, 1970, p 249)
Crow based his cost concept on genetic death, which remains a source of confusion to this day.
Genetic Load involves fitness differences. If you speak in terms of differences in fitness (or percentage difference in fitness), then you are using a genetic load concept, rather than genetic death or required reproductive excess. (Note: In the equations below, fitness is denoted by the variable w.)
Crow 1968 equates Cost and Load in the following places:
Implicitly in his equation 5:
Explicitly in his equation 8:
Explicitly in his equation 11:
Crow 1970 equates Cost and Load in the following places:
Implicitly in his second equation on page 246:
Implicitly in his equation 5.12.6:
Explicitly in his equation 5.12.8:
And in the following sentences:
Or more probably, if there is to be a gene substitution every 100 generations, the average fitness will be lowered by .09. (Crow, 1970, p 248)
Notice the dependence of the total load on the initial frequency. Most of the cost is in the early generations when the favored gene is still rare. (Crow, 1970, p 248)
The load becomes greater (and the algebra more difficult) when a diploid species is considered. (Crow, 1970, p 248)
We have no way, from this measurement alone, to determine whether or not Haldane’s 10% is a reasonable value of the differential fitness [a.k.a. load] that can be applied to evolutionary change. (Crow, 1970, p 250)
Since 1968 I have preferred "cost" rather than "load" for the same reason as ReMine does. .... I might add that Kimura preferred the word load; this is one of the very few places where we disagreed. …. In this [1970] paper we used Kimura's preferred vocabulary [“load”]. As I said above, I preferred cost rather than load. …. Since 1968 I have preferred to use cost when dealing with population change (dynamics) and to use load for equilibrium situations (statics). (Crow's post)
It is irrelevant here what Crow "preferred" or that his "reasons" are the same as mine, or that he and Kimura "disagreed", or anything else in his head. The only relevant thing is what Crow actually published, and whether it was correct, clear, and unambiguous.
Equating cost and load (and using the two as though they were the same thing) has been a continual source of confusion. Crow did that, and never corrected it. He often equated cost and load in his 1968 and 1970 papers, and his other papers never renounced it. Crow's co-author, Kimura elevated "load" to a predominant way of discussing the cost of substitution, and it remains a confusion factor to this day. At some point Crow may have "preferred" – in his head – the term "cost" rather than "load", but nonetheless he did not publish a clarification. He allowed the confusion to continue. My paper separates the two concepts and explicitly jettisons genetic load as a source of confusion. Crow didn't do that.
ReMine wrote: "My cost concept applies in ANY circumstances (such as non-constant population size; Crow's doesn't – his applies only under a constant population size."
See the formula near the top of page 246 in the 1970 book. wA differs according to whether the population is increasing or decreasing, and need not be constant. Since I was interested in relative rather than absolute values, I let the wAs in the numerator and denominator cancel in the next formula, 5.12.1. (Crow's post)
He is referring to this formula: 
,
which is
a genetic load equation. For example, if there are no fitness differences
and the population size increases, then that equation is zero, but the increase obviously requires
reproductive excess. If we now allow fitness differences too, then the equation
still underestimates the required reproductive excess. In other words,
there is a disconnect between that equation (which is a Load equation) and the
concept of 'required reproductive excess.' The disconnect is both conceptual
and quantitative. That equation does not
properly handle changes in population size.
Moreover, that equation is irrelevant, because it is not how Crow defined the cost of substitution – which occurs after the above formula, at equation 5.12.1 and which is defined only for constant population size.
Crow's cost concept cannot handle changes in population size. Mine can.
ReMine wrote: "My cost concept applies to ANY model (including continuous generation models); Crow's doesn't – his is defined only under discrete-generation models."
I assumed that a discrete model could be approximated by a continuous model and replaced summation by integration. There are several ways to consider a continuous, age-structured population (by using Malthusian Parameters, for example), but for slow selection they all would give about the same result, as I believe ReMine said. (Crow's post)
Notice his requirement for “slow selection” (s<<1). As shown in my paper, genetic death and genetic load give inaccurate results in continuous generation models when the substitution is not slow. (Hint: It has to do with an unavoidable reproductive 'time delay' between birth and parenthood. Since neither genetic death nor genetic load speak anything about a 'time delay', they lead to increasing inaccuracy as the substitution speed increases.) This is another error wrought by the concepts of genetic death and genetic load.
Crow's cost concept leads to erroneous handling of continuous generation models. They cannot be handled correctly until you abandon genetic death and genetic load, which Crow’s papers never did. My cost concept identifies that error, and corrects it.
Crow states:
"If the selection is acting on fertility there must be enough variability in progeny number so that
, which is equal to
Crow's focus on "enough variability in progeny number" is another needless confusion factor that lingers today. Correctly understood, the cost of substitution is not about "variability," but something else entirely. (And clarified in my paper.)
Crow's post claims:
In the 1970 book, the basic equations 5.12.1 for haploids and 5.12.7 for diploids are in terms of reproductive excess and do not involve any concept of genetic load.
Crow's post mis-states what his papers actually did do (and didn't do). As shown further above, the derivations for his cost equations always began with (and therefore ended with) calculations of genetic death and/or genetic load. His equations were not derived based on reproductive requirements, and he never supplied a physical rational for translating (or converting) from his derivations into reproductive requirements. Moreover, his papers did not make his above assertion. We must take what his papers actually said, not how he would attempt to re-phrase them today.
In both his papers, Crow handles diploids by switching over to defining Cost in terms of Genetic Load (as documented above). This maneuver escapes him from having to physically explain "genetic death" in diploids, which is quite awkward. In diploids, what is a "genetic death" physically? For example, under a recessive substitution, how is a heterozygote death tallied as opposed to a homozygote death? If an individual contains both the old- and new- allele (a heterozygote), then does it count as a genetic death, or not? Such fundamental confusions were never addressed in the literature, because "genetic death", as a rational physical concept, quickly becomes incomprehensible once you get past single substitutions in haploids.
Crow's derivation requires selection coefficients approaching zero (s<<1), which means extremely slow substitutions, and the lowest total cost of substitution.
Moreover, Crow did not derive his equations based on a requirement for reproductive excess. My paper is the first to accomplish that. My derivation arises directly from first principles, without any use of genetic death or genetic load. Also, my derivation is valid for any selection coefficients (including large selection coefficients), and non-constant population size. No one else did that.
ReMine wrote: "My cost concept applies to ANY substitution (including neutral substitutions). Crow's doesn't – his cost concept applies only to beneficial substitutions, and gives false answers for neutral substitutions."
I didn't discuss neutral changes. My treatment was deterministic; neutral changes require a stochastic model. ReMine is correct in saying that even a neutral change requires some reproductive excess. This was a topic in several papers in the 1970s. (Crow's post)
In effect, Crow agrees that his cost concept does not apply to neutral (or harmful) substitutions. Mine applies to any substitution. Our concepts are different. QED
The predominant view today is that neutral mutations have "zero" cost of substitution. My paper corrects that error, and shows that neutral mutations actually have a high cost of substitution. Ewens and Crow agree my paper is correct, but they say a clarification isn't needed.
They are contradicted again by peer-reviewers at Heredity. Ranjan Chaudhuri wrote: "The author also, rather controversially, concludes that there is a cost associated with a neutral allele spreading by genetic drift." John Brookfield concurred, saying "I am particularly unhappy about the concept of a cost being applied where any genetic variation in populations is neutral." In other words, these reviewers demonstrate that clarifications are needed concerning the cost of neutral substitution. (Note: I eventually got editor Brookfield to calmly read my paper and acknowledge that my point is clear and correct.)
That traditional error was due, in part, to conflating "cost" and "load", as though they were the same thing, a habit that is a continual source of confusion. The "neutral substitutional load" was debated in 1972-73, by Ewens and Nei, but never resolved, precisely because of that confusion. Ewens argued that neutral mutations have a high "neutral substitutional load". Nei argued it is zero, and the consensus today is that Nei won the debate.
Peer-reviewer Ewens claims his correct answer was known and widely accepted by his colleagues in the 1970s, and therefore my paper is not needed. Again he is contradicted by another peer-reviewer, John Brookfield, who wrote in his review of my paper, "I knew about the papers from Ewens in 1972, and the fact is that he simply made a mistake – there is, by definition, no load associated with changes in frequency of a neutral allele." Brookfield got that right; the neutral substitutional load is zero, by definition. But the field is wrong to claim there is no cost for neutral substitution. Load and cost are different things.
A clarification is needed in this field, and my paper supplies it. Ewens and Crow are wrong to reject it.
|
I defined cost as required reproductive excess in my 1968 article and in my [1970] textbook with Kimura (pages 244-251). (Crow's post) |
I defy you to find that assertion in Crow 1968 or 1970. Crow cannot now add new assertions to his papers. He cannot now re-write his works retroactively. We can only use what Crow's papers actually stated. As shown above, his cost definition was explicitly an equation, and was explicitly also genetic death. And often he explicitly equated cost with genetic load in the general case. But he never defined cost as a required reproductive excess.
Indeed, his papers seldom used the term "required reproductive excess". I here document how Crow's papers use "reproductive excess".
To be concrete, we may think of a population with a reproductive excess, but which because of mortality has an adult population that is roughly constant from generation to generation. [Note: That wording is essentially identical in Crow 1968 & 1970.]
That is not a definition, much less a definition of cost. It merely used the phrase "reproductive excess" in a rather ordinary and unrevealing manner.
Next comes what Crow cites as his justification for rejecting my paper.
In this population, if sq0 = .2 and therefore 1 – sq0 = .8, the ratio of eliminated to non-eliminated is .2/.8 or .25. In other words, to have this much selection there must be a reproductive capacity of 25 per cent in excess of the adult number if the population size is to be maintained. (Crow, 1968, p 168)
That is not a definition, much less a definition of "cost". Definitions are general-purpose, whereas he is discussing a special case. It is a special case, with an aside comment concerning reproductive capacity. Or so it appears to be. Is the 'aside comment' merely a happy coincidence of his specially chosen case? Is it a coincidence due to the particular circumstances he chose: sq0 = .2, during a single substitution, in haploids, with a constant population size? [Answer: Yes.] But Crow's papers do not say. He left it ambiguous.
Moreover, Crow's comment about reproductive capacity is qualified by the phrase, "if the population size is to be maintained." His readers immediately recognize there is no inherent reason the population size must remain constant, so they are left to conclude his notion about reproductive capacity is flimsy, with little traction in the real world – a mere byproduct of Crow's specially chosen case. Crow's papers were ambiguous about what is important and central, versus what is unimportant and peripheral, and that caused confusion. Constant population size is required in order for genetic death and genetic load to yield correct numbers. But constant population size is completely unnecessary once the cost concept is rebuilt on clearer foundations, which Crow never did. In Crow's papers, constant population size is another confusion factor obscuring the role of reproductive requirements.
Most importantly, Crow's phrase, "In other words," conceals a grand gap in logic. With that glib phrase – "In other words" – he pretends to translate from genetic death (i.e. elimination of the unfit) into some notion about "reproductive capacity." Those two are worlds apart conceptually, and the translation is not remotely obvious. (For his specially chosen case, it would require him to answer the genetic death challenge through step 7.) My paper shows the necessary physical reasoning is already awkward and convoluted in the simple cases, and quickly becomes incomprehensible for complicated cases (such as diploids with multiple concurrent substitutions). Crow does not supply the physical reasoning, and the grand gap in logic cannot be bridged. So readers fall back to fundamental definitions – the cost defined in terms of genetic death.
In Crow's remaining few examples, his grand gap in logic is incomprehensible. Here is an example:
That is to say, the total number of eliminated individuals is 10 to 100 times the number of adults in any single generation. Haldane suggested 30 as a representative number. With this cost value, a species that effected one gene substitution every 300 generations would have to have an average of 10 percent reproductive excess during the process. (Crow, 1968, p 172)
In Crow's examples he starts with genetic deaths (e.g. "the total number of eliminated individuals"), specifically referring to it as "this cost value", and then he slips into a notion of reproductive excess, without supplying any physical justification. In this example, the missing logic is incomprehensible, because it deals with diploids and multiple concurrent substitutions. I challenge anyone to rationalize genetic death under those circumstances. Crow's grand leap in logic is pure confusion, unsupported by physical reasoning.
Crow's further uses of reproductive excess follow that same mistake. That is, Crow's approach requires a translation (or conversion) between genetic death (or genetic load), and some notion about reproductive capacity. Yet in general, there exists no such translation. The physical reasoning behind Crow's argument was never there, and his glib phrase "In other words" does not supply it.
My solution is to completely jettison genetic death, genetic load, and constant population size as unnecessary sources of confusion. and to cleanly define the cost of substitution so it's meaning is consistent throughout a given cost argument. So there is no need for translation (or conversion), and the grand gap in logic is eliminated.
ReMine wrote: "Then we have the testimony of history (in the literature and on the internet). Countless researchers still embrace Crow's notions of "genetic death" and "genetic load" – which are entirely unnecessary sources of false "solutions" to Haldane's dilemma. Crow has had many opportunities, and many decades, to clear up the confusion. If Crow had known my material in 1968 (as he suggests) then he was profoundly negligent for allowing the errors to thrive. The most charitable conclusion is that Crow did not really know my material."
I really thought that in the 1970 book I had adequately explained the problem. Perhaps I should have shouted from the housetops that Haldane's dilemma should be interpreted as reproductive excess, but I didn't. Let me emphasize that the change of interpretation from Haldane's load to reproductive excess is a small improvement and should not be regarded as comparable in importance to Haldane's original very clever idea. Compared to this, the change in emphasis from genetic deaths to reproductive excess isn't such a big deal. ....
I don't want to say that what I wrote 35 years ago is a model of clarity and consistency and couldn't be improved on. (Crow's post)
Crow acknowledged negligence, when he wrote, "Perhaps I should have ... but I didn't.” He did not clearly define the role of reproductive excess.
But that is only half his negligence. Crow fostered confusion factors, and did nothing to eliminate them. He claims his papers capture the "essence" of my paper, and that there is only a "semantic" difference. That suggestion is simply outrageous. My paper and Crow's are not remotely the same.
Crow belittles the clarifications as “a small improvement” of little regard, and “isn’t such a big deal”. That mirrors his rationale for rejecting my paper, that is, he views my paper as not a sufficient improvement over his own papers. In reality, it is a big deal because the confusion prevents people from seeing Haldane’s very clever idea.
I don't think the Haldane principle is useful… I doubt its usefulness in the rough and tumble of the real world of evolution and natural selection. (Crow's post)
That statement gives a motive for decades of negligence – he doesn’t think Haldane’s principle is “useful”, so why bother clearing it up? In effect, Crow decided the matter for us, by allowing confusion to thrive for decades. Haldane’s cost principle, when garbled by confusion factors, is indeed useless.
But Crow's statement also reveals a deeper misunderstanding. My paper is a clarification of Cost Theory, re-founding it as an important tool for studying the plausibility of evolutionary scenarios. My paper clarifies the fundamental cost concept, and how the various costs interact – such as the cost of mutation, the cost of segregation, the cost of substitution, and several others. I predict Cost Theory will have a useful role in population genetics, including the study of substitution rates. In effect, Crow is doubting the "usefulness" of Cost Theory and evolutionary genetics. That is, how can he doubt the usefulness of Haldane's cost principle "in the rough and tumble of the real world" without simultaneously casting the same doubt on evolutionary genetics? This is a point I often make. The evolutionists' attacks on Haldane's cost argument are, in reality, attacks on evolutionary genetics itself. They cannot throw out the one without substantially throwing out the other. These matters cannot be disentangled with a glib phrase and sweep of the hand, as Crow attempts to do.
There is a large literature subsequent to my writings that I did not participate in, and there has been much discussion of more difficult aspects. Nei and Ewens discussed the cost of substituting a neutral allele. Ewen's considered a finite population, as did Kimura. Nei in 1971 independently thought of the idea of reproductive excess. Felsenstein discussed the situation where the substituted allele is intrinsically favorable versus that when the environment is deteriorating. Kimura considered a situation in which the mutant slowly changes from deleterious to selectively favored. (Crow's post)
Crow listed various authors and their works. That is a tactic evolutionists often use to brush aside Haldane’s Dilemma. That is, they overwhelm you with many confusions, then posture like Haldane’s principle is “useless.” Ironically, for each of the examples Crow listed, my paper brings needed clarification:
In other words, Crow's listed examples are all reasons my paper should be published.
This is the latest chapter in a long scandal known as Haldane's Dilemma. Crow and Ewens have no legitimate grounds for suppressing my clarification of it.
1 Crow's post was relayed through Nikolas Voss, who works down the hall from Dr. Crow. An email with Nikolas Voss at his U-WI email address, verified that he is who he claims to be. I can also testify that the post looks authentically like Dr. Crow's wording, and the post is similar to sentiments expressed to me during the peer-review process, though with greater detail in his post. I believe the post to be authentically from Dr. Crow.
2 Crow's post cites his two papers:
3 The advances given in my paper can only be hinted here, and will become known when the paper is published.
4 Crow's post confirms what I posted previously: here, here, here, and here.
I did review ReMine's paper for Theoretical Population Biology. I have forgotten most of the paper, but I recommended rejection on the grounds that, although the paper was essentially correct and made some interesting points, there was not enough new theoretical content to merit publication in this highly technical journal. Another reviewer, Warren Ewens, reached essentially the same conclusion. (Crow's post)
5 I had not expected that defense from Crow, since it is such a bad one. Like most readers of Crow's papers, I read Crow as defining "cost" in terms of genetic death and/or genetic load. If he now claims his cost concept is "essentially mathematical", (defined as an equation), then he is abandoning any clear physical definition, and consigning his papers to ambiguity.
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