Haldane defined the cost of substitution in terms of "genetic death."¹ As a physical concept, genetic death focuses on the elimination of the old-allele, by the elimination or death of individuals containing it. The professional literature continues that tradition even today.  Therefore, my book, The Biotic Message, introduces the subject using the traditional approach, in terms of genetic death. My book then clarifies the real meaning of the cost of substitution, which is fundamentally not about death, genetic or otherwise. 

At the time, I believed that starting with the traditional approach was obligatory for sake of scholarship, and that some use of the traditional term, "genetic death," would be helpful to readers, who were sure to encounter that terminology in the technical literature.  I believed it helpful to construct a conceptual bridge between the traditional view (genetic death) and the clarified view given in my book.  In short, my book de-emphasizes genetic death, and advances a clearer interpretation of the cost of substitution. Quite simply, the issue is not the elimination (or death) of the old-allele, but rather the growth of the new-allele.  Evolution requires each new-allele to grow from one copy to, say, a million copies. This growth rate is limited, by the finite reproduction rate of the species, and this limit is precisely quantified by the cost of substitution. Put simply, it's not about death, it's about the required reproduction rate. Conceptually there is a huge difference. 

After my book came out, and many further discussions with evolutionists, I found that genetic death is rigidly held and central to their confusions about the cost of substitution and Haldane's Dilemma. I found that genetic death is not only unnecessary to understanding, it is downright harmful to understanding.  I found that the cost of substitution can be, and should be, defined without any mention of genetic death or elimination of the old-allele.  So I went further, and without using the genetic death concept, I re-defined the cost concept, and cleanly re-derived all of Haldane's equations for the cost of substitution, and rebuilt Haldane's argument on a clearer foundation. This material, and much additional material, was placed into my recent paper, which is now acknowledged as correct by leading evolutionary geneticists, James Crow and Warren Ewens. 

In other words, I discovered it is not enough to merely de-emphasize the genetic death concept, we must actively oppose it.  I claim the genetic death concept should be abandoned, because: (a) it is unnecessary, and (b) it is a fountain of confusions. 

Yet I remain unapologetic about the approach taken in my book. It was a useful step forward at the time. If I had offered my clarified cost concept out-of-the-blue, without any connection to Haldane's cost concept, then opponents would claim there was no connection between the two, and that my cost concept was therefore irrelevant to Haldane's Dilemma. In fact, that is what most evolutionists tried to claim. (Ironically, Crow and Ewens would later acknowledge my cost concept is correct, but they refused to publish my paper on it, on the grounds that they and their colleagues already knew my material in the 1970s.)

As I said, my book provides a conceptual bridge between the traditional cost concept and my cost concept. The derivation given in my book's Appendix aids that goal. I next comment on that derivation. 

The derivation –

My derivation had to show that my cost concept is a clarification that is truly and deeply connected to the traditional Haldane-Crow cost concept. To help solve that difficulty, my derivation begins and ends with something recognizable from traditional literature. That is, it begins with genetic death, and ends with Haldane's classic formula for the total cost of substitution – but in-between a transformation occurs, whereby the derivation is explained entirely in terms of my clarified cost concept. 

Recall the classic perception diagram (shown at right) – you look at it one way, and it's a side view of a large-nose old woman, but you look at it another way, and it's a three-quarters rear view of a beautiful young woman. That was a goal with this derivation. I constructed the derivation so you could see "genetic death" and see my cost concept in one-and-the-same simple example. 

Another goal was to keep it simple, and avoid all complexities that are not essential to understanding the core issue. Make the fundamentals understandable to the widest possible audience. So the derivation sets aside our notions of complex genetics (such as diploidy, and recessive mutations).² 

"I will here derive the fundamental formula in a manner that avoids genetics, selection, and the environment. The derivation focuses instead on survivors and their reproductive capacity. By so doing, I will show that the cost of substitution is simple in concept and unavoidable."  (The Biotic Message, page 499)

That is an important statement, because many evolutionists erroneously claim the cost of substitution is reduced by soft selection (e.g. Bruce Wallace), or by the environment (e.g. Joe Felsenstein). My approach shows they are mistaken, and that the cost is unavoidable and cannot be reduced one iota by their claims. 

Assume the population size S is constant (which, by the way, is a requirement for making anyone's concept of "genetic death" work properly). Under this assumption, in any given generation, the net reduction of the old-type individuals equals the net increase (or excess births) of the new-type individuals. 

The population of size S is composed of Q individuals of the old-type, plus P individuals who possess a new mutation that will be substituted into the population. So S=P+Q. (Note: The letters P and Q were chosen to remind us of p and q, the traditional notation for allele frequencies, except the capitalization indicates that P and Q are the actual number of individuals, not frequencies.) 

The derivation begins by considering a substitution in one generation. In that scenario, Q individuals die in one generation, there are Q "genetic deaths". And the P individuals are required to produce all the replacements. In this scenario, Q is the 'net reduction' in Q; and DP is the net increase in P; and because the population size is constant, those are equal: Q=DP. In this scenario the cost of substitution is:

Cost = Q/P      (Equation 1)
        = DP/P    (Equation 2)

Equation 1 includes the notion of Q deaths. Whereas Equation 2 mentions no such thing, and is stated purely in terms of the required reproduction rate, (or more precisely, the required excess reproduction rate).  This is where the 'large-nose old woman' turns into a 'beautiful young woman' – this is where a change in perception begins. Again, that was a key purpose of this derivation, to show a clearer interpretation of an old concept. 

More details: My derivation uses the simplest of all scenarios. (Haploids with constant population size and a single substitution in one generation.) This scenario also has the unique property of displaying both the Hoyle-Feller notion of genetic death, and Crow's cost concept (which is based on a different concept of genetic death). In other words, my derivation displays these various notions of cost together at the same time.  The Hoyle & Feller notion of genetic death is physically concrete and understandable. They define the genetic death in a given generation as the net reduction in the number of old-type individuals. In a one-generation substitution, that net reduction would be Q. Unfortunately, Hoyle and Feller normalized Q by the population size S, thereby calculating the cost of substitution as Q/S, (instead of Q/P), which always gives a cost of approximately 1, not matter how fast the substitution occurs. (Whereas Haldane calculated an average figure of 30 for extremely slow substitutions.) They thereby concluded that Haldane's cost concept is mistaken and useless. In other words, they used a concrete physical definition of genetic death (which was a desirable goal), but unfortunately they viewed genetic death as the central issue, when it isn't.  James Crow, on the other hand, defined the cost of substitution in a given generation as the eliminated individuals divided by the non-eliminated individuals. (His cost definition twice uses genetic death, once in the numerator, and again in the denominator.) In this scenario that would be Q/( S-Q). Crow's cost concept, applied in this scenario, happens to give the same mathematical result as my cost concept. However, that quickly changes when we move away from such a simple scenario. For example, when the substitution takes more than one generation, then Crow's concept of "genetic death" is no longer physically concrete, and becomes more abstract.  Crow's concept of genetic death is physically awkward, and quickly becomes incomprehensible for scenarios that include diploidy, recessive mutations, and multiple-simultaneous substitutions.  Again, my derivation was designed to display my cost concept, and the traditional cost concepts (based on genetic death) at the same time. To show their equivalence, at least for this simplest of all scenarios.

For example, in a population size of one million individuals, where one individual has the new mutation, the substitution in one generation requires an extra reproduction rate of 999,999 – this is the cost of substitution. The cost of continuity is always 1, and the cost of substitution here is 999,999, so the full cost of this scenario is (at least) one million. The species must actually supply a reproduction rate of at least one million for this scenario to be plausible. If the new trait is to begin with one individual, and end with one-million individuals one generation later, then a reproduction rate of one-million is required. It is unavoidable. The cost of this scenario is not reduced one iota by soft selection or by the environment. Such a high reproduction rate is impossible for earthly species, so the scenario is rejected as implausible. This is how a cost argument operates. We calculate the reproduction rate required by a given scenario (this is the cost), then we compare it with the species' actual reproduction rate (this the payment). If the species cannot 'pay the cost', then the scenario is not plausible.  

The math derivation is also explained in text, which further helps the conceptual bridge to a clearer understanding. First the text introduces the traditional focus on the old-type individuals who are going to become 'genetic deaths', and tells about them. 

"To substitute the new trait, all the Q individuals must have their line of inheritance terminated, either in this generation or in some future generation. We will here simply forget about their actions, as they have no effect on our analysis." (The Biotic Message, page 499, emphasis added)

The text is saying do not focus on the old-type individuals (the genetic deaths) as they and their actions have no effect on our analysis. This is in contra-distinction to the cost-literature which focuses intensely on the old-type individuals, what their traits are, how they live, when and how they die – which is all needless confusion. Most of the traditional "solutions" to Haldane's Dilemma are erroneous due to that confused focus. 

The text then re-focuses the reader's attention on the core issue:

"If evolution is to be an on-going, long-term process, then ultimately the entire population must be regenerated by the few P individuals." (p 499)

The text gives the traditional cost definition, and then immediately re-interprets what it means. 

"The substitution incurs a cost defined in terms of the genetic deaths per survivor. [Note: that was the traditional cost definition. Next comes a clarification.] It is the excess reproduction (in births per parent) that must be produced for the specific purpose of replacing the genetic deaths." (p 499, note and emphasis added)

The text then gives the cost for that single-generation substitution: Cost = Q/P. The P individuals are identified as the "survivors", as opposed to the Q individuals who will be eliminated as genetic deaths. Thus that formula, Cost=Q/P, is seen to have units of "genetic deaths per survivor" – and equals DP/P, which has the units of reproduction rate (or more precisely, excess reproduction rate). In other words, the 'conceptual bridge' is set up to include the units. The traditional goof-ball units of "genetic deaths per survivor" are better interpreted as units of excess reproduction rate. And those two formulas occur right next to each other (as in Equations 1 and 2) for easy comparison. 

The text then reinforces the true meaning of it all:

"In this case, the cost is large because the population is regenerated in one generation, by only P individuals. This small number of individuals bears the entire reproductive cost." (p 499, emphasis added)

The above matters from the Appendix are reinforced in the Chapter, which repeatedly emphasizes that the survivors must pay the 'cost', not those who are eliminated. This is in contra-distinction to the cost-literature, which views those who die as "paying the cost" by dying. The Chapter repeatedly focuses on, and emphasizes, the reproductive requirements – in contra-distinction to the cost-literature. 

Notice another key point. My discussion was phrased so as to never mention whether the new substitution is beneficial, neutral, or harmful – because it does not matter. As I said, my goal was to keep it simple, so as to display the fundamental principles with the least amount of distraction: If anything is claimed to increase in number of copies (through reproductive means), then reproductive excess is required. Absolutely, positively, no exceptions. It is the task of cost theory to calculate that requirement, and compare it with the given species reproduction rate. 

The derivation continues onward to show that the cost can be reduced by slowing the substitution rate. That is, the scenario is expanded to include substitutions that take many generations. The slower the substitution, the lower the cost. It shows that for an infinitely slow substitution, we arrive at Haldane's classic formula, [the minimum possible total cost of substitution = natural_logarithm(1/p₀), where p₀ is the starting frequency of the substitution]. In other words, my cost concept yields the traditional cost equation, but with a clearer interpretation. 

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¹    More precisely, Haldane (1957) defined the cost of substitution in terms of "selective death", which quickly was re-named as "genetic death", which is the term more commonly seen in the literature. 

²    Also, "genetic death" quickly becomes incomprehensible and unworkable as we move away from the simplest scenarios. Whereas my cost concept is straightforward and works under any scenario.  In other words, the derivation given in my book shows an equivalence (between genetic death and my cost concept) only for the simplest scenarios. There exists no general-purpose equivalence between the two, just as there exists no general-purpose notion of genetic death. 

2/23/2006 - by Walter ReMine