Re: Desiderata for a "cost" concept

In article <e74p1g$2uup$1@xxxxxxxxxxxxxxxxxxx>,
Perplexed in Peoria <jimmenegay@xxxxxxxxxxxxx> wrote:
What we are (or ought to be) seeking is a notion of the "cost of
substitution" or the "cost of evolution" which permits us to place
an upper limit on the speed of adaptive evolution. (Assuming, of
course, that a limit really exists.) The metaphor is that a certain
amount of evolution carries with it a certain 'cost' and that a
species cannot accomplish this amount of evolution in a very short
time because it is unable to 'make the payments' fast enough.

One issue is what imposes the cost, and what happens if the cost is too
high -- does the population go extinct, or does the population-genetic
model instead go extinct?

And that supplies our primary desideratum for a cost concept - that
it be useful in arguments which explain a limitation on the speed
of adaptive evolution. The other desiderata flow from this primary
one. So what are the other desiderata?

One is that it should be additive across loci. That is, if we know
the cost for one substitution, and we also know the cost for another
substitution at another locus, then we should be able to compute the
total cost of the two substitutions by simply adding. It is not
clear to me that ReMine's definition of 'cost' has this property -
it certainly is not manifest that it does. In Remine's view, the
cost of the first substitution is related to the reproductive excess
exhibited by the organisms containing the "good" allele at that locus.
The cost of the second substitution is related to the reproductive
excess of a (possibly overlapping) group of organisms carrying the
"good" allele at the second locus. It is not clear to me at all
that we can simply add these two different kinds of cost and get a
meaningful result.
.....
A related desideratum is that the total 'cost' of two substitutions
ought to be independent of the order in which the substitutions take
place, or whether they take place concurrently. Remine's concept
does not obviously meet this requirement. The amount of reproductive
excess provided by a particular substitution may depend on whether
the other substitution has already been accomplished.

These are good points.

Another desideratum is that the cost concept should be capable of
incorporating explanation of all of the other processes that might
limit the maximum rate of adaptive evolution - things like genetic
load, etc. ReMine's concept can certainly be stretched in that
direction - it is essentially just a different way of looking at
Crow's concept. But Remine's argument that his concept is a clarification
of Crow's concept is misleading. The clarity that he claims disappears
when you bring things like genetic loads into the picture.

But Felsenstein's paper does not address my desiderata either. It
doesn't try to. What it instead attempts to do is to explain in what
sense adaptive evolution can be said to be "costly". In the case of
adaptation to a changing environment (so as to maintain, but not
permanently improve fitness), Felsenstein shows that a certain amount
of reproductive excess is needed just to avoid extinction. He claims
that this is the true 'cost of substitution' though the word 'cost'
seems inappropriate here. And he further states (correctly) that
for positive adaptations in an unchanging environment, there is no
real 'cost' in this sense. True enough, but this is not the sense in
which I (and ReMine) want the word 'cost' to be used.

It is true that in my paper, the reproductive excess is not additive:
when there are twice as many substitutions per unit time, you do not need
twice as much reproductive excess to avoid extinction. But it's close:
the required excess d is p0^(-1/k)-1, where k is the number of
generations between substitutions. If we instead look at ln(1+d),
which will be close to d when d is small, it is additive. It needs to
be -ln(p0)/k (make a slight rearrangement of my equation (9)). So
when k is half as big (twice as many substitutions per unit time) ln(1+d)
needs to be twice as big. Note that ln(1+d) turns out to be the same as
Haldane's cost.

Note that for my definition you have a constant rate of substitutions.
Just having one, or two does not impose a requirement for any finite
amount of reproductive excess as the population can survive given even a
very small reproductive excess.

I don't think I "claim that this is the true 'cost of substitution'".
I claim that it is one way of defining a cost that ends up meaning something.
If the "cost" is held to be the reproductive excess necessary to prevent
extinction in the face of these substitutions, then it is the correct way
to compute that.

Let's look at whether Remine's cost is additive. In my notation,
Remine gives the "minimum total" cost for one substitution as

k ( (1/p0)^(1/k) - 1)

where k is the "duration" of the substitution. If in twice as much
time, we have twice as many substitutions, then if that means k is the
same but the haplotype that has both mutations has initial frequency
p0^2, as it would if there were no linkage disequilibrium, then
the total cost he calculates would be

k ( (1/(p0^2))^(1/k) - 1) = k ( (p0)^(-2/k) - 1)

For example, if p0 = 10^(-6) and k = 1000, his cost for one substitution
is 1000*(10^(6/1000)-1) = 13.9113857. With 2 substitutions it is
28.016298. That is nearly double. In fact, if C is Remine's minimum cost,
ln(1+C/k) will be additive. To be consistent with his framework I
think you also have to assume the population size remains constant in this
calculation, at least if it does his scheme gives this result.

But in doing this I am assuming k remains the same as we consider the
two substitutions. If they occur one after the other and each completes in
k generations, this is reasonable. But if they overlap, as would be
the case when the initial haplotype frequency of the double mutant is
(p0)^2, then I am not sure what Remine would count as being k. Perhaps
it would stay the same.

I can see that
Felsenstein is talking about a different kind of 'cost' than I am.
ReMine apparently cannot see this, and therefore claims that Felsenstein
is simply wrong.

Yup. He does that all the time. If several people have calculated
different things called costs, and one is him, then all but one are
wrong or confused in his view. If one is not him, then all are confused
and allowing error to thrive. Yes, he cannot see that there could be
different definitions of cost. This has been pointed out to him many
times, and he never gets it. Absolutely. Positively. No exceptions.

----
Joe Felsenstein joe@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Genome Sciences and Department of Biology,
University of Washington, Box 357730, Seattle, WA 98195-7730 USA

.

Relevant Pages

  • Re: The Cost of Substitution
    ... ReMine's paper "Cost Theory and the Cost of Substitution ... I agree with ReMine that the cost of substitution cannot be zero. ... substitution requires reproductive excess. ...
    (sci.bio.evolution)
  • Re: Haldanes Dilemma
    ... It clarifies and generalizes the cost concept. ... When stated as generations per substitution, ... _Ecological Genetics_ does the best ... environmental-change scenario is unnecessary and does favor evolution. ...
    (sci.bio.evolution)
  • Re: The Cost of Substitution
    ... The cost of substitution remains EXACTLY deductive from what Felsenstein ... disallows extinction while providing a single measure of zero reproductive ...
    (sci.bio.evolution)
  • Re: Reproductive Excess: Is Required
    ... >> Then can we now agree that Felsenstein's cost concept ... Felsenstein will clarify -- and makes abundantly clear (some things ... Dilemma (and the more severe limitation on substitution rate calculated ... substitutions require no reproductive excess. ...
    (sci.bio.evolution)
  • RE: The Cost of Substitution
    ... ReMine's paper "Cost Theory and the Cost of Substitution ... I agree with ReMine that the cost of substitution cannot be zero. ... substitution requires reproductive excess. ...
    (sci.bio.evolution)