Re: Kin Selection contradiction?

From: Guy Hoelzer (hoelzer_at_unr.edu)
Date: 06/24/04 

Date: Thu, 24 Jun 2004 04:29:22 +0000 (UTC)

Jim,

in article cbc9qk$276c$1@darwin.ediacara.org, Perplexed in Peoria at
jimmenegay@sbcglobal.net wrote on 6/23/04 9:06 AM:

> "Guy Hoelzer" <hoelzer@unr.edu> wrote in message
> news:cbaj57$1kj0$1@darwin.ediacara.org...
>> Jim,
>>
>> in article cba42o$1eub$1@darwin.ediacara.org, Perplexed in Peoria at
>> jimmenegay@sbcglobal.net wrote on 6/22/04 1:16 PM:
>>
>>> "Guy Hoelzer" <hoelzer@unr.edu> wrote in message
>>> news:cb7rtf$mn0$1@darwin.ediacara.org...
>>>> in article car9d1$2lm6$1@darwin.ediacara.org, Perplexed in Peoria at
>>>> jimmenegay@sbcglobal.net wrote on 6/16/04 10:15 PM: Whatever method is
>>>> used, the result has to be an "r" with the following property: Randomly
>>>> choose one of the two genes at any locus in the donor. Suppose that the
>>>> frequency of this allele in the general population is "p". Now, randomly
>>>> choose one of the two genes at the same locus in the recipient. It must be
>>>> the case that the probability that the two randomly selected genes are
>>>> identical is (r + (1 - r)p). That is, there is a probability r that they
>>>> are identical IBD, but if not, then there is still a probability p that
>>>> they are identical for other reasons - because the allele is fairly common
>>>> in the population.
>>>>
>>>> I think there is a big practical problem with applying any models relying
>>>> on IBD calculations, because our data on genealogical history is virtually
>>>> always very shallow and incomplete.
>>>
>>> No doubt. I have no idea how field workers go about applying the model.
>>> Though I would imagine that the problems with estimating "b" and "c"
>>> are much larger than the problems of estimating "r".
>>
>> Well, "r" can never be known and can only be estimated with an
>> underestimation bias. The degree of the bias depends on the amount of
>> unknown inbreeding in the common ancestry of two individuals. At least "b"
>> and "c" might be estimable without bias. Nevertheless, my biggest problem
>> with the kin selection literature has to do with the uncritical (even
>> unconscious) acceptance of kin selection as THE explanation for altruism-like
>> behaviors observed in nature. I think the difficulty associated with
>> estimating these parameters has a lot to do with giving in to this elegant
>> model.
>
> I disagree about the effect of unknown inbreeding below.
>
> I fully agree that most altruism-like behaviors in nature (excluding
> parental care, and perhaps the social insects) are based on reciprocity
> rather than on the unilateral altruism covered by the Rule.

It's good to know that our estimates of reality are close.
 
> But this conversation began because, IMO, you made some false statements
> about the model - specifically that the applicability of rb>c depends
> upon the frequency of the altruistic allele. Or at least that is what
> you seemed to say. I am trying to defend the model against distortion,
> not to defend the importance or applicability of the model. You would
> have to have that discussion with someone who knows more "natural
> history". I am just an amateur OOL person who happened to get interested
> in Hamilton's Rule because it seemed to be controversial in this group.
> So, ultimately, I suppose this conversation can be blamed on Edser :-)
>
>>>> For example, all individual organisms (across all species) are probably
>>>> genealogically related, but... There is also the problem of genealogical
>>>> relatedness in the face of mutation (common decent without identity).
>>>
>>> If you believe that these issues are "big practical problems", then I
>>> suspect that you don't yet understand the model. The effect on r of
>>> pushing the calculation back additional generations is small in a sexual
>>> population.
>>
>> The problem is not just about pushing the estimate of "r" back additional
>> generations. If that were the case, then I would agree that say 3
>> generations would be sufficient in most cases. Putting aside the fact that
>> we rarely have information going back more than one generation, the greater
>> issue has to do with inbreeding. The parameter "r" can take on any value in
>> the face of inbreeding. For example, full sibs actually have 0.5 >= r <= 1.
>> In fact, it is very common to have levels of inbreeding that make our naïve
>> estimates of "r" significantly lower than they actually are. Truth be told,
>> r=1 for every pair of individuals (even from different species) if you were
>> to consider all the data (and ignore mutation/divergence). Do you have a
>> justifiable rule for how ignorant we ought to be when we try to estimate "r"
>> in order to make the false estimate useful in understanding kin selection in
>> nature?
>
> It is clear that you still don't understand what "r" is. The history of the
> population, and the fact that it may be inbred in the recent or distant
> past is totally irrelevant. The only way inbreeding could be relevant would
> be if the population routinely breeds with close relatives for reasons other
> than small population.
>
> Forget IBD for a moment. Let p be the frequency of the altruistic allele
> in the general population. Let P be the frequency of the allele in the
> recipients. Define r to be that value which satisfies the equation
> P = r + (1-r)p
> That is, r represents, in a mathematically wierd way, the degree to which
> P exceeds p.

As I said before, I like this way of thinking about the model, at least in
some ways. I think you have internalized Maynard-Smith's version of
Hamilton's model. Those two great thinkers thought about things in very
different ways, IMHO. I am not sure that Hamilton ever verified that the
Maynard-Smith version was consistent with his thinking. IMHO Hamilton
thought of "r" primarily as a measure of genealogical relationship, which is
not what it means in your equation above. My comment above about "0.5 >= r
<= 1" was specifically referring to the genealogical "r". Nevertheless, I
will go with the Maynard-Smith version of things for our dialogue.

> This value r can be calculated (within sampling error) in the field if you
> take DNA samples from a random sample of the donor-recipient pairs. Of
> course, you have to normalize against the results for random pairs that
> are not donor-recipient.
> Or, if you have been observing the population for a few generations and
> you know genealogies of individuals, you can estimate r by truncated
> IBD. Or use truncated IBD if you are not a fieldworker and you just
> want to understand a reason why P>p might be expected.
>
> If you wish to understand why inbreeding is not important, perform
> the following thought experiment. Imagine a population derived
> from a single breeding pair which has grown to a population of
> 64 with the population doubling each generation. But, to make
> sure that we have variation for altruism, make both of the original
> pair heterozygous. Assume that mating is random and monogamous.
> "p" is 1/2. Assume the altruism is directed to full sibs. I think
> that you will find that r is not much larger than .5 and certainly
> less than .6. Or, for variety, start with p = 3/4 or 1/4.
> "r" still will be less than .6 AFAICS

You assumed something like random mating in an exponentially growing
population, which basically the same as assuming that inbreeding is not
occurring. It is certainly not surprising to conclude that inbreeding does
not affect "r" when you assume that inbreeding does not occur. To
illustrate why inbreeding IS important, consider a hypothetical population
in which sib-sib mating is the norm. Such a population quickly loses its
heterozygosity and becomes constituted by families filled with altruists and
families lacking altruists. Now P>>p, and your "r" value is correspondingly
much higher. So "r" is sensitive to inbreeding.
 
>>> And the chance that a mutation has destroyed an altruistic
>>> allele within those few generations is also small.
>>
>> In most cases that is probably true. Of course, we haven't even touched on
>> the most common criticism of Hamilton's kin selection model, which is that
>> altruistic behavior is probably not directly caused by a genetic mutation
>> most of the time.
>
> I assume you mean that it is not caused by a single gene. (If you meant
> something different by "mutation", you will have to explain.)

I use the term "mutation" in a more general sense here. For example, a
chromosomal inversion might be involved, which does not disrupt the function
of any particular gene. I am not trying to argue for the importance of
non-genic mutations. I am merely allowing for effects of a broader class of
mutations that would exhibit Mendelian segregation as assumed under
Hamilton's model.
 
>> It is probably far more complicated than that, and it is
>> not clear to what extent the Hamilton's rule would work outside of the
>> genetic framework. Here again is where my real problem lies. The vast
>> majority of the empirical literature claiming to support Hamilton's model
>> never addresses the issue of genetic control over altruism.
>
> Well, it turns out that non-additive epistasis is not a problem in
> this model, for the usual reasons. If you doubt this, just make the
> penetrance factor "f" below depend on both "p" and the frequency
> of some other gene(s). For a derivation of the rule, I think we
> are justified in treating each of these frequencies as a constant.
> Though perhaps we are now adding a weak selection assumption.
> However, pleiotropy turns out, surprisingly, to be more problematic.
> My toy derivation of the Rule, sketched below, doesn't handle it,
> for reasons sketched in my reply to Bill Morse. I don't know whether
> Hamilton [1964] handles it.

I don't think Hamilton ever challenged his model with such mechanistic
detail. I wasn't specifically implying anything about non-additive
epistasis in my comment. In fact, I was thinking more about non-genetic
effects altogether. Still, it is nice to know that the model appears robust
to non-additive epistasis.
 
>>>>> Or, if like McGinn, you have an intuition that geneological history cannot
>>>>> be causal in this situation, ignore the IBD above. "r" is simply a
>>>>> measure of how much more likely than "p" it is that the two genes are
>>>>> identical for whatever reason. The key thing is that the formula (r +
>>>>> (1-r)p) gives the probability that the alleles are "shared".
>>>>>
>>>> Hmm. There are some things about this formulation that I like, and some
>>>> problems I see. Can you please save me a little research time and tell us
>>>> where you come by the formula (r + (1-r)p)? Is this your interpretation of
>>>> Hamilton, or has it been published?
>>>
>>> It is a straightforward interpretation of the verbal explanation given in
>>> Maynard Smith's "Evolutionary Genetics" (2nd ed. p169)
>>>
>>> Now we can picture the genome of the recipient as consisting of two
>>> parts:
>>> 1. a fraction r containing genes IBD to genes in the actor; and
>>> 2. a fraction (1-r) consisting of genes that are a random sample
>>> of genes in the population.
>>
>> This is a very familiar modeling trick. The same thing is done when modeling
>> inbreeding for other purposes. It is, however, just a trick that makes the
>> math work out easily. The flaw becomes clear when you recognize that any
>> random sample of the gene pool will potentially contain gene copies that are
>> IBD with the target, so the fractions are not mutually exclusive. Given your
>> definition of "r", "(1-r)" must be the fraction of the recipient's genome
>> containing genes that are NOT IBD, which is different from "a random sample
>> of genes in the population."
>
> I think that my argument above including the phrase "forget about IBD
> for a moment" addresses this concern.

I agree that this second way in which you defined your parameters is more
logical. However, that does not validate the way you defined them at first,
which I still argue had a logical flaw. In fact, I think these two
definition sets are inconsistent with one another, so you should decide
which one you want to use and stick with it.
 
>>> As a "proof" that this is the correct interpretation, you can derive
>>> "rb>c" from this formula.
>>> 1. Assume b and c are the benefits and costs of a single altruistic
>>> action.
>>> 2. Assume a "penetrance factor" f (which may depend on r!) gives the
>>> number of times an allele causes altruism during its organism's
>>> lifetime. Assume that homozygous altruists share the cost between
>>> alleles. The factor f allows us to ignore whether the allele is
>>> dominant or recessive. For example, if the allele is recessive,
>>> then f will be small when p is small. But if the allele is
>>> dominant, then f will start large and then fall to half the
>>> original value, due to "sharing the credit" between homozygous
>>> gene instances.
>>> 3. Calculate the total benefit received by recipients. Divide this
>>> up between carriers and non-carriers, with the benefit split in
>>> heterozygotes. Now calculate the "per capita" benefit to each
>>> haploid genome of the two types.
>>> 4. Calculate the total and per capita costs of altruism.
>>> 5. Compare the per capita benefit for non-carriers to the per
>>> capita (benefit - cost) for carriers. You will notice that
>>> the variable factor f cancels out, as does the allele frequency
>>> p. You are left with the fact that carriers receive enough
>>> extra altruism to compensate for the costs of acting altruistically
>>> when rb>c.
>>> Try it. As I wrote:
>>
>> This is very clever. Given my logical analysis above, your "proof" would
>> seem to reveal cryptic ambiguities in Hamilton's original thesis. BTW, have
>> you agreed in the past that all of this goes out the window for deterministic
>> reasons when there is only one copy of the allele around, because then
>> altruism only costs the allele fitness points?
>
> I understand what you are saying. Clearly, the only altruist in the
> population cannot also be a recipient, and hence can't be more fit than the
> rest of the population. (I could point out that you are assuming that the
> allele is dominant, but that is not my real objection to your point.)
>
> Frankly, I consider this a minor quibble - not much more forceful than if you
> had pointed out that a rabbit's speed is simply a drain on its metabolism if
> it never encounters a fox. But, I admit that what you say is true. I'm not
> going to try to convince you of the validity of my earlier response to your
> point.

So I guess that you concede that the validity of Hamilton's rule depends
"p", at least at this singular point. I suspect you would even concede that
the effect of "p" would be observed at very low values of "p" when there is
more than one copy of the altruism allele around. So, we only disagree
about how far this effect will reach as you increase the value of "p", which
I have been arguing depends strongly on population structure (and I would
add population size).
 
>>>>> Why is that particular formula so important? Well, when you do the math,
>>>>> you will see that the average fitness of allele carriers will be greater
>>>>> than non-carriers, as long as the carriers direct their altruism to
>>>>> recipients of relatedness "r". That is, average fitness of carriers will
>>>>> be higher as long as rb>c. And the parameter "p" nicely cancels out of
>>>>> the equations. Hamilton's rule is independent of p. As long as "r" has
>>>>> the meaning above.
>>>>>
>>>> My criticism here is the same as one that I have been posting earlier in
>>>> this thread. Your conclusion is sensitive to the implicit assumption that
>>>> the population is large (effectively infinite) and well mixed. This
>>>> combination does not generally exist in nature, because the larger a system
>>>> is the harder it is to mix up.
>>>
>>> Not my conclusion. Hamilton's. But I think you exagerate the
>>> sensitivity.
>>
>> Like you, I would say "just try it." If you do a simple-minded model of
>> agents interacting in space I think you will quickly see the sensitivity.
>>
>>> I think that all that is required is that the "well-mixed" or effectively
>>> random mating breeding population is larger than the local
>>> socially-interacting clique of each individual.
>>
>> Interesting. I would like to see the argument to this conclusion; or is this
>> your "gut feeling" at this point. [BTW, I do not intent to impune "gut
>> feelings." I admire people that are in touch with their guts.]
>
> Assume that the population is divided into "states", within which mating
> is random (well mixed). Assume that states are subdivided into "zip-codes",
> within which organisms interact. A state can have one zip-code, or several.
> Let "p" vary between states, but be constant within a state. Perform
> the derivation of the Rule separately for each state. Since p is
> constant within a state, the derivation still works in each case. Note
> that this would no longer be valid if a single zip-code covered two
> different states with different "p" values.

I'm beginning to lose the thread of our argument, but I think the assumption
of constant "p" values within subpopulations subverts the problem I was
pointing to. Indeed, that assumption is inconsistent with Hamilton's model,
which is about predicting changes in "p". If you loosen up your model a
little to allow for dynamic "p" values, then each of your subpopulations can
evolve deterministically in different directions under their own versions of
Hamilton's rule. What would this mean for the evolution of the whole
system? I don't think that Hamilton's rule would be a very precise guide at
the global scale.
 
>>> And that "r" is pretty much the same everywhere.
>>
>> As I have indicated before, I think this is a very unrealistic assumption if
>> you mean that "r" is the same everywhere for any particular individual. If
>> you mean that the distribution of "r" is the same in the neighborhood of
>> every individual, then this may be a reasonable approximation for some
>> systems.
>
> The altruistic gene "directs the altruism" toward kin or neighbors. I am
> assuming that the gene produces the same biased effects everywhere, and,
> as you say, that neighborhoods are similar in their kinship structure
> everywhere.
>
>>> It is not important whether "p" is pretty much the same everywhere.
>>
>> What if I moved the pieces on the board of life so that each copy of the
>> altruism allele is surrounded by a sea of purely selfish individuals? Note
>> that this can only be done when "p" is small.
>
> But this means you have just set "r" to zero (or actually, to a small negative
> number)! Everywhere. So the Rule still works.

Sorry. I forgot about your definition of "r". I was still using it in the
genealogical sense of KIN selection.
 
>> If "p" is large, then I could still change the balance of Hamilton's rule
>> when "rb" is only slightly larger than "c" by isolating many copies of the
>> altruism allele in local seas of selfishness.
>
> Your thought experiments strike me as artificial, and hence not particularly
> relevant. Maybe you are trying to tease out hidden assumptions. I think
> that the assumption that you are attacking here is an explicit one - that
> "r" is constant. In this case, you are still changing "r".

So your "r" is critically dependent upon population structure. Right?
Didn't you once argue that Hamilton's rule did not depend upon population
structure?
 
>>>> It also gets harder and harder to effectively mix the finite population
>>>> when "p" is either very small or very large, because in either case there
>>>> are very few quanta of the rare allele.
>>>
>>> True enough, but now you are criticizing how deterministic Hamilton's rule
>>> is, rather than claiming that it is incorrect (biased) at extreme
>>> frequencies.
>>
>> Note my arguments above about bias when "p" is small and/or unevenly
>> distributed.
>
> Noted. I am discounting your small p argument, and accepting your
> uneven distribution argument only if you have interactors that, for
> some reason, don't interbreed. Perhaps modern human India, Bosnia,
> or Northern Ireland fit your requirements. Hmmm. Now that I think
> about it, perhaps my zip-code analysis is extraneous.

What do you think of my "flipside" argument above regarding systems
constrained to sib-sib mating?

Regards,

Guy

Relevant Pages

  • Re: Hamiltons rule
    ... >> - If the altruism locus is purely dominant, ... That doesn't matter as long as the donor D line never drops ... >> or of how many loci are involved, it remains the case that the recipient ... > favor the altruism allele. ...
    (sci.bio.evolution)
  • Re: Underestimating r
    ... >> relatedness in family-structured populations with inbreeding. ... Inbreeding and the evolution of altruism under kin ... Unless populations are "usually" inbred, they may have evolved to use ... > amount of 'background' inbreeding in the entire competing population. ...
    (sci.bio.evolution)
  • Re: Hamiltons rule
    ... slightly higher fitness (unless maybe the environment is different ... > I think we have pretty different expectations of the strength of selection ... cooperation and altruism ... > focusing on the advantage of being in a family with the altruism allele. ...
    (sci.bio.evolution)
  • Re: Hamiltons Rule: light at the end of a LONG tunnel?
    ... > I think you are confounding Hamilton's kin selection model with Trivers' ... > reciprocal altruism model. ... > certainly further facilitate an increase in the altruism allele through the ... > mechanism of reciprocal altruism, is also not accounted for in the precise ...
    (sci.bio.evolution)
  • Re: Hamiltons Rule: light at the end of a LONG tunnel?
    ... >>> I think you are confounding Hamilton's kin selection model with Trivers' ... >>> reciprocal altruism model. ... >>> certainly further facilitate an increase in the altruism allele through the ... >>> mechanism of reciprocal altruism, is also not accounted for in the precise ...
    (sci.bio.evolution)