Logic of kin selection

From: Joe Felsenstein (joe_at_removethispart.gs.washington.edu)
Date: 01/26/05 

Date: Wed, 26 Jan 2005 11:41:09 -0500 (EST)

This new thread is intended to explore a simple situation where kin
selection is occurring. We can check whether Hamilton's conditions for
selection for an altruistic allele are correct in that case, or whether
there is an problem with Hamilton's argument, as Jim McGinn argues.

The situation is not a totally general one, but it is one to which
Hamilton's rule is argued to apply. If the rule won't work here, it is
in trouble generally. As recent controversies seem to have narrowed down
the area of disagreement between McGinn and myself and others, I hope that
we can go over this case in detail, to see whether there is a problem and,
if so, where.

Consider a diploid species with discrete generations, infinite population
size, and an allele A that predisposes the individual to an altruistic
behavior towards its neighbor. For simplicity let these be pairs of
neighbors, who are in two roles, role #1 and role #2. An individual
finding itself in role #1 has the opportunity to display the altruistic
behavior (the one in role #2 cannot).

The allele A is assumed to be rare in the population. That means that
we can ignore the presence of AA homozygotes and just have a fraction
2p of Aa individuals and 1-2p of aa individuals. Thus there are
four possible situations:
                             Role #1 Role #2
                             ------- -------
                               Aa Aa
                               Aa aa
                               aa Aa
                               aa aa

The two individuals are neighbors. Let's assume that 50% of the time,
they are also siblings, the rest of the time being unrelated. In that
case, the frequencies of the four types of pairs (these are not matings,
but pairings in a situation that allows this behavior) are then:
                            
              (Aa, Aa) (1/4)(2p)(1-2p) + (1/2)(2p)^2
              (Aa, aa) (1/4)(2p)(1-2p) + (1/2)(2p)(1-2p)
              (aa, Aa) (1/4)(2p)(1-2p) + (1/2)(2p)(1-2p)
              (aa, aa) (1/4)(2p)(1-2p) + (1-2p)^2

These expressions are gotten by taking all cases, and mixing together
half from ones where both individuals come from the same two parents, and
half where they are unrelated. In doing this we can ignore cases where both
parents of the sibs are Aa as we are going to toss out terms of order
p^2 anyway.

The result is that the frequencies of the four situations, after we
toss out the p^2 terms, are

            (Aa, Aa) (1/2)p
            (Aa, aa) (3/2)p
            (aa, Aa) (3/2)p
            (aa, aa) 1 - (7/2)p

I have to break off, but will continue in the next posting by working out
the fitnesses of everybody in a simple case where Aa carries out an
altruistic behavior when in role #1, and the individual in role #2
benefits from it irrespective of genotype. Then we will see whether
Hamilton's conditions tell us what the conditions are for the A allele
to increase in the population when rare. If Hamilton made some elementary
mistake then these conditions may conflict with his.

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

Relevant Pages

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