# The genetical evolution of social behaviour – I

By Razib Khan | February 4, 2008 9:07 am

The fitness of an individual is the sum of his basic unit of his personal genotype and the total of effects on due to neighbors which will depend on their genotypes:

a = 1 + δa + e°, where sybmol the personal effect of δa into any aggregate, and ° will represent its exclusion. (1)

Hamilton then asks you to imagine two individuals, A & B, and their relation on a locus with represent to their identity by descent. The sum of the probabilities of state is represented like so:

c2 + c1 + c0 = 1

Where the subscript indicate the number of copies with match across the two individuals, and the probabilities must sum up to 1. From these relations Hamilton surmises that one can derive r, the coefficient of relationship, and this value is essential in understanding the distribution of effects when an actor operates upon another individual. From a genetic viewpoint when fitness is at issue r is of the essence. From this Hamilton constructs a vector,

{ δar }A , which sums up the affect of A upon a range of individuals whose genetic relationship is measured by the r between A and the said individual.

Next Hamilton wants you to consider a set of genes at the locus, p1, p2, p3pn, so that the array may now be thought of as:

{ δar }ij, where ij are now to represent genotypic values of A. Now, the total effect on fitness due to A can be considered:

Σ (evaluate across all r) (δar)A= Σ (evaluate across all r) r ar)A + Σ (evaluate across all r) (1 – r) (δar)A

This equation is decomposing the effect of A upon genes which are identical by descent and those which are not identical by descent using the coefficient of relationship. Hamilton, in another flurry of opaque formal transitions then rewrites the above as:

δTA = δRA + δSA

This sums up the total effect of genes by A upon those identical by descent (that is, related), δRA, and those not identifical by descent, δSA. Now I’m going to skip some algebraic manipulation, and translate these into the affect upon an allele frequency, pi, over time:

Δpi = { pi / ( R.. + δS.. ) } ( Ri.R.. )

At this point, it is important to note that R..represents the inclusive fitness as such, while Hamilton terms δS.. as the dilution effect. Remember, the latter are genes which are not identical by descent. Hamilton states that the sign of the inclusive fitness determines the direction of the change in gene frequencies, while δS.. influences the magnitude. The periods are placeholders for i & j, note that in this case one of the inclusive fitness variables does have i, indicating identity with the allele which we are tracking over time. After some algebraic manipulation Hamilton "proves" that the inclusive fitness is always maximized over time. I place quotations marks because Hamilton himself acknowledges "artificialities" in the model, for example, he uses weak selection to approximate zero selection because he isn’t changing the coefficients of relatedness over the generations through the iterations. His apologia is that selection as such should be weak, and that large effect mutations are ludicrous by definition. History is not on his side in this case! In any case, Hamilton makes it clear in the text that his goal is emulate the "classical model" of his time, which focused upon the spread of an allele via individual selection without an assumption of inclusive fitness, and that constrained and shaped his exposition and ends. In the next section, after a little algebra Hamilton constructs another equation where r is included more explicitly:

Δpi = ( pi r ) ( δTi. δT.. ) / ( 1 + – δT.. )

Remember that r manifests like so:

0.500 parent-offspring
0.250 grandparent-grandchild
0.125 great grandparent-great grandchild
1.000 identical twins
0.500 full siblings
0.250 half siblings
0.125 first cousins

In short, the rate of change of an allele in this case may be modulated by the relationship across which the allele operates. Hamilton says that "the advantages conferred by the ‘classical’ gene to its carriers are such that the gene spreads at a certain rate the present result tells us that in exactly similar circumstances another gene which conferred similar advantages to the sibs of the carriers would progress at exactly half this rate." Does this sound familiar? Recall the idea that the "gay gene" could spread because aid given to nieces and nephews at the expense of individual fitness; the implausibility of this sort of evolutionary action is simply due to the fact that with an r of 1/4 between aunts & uncles and nieces & nephews the fitness enough would have to be incredibly large on the order of multiples.

There are some further details I’ve left out of this "exposition," as it is. But I think I provided a taste of the general line of thinking that Hamilton is proceeding along. If you’re curious I highly recommend that you obtain a copy of The genetical evolution of social behaviour – I. The biographical introduction to these two chapters are extremely informative and illuminating; and give you a heads up on the weak points in the papers.

HAMILTON, W. (1964). The genetical evolution of social behaviour. I. Journal of Theoretical Biology, 7(1), 1-16. DOI: 10.1016/0022-5193(64)90038-4

CATEGORIZED UNDER: Genetics

1. DGS

Thanks for recommending this paper, I hadn’t read it directly… you helped me nail down the meaning of his dilution effect, my conception was equal to yours but I didn’t have 100% confidence in it.
I really liked how he brought the spatial component into it. It is a natural extension once you’ve started to look at the implications of “viscous” populations, it leads to conceptions like the figure in p.15, and together with his consideration of what would happen to a new mutation (p.14), to applications of game theory. It’s all linked.
The clumsiness of the delivery of the theory is far outweighed I think by the “but of course!”-ness of his thoughts. Using a relatedness vector to weight effects of other individuals seems straightforward to us now, but at the time, I have to think not. I think it was in E.O. Wilson’s autobiography where he wrote about reading these two papers on a train trip and thinking jealously, “but of course, he’s solved it all”.
I also like his statement (p.16) regarding lack of counter-selection, even when lack of direct selection exists. This is sometimes missing in evolutionary analyses, examining the nature of selection against a trait when selection for it is weak or lacking.

2. The clumsiness of the delivery of the theory is far outweighed I think by the “but of course!”-ness of his thoughts.
agreed! it’s a hard paper, but i think worth it (i don’t know if the notation was more common in the 1950s or something, i haven’t read moran’s original papers).

3. It is a hard paper! Having spent several hours struggling with it, I can quite understand the referees who asked for the maths to be split out from the rest of the paper. And they certainly did Hamilton a favour by doing so: I suspect fewer people would have been willing to read the combined paper with all the maths up front. I would probably need to set aside at least a week if I was to understand all of the mathematics.
As you said, the notation is pretty obscure, but he might have borrowed that from others. It would probably have helped to have copies of his references at hand. (That’s what the web is for isn’t it? Roll on open access!)

4. jfc kingdom, proceedings of the cambridge philosophical society 57, 574 (1961)
cc li and l sacks, biometrics, 10, 347, (1954)
o kempthorne, an introduction to genetical statistics, (wiley, new york, 1957)
jbs haldane and sd jayakar, journal of genetics, 58, 81 (1962)
pap moran, the statistical process of evolutionary theory, p 54 (clarendon press, 1962)
pag scheuer and sph mandel, heredity 31, 519 (1959)
hp mulholland and cab smith, american mathematical monthly 66, 673
jfc kingman, quarterly journal of mathematics, 12, 78 (1961)
jbs haldane, transactions of the cambridge philosophical society, 23, 19 (1973)
i think it would be in moran. that’s the treatment he’s trying to emulate.

5. windy

no biggie, but “the fitness enough would have to be incredibly large” fitness gain or something?

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