A puzzle for Wagnerian geneticists (if such exist): what is the coefficient of relationship between Siegfried and Brünnhilde?

For the benefit of non-Wagnerians, Siegfried is the son of Siegmund and Sieglinde, the twin offspring of Wotan (in the guise of Wälse) and an unnamed mortal woman. Brünnhilde is the daughter of Wotan and the Earth goddess Erda. So far as we know, Wotan, Erda and the mortal woman are not inbred or related to each other.

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I was reading about kin selection, Hamilton’s Rule, etc, while listening to Wagner (as one does), and it occurred to me that it would be amusing to work out the coefficient of relationship between Siegfried and Brünnhilde. This would be easy enough if they were both outbred, but Siegfried is inbred (which incidentally may explain why he is such a moron), and elementary books on genetics (e.g. John Maynard Smith’s Evolutionary Genetics) usually dodge the complications of inbreeding. But I thought it would be easy enough to find a fuller treatment somewhere, and I looked at a few other books. This proved a frustrating experience, as there is a confusing diversity of terminology for measures of relationship used by different authors.

However, I thought I had hit paydirt in E.O Wilson’s Sociobiology, which gives a general formula for the coefficient of relationship between two individuals, allowing for inbreeding, in the form: rAB = 2fAB/root[(fA+1)(fB+1)], where A and B are two individuals, fAB is the coefficient of kinship between them, fA is the coefficient of inbreeding of A, and fB is the coefficient of inbreeding of B. The coefficient of kinship is the probability that a gene randomly selected from A is identical by descent (ibd) to a randomly selected gene at the same locus in B. This is a symmetrical relationship (fAB = fBA), and it is possible to calculate it whenever the ancestry of two individuals can be traced back to their closest common ancestors, who are either outbred or have a known level of inbreeding. In the case of Siegfried and Brünnhilde, fAB = 1/8. The coefficient of inbreeding for an individual is the probability that a gene randomly selected from that individual is ibd to his other gene at the same locus, in other words, homozygous at that locus by virtue of descent from a recent common ancestor. Since the two genes at the same locus in an individual are randomly selected from the individual’s parents (one from each parent), the coefficient of inbreeding for an individual is equal to the coefficient of kinship between the individual’s parents. Siegmund and Sieglinde are full siblings, with a coefficient of kinship of ¼, so the coefficient of inbreeding for Siegfried is also ¼. Brünnhilde, on the other hand, is outbred, with a coefficient of inbreeding of 0. Using Wilson’s formula (which is taken from Sewall Wright), if Siegfried is A and Brünnhilde B, rAB = 2x1/8/root1.25 = approx .225.

So far so good. However, Wilson gave no explanation of the basis for his formula, and I was curious to know more. The good news is that in Robert Trivers’s Social Evolution I found a plausible justification for a formula of relationship allowing for inbreeding. The bad news is that it is different from Wilson’s! Trivers’s formula for the ’degree of relatedness’ is 2fAB/(fA+1), where fA and fAB have the same meaning as before, and where A is the individual performing an action, and B is the individual receiving its effect. So if the roles are reversed, the formula would be 2fAB/(fB+1), which takes a different value from 2fAB/(fA+1) if A and B have different coefficients of inbreeding. Both versions also have different values from Wilson’s formula in this case. Of course, different authors may reasonably use different formulae for different purposes, but Wilson and Trivers use their different (and sometimes conflicting) formulae for the same purpose, namely as the value of r in Hamilton’s Rule.

So which is right, and why? It took me a while to get to the bottom of this, but to cut a long story short(ish), when Hamilton originally introduced Hamilton’s Rule in 1964 he proposed that Sewall Wright’s coefficient of relationship would usually be the appropriate value for ’r’ in his Rule. Then in the early 1970s he proposed a modification (the formula used by Trivers) which gives different values in the case of inbreeding (see Narrow Roads of Gene Land, vol. 1, p. 179 and 272-3). The formula still depends on various assumptions for its validity. Notably, it assumes additive gene effects, and it disregards the effects of selection on the relative frequency of genes within pedigrees. This means that Hamilton’s Rule is only an approximation (as was recognised from the outset in 1964). To see this point, consider a gene that causes an individual to dispense altruism to his nephews at the expense of his personal fitness. Such a gene can only be found in the nephews (at higher frequencies than in the general population) if it is also found in one of their parents (the brother or sister of the altruist). But this means it will reduce their parent’s fitness, so there will be fewer nephews with the gene in question than might be expected. (Putting it another way, brothers or sisters of the altruist who do not carry the gene for altruism will have more offspring than those who do.) To allow for this would require the formula to be modified for every different kind of pedigree, so disregarding selection is a necessary simplification, OK?

Subject to these assumptions, Hamilton and Trivers both give justifications of the formula which are quite persuasive, but not strictly proofs. I will attempt a more direct justification. (I also assume that all the parties involved - gods as well as humans - are diploids).

Hamilton’s Rule can be expressed in the form br > c, where b is the fitness benefit to the recipient of an action, c is the fitness cost to the actor, and r is the appropriate measure of relationship between actor and recipient. What we want is a formula for r which makes the Rule true in cases involving inbreeding. The Rule will be true if the increase in the number of ibd genes for altruism passed on by the recipient is greater than the reduction in the number of ibd genes for altruism passed on by the actor. The increase or decrease in the number of ibd genes passed on is the increase or decrease in the number of offspring times the expected number of ibd altruism genes per offspring. The increase in the number of offspring of the recipient due to the gene for altruism [Note 1] is expressed by b, while the decrease in the number of offspring of the actor is expressed by c. Since both parties are diploids (by assumption) the expected number of ibd altruism genes per offspring will be half the number in the parent, so the Rule br > c takes the form b½(expected number of ibd altruism genes in recipient) > c½(expected number of ibd altruism genes in actor). The coefficient r is therefore equal to (expected number of ibd altruism genes in recipient)/(expected number of ibd altruism genes in actor). The probability that a randomly chosen gene in the recipient is ibd to a given gene at the same locus in the actor is given by the coefficient of kinship (fAB) between them. [Note 2] Since there are two genes at that locus in the recipient, the average expected number of genes in the recipient ibd with a given gene in the actor will be 2fAB. [Note 3] As to the number of ibd genes for altruism in the actor, we are considering the effect of a particular gene for altruism, so we know that there is at least one copy, and the probability that the other gene at the same locus is ibd with that gene is given by the coefficient of inbreeding, [Note 4] which is fA for the actor. The expected number of ibd genes for altruism in the actor is therefore fA+1. Putting these results together, we have r = 2fAB/(fA+1). QED.

Returning to Siegfried and Brünnhilde, if Siegfried is the actor r = .25/1.25 = .2, and if Brünnhilde is the actor r = .25/1 = .25. We might therefore expect Brünnhilde to behave better towards Siegfried than vice versa, which is indeed the case. (OK, even allowing for Hagen’s trickery, Siegfried was still a shit. What kind of ‘hero‘ takes a ring from a woman by force and then lies about it?)

Note 1
If there is more than one ibd gene for altruism in the actor, by assumption their effect is additive. We can therefore treat each gene as having its own distinct effect without worrying about the effect of the other one, if any.

Note 2
As usually defined, the coefficient of kinship is the probability that a gene randomly selected from A is identical by descent to a randomly selected gene at the same locus in B. But in this case we are selecting a gene in A which we know to be a gene for altruism, so the selection is not strictly random. Does this not invalidate the argument? I think the answer is that what matters is not whether the selection is strictly random, but whether the method of selection affects the probabilities. In this case the fact that we are selecting a gene for altruism does affect the probabilities, via the possible effects of selection, but by assumption we have agreed to disregard these.

Note 3
We are considering the average expected number of ibd genes in the recipient. For this purpose it is legitimate to calculate the expected number simply as probability per copy at a locus (i.e. average number in a large sample of copies) times number of copies at the same locus (i.e. 2 in the case of diploids). This gives the expected number per individual as 2fAB, whether or not the individual is inbred. This should not be confused with the question whether any (one or more) genes in the recipient are ibd with a selected gene in the actor. For outbred recipients, since each individual has at most one copy of the gene, the probability that he has any copy of the gene is the same as the probable number of copies per recipient, i.e. 2fAB. In the case of inbred recipients the probability would be somewhat lower than 2fAB (in fact 2fAB-fB), because some recipients would have two copies, and the same total number of ibd genes in the population of recipients would be concentrated in fewer individuals. Where both parties are outbred diploids, the coefficient of relationship reduces to 2fAB and can fairly be described as the probability that a randomly selected gene in one is shared by the other as a result of common ancestry, or equivalently as the proportion of all their genes which they share as a result of common ancestry, e.g. for full siblings ½, or for uncle and nephew ¼. Where one or both parties are inbred (or where they have a different ploidy, as with haplodiploid insects) these expressions are ambiguous and should be avoided. I have gone into this tedious detail because I have found many authors confusing in their terminology, and it took me some effort to work things out.

Note 4
The same qualm about non-random selection arises here: see Note 2.