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R! How can relatedness be negative?

Hamilton's equation for predicting the evolution of altruism is widely misunderstood. A simple diagram from a classic paper can help.
Source: Jesus and Mo

You sank in the opinion of your fellow-men... by leaving your money in a capricious manner without strict regard to degrees of kin...
There wasn't much good i' being so rich... if she'd got none but husband's kin to leave it to.

- The Mill on the Floss

This week I will discuss two classic papers on how relatedness affects the evolution of social behavior. Altruism towards relatives is widely recognized, but W.D. Hamilton was apparently the first to make quantitative predictions of how relatedness would affect the evolution of altruism. In 1964, he published two papers on "The genetical evolution of social behavior"? (J. Theor. Biol. 7:1-52). Hamilton's rule c < r b is now widely known, but also widely misunderstood. The rule states that a gene causing some altruistic behavior (donating blood, say) may spread if the cost of the activity is low enough, and if it preferentially benefits others who carry the same gene, typically because they are genetically related. The cost to the donor and benefit to the recipient are c and b, both measured in fitness units (average increase or decrease in reproduction, due to the altruistic activity). But what is r?

Wikipedia says r is the "coefficient of relatedness."? This is consistent with J.B.S. Haldane's famous joke: "I would lay down my life for two brothers [1/2] or eight cousins [1/8]."? But this definition is not necessarily right. By the correct definition, r can even be negative!

The most intuitive explanation of Hamilton's r that I've seen is in Alan Grafen's "A geometric view of relatedness"? (Oxford Surveys in Evolutionary Biology 2:28). He plotted the frequency of the altruism gene in an actor A, in a group of potential beneficiaries B (each of whom may receive a benefit b, at a cost c to A), and in the population P with which A competes for resources. (The importance of this definition of P will be explained later.) The frequency of the gene in a group or population is the fraction of the group having two copies (one each from mother and father), plus half the fraction having one copy. This frequency can range from 0 to 1. The frequency of the gene in an individual, such as A, is either 0, 0.5 (a copy from one parent only), or 1 (copies from both parents).
Grafen defined Hamilton's r as the fraction of the way along the line from P to A where B is found. In the drawing, B is halfway from P to A , so r is 0.5.

If altruism depends on two or more genes, you can use a two- or three-dimensional version of this graph, but things get complicated if B isn't somewhere on the line connecting P and A.

Does this definition make Hamilton's rule work? If group B produces 10 seeds -- for now, assume we're talking about plants -- that will move the frequency of the altruism gene in the population P in the direction of A. How far? Half as far as if A had produced 10 seeds. B would have to produce 20 seeds to have the same evolutionary effect as A producing 10 seeds. So, a gene that causes A to make 10 fewer seeds (from not shading a neighboring plant B, say) will become more common only if it increases B's seed production by more than 20. Hamilton's rule works with this definition of r. The gene spreads if and only if c < 0.5 b.

Is Grafen's definition of Hamilton's r also consistent with genealogical definitions of relatedness? Sometimes. Suppose group B grew from seeds produced by plant A last year, using a random sample of pollen from the whole population P. In other words, assume mating is random. Each B plant has one copy of each gene from A and one from P. So the frequency of the altruism gene in group B will be halfway between P and A, as graphed above, and therefore r is 0.5. This is the usual relatedness between parent and offspring.

So far so good. But r can take on surprising values, including negative ones, which are inconsistent with our usual understanding of genetic relatedness. For example, suppose mating isn't random. If individuals are more likely to mate with those that are genetically similar, then B could be genetically closer to A than to P, and r would be > 0.5. In that case, the altruism gene would become more common even if the cost c were more than half the benefit b.

Even stranger, an animal might be able to identify some group B -- maybe we should call it V for victim -- that is less similar to them itself than the population as a whole is. Then r is negative, and a gene to spitefully reduce the reproduction of that group, even at some cost to its own reproduction, could spread.

Positive values of r depend on benefits from A going only to a group B that is more likely to share A's altruism gene(s) than the population P does. How might this happen? Animals may recognize kin in various ways and direct help preferentially to them. This may even be true of some plants and microbes.

But is kin recognition essential for r to be >0? Plants that drop seeds on the ground tend to be surrounded by their own seedlings. Bacteria that reproduce by dividing may be surrounded by clonemates. Wouldn't this automatically make potential recipients of altruism more likely to have the same altruism gene(s), relative to the overall population?

No. Remember that P is the population with which A competes for resources. This definition isn't arbitrary. Up to this point, we've implicitly assumed no competition. That is, reproduction by B doesn't have any negative effect on reproduction by A. But Imagine a plant A surrounded by its own seedlings. They are closely related, but they are also competing for the same soil resources, and for light. These resources can support only a limited number of plants. Once that limit is reached, any reproduction by B reduces reproduction by A (which is more similar to itself than to B) by the same amount. So, if competition is strictly local, we don't expect altruism to evolve. This requires that r=0 and therefore P=B. This is turn implies that P is the local, competing population, as defined above.

If seeds are widely dispersed by the wind, then P will be more widely distributed, so P will be less similar to A, potentially increasing r. But wind dispersal may also bring less-related seeds into the neighborhood B, decreasing r. These two effects tend to cancel each other, so limited dispersal does not automatically increase r. It depends on the timing of dispersal relative to competition and potential altruism.

Here's one last example. B is the population of rhizobium bacteria in root nodules on a given plant, with a half of the nodules occupied by each of two bacterial strains. Both strains benefit from associating with a healthy plant, but only A has a gene that makes it invest in the expensive process of nitrogen fixation. By helping the plant to photosynthesize, nitrogen fixation indirectly helps all rhizobia infecting that individual plant. The frequency of the beneficial gene in beneficiary group B is 0.5. What is P? Two possible answers are graphed below.
If the bacterial population in the root nodules is a random sample of the population, then P=B and r=0. But what if the beneficial gene is much more common in nodules of this plant than in those of other plants nearby, or in the soil? (Perhaps it's the result of a recent mutation that occurred near this particular plant.) The frequency of the altruism gene in overall population P could be close to 0 and r would be about 0.5. Then, if the cost c of nitrogen fixation is less than 1/2 the benefit b it generates for members of B, the frequency of the gene will increase.

However, B would tend to approach P over time, to the extent that the bacterial population in a given plant is a random sample of the local population. Therefore, we don't think the collective benefits to rhizobia from associating with a healthier plant have much to do with the evolutionary persistence of nitrogen fixation by rhizobia. Instead, we think that rhizobia fix nitrogen to hold off plant sanctions directed at individual nodules that fix little or no nitrogen. With one rhizobium genotype per nodule, B=A, so r=1. The benefits to millions of clonemates from averting host sanctions outweigh the cost to a rhizobium cell of fixing nitrogen instead of using the same resources to reproduce.

There is no evidence that Hamilton associated with pirates.

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