Dynamical models of multilevel selection: Another problem with Kin selection

First off, if you haven’t seen it check out the American Museum’s on line collection of photographs. I haven’t had a chance to really explore the hundreds of thousands of photos they have, but I am certain there are some real gems in there.

G. G. Simpson

One of the photos from the American Museum of Natural History. This is G. G. Simpson at his desk, at the museum. AMNH has over a million photographs on line.

It turns out I am not quite done “dissing” kin selection, although my discussion this time is nothing I would have thought of as a problem. What I want to talk about is a pair of papers that appeared in Evolution in a special section on multilevel selection that I edited.

The first of these is my own paper on direct fitness and contextual analysis (Goodnight. 2013 Evolution 67, 1539-48). In this paper I work through the relationship between direct fitness and contextual analysis. It turns out that both of these approaches are using multiple regression to analyze selection. In direct fitness the equation is:

dyamical models eq 1

Where W is absolute fitness, (ind) is the individual trait, (grp) is the trait in interacting partners, and x is a measure of genotype. Without loss of generality I converted absolute fitness to relative fitness (come on guys, working with absolute fitness is for chumps!), and I recognize that because these models are so naïve there must be a function relating genotype to phenotype. Thus, there is a value dynamical models eq 2that relates a change in phenotype to a change in genotype. So multiplying through by dynamical models eq 2we get the equation for contextual analysis:

dynamical models eq 3

which is really the same equation, but to me much more aesthetically pleasing for two reasons. First, as I said, working with absolute fitness is for chumps (AND it makes a difference for contextual analysis), and second, get real, we cannot measure “genotype”, hell, I don’t even know what that means, whereas I have a very clear idea of what I mean when I measure the phenotype.

Anyway, be that as it may, the end result is that the difference between the direct fitness approach (or neighborhood modulated fitness approach) and the multilevel selection approach of contextual analysis does not lie in the equations they use. Rather it lies in HOW the equations are used. In the direct fitness approach the equation is solved for the point where dW/dx = 0. Mathematically this has to be one of three types of points, a fitness maximum, a fitness minimum, or an inflection point. Simple inspection can distinguish between these three possibilities (or second derivatives if you prefer). In contrast, in contextual analysis the slope is analyzed at the point  where the population is currently residing, and dw/dz becomes a measure of the rate of change in relative fitness as a result of multilevel selection. In any case, it is quite reasonable to argue that kin selection and multilevel selection are very similar if not the same thing.

Next, we turn to Simon, Fletcher and Doebeli (2013 Evolution 67, 1561-72.). This is a dynamical model of two level selection using a continuous-time Markov chain, and a companion deterministic partial differential equation model. One of the first things I got out of this model is that Burt Simon is a better mathematician than I am, but as far as my little mind is capable of understanding such things, this model is quite complete, and an excellent general model of multilevel selection. Without going into details they develop a pair of partial differential equations, one in which it is assumed that there is not change in the number or types of groups, basically the frequency of individual types is allowed to change, but the overall change in group types is zero:

dynamical models eq 4

where αi is the growth rate (births-deaths) if the ith type of individual, xi is the trait value of the ith individual, and t is time. They then go on to argue that there are group level processes (group extinction, recolonization, fusion, fission, differential growth) that enter in to the equation.   On the other hand, if no changes in individual fitness are allowed then:

Dynamical models eq 6

Thus, and without going into detail, they then show that the overall change in the trait is:

dynamical model eq 5a

Please remember I am not doing this model justice, so, either believe that what I say is true, or read it yourself. (Word of advice: As Reagan, citing an old Russian proverb, said: “trust, but verify”) (Comment 2, I have no idea why these equations are so ugly.  Click on them for a clearer view).

The result of this is that they argue that if a selective event changes only the αi – the growth rate of the ith type without affecting the distribution of group types then only individual selection is acting, if the selective event changes the distribution of group types with out affecting the growth rate of individual types it is a pure group selection event, and finally if both change it is a multilevel selection event.

They then go through two examples that show the logic of what they are talking about, and eventually ask whether inclusive fitness, that is there in all cases a function bi that can be found that successfully combines individual fitness effects and group fitness effects. The answer to this is no. They point out that the two level approach can be solved directly, but the one level approach necessarily requires the prior solution to the two level approach. In their words, the reductionist approach is not “dynamically sufficient”, and there is a real difference between multilevel selection and kin selection models.

This is an interesting conundrum. On the one hand, the non-dynamical models of kin selection and contextual analysis arguably suggest that the two processes are the same. A dynamical model indicates that they are not the same. Who is right?

The answer seems a bit complex. First off, Goodnight and Simon et al. actually have different definitions of group selection. The Goodnight definition is that group selection is acting when the fitness of an individual is a function of group membership. The Simon et al. definition is that group selection is acting when the outcome of selection depends on group level fitness effects. However, I don’t think this is the problem. I think the bigger problem is that direct fitness and contextual analysis are statistical models that measure the conditions at the current value of the population. Contextual analysis works here because it is measuring the regression slopes at the current population values. It is certainly possible to imagine a system that overall had multilevel selection acting, but at a particular set of gene frequencies (or what have you) group selection was not acting at that moment. Thus, at least in theory, the strength of selection at the two levels may change from generation to generation, and selection at one level might even disappear briefly. This rather minor problem for contextual analysis is a huge problem for kin selection. That is, another way of saying the complaint about these regression models is that there are non-linearities built into multilevel selection. I suspect that if you could force the models to be linear that the manipulation of equating inclusive fitness with multilevel selection in a dynamical model just might work. However, because the two levels will inevitably have non linearities, and in most cases will in some way interact, the linear approximation of kin selection models are doomed to failure.

In other words, kin selection practitioners are guilty of one of the basic errors that all undergraduate statistics students are taught. They are extrapolating from the current population conditions to some point in the far distant frequency space. In short, they are extrapolating beyond their data.



2 Responses to “Dynamical models of multilevel selection: Another problem with Kin selection”

  1. Bjorn:

    Actually the other way around I would think, that is a phenotype genotype map. It is saying that a given change in phenotype translates to a change in genotype, and there is only one genotype per phenotype. Doesn’t make much sense.

    I think the bigger problem is the dw/dx of kin selection. This is saying that a given gene has a single identifiable effect on fitness, and a change in gene value has a differentiable change in fitness. One could argue they mean something more than a gene, but remember x is a single value, so it is either the value of the gene, or a summary statistic of a set of genes (the whole genome)? The only good summary statistic I know for the whole genome is called the phenotype.

    I am not comfortable with either dw/dx, or dx/dz since they both imply a genetic world that is much simpler than can possibly be true. The dw/dx was thrust on me and it is an over-simplistic concept that calls for an overly-simplistic solution. That is exactly what dx/dz: a simplistic solution to a problem brought on by over-simplification.

  2. Does multiplying with dx/dz imply anything about the genotype-phenotype map, like it is one-to-one, or can it be anything?

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