R. A. Fisher

Peter Ralph and Graham Coop have an interesting paper in the current Genetics, titled, "Parallel Adaptation: One or Many Waves of Advance of an Advantageous Allele?" [1]

Fisher [2] famously considered the case in which an advantageous allele is dispersing through a spatially dispersed population, showing that the dispersal forms a "wave of advance". This work was the foundation for a lot of progress in understanding spatial dynamics of organisms.

As I discussed in 2008 ("Overstating the obvious"), one of the consequences of the Fisher wave model for human evolution is that advantageous alleles will spread very slowly through the population. During the course of the Holocene, a strongly selected mutation might move only across a radius of a thousand or so kilometers. That provides one explanation for why new advantageous alleles haven't spread very far beyond their points of origin -- they just haven't had time yet.

Another reason why an allele might not have spread widely is interference from other alleles with similar effects. I mentioned this process last year ("Spatial variation and near-fixed selected alleles"):

Greg Cochran and I have been discussing this idea for some time. We call it the "Stooge effect". Think of the Three Stooges all trying to run through a door at the same time and getting stuck in the middle. That's what these genes are doing -- all of them are competing to respond to selection, but each is slowed by the presence of the others.

Ralph and Coop have cleverly combined the "Stooge effect" phenomenon with spatial dispersal. They suppose a case in which two separate advantageous mutations arise in different geographic locations, each affecting the same trait. Each begins to spread independently as a Fisher wave of advance. What happens when they meet?

As they show, the dynamics in this case give rise to a static equilibrium -- once the "waves of advance" meet, they stop moving, forming a stable boundary. A new favorable mutation makes headway only so long as it has no equally favorable mutation to compete against.

I like the way they used both analytical approaches and simulations to come to this outcome. The appearance of stable boundaries in a reaction-diffusion system has long been known (demonstrated first by Alan Turing, actually!). But to my knowledge, no one has considered this specific case from an analytical perspective.

The Fisher equation is not all that simple for most students to work with. If you become familiar with the equation, you will notice the key aspect is that it has two separate components -- a logistic (or reaction) component representing the increase in frequency at a single point in space, and a diffusion component representing the dispersal across space.

The muscle of the dispersal process comes from the logistic component. Without the intrinsic growth of the selected allele, the dispersal of individuals along the boundary would not carry many copies of the selected allele into new geographic areas. If the local selective advantage dies, the wave of advance rapidly stalls. A static equilibrium arises, with the frequency of the selected allele forming a cline that correlates with the local selection pressure.

Ralph and Coop's model approximates this case, in a dynamical sense. Each new selected mutation forms an increasing zone in which the selective advantage of other mutations is zero. When those other mutations encounter this zone, they form a stable cline. The cline is stable in the short term, but the diffusion component still disperses copies of an allele; they just lack the muscle to continue their deterministic expansion.

The most interesting simulations by Ralph and Coop show the two-dimensional case, in which the stable boundaries emerge in a "tesselation" pattern.

Figure 6 from Ralph and Coop (2010), showing "tesselations" in 2-d simulations of waves of advance.

The lower three panes in the figure show the stability of the boundaries between the selected alleles. They proceed to fixation locally, but their dispersal stops where they come into contact with other adaptive alleles. Over the very long term, the population will mix -- the diffusion process will slowly carry all these alleles throughout the species' range. Look at the process after a million generations and the entire zone will be gray. But this dispersal occurs at the neutral rate, where the diffusion term is the only factor driving the dispersal.

What about humans?

My graduate student Zach Throckmorton and I have been working in this area for a while now. One of the things that impresses us is the way that much more interesting dynamics can emerge when you alter the assumptions. I learned some of this stuff by talking to Frank Livingstone, who gave a lot of thought to these issues of spatial dispersal and selection as applied to malaria resistance alleles.

In particular, Frank thought about the case where one allele has a slightly larger advantage than another. In some contexts, this allows the "better" allele to overtake and swamp the expansion of the "weaker" (but nonetheless adaptive) one. In others, the two come to a near standstill, one displacing the other only very gradually. Much depends on the timing of the two mutations and the local conditions controlling their initial dispersal.

Ralph and Coop briefly consider this case in their paper, noting that the difference in fitness advantage of two alleles will allow one to advance into the range of the other, albeit at a slower rate. In humans, we may be seeing a smaller subset of cases, where one or more of the alleles have not yet established a wavefront. In these cases, the arrival of another wave can disrupt the spatial pattern of the rarer allele. The diploid case gives rise to the possibility of more complex epistases. Well-defined boundaries between selected alleles are rare, and where they occur (as may be the case with HbC and HbS in Africa), many have focused on negative epistasis as an explanation.

Also, alleles are unlikely to substitute perfectly for each other. In many cases, they may work synergistically -- individuals carrying two selected alleles that affect the same function may outperform those carrying only one such allele. At some point, new selected mutations may start to have diminishing returns, even on a trait like skin pigmentation where dozens of alleles may have been selected in widespread human populations. So the current distribution may to some extent be "frozen", but by a more complicated dynamic than the simple intersection of waves of advance.

As Coop and colleagues showed last year [3], and we discussed in 2007 [4], there are really only few genes that have approached local fixation in recent human evolution. The current spatial pattern of recently selected alleles doesn't look like a tesselation with many alleles near local fixation. Over most of the Old World, it looks like populations have a very large number of very new alleles, far from fixation, and few up over 70 percent in frequency.

So the specific scenario in this paper by itself probably does not explain the overall empirical pattern in humans. But if we consider the current pattern as a transient, approximating the early stages of dispersal for many selected alleles, we may not be terribly far off the mark.

Mutation-limited evolution

This is a long dense paper and there's a lot in it. One further aspect of the paper that I think is essential is the way that Ralph and Coop reiterate the basic point that more people means more mutations. In their case, they focus on population density over space (population number, when you multiply them) as a constraint on the number of possible adaptive mutations. They apply this idea as a hypothesis to account for parallel adaptations that may have emerged in recent human evolution.

Multiple mutational origins are likely if the characteristic length is shorter than the physical dimensions of the region. Eurasia measures >8000 km across, and so Table 1 suggests that multiple origins at a single base pair are very unlikely at the lower population density. On the other hand, if the mutational target is large, then multiple origins are likely at low densities, while at high densities independent origins are ubiquitous. The complementary cases of (rho = 2, µ = 10^–8) and (rho = 0.002, µ = 10^–5) give identical characteristic lengths of 3000 km, although the timescale on which the mutations spread differs. Thus for these two parameter combinations we can expect a few mutations to dominate within continents and for multiple mutations to be common in a population spread across an area the size of Eurasia. Obviously these calculations are very crude, as population densities vary through space and time, and dispersal across continents is not simply a function of geographic distance and individual dispersal. Nevertheless, these calculations suggest that it is plausible that for adaptive traits with reasonable mutational targets (e.g., a change anywhere within a gene or pathway) even low population densities can lead to parallel adaptation across an area the size of Eurasia, and higher densities almost certainly will.

We note that as human population densities have increased dramatically over time, so too has the probability of parallel adaptation. It is interesting therefore to note that a number of recent human adaptations (e.g., sickle cell alleles) involve repeated changes at very small mutational targets in relatively small geographic areas, while older adaptations from single changes (e.g., skin pigmentation) are more broadly spread.

They are describing a scenario in which small human populations would have been mutation-limited -- that is, the number of new mutations is small, making it unlikely that adaptive mutations will happen in any given generation. In such populations, the rate of adaptation is limited by the availability of new mutations. In an extreme -- in the very small effective sizes of Pleistocene human populations -- the rate of adaptation may be extremely slow and regional populations may come to differ at many weakly selected loci, which spread very slowly.

As the population grows, strongly adaptive mutations become more and more likely to happen somewhere in the species' range. Yet they are still relatively rare -- meaning that they have an opportunity to spread fairly far before encountering another equally strongly selected mutation affecting the same trait.

This process can give rise to very large differences on a continental scale, even when the selection pressures in different regions do not differ. In humans, the dispersal of selected alleles across space may have been significantly accelerated by actual dispersals of populations. It is not a mere coincidence that very widespread alleles in Eurasia also tend to be much older than 20,000 years old -- long-distance dispersals prior to that time had a higher chance of leaving a lasting influence on subsequent populations.

But as the population gets bigger and bigger, parallel mutations are more and more likely to happen. As Ralph and Coop point out, at the extreme of large population size and likely mutations, you shouldn't see any new mutations emerging and spreading over very large areas. Any of these mutations would be very likely to encounter other new mutations that do the same thing.

Is this likely in humans? Clearly some mutations have happened recurrently. Making a broken gene is easy -- there's a large mutational target, since a large fraction of nonsynonymous substitutions might do the job. So if there's a net selective advantage to breaking a gene, we ought to see that happen recurrently in human populations.

In contrast, if the mutational target is very small, then mutations will still be rare even in a very large population. If only one base change can have an adaptive effect, that precise change will happen less than once in 10⁹ births (remember that not just any mutation at a site, but some particular mutation is what we may need). If a rare duplication or gene conversion is the necessary change, then it may be much rarer.

Looking across the last few million years, when human population numbers were much smaller than the Holocene, we can be pretty sure that some aspects of our evolution were mutation-limited. The changes that took hold in our ancestors were the ones that happened, and that survived the winnowing of genetic drift. Many changes that would have been adaptive didn't happen in our ancestors. They just weren't lucky enough.

But some of those changes would still be adaptive now, if we could get them. And we have had much larger numbers in the last 10,000 years. Homo erectus needed these mutations, but we only now are seeing them selected in the human population.

Malaria adaptation

Hemoglobinopathies are among the cases of easy mutations -- where breaking a gene is adaptive. It's not just any broken version of alpha- or beta-globin that does the job, though. The hemoglobin needs to be impaired in certain ways to impede the parasites while maintaining blood function. This provides many of the classic cases of human adaptation, and Ralph and Coop turn to this system for examples of parallel adaptation:

The sickle cell allele HbS at the β-globin gene in humans provides a particularly interesting case of putative parallel adaptation. The HbS allele (β6 Glu-Val) has been driven to intermediate frequencies by selection within the past 10,000 years due to increased resistance to malaria of heterozygotes for the allele (HALDANE 1949; ALLISON 1954; CURRAT et al. 2002; KWIATKOWSKI 2005). The HbS allele is present on at least four major distinct haplotypes in Africa, each at intermediate frequency within a different geographic region; the haplotypes are named after the population sample where they were first discovered (Central African Republic, Senegal, Benin, and Cameroon). This is consistent with multiple origins of this single-base-pair change. Note that a distinct, malaria resistance allele, HbC (β6 Glu-Lys), has also arisen in Africa at the same codon as the HbS allele (TRABUCHET et al. 1991; AGARWAL et al. 2000; WOOD et al. 2005a), increasing our confidence that the mutational input was high enough to allow multiple types to arise. However, FLINT et al. (1998) thought the hypothesis of multiple new mutations arising at a single base pair was extremely unlikely and proposed that it was more likely that gene conversion had spread a single mutation across multiple haplotypes.

The theory we have developed can be used to assess the plausibility of the multiple mutational origins of the sickle cell allele, by exhibiting parameter combinations that yield characteristic lengths consistent with the separation of the sample locations. [Recall that the wave of advance, and thus also our model, works in the case of heterozygote advantage (ARONSON and WEINBERGER 1975).] The different HbS haplotypes co-occur within a few thousand kilometers of each other (see Table 5 of FLINT et al. 1998) (noting that these locations are unlikely to reflect the geographic mutational origins, and mutations will have been spread by large population movements). As the HbS changes occur at a single base pair, the mutation rate would have been 10^–8, and we take an s = 0.05 (as in CURRAT et al. 2002). If human dispersal at that time was well approximated by a Gaussian kernel with sigma = 100 km, then a characteristic length of 1000 km would require an effective density of individuals of rho = 25 km^–2, while if sigma = 10 km, then we would require only rho = 2.5 km^–2. This latter set of parameters does not seem unrealistic, considering our knowledge of population density and dispersal parameters, so our model suggests that the hypothesis of multiple origins is not unreasonable.

I think they've got the basic idea correct here, but there are some additional details to consider. The distribution of HbE is not quite so easy to understand if parallel mutations are really so likely, and of course there is the negative epistasis of different alleles (and the thalassemias) which impacts their dispersal ability when they become moderately common. The dynamic may be of similar form to the one described here, but boundaries between alleles may be reinforced by the fitness costs of carrying multiple ones.

This situation raises the issue of path dependence. Some mutations have "first mover" advantages. Once they are common, other adaptive mutations may still occur -- even mutations that are better from the standpoint of fitness -- but be lost or grow very slowly because their net fitness advantage over the common mutant is slight. Where HbE is common, new HbS alleles are unlikely to invade quickly. Where HbS is common, new HbE mutants are similarly unlikely to invade -- even though HbE has a higher fitness.

Network effects among genes may also dominate the spatial dynamics. HbS spread most widely in the context of populations that were already Duffy null, and in which G6PD deficiency was rapidly increasing. The first conditioned the parasite environment -- P. vivax had a strong disadvantage in Duffy null populations, P. falciparum made up most of the parasite load. G6PD deficiency should have impacted the relative advantage of HbS, more and more as it became more common. Those are two loci among many that alter malaria dynamics in Africa compared to South and Southeast Asia.

Conclusions

There is much more to say about this paper -- it's 22 journal pages. But I think I've given an impression of what's there and how the ideas may impact our interpretation of recent human evolution. Many of the central concepts were presaged by earlier work in 2007 and 2008, as reviewed here on the blog. The new analytical and simulation work, I really like.

Hopefully we can get out some shorter papers that will focus on aspects of these problems as applied to humans. A message that comes across very clearly in our work and this new paper is that different time periods in our evolutionary history must have had very different selection dynamics. Pleistocene humans were not only in a different ecology than us, they experienced a radically lower potential for adaptation.

References

Chapter 2 of R. A. Fisher's Genetical Theory of Natural Selection is remarkable for many reasons. In it, he presents a model of selection in an age-structured population, the concept of reproductive value, and the Fundamental Theorem. Toward the end of the chapter, he discusses "The Nature of Adaptation," presenting a geometric model to justify the assertion that the probability of favorable genetic changes declines as the effect size of those changes increases.

In order to consider in outline the consequences to the organic world of the progressive increase of fitness of each species of organism, it is necessary to consider the abstract nature of the relationship which we term 'adaptation.' This is the more necessary since any simple example of adaptation, such as the lengthened neck and legs of the giraffe as an adaptation to browsing on high levels of foliage, or the conformity in average tint of an animal to its natural background, lose, by the very simplicity of statement, a great part of the meaning which the world really conveys. For the more complex the adaptation, the more numerous the different features of conformity, the more essentially adaptive the situation is recognized to be. An organism is regarded as adapted to a particular situation, or to the totality of situations which constitute its environment, only in so far as we can imagine an assemblage of slightly different situations, or environments, to which the animal would on the whole be less well adapted; and equally only in so far as we can imagine an assemblage of slightly different organic forms, which would be less well adapted to that environment (38).

I've highlighted that last sentence, which is saying that organisms fit their environments in possibly many different ways, so that their fitness is not actually tied in any single feature, such as "tint," but is instead an optimum of many features, with respect to any single factor the organism may be more or less well adapted. The rest of that paragraph continues on to make the same point.

Then:

The statistical requirements of the situation, in which one thing [the organism] is made to conform to another [the environment] in a large number of different respects, may be illustrated geometrically. The degree of conformity may be represented by the closeness with which a point A approaches a fixed point O. In space of three dimensions we can only represent conformity in three different respects, but even with only these the general character of the situation may be represented. The possible positions representing adaptations superior to that represented by A will be enclosed by a sphere passing through A and centered at O.

This is really very simple, and the geometric model reveals an interesting switch. Suppose that we imagine an organism as a set of a traits, each of which lies at some distance d₁, d₂, d₃, ..., d_a from the optimum value for that trait, O. We could imagine adaptation as a stepwise process, in which a any one of the a traits may change, and only those changes that reduce d_a will potentially be selected.

But there's no reason at all why we should consider every trait as an independent entity. Suppose that a single genetic change could improve two traits, or three, or even all the traits at the same time.

Or, more interesting, a change might greatly improve trait 1, while making all the rest of the traits marginally worse. Without a word, Fisher has removed the issue of adaptation from the fit of many separate variables, to a single distance in multidimensional space -- a change from cartesian to polar coordinates, as it were.

If A is shifted through a fixed distance, r, in any direction its translation will improve the adaptation if it is carried to a point within this sphere, but will impair it if the new position is outside.

The geometric model really assumes very little. It tells us nothing at all about the relationship between fitness and any particular phenotype, aside from assuming that (1) the relation is continuous within the sphere centered on O with radius A, and (2) there are no "holes" of low fitness within that sphere. This is not a fitness landscape. Fisher's view, as will become clear, was that species are well-adapted in nature. He assumes that the distance between A and O is always rather small in comparison to the kinds of phenotypic effects that mutations might cause. So in assuming that the fitness function is continuous, and that the area is small, he more or less automatically arrived at the assumption that there's only one peak at O.

The next part is the famous one that people remember:

If r is very small it may be perceived that the chances of these two events are approximately equal, and the chance of an improvement tends to the limit 1/2 as r tends to zero; but if r is as great as the diameter of the sphere or greater, there is no longer any chance whatever of improvement, for all points within the sphere are less than this distance from A.

In this model, small changes are roughly 50-50 beneficial versus deleterious, but since their effect is very small, it hardly matters. Big changes are much less likely to be beneficial -- and if they exceed twice the distance from A to O, they can never be beneficial.

After this quote, he gives an exact expression for the probability that a given change r is beneficial ((1/2)(1-(r/diameter))), but this is limited to the three-dimensional model. Then, there's another interesting one-sentence switch:

The chance of improvement thus decreases steadily from its limiting value 1/2 when r is near zero, to zero when r equals d. Since A in our representation may signify either the organism or its environment, we should conclude that a change on either side has, when this change is extremely minute, an almost equal chance of effective improvement or the reverse; while for greater changes the chance of improvement diminishes progressively, becoming zero, or at least negligible, for changes of a sufficiently pronounced character.

Remember that A represents the current "degree of conformity" of the organism to its "particular situation." This degree of conformity might be changed either by changing the organism or its environment -- the implication fo the last confusing sentence from the first paragraph, above.

The point O is not an objective location (as it would be if we assumed it is the optimum within the current environment). Instead, it is a geometric abstraction that exists only in so far as it bears a relation to A in the present multidimensional space of "adaptation". We may change the distance from A to O either by changing the organism or by changing its environment. Fisher refers explicitly to this "assemblage of slightly different situations" with respect to both -- and again, the concept of "slightly different" underlies the assumption that the fitness function within the sphere is continuous.

It seems to me that the idea of "niche construction" falls very easily within Fisher's model. An organism that systematically alters its environment may thereby change its level of adaptation. So even though mutation is an obvious candidate for a process described by the model, I've continued to refer to "change" as a nonspecific term for the unit of adaptation.

The representation in three dimensions is evidently inadequate; for even a single organ, in cases in which we know enough to appreciate the relation between structure and function, as is, broadly speaking the case with the eye in vertebrates, often shows this conformity in many more than three respects. It is of interest therefore, that if in our geometrical problem the number of dimensions be increased, the form of the relationship between the magnitude of the change r. and the probability of improvement, tends to a limit which is represented in Fig. 3. The primary facts of the three dimensional problem are conserved in that the chance of improvement, for very small displacements tends to the limiting value 1/2, while it falls off rapidly for increasing displacements, attaining exceedingly small values, however, when the number of dimesnions is large, even while r is still small compared to d.

Here we see the problem of universal pleiotropy emerging. As the number of dimensions of adaptability increases, the probability that one change will be a net increase in fitness, considering all dimensions, must decrease. This probability remains larger when the distance between A and O is larger, but declines as the number of dimensions increases.

However, what we would call "universal pleiotropy" is intrinsic to Fisher's assumption that the direction of changes in this multidimensional space is random. If changes may occur in any direction, this is the same as asserting that a mutation may induce correlated changes between any set of phenotypes. With thousands of genes, and therefore thousands of dimensions of genetic "adaptedness", we might guess that the dimensionality is high enough to make this assumption useful.

If on the other hand, the direction of changes is constrained in some way, then the dimensionality of the space accordingly is smaller by some degree. This is the rough equivalent of modularity in Fisher's model: if we say that some genetic correlations cannot be changed, we are saying that the phenotypic structure of the organism is modularized.

The remainder of the section discusses the general case in spaces with higher than three dimensions. The basic point is that the probability that a change of a given size will be adaptive increases with the distance from A to O, decreases with the effect size r, and also decreases with the number of dimensions.

Fisher finishes with a paragraph that, had he begun the section with it, might have made everything much clearer:

The conformity of these statistical requirements with common experience will be perceived by comparison with the mechanical adaptation of an instrument, such as the microscope, when adjusted for distinct vision. If we imagine a derangement of the system by moving a little each of the lenses, either longitudinally or transversely, or by twisting through an angle, by altering the refractive index and transparency of the different components, or the curvature, or the polish of the interfaces, it is sufficiently obvious that any large derangement will have a very small probability of improving the adjustment, while in the case of alterations much less than the smallest of those intentionally effected by the maker or the operator, the chance of improvement should be almost exactly half.

But there is an obvious objection: if you twist a knob on a microscope, it may move one of the lenses, but it isn't going to polish it. It's not possible to effect simultaneous changes on distinct elements of the microscope, because a microscope is in fact modular in its construction and controls. That doesn't disprove Fisher's model, but at least in the microscope case, the possible changes are not random in direction, but are constrained to "Cartesian"-like independent axes.

Fisher's general idea is different from the fitness landscape, in the sense used by Sewall Wright. Fisher assumes a single adaptive optimum; Wright espoused the possibility of a "rugged" landscape in which large genetic changes might diverge from the nearest local optimum yet place the population near an alternate (possibly higher) local optimum. But the geometric model shares many properties with the fitness landscape, including its assumptions about hill-climbing toward the local optimum.

I pointed to Sergey Gavrilets work a couple of weeks ago. Fisher's geometric model is most similar to the second meaning of fitness landscape, which pertains to the mean fitness of a population given a discrete genotype -- or in this case, a discrete phenotype-environment combination. Fisher's model is entirely about the relationship of two equilibria: the population mean fitness before and after a given change. It does not deal with the dynamics of the transition from one state to another.

References:

Fisher RA. 1930. The Genetical Theory of Natural Selection. Clarendon Press, Oxford.

I went looking for Lowie, because I was curious about the introduction of the diffusion concept into cultural anthropology. The mathematical description of diffusion, originally developed in thermodynamics, became important in statistical genetics during the first half of the twentieth century. In particular, R. A. Fisher introduced diffusion methods to examine the effects of natural selection in his 1922 paper, "On the dominance ratio." Diffusion methods made it possible to derive analytical approximations for many interesting biological parameters, and also came to underlie models of population dynamics outside the field of genetics.

So I wondered: How did the use of diffusion in cultural anthropology compare to the introduction of the diffusion concept in genetics? Kroeber's systematization of the concept of cultural diffusion was certainly later than Fisher and Wright's major works on diffusion theory.

It turns out that "diffusion" was initiated into cultural anthropology by E. B. Tylor himself. The OED has the earliest anthropological use of the term in Primitive Culture:

How good a working analogy there really is between the diffusion of plants and animals and the diffusion of civilization, comes well into view when we notice how far the same causes have produced both at once.

Later, Boas also makes extensive use of the concept of diffusion, in essentially the modern sense as an alternative explanation to independent invention for a cultural trait. For example, in his 1891 article, "Dissemination of tales among the natives of North America," he compares the transfer of myths and stories among groups in the New World to that in the Old. He writes:

Then, we may ask, is there no criterion which we may use for deciding the question whether a tale is of independent origin, or whether its occurrence at a certain place is due to diffusion? I believe we may safely assume that, wherever a story which consists of the same combination of several elements is found in two regions, we must conclude that its occurrence in both is due to diffusion. The more complex the story is, which the countries under consideration have in common, the more this conclusion will be justified (Boas 1891:13-14).

So by the time Lowie was writing his Culture and Ethnology, the concept of cultural diffusion was well-established, and the Boasian school was concerned with classifying culture similarity in terms of diffusion.

Cultural problems tended to involve the spread of relatively large quantities of information -- reflected by the Boas quote above and its concern with the "combination of several elements." We can interpret this focus as a statistical consequence of observing culture: With so many potential observations, diffusion is difficult to distinguish from the null hypothesis of parallel development (chance similarity) unless the similarities are sufficiently detailed (involve a threshold of information) to prove otherwise.

I have not yet found anywhere that this concept of assessing diffusion by "information content" was formalized beyond verbal descriptions like Boas' above. I will review Kroeber's contribution later; he does not provide any formalism at all.

What I want to point out here is that this concept of diffusion is delimited quite differently from the use of diffusion models in genetics. Fisher and Wright initially introduced diffusion methods to deal with the effects of random changes of gene frequencies. In the case of Boasian cultural diffusion, random change will almost always fall short of the information content necessary to identify specific resemblances in a cross-cultural context.

I can imagine datasets on cultural traits that would be sufficient to test the hypothesis of undirected (non-selected) diffusion. For example, phonological data on dialects is often detailed and coded to geographic locations. If we approached these data with a diffusion model, they would in many cases be sufficient to demonstrate departures from the null model of undirected diffusion.

But most observations from ethnography are not of this character. For this reason, cultural diffusion is a priori a phenomenon involving direction by some selective mechanism. In genetics, this would be natural selection. In cultural variation, the selective mechanism may be less clear -- some combination of conscious decision, customs concerning borrowed behaviors, and functional efficacy may be involved.

In the case of natural selection driving a gene substitution through space, Fisher's model of a wave of diffusion assumed only a single parameter determining intrinsic increase -- the "reaction" term in a reaction-diffusion equation. This was sufficient because the only relevant difference between the selected and non-selected alleles was fitness.

The case of cultural diffusion, in contrast, makes it tempting to suggest that there might be many independent terms contributing to the spatial dispersion of a cultural trait. Traits might be different in that some are more transmissible than others; thereby spreading more widely. Some might have greater functional advantages than others. Some ideas might be impeded because they conflict with widespread taboos; others might be facilitated by the same factors.

I write all this because I'm curious about why there was not a more formal development of diffusion in the context of cultural theory. There was every potential of it: Boas did not begin with a formalization, but the ideas critical to a formal theory are present in his description. But the development of the field quickly went in the opposite direction -- away from formalization and toward description. This was despite the fact that the concept of diffusion became incredibly important in the conflict between the Boasian school of ethnology and the cultural evolutionism of Leslie White and others.

Others (non-anthropologists) did develop more formal theory, but this had little (if any) impact within anthropology. As I continue, I'll go back to the early cultural evolutionists Tylor and Morgan, and trace their influence on 20th century neo-evolutionists. Additionally, no account of cultural diffusion can omit the importance of the Kulturkreis school and its concept of the culture area.