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NOTE: I had a couple typos in my equations in the original. This is updated and fixed, and hopefully totally correct. Thanks to bioIgnoramous for pointing it out.

Over at Scienceblogs, people are talking about waves. Of course, everyone thinks that waves are in the domain of physics, and people always forget about one of my favorite subjects: waves of advance. Way back in the day, RA Fisher wondered what might happen if genes had to spread not just locally but across space, and he published his findings in a landmark article called The Wave of Advance of Advantageous Genes. This paper was not just important for its contributions to population genetics, but because of fundamental contributions to applied mathematics. As far as I can tell, Fisher and the great mathematician Kolmogorov published similar findings on this same subject in the same year. To that end, these kinds of waves of advance are often referred to as “Fisherian” waves.

What was Fisher’s model, what did he find, and how has it been extended?

Fisher assumed (without much justification) that genes should diffuse through a population, much like dye diffusing through a glass of water. In addition, at each point the dynamics would be affected by natural selection. This led Fisher to writing down a partial differential equation:

Where p is the frequency of the advantageous allele, t is time, and x is space. The parameter sigma^2 is the varaiance in the parent-offspring distance, and s is the selective advantage of one allele over the other. He then assumed that the solution had the form of a wave of stationary shape, and was able to derive a necessary condition on the velocity, namely that the velocity is at least

This showed that even though a gene can sweep through a local population relatively quickly, e.g. ~900 generations to go from a frequency of of .01 to .99 with a selective advantage of 1%, it will take a while for it to spread spatially. That same gene will take an additional 250 generations to fix, say, 50 meters away, assuming parent-offspring variance of 2 meters^2 per unit time. An interesting observation, at least.

But what’s particularly interesting are some extensions of this work. One of the major extensions comes when the “reaction” term is changed to something just a little bit more complex:

Where the p with the triangle on it, called “p-hat” by those in the know, is an internal equilibrium. What kind of biology might be relevant to this situation? Well, that equation is one way of expressing the famous allee effect in ecology, which describes populations who have lowered growth rates both when the population is large (too much competition, for example) and when it is small (it’s hard to find a mate). In evolution, it was thought that that equation could describe the phenomenon of hybrid zones. In another landmark paper, NH Barton described The Dynamics of Hybrid Zones (unfortunately not open access). Now, to be quite honest, I have no idea what Barton is doing in this paper—he’s using methods from physics that I’m not really familiar with. Nonetheless, he proved that if the internal equilibrium is too large, specifically if p-hat is bigger than 1/2, then the wave of advance stops dead in its tracks. It can also be stopped by steep changes in population density, among other things. This result is pretty cool: it seemed to explain why some hybrid zones moved, and why some hybrid zones stayed in their placee.

Unfortuantely, hybrid zones probably AREN’T described by the simple dynamics above. But we can use that simple equation to learn a lot of stuff about them, nonetheless!

• Category: Science 
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…well. I took a bit longer than I had intended in getting to work on this next post, but it’s okay. When I left you hanging a few weeks ago, I mentioned that the idea of reinforcement (that is, selection for increased pre-mating isolation in the presence of post-mating isolation) was out of favor for a while, but it was brought back in a big way by a certain paper. I’ll spend some time talking about that paper here.

So, of course, the paper I’m talking about is Coyne and Orr 1989 (open access). Maybe it’s just because of the way I was taught speciation, but this paper arguably opened the flood gates for a lot of modern research on speciation. First of all, Coyne and Orr came up with a useful way to quantify reproductive isolation. This allowed them to undertake one of the first meta-analyses of speciation ever, and they came up with some rather interesting data. The most important stuff is in figure 5:

Figure 5 shows that among both “allopatric” and “sympatric” taxa, pre-zygotic isolation increases with time, but among sympatic taxa, it increases faster! This is in contrast to the rates of increase of post-zygotic isolation, which are the same between allopatric and sympatric taxa. This definitely argues in favor of a reinforcement hypothesis.

Anyway, there are many other important points to this paper, primarily that there is a correlation between genetic distance and reproductive isolation. This is precisely what we expect under the current model of speciation, which I will hopefully talk about in the future.

As an administrative note, I figured initially that I would do a series of posts, which would build on top of each other, but I’ll probably put that off. I’ll just be blogging about things I find that are interesting, which can come a lot quicker! Hopefully you’ll see more of me in the near future.

• Category: Science 
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Well, Razib said not to do an intro post, but I figured I should at least say that I’m new ’round these parts, and give a bit of background on my interests.

Everyone knows that evolution is a continuous process, where one population is descended from an ancestral population by a string of intermediates who were capable of interbreeding. Nevertheless, evolution seems to inevitably result in discrete units, which we call species. It would seem that since the beginning, evolution by natural selection was proposed as the causal mechanism being “The Origin of Species”. But the general agreement is that Darwin mostly explained the origin of biodiversity, not the origin of species. Part of this problem is that it is incredibly difficult to define a species; we all feel like we can tell when two different populations are species, subspecies, races, or whatever you want, but when it comes down to a hard and fast definition, it is incredibly difficult. This has been the life’s work of many biologists and philosophers, including scienceblogger John Wilkins. I’m not going to propose that I know the true definition of a species, but rather that there is one definition that is the most conducive to empirical work regarding species.

That definition is the biological species concept (what, did you think I would suggest anything else?). The BSC was first fully articulated by Ernst Mayr in his classic (but in my opinion somewhat boring) book, Systematics and Origin of Species. The current standard definition is that

species are groups of interbreeding natural populations that are reproductively isolated from other such groups.

This definition shows of precisely why the BSC is a useful definition for research: it tells that if we want to understand the origin of species, we ought to understand the origin of reproductive isolation! This has been the key motivation in the study of speciation for many years now, and is typically broken down into two components.

First of all, there is post-zygotic isolation. This is the kind of reproductive isolation people refer to when they say that horses and donkeys aren’t the same species because mules are sterile. However, besides hybrid sterility, there are other forms of post-zygotic isolation, including hybrid inviability and hybrids being less fit. The latter is a particularly interesting case, as we have several interesting (and oftentimes bizarre) stories of hybrids showcasing phenotypes intermediate between the parental phenotypes. An interesting example involves the hybridization of two populations of blackcap with different migratory directions in the lab. The hybrids were actually shown to migrate a direction intermediate to the directions of the parental populations. The fitness consequences should be obvious should this mating be widespread in the the wild.

Pre-zygotic isolation most often refers to pre-mating isolation. For example, various Drosophila species won’t mate because they don’t understand each others’ mating dances. This is relatively easy to understand, but the origin of pre-zygotic isolation was the cause of one of the biggest debates in the history of the study of speciation. One one side, there were those that argued that pre-zygotic isolation would accumulate faster if two incipient species came into secondary contact than it would if they remained allopatric. This idea, called reinforcement, has a satisfying logic to it: if there are fitness consequences for creating hybrids, perhaps there would be selection to minimize the amount of hybrids that are made. Unfortunately, for many years, reinforcement did not seem plausible. That is, until a very important paper was published…

I’ll talk about that paper next time (might be a while, though… I’m in the middle of finals right now).

• Category: Science 
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