Closing lecture with David Nelson
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Population waves have played a crucial role in evolutionary history, as in the 'out of Africa' hypothesis for human ancestry. Population geneticists and physicists are now developing methods for understanding how mutations, number fluctuations and selective advantages play out in such situations. Once the behavior of pioneer organisms at frontiers is understood, genetic markers can be used to infer information about growth, ancestral population size and colonization pathways. Insights into the nature of competition and cooperation at frontiers are possible. Neutral mutations optimally positioned at the front of a growing population wave can increase their abundance by 'surfing' on the population wave. In addition, obstacles such as lakes, deserts and mountains alter migration fronts and organism geneologies in important and interesting ways, which can be illuminated by a kind of 'Huygens Principle' for biological waves. Experimental and theoretical studies of these effects will be presented, using bacteria and yeast as model systems.
 Okay, welcome to the last talk of this conference. It's my great pleasure to introduce David Nelson from Harvard. So David got his PhD in 1975 at Cornell University on randomization group theory on critical phenomena with Michael Fisher. And then basically from 1975, he had been at Harvard and was presently the author of Solomon Professor of Biophysics and Professor of Physics and Applied Physics. And I think he is a wonderful example that fundamental physics and applied physics don't have to be contradictions. He has various honors. The Badim Prize for Superconductivity, the Buckley Prize for Soft Matter, and it is very short curriculum vitae on the web, which has about 10 items. He lists as special honors two talks in the Netherlands. So the Lawrence Visiting Professorship in Leiden in 2006 and also the Cuffley Lancetorship in Delft in 2010. So I think it's very nice that you list two items in the Netherlands and I'm glad that you're back. We are looking forward to your talk on gene surfing and migrations in structured environment. Well, it's a great pleasure to be back in the Netherlands. I first came in 1974, I think. I was a student in a wonderful summer school on fundamental problems and statistical mechanics in Wageningen. And it was a delightful introduction to the Netherlands and I've been coming back very frequently. So what I'd like to do to motivate this talk, especially for physicists in the audience, is to summarize a remarkable recent development in population genetics. Some of you may have heard of the National Genographic Project, which sequences the genes of indigenous populations, typically the genes in mitochondrial DNA, and deduces various arrival times for the human migration out of Africa. Probably were several such migrations, but one of them started about 130,000 years ago. And you notice there are various arrival times in Asia and Australia and Europe. And looking at this from a perspective of, say, physics or chemistry or even biology, there's two striking things. One is that there's no error bars in this cartoon. I can assure you there are error bars and they're large. And secondly, regarded as an experiment, this is an experiment that's not easy to repeat. You probably wouldn't want to repeat it. And so what we got interested in, Oscar Halacheck and I a couple years ago, was in a kind of simulation of this kind of range expansion, but on a modest link scale in a laboratory petri dish. And the mitochondrial DNA that serve as markers of human range expansions probably had a common ancestor with bacteria like the E. coli, whose migration I'm going to show you here. These are rod-shaped organisms. There are more E. coli in all of our intestines and there are human beings on the surface of the earth. And they go forward in time as well. And I'll try to show you that this has really important implications for population genetics. And the nice thing about these kinds of experiments is, of course, they can be repeated. And if you don't mind a change in link scale of 10 to the ninth, whereas in 500 generations, humans or large mammals or cane toads invading Australia, they might expand 10,000 kilometers. These guys in the same number of generations stay on the petri dish, the only go a centimeter. And the aspect of population genetics is illustrated in this range expansion, which is worked by Kevin Foster on another bacteria called pseudomonas aerigenosa. And here, there's a mixture of two types of bacteria. And they are identical as he could make them, but they genetically demix as you go on the outward, radial migration from the central area, which we call the homeland. And you can see that there's an interesting question here. Why do they demix? It's not like oil and water demixing. They are chemically and genetically identical. The only difference is a fluorescent tag. They were similar, as I said, as we could make them. And it's the descendants that are demixing. And what you're seeing here is an effect of fluctuations at the frontier, a phenomenon in population genetics called genetic drift. And you could ask the following question in the spirit of these human range expansions. If you went around the edge and, say, counted the number of sectors in this inflating boundary, turns out to lock in eventually after a couple days or so in this migration. And for the few other properties in the boundary, could you deduce the size of the homeland? Because we have a record here of the entire range expansion, which can be replicated, as I said, 40 times or so, as we've done in the laboratory. That's not what I'm going to talk about so much today. That's published work, which you could read about if you're interested. I'm going to talk about a variant of these range expansions, which is range expansions in structured environments. So we're going to be looking at populations with frontiers, just like we saw there, because of course our world is not a featureless landscape. And various geographical features can influence biology and population genetics. So here's a famous example of these salamanders migrating from North to South in California. It's believed they got separated by these ranges of mountains. And they gradually went their own way on the two sides of this migration event. And finally, when they came back together, they could no longer mate. They were two distinct species, which illustrates how geography can affect what's going on. So I want to talk about how range expansions in an inhomogeneous environment influence the kind of genetic diversity that we saw in those bacteria. And we'll see that eventually we'll lead to some simplified models of spatial structure, migration around obstacles that we think of as lakes or deserts or they could be mountain ranges. And since I'm in the Netherlands, I will use Huygens principle to analyze these range expansions in, I hope, an interesting way. So then we'll also talk about population genetics. So that's where we're headed. And these are my collaborators, an amazingly energetic postdoc who used to be a theoretical physicist. He now does both theory, simulation, and also experiment. And my colleague in molecular and cellular biology, a very imaginative colleague named Andrew Murray. Okay, so first I want to discuss what we call the elephant in the room. The elephant in the room here has to do with the nature of physics and biology. Here I am, I was trained as a theoretical physicist, as you heard with Michael Fisher at Cornell. And I'm giving a talk that basically is heavily biological. And there's actually an anecdote when the human genome was sequenced or the first draft was announced at a press conference. Of course, Bill Clinton was there, our president at the time, and he had an exchange with his science advisor who was a physicist. And he said, okay, you physicists have had your century, this was the year 2000, what are you going to do next? And I personally don't believe that. And if you start talking about centuries, you can say, well, when did chemists have their century and so forth? And I looked on the internet and there is a quote that says, if in scientific terms the 20th century has been the century of physics, then the 21st will surely be the century of biology. Who do you think actually made that quote? Anybody have a suggestion? It wasn't Bill Clinton, it was Hillary. In her book, it takes a village. Now, why she put that in there when she's talking about how lessons children teach us, I don't know, but we could also ask, is this really true? Now, I'm going to have a show of hands. How many in this audience have ever used the Google Ngram viewer? Anybody played around with that? There's a couple people. It's an interesting piece of software and it allows you to take, based on Google's scanning, all the books they could be allowed to scan until various people stop them in various languages. And so what I did, just to give you a sense of how this Ngram viewer works, and since we're in the Benelux Hall, is I competed the word frequency of Belgium, Netherlands, and Luxembourg for books written in English over the last 200 years. And it's kind of interesting, you'll be happy to know that the Netherlands comes out on top eventually. This spike is presumably due to World War I, when Belgium was in the news and books and so forth. Luxembourg is at least noticeable down here. But we could also, following up on Hillary's observation, compete physics and biology. So interestingly, here's physics in blue and it was around. Biology was sort of flat lined down here at zero. And about the time of Darwin, it became its own independent science and it's rising and the gap seems to be narrowing. So something we might want to think about. On the other hand, it's also instructive to compete physics and biology against DNA for books written in English. And notice that DNA beats everybody. And I think that's a nice observation because DNA, to understand it, you need to know polymer physics. You need to know chemistry of nucleotide binding and hydrogen bonds. And of course, you need to know biology. So it's an intrinsically interdisciplinary molecule. And the message I would take away from some of the biology or the biophysics or physical biology that we're hearing at this wonderful conference is that perhaps in fact, the 21st century would be the century of DNA as revealed by interdisciplinary research. So with that, link the prelude. Let's get down to business. I know that Tom Schmizu is in the audience and he actually supplied some strains of E. Coli for these range expansions. I'll just remind you what was on that first slide in the context of E. Coli. These were two nearly identical E. Coli's except the main chromosomes were the same. These little plasmids were more or less the same except there was one change in the amino acid of a fluorescently expressed protein that changed the color, say, from yellow to cyan. And the experiment was so simple that even I, a card-carrying theoretical physicist, could do it. Namely mix these things up overnight in 50-50 proportions, let's say, pipette them down in a little two microliter aliquot. The carrier fluid dried out, leaving behind these bacteria, wait four days, and this is one quarter of a coin-sized colony in white light that appeared. But this looks kind of dull unless you really love microbiology. But in fact, under fluorescent excitation, it looks like that. And here's a more global view. We changed colors just to make it a little more visible. This is the homeland. And these are these sectors radiating out. I'm not going to talk so much about them until the very end. I want to focus now on the homeland and see if we could use the homeland and exploit its properties to make a structured environment. And in fact, here are some homelands displayed at various densities, 50-50 concentrations, with different densities of the initial inoculant on the petri dish, and going from 25 to 250 to 2,500 of each of the founder colonies. These colonies in the interior in this homeland are, initially, the founders are separated, and they come crashing together to make these beautiful fractal patterns. What I'd like to do is to regard these homelands as a kind of ecological landscape occurring on a millimeters scale through which we could have populations migrate. And so here's a blow-up of this little region here. It looks like one of these maps that you might find in the Game of Thrones or something in the back of the book. And you can imagine that these are inlets and these are land masses and ask, what would it be like to live on such a complicated set of estuaries and fjords? Of course, we need to have something to live there. And what we decided to study was the spread of a virus. The particular virus was T7. It infects E. coli. It's a well-studied virus. Here are some of its properties. And basically what it does, here's a, has an icosahedral symmetry. This virus seems to know about the platonic solids. And one, as many of you know, when it injects its small, modest complement of DNA into a bacterium, it takes over the reproductive machinery and reproduces itself, maybe 30 or 50-fold. This is its genome with its various genes, not too many. They're indicated, I think, by flags in this picture. And the, here is a picture. It's a different virus. If you don't like getting bitten by mosquitoes in dense forests and so forth, think of this poor bacterium being attacked, so to speak, by all these viruses, trying to use their sort of hyperderminal needle-like projections to get the DNA inside. This is a movie. Wolfram actually made this a couple, a month or two after he arrived. And it's actually kind of fun to watch because the E. coli are growing up. And you can see them elongating, they're rod-shaped, they undergo cell division. They're sort of happily about to collide these little clonal founders. And everything seems to be fine, except he mixed in a little virus into the agar, took a while for the virus to diffuse in, and you can see it's a disaster. All these bacteria are exploding. And then once a little plaque, which is the region of exploded bacteria and excess virus, is produced, the virus is diffused to a fresh region of bacterial lawn and explode over there as well. And you can think of this as a little chemical reaction where the virus jumps onto a bacterium and may jump on and off irreversibly, producing some infected thing. And once it gets into the interior with some other rate constant K2, it multiplies itself by a factor Y. Y is just an integer like 30 or 40. It's the proliferative power of these viruses. And this is T7, a well-studied example. Yinn and Magaskal studied these objects as kind of population waves. And what happens is you end up with a concentration of viruses, which obeys a piece of this equation that looks like a diffusion, just ordinary diffusion. This is the diffusion constant of the virus in some bacterial lawn as it meanders around, exploding things, reproducing itself at a rate A. And K is just the carrying capacity here. And so the question is, I don't know how many of you have encountered this parabolic but nonlinear differential equation before. Not many physicists have usually run across it, but it has wonderful properties that I'll show you. If you want to know about the properties, one way is to read a beautiful review article by Wim van Sarloes, now director of the foam. However, this does have almost 200 pages. It's extremely methodical and thorough. Or you could look at the next slide. Whoops. Let me go back here. Next slide is not here. Here it is. Okay, so let's take a look at this next slide. This is population waves for dummies. So here's the idea, parabolic differential equation in one dimension. Over short times, it just rises up. Let me go back here to here. Rises up exponentially at the origin, spreads out like the square root of time. That's the result of the linear term and this diffusive term. So that's pretty standard. But remarkably, once this nonlinear term gets into play when it reaches the carrying capacity over some region in this one-dimensional illustration, this wave equation, this becomes a wave-like. There's actually a propagating front velocity. It's not a soliton per se, but it has a velocity. And the velocity depends in this way on the diffusion constant and the growth. And there's also a well-defined interface width. So it's not the conventional wave of crests and troughs, as we're familiar with in linear wave mechanics. It's a nonlinear phenomena, but it still has a constant wave velocity. You can actually model these viral plaques by something precisely like this. And so here we have an example of infection in the homeland by this virus. Several actually are going to take off here. So I inoculate. And you can see the time scale is ours, I think, in the inset. This is what's going on. And I also notice that there's refraction. That is, there's circular waves that are propagating out, these fissure waves. But eventually, the waves on the edge start to go faster. So look up here. There's an inhomogeneous wave speed. Here it's approximately a flat front, but you'll see it start curving around. And so we seem to have a viral analog of refraction. Why would a wave want to do that? One of these population waves. Here's another example, and this is now getting to our structured environment, where Wolfram took a homeland with yellow and red organisms. They haven't quite grown together. That's why you see these black gaps. And now he introduced a virus, this is not shown to scale, on the right side there around three o'clock. And there it goes. You notice it's preferentially attacking the yellow susceptible strain, but sort of avoiding at least initially the resistant strain of this bacterial lawn. This virus, however, can mutate so fast it eventually will attack the red as well. So this is a kind of model of epidemiology where you can do replicates, 40 replicates on a petri dish, and try to study what's going on with these population waves in inhomogeneous, structured environments. So you can see it eventually attacks the red, but preferentially at least initially goes toward the yellow. And there's a lot of interesting science here. It's related to percolation, related to waves in inhomogeneous media. There's a great tradition in the Netherlands of studying anomalous backscattering of light and through inhomogeneous, random frozen dielectric landscapes, spatially dependent dielectric constants. This is something similar, but with a very different kind of wave. So that's what I want to talk about. And what we eventually decided to do was to first simplify the problem by printing our own bacterial lawns. And so what Wolfram realized was that he could take an ordinary inkjet printer like this one. Its original purpose in life was to print labels on CDs and then persuade it to print bacteria of different shapes and colors. And so here's the E in E. coli. And you can see it's preferentially, the yellow part of this E in E. coli is being chewed up, turned into plaque, which has just exploded cells and the virus that came with it. And we first thought we would look at what happens when these obstacles could regard them as lakes and deserts in an ecological context would confront a wave of viral plaque spreading. So that's the idea. And so first thing we looked at was a diamond. And what's shown down here at the bottom is a piece of virus-soaked filter paper, which allows us to launch a linear wave when you lay it down on this petri dish. So it's a lawn of yellow resistant, of susceptible bacteria and orange resistant bacteria. And now let's watch this go around. Notice that's an incipient cusp that seems to form on the far side of the diamond. So that's one of the interesting phenomena that we'd like to understand. And it turns out that for us anyway, the way to do it is to think in a kind of inspiration from Christian Huygens. So you know more about his work, I'm sure, than I do, but he's famous in all levels of education and science in the United States. Not quite so famous as Fermat, but he also had nice, interesting ideas. And so I'd like to ask in the context of these population waves, what can we learn from these gentlemen other than the fact that the French flag is the Dutch flag rotated by 90 degrees? Well, here's Huygens' principle. And he thought of these little wavelets, and he superimposed them and got beautiful diffraction and refraction. So I'll talk about refraction here. And we could regard these as individual founder clonal bacteria or virus plaques all along this line going into a different medium with a different velocity. And if you do this nice superposition, a la Huygens, then you in fact get a bending, a refraction. As you would a la Fermat, who formulated a principle of least time. And we'll see how that comes into play in a minute. But the formal realization of this principle of least time, it's an elementary exercise in the calculus of variations. You had a complicated medium here, a path parameterized by arc length. Take my word for it. This equation down here, where I have a position dependent velocity, describes the principle of least time. The question is, does that work for these nonlinear wave equations? I'm not saying there's interference that would involve crests and troughs. These are not crests and troughs. These are like a soliton or a wave front coming by. But maybe a similar principle could be applied. And if I have a diamond, we could implement Huygens and Fermat by saying, OK, here's a little virus or an infected cell right at the front. So here's a front. That's going to be sweeping by this diamond of size L. And we can ask this person, the following hypothetical question, not person, but cell or virus, where did your ancestors come from? And clearly, the ancestors that were most likely to get there first, summarized up here, it's survival of the fastest. And so the ancestor probably came from here. On the other hand, on this little point on the front, this emerging front whose shape we're trying to calculate, presumably there's a better, shortest path, and it's coming up straight here. And then it's bending somewhere on the arc of this circle. So that's the idea. And notice it leads naturally to a cusp at the tip of the diamond. Now we have to be a little careful, but we can go through. And this is a little calculation I did of the shape of the cusp downstream of the diamond. This is the angle, the opening angle of the cusp. And it gradually flattens, and it starts off for a diamond like a 90-degree cusp, and it gradually flattens to zero like one over time. So it's not exponential. It's pretty slow. So that's one interesting observation. This is like geometrical optics. Geometrical optics works when the scale of the variation or the objects that you're refracting around are large compared to the wavelength of light, for example. Here the requirement is that the object be large compared to the width of that fissure wave, the front, the frontier width of it that I was talking about earlier. However, I want to mention this will come in at the very end. We have to be careful. Diamonds' Fermat neglects the discreteness of viruses and cells. And it's a good, it's reasonable first approximation, but the idea is that here's a simple computer simulation of an infected cell giving rise to another infected cell and so forth. And in this simple lattice context, you could program this on your laptop yourself. You can actually look at what happens when you go buy a diamond, and you can see here now the lineages, the actual lineages of who begat whom or who destroyed whom coming up here. So imagine this is a viral lineage, for example. Here's the cusp, but you can see that the front is rough. That's one difference. That's an important difference. But also, the paths are not straight lines. They're wiggling. They're jiggling around due to a phenomena called genetic drift in population genetics. Here there's no genetic differences that I'm tracking, but these patterns that look like the Nile River Delta and so forth are lineages that sort of are reminiscent of a kind of Feynman-Path integral approach to understanding complicated dynamical problems. And in general, you'd have to sum over paths, but there's certainly something very interesting, but vaguely reminiscent of Hoigens and Fermat. There's a curved path here, and except for these fluctuations, which are quite interesting, I think, it might be a good first cut at what's going on. Just like geometrical optics is a good first cut for light phenomena, but although it neglects the way of nature and so forth. So here's we have our Fisher equation. And what we can do actually to sort of test Hoigens and Fermat is to solve this Fisher equation with an obstacle. And what we do is we take the growth rate to be zero inside the obstacle, although the viruses can still diffuse. And so here I have a diamond-shaped object. I don't know why we got obsessed with diamonds, but there's a cusp that heals. And if you think the diamond is special, over here is an ellipse, or I think it was originally a circle. And now there's again a cusp. And the cusp we've checked numerically, it does seem in experiments to heal like one over time. And of course, understanding as we think we now do, single obstacles, ultimately we want to look at multiple obstacles. So here's an interesting array of squares or diamonds. The purple here shows where the wave would have gone in the absence of the obstacles. This is like a lattice of square, like the wind tree model of statistical mechanics. It's a lattice of lakes. And this is the front, I can get this to work. This is where the front would have been in the absence of diamonds. This is where it actually is. And you can see it's slowing down. So there's an index of refraction that describes these complicated media, at least under some circumstances. On the other hand, if I move the squares a little further apart, as Wolfram did, now you can see that you get cusps, but the cusps heal, and the overall velocity is unchanged. This is a big difference from wave phenomena where there would be backscattering and we would probably slow down a bit, even in this expanded geometry. So we're still trying to track that down, but at least under some circumstances can be a reduced velocity, and hence the bending and an index of refraction. So now I want to conclude by talking about population genetics. And ideally we would have been able to make viruses that were different colors and track the different viral species. We weren't successful, at least yet, in doing that. Maybe someone in the audience could do it. But what we decided to do instead was to now, instead of having viruses, grow through bacterial lawns to let bacteria just grow on a petri dish, just like in the original experiment that I showed you. There's, again, a bacteria soaked filter paper, three different colors. There's a great bit of chaos and fighting for dominance or whatever you want to call it with these approximately neutral strains, red, blue, and yellow in the early days. But eventually there's local fixation, genetic demixing, of these three colors. And now I want to ask what happens when we add an obstacle. So obstacle, in this case, was simply a circular, very fine mesh piece of filter paper that you can vaguely see here and see here as a kind of black circle. And nutrients couldn't get up through the agar to feed these bacteria that were coming down from above. So if I take a look here, you can see the effect of the obstacle and the population genetics. It's not very visible, but there was a cusp, same cusp we had before, but right at the cusp, at six o'clock, two genetic variants come together. And this only happens when they're neutral. If there was a selective advantage, this meeting would have been shifted to seven o'clock over here, or five o'clock, let's say. So this is a kind of selective advantage meter. And notice that this boundary is much straighter than the wiggles that you have over here. So there's a scar in the genetic landscape of this obstacle, this lake-like obstacle, that seems to persist for a very long time. And here's a radial range expansion version of the same thing. Again in this case, the yellow and the blue strains meet at least approximately at six o'clock, maybe a little more towards seven. And this shows that these obstacles could really have, we think, a pretty important role in the population genetics. And as I said, Wolfram is trained as a theoretical physicist. So when he got these beautiful results, he just went and simulated. And what he did was to simulate not three colors, but essentially an infinite number of colors. One color, every bacteria has its own distinct genome. And that's true of humans, to an excellent approximation. And it can be implemented actually using something called the brain-dote technology in molecular biology, which we may or may not do ourselves. But here's a whole bunch of colors migrating around a elliptical object. And you see pretty much the same thing. You'll see a cusp when things come together right now, almost a vertical slope. And then with these fluctuations, we see the blue and the green, a genetic boundary that is inherited by the passage around the obstacle. And you can think of this as a kind of selection by geometry. Because notice the very fortunate bacteria that just grazed the ellipse up here, the green, and then this blue strain, which almost got squeezed out, basically had this open range downstream, and were able to sort of take over. And the other thing that we're following up on, which we think is quite interesting, is here's the experiment again. Here's a simulation going around a diamond-shaped obstacle, so you can see that the shape doesn't matter that much. Here's the cusp, there's the boundary, in this case purple and green. And there are the lineages. And notice that if you go up here to the frontier, and a la Heuagins and Fermat, you ask, where did your ancestors come from? Most of the time, those ancestors just grazed the corner of this little rhombus or this diamond. So it's a kind of selection, it's like a population bottleneck, but it's a half-space only that is the nature of the constriction. And we think it's very interesting to try to do sort of coalescence theory and look at these lineages and how they branch, and how that branching is influenced by these obstacles. So I'd like to conclude by with some summaries, and then one last nice experiment. This is what I was just talking about. There are these unlucky genotypes that go smashing into the obstacle. So this light blue turquoise, it's all over for them, they couldn't go anywhere. At least they have to only can get across by diffusion if they could diffuse at all. And the fortunate types that actually go just grazing along the edges, obstacles, reduced genetic diversity by these unfortunate extinct strains here, there are the lucky genotypes as well and the cusps eventually heal, but the sector boundary seems to persist indefinitely. I want to end with a beautiful experiment, it's more of a facetious experiment, which Wolfram did and he can print anything and so why not print a map of the world? So here the continents are a susceptible strain of bacteria and the oceans are a resistant strain, we're at inoculate T7, well where else but the Rift Valley. And so here is an inoculation in this artificial geography in a petri dish. And I'm not saying nor I think would Wolfram say that human range expansions can be compared to viruses, but it does show that geography matters. Here's Wolfram who had many of the key ideas here and I'd like to close and thank you for your attention.
 Okay, welcome to the last talk of this conference.  It's my great pleasure to introduce David Nelson from Harvard.  So David got his PhD in 1975 at Cornell University on randomization group theory on critical  phenomena with Michael Fisher.  And then basically from 1975, he had been at Harvard and was presently the author of  Solomon Professor of Biophysics and Professor of Physics and Applied Physics.  And I think he is a wonderful example that fundamental physics and applied physics don't  have to be contradictions.  He has various honors.  The Badim Prize for Superconductivity, the Buckley Prize for Soft Matter, and it is very  short curriculum vitae on the web, which has about 10 items.  He lists as special honors two talks in the Netherlands.  So the Lawrence Visiting Professorship in Leiden in 2006 and also the Cuffley Lancetorship  in Delft in 2010.  So I think it's very nice that you list two items in the Netherlands and I'm glad that  you're back.  We are looking forward to your talk on gene surfing and migrations in structured environment.  Well, it's a great pleasure to be back in the Netherlands.  I first came in 1974, I think.  I was a student in a wonderful summer school on fundamental problems and statistical mechanics  in Wageningen.  And it was a delightful introduction to the Netherlands and I've been coming back very  frequently.  So what I'd like to do to motivate this talk, especially for physicists in the audience,  is to summarize a remarkable recent development in population genetics.  Some of you may have heard of the National Genographic Project, which sequences the genes  of indigenous populations, typically the genes in mitochondrial DNA, and deduces various  arrival times for the human migration out of Africa.  Probably were several such migrations, but one of them started about 130,000 years ago.  And you notice there are various arrival times in Asia and Australia and Europe.  And looking at this from a perspective of, say, physics or chemistry or even biology,  there's two striking things.  One is that there's no error bars in this cartoon.  I can assure you there are error bars and they're large.  And secondly, regarded as an experiment, this is an experiment that's not easy to repeat.  You probably wouldn't want to repeat it.  And so what we got interested in, Oscar Halacheck and I a couple years ago, was in a kind of  simulation of this kind of range expansion, but on a modest link scale in a laboratory  petri dish.  And the mitochondrial DNA that serve as markers of human range expansions probably had a common  ancestor with bacteria like the E. coli, whose migration I'm going to show you here.  These are rod-shaped organisms.  There are more E. coli in all of our intestines and there are human beings on the surface  of the earth.  And they go forward in time as well.  And I'll try to show you that this has really important implications for population genetics.  And the nice thing about these kinds of experiments is, of course, they can be repeated.  And if you don't mind a change in link scale of 10 to the ninth, whereas in 500 generations,  humans or large mammals or cane toads invading Australia, they might expand 10,000 kilometers.  These guys in the same number of generations stay on the petri dish, the only go a centimeter.  And the aspect of population genetics is illustrated in this range expansion, which is worked by  Kevin Foster on another bacteria called pseudomonas aerigenosa.  And here, there's a mixture of two types of bacteria.  And they are identical as he could make them, but they genetically demix as you go on the  outward, radial migration from the central area, which we call the homeland.  And you can see that there's an interesting question here.  Why do they demix?  It's not like oil and water demixing.  They are chemically and genetically identical.  The only difference is a fluorescent tag.  They were similar, as I said, as we could make them.  And it's the descendants that are demixing.  And what you're seeing here is an effect of fluctuations at the frontier, a phenomenon  in population genetics called genetic drift.  And you could ask the following question in the spirit of these human range expansions.  If you went around the edge and, say, counted the number of sectors in this inflating boundary,  turns out to lock in eventually after a couple days or so in this migration.  And for the few other properties in the boundary, could you deduce the size of the homeland?  Because we have a record here of the entire range expansion, which can be replicated,  as I said, 40 times or so, as we've done in the laboratory.  That's not what I'm going to talk about so much today.  That's published work, which you could read about if you're interested.  I'm going to talk about a variant of these range expansions, which is range expansions  in structured environments.  So we're going to be looking at populations with frontiers, just like we saw there, because  of course our world is not a featureless landscape.  And various geographical features can influence biology and population genetics.  So here's a famous example of these salamanders migrating from North to South in California.  It's believed they got separated by these ranges of mountains.  And they gradually went their own way on the two sides of this migration event.  And finally, when they came back together, they could no longer mate.  They were two distinct species, which illustrates how geography can affect what's going on.  So I want to talk about how range expansions in an inhomogeneous environment influence  the kind of genetic diversity that we saw in those bacteria.  And we'll see that eventually we'll lead to some simplified models of spatial structure,  migration around obstacles that we think of as lakes or deserts or they could be mountain  ranges.  And since I'm in the Netherlands, I will use Huygens principle to analyze these range  expansions in, I hope, an interesting way.  So then we'll also talk about population genetics.  So that's where we're headed.  And these are my collaborators, an amazingly energetic postdoc who used to be a theoretical  physicist.  He now does both theory, simulation, and also experiment.  And my colleague in molecular and cellular biology, a very imaginative colleague named  Andrew Murray.  Okay, so first I want to discuss what we call the elephant in the room.  The elephant in the room here has to do with the nature of physics and biology.  Here I am, I was trained as a theoretical physicist, as you heard with Michael Fisher  at Cornell.  And I'm giving a talk that basically is heavily biological.  And there's actually an anecdote when the human genome was sequenced or the first draft  was announced at a press conference.  Of course, Bill Clinton was there, our president at the time, and he had an exchange with his  science advisor who was a physicist.  And he said, okay, you physicists have had your century, this was the year 2000, what  are you going to do next?  And I personally don't believe that.  And if you start talking about centuries, you can say, well, when did chemists have  their century and so forth?  And I looked on the internet and there is a quote that says, if in scientific terms the  20th century has been the century of physics, then the 21st will surely be the century of  biology.  Who do you think actually made that quote?  Anybody have a suggestion?  It wasn't Bill Clinton, it was Hillary.  In her book, it takes a village.  Now, why she put that in there when she's talking about how lessons children teach us,  I don't know, but we could also ask, is this really true?  Now, I'm going to have a show of hands.  How many in this audience have ever used the Google Ngram viewer?  Anybody played around with that?  There's a couple people.  It's an interesting piece of software and it allows you to take, based on Google's scanning,  all the books they could be allowed to scan until various people stop them in various languages.  And so what I did, just to give you a sense of how this Ngram viewer works, and since we're  in the Benelux Hall, is I competed the word frequency of Belgium, Netherlands, and Luxembourg  for books written in English over the last 200 years.  And it's kind of interesting, you'll be happy to know that the Netherlands comes out on  top eventually.  This spike is presumably due to World War I, when Belgium was in the news and books  and so forth.  Luxembourg is at least noticeable down here.  But we could also, following up on Hillary's observation, compete physics and biology.  So interestingly, here's physics in blue and it was around.  Biology was sort of flat lined down here at zero.  And about the time of Darwin, it became its own independent science and it's rising and  the gap seems to be narrowing.  So something we might want to think about.  On the other hand, it's also instructive to compete physics and biology against DNA for  books written in English.  And notice that DNA beats everybody.  And I think that's a nice observation because DNA, to understand it, you need to know polymer  physics.  You need to know chemistry of nucleotide binding and hydrogen bonds.  And of course, you need to know biology.  So it's an intrinsically interdisciplinary molecule.  And the message I would take away from some of the biology or the biophysics or physical  biology that we're hearing at this wonderful conference is that perhaps in fact, the 21st  century would be the century of DNA as revealed by interdisciplinary research.  So with that, link the prelude.  Let's get down to business.  I know that Tom Schmizu is in the audience and he actually supplied some strains of E.  Coli for these range expansions.  I'll just remind you what was on that first slide in the context of E. Coli.  These were two nearly identical E. Coli's except the main chromosomes were the same.  These little plasmids were more or less the same except there was one change in the amino  acid of a fluorescently expressed protein that changed the color, say, from yellow to  cyan.  And the experiment was so simple that even I, a card-carrying theoretical physicist,  could do it.  Namely mix these things up overnight in 50-50 proportions, let's say, pipette them down  in a little two microliter aliquot.  The carrier fluid dried out, leaving behind these bacteria, wait four days, and this is  one quarter of a coin-sized colony in white light that appeared.  But this looks kind of dull unless you really love microbiology.  But in fact, under fluorescent excitation, it looks like that.  And here's a more global view.  We changed colors just to make it a little more visible.  This is the homeland.  And these are these sectors radiating out.  I'm not going to talk so much about them until the very end.  I want to focus now on the homeland and see if we could use the homeland and exploit its  properties to make a structured environment.  And in fact, here are some homelands displayed at various densities, 50-50 concentrations,  with different densities of the initial inoculant on the petri dish, and going from 25 to 250  to 2,500 of each of the founder colonies.  These colonies in the interior in this homeland are, initially, the founders are separated,  and they come crashing together to make these beautiful fractal patterns.  What I'd like to do is to regard these homelands as a kind of ecological landscape occurring  on a millimeters scale through which we could have populations migrate.  And so here's a blow-up of this little region here.  It looks like one of these maps that you might find in the Game of Thrones or something in  the back of the book.  And you can imagine that these are inlets and these are land masses and ask, what would  it be like to live on such a complicated set of estuaries and fjords?  Of course, we need to have something to live there.  And what we decided to study was the spread of a virus.  The particular virus was T7.  It infects E. coli.  It's a well-studied virus.  Here are some of its properties.  And basically what it does, here's a, has an icosahedral symmetry.  This virus seems to know about the platonic solids.  And one, as many of you know, when it injects its small, modest complement of DNA into a  bacterium, it takes over the reproductive machinery and reproduces itself, maybe 30  or 50-fold.  This is its genome with its various genes, not too many.  They're indicated, I think, by flags in this picture.  And the, here is a picture.  It's a different virus.  If you don't like getting bitten by mosquitoes in dense forests and so forth, think of this  poor bacterium being attacked, so to speak, by all these viruses, trying to use their  sort of hyperderminal needle-like projections to get the DNA inside.  This is a movie.  Wolfram actually made this a couple, a month or two after he arrived.  And it's actually kind of fun to watch because the E. coli are growing up.  And you can see them elongating, they're rod-shaped, they undergo cell division.  They're sort of happily about to collide these little clonal founders.  And everything seems to be fine, except he mixed in a little virus into the agar, took  a while for the virus to diffuse in, and you can see it's a disaster.  All these bacteria are exploding.  And then once a little plaque, which is the region of exploded bacteria and excess virus,  is produced, the virus is diffused to a fresh region of bacterial lawn and explode over there  as well.  And you can think of this as a little chemical reaction where the virus jumps onto a bacterium  and may jump on and off irreversibly, producing some infected thing.  And once it gets into the interior with some other rate constant K2, it multiplies itself  by a factor Y. Y is just an integer like 30 or 40.  It's the proliferative power of these viruses.  And this is T7, a well-studied example.  Yinn and Magaskal studied these objects as kind of population waves.  And what happens is you end up with a concentration of viruses, which obeys a piece of this equation  that looks like a diffusion, just ordinary diffusion.  This is the diffusion constant of the virus in some bacterial lawn as it meanders around,  exploding things, reproducing itself at a rate A. And K is just the carrying capacity  here.  And so the question is, I don't know how many of you have encountered this parabolic but  nonlinear differential equation before.  Not many physicists have usually run across it, but it has wonderful properties that I'll  show you.  If you want to know about the properties, one way is to read a beautiful review article  by Wim van Sarloes, now director of the foam.  However, this does have almost 200 pages.  It's extremely methodical and thorough.  Or you could look at the next slide.  Whoops.  Let me go back here.  Next slide is not here.  Here it is.  Okay, so let's take a look at this next slide.  This is population waves for dummies.  So here's the idea, parabolic differential equation in one dimension.  Over short times, it just rises up.  Let me go back here to here.  Rises up exponentially at the origin, spreads out like the square root of time.  That's the result of the linear term and this diffusive term.  So that's pretty standard.  But remarkably, once this nonlinear term gets into play when it reaches the carrying capacity  over some region in this one-dimensional illustration, this wave equation, this becomes  a wave-like.  There's actually a propagating front velocity.  It's not a soliton per se, but it has a velocity.  And the velocity depends in this way on the diffusion constant and the growth.  And there's also a well-defined interface width.  So it's not the conventional wave of crests and troughs, as we're familiar with in linear  wave mechanics.  It's a nonlinear phenomena, but it still has a constant wave velocity.  You can actually model these viral plaques by something precisely like this.  And so here we have an example of infection in the homeland by this virus.  Several actually are going to take off here.  So I inoculate.  And you can see the time scale is ours, I think, in the inset.  This is what's going on.  And I also notice that there's refraction.  That is, there's circular waves that are propagating out, these fissure waves.  But eventually, the waves on the edge start to go faster.  So look up here.  There's an inhomogeneous wave speed.  Here it's approximately a flat front, but you'll see it start curving around.  And so we seem to have a viral analog of refraction.  Why would a wave want to do that?  One of these population waves.  Here's another example, and this is now getting to our structured environment, where Wolfram  took a homeland with yellow and red organisms.  They haven't quite grown together.  That's why you see these black gaps.  And now he introduced a virus, this is not shown to scale, on the right side there around  three o'clock.  And there it goes.  You notice it's preferentially attacking the yellow susceptible strain, but sort of avoiding  at least initially the resistant strain of this bacterial lawn.  This virus, however, can mutate so fast it eventually will attack the red as well.  So this is a kind of model of epidemiology where you can do replicates, 40 replicates  on a petri dish, and try to study what's going on with these population waves in inhomogeneous,  structured environments.  So you can see it eventually attacks the red, but preferentially at least initially goes  toward the yellow.  And there's a lot of interesting science here.  It's related to percolation, related to waves in inhomogeneous media.  There's a great tradition in the Netherlands of studying anomalous backscattering of light  and through inhomogeneous, random frozen dielectric landscapes, spatially dependent dielectric  constants.  This is something similar, but with a very different kind of wave.  So that's what I want to talk about.  And what we eventually decided to do was to first simplify the problem by printing our  own bacterial lawns.  And so what Wolfram realized was that he could take an ordinary inkjet printer like this  one.  Its original purpose in life was to print labels on CDs and then persuade it to print  bacteria of different shapes and colors.  And so here's the E in E. coli.  And you can see it's preferentially, the yellow part of this E in E. coli is being chewed  up, turned into plaque, which has just exploded cells and the virus that came with it.  And we first thought we would look at what happens when these obstacles could regard  them as lakes and deserts in an ecological context would confront a wave of viral plaque  spreading.  So that's the idea.  And so first thing we looked at was a diamond.  And what's shown down here at the bottom is a piece of virus-soaked filter paper, which  allows us to launch a linear wave when you lay it down on this petri dish.  So it's a lawn of yellow resistant, of susceptible bacteria and orange resistant bacteria.  And now let's watch this go around.  Notice that's an incipient cusp that seems to form on the far side of the diamond.  So that's one of the interesting phenomena that we'd like to understand.  And it turns out that for us anyway, the way to do it is to think in a kind of inspiration  from Christian Huygens.  So you know more about his work, I'm sure, than I do, but he's famous in all levels of  education and science in the United States.  Not quite so famous as Fermat, but he also had nice, interesting ideas.  And so I'd like to ask in the context of these population waves, what can we learn from these  gentlemen other than the fact that the French flag is the Dutch flag rotated by 90 degrees?  Well, here's Huygens' principle.  And he thought of these little wavelets, and he superimposed them and got beautiful diffraction  and refraction.  So I'll talk about refraction here.  And we could regard these as individual founder clonal bacteria or virus plaques all along  this line going into a different medium with a different velocity.  And if you do this nice superposition, a la Huygens, then you in fact get a bending, a  refraction.  As you would a la Fermat, who formulated a principle of least time.  And we'll see how that comes into play in a minute.  But the formal realization of this principle of least time, it's an elementary exercise  in the calculus of variations.  You had a complicated medium here, a path parameterized by arc length.  Take my word for it.  This equation down here, where I have a position dependent velocity, describes the principle  of least time.  The question is, does that work for these nonlinear wave equations?  I'm not saying there's interference that would involve crests and troughs.  These are not crests and troughs.  These are like a soliton or a wave front coming by.  But maybe a similar principle could be applied.  And if I have a diamond, we could implement Huygens and Fermat by saying, OK, here's a  little virus or an infected cell right at the front.  So here's a front.  That's going to be sweeping by this diamond of size L. And we can ask this person, the  following hypothetical question, not person, but cell or virus, where did your ancestors  come from?  And clearly, the ancestors that were most likely to get there first, summarized up here,  it's survival of the fastest.  And so the ancestor probably came from here.  On the other hand, on this little point on the front, this emerging front whose shape  we're trying to calculate, presumably there's a better, shortest path, and it's coming up  straight here.  And then it's bending somewhere on the arc of this circle.  So that's the idea.  And notice it leads naturally to a cusp at the tip of the diamond.  Now we have to be a little careful, but we can go through.  And this is a little calculation I did of the shape of the cusp downstream of the diamond.  This is the angle, the opening angle of the cusp.  And it gradually flattens, and it starts off for a diamond like a 90-degree cusp, and it  gradually flattens to zero like one over time.  So it's not exponential.  It's pretty slow.  So that's one interesting observation.  This is like geometrical optics.  Geometrical optics works when the scale of the variation or the objects that you're refracting  around are large compared to the wavelength of light, for example.  Here the requirement is that the object be large compared to the width of that fissure  wave, the front, the frontier width of it that I was talking about earlier.  However, I want to mention this will come in at the very end.  We have to be careful.  Diamonds' Fermat neglects the discreteness of viruses and cells.  And it's a good, it's reasonable first approximation, but the idea is that here's a simple computer  simulation of an infected cell giving rise to another infected cell and so forth.  And in this simple lattice context, you could program this on your laptop yourself.  You can actually look at what happens when you go buy a diamond, and you can see here  now the lineages, the actual lineages of who begat whom or who destroyed whom coming up  here.  So imagine this is a viral lineage, for example.  Here's the cusp, but you can see that the front is rough.  That's one difference.  That's an important difference.  But also, the paths are not straight lines.  They're wiggling.  They're jiggling around due to a phenomena called genetic drift in population genetics.  Here there's no genetic differences that I'm tracking, but these patterns that look like  the Nile River Delta and so forth are lineages that sort of are reminiscent of a kind of  Feynman-Path integral approach to understanding complicated dynamical problems.  And in general, you'd have to sum over paths, but there's certainly something very interesting,  but vaguely reminiscent of Hoigens and Fermat.  There's a curved path here, and except for these fluctuations, which are quite interesting,  I think, it might be a good first cut at what's going on.  Just like geometrical optics is a good first cut for light phenomena, but although it neglects  the way of nature and so forth.  So here's we have our Fisher equation.  And what we can do actually to sort of test Hoigens and Fermat is to solve this Fisher  equation with an obstacle.  And what we do is we take the growth rate to be zero inside the obstacle, although the  viruses can still diffuse.  And so here I have a diamond-shaped object.  I don't know why we got obsessed with diamonds, but there's a cusp that heals.  And if you think the diamond is special, over here is an ellipse, or I think it was originally  a circle.  And now there's again a cusp.  And the cusp we've checked numerically, it does seem in experiments to heal like one  over time.  And of course, understanding as we think we now do, single obstacles, ultimately we want  to look at multiple obstacles.  So here's an interesting array of squares or diamonds.  The purple here shows where the wave would have gone in the absence of the obstacles.  This is like a lattice of square, like the wind tree model of statistical mechanics.  It's a lattice of lakes.  And this is the front, I can get this to work.  This is where the front would have been in the absence of diamonds.  This is where it actually is.  And you can see it's slowing down.  So there's an index of refraction that describes these complicated media, at least under some  circumstances.  On the other hand, if I move the squares a little further apart, as Wolfram did, now  you can see that you get cusps, but the cusps heal, and the overall velocity is unchanged.  This is a big difference from wave phenomena where there would be backscattering and we  would probably slow down a bit, even in this expanded geometry.  So we're still trying to track that down, but at least under some circumstances can  be a reduced velocity, and hence the bending and an index of refraction.  So now I want to conclude by talking about population genetics.  And ideally we would have been able to make viruses that were different colors and track  the different viral species.  We weren't successful, at least yet, in doing that.  Maybe someone in the audience could do it.  But what we decided to do instead was to now, instead of having viruses, grow through bacterial  lawns to let bacteria just grow on a petri dish, just like in the original experiment  that I showed you.  There's, again, a bacteria soaked filter paper, three different colors.  There's a great bit of chaos and fighting for dominance or whatever you want to call  it with these approximately neutral strains, red, blue, and yellow in the early days.  But eventually there's local fixation, genetic demixing, of these three colors.  And now I want to ask what happens when we add an obstacle.  So obstacle, in this case, was simply a circular, very fine mesh piece of filter paper that  you can vaguely see here and see here as a kind of black circle.  And nutrients couldn't get up through the agar to feed these bacteria that were coming  down from above.  So if I take a look here, you can see the effect of the obstacle and the population  genetics.  It's not very visible, but there was a cusp, same cusp we had before, but right at the  cusp, at six o'clock, two genetic variants come together.  And this only happens when they're neutral.  If there was a selective advantage, this meeting would have been shifted to seven o'clock  over here, or five o'clock, let's say.  So this is a kind of selective advantage meter.  And notice that this boundary is much straighter than the wiggles that you have over here.  So there's a scar in the genetic landscape of this obstacle, this lake-like obstacle,  that seems to persist for a very long time.  And here's a radial range expansion version of the same thing.  Again in this case, the yellow and the blue strains meet at least approximately at six  o'clock, maybe a little more towards seven.  And this shows that these obstacles could really have, we think, a pretty important  role in the population genetics.  And as I said, Wolfram is trained as a theoretical physicist.  So when he got these beautiful results, he just went and simulated.  And what he did was to simulate not three colors, but essentially an infinite number  of colors.  One color, every bacteria has its own distinct genome.  And that's true of humans, to an excellent approximation.  And it can be implemented actually using something called the brain-dote technology in molecular  biology, which we may or may not do ourselves.  But here's a whole bunch of colors migrating around a elliptical object.  And you see pretty much the same thing.  You'll see a cusp when things come together right now, almost a vertical slope.  And then with these fluctuations, we see the blue and the green, a genetic boundary that  is inherited by the passage around the obstacle.  And you can think of this as a kind of selection by geometry.  Because notice the very fortunate bacteria that just grazed the ellipse up here, the  green, and then this blue strain, which almost got squeezed out, basically had this open range  downstream, and were able to sort of take over.  And the other thing that we're following up on, which we think is quite interesting,  is here's the experiment again.  Here's a simulation going around a diamond-shaped obstacle, so you can see that the shape doesn't  matter that much.  Here's the cusp, there's the boundary, in this case purple and green.  And there are the lineages.  And notice that if you go up here to the frontier, and a la Heuagins and Fermat, you ask, where  did your ancestors come from?  Most of the time, those ancestors just grazed the corner of this little rhombus or this  diamond.  So it's a kind of selection, it's like a population bottleneck, but it's a half-space only that  is the nature of the constriction.  And we think it's very interesting to try to do sort of coalescence theory and look at  these lineages and how they branch, and how that branching is influenced by these obstacles.  So I'd like to conclude by with some summaries, and then one last nice experiment.  This is what I was just talking about.  There are these unlucky genotypes that go smashing into the obstacle.  So this light blue turquoise, it's all over for them, they couldn't go anywhere.  At least they have to only can get across by diffusion if they could diffuse at all.  And the fortunate types that actually go just grazing along the edges, obstacles, reduced  genetic diversity by these unfortunate extinct strains here, there are the lucky genotypes  as well and the cusps eventually heal, but the sector boundary seems to persist indefinitely.  I want to end with a beautiful experiment, it's more of a facetious experiment, which  Wolfram did and he can print anything and so why not print a map of the world?  So here the continents are a susceptible strain of bacteria and the oceans are a resistant  strain, we're at inoculate T7, well where else but the Rift Valley.  And so here is an inoculation in this artificial geography in a petri dish.  And I'm not saying nor I think would Wolfram say that human range expansions can be compared  to viruses, but it does show that geography matters.  Here's Wolfram who had many of the key ideas here and I'd like to close and thank you for  your attention.
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