Closing lecture with David Nelson
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Transcript: English(auto-generated)
00:04
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 granularization group theory on critical phenomena with Michael Fisher.
00:21
And then basically from 1975 he had been at Harvard and was presently the Arthur 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 honours, the Badim Prize for Superconductivity,
00:42
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 honours two talks in the Netherlands so the Lawrence visiting professorship in Leiden in 2006
01:00
and also the Cuffely lectureship 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 are back. We are looking forward to your talk on gene surfing and migrations in structured environment.
01:24
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
01:43
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 summarise a remarkable recent development in population genetics.
02:00
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.
02:21
There 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
02:41
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.
03:00
You probably wouldn't want to repeat it. And so what we got interested in, Oscar Halacek and I a couple of years ago was in a kind of simulation of this kind of range expansion but on a modest length scale in a laboratory Petri dish
03:21
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
03:41
than 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 length scale of ten to the ninth
04:04
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. They only go a centimeter.
04:21
And the aspect of population genetics is illustrated in this range expansion which is a work by Kevin Foster on another bacteria called Pseudomonas aeruginosa and here there's a mixture of two types of bacteria
04:43
and they are identical as he could make them but they genetically demix as you go on this outward radial migration from this central area which we call the homeland. And you can see that there's an interesting question here.
05:01
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.
05:22
A phenomena 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
05:44
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.
06:02
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
06:21
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
06:40
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
07:03
influenced the kind of genetic diversity that we saw in those bacteria and we'll see that eventually we led to some simplified models of spatial structure, migration around obstacles that we think of as lakes or deserts or they could be mountain ranges
07:21
and since I'm in the Netherlands I will use Huygens principle to analyze these range expansions and I hope in 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.
07:43
He now does both theory, simulation and also experiment and my colleague in cellular biology, a very imaginative colleague named Andrew Murray. Okay, so first I want to discuss what we call the elephant in the room.
08:02
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
08:22
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?
08:42
And I personally don't believe that and if you start talking about centuries you can say, well, when did the 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.
09:03
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?
09:25
Now could I 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 of all the books they could be allowed to scan
09:45
until various people stopped 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
10:05
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.
10:22
Luxembourg is at least noticeable down here. But we could also, following up on Hilary's observation, compete physics and biology. So, interestingly, here's physics in blue, and it was around,
10:42
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,
11:06
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,
11:25
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,
11:40
is that perhaps, in fact, the 21st century would be the century of DNA, as revealed by interdisciplinary research. So with that lengthy prelude, let's get down to business. I know that Tom Shimizu 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.
12:04
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.
12:23
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 2 microliter aliquot,
12:42
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,
13:02
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,
13:21
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, but different densities of the initial inoculant
13:43
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
14:02
to make these beautiful fractal patterns, and what I'd of course 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.
14:24
And so here's a blow up of this little region here, and 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
14:42
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,
15:03
it's a well studied virus, here are some of its properties, and basically what it does, it has an icosahedral symmetry, this virus seems to know about the platonic solids, and as many of you know, when it injects its small modest complement of DNA
15:22
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, indicated I think by flags in this picture, and here is a picture, it's a different virus,
15:42
but 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 hyperdermal needle-like projections to get the DNA inside.
16:01
This is a movie, Wolfram actually made this 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,
16:24
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
16:42
and excess virus is produced, the virus is diffused to a fresh region of bacterial lawn and explode over there as well. 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,
17:04
but 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,
17:21
and this is T7, a well-studied example. Yinnen Magasko 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,
17:42
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 non-linear differential equation before.
18:03
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 Sarlu's, now director of the foam. However, this does have almost 200 pages.
18:22
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.
18:45
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.
19:02
That's the result of the linear term and this diffusive term, so that's pretty standard. But remarkably, once this non-linear term gets into play when it reaches the carrying capacity over some region in this one-dimensional illustration, this wave equation, this becomes wave-like.
19:21
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.
19:43
It's a non-linear phenomena, but it still has a constant wave velocity. And 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.
20:02
Several actually are going to take off here. So I inoculate, and you can see the time scale is hours, 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
20:22
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,
20:40
and so we seem to have a viral analogue 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.
21:01
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 3 o'clock, and there it goes. Notice it's preferentially attacking the yellow susceptible strain,
21:25
but it's 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,
21:42
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.
22:03
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 through inhomogeneous random frozen dielectric landscapes,
22:20
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,
22:46
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,
23:02
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,
23:25
you 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 the 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,
23:48
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.
24:01
And now let's watch this go around. Notice there'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,
24:22
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.
24:47
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?
25:02
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,
25:21
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'll in fact get a bending, a refraction, as you would a la Fermat, who formulated a principle of least time.
25:42
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 homedium here, a path parameterized by arc length.
26:01
Take my word for it, this equation down here, where I have a position-dependent velocity, describes the principle of least time. And the question is, does that work for these non-linear wave equations? I'm not saying there's interference. That would involve crests and troughs. These are not crests and troughs.
26:21
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, okay, here's a little virus or an infected cell right at the front.
26:40
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, as summarized up here, it's survival of the fastest.
27:04
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.
27:23
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. 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,
27:44
and it gradually flattens, and it starts off for a diamond like a 90-degree cusp, and then 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.
28:01
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.
28:22
However, I want to mention, this will come at the very end, we have to be careful. Huygens Fermat neglects the discreteness of viruses and cells. It's a 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,
28:44
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.
29:02
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.
29:22
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 are sort of reminiscent of a kind of Feynman path integral approach to understanding complicated dynamical problems,
29:40
and in general you'd have to sum over paths, but there's certainly something very interesting, but vaguely reminiscent of Huygens and Fermat. Notice 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,
30:08
although it neglects the wave nature and so forth. Okay, so here we have our Fisher equation, and what we can do, actually, to sort of test Huygens and Fermat
30:21
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 virus is still diffused, 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,
30:42
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,
31:02
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. So this is like a lattice of a square, it's like the wind tree model of statistical mechanics,
31:23
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,
31:43
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.
32:03
This is a big difference from wave phenomena, where there would be back scattering, and we would have probably slowed down a bit. Even in this expanded geometry, so we're still trying to track that down, but there, at least under some circumstances, can be a reduced velocity,
32:21
and hence 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 bacteria that were different, sorry, viruses, that were different colors, and track the different viral species.
32:42
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, and this is, again, a bacteria-soaked filter paper,
33:04
three different colors. There's a great bit of chaos, and fighting for dominance, or whatever you want to call it, of these approximately neutral strains, red, blue, and yellow, in the early days, but eventually there's local fixation,
33:20
genetic de-mixing, 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,
33:42
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, and so if I take a look here, you can see the effect of the obstacle on the population genetics.
34:03
It's not very visible, but there was a cusp, the 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,
34:25
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,
34:42
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,
35:02
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, and 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,
35:22
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 brainboat technology in molecular biology, which we may or may not do ourselves, but here's a whole bunch of colors
35:42
migrating around an 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,
36:02
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
36:20
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,
36:41
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. There's the cusp.
37:00
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 Huygens and Fermat, you ask, where did your ancestors come from? Most of the time, those ancestors just grazed the corner
37:24
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
37:42
and look at these lineages and how they branch and how that branching is influenced by these obstacles. So I'd like to conclude with some summaries, and then in one last nice experiment, this is what I was just talking about.
38:02
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 only can get across by diffusion if they could diffuse at all. Then the fortunate types that actually go just grazing along the edges,
38:24
obstacles reduce 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.
38:43
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,
39:04
and the oceans are a resistant strain. Where to inoculate T7? Well, where else but the Rift Valley? So here is an inoculation in this artificial geography in a petri dish,
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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. There's Wolfram, who had many of the key ideas here, and I'd like to close, and thank you for your attention.