Scientist Stories: Uri Alon, Geometry of Phenotype Space

[No speech]In this case, each point here is not a different species, it's a different individual ant from a colony of leafcutter ants measured by EO Wilson. The parameters are, the traits are the size of the ant and the relative size of the gland that makes the pheromone trail. And he also measured what these ants do. And there's 3 major tests: staying at home, going to conferences, like the foragers, like me now, and the soldiers. Literature, there's a question why— and most ants with different morphologies, very different morphologies here. In fact, ant people ask, like E.O. Wilson, why not have 3 clusters of ants, 3 distinct morphologies, so you have 3 optimal morphologies? Why do we need the ones in the middle? And I find this question very good as a metaphor for later, for talking about division of labor in cells inside a tissue. So we'll get to that later. And the answer is that the ants in the middle, they're suboptimal at the 3 tasks, but they can be quickly recruited. So if your nest is attacked, you don't have time to make the new soldiers, but those ones in the middle can be recruited. It gives you dynamic flexibility. So if the time scale of changes is larger than when you can produce new phenotypes. And we asked also about these archetypes. What happens if they move? Do they move on evolutionary time scales, or do they need to follow a rotation? So we wanted to ask, Does this work also on time scales of geological type, like 300 million years? And we turned to ammonite fossils as a system where there's— you can look at their geometry and characterize it very well. They're described as two logarithmic spirals, one inside the other, as David Raup showed in the '60s, really nice morphology. And there's data because they're used to date rocks and just because they're beautiful too. A lot of work about them and a lot of extinctions too happened. It's an interesting evolutionary example. Each point here is a genus of ammonites and axes are about the two logarithmic spirals. This axis of their W is the amount that the logarithmic spiral grows, the rate it grows with each revolution, the fraction that it grows. D is the ratio between the inner and outer logarithmic spirals because they look like this, the opening. You can see that half of morphospace is empty, right? This A region. It's certainly feasible. There are species that live there. They're open kind of things, but they're not ammonites. I drew here the archetypes that you can infer. They look like a rope, a disc, and a horn. Now, there was a mass extinction where only two genera survived. Everything else was wiped out. And in about 10 million years, they diversified to fill up essentially the same triangle. And then there was another mass extinction where only one genus survived and again filled out the same triangle. And then 65 million years ago was the dinosaur extinction, and basically they didn't manage to do that trick again. There's only now, there's only nautilus species. I don't know why, maybe because mammals filled up other niches. It's not clear. But this argues for us that the archetypes are constant over hundreds of millions of years. And you can look at the shapes, and they're simple enough to actually make hypotheses about what the task might be. So the rope one is really close. It's almost identical to the point. You can calculate. Of maximum volume per shell material because the shell has a constant thickness per radius. So that's, we think, has to do with some kind of economy. And drag is the other archetype. Chamberlain measured plexiglass models like drag coefficients. So we think this is for swimming, that's for economy, and the other one is close to keeping your— it has to do with rapid growth. It's important for evading predators. And you can find out what the function is that maximizes that. Those are the performance contours. You see they're not very circular, but you still get a kind of a triangle. All right. You can also think about molecules. So here, it's Vit Lustig, who was here at the Institute of Advanced Studies for a while. He looked at this protein, RuBisCO, that takes carbon from CO2 and turns it into sugar. That's why we all are made of carbon, thanks to this enzyme. And from 30 different plants, you can measure the 4 kinetic coefficients, which is the catalytic efficiency and the halfway point for carbon dioxide and for oxygen. So it's a 4-dimensional morphospace at each point from a different plant. And they fall on a line. And the line has to do with environments with higher and lower CO2 or CO2 concentration. So you can also think of individual molecules, I think. I mean, I'm really interested in this direction if anyone here is interested in molecules and their properties and asking the different tasks of a given protein, you can say. This is very interesting to me, and I don't know much about it. Okay, and then it occurred to us after— so I mean, I've been thinking about this for about 5 years, and it took a few years to realize that we can also apply this to not only to morphology but also to gene expression in cells. Now, you might think that gene expression in cells is unlike bird beaks, and because your bird is born with a beak and can't change it. But genes constantly— cells constantly change their gene expression to match conditions, right? But still, if you think about a small element of time, like a second, and maybe the cell can make 1,000 proteins in that second, it still needs to decide what to make. Because if a cell has different tasks like growing and surviving, for example, you can't be optimal at both tasks with the same kinds of proteins, right? You need ribosomes to grow, stress proteins to survive. So we thought to look whether we can see these polytopes, polyhedra, in gene expression. So we started with the kind of experiments that we do. We made— in order to look at questions like this, but this was much before, we made a library of 2,000 E. coli that are all identical except that each one turns fluorescence in proportion to its rate of producing or promoter activity of gene number i. And that's— we did that by taking Green fluorescent protein gene from jellyfish with standard technique, and gluing it in front of the regulatory region, piece of DNA from E. coli that controls the sugar gene. Now you add sugar and bacteria turn green. This is the amazing plug-and-play feature of biology that I can't stop but be amazed by. When we grow the bacteria in this robotic system where we can measure the fluorescence very accurately anyway, as a function of time. And this is 4 different conditions and reasonable accuracy and reproducibility. So when you look— oh, there's a question. Thanks. Yeah. [Speaker:AUDIENCE] Are you looking at single bacteria at a time, or are you looking at groups of bacteria? [Speaker:PROFESSOR] So the question is, am I looking at single bacteria or groups? In this experiment, it's groups, millions in the same— [Speaker:AUDIENCE] Isn't that a problem? [Speaker:PROFESSOR] It is a problem, yeah. And so later I'll show you data for single cells too. So what is the trait space here? It's the fraction of the total gene expression devoted to gene number i. Let's see, ribosomes, this is percent. So 0.8% in this given time point is for this gene. This is an iron storage gene. This is oxidative stress gene. So there's 2,000 of these, 2,000-dimensional space. 90% of the variance is limited to 200 genes. These are the top genes. So we look at this 200-dimensional space. I'll tell you about the rest later. So that's the majority of resource allocation. And E. coli in different conditions, the data falls on a line. It doesn't matter any two genes of those 200 you take, you can see the line. You don't need principal component analysis or anything. When you put them in a nice fresh medium, they're up there. And when they finish up their food, they're down here, or when they're stressed. We look at the genes and it's clear what the tasks are. It's growth and survival. And cells, what they're doing is, you might say, interpreting their environment in terms of one number between 0 and 1, telling them where they should be in the next time unit between growing and surviving. That's all they need to do. And we know the molecular mechanism for getting a line, how you get a line in this E. coli. And it's a very simple molecular mechanism, so I'm just going to tell you about it, is that each of the 2,000 promoters in E. coli gets activated by RNA polymerase. You know, RNA polymerase is what makes RNA and makes the genes. But in order to identify the DNA, it needs a subunit called sigma. Sigma-70 subunit is the one for growth, and it's recognized by certain sigma 70 site. But there's another sigma factor, sigma S. These are proteins, and they compete. And sigma S is what gives the specificity to sigma S site. And so the total transcription is a transcription of RNA polymerase bound to sigma 70 plus the transcription of RNA polymerase bound to sigma S. And so this number between 0 and 1 is just RNA polymerase bound to sigma 70 divided by RNA polymerase bound to sigma 70 plus RNA polymerase bound to sigma S. And that's a number between 0 and 1. All the cell needs to do is control the relative concentrations of these two proteins, which indeed it does with the complicated signal transduction system that controls their stability quickly. So in general, if you want to get a triangle or tetrahedron, I can tell you that one molecular mechanism is to have a factor where 3, 4, k factors compete over something. You can get a simplex in gene expression. That's at least one mechanism. Yeah? [Speaker:AUDIENCE] I was just wondering if this behavior can also anticipate and predict —preemptively change before the condition has developed? So the question is, does E. coli predict the future? Yeah. Preemptively, yeah. And the answer is yes. That's not our work. It's work by Sahripilpel and by Tavazoi. So there are conditions where E. coli, let's say, feels it's going into 37 degrees. That must be inside the human body. Let's upregulate. Acid response genes because soon it'll be in the stomach. And you can break that by evolving it at 37— and that's suboptimal because in fact by doing that it doesn't grow as well for those— and you can break that by evolving it at 37 degrees without acid very quickly. So it does have a Pavlovian effect. It hears the bell of 37 degrees and starts preparing for the digestive juices. And we also know, and this is something we did do, if you take E. coli, the growing— evolve it in a nice comfortable environment, and you put it in a given condition, and compare it to E. coli that we evolved for, let's say, 200 generations in a tough environment, and put it in the same conditions, these two strains interpret the same conditions differently. The one that grew up in the nice food thinks it's better to express more growth genes, and the ones— thinks, right? The ones that evolved in stressful condition is ready for Ready. So it's quickly evolution can tune the perception. Did I answer your question? Okay, good. I want to talk now generally to— because I want to address gene expression in single cells and other situations, a little bit about how we look at high-dimensional data in biology and I guess also other fields of science, except in biology we have the advantage that natural selection is working to make things simple for us. That's my view anyway. Looking at high-dimensional data is very, very complicated. And of course, we can't visualize it. So there's different approaches. One of the best-known approaches is data clustering, where it works if you think the data is in distinct clouds. And then the different algorithms can discover it and say these are red data points, blue data points, et cetera. [LAUGHTER] But what if your data is continuously distributed in a triangle, let's say? Clustering would give you arbitrary results. It would just depend on the clumping of the data. And also, if you have a continuous situation, you want other description, not just the centroid of the cluster, but rather the distance from the archetypes. That's more meaningful. So we worked, and we developed an algorithm, and it's available online on our website, and I'll give you the reference soon. Where we take data and we try to infer or to convince ourselves or to statistically test whether it's described by a polytope, by a line, triangle, tetrahedron, 4-dimensional simplex, etc. And we borrowed algorithms that try to do this in different fields. There's remote sensing, satellite imaging, and analytical chemistry, and there's different algorithms, in fact, 6 different algorithms at least. The way they work is that you have your data, for example, okay? And you put points at infinity that are connected by springs, k points. And you start simulating and annealing it until they enclose most of the data, let's say. And that gives you— and then you do it 100 times or something. Or you look at the convex hull of the data and you choose k points on the convex hull that explain most of the variance in terms of weighted averages of those k points. So those are two algorithms, one of them called PCHA by Moro Proo. And we want all these algorithms to give us essentially similar results. And how do we choose k? How do we know how many vertices there are? So typically what happens is that data starts to be described well by some k, let's say tetrahedron, 4, and then it's well described by 5, 6, 7. That gives you more and more resolution. And we do all kinds of model— comparisons criteria to see how much more information you get by adding more vertices, etc. So there's a whole statistics, and I think this could be done much better, and maybe one of you can help me to do this better, but we have something that we're satisfied with. So, for example, just to show you data on tumors, unfortunately there's many tumors in the world, and in the hospital a lot of times today, you take this tumor and you can place it in a— 20,000-dimensional space of gene expression by doing these RNA-seq or microarrays. So these, again, the traits are log expression of gene i, typically. That's log because you have a wide— some genes are low, some genes are high, maybe several orders of magnitude different. So you do a log to compare them. And log turns out to be the transformation that works also for tetrahedrons for some reason. I don't know why. All right. So it's like relative changes. And then you get— you say, OK, these genes are a certain type, let's say red, and this is helpful to determine treatment sometimes even. Clustering is the state of the art, I think, or similar techniques. So— [Speaker:AUDIENCE] So the plots before, were those also log scales for the expression? You showed the bacteria E. coli analysis. [Speaker:JOSHUA Elkington] E. coli analysis— no, it wasn't log space. It was regular. But I have to think about what happens— I don't remember if we did log space. I think we did. Yeah. [Speaker:AUDIENCE] Can you just explain this entire thing depends completely on the coordinates, but if you're going to use this to determine the linear structure— [Speaker:PROFESSOR] I agree. So again, I think your question and your question— this depends very much on the coordinates. And so when we looked at morphological data sets, we used the coordinates that— I think experts need to tell us what the coordinates are. And in gene expression— [INAUDIBLE] Good point. So another way to look at it is, can we find coordinates, a transformation of the coordinates that gives us these shapes? But if you change coordinate systems, you deform it, but you still have the vertices. And the vertices is what's important for me. Yeah. So the whole thing underlying everything is the fact that you have these hidden soil structures for the—— that you have? How far you are from an archetype is your fitness, you know, where the metric is a Euclidean metric, let's say. So in the end, it's a question of— you see my point, right? No, I don't see your point. I'm going to repeat this point anyway. So yeah, yeah, there has to be something physiological about the fact that that's what matters physiologically, and that distance I agree with you. So if we discover that log space gives us the simplexes— of course, it could be simplexes for a different reason, and we're hallucinating a delusion. It could be. But let's continue with the belief that this is the underlying reason. That means that this metric is, in fact, you can say, in log space. It's a metric in log space. That means that the fitness decays as the— if you change the protein by a factor of 2, —then another factor of 2, you get similar reductions in performance. And that might be physiological, as you say, perhaps. Perhaps it just doesn't matter very much exactly what the transformation— I don't know. Anyway, we're using these algorithms. We think that a tetrahedron is a good description of this data, and also higher-order simplexes. And just to show you what I mean is, in order to visualize it, I did principal component analysis here. So these are— this is what you look like in two principal components. And what we want to see is these straight lines in the data, these edges in the data. I don't know if you can see these gray points. Can you see them? And you back there? Can you see? So if the data has these edges, that's what the algorithms use to extrapolate out and find the archetypes. These archetypes are points in the data in the 20,000-dimensional space. They're like idealized gene expression profiles. They don't exist. You see, they're outside of the data. And if you go to 5-dimensional archetypes, it's— okay, some of the archetypes split up, and these ellipses are what happens when you do data resampling. So you take data with returns, and the archetypes— why do we do that? We want to see if the archetypes are sensitive to outliers. So they don't move much. That's these ellipses. That's the standard deviation of other— so we do all these statistical tests to compare it to randomized data to see how snugly the polytope fits the data. So there's a lot of statistics. But you don't— we want to just see— we want to see by eyes that it looks fine. You don't need the statistics really. Statistics, I think, are more like a rhetorical device. But, OK, now do these archetypes mean something in terms of biology? So as I said, we want to find out if the points closest to an archetype have something special about them. And this data has different layers of meaning on it, like for instance, clinical identification of the tumor. So the blue points here are ones where pathologists say this is a basal kind of breast tumor. Tumor. And so that's one feature. And I plot here the probability being blue as a function of distance from archetype 1. Okay? And what we're looking for is features that are maximally prevalent closest to the archetype. That gives us clues what the task might be. It could be tissue composition, because these are heterogeneous samples. It could be— when we look at the genes that have to do with archetype 1, we see very clear biological functions too. They have to do with growth, division. Dealing with the immune system, and dealing with extracellular matrix. So, it could be, again, that this is some artifact. It could be because cancer is such a messy disease and heterogeneous, and you can't make sense of anything, at least not in this way. Another way to— another fairy tale, you can say, or something, is that cancer is also evolving in the body, and therefore, it has different tasks, right, like growing, surviving, et cetera. And different niches where the cancer is in one second weigh those different tasks differently. And so you can have specialists near the corners and generalists in the middle. And if that's the case, what we're testing now is the drugs that target the function that we infer should be most effective for the cancers that are closest to the relevant archetype. See what I mean? So that's something we're testing right now. So yeah, there was a question. Yeah, but why do you think, especially with this 20-dimensional, 20,000-dimensional, high-dimensional set of phenotypic features, that there should be corners? Isn't this more like— one could imagine that there are messa-like states where many gene expression states are OK and have equivalent fitness that would wash out OK, so your question is maybe it makes sense that there's many equivalent gene expression states that have the same fitness, right? Yeah. So I want to say there's something very fundamental that this multi-objective situation does to reduce the probability of large neutral spaces. I think this is very important, so I'm going to take a second to explain what I mean. So a fitness function— The maximum— the top of the fitness function is actually a huge region that's called the neutral space. No matter where you are, you have the same fitness. But performance functions, okay, suppose I have performance functions, right? So performance is something. If it has a big— this is now instead of a point, it has a big region that's maximal. So that's a neutral region for performance. And I have another performance function, instead of a maximum, it has to get a neutral region. The Pareto front chooses the point in this neutral region which is closest to to the other region. So it breaks the symmetry once you have two tasks or more. And you get a Pareto front like this. So if you have multiple tasks, one has one neutral region, another one has another. And where they intersect, you can say, is much, much smaller. And if you— did I explain your question? Answer your question? Now the question— another thing you can say, 20,000 genes, so much biological function. Cells do more than four things. Why only four archetypes? So again, I think it's a matter of resolution. The way I think about it is the probability of having more than a few points that have large and comparable impact on fitness is small. So if you have 20, it's more— so the other tasks, you need to look at finer, finer resolutions, find more and more archetypes. And the archetypes do split up like that a little bit. So these are the 4, let's say, most impactful They have the highest gravitational pull on the data, you might say. And so you need more and more resolution to see the other tests. These data are not, not yet. Soon individual cells. They're all mixtures. And so part of the story might be tissue heterogeneity, which also can give you— A related question? So assuming that all tumors, cells in a tumor have a similar task. Yeah. Yeah. Yeah, this is not single cell yet. I mean, that data is coming out. I'll show you in the next slide single cells. Can you wait? Yeah, and we'll see what happens with single cells, because it's very important. Some— so this is this tumor cancer genome. A lot of different cancers have similar situation. Some cancers just look like a cloud. This doesn't always work, right? Sometimes it's a cloud. You can't fit it reliably. It doesn't always work. But for most of the cases, it does. And we find that some of these archetypes are shared between tumors. So there's like tasks that are specific to one kind of cancer, tasks that are shared. We're interested in basically the space of phenotypes, archetypes, you might say. Now we come to the single cell. Finally. So, as those of you here are— oh, there's a huge technological advance. So when I was a postdoc with Stan, I witnessed—— and with RNA analyzing these arrays of the colon cancer cells, right, we witnessed the ability to, instead of measuring one gene at a time, to go to these 20,000 genes. That was a big step, right? And that actually opened up a lot of physicists coming into— mathematicians coming into biology because that's high-dimensional data. We thought maybe we can do something. Now there's another step, but in order to do that, you had to grind up millions of cells, right? Now the technology has advanced so you can look at a single cell And it's kind of noisy, but you can still get data for— it's well, higher resolution if you look at, let's say, 80 genes, and lower resolution if you look at all the 20,000. You can only get a sampling, but still, now you can look at the diversity of cells that look identical. Let's say the different, in this case, intestinal cells. They look identical under the microscope. But in terms of the gene expression, we can ask, just like the ants, what happened. And what we find is that when you think of a cell type, these are intestinal stem cells, and we You see something very similar for, let's say, dendritic cells. Think of a cell type. Cell types have— also have multiple functions. Yeah, so, for instance, a liver cell needs to detoxify blood, make albumin, glycogens, produce glucose, so a lot of things. So, but they're all hepatocytes. So, what is the cell type? Is it a cluster in gene expression space? So, we— what we find is that it looks like a polytope in gene expression space. So, for example, in these— each point here is a different— individual cell from the same organ. And you can see— so this tetrahedron, when you project on two dimensions, you get these kite shapes. It depends if the other vertex is inside or outside. The biology is very clear. Archetype 1 is stemness, which is renewal. And the other three ones are differentiation. It's turning on differentiated genes. Basically becoming the kind of cells of the intestine, and stopping division, and global reduction in gene expression in general. And over time— this is a snapshot. One limitation is we can't see cells over time, of course, yet, in many genes. But we do know the height above the intestine crypt. We know something that's a proxy for time. Over time, the cells somehow move from one parallel to the other plane. So I really want to know what the dynamics are here. Still can't see that clearly. All right, so we propose that cell types might be polytopes, and that can give you the tasks of the cells. It might relate to their position in the organ, et cetera. Yeah? [INAUDIBLE] OK, the question is, is there a difference between protein and RNA? And I don't know because I don't have data for proteins. But still— OK. All right, I'm going to end soon, very soon. And to end, just tell you one more issue that's puzzling me right now that I don't know the answer. I would like your help in thinking about, maybe I should say. So, this all began when I met Kathy Cavanaugh, who's a zoologist, looks at— and she showed me data about phalanges, these bones here. If you look at your fingers, you have these bones called phalanges. And she showed me a line, basically, of phalange proportions, and that started this whole thing. So, the— Fourth, I think the fourth toe of the bird, third toe of the bird has four phalanges. We only have two here. And this bone, it's a tip. They have four. And you can look at the ratio of the areas of the bone. Again, that's the coordinates in your honor. The ratio of 4 divided by 1, 3 divided by 1, and 2 divided by 1. So it's a three-dimensional space of dimensionless coordinates. And in that space, each point is a bird species. And you look at the different bird species and they fall on a plane, and within that plane they fall in a triangle. And now you, based on what I told you, you understand what's going on. And you look which birds are here— parrots and stuff. This is the archetype for grasping a branch. It's equal-sized phalanges. And this is the archetype of the ostriches and stuff for walking. It's long, short, short, or nothing. And up there, it's for scratching or or raptoring because of the long fourth phalanx. And that's the biomechanical optima too, equal size phalanges for grasping a stick, et cetera. And if you remember anything from this lecture, now you understand the difference between your hands and your feet, right? So our hands are equal size for grasping, and our feet have this long foot and the short toe, which is the biomechanical optimum for walking. Okay, but what I want to ask is if we look at the chicken here, it's one point here. That's the average over all the chickens in the sample. Now if you look at individual chicks, individual chickens, they're different from each other, right? So they form a cloud around the average point. Does this cloud— is it symmetric in all dimensions, directions, or is it more like flattened along the Pareto front that describes other species? Cathy did that experiment with 100 chicken chicks and 100 zebra finch cheeks, and the answer is that there's variation between— there's sizable variation, but that variation is flattened in the direction that describes different species. So here is something that— to understand. Excuse me. And one thing is that the way I understand natural variation between individuals, which comes of course from mutations, what's called polymorphism, the differences, each one of us is different in about 1 million places in our genome. And also the noise, the way that is translated to pattern formation, those two areas. So the system is designed such that you get a variation, and I think the reason for this variation is that chickens in fact have multiple niches that vary in the importance of these three tasks: scratching, walking, and grasping. And so the purpose of this variation is to cover a certain region of parameter space, but the system is built in a way so that both the mutations and the noise keep you on this plane. And how does that work? And that's a puzzle for me. And one thing we see in simulations is that in order for this to work, one way for this to work is that the effects of the naturally occurring mutations of polymorphisms, which is a big question also in human phenotypes, we want to understand what the differences between us mean in terms of our phenotype. Phenotype. If they are— if the polymorphism is not just mutations, unfortunate mutations that push you off the front, those can happen. And in fact, Kathy Kavanagh showed that if you perturb chick development in different ways, 80% of perturbations keep you on the front and 20% push you off. So it's certainly possible to get phalange proportions that are different. It's not impossible. It's possible. You can do it experimentally. But the ones that are commonly found, the mutations commonly found in the population, are ones that push the phenotype in the direction of the plane, and that you have additivity in the sense that if you get from your parents two of them, they add up, or at least their epistasis, their nonlinear interactions, are limited to the direction of the plane. So if you have those two, like a lot of additivity and polymorphisms that push you along the plane, you can get this situation. And in our simulations, natural selection actually with multiple tasks moves those polymorphisms into that solution. It evolves to that solution. And then what you have, each individual has, let's say, 50 mutations pointing this way, 50 mutations pointing this way. In total, it's a kind of a Gaussian distribution. But if you want to evolve now, you can quickly purify out the ones that are pointing in the wrong direction and zip up, as long as you're evolving to a niche that has the same tasks. So you can quickly evolve, change proportions. But evolving off the plane, like if there's a new task, is very difficult because there's little genetic variation in that direction. So this field of evolutionary medicine, for example, that we complain about our back not being optimized for laptops, is concerned with things like this. And the question of what morphological— what differential equations that give you patterns can also give you patterns, but such when you change parameters keep you on a triangle, is a theoretical question we're trying— in general, is a theoretical question we're trying to understand. All right, so to summarize, those trade-offs can lead to simplicity in phenotype space. And you can expect shapes that have corners, number of corners, the number of main tasks that are important for fitness. And from this analysis, you can understand what the tasks are by looking for what's special about the organisms closest to each archetype. And if you want to analyze data like this, you can look at this algorithm where you just put your data in, and it gives you the best fit polytopes and statistical significance and analyze what features are maximal near-age archetypes so that you can guess what the tasks are. And Cathy Kavanagh here is a partner in this work. She's at UMass. And this group of graduate students, Yael Korem, Hila Shefteh-Lavimayo, Jean Hauser, Yuval Hart, and Pablo. Secondly, they're all physicists except for Jeanne, who's a computational biologist, who participated in this study. So thank you very much.