 Let's see if I'm loud enough. Okay, so I'm a theoretical physicist and I work on a variety of probes and population dynamics ranging from invasive species to cytoskeleton. But today I'm going to talk about all about microbes and of course as you know microbes affect all aspects of our lives and they do it by working together, right? So any of the function you will hear is a product of many species from ten to a hundred, potentially a thousand species working together to provide a particular ecological function. And therefore people have been interested for a long time to understand how the microbes interact, right? You can construct interaction networks based on the metabolic exchanges, based on time series data, based on experiments or you can look at cross-sectional samples and infer correlations between microbes and interpret those interactions, right? As a result of this, you typically end up with a network which is a human microbiome based on these cross-sectional correlations and you can't see the links very well but believe me this network looks like a nice hairball and that's a typical result you get from these studies. So the goal of my talk is very simple, I want to convince you that all of these interactions cannot be happening at the same point in space at the same time, right? And why that's the case? Well, because microbes have a finite size, right? And you can't pack too many cells next to each other so within any spatial location on a very small subset of this network going to be realized. And things are worse than that because spatial structure is not random. So here is an example of spatial structure obtained by Gary Barisi's lab who was able to fluorescently label up to 20 most abundant species in the oral microbiome and you see that microbes from this got patches and the number of interactions in different species is very small compared to what you would imagine in an alumnus population. So this spatial structure forms for a variety of reasons. It could be a product of ecological succession, you know, different organisms could chem attacks towards each other, they actually have special receptors to bind to each other. But apart from all the biological, complicated reasons, this structure arises just from the fact that if a blue cell divides, well, it's right next to another blue cell. And so over time you end up with a patch of distribution where similar cells are next to each other, right? And so my goal is trying to understand how this very simple process of, you know, limited dispersal affects microbial interaction, right? Cells have been studying this process in a simple laboratory environment by doing the following experiment, right? You take your favorite organism and design two strains that could be interacting, but for the simple example, they're not interacting, in fact, they're identical except they have a different fluorescent marker. You mix them together, put in a pitch redish and then wait for a week and you have a nice colony. This colony doesn't have any features because it doesn't matter whether you grew one strain or two strains, they're identical. However, if you look under fluorescent microscope, you end up with an interesting pattern, or the pattern of sectors, where different strains tend to separate from each other. And again, that's simply just the fact that, you know, when blue cell divides, it's next to another blue cell, right? We can also think about it maybe more theoretically by making a connection with population genetics. And again, to do that, you can imagine looking along this arrow, which is kind of tracing the history of this expansion, and I point out that in these experiments, most of the growth happens outside of the colony where there are enough nutrients. And so in the beginning, you have a mixed population, however, as the expansion proceeds, there are fluctuations, which in population genetics and all in the genetic drift. And these fluctuations eventually cause one of the genotypes to go extinct and the other one to reach fixation. So as a result, you end up with the formation of a sector of a single color. Right? So these patterns occur for a variety of organisms. Here, I just showed a few examples, right? So on top, you see E. coli, which is now expanding in different geometries. They've been expanded from a droplet. There was a linear inoculation that grows up and down. There is our favorite lab yeast. There is 2-demonus. And these patterns have been found for aspombi, so satelus, and many other organisms. It's also very straightforward to simulate these dynamics. All you need to do is somehow populate on the lattice where you have sides. Each side may have an organism, and there's migration between sides. And in no time, you can reproduce patterns like that in simulations, which are very simple simulations without any complexity that reproduce biological experiments. And you can also do more complicated simulations where cells mechanically in a rack, they exchange nutrients. But regardless of the complexity, the robust behavior of sectoring and genetic drift is preserved, right? In addition to being able to simulate these patterns very easily, one can actually solve them entirely analytically. One can compute almost every property one wants about these expansions. I just as an example, I'm showing you that one can calculate how the number of the sectoring domains depends on migration rate, velocity of the expansion or size of the colony. And these predictions are in quantitative agreement with experiments and simulations. So these patterns affect many processes in microbial populations because almost all evolution of ecological processes are going to be confined to the boundaries between the different strains, rather than being happening everywhere in a population. And I've spent quite some time trying to figure out how these patterns affect species competition, mutational processes, horizontal gene transfer. But today I'm going to focus just on microbial interactions. And I want to answer a few questions. The first couple of questions that I'm going to answer in quite some detail talk about how spatial structure affects interaction networks and then talk about the diffusion of metabolites in this colony. And then as we have time, I'm going to address maybe some more interesting questions which are part of the unpublished work, which is how does the geometry of the habitat affects interactions. What can we say about hybrid interactions that involve more than two species? And then maybe we can identify some design principles in microbial communities and maybe even try to infer interaction patterns purely from the spatial patterns of microbial abundances. So let me begin with the first question. And to do that, we're going to begin with a very simple system, a cross-fitting of two strains. It's a common motif in more complex interaction networks as well as probably a key element in synthetic microbial communities that we would like to design. And for that reason, probably Winning-Shu has introduced this system about a decade ago, trying to understand how simple heterotrophic cooperation works. So here are the two species that exchange an amine acid and they really require each other for growth. We're going to label the species A and B because we're physicists and then we're going to denote the relative fraction population by Fs and these fractions sum up to one. To model the interaction between them, you have to define their fitness and their fitness must depend on the abundance of the partner in the population and in the simplest possible model, you can say that the fitness of species A is going to be some basic growth rate plus a term that is proportional to the abundance of the partner organism, right? So the fitness of A depends on the abundance of B and vice versa. So if you define a fitness function like that, then in a well-mixed population, you end up with a following dynamics where every species is going to invade when rare and they're going to stabilize at a certain fraction, F star, right? This fraction is given by the relative benefit that organisms get from crossfeeding and the rate at which they converge to this steady state is given by the strength of the interaction, right? And here's a simple mathematical expression that one can obtain from the evolutionary dynamics, right? In addition to describing mutualism, the same model can also account for by-stability and competitive exclusion and that's pretty much the same diagram that Jeff Gore was showing a couple of days ago where, you know, if both alphas are positive, you end up with mutualism, if they're negative, you have by-stability and if they have different signs, then we have competitive exclusion, right? And again, so what I'm going to try to do next in the next five minutes is try to tell you how this diagram changes if we put this interaction in the spatial population, right? So the first step is very simple. You can just put this population and simulate this dynamics and, for example, you can look at the spatial patterns as you change the strength of interaction, right? And so here what you see is a population that was initially occupying this low edge and was allowed to expand outward, right? And if you look at these pictures, you immediately see that cross-feeding can overcome the spatial separation to domains, right? So the bigger the cross-fitting, the smaller the domains that we see. However, this is not a simply quantitative effect. In fact, there's a qualitative difference between small and large cross-fitting and we can see just by running simulations longer and what you find is that if the cross-fitting is too small or is below some threshold, then the populations will completely face separate and eventually one of the species is going to take over while the cross-fitting is strong enough then the mixing is going to... Yeah. Okay. So the dynamics of simulations, we have lattice of sites. Each site has n organisms. They can migrate between them and we do a Moran type of update, right? So one particle is always killed, one is born and the probability of birth is depends on the local fitness and the local fitness in this case is computed according to those formulas where it's proportional to the fraction of the partner species, right? So in these guys, it's effective. So I showed you both 2D. These guys are 1D where we simulate just the growing edge of the colony because that's where all dynamics happens. But we will get to some 2D simulations later as well and I'll try to make clear which one is which. You can also do 2D, but again, as long as you expand in one direction, it's going to have the same results. Okay. So I'm going to... Instead of showing a bunch of these pictures, I'm going to build with more quantitative and I want to define a few indices that are going to characterize how these... How about characterize spatial structure? The first index you can define is somehow a mixed index which is analogous to heterozygosity that we have in a certain genetics and just tells you the probability to find two organisms, two different organisms of the same spatial location, right? So in the simple model I'm talking about now, this mixed index actually tells you about fitness because if, I know, age is zero, there is no cross-feeding because species have to be present at the same location. As we move to more complicated models that for example include diffusion of metabolites, this will no longer be the case because even though these species are locally demixed, they could still exchange nutrients and so it's nice to define a more general quantity which I'm going to call community productivity and it basically measures the fitness gained due to cross-feeding, right? So it's your fitness minus the fitness without cross-feeding normalized by the fitness you would get in a well-mixed population, right? So I have my definition of W's, right? So, you know, W naught plus alpha beta so I can compute the fitness at every spatial location using my definition of fitness. So I mean in simulations you define a fitness function and you run simulations, right? So of course you can always compute fitness in your simulations. No, it's a Moran model, right? So in Moran model you can have more complicated models where you have no carrying capacity, growth rate, competition. In this case, it's just one number because it affects, I mean, you can think of it as a growth rate, right? Yeah? Sorry? Well, that's the probability, right? Because if you know, if for example F's, if you have 50-50 ratio, right, you only have 50% chance to take two different organisms, right? And that's why the factor of two is there. It's really a kind of, it's an exact formula. So really the probability to get two different organisms. So this H is gonna go between zero and a half. And if we didn't have a factor of two, it's gonna go between zero and a quarter, which is not quite right. Yeah, so we don't have nearest neighbor. We have many organisms per site. And within each site, it's one, whatever, W naught plus alpha times F of the partner. Right? Okay. Any other questions? So again, as I said, in this very simple model, there is a very simple relationship between these two quantities. And it's just, you know, the productivity just proportional to the to the mixed index because it's all interactions are local, right? Okay, so this measure, I can show you pictures like this where I can quantify the community productivity as the function of the interaction strength. And as I said earlier, I have a phase transition where for strong interactions, I have positive community productivity. But below also on threshold, the productivity drops to zero, right? So this was, this is the data for symmetric interactions. So we can also of course do asymmetric interactions. And what turns out is that the symmetric interactions are significantly more stable than asymmetric ones, right? So here we see the heat map of community productivity. The white regions correspond to productive cross-feeding while the black regions correspond to competitive exclusion, right? And so I also show you the well-mixed diagram here so you can compare and see that this entire region of coexistence was squeezed to a much smaller region shown here in white. And for the physicists in the audience, of course, we see this picture with slightly different eyes. You know, these lines over here are lines of phase transitions and these phase transitions can be classified in a relatively small number of universality classes, right? In this case, most of them belong to the so-called directed percolation transitions, but this very special point is a different universality class called DP2. And the crossover between these universality classes, of course, determines the shape of this phase diagram. I know we see as a go forward that this idea of universality classes and phase transitions can sometimes give you significant insight into the dynamics of these populations. Around the same time as we were thinking about these ideas, Winning-Schrupp was also thinking about it together with her post-doc, Babak Momeni, and they've done experiments to demonstrate precisely that. So they have engineered a few strains of yeast and they indeed showed that cross-feeding persists only if it's sufficiently strong and populations that coexist in a well-mixed culture fail to do so in spatial populations. A few years later, Melanie Miller in Andrew Murray's lab has done a much more thorough experiment so where she basically tried to tune the degree of cross-feeding by adding amino acids to the media. So she can have these two amino acids that the species cross-feed and if she doesn't add them to the media, then of course, the mutualism is obligate. However, if she adds enough of these amino acids then the benefit of cross-feeding drops to zero and in that case, you have demixing, right? So in shape, no, made this entire diagram, I'm gonna simplify it on this picture. So basically the summary is that in the experiment she indeed found that there are these four regions, there's a region of coexistence, there's a region of almost neutral dynamics where there's demixing and the regions of competitive exclusions. Okay, so I have now addressed the first question and I wanna move to the second one which is talk about public good diffusivities. And the diffusion you can think of, you can imagine it's gonna be very important because all my main assumption was that the interactions have to be local, right? You have to have both species at the same time but you can imagine in a situation like this if the nutrients can diffuse across the boundaries then the species do not have to be at the same point and they can still productively cross-feed. So what's the following question? Well, how does community productivity depend on the diffusion of metabolites? Okay, so I mean, again, now you modify the simulations so now we don't have these dynamics, now the species secrete public goods, they consume them, they produce them, the public goods diffuse with a different diffusion constant and now the fitness depends not on the fraction of the partner species but rather on the local concentration of the public goods. So if we do these simulations we find that actually if we increase the diffusion constant then the productivity of the community goes down and below a certain diffusivity it drops precisely to zero which is a little bit different from what we expected and to see why this is happening we looked at the spatial structure and we found that as we change the diffusivity the communities transition from being mixed to being demixed, all right? Moreover, the demixing, so the length scale of domains grows much faster than the length of which nutrients can be exchanged. So even though as you increase the diffusion constant you transport nutrients further along but this process cannot catch up with the growth of the domain length, all right? To understand why this is happening again it's sufficient to look what happens in the main boundary and this is probably one of the most complicated figures and the stocks are looking to try to work through it slowly. So here these thin lines they show you the concentration of the species, right? So here it has species A, here it has species B and this is a transition zone. The thicker lines show you the concentration of the nutrients or the public goods of the exchange and these lines are smoother because of the diffusion constant that smoothies the variation. Now the differences in the public good concentration tells you the local strengths of selection for coexistence and as you can see here it's pretty high away from the boundary but right at the boundary you have both nutrients present and the selection for coexistence is actually zero. And now you can see that because natural selection can only operate when both species are present it is confined to this very narrow region of the species boundary where the selection is small and one can actually estimate how small it is by a very simple argument the selection coefficient here is gonna be given by a selection coefficient outside times the ratio of the boundary with to the length scale which an nutrient diffuses. So this is a very simple argument tells you that diffusion does not really have any, no new fundamental role in the system it simply renormalizes the selection coefficient and makes it smaller, right? And this is not the way the argument we can use it for example to compute how the critical diffusivity when the productivity collapses depends on the migration rate, population size and selection and indeed absorb the same exponents in simulations. We can also now use the fact that the diffusion does not change the phase transition in the system and immediately argue that for any phase transition we will expect that as we approach the critical point the length scale, in this case the main length scale has to diverge with some exponent given by the universality class of the phase transition and this is indeed the behavior that we see, right? As we approach the critical diffusivity we see that the main length diverges. And again this is something you would not probably guess without the theory of phase transitions and indeed people have been trying to use other approaches like dimensional analysis and the scaling and predicted that for example the main length's gonna go as d to some power which is of course inconsistent with this picture. Okay, so I wanna maybe mention one last thing about the public good diffusion, right? So I've told you that diffusion does not gonna rescue mutualism, it's actually gonna make things worse. But then we were very inspired by this paper that looked at other public goods in this case Hedera force among different organisms and what they found is that even though all Hedera force have a very simple function of side given by this pair of OH groups, they have amazingly diverse set of shapes and they were able to compute the diffusion constants for these different shapes and again they have similar molecular weight but they have very different stickiness to the environment which affects the diffusion constant and they found that diffusion constants can vary by orders of magnitude between different microbes. So I'm not gonna show any data from our work but we explored the effect of whether a species can get a benefit by changing its diffusion constant and we found that diffusion constants are under very strong selection. However, there is no simple answer like I wanna have a very high diffusion constant and I have a very low diffusion constant because the answer depends on the nonlinearities of the dynamics. In particular, whether you have diminishing returns from the public goods or you have somehow accelerating returns and other nonlinearities in production and consumption you may wanna choose different diffusion constant either faster or slower than your partner and so as a result of these dynamics it's quite possible that the diverse chemical shapes could be indeed optimal for a given ecological dynamics. Okay, so by now I've answered the first two questions at least to some extent. I've told you about the role of spatial structure that it can for example remove some interactions in the microbial, basically it can remove cross feeding so you might think that you have this interaction but if you put things in space the interactions don't exist. In a few slides I'm gonna tell you that spatial structure can also create new interactions and I basically told you that diffusion is not a fundamentally new thing it's basically just renormalizes the parameters in the model. So now I wanna move on to so maybe some more interesting questions about the geometry and the high number of species, right? So the simplest of these questions is the question of geometry and dimensionality because in physics we understand it quite a bit and the general answer is that the high number of dimensions you have the more the most similar the dynamics is to a well-mixed population. So why do we have different dimensionalities? Well, so let's think about the growth in a colony where you grow only at the very edge and that's effectively one-dimensional population growth or you can think about growth on top of a biofilm in which case the dynamics is quite two-dimensional, right? And in many cases the difference between one and two dimensions is just quantitative but there are a few examples where one really gets very different qualitative behavior one of them is this symmetric mutualism so in one dimension as I showed you earlier there is this phase transition where below critical strength of cross-fitting the cross-fitting is completely lost this doesn't happen in two dimensions in two dimensions if you tune the cross-fitting arbitrarily small the species maintain their coexistence. However this is a fragile point and if you introduce any asymmetry between the cross-fitting that you recover the phase transition, right? Now the more challenging question than spatial structure is the idea of high number of species, right? And I have this network of the human microbiome but I actually have no idea how I'm gonna model it and if I'm gonna ask simple questions well how does the genetic drift affect these interactions? I don't know how to answer it because I can put some random matrices that Pankaj did on Wednesday but I never know whether the properties I see are from the specific assumptions that I put into the model or from the topology of the interaction network. I don't think we have a very satisfactory answer to how to proceed but we come up with I mean at least maybe a step forward what we call topological model, right? So let me give you a basic idea so we have some habitat, right? And this habitat you're gonna have different species and different number of species in different location. So for example this location has only two species and so what I'm gonna do I'm gonna simulate the dynamics of just these two and say for blue and red there is no interaction so I'm gonna evolve them neutrally. Now in a different patch I have three species so now I'm gonna evolve them according to the dynamics for these three species, right? And I'm gonna say that there are two, I mean I'm gonna basically evolve them according to a well-mixed prediction so if for example these guys are strongly mutualistic then I'm gonna evolve them towards say an equilibrium and equal fractions. And I'm gonna do it with some probability I was gonna capture the strength of the interaction so in this case I'm gonna set them to be one third, one third, one third of probability S and with one minus that I'm gonna evolve them under neutral dynamics sort of binomial sampling, right? So again this model is not perfect but it allows us to quantify the strength of the interaction relative to genetic drift. You know these models give you very similar results if you compare them to mechanistic models, right? So here again my plot of one and two dimensions I mean the actual numbers you see here are very different but like the type of the phase transition is exactly the same it happens in one deep it doesn't happen in 2D. So the models again they're not exactly the same but I think they capture the core the core aspects of the dynamics, okay? So now with these models I can ask some interesting questions. So the first question I'm gonna ask, well what happens to higher order interactions, right? And I'm gonna take a specific type of high order interaction which is I'm gonna call collective mutualism so imagine where you have a community where all three species have to be present to accomplish a particular goal, right? So it's not a combination of some pairwise interaction where you really have to have all three if you have only two, nothing really happens, right? And I can of course do it to four, five and higher number of species. And when I change the number of species I see that these are very different dynamics, right? For two species I always have coexistence and community is always productive no matter what the strengths of the mutualism but as I increase the number of species I have now have a phase transition moreover the critical strengths of interaction increases dramatically as I increase the number of species, right? We can understand the difference between high number of species and two species by looking at the spatial structure and again two species are very special because if you look at the boundaries between the species they're active and what I mean by that is that if you have a migration between red and blue you're gonna create a mixed patch and that mixed patch is now productive because it has both species. If you look at three and more species again most boundaries for example this one are between domains of two colors and if you have a migration across that domain you're not gonna create a patch with three species. So most boundaries are inactive and only when three boundaries meet for example here then you end up creating productive domains and that changes the universality class of the phase transitions again. So here you see the spatial patterns as a function of the number of species and the selection. The first thing we see is that so the black squares show that one of the species went extinct and you see that for each number of species you have to increase the selection by about an order of magnitude but you can also see these plots, right? So let's show you how the productivity changes within the close to the phase transition for high number of species it has the same behavior consistent with the directed percolation universality class but it's a different behavior for two species. And that of course reflects the fact that you are much more likely to have two species interacting than high number of species, right? For example for five species you only have coexistence almost when you deterministically every time update them to be at equal fractions. Well given that how the interactions seem to be unstable but you might somehow one design communities that have them the next question is can one stabilize them by adding pairwise interaction to the dynamics? Okay and the most natural interaction you may think about is adding pairwise interactions that would help these two different species, right? So if you do that and you say you begin without any pairwise interaction which I call reciprocal and just have the collective mutualism that's the result I showed you earlier you have this critical threshold and then if you start adding some pairwise interactions you see that first the threshold shifts and then it disappears and with enough pairwise interactions you can actually stabilize collective dynamics. Now whether this happens in nature I don't know but I can tell you a story. So in fact it's very hard to find a community where you have this high order interactions and that's not surprising because we think that they're unstable. One of such communities is shown here this is a community that has to live off beauty rate without oxygen and that's a very poor energy source and it grows only when all three members are present because of some metabolic interactions that are involved in this community. This community is so poor that it's doubling time approaches a year compared to 20 minutes for E. coli so it grows extremely slowly so you can see the time here in days and it goes to like 2000 days. And then if people looked at the oxotrophis in this community they found that each of the partners makes a unique and amino acid. For example the first one makes these two amino acids the second one is the only one who makes histidine and lysine and the final partner is the only organism that makes serine. So again this is not the main ways to prove but it's quite possible that this pattern of cross-feeding might be useful for the organisms to stabilize the spatial structure in the community. Okay, so I also promised you to show that spatial structure can create new interactions and this is an example of that. So here I'm plotting the probability to find three species in a patch as a function of the strengths of the mutualism. So again the blue one is the collective one we have a threshold and then a higher probability but the same dynamics also occurs in the reciprocal mutualism. And I emphasize that if I had this interaction in a well-mixed culture I would not get that because I only impose pairwise cooperation so if I put all three in a test tube actually any fraction of the three including for example the blue guy going extinct would be equally how likely the species would have the same fitness. So in a well-mixed population you would end up with one of the species goes extinct stochastically but in a spatial structure there is really a selection towards coexistence of equal fractions. Okay, so now moving on for maybe a different topology of interactions the next thing you can ask, well what about a cyclic interaction which is kind of maybe the most opposite to the reciprocal interaction I showed you earlier where species A goes B and B helps C, all right. Again it involves few interactions to engineer so it's maybe more useful. Well this interaction can also stabilize how the cooperation, so again this is without any cyclic interaction and this is with cyclic interaction you see that even a small amount of cyclicity already removes the phase transition and now you're stable throughout the strengths of the mutualism however there's a trade-off even though you become more stable at low values of S you are less productive at high values of S. And this becomes even more clear if we just compare a pure cyclic interaction to a pure collective interaction so the cyclic interaction is always stable however it never becomes very productive. So to understand this difference we can of course look at the spatial structure and here is the spatial structure and again it may not be very illuminating but I wanna point out that the productivity and stability come from different spatial length scales. So the productivity required that three organisms occur at the same spatial point while stability just required that all three occur somewhere in the ecosystem and so we can make it more quantitative by measuring the probability to find species across different length scales. So what do we expect? Well at very large length scales the probability to find all three species has to approach one. The length scale whether it happens is known as the main size of the correlation length. Now the productivity is determined by the interaction length which could be a lattice side in my simulations or the diffusion range in a microbial community and so the value of the productivity depends on how steeply this function decays from one and that decays usually a power law where the exponent is given by the universality class of the phase transition. So again if you look for the actual data so here we have these plots for different types of mutualism and we see that for the psychic interaction the exponent is about a factor of two larger so you lose productivity much more rapidly compared to other topologies. I also wanna point out that this plot is done with just a few, I mean 200 by 200 lattice sides so you don't have to have all the data to infer that you have different types of interactions among these communities and this is perhaps spatial patterns are perhaps the only way to study interactions without perturbing the system because if you take them in a lab you never know whether the interactions are gonna be different from the natural environment because of the different conditions or because you didn't take all the species present in the natural habitat. So perhaps you can use these spatial patterns to at least say something about the interactions present in the ecosystem. So this I wanna summarize, I begin to summarize so I begin by telling you about genetic drift I showed you that it can destroy cross-fitting that diffusivity does not really help and then I moved on to talk about high number of species dimensionality as well as different topologies and how they change the spatial structure which in turn affects productivity and stability of the community. So the earlier part of this work was done very long time ago in collaboration with David Nelson, Kevin Foster, Giorgio Zavio, Andrew Mare, Oscar Howard, Jack and Melanie Miller and all of the new work was done by Rajita Menon in my group and I wanna conclude with a very simple single take home message, right? So when we think about natural dynamics or design of microbial communities we often think that we can go from interaction to the community functions and that may very well work in the wellings populations but if you have spatial structure I argue that there's an intermediate establishes conceptually very important. You have to think about whether your interactions are consistent with the spatial structure and if you're designing some community function you might wanna first think about what kind of spatial structure would facilitate that design and particularly you may choose different topologies of the interaction depending on whether you're designing a stable or productive community because in our case we see that cyclic and reciprocal interactions achieve one of those two goals of the expense of the other. With this I wanna finish and I think I should still have some time for questions.