 I think participants are in and we are brought up. Yes, we can now. Please go ahead. Should, sorry, Atish. Yeah, okay. Okay, I can start. Okay, so welcome you all to ICTP colloquium. We are very happy to have a net wind green today from Princeton University talking to us about modeling microbial diversity. He's currently Howard prior professor of the life sciences at Princeton University. He's a member of the department of molecular biology and Lewis signal Institute for integrated genomics. There is currently also the director of graduate studies and he's the associate director of Princeton Center for theoretical physics and also associated faculty in the department of physics. And I will leave it to Jacopo to introduce the speaker about his scientific. Yes, so welcome everybody was from me so the research of net wind green has found several topics and gave important contribution in quantitative biology, and in particular to the study of intracellular networks of bacteria. But the topics of today is related to ecology and in particular to the modeling of understanding of them to understand diversity in microbial communities and the quantitative and theoretical approaches to his probe to his problem and to try to answer these questions. Before I leave the floor to me doing for the talk I'd like to remind a few rules for how to interact with the speaker. So if you're following from zoom, you can either post your question in the q amp a box, or you can raise hand using participants three dots at his hand. You can ask questions or if you're following from YouTube you can type the question in the chat and I'll be reading it for you. So with that, let me thank you again, net for agreeing in this colloquium, and please. Thanks very much. Okay, can we see everything see the full screen. Yeah, well, I've been looking forward to this opportunity to talk to you today about an area of interest for me and my group and I wanted to acknowledge at the onset, the people who actually do the work in the group on microbial diversity. All of these folks were at Princeton at some stage and have moved on in some cases to other places. I want to start by actually casting my broad view of what the questions that we're interested in here. And so, let me begin by pointing out that there's something amazing about life we got we get used to it. But when we go outside, wherever it is in the tropical rainforests if you look around and you know something about tree species, you find that in a very small area of land, you can find 300 different species of trees. We're used to that we don't see the all not all trees are exactly the same there's variations among the trees and this becomes even more extreme if we start thinking about the diversity of the microbial world if you go outside and you dig in the soil and these days because we're able to sequence directly without having to necessarily grow the organisms in laboratory. People have made estimates that there are somewhere between 2000 and maybe up to 20,000 different genomes that's not just different cells, there's, there's trillions of different cells in out out there but the, this is different what would be different species if we were thinking in animals or plants. And so it's, you know, it's a remarkable diversity of nature. And all of them are somehow managing to coexist in environments that seem on the face of it to be extremely similar to be extremely uniform one gram of soil, one small piece of an island. So, from, from the point of view of just being a human and living on the earth this is a wonderful thing but, but from the point of view of a physicist it's actually very disturbing. And, and I go back to one of the first things that I learned as a graduate student in physics. There were a series of lectures by a very famous physicist who had been there at the founding of quantum mechanics and he said, you know, a good physicist should be able to estimate anything in the universe to within an order of magnitude. And he then proceeded to do so for essentially from black holes to to to molecules in the cells. And I have to say he would be very disappointed because we don't yet have the ability to estimate these numbers a priori to within an order of magnitude we don't know why there maybe should only be 100 different organisms in a gram of soil, or why in principle there couldn't be 100,000 we, we don't have the ability to to make this estimate a priori. And I have to say that as a theoretical physicist the situation is even worse because when I turn to theory. It doesn't help at all it actually makes the situation worse because the simple theories that people have developed for trying to understand competition among organisms, particularly for resources has a very strong statement that you can't have two species competing for the same limiting resource can't stably coexist. So if there shouldn't even be two species present let alone 10,000. Now, you can get more species than that in these models if you have more resources, but it's possible to understand for example maybe there are 50 different sources for carbon sugar different sugars in the environment or maybe 100, but 10,000 is really getting up there and it just, it's not credible that this is exactly what's happening and so, so there's a real puzzle here and this is, this has been around for quite some time in the field going back to Hutchinson who was the father of of modern ecology. Thinking about the case now of plankton, these small organisms living out in the ocean he was, he was very clearly puzzled by this and I'll just read what he said a limited range of resources supports an unexpectedly wide range of plankton species, apparently flouting the competitive exclusion principle so he knew well about the theoretical work, and he said but this is clearly wrong in some way there's far too many species out there to account for the different sources of nitrogen or carbon or even light in the environment. And so this paradox has led to a great deal of thought and a great deal of work, and there are many interesting ideas about what could be leading to the great diversity we see out in nature. And so possible solutions include the idea of cross feeding, even if there's one primary source, for example, I feed some bacteria, a particular sugar, well one bacteria eats that sugar, but it converts it into other molecules and some of those leak out of the cell and the other bacteria eat those secondary molecules and so on. And that certainly does happen in even in experimental situations it's very clear that there's a certain amount of cross feeding. But again, the number of different molecules that are being cross fed is probably small compared to the number of species that are coexisting. Other ideas are the idea that well these theories are about steady state but maybe the situation is never steady it has intrinsic instabilities to oscillations or chaos. But even if internally the system would be stable the world is not stable there are always temporal variations associated with seasons or weather that are constantly changing the playing field. And furthermore, these these models are about a well mixed environment where all of the sort of species see the same in conditions, and of course the real world is spatially structured. So even on this little island to their hills and valleys and maybe a little bit of different soil or in the gram of soil, even though it looks small, maybe there's different levels of moisture gradients of things like temperature, salt or light. And of course, finally, it's not all about competition for resources there's also predation there's for bacteria, they're getting eaten at one end by larger organisms, but they're also have their own viruses as we're very much aware of these days. Viruses are not just a human challenge, but even bacteria have their own viruses. And so they have different pressures on these organisms. But I have to say that none of these possible solutions have are clearly the answer. Each one has perhaps some role and I suspect that all of these are playing a role at least under some circumstances, but I think it's possibly important to take a step back and think about what this problem maybe from a from a fresh perspective, particularly in the context of microorganisms. And so here's a, you know, starting with a new page, a blank page what what might we, we think about and I, my background is in physics but I have lots of friends now in biology and some of them by my history turn out to be study bacteria and bacteria do lots of wonderful things they swim they find food they process the food they reproduce. But what I've learned from my colleagues is that these bacteria are always working relatively close to some biophysical limit. They're a very limited proteome they can't just make more proteins to do something, because they're working as hard as they can just to make the proteins they've got. And as a result, they're always trade offs. The bacteria is good at one thing it's making a lot of proteins to, you know, consume one nutrient, then it doesn't have the resources to make the proteins necessary to consume another resource. And so the idea that my group and I wanted to pursue was the idea that perhaps trade offs could play some role in increasing diversity. And so what I'm going to do now is I'm going to show you this very simple model kind of I have to admit absurdly simple model for what happens in a real ecology of microbes. But introducing one new thing to these old resource competition models, which is the idea of trade offs. And so here's just a, you know, our view of what could be happening with competition among different species over number of resources. Think of a certain number of resources say P resources and, and for those of you who know sugars you know this could be glucose. This could be galactose fructose there's a few different sugars, and these would all be necessary sources of carbon organisms have to take in some kind of carbon in order to be able to grow and survive. And this species is going to be described entirely by a little vector which says, well, how good is it at eating glucose and how good is it at eating galactose and how good is it at eating fructose. And that's just going to define the species. And the other thing we're going to add it to this is that somehow there's a trade off that if you're good at eating glucose you can't be that good at eating these other sugars and vice versa. So imagine in your mind some some in situation like this, this would be a little colony of different species living together, or it could be in an laboratory, a chemostat, which is a vessel where nutrients are dripped in and cells and whatever's left is out and establishing some kind of steady state environment of a mixed community. And so imagine in this community for the moment that they're just two different species present these are these little brown ovals and these little gold ovals. And they're competing for the resources and the experimentalist is dripping in these resources, this is maybe glucose that's coming in, this is galactose this is fructose coming in in slightly different rates. And I should say, cells need more than just carbon but we're going to provide plenty of nitrogen sources and phosphorus sources and so on. So they're going to be limited by the supply of these sugars that are coming in. And now I'm going to consider well, you know what's different between these species and what's going to be different is their, their focus, which nutrients they decide to devote their internal resources to consuming. And so this species here this brown one, it's a specialist, it's devoted all its resources to importing this particular sugar the green one. And they're really good at that if they're that sugar is around it will take it up very very quickly. But as a result of the trade off, it's no good at importing the others and the and the trade off could be implemented just as a simple sum here that maybe there's a cost and in the moment. Soon we'll ignore this cost set this to one without loss of generality but principle it could cost you different amounts to import the different sugars, but all of these are going to add up to a constant. There's a certain amount of resources of proteins that the cells make that it can devote to taking up sugar. And if this guy is a specialist doing only one thing it can't do the other, but there's other possible strategy so for example this species is more of a generalist. It devotes some of its resources to taking up the green sugar, but some amount to importing red sugar some amount of putting the blue sugar. So it kind of hedges its bets. And now if we take this as a sum and for the moment set these costs to one, because they add up to a constant. There's a simple manifold here if there were three different resources and therefore three different rates three different values of alpha, they're all going to live on this little triangle where the, the sum of these values adds up to a constant. And so the specialist would be here at this, this vertex. The generalist would be closer to the middle. And we can actually also plot the supply in the same space just by the ratio of which sugars are being supplied. So both the strategies and the supply can be plotted onto this little manifold. So this is a good moment this is this is the model that we're going to study it doesn't get more complicated than this. And this would be a good time. If there are any questions and be happy to answer them. So I don't see any question in the in the chat. Great, there will be. There will be other opportunities I will regularly pause for questions and this is this is fine point I will, I will can, I will continue on to talk about the equations now associated with the dynamics of that model. So, there is a question. Okay, great questions are good. Why, why can you assume w equal to one. So that was the cost of a resource. Yeah. It turns out that as long as the cost of the resource is the same to all of the different species, as long as you know, the cost of making importers for glucose is the same whether you're an E. Or an E coli or a bacillus subtilis. Then I will just state without proof at the moment that without loss of generality, we can rescale the effectively the alphas by the W's. And as a result we can replot things in the same way with a with a slightly different interpretation of what alpha means. So, but so there will be there's a paper that you can read and this is in the supplementary material. But again without loss of generality we can rescale things to absorb those W's. And I think the question also referred to why does not depend on the resource. The W's can certainly depend on the on the which resource so there will be a different w for each resource that will effectively rescale the alpha for that resource but again as long as it's the same W independent of which bacteria you are, then it can be absorbed. Good. Let me let me push on. So, let's consider now in this in this chemostat setting on this continuous culture setting. What's the dynamics for the nutrients. And ultimately we will be be setting this, for example, to zero in steady state but the. How do I the level of a sugar change while there's a supply is an experimentalist some amount of sugar is being dripped into the to the chemostat. And then it there's a consumption term so all of the species that are present they're labeled by Sigma have some population. And that population has some rate per individual of taking up that resource proportional to some function of the concentration of the resource and here we've used something very simple a saturating term that's linear at small levels and saturates. It turns out for everything I'm going to say it doesn't matter exactly what form we take here, as long as it's monotonic as long as the more resource, there is the faster you take it up. And in principle, in a chemostat things are being dripped out as well to keep the same volume and there could be some loss of nutrients that that are not consumed to get dripped out proportional to this steady state concentration. Okay, so that's the dynamics of the nutrient. What about the growth of a species. Well, each individual is just simply taking up resources and the rate at which it's going to grow and divide is based on how fast it can take up resources. So, for each each species there's a sum over the different resources. Here, again, there's a parameter V which is the value of a resource it might be that you grow a little bit faster per per glucose molecule and per fructose molecule, but in the same way as the fructose could be absorbed into a redefinition of the Alpha so can the V so these will go away in a moment. And this is just telling you how fast you're taking up the resources so you grow at a rate proportional to the total uptake of carbon from for each each each species. And then the population dynamics is essentially trivial. The overall population of species Sigma grows is proportional to how many there were. And then the difference between how the growth rate and the death rate here in a chemist at the cells don't really die they just get lost by dilution that is the as nutrients are being dripped in medium is being dripped out that contains cells, and so cells are being lost that way. And now we can implement these these simplifications I mentioned, at least for the purpose of showing you the analytics we can do this with or without the animal that these assumptions on numerically. Indeed in a chemist that that's limited for carbon essentially the cells eat all the carbon and so nothing is left to drip out so we can, we can eliminate this loss term. So the separation of time scales. The cells take up nutrients and process them very fast compared to their overall growth rate so effectively the nutrient concentration is always at steady state as a fast variable slave to the slower variable of the population sizes so we can separate time scales. And then without loss of generality, we can set these, these parameters all to one, the, the, the cost of importing particular sugar, the half saturation value, and the, and the benefit, and that leaves us a close out of closed equations for the nutrients which is super simple conceptually the idea is that for the population, the rate of growth of a population is proportional to how many there are, and the difference between the growth rate and the dilution rate, and the growth rate is simply set by how fast you're importing a given nutrient by how much is present, and how much is present is determined by the supply divided by the rate at which it's being taken up by all of the species present. And so this is exactly what would have been written down in had been written down in resource competition models for many, for many years. And all we're going to add to that is that there's some, some role that there's a trade off that each species has a limit on its ability to import these different nutrients, so that the sum of the alphas are always a constant. So now we can look at the behavior of these populations, and I'm going to show you some examples of that. So here, back on this little manifold where remember this is the, in this space I can plot the strategies, the specialists that eat only one sugar live at the corner, something that was a perfect generalist that ate everything would be in the middle, and I'm going to give you two different species, trying to live together or being thrown together. And in this case, the supply is actually going to be balanced. It's, it's one third one third one third of three different nutrients. And if you put these together and give them that supply. So what happens is that one of them wins in fact the blue one which is itself, almost a generalist, almost in the middle here will take over drive to extinction, the gold one and of course this is not very diverse, it's only one species survives. So what happens. Now if I add another one that's maybe also a little closer to the middle but not quite as close. And indeed, the blue one, which is nearer to the center here is still dominating in fact it's driving the other ones to extinction, the rate at which the close to extinction is a little different. So say well you know this isn't looking very diverse I can add this red one over here. Again, this changes the rates and all of this of course is very consistent with this competitive exclusion principle I have only have one species surviving, even though I had three species in principle I could have had more. So, so you know I can move this red one around here and ask what happens. And then something happens. Right. So now I've got four species and three resources, but actually all four species are surviving their coexisting. So they're supposed to violate this competitive exclusion principle they're not supposed to be more species surviving than resources and so I say, Well, you know maybe I've done something tricky, maybe I have balanced these in some very mathematically fine tuned way that gets around this principle so you say, You know you don't trust me. So you shouldn't. And I can add something to maybe unbalance this I put a purple one right down in this corner something like a specialist and see what happens. But, okay it didn't go away. So adding some perturbation didn't didn't make the diversity go away. I've now got five species coexisting on three resources. I can be a little bold and I can throw in a bunch of different strategies and ask, Okay, what happens to them. And now you see that they're all surviving. And in fact this scale is such that they're all surviving at relatively equal similar levels. So there. So there's not that one of them is very dominant over the others even though this blue one was was driving the others to extinction when there were a lot of them. Now when there are a lot of them present. The blue one is nothing special. It's just one of the one of the crowd. And so I can go a little bit crazy and I can say come on now. You know, how about that gram of soil that has 10,000 different species present. Maybe I throw in hundreds of species into this. And what happens is that they all coexist. So, at the moment this is looks a little bit like a magic trick that I've done something. And now as a as a good magician I should just move on to another trick. Leave you wondering, but okay but this is this is this isn't this is physics or science at least it's not, it's not magic and, and in fact, you know this is the moment you should, you should enjoy, because as soon as you hear how the trick is done by the magician, it no longer seems that interesting seems relatively trivial. And I'm afraid that will be true here as well. But again where we're not magicians we have to we have to explain our tricks. And so, so to explain this one I want to consider something a little bit different in the same little manifold here. So what happens if I give you three species I'm just going to give you three species and they're going to stay the same three these three stab strategies. And now what I'm going to do is I'm going to vary the supply. So up here I'm giving you just one sugar, and ask, Okay, what what what happens if I just give you this one sugar and this blue color means that only this species survives. I'll give you anything in this, this quadrant here or here, just one species survives. And then in this these regions, two of the species survives so the blue and the green would survive if the supply is here, and so on. But then there's this gray region in the middle, and the gray region in the middle, if I put the supply in there all three of them survive. And, and now I can look over here and say well alright so what were the nutrient concentrations at steady state under those conditions and all the around the outside. There's some ratio of the different nutrients, different from what I supplied because it's the supply and then there's the rate at which they're being consumed that give the steady state. So what special happens inside this triangle, it's all the same gray color. And what this means is that even though in this region, different points here do correspond to different ratios of the supply. Everywhere in here at steady state, the nutrients at steady state are equal. So it's, it's one third one third one third even though you know I didn't supply them in equal ratios. So somehow what's happened is that these species that are present have have leveled the playing field, I've supplied things with you know little more glucose than fructose a little more than galactose. But the species have consumed them at a rate that makes the concentrations at steady state equal. And the point is that if the steady state concentrations are equal, then it doesn't matter how you devote your resources to taking them up. You could eat all of one or equal amounts of all three or any kind of ratio, but because this the nutrients are an equal abundance, your strategy no longer matters to your growth rate. So, by leveling the playing field, they've made it possible for any species that's present to co to coexist and grow at the same rate. And so, what you end up with is a very simple principle, at least in this simple ecosystem that we've developed. And that is that if the supply lies within the convex hull if I stretched a rubber band around this. If the supply lies within the convex hull of the species that are present, then the steady state of this system has an equal supply of each nutrient, and everything can coexist. But if the supply is outside of the convex hull, then these nutrients are no longer equal, and we end up with only a small number of co existing species satisfying competitive exclusion. For example, in this case, I can have lots of species but if the supply is outside the convex hull, then only one of them in this case survives it's actually the one that's closest to the supply. But all I have to do is move, add one more species here's this this gold one add one more such that now the supply is inside the convex hull here. And now the nutrients at steady state, all reach equality, the playing field is leveled, and everybody gets to coexist. And so that's, that's, you know, that's what happens the math behind that is very simple. And it's just a matter of having reached this, whether the system can self organize to this fixed point, where all the nutrients are equal and in fact it's, you know, I'm not sure you can see this by zoom but the, the issue of the convex hull is really whether the total amount of processing or the total amount of say, so transporters that are available can match the supply so the supply is some vector think about it as some vector in this direction. And all of the species present, they have their own vectors of their, their strategies. And the question is, can I add up a positive values of all of these vectors to equal that one. And the answer is, well, yes, if that vector lies within the convex hull of these other vector points, then I can add up different combinations of these to equal this one. And that means that the supply divided by the consumption can all be equal and all the steady states can be equal. But if this supply is not inside the convex how no positive combinations of these vectors will equal that one. And I can't reach this fixed point of equal steady state nutrients. So that's the, that's the geometric intuition for this result. Okay. So, what I want to point out though is that this has some potential connection to ecology because ecology has these things called Keystone species where you have some food web and various things eat other various things. And there are some species that if you disrupt them take them out. Then the whole food web falls apart. And, and in fact, if you think about it this system, this gold species plays the role of a Keystone species if it's present there's great diversity, you know, healthy food web, but you take out that one species, and the it really crashes and you end up with just one species left that survives. And, and so unlike the case in ecological situations where sometimes this is at the base of a, of a food web where well of course if I kill all the grass. There are no rabbits and if there are no rabbits there can't be any foxes. But here, the Keystone species is essentially a set by the, the set of other species that are present the gold one is a Keystone because it so happens, where the other species are present there was nothing about it that said oh this is the base of the food web it's sort of a dynamical emergence of a Keystone species. So, another opportunity for people to ask questions. I'm happy to, to take a moment and answer anything that has come up. Yes, there are a few people that raise their hand but I don't know how to give them the word so maximum if you could check. Meanwhile, I can read some questions from the chat. So, one question from Simon Levin is, are luxury, luxury uptake and storage a possibility. Are luxury uptake and storage of nutrients a possibility. So, in, in this, in this model and I think very realistically in a chemostat that for example is limited for a particular nutrient carbon in this case but it could be nitrogen or phosphorus. You know no advantage for the organisms to store any of these nutrients in fact that's a very, it's a big mistake to store those because then they, they actually lose their, their exponential growth. At the same time, you don't want to delay any kind of exponential growth, you want to use those resources as quickly as possible to keep your population high so that then you can continue to take up nutrients at the highest rate. So as long as you're in a, in a limited for a particular nutrient. The best strategy is to use that nutrient as quickly as you can to increase your own population. That's the way that things don't store thing I mean, for example, if the if you're limited for carbon and it may make sense to store nitrogen because who knows in the real world. You might then run out of nitrogen later, but for the limiting nutrient it certainly should be used immediately for growth. This is a question from Chen Liao, who is asking this model as well as the classical resource consumer model uses the assumption of substitutable resources. How realistic is this assumption and how much it will affect the conclusions. There are many resources that are definitely substitutable. So for example, giving carbon sources, you know, those are, those are very substitutable in the sense that what the cells get from those are, you know, carbon and energy, and those are just sort of universal quantities. But of course, you know, whether something is substitutable or not, as we can see in this model does depend on what your, what your repertoire is. So whereas in principle, all of these sugars are substitutable for a specialist that only can consume glucose, the nutrients are not substitutable. I think the answer is, yes, there are lots of nutrients that are sort of substitutable in sense of what they're providing elements energy. But if you think about it from the point of view of the organism the these these nutrients may not be substitutable because of its particular repertoire of uptake and consumption. And then there are a few questions from different people that sort of summarize in the same question. And the question is, basically what happens if the species are not equally able to convert resources into growth, or if there are the trade off is inexact. So if there is variability among species in their ability to convert resources to biomass. So, so within the simple model, if some species has, you know, an advantage it has for, for example, you know, has discovered something and it can do do something better than anyone else. Then it will eventually drive the others to extinction. But the rate for that can be quite slow. And so if there are other sort of sources of fluctuation in the system. So you can end up with high diversity, despite these these imbalances, and I will be addressing that in the next few slides. So let me come back. So I think I'll push ahead now. And, and come back with more questions later. So what I've argued is that essentially the system tunes itself to a point where competition is is neutral. The idea that the supply was different but because of the species present. They were able to reach us, reach a steady state nutrient concentration where the nutrients were equally abundant. And therefore there was a kind of kind of neutral competition among the species that are present. And that hearkens to one very important train of thought or our class of ideas about diversity in nature, which is that that maybe these competitions we see are really neutral. So here's just a schematic of what, you know, a toy version of the neutral theory of evolution. The idea is that if I'm looking at trees on an island, there are various different trees, but basically all trees are the same. And so the only thing that's setting the different abundances of the different trees are stochastic events. So a tree will die. And then it will be replaced at a rate that's proportional to the presence of trees because they're all, each one is providing seeds and one of them has a chance will land here. And so there will be these these stochastic birth death processes. Now this is all there is eventually, you know, species go to extinction, but diversity is then maintained for example by some slow rate of immigration that there's a tree being replaced by one of these existing trees, a tree from another island provides a seed. And, and, and this kind of model seems sort of absurdly simple and yet it actually fits data extremely well in many contexts and I just showed a few examples from a from a review, and curves of the of rank abundance sort of the first ranked species has this abundance and the second one this one and so on. And these curves actually they're no I didn't show the fits but these can be fit extremely well by this neutral theory. And this is, you know, very generic, whether this is trees or birds or bats. And so there's been a, you know, this is a puzzle why this neutral theory works so well. But you can ask, well, you know, in our model, which is very clearly not intrinsically a neutral model that difference. She sees do different things with the nutrients. You know, how, how well does our outcomes match neutral theory. And the answer is that they match neutral theory ridiculously well so here are some results for these kind of rank abundance curves for the different different populations of species in our model. And it's even hard to remember which one is the neutral model green is the neutral model and red is what comes out of the, the very simple resource competition model that I showed you. And so these are essentially indistinguishable. And so, I kind of like the point here that you have a kind of reconciliation in our model world at least between different two different schools of thought there's the, there's a neutral theory that says, you know, everything is equal that doesn't matter all bacteria equal all trees are equal. That's why you get these distributions. And then there's another school of thought that says, look, I come and I do a measurement of, and I, this tree clearly has differences it likes this amount of son it likes these nutrients. This bacteria, you know, takes up this sugar but not that sugar. And, and they fight with each other over you telling each other that they're wrong but of course in our world they're both right. Right, because it's true that if you measure the individual species in our model they're very different they they this one has a niche it likes glucose and not fructose. But collectively, when you put together these different very distinct strategies. They dynamically create an emergent and neutral environment where they all can live. And so it is in this in this world at least the niche theorists and the neutral theorists can get along and everybody is right. So, so then we can come back to some of these, these questions of like okay well this was a very, very simple model you made a number of assumptions here. What what which of these assumptions are important in which are not. And so, let me push on because we had questions just recently. Let's talk about well what's how robust is this coexistence and, and first you could consider robustness against population disturbances, but I hope that's clear that these are these are these are fixed points in the sense that diversity is stable that I could start with different amounts of the species or perturb them, and I will get a diverse solution, although you can see from the from the idea that all that has to happen is that the supply vector has to be equal to the sum of these different species vectors that there are many different solutions. If I have more species present than dimensions, they're different solutions but they will all in principle allow for coexistence and be diverse. So more interesting maybe is this question of, of whether it's robust against fluctuations in nutrient availability up to now I've said, we're doing an experiment we drip in these nutrients at a certain rate, but the the real world of course, you know nutrients come and go or they come in at different ratios and and how does that affect the behavior. And, and, and first of course, the convex hull is is the condition for diversity so if the nutrient stays within the convex hull you always get coexistence, but what about a nutrient that kind of fluctuates in and out of the convex hull. And we find something very simple there which is that it's really only the time average of the nutrient that matters so if I have a, an average nutrient that lives outside the convex hull, even though sometimes it's inside sometimes it's outside. If on average it's outside this system will will lose diversity, whereas in my best sort of similar arguments before, if the average of the nutrient lives within the convex hull, then all the species will coexist, but you can see they're fluctuating around now. And this is because this nutrient supply is is stochastically moving in and out of this convex hull and sometimes there'll be some loss of diversity but it will recover when the nutrient comes back inside so so that that is relatively robust against these variations in nutrient, but then there's a there is a critical question about what about if this trade off is not exactly satisfied. And that turns out to be an important perturbation. So if you if you study simply a deterministic version of the model where there's no noise. Nothing else happening, and I let one species be have a bigger budget than the others, then it will try to tend to drive the others to extinction. But I would argue that the real world we don't sit around long enough to wait for this to happen that there are always other fluctuations going on. And so, for example, if you look at a kind of version of this neutral model where we now introduce stochastic births and deaths, and some stochastic rate of immigration, then deviating from these exact trade offs by this this is a measure of how much variation there is in the in the overall budgets that the different species have. If these trade if these trade offs are approximately satisfied so that basically bacteria are more or less have the same protein budget that they're available to to a devote to up taking these carbon sources, then you will end up with results that are indistinguishable from the case of exact trade offs you'll get these neutral curves, and only when you start to have very large variations so that one species will be 10 or 100% larger budgets than others do you start to lose this diversity. And I would argue that the bacteria really are working near biophysical limits they're not in a situation where they can simply devote 10% more of their of their resources to up taking particular nutrients, because then they lose the ability to perform other functions like replicating their DNA or dividing and so on. So, so I wanted to just move on briefly and talk about some other generalizations of the model that have been pursued we've been pursuing recently that also speak, I would say to this issue of, of inexactness of trade offs. So, so, before doing that I wanted to, you know, draw some conclusions from, from what I've said so far, I guess, I'm right on time. So, what are the features of this model that allowed for coexistence and, and they're essentially to one is that the organisms are taking part in shaping their environment. And this was not done in the context of an environment where the, where the species take up the resources and therefore control how much of the resource is present. If I simply provided a fixed supply to these species, and kept that, you know, more glucose than fructose and more fructose and galactose. Then it's very simple that the species that ate the most abundant nutrient would simply do the best. And what happens is the species starts that way, for example, the species that eats the glucose starts doing the best but that increases the number of the organisms that eat the glucose. They consume the glucose faster and that lowers the level of the glucose so it's critical that these species are playing a role in shaping their own environment. And of course, as I showed in the last slide it's also the species all have to be subject to the same trade offs if there's something to be sees that somehow miraculously is better at everything, then then that would drive the others to extinction. I've highlighted a couple of, you know, at least entertaining similarities with natural ecosystems that this simple model does give rise to dynamically emerging keystone species. And that because the feedback onto the environment basically levels the playing field we end up with population statistics that closely replicate neutral theory. So what else you know what's missing well there's a ton of things in the real world that are missing and we focus on a couple of them. So going beyond the chemostat, one of the things that's certainly essential to understand diversity in the real world is to think about spatial structure. And so one thing we've done is to look at a version of the model, where instead of everything being well mixed, we actually imagine something like bacteria growing and like biofilms on a surface where they have little territories. They're competing with each other at the boundaries of these territories. And so here's, here's what the situation looks like the same species with strategies the same uptake of nutrients the same overall nutrient of supply, you know, raining down for on this, these species like mana from, from heaven. But here, the different species occupy different territories and this is in in one dimension. So, and actually with periodic boundary conditions so imagine a ring, and I have you know, species one lives here around these these degrees and species two lives here and so on. And what they do is they take up the nutrients that come down with the nutrients diffuse around and so nutrient that rains down for example, on on this population. Some of it will be consumed but if it's a mismatch for example if this species only consumes you know nutrient one, then it will the nutrient two and three that rained down here will leak out to its neighbors on the on the on the opposite sides. And as a result, there's some kind of competition and it also matters who your neighbor is. Maybe you have a neighbor that you get along with well you eat one sugar they eat another, you leak out the others and support each other. But then there's the possibility that of neighbors who you know compete for the same resource. And so what we find there is something kind of in between. What what we saw before with and without the supply in the convex hull that is for particular sped of species at a particular supply, some of the species actually end up going extinct, while the others can coexist. You know well beyond competitive exclusion so there's still three nutrients and many species coexisting. It's no longer a simple case that all the nutrients become equally abundant in fact the nutrients very, very dramatically in space. And yet, you know, all of these species do manage at least many of them coexist and what we were able to understand here is that there are some species that are just very, very bad neighbors. And these are ones whose strategy lies between the supply and the central point here. They're essentially very greedy they managed to take up the, the largest amount of the supplied resources and leak as little as possible to their neighbors. And so basically if one of these species is your neighbor, you're not getting much from them, even though you're leaking nutrients to them, and they will tend to drive you to extinction. So this is a little bit like the idea in ecology of if there's some species that is able to grow at a lower concentration of nutrient than any others, it can drive them to extinction. It's related to that idea and we call these oligotrophs because they managed to sort of eat everything in just the right ratio to minimize that they're sharing with their neighbors, they're not good sharers. And I should point out that this system is now robust to variations in the, in the enzyme or the, the resource budget so you can add small amounts of advantage or disadvantage to these different species, and you largely get the same outcome that you will get coexistence beyond competitive exclusion. So as soon as you go to a spatial case, you're no longer required to have you sort of be in this mathematically precise tradeoffs in order to get coexistence beyond competitive exclusion. So that's one, one direction that I think is interesting and another one is a question of well what about seasonality. I mean that's very much a part of our, of our world that, you know, winter is here, we're actually supposed to get a foot of snow it's very different environment than in summer, and that's certainly true for microbes they are very much subject to changing environmental conditions. And so, so here we can add the idea of seasonal ecosystems, maybe in a very, I hope, experimentally friendly way of just serial dilution of these, instead of growing cells in a chemostat supply nutrients, let the cells grow, take some of those cells out and put them in a new in a new batch of nutrients and keep repeating that and so some kind of simple procedure of, of growth and dilution and what happens do I reach some kind of steady state ratios of the different species that are present, even though, in fact that each in each batch they will grow and then and then and then reach some steady state. Do I find that the ratios say at the end of the batches approach some, some steady state, and the answer is yes. I'm here, for example, with a, you know, in just two nutrients, I find some kind of steady state solution, but suddenly something does happen that's different in the chemostat, it didn't matter the overall supply if I just supplied 10 times as much nutrients. I would get 10 times as many cells with no, no other differences but here, suddenly it matters what the amount of the supply is so I have coexistence in the serial dilution but if I actually give them more food. Each batch that instead of getting coexistence, I get loss of diversity and one species takes over, and then I could add more nutrients and the diversity comes back. It just seems, again, rather magical and sort of does have relevance to questions of well what happens when you know you supply extra nutrients in the Arctic tundra with with global warming if there's that's going on are we going to get more diversity or less diversity, even in this simple model it turns out to be quite a quite a subtle behavior, but we can actually track this down to something we call an early bird effect where if species can grow quickly early on. They increase their population and then can consume more of the other nutrients, even the ones that they're not specialists on. And so really, we can understand a lot of this in the context of the question of whether there's some species that does well early on in in the season. And I encourage you to check that out. So really I've come to the end here. Just want to summarize. Again, I think, you know, if there's a take home message here, thinking about diversity in microbes and maybe in other contexts. It's that organisms play a huge role in shaping their environment and we have to take that into account. And that potentially we should also be thinking of trade offs that when we look at the species they're very different, but maybe there's no such thing as sort of a free lunch. If you're going to be good at eating one thing you're not going to be able to devote those resources somewhere else. We saw some similarities with natural ecosystems terms of keystones and neutral theory. And we're working hard to make our, our models more realistic. And there are many things that we still have to do in those in those in that regard. But for right now I just want to say thank you. Thank our sources of funding and and here are some references for those who might be interested. Thank you all for the very nice talk that are actually many questions in the Q&A in the chat. And I ask people that wants to ask the question that they can use the raisin tool, and I give them the possibility to talk. So there is one question from Simon Levy. Simon I think you can talk. Yeah, can you hear me now. Hi Simon. How are you. Thank you very much. Thank you in Italy that we work on the same campus. So, so Ned it's, it's not a surprise that if species can shape their environments this introduces, in effect, other sorts of limiting factors. For example, species could produce antibiotics. And species can shade other trees and they, they, David Pimentel wrote some paper some years ago I don't know whether I've mentioned those two, in which he suggested that natural selection would favor rare species. In that they most of the date in the following sense, most of their interactions will be with common species and therefore they will evolve to become better competitors against the indigenous species. Whereas those species that are, that are already at high abundance will mainly be selected for intraspecific competition, and then ultimately this may drive species towards having similar payoff functions and create a situation much like you described, I think. I'm wondering if there's something like that that's going on here and related to that also is the question as to whether the equilibria you get our stable equilibria since everybody's getting the same payoff, or if it all depends on initial conditions in some way, as to what the relative abundances of types are in equilibrium. Is it a neutral equilibrium. So let me, let me answer the second question first because I can actually speak, I think with some authority to that because it's a question about the model. And the idea there is that, you know, there's not a single solution that achieves this equal playing field. Very much the same statement that the level playing field is a statement that the abundance of the different species that are present have to be such that when I add up their little consumption vectors that they equal this factor. And it's clear that, you know, this vector lives in three dimensions. If I have exactly three of them there's one such solution and that would then be a very stable fixed point, right, of these three, and I could perturb those populations, and it would always come back to the same populations. And as soon as I introduce a fourth one. Now there's a degeneracy. Right there I have four vectors trying to add up with three constraints. And now there's a there's a line of possible solutions. And so if I perturb, I'll come somewhere back on that line, but it could be could be anywhere. And the more species I have the higher the degenerate manifold. So that's consistent. That's consistent with my question which is, if you look at the careful statement of the competitive exclusion principle, it says there is no stable equilibrium, meaning an asymptotically stable equilibrium involved involving more than three species so that doesn't sound like a violation but that has been observed. But there are no there are statements in competitive exclusion that you cannot. Well, okay, I mean these are, these are stable. Okay, I'm going to take that offline with you Simon. Okay. And another question of whether, you know, whether this kind of trade offs could be driven by evolution in in. Well, first off, I would say, yeah, absolutely. I mean that evolution is driving these bacteria to be better and better at what they do to be more and more efficient users of their proteomes. And that's exactly why there's no species that can simply be better at everything else because evolution, if that was true it would have evolved taken over but then, you know, branched into more species everything would be get get better up to some kind of a physical limit, whether that could also be, you know, argued in terms of like things going beyond resource competition like antibiotic production predation all of these other things I think that's a really interesting question. We haven't tried to build that into the model, but I think that's a great direction to go in to add some of these other species species interactions. And there's framework again that there's some kind of limited budget available for for killing your neighbor as well as eating his lunch. Thank you. Great. There is a question from aditia. Thank you. So I was wondering. So I guess I was wondering about so do you know about the statistical properties of coexistence if you increase the number of resources or the number of species for so I guess precisely what my question is, is how does the fraction of the simplex occupied by the convex hull scale with the dimension of the space and or with the number of species. So it's actually pretty simple that as soon as you get to the point of the state of coexistence in other words as soon as the supply is within the convex hull. Then you, you, you, well, then if they're the large number of species, you really reach true neutrality. But that is, you know, the, the, in some sense is answering Simon's question as well, that if the degenerate manifold is big enough, so that there's enough degeneracy, then you really the, the populations really don't know about the constraints very well and they can, you know, if you introduce any kind of stochasticity, they really diffuse randomly so it looks just like a neutral model. But as you get fewer and fewer species compared to the number of resources, then they start to know about the constraints so again if there's three nutrients and three species, then there's a single fixed point solution where you know that there's one solution where these three vectors add up to some of these three vectors add up to this one. And then it's not at all neutral then then the system is kind of homeostatic I changed the value, the population of one and the system, you know, returns to that fixed point. So there's some, well, maybe an interesting question how that crosses over from this behavior of there being only a single solution, definitely not neutral in the sense that there's no variation of these populations. And as I throw in more and more species, the, the dynamics then the stochastic dynamics comes closer and closer to the neutral model. That's about all I can, I can say in response to that we haven't looked at that in a more, you know, rigorous way. Thank you. Great. There are a few other questions in the chat. So there is a question from Rodrigo, who is asking whether you think that the linear shape of the trade off plays an important role. Yeah. I mean, I think that that is an interesting question in sort of mathematically speaking, if, if for example, you know, it caught the more you had of a certain transporter the more expensive that got. You know, that that would certainly drive you in one way toward, you know, you wouldn't be a special, there would be no specialists, right, because that would be a foolish strategy you would, you would have very little ability to import, because there was a non linearly increasing cost. Similarly, if there was a non linearly decreasing cost, I would expect everything to become specialists, you would you'd focus on only importing one one sugar because it would be cheaper. That would certainly I would say break the kind of diversity, a general diversity we see, but I also have to say, there's not a particular reason to believe that there are strong non linearities here because the real, the real dominant costs these organisms is simply compressing those proteins. And that really is primarily a, you know, you know, what each protein costs you a certain amount to make. And so, you know, making more of them is simply linearly increasing making two different types of proteins is just the sum of the costs of the basic individuals. The caveat is, you do have to carry around the genes and I think this is an interesting question like the relatively small cost to carry a gene or not carry a gene. But that could be important. And why perhaps we see more cases where there are real specialists out there that simply don't import or process any type of sugar because then they have, they have the saving of, well they don't need the genes for that and they replicate their DNA is a little lower cost. But it's an interesting question that way. Great. So, I think we have time for one more question. So, Zang D is asking, is it possible to do any experiment to verify the model. Yeah. So, it wouldn't, it's not something that I would immediately say, you know, you should go into the lab and start testing because the real bacteria are much more complicated than our, than our model organisms, right. They, for example, they regulate the levels of expression, rather than simply making, you know, certain ratios of these transporters, they can respond to the nutrients that are in their environment and that's certainly one of the directions. We're going in now is extending these models to include regulation. That said, in this, in the, in this elive paper I think it's at the supplement we actually look at some existing experiments on on E. Coli, and under a range of circumstances, they actually seem to behave very close to the, to what we are proposing for organisms that they, they really do seem to have a fixed ratio of where they're importing different sugars. And so it's, it's quite possible to imagine an experiment, you know, based just on E. Coli, where by genetically engineering them you could have a few different strains that you know, sugar preferences and look at, for example, whether with it by putting supply either that would be live within or without the convex hull, whether you go from this diverse solution to this loss of diversity. So I would imagine not trying to put in hundreds of different species, but rather putting in a few carefully engineered versions of a well understood species like E. Coli. I think these experiments are actually are actually feasible and and would be quite interesting. And so thank you for that question. Great. So there are other questions, but I think we are out of time. So, I'll thank again all the participants for all the questions and interaction and networking for giving this very nice presentation. I also remind all the diploma students that at 530, so in about 15 minutes, there is a meeting in a separate Zoom meeting room with the speaker. So thanks again everyone for being with us and thanks for the presentation. Thank you. Bye all.