 Usually people ask, can you hear me? But now I should ask whether I am too loud in the back. Well, OK, I'm not too loud. Can you hear me? Good. All right, so I will tell you today about some very recent work. We just submitted a preprint in July and also some of the unpublished work. And since I'm almost certainly run out of time, let me introduce the main heroes of today's paper upfront. This work was done in collaboration with my current PhD student, Zihang Wang at Urbana-Champaign. And my former student, Dakshit Goyal, who was at MIT for a while, and now just moved to his independent faculty position in Bangalore. And I am looking forward to visiting him there. And before I jump to the topic of my presentation, let me briefly summarize. We had an interesting discussion about ecology and physiology yesterday night. And I will use the sort of, Terry was driving this discussion in a lot of ways. And I will use his own quote from his own talk, which I heard about a month ago, where he was arguing that, well, we all know the success of the modern physics. And it started with astronomy. And astronomy started with astrology, observation of stars. And then there were Tiho Brahe, who did very careful calculations of the positions of the planets in the sky. And then Kepler sort of summarized it into empirical laws. And then Newton made the laws somewhat more mechanistic and so on. And we all kind of live with consequences of this. And the point he made is that this transition from astrology to astronomy should be mirrored in transition of biology to something like bioenemy. Bionomy is a biology transformed from just a lot of bunch of observations to something empirical laws. And then finally, some fundamental mechanistic laws. So I just want to extend this joke to the topic of this conference. So we all know that we gathered here to discuss ecology. And of course, ecology becomes economy, right? Which is a good thing. We also talk about the economy of the cells, so it makes sense. But physiology becomes physiognomy, which is a science of deducing the person's personal traits from the facial characteristics. So I'll leave it at that. Now, and let me again, as a kind of now jumping to the topic of my talk, I'll share my frustration as a sort of modeler and a quantitative person with a physics background who started working on microbial communities about eight years ago, give or take. And I've been kind of very frustrated by the very limited success of the modeling to make something predictive. Let me be more precise what I mean by this. In my kind of mind, a perfect application of modeling would be, well, experimentally measured some parameters of microbes. We modeled what will happen when we mix them together. We modeled which microbes are more competitive than others. And we predicted the success or failure of the assembly of microbial communities either before the experiment was done or immediately after the experiment was done. Maybe we could advise the experimentalist about how to best assemble their communities, what kind of nutrient ratios, what kind of parameters you choose from this for their system. That would be kind of an engineering success of the microbial community modeling as an engineering science. But in my experience, I haven't seen a single successful application of this sort of dream story. So there has not been a priori predicted prediction of an outcome of a particular experiment. Again, give or take, maybe there were a few very handful of successful examples, but it definitely didn't reach the mainstream. And you may ask why, because in principle we have known the basic equations, basic gross laws if you want of the microbes in nutrient-dominated environments for a while since 1940s and they are given by the mono-equation. And for better or worse, most of the modeling effort has been focused on chemostats on describing what happens in a steady state of a chemostat. And when you are considering a chemostat, the whole game is about the consideration of a resource R. So the microbes compete with each other which of them can drive the consideration of the resource lower. And the one who can drive it the lowest wins. And if there are multiple resources then there could be multiple winners. So the essential kind of feature of a chemostat is that mostly in a chemostat you are operating with R much smaller than the mono-constant K in the denominator of this formula which means that the equation becomes kind of linear in R which is another good thing. But there is a problem. There is an experimental problem. And by the way, just also to put what I'm talking about in the context of Martina's talk yesterday she talked about oligotrophs versus copiotrophs. Oligotrophs are the microbes adapted to live at a very low nutrient concentrations and grow slowly. Whereas copiotrophs are microbes adapted to live at high nutrient concentrations, grow fast and then of course collapse because you cannot grow fast forever. So chemostats are a good system to study oligotrophs but it's not a very good system to study copiotrophs. So one of the reasons why even you may say, okay, why don't we study oligotrophs and make some successful predictions about how to assemble them together? The problem is in this mono-constant K. The K first of all, it's notoriously hard to measure experimentally. If you approach your experiment list friend that ask him or her to measure K, they will immediately shift the conversation to weather. They just don't want to deal with it. And they have a point because it's hard and sometimes very impossible. It also varies a lot between strains of the same species. Somewhat depends on a gross history of the strain. Here is a table which I got from an influential and well-sighted review by Tom Aglian collaborators back in 1998, equal eye grown on glucose. What could be better studied than equal eye grown for glucose? And I want you to pay attention to this column which is this K constant, the mono-constant. Depending on the cultivation method and on the strain and on experimental technique, it varies by three orders of magnitude between 33 and 100,000. Actually almost four orders of magnitude. So even if you kind of say those are outliers still, we are nowhere within this factor two which I consider to be a gold standard for biology. Biologists can typically measure everything within factor two if it's more than factor two then it's not usable for modeling. So what I will argue is that we actually have been kind of trying to model the wrong type of system. And I will show you how we among others developed the methods to model the copiatrophs operating in a boom and bust environments instead of oligotrophs operating in a low nutrient environments in the chemostat. So I already said that we are modeling the chemostats mostly because it's simple. Mostly because we want to deal with a steady state and not with a full blown dynamics of microbial growth. But if you come to a typical lab of your experimentalist friend, they will do a serial dilution batch growth experiments because they are simple to do. You just grow the mugs in a test tube with a given mix of nutrients, wait until they eat them all up, dilute by a given factor, could be a hundred, could be a thousand, could be 10, and then repeat the growth after multiple cycles you reach some sort of dynamical steady state. And this is not only limited to the lab, we know that many of the natural ecosystems operate in this boom and bust cycle phase. And basically the point I want to make is whenever the licorice are present probably there is also this boom and bust type of phase. So unlike K in chemostats, the main parameters which are needed to quantify the boom and bust dynamics are easy to measure. In fact, Monod himself when he discovered the effect of dioxi shift of the microbe from growth on one nutrient source onto another. In this figure, this figure contains a summary of all the methods you need to measure the main three parameters of the, which you need to model the system. Well, you need to know the maximum growth rate in each of the two environments. Here it is, the environments are glucose first, used first and orbital use second. So you just need to feed the exponential growth rate to two of the regimes, which are easier when you plot it on a log scale. You need to quantify the lag phase, which is this flat plateau between when the microbe is shifting from one nutrient to another, which is also not very hard to quantify. You need to know by how much you dilute. Well, here he didn't dilute at all, but in the serial dilution experiments, you had experiments know how much dilution, how much fresh resources you had to the, well, how much fresh resources you had and how much of the old culture you had to the new test tube. And finally, if you are talking about a hierarchical utilization of the resources or dioxy, you need to know the order in which the microbes use nutrients, or if you are talking about the microbes which are co-utilized resources, eat multiple resources at the same time, you need to know in which ratios they use them. All of these parameters can be inferred from those three curves. So in particular here, he mixed glucose and sorbitol in three different ratios and watched how the position of the lag shifts. And because the more glucose he added to the mix relative to sorbitol, the higher up when this position of the lag phase, which means that for a first, in the first phase, the community is eating glucose and then it switches to sorbitol. And you can infer it by just observing several mixtures of nutrients and just looking at the optical density growth in those mixtures. So I will describe you today some of the sort of mathematical methods or ways to think about modeling boom and bust environments. And the use case scenario which I have in mind is that you have an experimentalist who walks to your office and says, well, I have this dream team of microbes. I really like those, I don't know, five, six microbes to coexist together in a serial dilution experiment. Can you help me to make it a reality? So and he loves, he or she loves this, all of the members of the team so much that he's only happy if all of them are coexisting. Just one of them is being outcompeted. That is not acceptable. So what we can kind of do using the models I will describe in a moment, first of all, we'll determine whether those pieces can in principle coexist together. Maybe they borrowed one of those superbugs from Terry and Leonardo and then of course we are doomed because it will outcompet everybody else and I will say that, well, I know that you love this vibrio species very much but get rid of it otherwise it will not let others to live in your serial dilution environment. Second of all, we will design and we will actually have a very simple kind of linear algebra formula allowing you to calculate what are the ratios of nutrients you need to mix in your test tube so that your dream team will be alive and happy and live stably together. And as a bonus, we can even tell you what ratio of nutrients you need to mix in order to get the relative abundances here for a species whatever you want. If you want them to be present in equal concentrations we will tell you how to mix resources. If you want the species number two to be five types more abundant than species number one we can also do that. So that's kind of the pitch I want to make on the experiment design side and then in the second half of the talk I will tell you how we can use the techniques we developed to address this problem to answer some fundamental biological problems about which microbial nutrient metabolic strategies are better or worse than others like co-utilization versus dioxy shifts versus different types of hierarchies and so on. So that's kind of the outline of the talk. Now, let me introduce several of the parameters which are important for my description. So first of all, the main parameter which you need to pay attention to is when you are doing a serial dilution experiment the resources are getting depleted one at a time usually. So one of the resources will be depleted first then some other resource will be depleted second then some other resource will be depleted third and some other resource will be depleted the last. A priori this order of depletion times is a very complex function of which microbes are present in the test tube in which abundances and of course the ratio in which I mix the resources. So I mix a lot of resource number four, chances are that it will be depleted last even though it's not guaranteed. So those capital T, T1, T3, T2, T4 are the resource depletion times. And while the resources are being depleted the microbes, individual microbes are growing exponentially. I will ignore a little transition which happens when the exponential growth starts to slow down and transition into the lag phase. So I will just approximate everything by exponential growth, no growth and other exponential growth, no growth and so on. So it's like on a log scale it would be a piecewise linear piecewise linear trajectory of the population of each and every species. And of course in the end of the boom phase comes the bust phase which in a serial dilution environment means that you dilute whatever you got in the end of the cycle by a constant factor say 10 or 100. So this slide is mostly oriented for people who are familiar with consumer resource models and again Tillman in the 1980s formulated a very nice and elegant way of thinking about it geometrically. And again, the Tillman was thinking about chemostat so he was developing his techniques about how microbial communities can negotiate together the resource concentrations in the chemostat at which they will all coexist if it's possible. So I wanna say that this boom and bust environments nicely generalizes Tillman graphical method. The only difference, the main difference is that instead of the resource concentrations in the chemostat you have to think about those depletion times, times until the nutrient I is depleted. So let's just, let me walk you through this diagram here. So there are two resources, resource one and resource two and they are characterized by depletion times T1 and T2. And there is this diagonal line which separates two different depletion orders. Everywhere here the nutrient I will be depleted second and the nutrient II will be depleted first and in the upper part of the rectangle of the square the nutrient II will be depleted second and nutrient I will be depleted first. So T1 and then T2. And consider a microbe, this purple microbe which uses nutrients and the preferential use of this microbe is, the preferential use of this microbe is that it prefers to use the nutrient II that is the number one in this hierarchy and only then it switches to nutrient I if it is available. If you just put this microbe alone in a test tube and start serially diluting this environment after a few dilution cycles it will reach a steady state and during this steady state the microbe will grow by exactly the same factor by which it is diluted. This is the definition of the steady state. So if I know the dilution parameter D say 100 then I know and I know the gross rate of the microbe then I know how much, you know what time of the nutrient II I will have. This is by the way I'm kind of right now considering a situation where I only add the nutrient number II and then it grows on nutrient II there is this exponential factor by which it grew and it needs to match precisely the factor by which it is diluted. From this it follows that the time until depletion of the resource must be equal in a steady state might be equal to the log D divided by the gross rate of a microbe. So we know one point of this curve. So this purple curve corresponds to all the possible steady states of this system of one microbe. And if on the other hand you mix two nutrients then you are guaranteed to be on this part of this line. So if you only add the nutrient number I then there will be the same kind of log D divided by G1 here. And here it will be an intersection between two lines. What is very important is that for hierarchically utilizing microbe this line will be flat. Why it will be flat? Well if T2 is longer than T1, so we are in this part of the diagram and the microbe, my microbe prefers nutrient II to nutrient I, doesn't matter how much, what is the depletion time of the resource I is there it will not be used by a microbe at all. It will be just basically the coexistence of this microbe with other microbes will not depend on the depletion time of the nutrient I. Again, this part of the line only makes sense once I introduce another microbe. Maybe I introduce a blue microbe and this is also a hierarchically utilizing microbe which now prefers the nutrient I to nutrient II. So it has this vertical line and equivalent of the horizontal line here and you will see that those two lines intersect. So if two microbes coexist the environment can only be in the unique state which is marked by this dashed circle. So once I know that two microbes coexist I know exactly the depletion times of both nutrients and this is just a, this is the only way how those two microbes can coexist. There is a separate question whether those two microbes would actually coexist and it depends on several factors. One factor is whether this point will be actually dynamically stable. So if I actually prepare the system at this equilibrium point it might be that I perturb it a little bit by the abundances of microbes and they will kind of separate. And the other question is whether a given nutrient ratio R1 to R2 ratio will lead me to this point or not. Both questions are answerable using standard linear algebra techniques which are a little bit too convoluted to present here. So I just want you to trust or to read our paper how to first test for dynamical stability of this point and second of all to tell whether a given ratio of resources R1 and R2 will lead me to this point. So again, right now I did two important simplifications. One is that I only consider the hierarchically utilizing bugs like dioxically shifting bugs, but some bugs co-utilize resources. This whole thing, yes? There could be multiple stable attractors. In principle, you could have two things which intersect in two points and those would be two possible stable attractors. It turns out that for two bugs those attractors are, there could be multistability in the system but usually it will be either two bugs coexist or not only one bug coexist because one of those attractors will also be dynamically unstable. In more complicated environments there could be indeed a multistability indeed. So what I was saying here is that you can generalize this to co-utilization of resources. The only difference will be that this intersection point will be shifted. For instance, if the microbes grow slower on two resources than on each one of them then it will be shifted like this. If they will grow faster, it will be shifted inside this rectangle. So in fact, I use probably not very realistic situation where the growth on two resources is slower. Usually it is faster when it's co-utilization. The second thing you can generalize is to include time lags. The time lags is this time interval when the microbes don't grow when they shift from one resource to another. The only consequence of a time lag would be that this point will be shifted away from the diagonal because you may say that not only the bug will not utilize the resource number one if it is depleted at the same time as the resource number two, but it will also not utilize it for a little bit longer. Maybe by the time I am shifting to use the resource number one, it will be already gone. And if my environment is at this point, the purple microbe will not utilize the resource number one. Okay. So finally, just the final Tillman diagram and then I will move on. The Tillman diagrams give you a very kind of graphical interpretation of competition between bugs. So if there are two bugs, like this blue and a purple, and you see that the blue bug is entirely inside this line for the purple bug. That means that the blue bug is better. The closer you are to the origin, the more competitive you are. Because what I forgot to tell you is that if I am looking at say a purple bug, maybe I should go back here. So if I'm looking at a purple bug and my environment is located somewhere inside this curve, closer to the origin, at this point, this bug will be exponentially decaying from one cycle to another. So after several cycles, it will be gone. If the environment is located here, the bug will be growing. So from cycle to cycle, it will be more and more of it and eventually it will take over and it will push the environment to be on the steady state line. So you can kind of say that in this situation, the blue bug is absolutely better than the purple bug in any environment. So if I add the blue bug, the purple bug will disappear. So the purple bug is dead. Okay. Now let me now kind of, these diagrams are good when I have two resources, but already for three resources, it's very hard to draw them in a three dimensional. So you need to switch to the linear algebra. And when you switch to the linear algebra, you have to kind of change the notation a little bit. So before I was talking about the resource depletion times, now I want to think in terms of the lowercase t variables. The lowercase t variables, which we and others call temporal niches, are characterized as the time intervals when a particular combination of resources is present in the environment. So for instance here, during the temporal niche number one, all four resources are present and all the bugs grow at whatever growth rate they have, maximal growth rate they have when four resources are present in the environment. Then the nutrient number one disappears and in the temporal niche number two, everything but nutrient one is present. So nutrients two, three and four are present. Then nutrients three disappears and in this niche I have the nutrients two and four. And then finally in this niche, I will have only nutrient four. And the last niche, which we are not considering right now, but we should consider if we are also worried about the bugs slowly decaying or dying in a stationary phase, that will be like a negative growth rate. The final niche is when all resources are depleted and all the bugs are basically in a stationary state and they started to die, they started to decay. So of course the simple algebra is that there are four N resources, there are two to the power N possible temporal niche, two to the power N possible combinations of resources which can be present or absent. And if you exclude the one niche where all resources are depleted, therefore there is no growth or maybe a negative growth, then I have sort of two to the power N minus one temporal niches to consider. So when I know the depletion order of resources like one, three, two and four in this illustration, I can define the durations of niches and I can also define what would be the resource combinations present in each niche. So in order to see whether a given, again now coming back to my motivation question, so I told you that I want to know if somebody brought me a given set of microbes, whether I can make the environment for all of them to coexist. I don't know a priori in what order the resources will disappear, so there are N factorial possible depletion order and I need to check each one of them. It will be a drag if the number of resources is very large, but for a number of resources, five or six, that's not a big deal. I can go over all six factorial combinations and test each one of them individually. But there's also complexity, so the number of stable fixed points. Well, that will come as a byproduct since I check every possible depletion order, if I will find more than one depletion order feasible, that will be a hint that I have several candidates for a steady state, and then I need to see whether they will be a truly multi-stable states or maybe some of them will be dynamically unstable. A lot of times you will find that you can realize different depletion orders, but they will be realized for very different nutrient ratios. So maybe if I'm operating with this nutrient ratio where I feed my test tube with maybe five times more resource one than the resource two, I have this particular kind of region in this diagram and one particular depletion order, but if I am in the other regime, I'm having another depletion order. So this is all kind of open question, but this is a good point. Indeed, there could be a multi-stability and there could be a sort of first order phase transitions between those different stable states of the system. So again, the first kind of question I want to address is whether I can assemble this community at all, whether I have some equivalent of the super bug which will make the assembly impossible, or maybe it's not just one super bug, but just this combination of bugs cannot leave together no matter what I do. So here is some notation. So capital G is a gross rate of a bug alpha in the temporal niche I. If it's a hierarchically utilizing bug, that means that in a given niche, I need to look at the resource which has the highest resource in the hierarchy of this bug which hasn't been depleted yet and measure the gross rate of this bug. In co-utilization, you need to apply another formula. So imagine that I fixed the depletion order and first as a warm-up exercise, let's ignore lags. Then of course, I already explained you that when I just have one resource, the gross rate times the duration of the depletion until the resource is depleted must be equal to log D. When I now have kind of a multi-resource environment, then I need to make sure that as a combination of gross in all of the niches, I need to get exactly exponent of this must be equal to D or the sum of this must be equal to log D where D is my dilution factor. Now if I have lags, I decrease the duration of every niche by lag and lags are species specific. Some species have longer lags, some species have shorter lags, but it's still a simple linear algebra which becomes apparent if I kind of leave the variables which I don't know, those lowercase t on the left-hand side and put everything which I know like dilution factor, gross rates and lags on the right. And then you can say, well, this is a dilution factor, effective dilution factor of a bug which is different for different bugs because of the lags. And by the way here, I want to connect it to something slightly less artificial. Right now I was just having in mind this test tube experiment where I'm doing serial dilution, but in the environment where there are boom and bust cycles, different species will experience different magnitudes of collapse. So if I know the average magnitude of collapse or average log of the magnitude of collapse, I can use this as the right-hand side for the equation. So I don't want you to think that my model is limited to this artificial situation where I'm always kind of diluting by the same amount all the species. So it can be formally solved by just inverting the matrix G. I know the right-hand side, so I can always find the left-hand side provided that the matrix is invertible, which it usually is. The problem as always with ecology is that those are physical entities, those are times, and they have to be positive. So as soon as one of the times becomes negative, that means that the solution, even though it formally exists, it doesn't mean anything. It doesn't correspond to an assembly. And I need to test every one of those n factorial depletion orders in order to see whether the solution is at all possible. The determinant of G gives me some idea about how stable is the solution with respect to changes in depletion in dilution factors D alpha. If I, for instance, change the lags due to some reallocation of resources or anything like this, in what range I can change the lags so that my times will remain positive? The answer to this is proportional to the determinant of G. And again, dynamical stability is a separate method which I don't want to completely kill you because there is a linear algebra which tells me whether the solution is stable or not. Okay, now I will kind of rush through this. Maybe the only take-home message here is that we first kind of started to first get an idea, to get an idea of what would be the statistics of how likely I am to be able to assemble a random community of random species. We consider the question, what if I have n species, n resources, ns species, nr resources, and all species have identical hierarchies. They all use, I don't know, glucose, lactose, and then other resources below. So everybody uses the same. This is actually a problem which we can solve mathematically and we use some old 1960s mathematical result which turned out to be equivalent to finding whether three vector or nr vectors contain the origin inside their convex hull or not. It looks similar to what Ned Wingren and Anna Posfi did for resource sort of allocations in the chemostat community, but this similarity is superficial. We are not dealing with resource supplies or we are really dealing with growth rates of, we have as many vectors as we have resources here, not as many vectors as we have species in Posfi. Anyhow, the take home message is that you can calculate and use it for a benchmark, the likelihood that your dream team can be assembled in principle. And one kind of number I want you to remember which comes from this theory is that for large number of species and large number of resources, we know that in principle we can have as many species as we have resources. But now if I take a given number of species, the same number of resources, say six species and six resources, what is the likelihood that I can actually assemble this community? It's given by one half to the power five, six minus one, which is one over 32 for six resources, but once the number of resources gets really large, this number gets kind of lower and lower. So in order to assemble the community with some reasonable probability, let's say what if I want a 50-50 chances of assembling of my dream team on a given number of resources? I need to have twice as many resources as I have species. This is a factor two comes from this complicated combinatorial formula. I will not bore you in how it comes. But just a number to remember is that when you are trying to assemble your dream team, be prepared to have at least twice as many resources, otherwise with overwhelming probability, I will tell you that it will not be possible. Now, another kind of intuition which is not surprising from what we know about ecology is that whenever the species are gross rates on different resources have some trade-offs, in other words, if a species is good on resource one, it's typically bad on resource two, that increases the likelihood of the assembly, but the existence of a superbug like Vibrio, which Terry and Leonardo started, decreases the likelihood. If there is a species which is good on every resource, then the assembly will be highly unlikely. Okay, the separate question is now, imagine that I realize that the assembly is possible and I now want to know in what region of this resource supply, I need to mix resources in my test tube at the start of every dilution cycle, how should I mix them in order to stabilize the community in the feasible state which I want? This is actually, again, a question we can answer and it depends on the dilution factor. What is shown here is a community of three species on three resources which is feasible, meaning that we can assemble it. So each of those triangles gives you the range of resource considerations where this assembly will actually result in a coexistence of three species for different dilution factors. This is dilution factor of 10, this is a hundred and this is a thousand. So you see that it's, depending on the dilution factor you use, you need to use different combinations, different ratios of resources. And you can also quantify what is the range of resource ratios where I can assemble by taking kind of the ratio of this little triangle to the big triangle. And because I have sort of three resources, meaning I have two independent ratio of resources, I take square root here. So if I had four resources, I will take a cubic root. This is sort of the gives you the idea, the technical term for this is structural stability in the ecology field. It tells you how stable is my assembly to respect to changes in resource considerations. And the structural stability pair resource gives you roughly an idea of how careful I have to be when mixing resources. What kind of error I can have? Am I allowed to make a 10% error or a 1% error or something like this? Okay, so this was kind of the methods part. And I can see that I kind of probably killed the brain of many of you by dumping all those methods on you. Let's try to see if we can learn something from those methods which will be, yeah. There is a question. Oh yeah, yeah, yeah, absolutely. Are you assuming that the first resource that's a bug is consuming is the one that grants the highest growth rate? Yes and no, that's exactly what I will be talking about right now. That depends on the strategy, right? So you can have the hierarchical strategy which is what we call SMART where the resource hierarchy perfectly matches the growth rates. There is one where it matches for the top resource, for other resources it doesn't matter, it's random. And then there is a random one. And then finally there is this random hierarchy more as a strawman because of course the bugs are not really random. There is another question, Tim? Yes. Can you just switch back to your ternary diagram for a second? This one? Yeah, I know the one, next one. This one. So again, just tell me which slide is it? I will keep going back until. No, no, next, forward, forward. Forward, uh-huh. This one, yes. This one, okay. It seems like resource one is just a nuisance really. This is perfectly stable with resource two and three. Do you also see examples where it's like really in the, I mean resource one just, if it's not there it's still stable? Yes, let me see this. In principle, if I am on this boundary one of the species will hit zero. So everywhere on the boundary of a triangle one of the species will hit zero abundance. So by the way, if I want to have species in equal abundances one to one to one, I have to be at the center of the triangle. So indeed to get the equally abundant species I need to give very little of the resource one. But if I will completely eliminate resource number one one of the species will go visit because there is a competitive exclusion here. So I cannot have three species on two resources. Sorry, I realized from the question that there is something I don't understand as well. So in the solution you find basically, if I understand correctly, you solve for the feasibility of the times. Yes, that is first step, yes. The first step is the feasibility of the times. Now the feasibility of the times does not necessarily guarantee the feasibility of the solutions. Well, and that is exactly what I am solving here. Exactly. This is a structural stability, meaning that I can once I know the times I can now start engineering how to mix resource. I know that once I know that the solution is possible I need to look for it. And by the way, this is, you may say that, well, experimentalist may say that, well, I can do it without you. I'll just mix the resources in a one to one to one ratio and everything will be fine. Here, if you mix it in a one to one to one ratio this community will never be stabilized. To stabilize it you need to carefully do this calculation and find that you need to have, say, if you prefer to have a hundredfold dilution you have to be at the center of this red triangle. And again, the reason why one of the resources is, the reason why this triangle is touched one of the boundaries is that in this solution one of the species, I don't know which one specifically is only using resource number one. How is it possible even all species use all resources? Well, the resource number one is its top choice and it runs out last. So it will not have any reason to switch to anything else. It will just use your top choice until it disappears. And if you will make this resource to be absent then this species will be gone. If you will make a little bit of this resource, well, maybe you will not exactly, yeah. So anyhow, let's talk about something more fun. So this is, I also probably need to speed up. How much time I have? Yeah, sure, sure, sorry. Is there possible to incorporate like the creation of a new environment, a new niche, for example, by the suppression of byproducts? Excellent question. Like cross-fitting. Excellent question. This is the next step. Again, we start with something simple. We know that a lot of the switches, let's say classical glucose, the acetate switch in E. coli is due to E. coli generating an acetate. And then if it leaves alone that it will, can of course pick up the acetate it creates, but if somebody else in the community consumes acetate faster than, you know, while, if somebody in the community prefers acetate then the species will consume acetate and by the time E. coli is ready to switch the acetate will be gone. So indeed it is possible. It's more, slightly more involved because this complete decoupling between species abundances and resources stops being true. So some of the resources, the ones which are generated by other species start to depend on species abundances. So in particular, this predicting for which ratio of resources I can get a desired ratio of species abundances becomes harder in this case. But this is exactly the direction we are going, by the way. Okay, so about metabolic strategies. We have just a refresher. So when you are studying a community in a chemostat, there is almost a law in biologists that in a chemostat the resources are coutilized. If you can utilize a certain number of carbon sources you will consume all of them at low concentration, at low resource concentrations. And the model is describing and has been known since 1960s. And if you look at a recent theoretical literature from Pankaj Mehta to James Adwire to Michael Tichen of the Ned Wingreen, all the labs are working on this MacArthur model and its variants. Now, sequential utilization of resources in the boom and bust environments is sort of only relevant when you have a batch grows. And sort of the theoretical techniques are more or less developed in my lab and Jeff Gore's lab. So that is a relatively new frontier. So this is kind of a pitch that it's a nice interesting way of applying the ecology to those systems which otherwise would have been too complicated to study. Now, what do we know about those? Why microbes choose to co-utilize resources or to sequentially utilize resources? And when you are thinking about sequential utilization the next question is what kind of hierarchy is there? So do you utilize glucose first always or in what order different bugs choose their resources? So there has been a review from Hiroyuki Okano, Radger Hampson and Terry recent review and they kind of broke it into three pieces. They started with something which we understand well at least in the quali, how this hierarchy is regulated. And again, some of the models, I'm not saying that this is a model in this review but some of the, I followed some of the reference some of the models of this carbon hierarchy in E. coli uses 280 variables and 476 parameters. Then of course, they switch to physiological role when the species grows alone and whether or not this hierarchy is reflect some optimal strategy carried out by the bug. And there is some literature and some thoughts about it including the papers by Chaotang Lab and so on. And Terry and collaborators emphasize that the bugs don't always optimize the proton allocation and that is actually something which we want to understand why. And the last chapter was on ecological consequences of this different metabolic strategies. And that basically was the statement was that we don't understand it very well and we would like to see how the ecology and physiology play together in species decide in which metabolic strategy to use. Okay, so in our model by the way this is kind of informational slide. We use a recipe on how to generate the growth rates on either single resources or combination of resources when there is a co-utilization. We are using it from this paper from Terry's lab. And of course, about the hierarchical utilization I will tell you immediately. So we are considering, we wanted to compare and contrast several different metabolic strategies. Let's start with the first one which is most natural. We call it a smart sequential utilization. Once you know the maximal growth rate hierarchy the species matches the order in which it prefers nutrients to the hierarchy of growth rates. First uses a max growth rate, second uses a second max growth rate and so on. The second one is more of a straw man. We wanna see what will change if instead of being smart the species will be just random. What if the species just decide on which order they utilize completely without any pain, any attention to growth rate. That's as I say, it's a straw man. The next one has actually some motivation. Our previous paper on the topic, we found that when the dioxy, when dioxy species assemble together from a large pool the species which dominate this assembly are first of all they only care about their top choice resource. The other choices almost don't matter so they are not optimized for them. And second of all on this top choice they are smart. So out of the species which survive the assembly from a large pool almost all of them will have their top choice matching their fastest growth rate. The rest of the resources could be anything. So we kind of knew that this strategy is important in at least a random assembly from a large pool. We wanted to understand why. We call it a top smart. So again top choice, maximal growth rate, others are random. And finally the main comparison is between any kind of the sequential utilization and co-utilization because for a bunch of resources where co-utilization is favorable in terms of the growth rate it's actually not totally obvious why the species might adopt the sequential utilization strategy at all. So this is just a flashback to this older paper, 2021 paper. Now I wanna visualize it on a two dimensional plot but I need to walk you through what this plot is all about. The x-axis is feasibility. Just whether or not I can assemble a community of random community of species on a given set of resources and to make it sort of easier to visualize to what it would be in this case of random. In a set by random here I mean that what we can calculate analytically only what would be this probability if the old species use the same resource hierarchy. So the exactly identical resource hierarchy. So that will be our kind of strawman. This one over two to the power number of resources minus one. So that whenever you see one on the x-axis that means that our dream team with this strategy assembles as likely as a dream team with identical hierarchies. Then there is a dynamical stability which I haven't described at all. I just mentioned that just the fact that it assembles doesn't mean that it survives small fluctuations abundances. And then there is a structural stability in what fraction of resource ratios this assembly will actually be materialized. How careful I have to be in my resource ratios in order to achieve this assembly. And again since it's a three dimensional plot and three dimensional plots never work we condensed feasibility and dynamical stability in one axis. So now feasibility and dynamical stability will correspond to what is the likelihood that the assembly is possible and dynamically stable. And then on the y-axis would be how sensitive is the state to fluctuations in resources whether it will kind of be able to survive. So let's start again. Remember that the one that has been calculated for the bugs which use the same hierarchy. All the bugs use the same preferred glugos to lactose to whatever comes next. So let's start with red. The red here is co-utilized strategy. So this is co-utilization. This is exactly as likely to assemble as a random case and it has a certain dynamical stability, certain range which we cannot compare to anything. Let's now look at the smart sequential utilization. So here is a big news. So first of all the sequential strategy is much more stable to fluctuations than co-utilization. The reason for this is that when you assemble, successfully assemble a community of sequentially utilizing bugs not all the bugs use all the nutrients. Actually many of the nutrients do not impact the bugs abundance. Like we saw in this case of a triangle due to your question that some of the bugs are not sensitive to nutrients two and three. And that gives them much higher almost order of magnitude higher structural stability. So it is easier to assemble sequential utilizers than co-utilizers. But there was a problem that there was kind of a downward trends in assembly likelihood. So the more by the way the solid symbols are a community of six species. The light symbol is community of two species and everything in between. So two, three, four, five, six. Then the top smart actually beat everybody else in our simulation. Not only it was ten order more structurally stable than the co-utilization but it was also more likely to assemble. So we will see in a moment some other quantities characterized in this strategy. And then we decided to see what is the strawman? What if the bugs were completely stupid and random? Well that would be actually a disaster. Well structural stability it would be halfway between the smart and co-utilizers. Co-utilizers at the bottom of the food chain. But there will be a very big downward trend in the likelihood of assembly due to the fact that those random assemblies are almost never dynamically stable. This is what I wanted to show here. Again the color code is the same. Co-utilization, smart, top smart and random. The black are dynamically unstable state. What I am showing you is a spectrum of the Jacobian which needs to be below one in order for the community to be dynamically stable. And everything which is in black is dynamically unstable. This is kind of a number for you. So when you are co-utilizers you are almost 100% dynamically stable. Interestingly not 100% even though for the Kimostat the James O'Dwyer and company have shown that it's 100%. So now there is this small deviation away from structural stability. Sequential smart almost always dynamically stable. The more randomness you throw in your order the less dynamically stable you are up to the point where for six resources only 4% of random utilization orders are dynamically stable. Some rarely we even see some chaotic solutions but that is very rare, one out of 10,000 so we don't need to worry about it. Okay, and how much time I have if anything including questions? Okay I have one kind of story which might be curious. It's more like the future work. So what if you want to beat the competitive exclusion principle? What if you want to assemble a community on fewer nutrients than you have species? Say six species on four resources. If you supply resources at the same ratio every dilution cycle that is pretty much impossible in our model at least very, very rare. However what you can do is you can if you are really kind of have a robot to do the dilutions for you maybe you can ask a robot to give the nutrients in one ratio in one cycle then to give nutrients in different ratio in another cycle and yet another ratio in the next cycle and then you can do this multi-cycle solutions which will beat the competitive exclusion. Again, this idea was actually inspired by Bloxom and Jeff Gorse and other papers on BioArchive where they said that well the competitive exclusion principle should not hold in the fluctuating resources for communities with fluctuating resources and they put this limit two to the power N minus one which I already told you minus one is because we are not including the niche where all resources are being depleted assuming that nothing grows there but as soon as there is some decay of a stationary state it actually becomes two to the power N. In practice what Bloxom and Jeff achieved is by fluctuating resource so every dilution cycle you are given a random ratio of resources according to some rule they were able to stabilize roughly N square species on N resources. Just look at this data. The dashed line is competitive exclusion principle the exponential line is this exponential upper bound the circles are what they accomplished in the fluctuating environments and they can really assemble something like 25 species on seven resources if the resource ratios fluctuate. We asked another question. We asked what if we are not relying on randomness but what if we are asking robot to give resource in different ratios at different cycles? So this is kind of very preliminary plot. This is just the diagram here. The x-axis is the number of resources the y-axis is the number of species. What is inside a square is a likelihood of assembly and what is on the right here is just the standard model which I described you before where you always give the same ratio of resources at every cycle and you can see that the competitive exclusion here applies you cannot assemble all this black triangle down there means that you cannot assemble say five species on three resources, never, right? And even if you are trying to assemble five species on five resources the likelihood of this to happen is only 6% which is one over two to the power four or something. So now if you allow for this different ratio of resources at different dilution cycles which is not that hard to implement if you have a robot doing your dilutions for you then you can beat it and this red line is the theoretical upper bound by a block summon war two to the power n minus one. You almost approach it and there are some kind of black squares which are adjacent to a line but everywhere you can kind of push you can actually assemble five species on three resources with non-negotiable probability. And I think I will wrap it up here. So I just wanna repeat the text take home messages since it was quite a mathematically heavy talk I realized. So here are the messages. We can engineer community assembly in a serial dilution environments based on the measurements of growth rates and lags and nutrient hierarchies of different species which are relatively easy to measure. I'm very happy to talk to an experimentalist who has a robot especially who can do this measurements at scale. Bonus one is that we can make any desired relative abundances of species. Bonus two is we can in principle if it's necessary assemble more species than the resources by doing this multi-cycle dilution solutions. The second direction is more kind of biological not engineering what different metabolic strategies how to compare the success of different metabolic strategies. I already said that we want to here are the next steps. We wanna do a little ecology and evolution model in which we investigate the trade-off between bugs making the lags shorter but at the cost of reduced growth rate. This is allocation question. So this is a physiology question. And in a multi-resource environments you kind of need to predict which resource be depleted next and allocate the proteome to this resource otherwise you lose. So there can be kind of fun models of this type which we are kind of started to develop. Cross feeding just your question. So cross feeding is the next step like glucose acetate. And can we use the success of this top smart sequential utilization strategy as explanation for why species sometimes deviate from the optimal hierarchy. If you read the papers you will see plenty of examples where the smart hierarchy is not respected. There are plenty of cases where the growth rates on preferred, well not the top resource but below the top there are plenty of violations where the growth rate on the resource is higher but the resource is lower in the hierarchy. Can it be due to this top smart sequential strategy being easier to assemble and more stable? That's all I have to say. Thank you. Thanks a lot for the talk. So I would say we have time for two quick questions. So Rosana. So I was curious about the, if you ever observed a perfect co-utilization of resources because I would expect that even if the cell is growing on using multiple sources together then those sources will run out in a hierarchical manner. You are absolutely right by the way. You have observed a perfect co-utilization when they run out at the same time. No, no, no, no, that was, maybe I didn't explain it properly. It's this hierarchy of depletion times is relevant for any strategy. That's why we use the same model, the same technique to compare co-utilization and hierarchical. Moreover, as we know there could be short lags when one of the nutrients ran out and they transition to the next one. You still need to reallocate the proteome so you can include lags there as well. So the main difference is that when it's, in this geometrical picture, the hierarchical strategies, those lines, manners and lines horizontal in the co-utilization they are both non-horizontal. But what I need to know experimentally is the gross rate on any combination of resources, whatever it is, it could be in a hierarchical, there is one recipe in co-utilization, it's another recipe, if I know it I can model it. Yes. Maybe I... That's okay. So I want to take this opportunity to raise this, I think forever question of the dichotomy between ecology and physiology. It's great to see you work out, okay, from ecological standpoint, stability and all that with the hierarchy, with the sequential utilization of that. But from a physiological standpoint, we wrote in our review that we never understood why there is hierarchy, okay? And because bugs grow better on their own if they use caret of a hierarchy, right? Theoretically we understand and experimentally we can demonstrate, right? So the conventional notion of the hierarchy is better utilization of resources is wrong, okay? But then neglecting the important feature of what's actually going on. But then so my question is, so you think okay, you're providing a reason, oh, that's why they have hierarchy because there's ecological stability, okay? But then bugs are not only living in a fixed environment, they're also by themselves, okay? So how, what prevents them from just dropping hierarchy and so that it can grow better when they're on their own? That's a million dollar question. By the way, it's not super straightforward even from ecological perspective to say that, well, even from physiological perspective to say that co-utilization is always better because you need to take legs into account and if you, the growth rate, I agree with you, legs, well, legs are much more sensitive things in terms of reality, so it's better, okay? Okay, okay, okay, okay. But now of course even in a community, when they grow in a community, you have a stable community, but if one bug mutates, it invades. So we know if I get your Vibrio and I put it in my carefully designed community, it will go to hell, I'm pretty sure. So, right, yeah, well, one testable prediction, one testable prediction is that the bugs which live in a stable communities for a long time may exhibit more violation of this physiological optimalities and bugs which are changing all the time. Indeed, if you have a random partners every night, pardon my expression, but then you are kind of, you are doing different things, so. Yeah, this is a game theoretical question. Actually, a nice game theory is to make the bugs play, if indeed it's relatively easy to rewire the strategy, you can try to see what kind of game they will play, if each one will choose whatever is best for the bug given the choices other bugs make, kind of Nash equilibrium type of thing. Okay, we are really late, so just one short question. Yeah, I was just very fascinated by the last results you were showing where you changed the ratio of the resources, and so my question is, was there a strategy in how the resource ratio was changed or were they changed like just randomly between different dilutions? Oh no, no, it was very engineered. Random changes is what Jeff Gore and Bloxson did. The random change will bring you, sorry, I'm just trying to get this slide. So this is what the random gives you. The random changes give you n square bugs on n resources, and again, there is an optimal amount of noise if you change too much, this is a random. We engineer it, we say, well, we kind of design, we carry the thing, and we say, it repeats every, you know, let's say we use the three cycles, right? So we say, on one cycle, mix it in one, two to five, and the next cycle, three, five to four, in the last cycle, 10 to, I don't know, two and one. So that's our, and then repeat it over and over. We call them seasons, it's like, you know, four or three or four seasons, yeah. Okay, let's thank Sergei again. Okay, now.