 Okay, excellent. We are live again. Welcome back for the next lecture by Samir Swaiz from the University of Padua, who will talk about community patterns in consumer resource models. Please, Samir. Okay, so first of all, thank you for the organizer. It's really for me an opportunity to be here and to try to convey and to share with you some works that we have done in the past two years. So let me share the script, the slides. Okay, you should be able now to see slide, right. So this, what I'm going to present is mainly the work done by Leonardo in his PhD. I just got the PhD, you met Leonardo because he made some tutorial for you. And so these I want to thank you because really a lot of works he did from experiments to theory so he did really a great, great and incredible job. And then with Andre Giometto from Harvard, now he moved to another university and Amos Maritan that you met in the first lectures. And of course, to all the lab that always in the discussion is very important. So I want to start from the fundamental questions that is one of the important questions in ecology. So while we can observe so amazing biodiversity. And I think you already been exposed by these questions on the fact that it's actually not so trivial to understand the incredible diversity that we observe in the natural ecosystem. This is the one of the most famous cases that plant on some plant we have very few resources in the ocean. And yet we found thousands of different species. And this goes under the name of paradox of planting, but this is in general, this more broad questions about how it's possible to observe the so large number of species coexisting species even in the absence of of many resources. And actually in the last years and again you will have the opportunity to hear from Alvaro Sanchez his great work on that you can actually sample directly now from environment DNA and analyze the for example the microbial diversity that you find in a very different types of environment. And again, you can cultivate this this DNA in on in the lab, so in a controlled experiment and again just using very few resources, even more resources that you observe many species to MP 30 that can actually coexist for a long time. And this is surprising because in our ecological fundamental setting. We understand that the species that coexist should somehow occupy a niche and that's how species can coexist. So if you have a like you can do this experiment actually you take two species, one two microbial species and one of the two is better, let's say in up taking resources so it's as a fit, it's a higher fit in that environment. So what happens that at the end, if you if you wait long enough, then the fittest species and the one with the highest growth rate will invade the system and will exclude the other species. That's why somehow we need an understanding of how it's possible that so many species can coexist. Okay. Let's say a way to study such problem is in a theoretical way it's through the use of consumer source models in particular, one of the fundamental achievement in theoretical ecology of course for sure is a is the model proposed by my car tour in the 17. And so this model, as you can see is a model of the system of a couple differential equation, one for the species population in the noted by here by and, and of course, species grows. Okay, and grows by consuming a resource. Okay, so the resource concentration here is given by the sea. And R is simply the resource uptake rate and typically is considered as a model function. So this is a classic model function. So this, this term here, this alpha sigma I represent the metabolic strategies that has the metabolites that the species needs to actually uptake the resource from the environment. And Vi is the so called resource values that is the amount of energy that the bacteria can extract for some from such resources. So this overall part here is what the fine and the growth rate of the species. Why we have that rate that time that here we consider dependent on the species. On the other hand, we have the resource concentration that grows here we are thinking of a biotic resources and as I represent a constant resource supply rate. Of course, we have a minus due to the fact that this resource is used by the different species. And we can have also degradation rate here denoted by new. Okay, typically, we will consider this new equal to zero, but this is not the mean we can do that without losing the general generalization of our result. So in general, you can see that alpha somehow represent so a bipartite kind of networks that tell you which kind of metabolites the species user to consume the resource I. Okay, so if Alpha Sigma is zero, it means that the species Sigma cannot use cannot uptake resource of the type I. Okay. Okay, so through this model is easy to to see so these are the two equation that we retrieve the so called the competition is to some pretty simple that I think you have already heard that is to say so consider the stationary states of these two dynamics. Okay, so you can see that putting zero the first the above equation here we obtained such a condition. Okay. And this condition you can see is these are M equation because we are MS species of Sigma here goes from one to M so we have an equation and we have a P variable. Because this is a sum over P alpha so alpha. Now are the variables we have an M equation in P variable, but therefore if M, so if the number of species is greater than the number of resources we have more equation on variable so there is no solution for the system, except in the generic case. The system is soluble only if M is smaller or greater than P so if the number of species can be only smaller or equal to the number of the resources and this go on the name of competition exclusion principle that I mean is a celebrated result that still we have to fully understand. Okay, as an exercise, I proposed to you to show that this couple set of equation, if you use a biotic resources instead of a biotic resources that is to say instead of a constant supply rate, you have a supply rate that follow a logistic equation. Okay, and you consider a linear resource concentration so RC just depend on C, then in this case in the quasi station approximation so if you put the concentration dynamics to zero, and you look at the stationary state for the concentration and once you do that then you put back this result into the population in the question for the population. Okay, so you can see that you will retrieve, you will recover the generalized.cavoltaire model. Okay, so the generalized.cavoltaire model can be obtained as a special case in a quasi station approximation of the MacArthur model. So, it's not possible to violate the competition inclusion principle but as pointed out recently there is a very important physical constraint that we are missing that is the amount of energy devoted to resource uptake cannot be a limited. Okay, and in other words, there is a trade off between the metabolic strategies. And this was pointed out by possibly angry angry in a recent PLL. So, species has a total budget of energy that can spend in a metabolic in producing metabolites basically, and so there is a trade off between the different strategies that can be turned on. And in the assumption of this work, E was equal for all the species and there is a hard bound so this is actually the sum over all the metabolic strategies should be equal to this energy E. And they showed that in this case a coexistence of more species down resources is possible in a sum and I will show fine tune conditions. Okay, what are these conditions? Well, again, these are our consumer resource equation. And again, you can compute the stationary states. So, a first sample assumption of that work was that the data rate was species independent that typically they are very small. So from this condition, you can see that a solution for a star is is the following. Okay, so you can find a solution that is a function of the total energy budget. Then if you put back now a star in the second equation here. So again, you look at the stationarity. So you have the supply rate is equal to the sum and you put the solution of a star. Then you can see that this species or species can coexist independently on the number of resources if such a condition is met. Okay, so this is a condition that once we introduce the energy budget, if this is satisfied, then we have coexistence of more. We can have coexistence of more species than resources. Then if you rescale all the quantities, so we call x, the rescale population as hot, the rescale supply vector. Okay, I pointed out here that with this tilde here, I do know the fact that I have absorbed this why this efficiency. But okay, as hot is the rescale supply rate and we rescale the metabolic strategy so that all these quantities sum to sum up to one. Then you can see that this define basically a multidimensional simplex and in particular all the strategies and the supply rates light in this P minus one. Okay, dimensional simplex. Okay, so you can represent the geometrically the solution of the consumer resource model with the total energy budget. And in this, I represent here that the resources. So these are the axis for the resources and all the metabolic strategies of the species and the supply rate lie on this P minus one dimensional simplex. So in general, we will consider P equal three for three resources so that the space in which the metabolic strategy and the supply rate leaves is a two dimensional simplex and so we can visualize it very clearly. So let's now see what what what does it mean that this condition for coexistence is satisfied. Okay, so in fact we can have a geometrical interpretation that allow us to understand when the the sum, the sum here is is actually satisfied. So after the scaling I said this can be represented in in in this simplex and so now consider the points the color points represent the strategies. So red species, it's only nutrient two because you can see this is only in the in the vertex of of the of the resource number two. Okay, blue feeds equity upon the blue species feeds equity upon species one and the species two. While the species, the violet and the orange species can feed on all resources. Okay. The star is a supply rate. So there is a supply rate that is actually have a component that is different from zero in all the different resources. Okay, so they these conditions imply that the supply rate must lie inside the convex are composed by the metabolic strategies that is to say if you know I'm not was not so good to be the convex are but this this region is the convex are that is to say the region where you have as you limited limited by all the metabolic strategies. They start the supply rate. In order to satisfy to this condition must lie inside this convex are so in this case, the star and supply rate is outside the convex are and therefore you will have extinction and the competition exclusion will be satisfied again. And so in this case only at best three species typically less will coexist. So there will be at least one extinction in this case and the competition exclusion principle is recovered. If instead the supply rate lie inside the convex are then this condition is satisfied and in fact that we have the coexistence of all the four species. Okay, so the total energy budget is a fundamental and important and physical ingredient to understand and to allow species to exist. At the same time, we must say that in this condition. In this case, if we put a soft bound or if we just perturbed a little bit for example the budget based on the energy budget dependent on the species. We will retrieve again, we will retrieve again the competition exclusion principle so in some how this is a condition. It is a fine tuning condition. In general, we understand a very important factors that is this total energy budget but still we want to better understand how can species coexist or at least how can species organize to coexist so that they can actually survive even in the presence of your resources. Okay, so now another apparently unrelated aspect is the observation is an experimental observation. Actually that date backs that dates back from Mono in 1949 in his PhD thesis that the growth of of microbial species in presence of more than one resources actually display the so called the oxy shift. The oxy shift is the fact that you have you see there are different the slope in the growth of the species. And that's because basically the species in the presence of, for example, two resources first use his best his favorable resources and then once he has consumed all that resources, then there is a shift in his strategy so basically turn off some some metabolites and turn on the other one. So to start to feed on the second resources. And this leads to this bump and this different regime in the in the growth rate. So this is a very strong and infected and there are of course a lot of evidence since Mono in 1949 that the strategies of the metabolic strategies in bacteria are not fixed in time, but they change in time. Okay, so this is a very fundamental ingredient that we are losing. We are not considering in consumer resource models typically. So they are far not constant but are function of time. Okay, so that's what we have done basically so we wanted to in consider in the McCarty's consumer resource model, metabolic strategies that evolve in time. So this means to write an equation for for the metabolic strategies. Okay, so how how to write such such a question where we used a simple idea. Maybe the most simple idea that is to use an adaptive framework so that each species changes its metabolic strategies in order to increase its own growth rate. Okay, so this is a in a way the possibility that the species can adapt to the environment and select which kind of metabolic strategies you want to use. So again, if this is the growth rate, the adaptation so that the equation for for the metabolic strategies is simply in a simple way the gradient of the growth rate. So this in this way we optimize the growth rate and one of the Tao or lambda then also the velocity of this adaptation. Okay, then we will see how this is related with the parameter. But so now we are so we have a species dependent adaptation velocity. Okay, so this is now our new equation so we want to optimize, of course, the metabolic strategies but it's clear now we need to put a bound. Okay, because again there is no possibility of devote an unbounded amount of energy and if you just increase this alpha dot you will have a different increase when we put a soft bound on on species dependent soft bound on the on the on the energy that can be used to produce a metabolic strategies and so now we have to perform such optimization constraints. So it's possible. These are a general result if you have to do optimization with some constraint that you can implement. So the constraint in in the equation. So the idea here. Okay, here you can see there are. So this is the face space of just the two, two resources so we have a fifth alpha sigma one and alpha sigma two, and this line that is star sigma divides the plane into region, one that is the, the allowed region we can move inside this, this half plane because here the energy is less than the easy must start while we are not allowed to to cross such such line we cannot move in the in the other plane and to do that what we do is that during the the optimization so while performing the gradient we remove basically the perpendicular length of the gradient that is parallel to the gradient of the of the energy. Okay, so in this sense we have to perform this evolution by removing all the time this component so that it will allow to not always to move at best around the tangent of the school. Okay, so if you do that this is a I mean this is just to perform this calculation is not so easy and also not allowing the energy to be to negative the metabolic strategy not to be negative you end up with the these two condition become this equation here. Okay, so now we have an equation for the metabolic strategies. And in this equation is also contained the constraint on the on the on the metabolic budget on the metabolic trade off. So these are the new equation of the consumer resource model with adaptation so you can see we have an equation for the population the equation for the concentration and our equation for the metabolic strategies. Okay, so now let's see what we obtain what this model with adaptation can can display with kind of behavior. First of all, okay, so in the red line. It's a simple situation of the model and kind of the general setting with general parametrization and you can see that indeed. Okay, so this is the model. This is the data that I showed before. And actually using metal that the metabolic strategies allow to to reproduce the oxy shift in the growth. Okay, so here, if we think that there is a strong preference on my resources or we can think that we can turn it off and on to metabolic strategies we need to observe this kind of behavior. This is the oxy shift and I stress here that we are completely neglecting the particular molecular mechanism of the of the species metabolism, but simply we are, for example, putting a strong preference on one of the two resources. Okay, with the parameter VI, but this is not only qualitative. Okay, so actually they are all together with with Andrea they performed an experiment so an experiment where they have a series easy that eats block block block Galactose and then as a waste produces ethanol and and in once a Galactose is depleted the series easy eats the so feed on the wasted ethanol. Okay, so this is a firstly grows on Galactose then it grows on ethanol and you can see these these are eight replicas of the of the experiments in the growth so you can actually see very well this this the oxy shift. Okay, now if we try to describe it to this behavior with the with the model with the consumer resource model. So this is the best fit using Monte Carlo chain methods of the model with adaptive on the left on the left and with the fixed metabolic strategies. Of course, here we have to constrained so we can measure some parameter of the model independently from the, and we know from the biology that we have constrained on the parameters, let's say, and given such constraints we perform this this best fit of the model. And you can see that as expected that the consumer resource model without metabolic adaptation follow a simple and let's say with the one slope growth curve while with that strategy you really can see that we can quantitatively describe the experimental data. Okay, now, let me make a link. Interesting link is a suggestion rather than a proof with the metabolic theory of ecology. Okay, so the metabolic theory of ecology is a fascinating topic I think that almost told you something about that. You can. I mean, one of the most celebrated equation describing this metabolic theory of ecology is the so called the clever law that describe the relation between the mass of species and their metabolic rate to be okay, and indeed over several magnitudes, you have that such relation is a power law with exponent three three or four. Now, is discussed about this exponent. Of course, this is just average is a relationship so these are average mass and average metabolic rate, but this is quite a strong evidence in many in many different of the existence of such metabolic rate that is the fundamental to rate that govern many pattern in biology. So, if we assume that clever laws holds, then in our equation, our in our, in our physics of the physics of the model, we have a different rate, one given for the that rate another one given by the adaptation velocity and other given by the rate of metabolic production where all these rates will depend on the rider law and so finally will depend on the on the biomass on the mass of the of the of the species. Indeed, it's easy. I mean, we can be shown that if we assume that the metabolic theory of ecology holds, then we have that both the total energy budget and that rate scale as the biomass to the minus one over fourth. Okay. Now, in this condition, what we have as a consequence is that the ratio between the energy budget and that rate is species independent. Okay. And also, we have that that rate. I mean, the adaptation velocity can be written as a function of the debt rate. Okay, so this leads everything to one only one characteristic type of scale. Okay, so this is not mandatory. Okay. We can relax this hypothesis and if you want, I can in the question can ask me what happened if we relax this hypothesis. But now we are assuming this hypothesis so we are considering in the following that the ratio between the total energy budget and the rate rate is species independent. Okay. So, again, now we can write for our model a condition for existence. And in fact, we have that in this case, all species survive. Again, so these are just different little mathematical details. But the final point is that again the supply rate, the scale supply rate must be inside the convex hull of the metabolic strategies. Okay, so different derivation, but the same result. Okay, so now the point is that now the alpha depends on time. Okay, so here I forgot to explicitly make a time dependence here of the, of the, of the, so let me do it just to stress this point. So basically here, this alpha, okay, here the alpha are courageous. I can put the pants on time now. Okay. And, okay, so now, if so now this is the initial condition for example. Okay, so after a scaling, we set the initial condition alpha equal to zero and the supply rate is outside the convex hull. Okay, so we have four species, three resources, same condition as before, and the supply rate is outside the convex hull. So the question is, what about species coexistence? Okay. So, I remember to you that in the case of fixed static metabolic strategies, then in this case, if the convex hull, if the supply rate is outside the convex hull, then you can, as you can see, we have an initial of many species and only two in this case survive and CHAP is recovered. Let's see what happened now if we allow this maximization of growth rate constrained by total energy budget in our equation. Okay, so this is what happened. It happened that there is a dynamics of the metabolic strategies along this simplex and finally you have that all metabolic strategies are self-organized in a way that the supply rate now is in a stationary condition is inside the convex hull. And in fact, as you can see, all the species survive. As you can see, the wild species is the closest one, is the most abundant one, but it's not trivial because for example, you can see that this is not, I mean, the orange one is not the closest. I mean, it's not trivial to find based on the position of the metabolic strategies, the abundance of the species is an open problem, as far as I know, but we can see that the strategy is self-organized and coexistence is allowed. So we can now look at the other experiment. For example, you can think about perturbing the environment so that supply rate now is the star for a given time. And then you turn it off and you turn it on a new supply rate, like having different kind of source of resources from the environment and you change it. Okay. And in the case of fixed metabolic strategies, of course, this leads to a stress of the population dynamics that you see they start oscillates larger and larger until most of them will reach extinction and only a few of them. Actually, if you wait long enough, maybe none of them will survive. Okay, so let's see what happened the same condition for adaptive strategies. Okay, so what do you see is that adaptive strategies increase the community resilience and so stabilize the population dynamics of our microbial community that is able somehow to follow to adapt to this external environment. Okay. So we have done a lot of different tests of, for example, what happened in the presence of resources that are heavily degraded or what happened in the presence of very inefficient resources. And all the time you see that they just implementing this optimization principle with constraints, the community self-organized to have the best response to the kind of perturbation you implement to the community. So this was very cool. Okay, so finally, one can say, okay, but so here we have a new paradox. Everybody always survive. Okay. Well, the answer is, this is not true in the sense that depends on the velocity of the adaptation and the velocity of the perturbation. It is to say that, so here we are, we plotted the rank abundance curve. So this is the log of the stationary abundance of a community of 20 species and three resources for different adaptation velocity and also for no adaptation at all. So for no adaptation at all, of course you recover check. Okay, so you only have three species survive. But you see you do not on a little, the, a little adaptation, then more species survive, but not all the species survive only six. If you never increase the adaptation around the 13 species of life, and then if you increase again, the adaptation velocity then all the species of life. So the adaptation velocity is a fundamental control parameter in controlling the observed biodiversity of the system. Okay, so let's, let's first make up first part of my conclusion of my, of this presentation. We have introduced adaptive metabolic strategies that maximize species growth rate. And we have, we have served that this allows to describe the oxy shift growth pattern. Moreover, that this adaptation drive self organizational species toward coexistent pattern. And also that metabolic adaptation increased stability of the ecology community against environmental perturbation. Okay. Finally, we have seen that adaptation velocity is the parameter controlling the actual number of species that will coexist at stationality. Okay, so this is the first, the first conclusion that I hope you have a question about this first part maybe we can take it now. Yeah, there's a question in the chat. Can you read it? Yes, how would one design experiments to test the adaptive strategies prediction. Okay, so, okay. Our first test was that we have done was to perform the experiment about try to describe the oxy shift. Okay, so in this case, I mean, this is maybe, I mean, it's not a prediction that we know that we observed the oxy shift but if you wish you can actually see one experiment to test the strategy about adaptive strategy prediction would be to be able to cultivate microbial community with species, maybe engineered species characterized by different adaptation velocity or you actually fall some inefficiency in some species. And actually, what you, what you would expect is that being less efficient in adaptation and somehow measuring this velocity adaptation, you can actually test that which species survive which not we have done a kind of similar experiment and I will present in the second part. Of course, I mean, you may think I'm an experiment that's what we are doing right now so not an experiment is I don't want to speak out of of rigor but yes, we can actually do. I think many experiments to test this prediction. So second question was, would you say that this model solve the plant on paradox. Okay, same definitive words about the science is always, I mean, to demanding so I will not say that sort of the paradox of the plant on I would say that suggest strongly suggest if you wish that I mean quantitatively suggest that adaptation is a very important mechanism for species at least the microbial species to exist. In this regard, I would say yes. We what what I think we have understood that dynamic, I mean, having adaptation is a fundamental ingredient for having a high biodiversity even in the presence of your resources. Okay, if it's not the only explanation is not probably the only contribution to the solution of the paradox but for sure is one, one, one part one important part of that. Okay, so it's not I go to the second part, I hope to be able to be on time. I don't see a question. I didn't see. Can you read that I don't see the chat it disappeared. Have you checked whether these results are robust to noise. In, in, in, in, where so yes, we have checked that this result must know it depends what you what you put in the noise. So I will, at the end, maybe remind me this question will show you that if you perturbed the condition. So if you put the noise in the ratio between the energy budget and the debt rate so if you allow the ratio of the energy budget and the debt rate to not to be species dependent. So in this case, it's not fixed to a constant but may vary like also, I mean, like you, you, you consider a variance. Then in this case, this is in the long time to competition exclusion. So as soon as that condition is is not observed, if you look if you wait long enough, you will obtain competition exclusion but this will occur in time scale of the order of 10 to the eight to the nine depend on the variance. Okay, so this is an important point. The effect of the noise is like when you when you go from deterministic to to you add a noise to I mean you have a solving the state of course in the deterministic case maybe you are stable. In the, in the stochastic case, if you have that you pass the barrier but maybe in exponential time. Okay, so in this sense it's robust to noise. Of course we have you have a quantitatively different result. Are there any other questions. What happens if some resources are not substitutable. What does it mean. I would love to answer but I have to ask. He can and ask. Yeah, my question. I was thinking like okay we heard the other day from James so do I have no resources non substitutable resources. It means that like one species for example plankton it needs nitrogen so the adaptation of its strategy and it can adapt strategies but still this resources is needed. You cannot stop using nitrogen. So, could you, like, did you think what will happen to the model if this happens. Well, I think that this is in the model would means that I mean this is not to incorporate in the model so I don't know, in the sense that from one side you can think that there is some metabolic strategies from alpha sigma I so if I is nitrogen this must always be greater than zero. So you put constraint on the entries of this metabolic matrix, but how to put this not substitute your resources in the growth rate that this is not three I mean this is not I didn't think about okay so this is not really. So you can constrain the model to use, if you wish some resources, but I don't know. This model this moment, there is no, I mean preferential use of one of resources so there is no this possibility to constrain the use of one resources. Can I also question here. Yes, or. Oh, sure. Thank you. So my question is about the assumption of the grocery to maximization. I mean, think about a community with, you know, consist of the two species that has obligate interdependence, like, you know, amino acid off stroke. Because in that case that your model needs to be, you know, changed or, you know, added a term to account for the cases, you know, the mutualism and and not not the species want to optimize its own but the whole community. This is not. This is a good point. I mean, this is really something that we are doing now. We are not considering any kind of cooperation. And so, any kind of, I don't know, for example, cross feeding type of effect for sure this will have an important effect also on on on the possibility to coexist. As I said before, that is not the only mechanism. Okay, this is not that I mean with that we can explain everything. It's a this is right now it's a it's just adaptation in a company in a poor competitive communities, but for sure in the same framework actually you could and we are doing we will adding also cross feeding and other cooperative mechanism. So I'm just reading the there is a consider now two questions about physiological adaptation is physiological genetic changes or. So let me just go to the second part because I think that this is a lightning on this second part of the question in fact my second aspect is bridging try to bridge we try to bridge the cellular and ecological scale. So, in fact, there are evidence that the abundance of microbial species is strongly correlated with the metabolic function, so that you can actually predict community composition by assembling microbial species in metabolic blocks that are specializing in the metabolic function. And we have already seen that the metabolic adaptation is very important in determining the evolution of the population so. What we want to understand that is the function is of course the function reform by a species depends on the protein is producing. In the function that is the metabolic trade off depends on how the proteome of species is allocated. So now we would like to try to understand what's the origin of the metabolic budget and of the adaptation of the fact that these strategies the evolving time. So, how does a location of the proteome affect the dynamics of my property community so we take a step in a in a in a smaller scale and we look at so we consider a species and we consider the complexity of its proteome. That is the all the proteins that can be produced by the species. And it's known that the total proteome can be divided into three, let's say, three important functions or regions. So they that are denoted by q p and r so all the proteome dedicated to housekeeping function such as transcription factors are the feet are the feet q so feet q is the fraction of the proteome allocated for housekeeping function. And typically this is a hard cord so this cannot be so it's fixed. This cannot change in time because these are the minimal condition for for for the for bacteria to work. Then we have two other region to other families. One is the proteome allocated for nutrient uptake and the metabolism that is the one related to our case. We call this with by 5 p. And then there is the allocation of proteome for biomass synthesis that are ribosomes. Now 5 p and 5 r, as I will show, can vary but aren't trading a trade off. Okay, and this important relation between these different protein allocation and the growth rate. So this interdependence of cell growth and gene expression is a very similar work of similar work important work of the group of tennis one that was published in science in 2010. So in this work, they found the phenomenological slow. Okay, that describe the relation between the problem allocation, for example, here of the R sector and the growth rate of the species G. Okay, so in this case, you can see that so these are data from the experimenter in the Y axis. This is basically a proxy for this protein allocation for for the for the sector are five zero you see is the basically hardcore. So this is the mean you cannot devote less than five zero because these are needed to leave. Okay, so it's a compressible part of the fire. And then you can see that increasing the amount of of protein dedicated to the sector you increase the growth rate. Okay, so it's a linear relation. Okay, Katie is basically a measure of translational capacity. So how fast the microbial species express its genome, while raw is just a conversion factor. Okay, so this is not the details. On the other way, he also for the protein allocated in the peer fraction. So the one of the Metabola for the metabolites. Also, this is in a linear relation with the with the growth rate. Okay, so these are the two phenomenological law that they observed. It can be synthesized in this way. So we have that the fraction of protein dedicated to the mutual uptake is proportional to the growth rate. The similarities for the our sectors that is proportional to the growth rate while the housekeeping function is a compressible constant. And we have this condition, of course, the fraction, the total fraction should be one. Okay. So what we have done is to generalize first the this phenomenological laws for and res and our resources and MP species. Okay, so now we have another sources MP species. What we consider is that the P sector is subdivided. Okay. For the different resources. So ph sigma P one is the metabolites is the proteome allocated to uptake resource of type one ph sigma to P is the protein allocated to uptake resource to and so on. Okay. Okay, so in this case we just generalize the basically the same definition but considering the growth rate contribution due to the resource I this is this g sigma I. So, because we are focused on the on the on the protein dedicated to the P sector, we will for simplicity just the node despite fight so when I there is no separate script superscreen this is of the P sector. Okay, so an important assumption that we are doing is that the sum of the different contribution of the growth rate for the different resources is basically this is the sum gives the total growth rate. Okay, so the total growth rate is simply the sum of the growth rate contribution for each single resources. And if you do this, and you put this condition in this constraint, the normalization constraint you go you obtain this condition for the for the proteome allocation for species sigma to all the resources I. Okay, here. So you can see that this summer this is constrained by this capital fee. Okay, so this capital fee is the total proteome that can be allocated for the P sector basically for growth for me for me for the source of things. Okay, so this is just come from this water logical law. So now the model is quite easy to generalize in this sense because we have that. Okay, for each resources, we can have a problem allocation of using my one that will be proportional to the uptake rate of the species. And again, in turn this uptake rate that will contribute to the growth rate of species one following the, the, the rows that I just shown. Okay, and the total growth rate will be just the sum of the different uptake rate for all the different resources. And then we have a maintenance cost that is called the IQ that is similar to the death rate. Okay, so the new equation are simply this couple of the question here. So it's the same as before, but now g sigma directly depends on the proton allocation of the species. So this is explicitly all the, all the equation. So, so now we go a little bit quick because I have not much time, but I want to stress that this is really similar structure of the equation that I before presented the change this now we have actually a more microscopic understanding of the different parameters. Okay, because we are working at the, at the cellular scale, and also the new, the, there is the new constraint that the totally budget allocation now takes just the form of this product allocation constraint. Okay. And very interestingly, you see that this product allocation for the P sector is fixed, but this is equal to this last part. This is the third equation. And actually you can see that because our changing time also fight must be valuable. So you see that the product allocation for growth. So for, for this metabolic strategies must be dependent on time. So we don't have to suppose it we just came out from this cellular description of the product allocation. At this point, this is just should be so if the, this fight must be valuable, then we need to write some dynamic, dynamic equation for the fight that we correspond to the metabolic dynamic metabolic strategies. And again, we have an optimization of the growth rate, but with these new constraints. Okay, so the mathematics is the same as before. So I'm not going to, to much detail, but you just need to optimize it through the gradient by imposing the constraint that is given by this equation. And then you end up with this final equation here that is the same as before for the alpha. Okay, but now the file are the problem allocated for the P sector. Again, you can study the result of condition you can look at the stationary solution of the system. Okay, these are the stationary solution of the system. From this you find that one solution is given by this R star. Again, you can call the ratio that ratio they are with the capital theta. And then you have that years, you can write R star in this way. And then you can actually the constraint the constraint the biological constraint now. The constraint that has a biological meaning is that you can see that maintenance cost is proportional to the protein allocation to the P sector. And in this sense, if he is facing my crisis, we have to spend more energy to synthesize catalytic catalytic and ribosomal protein protein, and therefore the maintenance cost increases. Okay, so this makes sense. And this is just a condition. So this condition must be fulfilled for the equation for the stationary solution to make sense. So if this is not fulfilled, you don't have a stationary solution out there. This is not enough for having coexistence for adding coexistence again you do the similar as before you rescaling MS and file, and you obtain again that species coexist if the supply vector is within the context. Okay. And now five is the problem at the stationarity five star is the problem allocation of stationarity of the species I to resource to the system to resource. So again, now you can see now you have the student hope. So we start with our given initial problem allocation. In the first case, you if the supply vector is outside the convex are you would have extinction, but now protein allocation change so that the new metabolic strategies are the supply vector is within the complex are and you find, and you find coexistence. So this is what I was mentioning that Leonardo used to generate strains of the colleague and completed demon one common resources. And one of the strain was in general so to, we can change the problem allocation experimentally. And we can evaluate the outcome of competition as we do so and in fact we predict the outcome of the competition in this case. Okay, so we can really do experiment engineering the product allocation of species and try to predict the outcome of the competition. And again, you can see that the velocity of adaptation is again a fundamental parameter if they, as you can see here. This is the condition where the velocity where the rotation is high so the species can adapt and coexist. In this case, adaptation velocity is, is low. And in fact, you can have in this case you you you don't observe coexistence. Okay, so in this case you can see if ipsion is, if ipsion is large enough, you start to have extinction that is to say, if adaptation is low enough, you start to observe adaptation. So depending on on the rotation velocity again we have we have a coexistence or not. I wanted to a lighter and I'm very few slide and I concluded. Because this I, there was a previous talk was mentioning this so depending on the amount of supply rate even if adaptation is low. If the supply rate is very strong, then we can have a species to coexist. So if we have a strong influence of resources, then you have that in this case, more species than resources can coexist. For example, here you can see that if the supply rate is smaller, the adaptation is not enough and so only two species coexist. But we have always three resources only the yellow and the blue species coexist. But now if we increase two times the, the, the supply rate, yet only two species coexistationality but the extinction take more much more time. And then if we instead increase the fire time the supply rate, then you see that all the species now start to exist. Okay. And also this the amount that the magnitude of the supply rate is an important control parameter for coexistence in this case. So the conclusion of this part of the whole talk is that including problem including problem allocation models of competitors communities give us new insight on the on the dynamics of bacterial communities and the relationship between a problem and dynamics seems relevant to understand the origin of the high level of biodiversity. So here we have like a microscopic, let's say, explanation or not explanation, microscopic ingredients, microscopic process that have an impact on the on the on the coexisting pattern of the community. So it's just a starting point, but it's a starting point that bridge fire physiology of micron to a community ecology, so we are very excited about this. And because we think that this is really an opportunity because basically all the questions that probably will will ask now are open questions because this was just really done in the few in the last few months. And but this consumer problem resource model suggests that coexistence could be reached by self organization adjustment of cellular properties of this of the species itself. Okay, and with that, with that I take for the remaining time questions, and I want to thanks of course all the collaboration again especially I really want to thank Leonardo for his great work. And these are the two references of what I have presented the one is already published in plus computational biology, and one was just accepted like a week ago in the ISME journal so we'll be available soon in the ISME but you can find the work in the bio archive. So thank you very much. Thank you so much. So we can start accepting questions. There's one in the chat if you want to. Yes, so now I can look at the chapter. Okay, can adaptation velocity be interpreted as the lag phase to change the metabolic machinery this is a very, very interesting question and actually, this is one of the key missing aspect. So because we know I mean, not an expert of this but we have a whole machinery to model metabolic fluxes. So the metabolic networks to metabolic networks so would be very interesting but yes, I mean I don't have a quantitative answer to this but for sure qualitatively adaptation velocity must depend on on on on the lag phase to change the metabolic machinery. And sorry, can I add something so I made this question I'm asking hi. So do you think that this could originate trade offs like so let's say that one species is faster and another species is lower. And this could let's say, this is the outcome of the final composition because of these differences in the velocity of adaptation. Absolutely yes this is the understanding this is the overall understanding so all the time that we observe the coexistence basically we see that there are trade off in place. And let's say, attended we are doing in order to understand this trade off within that we need to go on the cellular scale, for example the problem allocation but also for example this is a very good path to look at different metabolic machinery and understanding this way the margin trade off between species and and so to understand that basically that this trade off margin from the from the cellular and the metabolic scale will have a strong impact on the species. This is exactly the point. I have a related to the question. So, I mean related to to the adaptation velocity. I mean, is there a way that we can quantify the, you know, adaptation velocity by comparing that with the data. But I think that's important because, you know, by quantifying the velocity we know, you know, if it's under out of the magnitude of the days or weeks or, you know, then we can know if that adaptation in the met model actually reflect the physiological change without the mutations or on the, you know, more longer evolutionary, you know, time scale with a lot of mutations. I mean, you do have any thoughts on that. Yes, I can offer a total. Again, I'm really, I'm really not the guy going to the lab so I might say so I don't know if Leonardo is connected want to correct me or suggest something but my my I think that my understanding is a way to test adaptation velocity can actually be done through the oxy shift course. Okay, so if for example, you take different species you can you cultivate independently into into different medium but with two different resources, and you can actually have an idea of the velocity of adaptation for for each of the species so I think that basically this was mean this is the main suggestion here so to have a model which parameter can actually be measuring the lab so in this in this way you can actually perform first experiment with only single species and then try to see when you put together if you want to put together you have expectation of what is going to happen as a test that may be basically the mechanism that you are thinking important. Okay, so that's what we have done in the experiment with with with Leonardo where basically we have we have cultivated. Yes, so we have basically cultivated two strains and one of the strains was producing basically useless protein was a lot of protein in not useful for for the resources for the medium which the species growth and in fact we saw that this leads to a competitive advantage in a quantitative way, as expected by the model. Yeah, yeah, thank you. Thank you for me. Thank you. Okay. I don't think I see any other no version. So if that's the case, thank you again Samir. Thank you. Very interesting research talk. And then we will continue. Thank you. Thank you. Bye bye. We will have the next lecture in about 10 minutes so everybody can take a break. Bye.