 So first, happy birthday, Thibault. So for me, it feels a bit like time travel into the past to be here, and I really remember my time here as being very formative. So I think of IHS as a luxury place for research, where the clock seemed to be stopped to leave the time to think. But it was also a pretty stressful time for me to be surrounded by so many bright minds. And I still remember our discussion at lunches, where you would ask me about my progress and the question that you had asked me at the previous lunch. And so then there was a discussion at the table where there were always pen and paper. And I could even find back traces of those discussions. And I must say it was not always trivial to converse those notes and those drawings into a mathematical equation. Anyway, so you know that since then I have changed research field. But in my mind, I didn't really change path because instead of addressing fundamental question about space like singularities and the origin of the universe, I am addressing no question about living systems. And I'm particularly interested by embryonic development and the role of the geometric arrangement of the cells in the embryo to code for the body plan. But today I will tell you about another research line, which is about microbes. So I think it's pretty impressive to think that in each of our bodies, so inside and outside of each of our bodies, there are about 100 times more microbes than there are stars in the Milky Way. In particular, the gut microbial communities are one of the most densely populated ecosystem known, which makes them very complex systems. So why? So microbial communities are important, in particular human-associated microbial communities, because they have been associated with, so yeah, I wanted also to give another number. So they are important basically because they are very numerous, so that's an indication. And it has been estimated that for each human cell in the body, there is one microbe. So we are literally working microbial communities. And also, from a more pragmatic point of view, it has been shown that for basically any kind of disease you can think of, from cancer to obesity, to even depression, an association between a unhealthy gut microbiota and those diseases. So there is still many puzzles to understand the causal relationship, but the association has been already observed. So from a theory, so there is, yeah, of using the long run, we hope to have personalized medicine based on our specific microbial communities, because it varies among people. From a perspective, a terrorist perspective, the long run goal in the field is to build predictive dynamical models of those microbial communities. So the type of question we want to address with these models are the following. So we want to understand how the community can be stable. So it's a big question to understand how so many different microbes interacting can form a stable system. But I won't be speaking about that today. We want to understand community structure, and I will come back to this later. But the basic idea is that we observe in across microbial communities that the composition is really dominated by rare species. So we have a few abandoned species, but the majority of the community is many different rare species. So why don't we could think naively that they would die and they would be killed by the strongest guys, but it's not the case. We want to understand the region of anterotype so very briefly, so plot doesn't show really well, but the idea is that if we represent in 2D the community composition, so if you choose wisely, you don't choose bacteria one, bacteria two, but you choose wisely a combination of those bacteria to represent this in the 2D in the plane. So each dot is a different individual, and we see that there are clusters occurring, saying that basically people belong to show different types of bacterial composition in the gut. And if you analyze this a bit in more detail, you see that these clusters correspond to different families of bacteria who are dominating the community. So then we want eventually to gain control, so in case you have, yeah, I don't want to go to the detail of the plot, but basically if you have a unhealthy composition, how should we perturb the system to go back to our healthy states? So before building models, basically we want to look at experimental data. So how do we do this? And for human gut microbiota, the way to go and was done by Lawrence David here, with whom I actually shared an office at some point, but I was not working on that, unfortunately at that time, but what he did for a year, he collected these tools for one year every day, and he sequenced the tool to have as a probe for the community composition of the gut flora. So the result is as follows. So for visualization purposes, I represented only five species at random, but he could distinguish about 100 different species across that year. And what we see basically is a stable state with fluctuation. So the fact that there are these straight lines is just sampling artifacts, and the fact that it's a log scale, so it's not very important, but we see basically fluctuation around the steady state. And so we did a literature search, and we could find only about 12 different time series of microbial communities that were taken for such a long time. But the communities were coming from different, they are very different types of microbial communities, so we have marine plankton, but we have also microbial communities coming from different body sites, and each time it looks like a steady state and with fluctuation around. Something that is also, so that's the time, the temporal dynamics. But we can also look at a snapshot, so it's not very visible, but basically what we plot, the second line is the abundance versus the rank. So the rank we just take at one time point, we look what's the most abundant bacteria, that's rank one, second most abundant rank two, et cetera. And we always see the same shape of this curve, and this is, so it has been rescaled here, so I'm sorry for that, but if we had put relative abundance, we could see on this curve that the vast majority of the bacteria account for less than 1% of the total composition. So this is called heavy-tailed rank abundance distribution, and it's not really understood why we observe this. So now in the theoretical ecology, people use Lotka-Volterra models to build dynamical models of these microbial communities. So I will give you a crash course on Lotka-Volterra models, but in view of the audience, I think I can be very fast. So the idea is that you have, it's like in an ecosystem, you have different species, so here we have rabbit and sheep, and these animals, they compete for the same resource, which is grass here, and we can transform that into a question. So very quickly, so we have R is the population of rabbits, so the rate of change of these rabbits. The first term is a growth term, so if we had only the growth term, we would get exponential growth, which is not realistic. Then you have the self-interaction, so the R-square term, and this is representing the fact that at some point the resources will be limited, so you will go to a steady state, and then you can incorporate the effect of the competition for the grass, that's also a negative term, which is coming from the sheep, and then you have the same equation for the sheep, but obviously the growth rate of the rabbits is higher because they reproduce like rabbits, and the interaction is more important on the rabbits from the sheep than reversely, because they are stronger. So this leads to nonlinear equations, and typically we will need to rely on numerical simulation because we cannot solve that analytically. Just a tiny specification, so if you have just one species, this would be modeled with the logistic equation. And now we go to a community with 100 species, so we generalize this type of equation, and we can also include, maybe I will use this, so we can include an immigration term, then we have the growth term, pairwise interaction between all the species, and because this is obviously influenced by the environment, we have extrinsic noise, so this term here is actually a noise term in the growth rate because it's linear in the growth rate, then we have here a noise term for the immigration process, and this represents the intrinsic noise which is due to the discrete nature of the microbes and the fact that the processes of birth and death are stochastic. So this is our basis to model microbial communities, and the question we address was, okay, everybody uses that, we have experimental data from microbial communities, can this generalized logcavolta model describe the experimental time series that I've shown you? And so to answer that question, we need to be more specific, so we need to characterize the noisy dynamics. So basically we have, as I told you, fluctuation around the steady state, and to characterize this, we did four different, we used four different characteristics, so I will go through them one by one. So the first one we consider, so we can, for each, that's supposed to be one species within a community, and we consider the size of the jump between two time points, and for each species, we computed the mean value of this jump, the absolute mean value of the jump, and we reported that as a function of the mean abundance of the species in a log-log plot, and what we see, so each point here corresponds to one species in the community, and we see that this fits well with a linear, so the exponent is almost one, and we see that in all our time series for all the communities. So this means that the fact that the dominant source of noise is the linear noise, which means that what is dominant in the more noisy thing in the system is the growth rate, and it makes sense we eat three times a day that this growth term is very noisy. And then we reported also, we computed the ratio between two successive time points so that we can capture the fact that these go up or down, and we made an Instagram of the ratio between two successive time points, so the mean of this distribution will always be one because we are around steady state, and to characterize the strength of the noise, we computed the variance of the distribution. So this ratio of the jump, they would fit well with a log-normal distribution, and we observed that the width of the distribution is around one, which means basically that you do big jumps, essentially. So from these two characteristics, we concluded that we need to use, the linear noise is dominant, and this noise should be large, which is not surprising, but then the next characteristic we looked at is we wanted to assess how much temporal structure there was in the time series, and for that we computed the power spectral density for each species, and we used the well-known fact that, okay, if you compute, so the temporal structure can be encoded by the autocorrelation, and we know that the flatter the autocorrelation is at the origin, the more temporal structure there is in your time series, and this correlates with the fact that you get a steeper power spectrum. So we can associate, and it's kind of a usage in the community to associate the slope of the power spectrum to associate a color. So if you have a white noise, you will have the flat power spectrum. If you have a pink noise, you will have a spectrum in one over S, so the slope is minus one. Brownian dynamics, you will have a slope of minus two, and we associate, this is really bad choice because the white noise is blue, the pink noise is pink, but the brown noise is green, so we use a scale where it's white, pink, brown, and the darker, the more temporal structure there is in the community. So for each community, so one dot is one species, and we see that like the slope, so the color of the noise for each species, when we plot that against the mean abundance of the species, we see that there is no correlation. So this might seem like not that interesting, why plotting that as a function of the mean abundance if we don't see any structure, but the thing is that if we know the in-silico time series with the GLV model, we see structure. So we see that the slope of the power spectral density correlates with the mean abundance, but not exactly, so it actually correlates with the mean abundance times the self-interaction, but self-interaction is something we don't have access to in the experimental data, we have access to the mean value of the species, but not the self-interaction. And this is actually due to the fact that there is a tradition in the field to always set the self-interaction to minus one. So if we put the self-interaction to minus one, we see that there is a destructor and it's not disturbed by the interaction between the slope and the mean abundance, basically we cannot explain what we have seen on the previous slide. So what we propose is that we need to use self-interaction which varies over the order of magnitude in the load-cavalta model to be able to reproduce the experimental characteristics. Okay, and the last characteristic we used is also related to a broad debate in theoretical ecology, and the debate is a community neutral, in a neutral, I'm not sure how to say, like is, do we have a neutral community or a community which is in the niche regime? So neutral theory says that the driving force in the ecosystem is stochasticity of the birth and death processes, while the niche theory states that the driving factor in setting the community structure is the interaction between the species. And we can test for neutrality of a time series by, I mean, there are several tests that exist and we have used two of them. So one is based on the, yeah, maybe it's a bit technical, so maybe we can save a bit of time if I go quickly through this and I don't go to the detail, but I'm happy to explain if someone is interested after. So we can use something based on the Kublack-Leibler divergence which is basically comparing two probability distributions and another one which is comparing, like which is testing the invariance under grouping of the community. So if you group species, like you say species one and species two, we put, we sum them together, we consider them as one group and species three, four, five, or we, like we change the grouping of the species. Neutral teres says that the result basically should be independent of this grouping because species are not special. There are no, I mean, it's not important the difference between the species. And basically for all, so all the time series, like we looked at that we found in the literature, we, for the two tests, we always obtain that it's in the niche regime. So like light stuff here. It doesn't look that good, so the colors don't go well so it's not that neat, but it's always niche and reddish for the other tests. So it's also a niche. So basically we propose a minimal model which is this logistic equation. So no interaction, no pairwise interaction. So the logistic equation for each species plus a large linear noise. And if we do, I mean we do in silico time series with this model, we can study the properties and they are really comparable with what we see experimentally. Even the niche character which could be think of being a bit surprising because here we don't make any difference. So we put the same growth rate and same self interaction for all species. We don't make difference between the species. But we have, so I'm just a little bit lying because what's something that we have imposed at first here is the rank abundance distribution. So you can, so to determine the steady state you can set this equal to zero so you don't take care of the noise. So you can, so we fix the growth rate in such a way that we observe. So the steady state is given by x i is g i over omega ee and minus sign. But basically we can choose the growth rate to impose this distribution of abundances. So that was a bit cheating. And this is saying that the species are not equal. So it's not surprising that we get niche even if there are no interaction between the species. So to summarize this, all like I put here, each column is different communities and we see always the same typical behavior of these communities. And okay, we, yeah, too much text, sorry. So basically at the observed time scale we can reproduce all these things with a logistic model. But so the message, we don't say that interaction are not important but we say that they are not important at the observed time scale. And we were still puzzled by this. We had to impose this rank abundance distribution which is not very natural. So we wanted to get that as an emergent phenomena. So that was like a follow-up stories. And basically, so for Lotka-Voldera model, it doesn't come up for free. Like this rank abundance, so the dominance of the rare species. But we know from kind of old work that these everyday distribution, they can come from self-organization. And more recently, it some approaches of ecosystem that were based on individual modeling approach. You get this heavy-tailed rank abundant distribution as an emergent phenomena. So we wanted to understand precisely what's important in the individual model approach to get this heavy-tailed distribution. And we realized that it was really the fact that the, so you simulate each microbes so you place the microbes on a grid basically and then you have kind of an automaton. This is this individual modeling approach. And the fact that you get this heavy-tailed comes from the finite size of the grid that you place the microbes on. So we wanted to mimic this finite size of the grid on the GLV model. And this is natural in the sense that we have access only to a limited amount of resources. So this, we added a maximal capacity which is mimicking the finite size of the grid and physically representing the fact that we have a global limit on the resources. So we split the term in our GLV equation. So basically the dead term and the growth terms. And we multiplied all the positive term by this global maximal capacity. So when you reach the maximal number of species possible in the system, all the growth term will be suppressed. And if we now do the simulation of our system like this, we get for free like as an emergent feature the fact that we have heavy-tailed abundance distribution. So basically to obtain, so to summarize, to obtain a predictive dynamical model, the next step is to go to more densely sample time series and to capture the interaction between the species because we know that bacteria, if you do, you grow them two species in the lab, you know they interact through the nutrients. So we know it plays an important role, but so far we don't see it in this experimental data. But the field is like exploding and there are many experiments being done. So I'm confident that we can progress a lot by having access to better time series and also it would be very nice to have access to experimental data about the, to measure the concentration of the nutrients and to get models where the nutrients are also explicitly modeled in the system and to have access to the spatial structure would be even better. And yeah, so there is still a lot of work to be done. So thank you for your attention. Thank you very much. The question is close to you. You mentioned 1972 paper by Bob May at the very beginning. Yes. Right? And that was the one about how, you know, how very can the group of competing species be and still be stable. Yeah, many in 1970. I remember reading this years ago and I've kind of forgotten the conclusion. Roughly speaking, it was that you can't have more species stably surviving than the number of... The number of nutrients, basically, I think. And what has happened to that conclusion in this way of looking at things? Has that disappeared entirely? Does nobody think about that anymore? Yes, yes, many people think about that problem. And so now I cannot reconstruct the story quickly in my mind, but so it has been overcome. I don't remember exactly how, but so the basic conclusion is that I think I have in mind that if you have like three nutrients and three species, I mean, you cannot have, if you have only three nutrients, you cannot have more than three species or something. I don't know. I cannot... What was the conclusion? Yeah, but... It's back to that random matrices. Yeah. Yeah, I think I cannot... But I mean, in the considerations that you were going through here, the question of what these guys are eating, how many different things they were eating, doesn't come in. No, because it's an effective model, where the pairwise interactions are supposed to represent how these species compete for the same resource. So typically, if some microbes they can produce some nutrients for, that can be helpful or the letter use for other species. So it's... Yeah, but I really cannot... In the end, the conclusion that you reached is that you can forget about their interactions, at least at the... At some scale, like at some level, we can model that effectively, but there are also papers showing that these pairwise interactions, they cannot capture all the types of nutrients interaction. So sometimes you need to go to higher order times. I think, so the conclusion of some papers is that you cannot use GLV to describe all complicated type of nutrients interaction that they can be between the species. But I think that if you include a higher order term, not pairwise, but third order or higher terms in the Lord Cavalter equation, this could be possible. But I've never really worked on the complexity, stability, this type of problem. But I know there are many people who did works on that very recent work, but yeah, I couldn't. Okay, I think in view of the time, let's thank...