 Okay. Welcome back everybody for this long session of today, Friday. We are now streaming live on the YouTube channel. It's a pleasure for us to have Ms. Resha Guhtal from the Indian Institute of Science to deliver a set of lectures. And the title of today's lecture is Stop Assisting Best Stability in Ecological Systems Part 1. So as always, you can raise your hand and you can type your questions in the chat. And Ms. Resha and myself will keep an eye on it and then depending on how we especially want to deal with that, you can answer immediately or at the end of the lecture. Okay. So with that, I leave you the floor. Looking forward to hear from you. Thank you. Thank you very much. And it's a great pleasure and honor to be delivering these talks at an ICTP meeting. I was very much hoping to visit ICTP, but unfortunately, as you all know, everything is virtual these days. The last time it was only one time I did really visit ICTP. That was when I was an MSc student and it was in 2002 in the summer. So hopefully, maybe again in the near future, I will do a real visit to ICTP. Okay. So welcome everyone. I'm going to share my slides now. So I would like to keep it interactive. So if you have a question, type it out. If it's something absolutely not clear, just I think you should just unmute and also speak. And that makes it easy for me as well. So I have some plan to cover topics, but I don't have to cover everything, as I just want to make sure everybody's with me. So the broad goal of the three talks that I have planned, these two sort of introduce you all to by stability in ecological systems and the role of stochasticity. How do we apply these models in the context of real data as well? So it's going to be a combination of theory and how do we look at real systems in the context of these theories that are sort of deeply inspired by non-linear dynamical literature, non-linear dynamical systems literature. Okay. So the three part of my talks are sort of planned this way. So in the first part, which is today, I'm going to talk about what we call mean field approach. So I would imagine that you would have heard the word mean field in some talks throughout these last two weeks of fascinating talks. So I'm going to look at a mean field approach in this first hour today. And how do we understand ecological systems, those with by stability using mean field approaches? Okay. So one specific question are the aim for today would be can we predict? So whenever we have this by stability, we have something called tipping points. Okay. Now the question, the main question or the main motivation for today's sort of discussion would be, how can we use these mean field approaches to predict or anticipate tipping points in ecological systems? So the second part, which would be I think next Tuesday, it will be on the same question, but looking at spatial dynamics. Okay. The real liquid ecosystems are spatially spread, spatially extended. What can we do something better than what we do today? Okay. And the part three would be something more general, more interesting, equally interesting, hopefully. So but you know, part three, how much of that I will cover will depend on how much I'm able to accomplish my goals for the part one and part two. Okay. So the main, so let me, this is the outline for today's talk. So the main motivation to study by stability comes from the idea of tipping points and abrupt transitions. So I will provide that motivation why we must be interested in by stability in ecological systems. I will then move on to describe simple mathematical theories of by stability and tipping points. And then I will introduce some mathematical techniques intuitively, you know, there will not be too many calculations and so on. And then how can we use these mathematical theories and intuition that we build to predict or anticipate tipping points in real ecological systems? And finally, the last part of my talk would be discussion on and the demonstration of empirical evidence for these mathematical predictions. Okay. So that's the broad goal. And I understand that, you know, Carla Stavor has already covered some aspects of some examples of, you know, what I'm going to present today. So hope there might be some repetition, some overlap. So please bear me with that if it is there is some overlap. But to make my talk self-contained, I have assumed that you may not have heard her talk or you may not have you may have forgotten some of the details from her talk. Okay. So here is an example of, you know, I'm going to give some examples, empirically documented examples of abrupt transitions in ecological systems. So this is an example of a large scale abrupt change, large scale desertification. So what you are seeing here is the northern Africa. If you look at various proxies for vegetation in this huge landscape, okay, let me see, let me switch on the pointer. Okay. If you look at this large landscape, if you look at proxies for this large landscape over the last 10,000 years, you find that preceding 5,000 years before present day, the vegetation sediments were dramatically different from what you see today. So this current phase, current state of northern Africa, which is the desert, was not always a desert. In fact, the sediments indicate that they were in an entirely different state of vegetation. In fact, it had a pretty decent level of vegetation in the landscape for several thousands of years before it suddenly tipped and became the current state. Here is another example of the opposite type and this is on a much more local scale. So what you're seeing here is not a continental scale, but really a scale of half a kilometer by half a kilometer area in China where over the last 60 to 65 years, the 60 year in this graph represents 2010 also, the grass cover in this area has changed. It used to be a low grass cover area and then it has now settled down to a larger grass cover area. So this was an example of a restoration of grassland ecosystem and it stayed in this low state for several decades before switching and becoming a moderately covered grassland state in the present days. Here is a third example of lake electrification. Data is obviously noisy from many ecological systems, but indicates some very interesting features. For the same amount, same values of drivers, you can have different values of the state variable. The state variable here is the some measure of fraction of the lake surface covered by vegetation. So what you see here is that you know what physicists would call hysteresis. When the lake underwent, when the lake had a very high level of phosphorus concentration, so that's those are the red dots in this right extreme and a very low amount of the vegetation. As this forest values came down, it continued to remain in that state of low vegetation. However, it increased and a reverse transition however happened across a different route entirely, different direction. This is called hysteresis in the using the language of that is also used in physics and magnetic systems. Here is another example, classically cited as that of you know abrupt transition, which is that of stock market crashes. So where the stock markets often are in, even when they are in a very state of boom, there could be sudden crashes in the indices. That is remarkable over a very, very short period of time. So sort of the, if I sort of summarize and look at all these transitions that happened, so they often are abrupt. They are abrupt changes in the state of complex systems and once the change has happened, it's not just abrupt, it actually remains in that new state once the change has happened. It's a persistent change as well. And typically, they seem to have happened for no obvious changes in the driver values. So in every one of the example I gave, no people don't quite know what was the driver that changed dramatically, that could have also caused an abrupt change in the state of ecosystems. So the basic idea is that even for gradual changes in the known drivers, system can respond in an abrupt way. And sometimes these changes are irreversible. For example, if vegetation is lost, if certain species are lost, you can not really recover them back. And even when you can recover, they could still be irreversible on timescales that humans sort of deal with. So in studying this now, these are the empirical phenomena I have just described, we're going to hear a whole bunch of terms that which you may or may not have heard, regime shift. So many of these sudden ecological changes from one ecological state or the other, they're also called regime shifts. They're also genetically called abrupt transitions or tipping point events and more terms like critical transitions. So you will hear some of these terms, I will sort of clarify as and when necessary. Stochastic transitions because they're often driven by large amount of stochasticity in the drivers. And the mathematical concept called bifurcation, which I'm assuming you may be familiar now with many, many of the stocks and hysteresis. So I'm going to use some of these terms and define them more precisely when necessary. So with these examples I showed you, this is the example of this large scale, continental scale, the certification. This is a relatively local scale, the recovery of a grassland and then a stock market crash. So what are the important questions that people in the ecology literature or people more broadly in the complex system literature are interested in? Okay, so one is how do we mathematically model these systems? Okay, so do you build a very detailed process-based model to understand the systems? Yes, that is one approach. Or can we sort of develop fairly simple heuristic mathematical models that only captures essential details? The second question that people have been interested in, can we really have predictions for these kind of transitions? Are there early warning signals before these transitions happen? If there were such warnings, then one can do something to stop these events from happening. For example, in this specific case, imagine hypothetically you are somewhere here and if you had been given data from here to here, likewise, if you were somewhere before this dotted line here, you had somewhere here, let us see if you had been given data, time series of cross cover, could you have anticipated this abrupt transition, likewise in the stock market, which of course has really huge applications? Okay, so that sort of sets motivation for studying these phenomenon and to study them, mathematicians, applied mathematicians, physicists, as well as ecologists, and many of them have applied mathematicians and physicists, turned ecologists. They have been using mathematical theories of bi-stability and tipping points. So I am going to describe those simple models now and see how we can model them and how do we try to do predictions of these models. So here is a very simple model of ecosystem collapse. So here in this model, the ecosystem is represented by a single variable. The ecosystem is really a large interacting system of species. However, one can think of ecosystem or sort of lump all of those into a single quantity called biomass density and one can think of how this biomass density is changing over time or what is the dynamics and the simplest models of these ecosystem dynamics have this concept called carrying capacity and an intrinsic growth rate R and the carrying capacity K. And as long as this intrinsic growth rate is R or is positive, the biomass density V will reach a carrying capacity K over a period of time. This is also called the logistic growth model. Okay, now ecosystems are under constant pressure both internally and externally. One such important one such important process is that of grazing. Grazing could be driven entirely by herbivores within the system. It could also be driven by livestock, that human settlements that are there nearby forests. And usually this is modeled as this sigmoidal rate function. So V square divided by V square plus V zero square, this is represented last due to grazing. This is a non-linear term. So basically this assumes that if the V is low, the grazing rate is small. But if it increases, it increases non-linearly and saturates to a value of C. So what is the, what are the equilibrium values of this, of this simple mathematical model of ecosystem? So equilibrium will be achieved with the logistic growth term will be equal to loss due to grazing. And if you calculate the equilibrium points, what you find for some values of R, K, this is what you will find from the X axis, the driver, which is the grazing rate C here. And Y axis is the steady state or the equilibrium biomass density. So what you find here is, when the grazing rates are low, of course, you will still have, for example, here I have chosen a value of K is equal to 10. So the steady state biomass density will be still close to the original carrying capacity. However, as we increase the grazing rate, that does reduce. But what is really interesting is there is a threshold value of this grazing rate. And once the system reaches this threshold value, the system will collapse into a low, low biomass density state. And now if you do a reversal, it doesn't go back at the same point, but the system will stay in this low biomass density state before undergoing a transition back to a high biomass density states. So to understand these systems, one can think of the simple intuitive picture, which is that of a ball rolling in the landscape. So what is this landscape? Think of X axis on the landscape as, let me see if I can also write. Think of this as the biomass density ecosystem state. And so wherever the ball settles, that becomes the stable state. So in this case, it will settle in this point or in here. So this is a low biomass density state and a high biomass density state. And they both can coexist. And depending on where you start this ball, it can go here or it can go here. So the simple intuitive landscape picture can actually capture how a system can have two stable states. So the one stable state is the low, the deep well here. And the other one is a shallower but a local minima here. So this is an example of what is called a bistability. So you can think of ecosystem stable states as a balling roll in a cup or wherever it settles in this rolling cup or rolling landscape. And then there's also this important concept of basin of attraction. So anywhere, if I drop the ball anywhere to the left of this line, we will have the ball rolling to this side. This is the basin of attraction for this low biomass density state. And this is the basin of attraction for the large biomass density state. And then this leads to the concept of resilience because there is always a possibility that this ball will switch over to other minima. So how resilient is the system is a question that's extremely important from an ecological point of view. So mathematically, how do we define this potential? It turns out that if you have a simple model where x basically is a dynamical variable, you can define x by x a simple integration of this rate function, some x 0 to x. So this function is precisely what I have plotted in the previous graph to obtain the potential landscape. So this potential landscape is not just an intuitive picture. For simple models, one can actually mathematically represent them as well. So I'm going to skip some of those. So no, clearly this landscape picture is quite useful because we can think of minima as corresponding to stable equilibrium, maxima as corresponding to unstable equilibrium. And it captures features like hysteresis, under fact that there is initial condition dependence in these systems. And also that system, many biological systems are irreversible or they take long time to reverse once they are in another state. So all this can be nicely captured from this potential landscape picture. So one can also do the following. Whatever I have been discussing so far is a deterministic picture. So however, what we can do is introduce some stochasticity to these models. So I'm calling them ad hoc stochasticity because what I have done here is I have taken a deterministic mean field model and then I have just added stochasticity, stochastic term to this. Basically here sigma v is a strength of stochasticity and eta v is a random number which is drawn from like our same distribution. And we also assume that the random numbers are uncorrelated over time. So we can incorporate these kind of simple stochasticity. Someone has to be a bit cautious while doing so. For example, you want to ensure that the biomass density is never really negative. So I'm not going to go into those kind of technical details here. So what happens when we introduce the stochasticity is we can now capture much more realistic features of dynamics. For example, here's an example of a vegetation system that is undergoing a collapse from some value of near the carrying capacity to a low value. In all more in this case, really value of zero or close to zero. So this is an example of simulation where a system has undergone a tipping from a moderate or high value of vegetation density to close to zero. On the other hand, you can also have a scenario where the system will sort of fluctuate between two stable states. There is one stable state here, other stable state here and the system can actually fluctuate between the two states depending on the nature of stochasticity. So this is where I want to clarify a few terms here. So in this diagram, this is sometimes called stability diagram. It's also called bifurcation diagram. So in this bifurcation diagram, this point where the green branch is ending or this point where the black branch is ending, those are called bifurcation points or also tipping points and critical points in the context of non-linear dynamical systems. And the transition that happened near this critical point, they are called critical transitions, abrupt transitions and if they have catastrophic consequences, you also call them catastrophic transitions. However, these transitions can also happen when you are not necessarily close to critical points. Imagine you're really, this is a critical point, this 2.6 in the specific case. However, if the system is somewhere near in the middle of this by stable region, so there is this region of by stability from values close to 1.5 to 2.6. If you are in the middle of this region, but I'm going to the stochasticity, the external stochasticity is high, even then a system can fluctuate between these two states and which means that even far from a tipping point, far from critical point, you can actually have abrupt transitions from one state to another state and these are called stochastic transitions and what is interesting is this can actually take you back and forth. It doesn't necessarily take you to one side, it can take you back and forth. So remember these classifications, of course, I will revise these. There are two types of transitions I just mentioned here, one is that of critical transitions or tipping point transitions. The other one is a stochastic transitions. So it's something that happens far from the tipping point or the critical point. So this provides, so whatever I have done so far is some very simple mathematical model which has single variable and it can sort of capture various properties of empirically observed phenomena. So we had in the empirical systems, we had observed abrupt transitions, right? We had observed hysteresis and these two simple empirically observed phenomena, they can be nicely captured with this simple model of vice stability. So that's the point of the my talk so far and then I have also introduced you to this concept of potentials and this was helpful to intuitively understand how do we think of the dynamics. So now can we go for, one purpose of mathematical is not just to reproduce empirically observed features. Of course, we do want to do that, that's the bare minimum but can we do something more? And something more in this context is can you provide, can you predict tipping points? Can you anticipate tipping points? Can you forecast tipping points? That's the question I am now going to address. So for example, in this context of bifurcation diagram, let's imagine there is a real system that exactly follows this bifurcation diagram. However, we want to know, we know that the grazing rates are increasing but I don't know if I am here or if I am here, where am I? Is there some way of knowing where is the current parameter value? Okay, that's the question that we are interested in now. I just want to check if there are any questions at the stage. So I am sort of halfway through my talk today. I have a question. Sure. So in that equation where you go to the equation slides. Sure. Here, what is the functional form of v? You always show a plot v versus c with some bistability and a shift card. So what's the mathematical functional form of v? Yeah. So basically v, I don't have that with me right now but if you set the condition for equilibrium, which would be that, you know, this logistic term becomes equal to grazing term, right? You will have a cubic equation. Oh, okay. Then from that bifurcation analysis, you got not okay. Exactly. You set this and you get a cubic equation to solve. You have, of course, v equal to 0 is 1 equilibrium. Yeah. And then you have a cubic equation to solve. Yeah. Yeah. And those are the roots of the cubic equation. Any other questions? Yeah. And this is the mathematical form you assume but is it also common in real data? That's an excellent question. So these are, you know, sort of, you know, one can think of them as somewhat like toy models. They are inspired by the terms that I have included. They are inspired by ecological processes. But, you know, depending on the ecosystem that I am thinking of, okay, the exact terms can be dramatically different. So the question is how useful are they? Okay. So my answer, so the one way to think about that is to go back to this potential picture. Any, sorry, can we think of another by stable ecological system? Let's say I don't know the equation, right? Yes. But I know that the ecosystem has by stability. As long as I can think of them in this potential well-formed. Okay. No, how do you know that your system has by stability? You are given just a time series with some abundance or land cover data. But I showed you that it does show, right, you know, it shows the abrupt transitions, it shows hysteresis. And both of these are consistent with the existence of by stability. But that's 5000 year long desert time series. Yeah, exactly. Yeah. I showed you several others. I also showed you one grassland data, right, which is much lesser like more than 50 to 60 years. And then we also there are also many, many more data sets, you know, I will show you some references towards the end. So obviously, you know, there is no one model that will capture all of those in one equation, right? So I'm using sort of, you know, what are called stylistic features and try to capture them using the models. So the idea is not to cap, you know, compare the models with data directly. But idea is to compare the predictions and the sort of, you know, with that, you know, with the what happens in data. Okay. So let me go to the next part now. Okay. So, so the question we are trying to go into address now is, you know, if somebody, you know, if you had time series data, could you somehow tell I'm far from, this is the tipping point, right? This is the bifurcation point. I'm a far from the bifurcation point or am I really close to this some way of finding signatures of that from the, let's say time series data of the type that's available in nature. Okay, that's a question we're addressing. So this is, so what you now need to do is look at these potential wells I showed you as a function of the grazing grade in the specific model I showed you. So for example, in this grazing rate, you know, we are quite far from the critical point here, this is much closer to the critical point, right? Now observe this landscape features, which I have computed from the mathematical model. This is the landscape far from the threshold point, is the landscape close to the threshold point or the bifurcation point. Okay. So the landscape here has, you know, is symmetrical, right? It also is relatively deep. In contrast, the landscape in this case, right, landscape where the ball is rolling is actually has two features. One is there is an asymmetric feature compared to this. It also is shallower. It's also shallow around the, you know, minima. So one can ask, do these features of the landscape, do they have effect on the observed dynamics of the data? So what I am now going to do is, whether these two features, the shallower landscape in this case, and also an asymmetric landscape, how do these affect the dynamics of the ecosystem? If the model was right. Okay. So obviously, we're going to assume the model is right. And we're studying within the context of these models. So again, I'm showing you this is, you know, far from the threshold. This is close to the threshold. And, you know, basically, we need to think of this as, you know, this ball rolling in this landscape. What happens? So if you know, now do simulations of this model, what you find is far from the threshold, you know, this, the vegetation biomass, of course, fluctuates the characteristic density around the, around the minima. And so does in this case, closer to the threshold. But, you know, one can visually compare these two time series and observe that there are notable differences. One is that in this case, the amplitude of these fluctuations are large. Right. And secondly, there are these, you know, sort of, you know, spikes that you're observing towards the lower values. In some sense, there are no analogs of that in this, you know, there is an asymmetry in the time series again that you're observing here, which is a consequence of asymmetry in this, you know, potential landscape. Okay. So, and so what this tells you is there is something about the dynamics that is fundamentally different between, you know, these two values. One is that there's another point, which I forgot to mention, sorry, you know, as you are, if you are in a deeper well as compared to a shallower well, the system will take much longer to return to the equilibrium if you are in a shallower well, if there was a perturbation, takes longer to come back. And in fact, that again is quite sort of evident in this diagram. Just look at, you know, there was this perturbation here, right. And it does come back. And then observe the perturbations here, they're, you know, much more closely spaced, the return to the equilibrium value is much more faster. Okay. So what happens is because of the shallower potential, the system responds slowly to perturbations. And one can actually measure this from what is called auto correlation function of the auto correlation, which in the time series, auto correlation coefficient. And secondly, because of the shallow potential again, the system now fluctuates lot more around the equilibrium value. And because of the asymmetric potential, the system also, the time series also shows increased asymmetry. So if we measure the ACF of time series, auto, auto regression coefficient, if you measure the variance in time series, which the skewness in time series, all of them we expect will increase as system is going from, you know, far away to the closer values. So let us plot this, let me demonstrate this principle. This is a very important principle. So here what we have done in this simulation is the following. As the time is increasing, the grazing rate or an equivalent parameter is being increased. System is going towards the, you know, tipping point, the green line is a driver value. Notice the driver value itself is gradually changing. The system is responding in an abrupt way here. It suddenly collapses at around 1000 units of time. So what do we expect if the theory that I have shown you correctly is correct. What we expect is that if I were to plot auto correlation at auto correlation, regressive coefficient, that should also increase. Right? Likewise, if I were to plot the variance, that should also increase. And likewise, if I were to plot the skewness, you know, skewness is a value of symmetry. It could be positive or negative. Basically, the magnitude of skewness basically must increase before the actual collapse happens. So all of these must show these kinds of trends before an actual collapse. So therefore, the idea is if you do observe these, maybe we are approaching abrupt transitions or critical transitions. So in fact, this is an example of simulation from the models. So the auto correlation at lag 1, standard deviation and skewness, they are all showing the expected trends. And these expected trends can therefore be used as, you know, indicators that I am approaching a tipping point. So that's the sort of, you know, how we sort of use this mathematical models to make some predictions about systems that might show, that might show abrupt transitions. Okay, now here is the important thing. So although I showed you one specific model, one specific equation, and then and simulated and analyzed all of this, the mathematical theory behind this is much more general. It only relies on the fact that the threshold point in ecological systems maps on to what is called bifurcations in our models. For example, these are bifurcation points. So basically as we are going towards bifurcation points, these features are sort of universally observed. So therefore, although I have used one specific model, these trends that I am showing you in this, although for this specific model are likely to be true in a large number of cases of abrupt transitions. So that's the theoretical prediction. If a person is approaching, if an ecosystem is approaching tipping points, by measuring these simple dynamical quantities, one may be able to anticipate that you are approaching critical points or tipping points. So that brings me, there is one paper, for those who are interested, in the statistical aspects of it, which I am not going to cover at all in this paper, there is a paper in class 1 in 2012. There are also a toolbox that actually applies these theoretical principles and provides you a statistical estimator of how good can you actually measure these in real data sets. So I'm not going to go into details of this statistical aspects. So now I have basically covered the mathematical theory of tipping points and how we can use them to offer early warning signals. So now the last part of my talk will be, are they really true in real world data? So let's ask the following, how do we ask this question? How do we look for empirical evidence? Imagine if we had a laboratory system. If we can somehow subject a system to tipping point in a laboratory and then push that beyond tipping point, do you actually find those strengths in the autocorrelation values? Do you actually find strengths in increasing variability? Do you find this warning signal before the transition happened in your laboratory system? That's one way to ask this question. Of course, in the, we eventually want to apply these to field systems. So, for example, if you take data, long-term time series data, and if you look at that they have actually undergone the transition, did this system actually exhibit trends that we have predicted from mathematical models? So I'm going to address these two questions from, I'm going to show you what people have found. Okay, so this is a work by John Rake and others. So what they did was they did simple experiments where they, when sort of grew, you know, populations of Daphneam in the laboratories. And what they also did was they subjected these Daphneam populations to increasing stress over time. Basically, increasing stress can be sort of, you know, out of, you know, mimicked by reducing their, you know, food that's given to them. Okay, and what they do is they basically study them under also a controlled condition where they don't do it, and a condition where they are deteriorating their food supply over time. And what they do is they study for early warning signals in time series, variance, queerness, and correlations in time. And what they do is because they had an empirical system, they were also able to estimate what is the tipping point. And in their experiments, they say, you know, it's a year long experiment, they subject these populations to increasing stress over time over a period of one year. And they estimate that they reach tipping point in their data on day 300. And what they then do is they analyze early warning signals. So what I'm showing you here is coefficient of variation and queerness. What they find, interestingly, this is, you know, the dotted line here represents when the tipping point in their experiment actually happened, the small gray line in the bottom here, that's the control data. And what you're the one subject to stress, they are showing these, you know, much bigger trends. So what they basically show is that they were able to find signals of the upcoming tipping point in their experiments, almost 100 days ahead of actual tipping point. And in fact, this effect was most visible among the four indicators in coefficient of variation and queerness. And of course, since then there have been more experimental verifications of these ideals. For example, in East population, this is an example of East population density, you know, in sort of stable conditions. And this is the dilution factor, again, you know, mimicking stress. And what's remarkable about this diagram is, you know, it looks almost like the mathematical model I showed you, right? It almost shows, it looks like the bifurcation diagram of the mathematical model I showed you. A stable fixed point, unstable fixed point, and another stable fixed point, which corresponds to extinction. And what they do in this, again, they subject these populations to stress. What they find is that the coefficient of variation, standard deviation, autocorrelation, they all increase. However, in this specific experiments, they did not find strong evidence for increasing queerness. So this is then there have been more experiments. And, you know, other microcosm experiments, I won't be able to show you all of them. But the point is that, you know, certain interesting features and the mathematical predictions are actually testable using experiments in this case. Okay. So in the last part of the talk, what I'm going to do is to find empirical evidence for these early warning signals, which are predictions of mathematical models in, you know, in the field, in the data from field. So I spoke about, I mentioned this ecosystem in China, right, where there was a grassland restoration. Okay. So what they found, what they found was that there was this low grass cover for about four decades. And then there is a high grass cover, which is the current state. And, you know, intermediately, there were these strong fluctuations that preceded before the actual transition happened. Okay. So what you find is that if you look at the histogram of this time series, it shows this nice bimodality. And typically, the bimodality is a strong signature of bistability in the underlying system. So this bimodality in the data here shows that maybe the system is indeed quite stable. And what we were also able to show in this paper was that the dynamics of the system sort of dramatically changes around year 40, using something called a change point method. Okay. So what we then did was, okay, now we know that this system has undergone this abrupt change in the grass cover from a low grass cover to high grass cover. Did it exhibit early warning signals that the mathematical theory predicts? So what we did was we sort of drew this line at which the transition happens when we were only looking at data before that, before this year 40. And here we are, we are calculating autocorrelation at lag 1 and we're also calculating spectral density ratio. So now don't bother the spectral density ratio because I haven't described that to you. So what we find is that interestingly, there is no evidence for this signature of critical slowing down in this data set. So one of the main predictions of the mathematical models, we did not find that to be true in this case, in this data set. However, if you look at variance standard deviation, that shows a very clear increasing trend. Likewise, if you look at skewness, we again find a very clear increasing trend. So in other words, there is a very strong evidence for rising variability. But you know, if you remember the mathematical theory predicts that all of these must actually increase, not only this. Okay. But our data shows we rise only one of them to be true. So how do we really reconcile this? How do we explain the fact that there was no critical slowing down before the abrupt transition? However, there was rising variability. Okay. How do we explain this? Okay. So you know, this is where I would like to remind the slide I showed you long back, which is distinction between critical transitions and stochastic transitions. So critical transitions are those which happen at this tipping point or the bifurcation point. So your system is very close to the tipping point and then a small amount of noise pushes it down. On the other hand, you can have stochastic transition where the system is actually far from the tipping point, but a large amount of noise can kick it to the other stage. So now what we did was we looked at all the signatures for critical transition as well as stochastic transition. This is critical transition, the sort of standard theory which I have presented so far. All of the four sort of show these signatures. However, if you have a stochastic transition where a large constant noise pushes system from one stage to the other, you don't find any of the signatures. Excuse me. However, there's a second type of stochastic transition where the strength of noise is actually increasing. If you consider this scenario, you do not find these two signatures. These two signatures measure slowing down of the ecosystem, slow response of ecosystems. However, you do find that the variance and skewness increases. And in fact, this precisely what we found in our data. Our data does not correspond to critical transition not to constant noise stochastic transition, but to the increasing noise stochastic transition. So therefore what we conclude is that the dryland transition that we analyzed is a case of stochastic transition where there was no critical slowing down, but there was a rising variability. One slide that I have skipped because of time is that this rising, this sort of counterintuitive pattern is driven by stochastic rainfall. The rainfall is increasingly becoming variable over time. That seems to explain this phenomenon. I don't know how much time do I have now. I have about seven minutes left. Let me now present one more study we did to look at these early warning signals. So can we apply these tools to anticipate financial market crashes? So our motivation for this came from many papers that look at critical transitions. So for example, if you read this famous and classic paper by Sheffar et al. in nature 2009, complex dynamical systems ranging from ecosystems to financial markets and climate have tipping points at which a sudden shift to contrasting dynamical regime maker. So basically financial systems are sort of often quoted as examples of tipping points, but is it therefore then are the predictions of tipping point models are they valid? That's what we try to understand. Specifically, we asked two questions. Do we find evidence for critical slowing down, which is basically increasing in the autocorrelation as markets approaches a crash? Do these markets exhibit increased variability prior to the crash? What we did was we took a data from a whole range of stock market indices. This is an example of one specific stock market is a famous Dow Jones index. What we know historically is that there are four well studied crashes, 1927, 1987, 1991, and then 2008. So what we did was we took each of these windows and we analyzed all the four metrics. So here also what you find is that the autocorrelation at lag 1, which is a measure of critical slowing down has no clear patterns. In fact, as we are approaching the crash, it actually suddenly reduces. Therefore, there is no critical slowing down in this financial markets. However, if you look at the variance in time series, that shows a very strong and clear trend. So there is a rising variability prior to stock market crash downs. Okay, this is not one market and one index we have stayed. We stayed this for a whole range of markets and whole range of crashes in 1929, 87, 2000, 2008. For all of them, we find that the autocorrelation at lag 1, the critical slowing down doesn't show expected trends, but the variance always shows the expected trends. I don't need to do any sort of statistics for this. You can just see by eye that the fluctuations in the stock market indices are dramatically increasing prior to crashes. And this is true not only for Dow Jones, we also found this for S&P, Aztec and other markets as well. Again, how do we explain this lack of critical slowing down, but with rising variability? Again, I want to remind you about critical and stochastic transitions. In stochastic transitions, we don't find autocorrelation at lag 1 increasing, but we do find variance. So most likely our data corresponds to stochastic transition. The stock market data might correspond to stochastically driven transitions rather than the transition that happened near the bifurcation points. However, I want to highlight that there were also many, many instances of false positives. If one were to rely only on variance, there were also a large number of false positives that occur. Okay, so therefore, one can't just rely on variance as a predictor. So it's only a tool that may provide some signal, but it's not a predictive tool. So therefore, we conclude even in the case of financial markets that they are not critical transitions, they don't happen near a bifurcation point. Their features are better explained by stochastic transitions. Okay, so with that sort of, I want to summarize the first trace talk, which is that there are many, many real world examples of abrupt transitions. And we mathematically model them as tipping points or bifurcation points. And it's important to understand that these models can capture critical transitions which happen near critical points, and also stochastic transitions that are strongly driven by stochastic forces. Okay, and these critical transitions, even stochastic transitions can have early warning signals. And the two classic ones are that of critical slowing down, which is rising auto-correlation and increasing fluctuations measured by variance, queerness and other metrics. And we have, there is no quite a bit of empirical evidence that these metrics may indeed work in real systems. And our specific analysis showed that stochastic transitions are better, you know, many ecosystems and including financial markets are probably thought of as stochastic transitions, because there is a lot of stochasticity in these systems. They are not slowly going towards bifurcation points, but probably they are strongly driven by many stochastic factors. And this also provides an interesting case that, you know, mathematical model models can not only explain and provide insights, they can also provide tools for the assessment of ecosystem resilience. For example, observing these metrics could sort of indicate to us how close one is to, you know, are we approaching low resilience states or are we approaching potential catastrophic transitions. That's a summary of my talk so far. Okay, so I will take more questions now. And in the next talk on Tuesday, I plan to discuss more about the spatial patterns. And before I forget, and you know, before I take on question, I want to thank and acknowledge my collaborators. I start working with Professor Jayaprakash on these problems. And I worked with Nikunj and Srinivas Bhagavan on the financial market problem and chaining on the in the grassland data. So thank you very much. So I'm happy to take questions now. Thank you. So we have a couple of questions from the chat. So how do I open the chat now? And should I close my screen sharing by the way? Yeah. Oh, I can open the chat. We can also cut the chat down. Okay. So one question is the asymmetry and shallowness correlated? Yes. So yes and no. So they're correlated in the sense that they happen simultaneously. Okay. Now, but they're not happening by the same causal factor. So if I were to explain this mathematically, so, okay, I think I have a slide that may even explain this sort of mathematically. Okay. So if I look at if you think of, you know, classic and simple models of catastrophic transitions. So they have this, you know, linear term. Okay, let me see if I can switch on the. Oh, okay. Can you see this? Okay. Okay. In this equation, I have, you know, this bifurcation model with noise. This is linear term r times u. And there is this cubic term, right? So the shallowness is can be entirely explained by the linear term. But the asymmetry requires you to use this cubic term. So you cannot obtain asymmetry without having a cubic term. And so abrupt transitions in complex systems are typically modeled by saddle node bifurcations, right? And saddle node bifurcation does require you to have this cubic term. So, yeah, so they are related, but they are correlated in the sense that they happen simultaneously, but the causal factors are different. And the second question I have is, you know, is it somehow related to curvature? Absolutely yes. So the shallowness is absolutely basically equivalent to curvature. So shallowness means reduced curvature. And that is what is causing reduced return rate to equilibrium. That is also what causes increased fluctuations. So they're all same basically, shallowness and curvature are basically and the third question is my similar question are some of the drivers more often correlated are they vary largely case by case before it's important to consider all. So yeah, I mean, I don't have a good answer for this question because this question, one has to look at a specific system and try to understand what drives or how are the drivers placed. For example, if you were to think of a vegetation system, typically rainfall and fire are very, very important drivers, right? So we need to know the mechanics of the system to sort of, sort of, you know, address this question on a case by case basis. And in all of the model analysis I showed you, they were all assuming that a single driver is changing. Single driver is changing slowly towards a bifurcation point or the stochasticity in the single driver is what is really causing it. Okay, the next question. Stock market indices are often fact-tailed. This leads to divergence of moments. Is it then meaningful to analyze their trends? Yeah, this is a good question. You know, I'm not an expert on financial stock markets. So I will not be able to provide a very, very good answer to this question. So my own interest was, you know, more of academic, specific academic question since it is often used as an example of tipping points, does it, and we have really, very, really good high-resolution long-term time series data. Does it show critical slurring down? Does it show the simple feature? Let's ignore the moments part. I agree. You know, moments are complicated. I fully agree. You know, even if you ignore the moments part that I showed you, can we calculate the auto correlation function and the properties of it, right? So I think that can be done without being worried about fact-tailed distribution parts. And we do not find evidence for the mathematical prediction that there is a critical slurring down. And of course, you know, when it comes to thinking a bit more about the fact-tailedness, I think that to think of, you know, mean and variance rather higher moments, you know, to really observe divergence, you need a, you technically need infinitely large data. But we are obviously analyzing data for a finite window. So I think in that limit, it might still be reasonable to calculate that. Robert too, do drastic transitions due to noise happen for specific values of noise? Is it a feature of stochasticism? So I think it is related to, you know, you need a, so basically, you know, if you're the noise term in your model as a Gaussian noise term, right? Technically, the smallest noise can also cause, you know, transitions between two stable states, any small amount of noise. But, you know, you may just have to wait so long that it's meaningless now, right? But if you now say, okay, you know, I want to observe transitions within a given time scale. And I'm going to be interested only in those timescales. Yes, you do need the minimum amount of noise before a transition can happen. And in fact, that's the point that, in fact, my very first paper in my PhD thesis, I try to address that, that specific question fairly, this slide that I'm showing right now. So we basically, we, in fact, we showed that assuming that the noise is bounded, you need to have a minimum amount of noise for you to sort of induce transitions. And you need a minimum amount of noise to also to fluctuate back and forth. And I think it is related to Catholicism. You're right about that too. Okay, there's one more question. Is it possible, at least in principle, to amenable to account for feedback mechanisms? For example, rainfall sustains its own raining rate. Driver is not an external driver. Yeah. So I'm sorry that, you know, in this talk today, I have absolutely not paid attention to mechanisms in some sense, right? I just used a model that showed the features I'm interested in. So what, so what I'm hoping to do in the next one, where I will discuss spatial models is that positive feedback mechanism is really important. And if you have weak positive mechanism, weak positive feedback, you do not have abrupt transitions. Only when you have a strong positive feedback is when you will actually observe, you know, abrupt transitions. And in fact, it is precisely because of the positive feedbacks that, you know, two states, you know, in this slide, right, you know, there are two states, right, you know, the green high vegetation state and the black low vegetation state, you know, so the green line and the black lines, they coexist because both of these forest states are stabilized by their own positive feedback mechanisms. And therefore, you know, it could be, for example, rainfall under rainforest, it could, for example, be the, you know, it could even be much more local scale. For example, the presence of trees, a patch of trees will enhance the water infiltration that will in turn help the tree to go better, better, that will in turn help the local establishment of seeds. And therefore, the entire patch is sustained. On the other extreme, if there is nothing to start with, it's very hard to germinate a new seed because the water doesn't sustain there for long enough for a seedling to arise. So, so the both the alternative stable states are often maintained by these kind of feedback mechanisms. That's a great question. Thank you. Okay, I think we, we had a rich discussion session and thank you again, Bishwesha for a nice lecture. Now we have time for a short break and we'll be back in about six to seven minutes for the next lecture by Andrea Rinovo. Thank you everyone. I'll see you again next week. Ciao. Ciao Antonio. Antonio, non occorre che io faccio altre prove, no? Pritanto, son qui... No, they, you know, the... Hai parola, sono pronto qua? Non bo? Sono... Perfetto. A dopo. A dopo. Ti prenderò. Okay, I think we are ready to start over again and it's a pleasure to have once more Andrea Rinaldo from EPFL for his third final lecture. Please, Andrea. Thank you so much, Antonio, and here let me, let me share my screen and I've had today a few issues with the connection with Wi-Fi, I hope it doesn't happen, but because I, you know, I happen to have a few kids back home so everybody's using the van, but anyways should be okay. So here's lecture three and let me close this up, yep, of the series that we had and let me see where we are. Remember what you may remember where we had discussed like in the first class in the context of neutral meta-community or meta-population model which essentially is something in which if you look now at the right plot only compared to be so-called the pure lattice model of interactions, you have a sequence of interactions in the sense that the dynamics proceeds by canceling at random like a color if you want or an individual or a species if you take a meta-community model, it won't change the main result in a particular place randomly chosen and you keep going with that and you replace it with either an non-existing color slash species slash individual with a certain a probability of a diversification ratio whatever you're going to call it or you replace it with some nearest neighbor or some neighbor which was chosen respect to a kernel of the domain of influence for which you choose the most numerous or in a mean field sense whether you choose it from anywhere else based simply on abundance and that of course entails neutrality in the sense that the preference is simply due to sheer numbers, sheer abundance and that's the essence of a neutrality assumption and the patterns that we've seen in there are quite different depending on a sole modification which is in this case the directional dispersal which is embedded in the network structure, the tree-like structure if you please or the open to any, I'm sorry, nearest direction from a sorry stuff like so let's see how we can exploit this is Ignacio Rodriguez Turbe, emeritus professor at Princeton currently now at Texas A&M after like a couple of decades in Princeton in which he said well why don't we assume that each branch of a network structure here becomes like a meta-population which is called a directly tributary area that is a local community within a meta-community. One of the points that I made in the first class was that the result that the directional dispersal which is embedded in a network structure for interaction that is that defines your nearest neighbors or a way neighbors in a selective manner is rather insensitive of whether you're talking about an individual or a meta-community based approach the result still stands and what later on we have done like field verification of the same result studies theoretical or empirical on migration fronts for instance that feel the structure of a directional dispersal or laboratory in fact experiments we learned in my lab in which was nothing neutral there was a living community the result still stands. So the idea is here is to assume that the large river system this elementary the unit on which you have reactions of physical biological chemical nature in this case biological could be a directly tributary area which is a local community carved within the community like a very each scale and we thought of doing so we referenced to something which is quite important we took on the the whole archive of local species diversity of fish populations in Mississippi Missouri river system and LSR means local species richness however defined essentially can count the observed number of species which you have. So in a DTA in one of the notes of the links in which you can partition I'll be showing you now you can extract that objectively and manipulated remotely acquire and objectively manipulate the structure of the Mississippi in fact acronym means annual average runoff production which is essentially runoff I mean the total volume of runoff that passes through any particular cross section that is the hydrology. Now it doesn't take an expert statistician but simply a not a particularly trained eye to see that there is a correlation between how much runoff you have in a particular patient in different directly tributary area that is the hydrology the hydrology control and the local species diversity but to formalize this what you can do we can extract objectively even humongously large networks to detail which is actually in fact one of the reasons why in fact that over the debate of critical self-organization that is why in fact certain recurrent characters embedded in a power law distribution of some aggregation structure of a catchment is the same regardless of a climate vegetation exposed lithology the scale etc there's something truly remarkable already the networks that generated files like or or datasets like this one in which you can characterize a structure from the scale of one meter or less to the scale of thousands of kilometers and in terms of area even more so that can be done I hinted that that briefly in the first class are not touching on it but I assume that this is something which when you can trust me we've been working for like 20 years in characterizing those network shapes like these ones so let me see how essentially the model proceed it's it's well I have a few notes here about them it's not going to be particularly but the idea is that it's likely experiment you've seen the neutral meta community or meta population experiment that I showed you in the animation I showed before that I showed also in lecture one so these the assumptions in this case is every DTA that is every unit is essentially saturated at its capacity that is no resource available to fish is assumed to be left unexploited now of course you're talking about some sort of an upper limit to the to the fish diversity but that's also quite remarkable how close I will show you this will be reproduced so the model dynamics procedures in the other case which you have seen that is at each time step you randomly select a fish unit selected for all the fish units in the whole system and you assume it to die and the resources that previously sustained that unit are freed and available after grabs for a new fish unit so we have certain probability which we term again the diversification ratio the rate the new unit the fish unit will will be a new species it will probably be one minus new in fact it's going to be one of the existing species that colonizes the spot and the cost of diversification ratio rate in this case could represent external introduction on non-native species we have seen how foreign invasions are so important ecologically for a variety of ecosystems or it could be immigration or re-immigration in fact of a new species from outside region you've seen the case of the example if you may recall from lecture two when we studied the breeding birds of uh of North America as a or the cancerous prairie uh uh species or bassius species in fact that um in fact the the concept persistence time local persistent local extinction has to be seen in the context of the geographical uh area which this is done so um uh with the probability one minus new the new unit can belong in fact to the to the species to be uh to be etc and the idea is that in this case you don't uh touch only nearest neighbors but essentially you characterize a probability pji ij i'm sorry that the empty unit in the i-th uh dta in the i-th node the directly inhibitory area whatever you're not calling is colonized by a species which is elsewhere in the j dta's that appear in the system through a kernel a dispersal kernel that you're having this says which uh specifies and measures the range of species colonization now what is important here is a non-neutral effect um which is uh however not dictated by uh by uh a calibration not dictated by uh uh biological dynamics but essentially dictated by hydrological controls because it's geomorphological so you assume that the habitat capacity determining the resource in the place is the fluvial habitat capacity which is established on the scale of the basis of a scaling geomorphic relations we had hinted at that characterized the fact that um you can decide based on certain metric properties which are remarkably available enough producing area of any river worldwide and are dictated by the aggregation structures essentially the habitat capacity in the place is essentially dictated by how much area you had behind your back how many nodes do you collect through the structure of a network again which is a given now what is interesting that the dispersal kernel has certain features which I shall not discuss but essentially what we have I want to study the particular back-to-back exponential which has a tradition in ecology one or another in fact we tested several of them um they were asked by the nature of yours in fact to uh to run a comparative analysis and we apparently convinced them because we got published but the idea is that for instance if you assume that two units two nodes to DTAs are uh ij what is the distance that separates them now that's interesting for colonizers because if you assume the colonizer are strong fish units right um the path the path that connects you uh i to j could be partly downstream and partly upstream or could be strictly downstream or strictly upstream depending on ij now the question is you may bias the path because if you are a weak juvenile fish for instance a small one you may be way more affected by the velocity that is the drift which is embedded in the stream flow direction that is the oriented nature of the graph then you could be for a strong adult big fish right so either way then you may bias and weigh in fact the downstream direction the upstream direction distances which are in the system which makes it reasonable and this is a tunable parameter in a sense but in a neutral case we kill it and we assume that all species are equally important at the at the at the per capita level so briefly as a result of what you're seeing here is in the frequency distribution of local species richness by letting the model run to stationary state now um it may be uh you may like it you may dislike it etc but for us it was totally remarkable how simply the nature of a connected uh of a connected system and the habitat size which is produced by scaling relations being embedded in geomorphological laws it's the aggregation structure um can allow us to reproduce wonderfully well um how the alpha diversity the distance to outlet i'm sorry this is i made a mistake i was anticipating what i explained here in fact so this is all the frequency distribution this is the how the alpha diversity unfolds from the outlet to the upstream distances with all your differences which you have in a certain place for instance um this is blown out because you may have the the data telling you that some freshwater tolerating uh coastal fish species in fact could or human disturbance in fact or or pollution for that matter um alter what you would have otherwise a distribution that is what the data are showing you that near original new links and the same applies to the same thing at the same time which is also remarkable how the frequency distribution with respect to the distance to be out the frequency of local species with respect to thing which is you count essentially the number of species that equally distance from the outlet you got some sort of a range which is reproduced without any tuning by the model signifying once more one of the main tenets of my classes that there is something inherent in the directional dispersal implied by the network structure which is the substrate for ecological interactions which is given the system granting the system uh reliability and predictability what is interesting also is that if you run the same exercise without changing the habitat capacity per every node embedded as proportional through um predicted geomorphological laws by the uh structure the aggregation structure which is essentially dictated by the total contributing area at any point deciding how big is your river and then what you see and you screw up completely in fact the uh the exercise and you see the hydrology controls embedded in any neutral model which is the simplest possible zero order approximation there's no description of an of the properties on which fish biologists spending lifetimes of scientific work and are mostly explained by the hierarchical size structure of the fluvial network and it's embedded topology which is also reinforcing what we have seen before now things become slightly more complicated if you go into not only starting local species richness by the correlation structure it is a beta diversity that is um what is the probability that the existence of a species in one place is uh matched by the probability of existence of a safe species at a certain distance distance being uh measured in so-called chemical distance that is along the network structure so this can be done you can generate equi probability maps which is the ratio between the number of common species that you have and the species in the central dta to see how the system behaves in fact so this um reinforces the uh the main tenet that I've been hammering on for quite a few times and I am now ready to um move on following uh what Marino Gatto has told you about the evolution of our thinking about spatially explicit epidemiology uh by the original ideas that motivated us to go to jump into covid 19 studies through the same technological tool in as much as some tail some some uh small factories that you had in the milan area we used to make um high fashion dresses converted the alliance of production into massed productions during the covid 19 it was a relatively jump easy jump for us but I'll spell you how in fact this uh thinking this way of thinking and this line of thinking that brought to the book um I talked to you about uh allowed us to move on to jump from fish biodiversity straightforwardly to the study of river networks as ecological corridors for waterborne disease let me see why now um here's a river network and let's assume that those dots are uh human settlements so essentially what we're saying is that um what if we thought notes are human communities where they are in fact with their population on their size etc where disease can spread and um you have a demography of a population of a of a demography of a disease based on the demographic evolution of susceptible individuals in fact individuals possibly recovered and possibly via the as you've seen in the case of shisto quite importantly um through the example that um uh that Marino has pointed out or which I briefly returning by mentioning his mentor because it's important for one of the tenants of my work that is you may in fact couple these with control variables which pertain again water controlled but that pertain the ecology of a disease that is if you have for instance intermediate obligatory um intermediate hosts for the development of a disease so um why this is interesting the example of a shisto I will just briefly show the main results that Marino showed you the help means and you see why this is important right and but the motivator I'm giving it um it's more um it's it's somewhat important that's where we carried out the field work that we carried out with my lab if I did Burkina Faso we had like a 20 year 20 year uh longer experience in um a collaboration in in cooperation with development so um it what is incredible in this area is that in sub-Saharan Africa you have something like in the debilitating disease which is not killing anyone so it's a neglected tropical disease you have like 15 million uh disability adjusted life years uh that do take into account and what you have in the particular regions you have different types of shisto as Marino has shown I'm just going through briefly because and that's what we had uh in the camp that we have in Burkina Faso we deployed and one graduate student in fact carried out his entire PhD thesis on that why this is interesting again it's a complex life cycle and it is interesting for us the existence the need to take into account hydrology controls this is the part which indeed pertains to how form and function of the river network operates because why because if you have like this marriage this fertilization of the successive stages of the of the uh of the disease need to hatch the eggs the best place in which they they are generated within the infected host and they have a worm so they the fat worms that generate the sarcaria or myrassidia effect in live in the and they excrete the three thesis they get into a water environment in which I have to they have to be infecting snails in a given time now that depends of course on shear stresses depends on the flow velocity depends on the habitat size it depends on whether the habitat suitability for the intermediate host is given so something in which as you've seen with marino uh we'll be back into the system so that's what you have and what you have seen and that's the point I'm making um the point I want to make here is that um pricing the planet as a environmental economist pointed out that is um if you don't account for a depreciation of natural capital um that means that um essentially the ecosystem services that are carried out that you cannot quantify precisely are worth zero so pricing the planet means that you have to give the monetary value to the services you may lose as an alternative like in this case making way for a commercial center by destroying a mangrove swamp um that means that in the in gross domestic product indicator GDP like indicators or well-being you will see the advantage in the following year of the of the commercial center of the benefits of it but you don't see the loss which is associated with the ecosystem services you lose from flood protection to uh to uh carmel sequestration to fish nursery areas and the likes so the idea and that's part of the sculptor's main point is that indicators that do not account for the depreciation of natural capital uh put development thinking uh stacked against nature in a sense uh against environmental thinking and the very idea of a misinterpreted kutznet's curve that is that if your GDP goes up uh down goes the inequalities in their place is essentially false as Thomas Piketty in fact has shown quite clearly in his wonderful book uh the capital in the 21st century what happens is that in reality indicators that omit the depreciation of natural capital are totally unsuitable for describing the wealth of nations and here is my point meta studies um that and i'm i'm really sure that marino hasn't spoken about it and then to use my discussion in fact showed that there is a clear relationship between the expansion of irrigation canals and some of the 15 000 small dams so we have seen i took this picture near our field site in Burkina Faso in fact uh the uh the construction of irrigation channels that were possible because of a 15 000 world bank funded uh small dams that uh littered Burkina Faso in fact had the consequences so the water resources developments came of a largely improved GDP over Burkina Faso but in a humongously large expansion of a habitat suitable for the intermediate host of the disease and thus the prevalence of the disease so the idea is that can we put a price tag on the learning impairing disability brought in by a disease of this kind complicated complicated and our permanent our inability to predict for instance the number the increased prevalence um in in unless you have significant and reasonable models of the expansion of the disease then um you know as a matter of fact they won't be able to put the price tag on and this is the same place and this is a picture so they took in the same place how can you keep uh little guys away from the water when you have ephemeral ponds generated by the system mind you that in this case as you're very well known because of the wonderful lectures that truly marino has put forth on the subject that the the those largely penetrate the skin uh in matter of second our student there was here only once put his hands in the water dropping the gloves when he dropped scissors just he totally just picked up in a second the scissors but it was enough to create to to get the system why this is important and related to what you have seen um in the previous slides for the fish diversity it's a picture you're seeing from marino because the set of equations and I'm not insisting on that is something which you have coupled um coupled uh uh odys like in this case is the mean word burden that you have like in the system here the prevalence of infection in the intermediate host in this case is why and these are the the essentially the densities of the sarcaria and mirasidia in node i of a of a dta that is of a single node in which you can characterize the system i'm not pretending that you follow the system but you realize that in here you have a number of extensions which are quite important they pertain for instance human mobility that is if you have a guy that migrates to go to a place to cultivate the feeding that uh uh because of the expansion of the irrigation network human mobility does affect in fact carrying away an exposure and the concentration of sarcaria that generates the burden of disease in an individual that's how the system expands so in a sense you realize that you're moving the study of diseases onto a plane which is completely different and the set of system is a classical system through which used to uh engineering environmental engineering tools and i want uh build on that in particular nor or what you have seen uh with marino about how you characterize the stability of the system and uh possible ideas you have on how to curve again this eigenvector analysis you're seeing with marino is telling you essentially how you can actually generate the patterns of disease and what i really like is the idea is that you cannot make discussion what happens for instance if at random it's an exercise we ran you remove 10 percent of a small dense thereby reducing the distance the mean distance to the nearest water body which is arguably the most important factor of completely geomorphological origins that generates the exposure so and that's the experiments we had in the place and i'm not building any further or uh other diseases that can be treated in this manner what i'll be concentrating here i'm getting back to the first slide in the last 20 minutes of my of my lecture and then i be delighted to answer your questions um you remember the little guy here on the banks of a magna river uh where we did field work on chronic effects cholera i'll be talking about epidemic cholera in a minute we're trying to convince me that it's impossible the mighty waters of the of a magna were the cause of cholera which originated in that region in fact evolutionary um and then from there irradiated worldwide in several ways of pandemics but most significantly 200 meters downstream of the largest derail disease hospital in the world in Bangladesh now um the point i'm making and i'll be carrying out to the end is that the inherent predictability that you grant the system by using directional dispersal embedded in the known a priori and calculated and treated objectively offline remotely acquired and objectively manipulated offline structure of river network grants an unprecedented predictability to disease of this kind and in particular this became absolutely vital when we uh uh started we were just working on that we had published the first spatial explicit model of um epidemic cholera uh in with reference to the outbreak that was in quazulu natal in south africa based on data that were uh uh collected much afterwards in hindsight but what happened is that on october uh uh in the week preceding october 10 2010 all of a sudden in a country where it was cholera free for more than 200 years you've got an outbreak which started propagating downstream the arctic bonita river in the heart of in the heart of the haitian island the island of haiti the part of the left not the dominica republic which is cutting half good example that jarred diamond um in fact in his collapse uh wonderful book was making an example of how in fact the umprobo scene and the bad management of resources uh explains um the uh uh uh not simply environmental factors determine in fact the fate of societies now what's interesting you see the number of cases jumped all of a sudden from day zero to the from 50 to 100 to 200 in places which are small places indeed and right downstream of a un camp of peacekeeping curves what is not only ironic sad and it's it's really uh uh affecting me uh very much is the fact that why we were peacekeeping troops in haiti because a few months earlier haiti the poorest country in the world had been struck by an earthquake that killed 300 000 it destroyed the little infrastructure where it was there uh uh sewer systems were non-existent and roads were destroyed people died um a civil uh infrastructure was demolished it was a it sits on one of those plate tectonics on which earthquakes can be particularly devastating on top of that we planted the disease because it was shown when it was mapped the genome that it was a nepalese strain of cholera when it is endemic brought in by asymptomatic uh peacekeeping forces anyways that was a fantastic exercise in a sense because in a completely naive population as that's a term that you use in these cases um they end um uh but is uh or you can assume safely that because uh no sign of cholera was there for almost 200 years but the entire population was susceptible to the disease and what happens is that then you had thousands of deaths you had the mortality initial mortality we was totally remarkable because there were roadblocks to treat people transported by poor means like uh on the shoulders of a of a younger in fact to be treated to center the roadblock to make it and say that after a year like eight percent of the population of the million people was affected and this picture I took into an hospital in leogand and you see what was essentially the treatment was even I have to say among the sanctity that I've seen in the Medecin Saint-Francaire hospitals organizing the haste in the place or the Cuban brigade that took up the north of the country to assist but essentially put people on the stretcher you cut a hole in the thing and you collect the stools like six times a day and what is totally remarkable that um you survive cholera easily you only have hydration bags to which you should be attached now let me show you the evolution of the daily cases in Haiti for about a year and a half then I think it comes blurred afterwards but that's quite interesting so you have an evolution of the daily cases in behalf of the hispaniola island that is a part of Haiti and I didn't put the data which came later on in fact for the two islands that still belong to the same place etc so this is how the disease in terms of simply reported number of cases with all the inherent errors of the Medecin assistant and that you had in there this is the city of Port-au-Prince here is a long scale of a number of cases if I'm asking you what do you see here well you see the rivers so you even by seeing the most gross indicator number of reported cases what you see in the place is that the avenues of the riverways where the pathogen in fact survives in the environment in the open waters in fact is what generates the system etc so essentially you can't have something in which you essentially can calculate the rate of change of new cases of cholera in every single place that that you have but you have to account a spatially explicit system we put settlements where they are connectors where they are and the likes not only that if you take the red curve is what happened in the first waves followed like in covid but for completely different reasons a second wave which is clearly related to factors like the tropical rainfall that you have in the place why well the easiest is not simply an overflow of sewers but simply the washout of open air defecation sites which you have to take into account and the fact that the freshly shaped bacteria the bacterium in fact like a single infected individuals expels and through the school through feces like a hundred times more bacteria that in concentration which are also magnitude larger than any any possible survival in the environment so it is the human human host in fact the main reason of a propagation whether symptomatic or asymptomatic whether he moves or not if you have a susceptible person moving on to the wrong place drinking the wrong water and getting back he brings back the disease it happened to me when I was in the market of our tip on it well I'll tell you when I show you a picture later on so the tools of the trade now is knowing ahead of time where settlements are where patterns of rainfall evolve and how the disease can be predicted under these conditions putting cities in human settlements where they are connected by the waterways as we can see them directly so the idea that in a system like that you have two different networks a network which you have like pathogens connecting nodes if they are downstream of a river system or and that's a key place you can have connections among nodes human communities in which with population each population H sub i this is node i in which the disease can diffuse and grow connected by a multiplex network of a different time of a different kind in this case human mobility as you now very well know from marino gato's lecture on covid 19 the spreader the mean mechanism whatever its shape it is and but we had seen even in my set of lectures what happens when we consider from a zebra mussel invasion of a mccp Missouri river system you see that you saw that at times unconnected flare-ups of of those development of those clusters of zebra mussels were generated away from the main backbone of a hydrodynamically generated dispersion why because of the ballast water in which veligar survived where we're taken away and and tucked to different place maybe hundreds of kilometers away from the same place the mechanism of generating the system of this kind so the tools of the trade in this case are the tools of the trade of geosciences of digital information systems or geographic information systems if you want that is we can and we could do it remotely when we predicted the evolution of the of the epidemics of cholera in the place which i'll be showing you in a minute because the digital terrain map from which you extract the river network as i hinted at in my introductory class is something we should do it's a standard exercise that that master students do where we are you can have pixel-based estimate of population density you can have modeling of human mobility which is something which requires some thought and some care in fact generated maybe simplified at times but i'll tell you what is the capability of attracting places like on the main point like all the plants that you have in this system and the set the tools of the trade i mean they different every time but they that's why i showed you before the ones that marino have shown you for schistos so essentially the state variables are susceptibles at node i at time t infected and node i at time t and the bacterial concentration in the reservoir of the i-th community evolved it because of the different factors which is the mortality the survival of the vibrio in the environment with a certain mortality rate of seeing really how long you get survived or transport in a certain proportion coming from connections that are hydraulic and hydrologic connections that depends on the various sizes of the of the water reservoir the local water reservoir which is important because essentially you can assume that one stool a single infected person has a certain probability distribution in terms of absolute number of bacteria shed which is again six orders of magnitude more than you have for the concentration of bacteria in the free living waters and that is the infection thing that generates the P you see P the infection per unit infected person which is here you have the force of infection depends on the local infections plus the of infections connected by a mobility matrix this is a mobility factor which you have in the system so of the i persons that live in a community i sub i infected person whether symptomatic or asymptomatic stay there and pollute the water or you may have persons that because of mobility and its matrix of fluxes generates the infection shed into the place to which you could possibly add like you have it here the rainfall runoff production of vibrios generated by the system again i cannot pretend to explain the details but you see how this is done for instance and you have seen in introduction to the disease ecology that marino put together in this case we assume the timescale of our prediction is one in which recovered persons i mean i put back into the susceptible compartment every time which is one of the role of the order of two to three years and this is the force of infection depends on the number of factors my scope here is not to it's get you curious about the strength but you see that the structure is exactly the same of the shisto is exactly the same of a msccp misery by a diversity model and it's exactly the same ones we have seen before so let me show you how the model works if you assume a piano network very important because if you have like in every node you have a population of the same capacity all of them prone to have the diffusion of the cholera that what you will have in the system right and why this interest is a calculated speed of the traveling wave of cholera under simplifying assumption which depends on the local reproduction number but if you assume as it is meaningful and reasonable the distribution of it in a topologically connected system of this kind you have that node distribution the population distribution is taken drawn from an distribution which is normally i mean almost universally a power log exponent minus two the zipped distribution what you have is that now what you see in the system that you have flare apps in a mechanism which is exactly the same and why because that's the effect of factors that can be remotely measured and objectively manipulated in this case the population size and certain effects of the delays that you have in the system spatially explicit have nothing to do with the disease and everything to do with the geomorphology and the economy of the system so i took to haiti a few times and what these pictures i took they are kind of blurred and i love pictures but the reason being is that my my glasses were shut because people could kill you to steal your camera because there's no police left in the place we have no sewers no streets to speak of and this also shows how large patterns of infection accompanied you consider this safe water bottled water with the guy that handles them uh by the neck in this case quite remarkable or there are places is taking in the again in the bangladesh counseling is taking which the water reservoir which could be a highly abstract uh phenomenal parameter of the system in the case of urban setting possible proportion to the population or in this case you see it or a very physical system like you have in bangladesh as i was telling you and this is the market or car food in the outskirts of of port de prince where i've been sitting in the system and and what i told you and was totally remarkable this lady which you're seeing here blurred against because of a of a safe glass beyond it in a matter of second bought a cabbage at this guy and i couldn't i was speechless when i saw it by showing a nokia 1900 proof of concept i'm sorry knocking i think handle the telephone as a proof of payment so in a place in which you have no sewer system as you see it water flows through the market in this case you have no roads you have no police but you do have a telephone which is a way less uh biased socially biased system as we have seen and this is how in fact human to human transmission this is public transportation in fact takes a place and that's the last thing that i want to show you that models and data in fact uh they are not perfect modeling things etc i mean you have Bayesian estimation parameters but the very fact that you're using specially distributed quantities make sure that in reality the distance within model and data is so small that operational decisions can be taken based on that and i'm skipping this part because it's too late is marino gatto and his idea that in especially explicit models of disease development in fact you can have even an eigenvector can represent the pattern of disease before it happens and quite interestingly you show also that the local reproduction numbers meaning the test for the potential for the outbreak to occur is neither necessary nor sufficient the condition for epidemic disease outbreak if you compare to real cases whenever you have spatial explicit system in which human mobility is a driver not is an embedded driver but it can be calculated and i skip this part i skip also this because i realized i chatted too much about how proliferating kidney in fish can be studied and i jumped from the last two uh fish diversity in that case and and the deadly infections in fish are in fact the proper into the channel network was all that i thought to speak about so my conclusions the whole the general conclusion is that the eco hydrological footprints of river networks as ecological corridors are demonstrated they are pretty strong in fact and from peaks of prevalent in waterborne disease infections to any kind of large scale patterns of species abundance and biodiversity or even the susceptibility to biological evasions that we're seeing it's only the water so in a way it's written is something which could be remotely acquired over over virtually six orders of magnitude and remark will be compelling so in a sense towards a fair distribution of water which is my punchline that is that attaching a price tag to certain things which weren't um material uh materially or observables in economic terms but they are absolutely vital because they don't have a way of predicting what will be the impact for each of the of a an expansion of water resources um uh uh exploitation patterns in the expansion of a disease for each cost so they open um the uh a quantitatively open to a quantitative evaluation on ecosystem services to rethink in a sense um social equality and i thank you uh with that thank you thank you very much on that um we can open the floor for questions you quick question yes please um in the in the model um towards the beginning when you show the alpha diversity increasing as you go downstream um is this is this increase and consequence of accumulation of species going down because they cannot migrate upwards or is it a consequence of just more populations interacting and getting uh going together uh they will reduce the number of canal downstream it's good question the kernel for for the special species are the same everywhere what makes the difference is the fact that habitat size and thereby the carrying capacity of the population of every six species changes uh with respect to be downstream direction because of a natural accumulation so it's an external factor which is dictated by the aggregated structure of a network that gives an inherent predictability even though i mean how could you possibly uh consider all fish species equally uh equally capable of dispersal for instance or insensitive to drift uh at the per capita level and yet neutral pattern um doesn't require neutral process that's what i'm saying so the neutral patterns are more general that's what in fact famously pervis and pacala could hold and if i if i may just a quick follow-up was that pattern completely monotonic i i noticed it was not it was not a straight line it was there was some no that's you have to see the two patterns together so one is the frequency of species distribution uh with respect to the distance to the outlet so essentially you count the number of sites which you have the same distance from the outlet it's a fairly complicated structure so essentially it opens up it closes up and uh and uh and the other one is essentially the uh simply they you simply measure the average that is the local species that are there so you have this different distance from the of course if you are interested in the catfish distribution the neutral model won't work right so you have to go into a serious model of the thing but if you look at large scale patterns for instance for uh conservation reasons that the the neutral pattern gives you robustness reliability and the capability to make decisions actually it's basically a conversation for conservation practice okay if you're not further question i'm totally a bit delighted to go because i have another meeting fairly soon sure sure absolutely if anybody if anybody is interested in any feedback and wants to digest these they can write me anytime thank you thank you very much for your kind of ability and for the beautiful side of lectures and uh uh we will uh return me again uh in a short while for the following lectures okay okay okay excellent we are live again uh we'll come back for the next lecture by Samir Suarez 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 joining us here 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 uh 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 he did from experiments to theory so he did really a great great and incredible job and then with Andrea Giometto from Harvard now he moved in another university and 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 from the let's say fundamental questions that is one of the of the important question ecology so wow you can observe so amazing biodiversity and i think you already have been exposed by these questions on the fact that it's actually not so trivial to understand the the incredible diversity that we observe in the natural ecosystem this is one of the most famous cases the plankton so in plankton 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 plankton but this is in 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 in the in the last years and again you will have the opportunity to to 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 in the lab so in a controlled experiment and again just using very few resources even more resources you observe many species 20 30 that can actually coexist for a long time and this is surprising because in in an 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 one of the two is better let's say in uptaking resources so it's a has a fit it's a higher fit in that environment so what happens is that at the end if you if you wait long enough then the the fittest species 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 resource models in particular one of the fundamental achievement in theoretical ecology of course for sure is a is the model proposed by McArthur in the 17th and so this model as you can see is a model of the system of a couple differential equation one for the species population denoted by here by n and and of course species grows okay and grows by consuming a resource okay so the resource concentration here is given by the c and r is simply the resource uptake rate and typically is considered as a monofunction so this is a classic monofunction 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 the amount of energy that the bacteria can extract for some from from such resources so this overall part here is what define the growth rate of the species while we have a death rate delta that here we consider dependent on the species and on the other hand we have the resource concentration that grows here we are thinking of abiotic 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 the gradation rate here denoted by mu okay typically we will consider this mu equal to zero but this is not i mean we can do that without losing the general generalization of our result so in general you can see that alpha somehow represents so a bipartite kind of networks that tell you which kind of metabolites the species use 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 equations that we retrieved the so-called the competition exclusion principle that i think you have already heard that is to say so consider the stationary states of these two dynamics 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 have m species so sigma here goes from one to m so we have m equation and we have a p variable okay because this is a sum over p alpha so alpha now are the variables we have m equation in p variable but therefore if m so if the number of species is greater than the number of resources we have the more equation than variable and so there is no solution for the system except in the generic case otherwise the system it's solvable only if m is smaller or greater than p so if the number of species can be only smaller or equal of the number of the resources and this is 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 you can see that you will retrieve you will recover the generalized dot-cable terra model okay so the generalized dot-cable terra model can be obtained as a special case in a quasi-station approximation of the mccartel model okay so it's not possible to violate the competition exclusion 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 unlimited okay and in other words there is a trade-off between the metabolic strategies and this was pointed out by and being in a recent pl 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 termed 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 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 assumption of that work was that the that rate was species independent that typically they are very small so from this condition you can see that a solution for r star 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 r star in the second equation here so again you look at the stationality so you have the supply rate is equal to the sum and you put the solution of r star then you can see that this species all 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 consist of more species than resources then if you rescale all the quantities so we call x the rescale population as hot the rescale the supply vector okay I pointed out here that with this tilde here I do not the fact that I have absorbed the this y this efficiency but okay as hot is the rescale supply rate and we rescale the metabolic strategy so that all these quantities sum up to one then you can see that this define basically a multi-dimensional simplex and in particular all the the strategies and the supply rates line in this b minus one okay dimensional simplex okay so you can represent the geometrically the solution of the consumer resource model with energy budget and in this I represent here the the resources so these are the axes 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 or 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 colored points represent the strategies so red species it's only nutrient two because you can see this is only in in the vertex of of the of the resource number two okay blue feeds equally upon the blue species feeds equally upon species one and species two while the species the 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 all the three resources okay so the these conditions imply that the supply rate must lie inside the convex hull composed by the metabolic strategies that is to say if you okay now i'm not was not so good to to be the convex hull but this this region is the convex hull that is to say the region where you have as you limited limited by all the metabolic strategies this star the supply rate in order to satisfy to this condition must lie inside this convex hull so in this case the star the supply rate is outside the convex hull and therefore you will have extinction and the competition exclusion will be satisfied again 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 hull 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 perturb a little bit for example the budget based on the energy budget depending 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 so of course in general we understand 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 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 what is the oxy shift the oxy shift is the fact that you have you see there are different 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 metabolic 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 growth rate so this is a very strong and in fact then 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 our function of time okay so that's what we have done basically so we wanted to uh in consider in the McCarty's consumer resource model uh metabolic strategies that evolve in time so this means to write an equation for uh for uh the the metabolic strategies okay so how how to to write such such equation well we 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 um uh in a way uh the possibility that the species can adapt to the to the environment uh and select which kind of of metabolic strategies you want to use so again if this is the growth rate the adaptation so the the the equation for for the metabolic strategies is simply in a simple way the the gradient of of the growth rate so this in this way we optimize the the the growth rate and the one over tau or lambda denotes the velocity of this adaptation okay then we will see how this is related with the parameter debat so now we are so we have a species dependent um adaptation velocity okay so this is now our new equation so we want uh uh 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 uh alpha dot you will have a different increase when we put a soft bound on on a species dependent soft bound on on the on the energy that can be used to produce metabolic metabolic strategies and so now we have to perform such optimization constraints so it's possible this is a 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 uh so this is uh uh the phase space of just the two uh two resources so we have a fifth alpha sigma one and alpha sigma two and this line the east star sigma divides the plane in two region one that is the the allowed region we can move inside this this uh half plane because here the uh energy is less than the east star why we are not allowed to to cross such such line we're not moving 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 uh components of the gradient that is parallel to the gradient of the uh of the energy okay so in this sense we have to to perform this uh evolution by removing all the time these components so that it uh these will allow to not always to move at best around the tangent of the school okay so if you do that this is uh i mean this is just to perform these calculations 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 uh 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 um 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 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 simulation of the model and you know kind of the general setting with general parametrization and you can see that indeed okay so this is the the the model this is the the data that I showed before and actually using a lot of metabolic strategies allow to to reproduce the oxy shift in the growth curve okay so here if we think that there is a strong preference on more resources or we can think that we can turn it off and on to metabolic strategies we indeed observe this kind of behavior this 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 Leonardo together with Andrea they performed an experiment so an experiment where they have cerevisia that eats glock glock galactose and then as a as a waste it produces ethanol and and in once a galactose is depleted the cerevisia eats the so feed on the wasted ethanol okay so this is 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 to describe it want to to this behavior with with the model with the consumer resource model so this is the best fit okay using Monte Carlo chain methods of the model with adaptive on the left on the left and with 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 we perform this this best fit of the model and you can see that as expected the consumer resource model without metabolic adaptation follow a simple and let's say with the one-slope growth curve while with the adaptive strategies you really can see that we can quantitatively describe the the experimental data okay now let me make a link an interesting link this 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 and you can i mean one of the most celebrated equation describing this metabolic theory of ecology is the so-called the clever law that described the relation between the mass of species and their metabolic rate b okay and indeed over several order of magnitudes you have that such relation is a power law with exponent three three over 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 fields of the existence of such a metabolic rate that is the fundamental trait that govern many patterns in biology so if we assume that clever laws holds then in our equation our in our in our physics of the by physics of the model we have different rate one given for the death rate another one given by the adaptation velocity another given by the rate of metabolic production well all these rates will depend on the clever 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 it can be shown that if we assume that the the metabolic theory of ecology holds then we have that both the total energy budget and the death 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 the death rate is species independent okay and also we have that the death rate uh i mean the the the the adaptation velocity can be written as a function of the death 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 you can ask me what's happening 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 death rate is a species independent okay so again now we can write for our model a condition for coexistence and in fact we have that in this case all species survive again so these are just different little mathematical details but the 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 uh derivation but the same result 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 okay let me see if i can put it depends on time now okay and okay so now if so now this is the initial condition for example okay so after scaling we set initial condition alpha t 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 actually you can see we have extinction of many species and only two in this case survive and chap is recovered the commercial principle 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 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 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 it's not the closest so i mean it's not trivial to find based on the position of the metabolic strategies the the abundance of the species these are open problem as far as i know but we can see that the strategy self-organized and coexistent is allowed okay so we can now look at the other other experiment for example you can think about perturbing the environment so that supply rate now is the star and for a 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 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 so 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 the constraints the community self-organized so 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 non-velocity of the perturbation that is to say that so here I 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 uh adaptation velocity and also for no adaptation at all so you know for no adaptation at all of course you recover check okay so you only have three species survive but you see you do not down a little the adaptation then more species survive but not all the species survive only six if you never increase the adaptation around 13 species survive and then if you increase again the adaptation velocity then all the species survive so the adaptation velocity is a fundamental control parameter in controlling the 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 the species growth rate and we have we observed that this allows to to describe the oxy shift growth pattern and moreover that this adaptation drive self-organization of 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 uh conclusion that I hope uh you have uh if there are questions 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 experiment to test the uh adaptive strategies prediction uh okay so first okay uh our first test was that we have done was uh to to to perform the experiment about uh try to describe the oxy shift okay so in this case uh I mean this maybe I mean it's not a prediction in the sense that we know that we observed the oxy shift but if you wish you can actually see one experiment to test the strategy about uh adaptive strategy prediction would be to be able to cultivate microbial community with species maybe engineered species characterized by different adaptation velocity or you you actually false some 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 but of course I mean you you may think of many experiments that's what we are doing right now so not an experiment at least I don't want to speak uh out of uh of rigor but uh yes we you can actually do uh I think many experiments to test this this prediction so second question was uh would you say that this model solve the planton paradox well okay same definitive words about the science is always uh I mean uh to to demanding so I would not say that solve the paradox of the planton I would say that suggest uh 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 yes in this regard I would say yes we what well 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 it's not probably the only contribution to to to to the solution of the paradox but for sure is one one one part one important part of that okay so if not I go to the second part I hope to to be able to to be on timer there's a 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 uh to noise in in in in where so yes we we have checked that these results are robust to noise depends where you what you put in the noise so I will at the end maybe you remind me this question I will show you that if you perturb the the condition so if you put the noise in the ratio between the energy budget and the deteriorated so if you allow the ratio of the energy budget and the deteriorate uh to uh 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 uh competition exclusion but this will occur in uh uh time scale of the order of 10 to the 8, 10 to the 9 depends on the variance okay so this is a important point the effect of the noise is like when you when you go from the deterministic to to you add a noise to I mean you have a solving state of course in the deterministic case maybe you are stable uh in the in the stochastic case if you have that you you pass the barrier but maybe in exponential time okay so in this sense it's a 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 uh what does it mean I would love to answer but I have to ask uh you can and uh ask yeah my question I was thinking like okay we heard the the other day from James O'Dwyer of non resource non-stitutable resources it means that like one species for example plankton it needs nitrogen so the adaptation of its strategy I mean it can adopt strategies but still these resources is needed it 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 uh would means that I mean this is not to incorporate in the model so I I 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 constrained on the entries of this metabolic matrix but how to put this non-stitutorial resources in the growth rate this is not I mean this is not I didn't think about okay so this is not trivial so you can constrain the model to use if you wish some resources but I don't in this model in this moment there is no I mean preferential use of one of resources so there is a 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 the assumption of the growth rate maximization I mean think about a community with um you know consists of the two species that have the obligate interdependent like you know amino acid offshore um so how does in that case that your model needs to be you know changed or you know added a term to account for um the cases you know the the mutualism and and not not the species want to optimize its own but the whole community okay yes 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 so that's why I said before that this 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 it's just adaptation in a complete in a poor competitive communities but for sure in the same framework actually you could and we are doing we want adding also cross feeding and other cooperative mechanism okay so I'm 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 enlightening on this second part of the question in fact my second aspect is bridging try to bridge we try to bridge the cellular and the ecological scale because 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 specialized in particular metabolic function and we have already seen that the metabolic adaptation is very important in in in determining the the the evolution of the population so what we want to understand is the function is of course the function performed by a species depends on the protein it is producing and so the balance between 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 okay so how does a location of the proteome affect the dynamics of microbial 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 the complexity of its proteome that is the all the proteins that can be produced by that 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 fee are the fee q so fee q is the fraction of the proteome allocated for housekeeping function and typically this is a hard chord 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 okay then we have two other region two other families one is the proteome allocated for nutrient uptake and the metabolism that is the one related to our case and we call this with by 5 p and then there is the allocation of proteome for biomass synthesis that are ribosomes okay now 5 p and 5 r as i will show can vary but are in trade in a trade off okay and this important relation between these different proteome allocation and the growth rate so this interdependence of cell growth and gene expression is a very similar work of a similar work important work of the group of tennis one that was published in science in 2010 so in this work they found phenomenological slow okay that described the relation between the proteome 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 proteome allocation for for the for the sector r 5 0 you see is the basically a hard core so this is the mean you cannot devote less than 5 0 because these are needed to leave okay so it's a compressible part of the 5 r and then you can see that increasing the the amount of of of proteome dedicated to this sector you increase the growth rate okay so it's a linear relation okay kt is basically a measure of translational capacity so how fast the microbial species expresses genome while raw is just a conversion factor okay so these are the details on the other way he also for the proteome allocated in the p fraction so the one of the metabolize for the metabolize 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 proteome dedicated to the mutual uptake is proportional to the growth rate the similar is for the r sectors that is proportional to the growth rate while the the housekeeping function is is uncompressible it's 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 n res nr resources and np species okay so now we have nr resources np species what we consider is that the p sector is subdivided okay for the different resources okay so phi sigma p one is the metabolize is the proteome allocated to uptake resource of type one phi sigma two p is the proteome allocated to uptake resource two 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 okay so because we are focused on the on the on the proteome dedicated to the p sector we will for simplicity just denote this fight phi so when i there is no uh superscript superscript 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 of 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 resource and if you do this and you put this condition in this constraint the normalization constraint you go you obtain this condition for the uh for the proteome uh allocation for uh species sigma to all the resources i okay here and you can see that this sum this is constrained by this capital phi okay so this capital phi is the total proteome that can be allocated for the p sector basically for growth for me for me for resource update okay so this is just come from this uh water and 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 proteome allocation of this sigma one that will be proportional to the uptake rate of the species and again and in in term this uptake rate that will contribute to the growth rate of species one following the the the laws that i just showed 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 that rate okay so the new equation are simply this coupled equation here so it's the same as before but now g sigma directly depends on the proteome allocation of the species so this is an explicitly all the all the equation so so now we'll 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 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 there is the new constraint the totally budget allocation now takes just the form of this proteome allocation constraint okay and very interestingly you see that this proteome allocation for the p sector is fixed but this is equal to this left part this is the third equation and actually you can see that because r changing time also phi must be variable so you see that the proteome allocation for growth so for for these 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 proteome allocation so at this point this is just should be so if this phi must be variable then we need to write some dynamic dynamic equation for the phi that correspond to the metabolic dynamic metabolic strategies and again we have 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 too 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 phi are the proteome allocated for the p sector so 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 and again you can call the ratio that ratio they are with the capital um theta and then you have that here you can write r star in this way and then you can actually the the the constraint the constraint the biological constraint now i mean this is a constraint that has a biological meaning is that you can see that maintenance cost is proportional to the proteome allocation total proteome allocation of the p sector and in this sense if phi is phi sigma increases we have to spend more energy to synthesize catalytic and ribosomal 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 having coexistence again you do the similar as before you rescaling m s and phi and you obtain again that species coexist if the supply vector is within the convex hull okay and now phi is the proteome at the stationality phi star is the proteome allocation stationality of the species i to resource to the three sigma to resource i so again now you can see now you have that through the hope we start with our given initial proteome allocation in the first case if the supply vector is outside the convex hull you would have extinction but now proteome allocation change so that the new metabolic strategies are the supply vector is within the convex hull and you find and you find coexistence so this is what I was mentioning Leonardo used two engineering strains of the collie and completed demo on one common resources and one of the strain was engineered so to we can change the proteome allocation experimentally and we can evaluate the outcome of the 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 proteome allocation on species and and and try to predict the outcome of the competition and again you can see that uh the the the velocity of the adaptation is again a fundamental parameter if they as you can see here this is a the condition where the velocity where the adaptation is high so the species can uh adapt and coexist all the species can coexist in this case adaptation velocity is uh is low and in fact you can have uh in this case you you you you don't observe coexistence okay so in this case you can see if ipsion is a ipsion is large enough you start to have extinction it is to say if adaptation is low enough you start to observe adaptation so depending on on the adaptation velocity again we have we have coexistence or not i wanted to align and i'm i'm i'm going very few slides and i concluded uh because this i there was a previous talk was mentioning this so depending on on the amount of of of supply rate even if adaptation is low but if supply rate is very strong then we can have a species to coexist so if we have a strong influx of 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 velocity is not enough and so only two species coexist yeah we have always three resources only the yellow and the and the blue species coexist but now if we increase two times uh the the the supply rate yet uh only two species coexistent rationality but the extinction take more much more time and then if we instead increase the file time the supply rate then you see that uh all the species now start to exist okay so 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 and overall talk is that including problem including problem allocation models of competitive communities give us new insight on the on the dynamics of our bacterial communities and the relationship between a problem allocation and population dynamics seems relevant to understand the origin of the high level of biodiversity so here we have like a microscopic let's say uh explanation or not explanation microscopic ingredients microscopic process that have an impact on the on the on the macroscopic coexisting pattern of the community so it's just a starting point but it's a starting point that bridge physiology of micron to a community ecology so we are very excited about this uh and because we think that this is really an opportunity because basically all the questions that probably we will uh we'll ask now are open questions because this was just uh really uh done in in the few in the last few months and but these consumer problem resource models suggest 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 uh I really want to thank Leonardo for his great work and these are the two references of the 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 it will 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 uh 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 our whole machinery to to model metabolic fluxes so the metabolic networks through metabolic networks so it would be very interesting but yes I mean I don't have a quantitative answer to this but for sure uh qualitatively adaptation velocity must depend 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 uh change the outcome of the final composition because of these differences in the velocity of adaptation absolutely yes this is the understand 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 this is the main let's say attempt that we are doing in order to understand this trade-off we think that we need to go at the cellular scale for example at the plot location but also for example this is a very good path to look at different uh uh uh lag phases in the metabolic machinery and understanding in this way the emergent trade-off between species and and so to understand uh basically that this trade-off emergent from the from the cellular and and the metabolic scale will have a for a strong impact on the species coexistence this is exactly the point okay thank you yeah i have i have a related question um so i mean related to to to the adaptation velocity but 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 um 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 hours then we can know if that adaptation in the metabolic model actually reflects 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 do you have any thoughts on that yeah thank you yes i can offer a thought 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 i think that my understanding this is a way to to to to to test adaptation velocity can actually be done through uh the oxy shift uh uh pools okay so if for example you take different species you cultivate you cultivate independently into into different medium but with two different resources and you can actually have an idea of the velocity of of adaptation for for each of the species so i think that basically this was i mean this is the main uh uh suggestion here so to to have a model which parameter can actually be measured in the lab so in this in this way you can actually perform the 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 and so test that maybe basically the the mechanism that you are thinking are important okay so that's what we have done in the experiment with with with Leonardo where basically we have we have uh cultivate uh yes so we have basically cultivate two strains and one of the strains was producing basically useless proteomes was allocating proteome in not useful for for the resources for the medium in which the species growth and in fact we saw that this leads to a competitive advantage in that quantitative way as expected by the model yeah yeah thank you thank you thank you okay i don't think i see any other no operation uh so if that's the case thank you again samir okay thank you it's very interesting in research talk thank you very much uh i will and have a good continuation thank you thank you bye bye we will have the next lecture in about uh 10 minutes so everybody can take a break bye welcome back everybody uh we can start again with our Friday session the next lecture that is Daniel Segre from New York Steel Boston with his second lecture about metabolism from his genomic skills to existence please Daniel thank you slides work yes okay great um hi everyone again and so we're gonna continue talking about metabolism in microbial communities so this is a perfect segue from the previous talk and i just want to remind you um in the previous time we were talking about the logic of the cell the metabolism as a resource allocation problem and just to remind you briefly briefly um the idea was to look at the complete metabolic network in an organism with knowledge of the nutrients that are coming in the biomass components that are needed to produce a new cell and new biomass and we introduce flux balance analysis where there are constraints on the concentration of each metabolite not changing in time so steady-state constraint there are assumptions of capacity limits for the nutrients that are coming in and for some reactions that may be known to be irreversible and we showed that one can use optimization for example finding the the state that is optimal for the cell to efficiently produce its biomass through max maximization of the growth rate and in general one can actually write this problem in the following way where there is a steady-state this expressed as this relationship between the vector of the fluxes multiplied by the stoichiometric matrix we saw last time and there are general capacity constraints in general what can put upper and lower bound to each flux and optimize any linear combination of the fluxes using linear programming we also showed the geometrical interpretation of this where there's a feasible space and we're really looking for an edge and vertex in this polyhedral cone this convex structure that will represent the point that maximizes our objective function now we want to dive straight into how we can apply this to the study of microbial consortia and I will first show why metabolism really matters for microbial consortia you saw some examples already but I want to walk you through the way we and others started thinking about this from the perspective of the complete knowledge of the metabolic capability of organisms and one slide I always like to show about this is the following so this is one of few instances I think where a lot of the interaction between different microbes are well characterized so this is a picture from a review by Colin Brander and colleagues showing different shapes these are different microbes that colonize in this case the human teeth so this is part of the human oral microbiome and what is stunning about this image is that really a lot of these links are known metabolic interactions between microbes some of these are the early colonizers and then there is this growing community that is we try to get rid of by brushing our teeth and what is known about this interaction some of these are known as contact interaction between different species and what is known also is that some of these interactions are related to metabolic exchange and as you heard before this is probably a very common way in which microbes can cooperate with each other or exchange with each other material in many different ways in this case for example a non-pathogen for pyromonas gingivalis can exchange different metabolites with another microbe in this biofield so this was you know this is interesting the question is how common are these interactions between microbes based on metabolic exchange can we model them using flex balance modeling and so on and I'll show you first how we um early on tested this idea that metabolism really plays a role in the formation of biofilms and microbial communities using exactly this uh known structure of the biofilm um this is an idea from a former student in my lab Barou Mazumdar who took this map and asked the following question so you can imagine having looking we knew the genomes of all of these microbes these were well characterized microbial strains so we could look at each of these strains and their intracellular internal capabilities in terms of what metabolic functions they had and at this stage we weren't ready to do yet genome scale modeling this flux balance analysis for each of these organisms but we took a much simpler approach which as I'll show you soon will was nevertheless quite quite insightful um and the idea was the following for each pair of microbes we could ask we could compute a metabolic distance so this is a pair way pairwise metabolic distance just based on the profiles of which reactions each organism contained so for example if you have two organisms here a and b um you know simple version of this reaction vector reaction content vector so for example organism a contains reaction one and reaction four organism b contains reactions three and four and based on these strings that are just binary strings which you can obtain from the annotated genomes of this organism you can compute a distance we computed a jacquard distance um to to quantify how different metabolically these two organisms are and of course you can do this for any pair of organisms and you'll have this matrix of similarity or dissimilarity based on the metabolic capabilities of this organism um so what was done then was to compute the average metabolic distance for different kinds of paths through this community so you can imagine taking paths that are we called order preserving so paths that only go upwards in the biofilm in what is known to be uh the layer structure of the biofilm from the early colonizers to the late colonizers so this would be an order preserving path because it only goes upwards and of course there are many other random paths that do not preserve the order of colonization that can jump up and down um between different species in the biofilm so in doing so we could then compute the average distance um between all pairs of organisms in both the random paths and the order preserving paths and what we observed was that there was a clear difference between the distribution of metabolic of average metabolic distances between the order preserving paths and the random paths and in particular the order preserving paths had a pairwise average pairwise distance between subsequent organism that was significantly smaller than the average pairwise metabolic distance for any possible random uh non-order reserving path um so this was an interesting indication that somehow if you look at the correct order of colonization there is something unique about the species species similarity in terms of metabolic functions and um as you may imagine one possible interpretation of this is that organisms that are um metabolically more similar to each other will have metabolites to share and will be able to gradually connect to each other metabolically building this biofilm now um this somehow is a there was an early indication that metabolism matters in microbial communities and in particular in this case in the order of colonization of the biofilm we observed by the way that if you look at the same um property of in pairwise average distance for non-metabolic genes you don't see this clear distinction between the two distributions um and of course there are different possible interpretation of what was found right this could reflect really the fact that organisms build on top of each other but could also reflect at least partially an environmental gradient for example maybe there is a more anaerobic environment at the bottom of the biofilm and increasingly aerobic as you uh as the biofilm grows and in that case perhaps the similarity in the similarity reflects uh adaptation to this gradient uh but there was something else that came out of this which uh felt was quite interesting this kind of paradox if you take this idea that metabolically seemingly similar organisms will tend to stay close to each other and build the biofilm um what would prevent this to collapse into a you know absolutely minimal distance boredom where all the organisms are really clustering in in a structure by uh by their similarity of course in that situation competition might dominate the capacity to exchange metabolites to synergy but it raises this interesting question of whether there is an optimal metabolic distance for metabolic synergy between uh complete difference between two metabolic networks and complete similarity and I don't I don't think this is a resolved problem I think there are interesting papers coming out recently on this question I want to show you how we start addressing this early on by using a concept that is called elementary flux modes and I don't have time and I'm going to hold the details of that but I'll just mention what is essential here which is this elementary flux modes are a way of um enumerating all the possible pathways in a metabolic network all the possible minimal unique pathways in a metabolic network um and you can take an organism um and um duplicated and ask the following question if I put two organisms together and just count the number of pathways that are possible when I have these two organisms together uh relative to the elementary modes that are present in organism one plus the number of elementary modes the number of these pathways that are present in organism two and if the two organisms are completely non overlapping and you can build artificial metabolic network to um engineer to have an arbitrary degree of overlap between these two networks so if these two networks have zero overlap then if you come compute this quantity the number of pathways of the system all together divided by the sum of each of the two individually then of course the sum of the pathways exactly the the sum of the pathways present in each organism and this quantity is one and at the opposite extreme if you take two organisms that are exactly the same have exactly the same metabolic capabilities uh when you compute this quantity each organism the two organs are the same so the elementary modes of the uh junction between these two organism is really the same of each organism alone and this quantity will have a value of half and what is interesting and you cannot really uh quantify this unless you actually do the calculations and we did these calculations you can find that there is and this is almost like an um analysis theorem right there there is a maximum here uh as one might expect but it's interesting that you can find this sweet spot of how much metabolic overlap will lead to the maximal number of new metabolic pathways that are embedded in this combined systems uh system of organs one organism two so this is just based on the topology it's a very simplistic calculation but it points to the possibility that there may be out there some um a sweet spot or some kind of uh ideal level of metabolic similarity that will lead to maximal enhancement of metabolic capabilities and of course this doesn't take into account uh competition it's just about how much new metabolism can be done by bringing two organisms together and as I mentioned last time I'm gonna pause occasionally but feel free to interrupt and if I'm not monitoring the chat but there is a question just someone please stop me and I'm happy to uh pause and address any question so from this early very um simple um presence absence analysis we really want to move to a stage where we can model computationally the dynamics of communities and as we saw in in multiple talks including the previous one one can look at uh a community as a set of entities right that can represented between you know in different different different levels of description this would represent maybe more like a lot of altera model where you have individual variables representing individual organisms you can try and model the community as um an ecological system in this way um there is the possibility which is really what flux balance can do and what we'll focus on where you can think of uh the circuits within each organism and try to predict the interactions between different species based on what you know about the intracellular circuits of each organism and as we'll discuss probably uh next time there is also the possibility of thinking but I want to hint to this now there is the possibility of thinking of a community as a soup of enzymes where perhaps uh for a complex community what really matters is what functions are present overall in the community and we can ask the question of whether or not compartment compartmentalization matters so is it important to know which functions are performed by which organism or is it possible and useful to think of a community as this overall conglomerate of metabolic functions so for now we will focus on this type of modeling where we really uh know the intracellular wiring we know the environment and we try to predict ecological interactions in the dynamics and the structure of the community and we'll also talk a little bit about design of the community and I see there is something in the chat if it's a question yeah do you want to ask this question uh yeah yeah thank you so I type in the box so my question is that for the first example of the order of the colonization in the bathroom so I saw the result is that the bacteria in the order of the past has smaller metabolic distance so my question is that does that mean that you know if there are two types of the the effects which is one is the environmental gradient another one is the ecological interactions could could change the dynamics so would that result mean that the uh environmental ingredients dominate over the ecological interactions uh because in my mind you know maybe I'm wrong just think that if the ecological interactions are more importantly to you know preserve the order then there might be a lot of cross-fitting between the metabolically more different bacteria that might be prevalent which is not the case in your data so I just want to add some some comments on that question yeah that's a very interesting question and I guess I think from this early analysis we don't don't really have enough information to to determine this what I thought was interesting here is that if you practically look at what organism what like the evidence we found there was that if you look at the organs that are close to each other in the biofield there seems to be a tendency toward a smaller metabolic distance and I agree that we don't really have enough information to you know to know whether to interpret this as okay there is a just driving force that is the gradient of environmental oxygen or nutrients and so on but there could be also a situation where each organism right modifies the environment for the next organism to occur and and then the question is and the possibility what I what I think might be happening is that uh organisms that are too different from each other they may not have enough um cross-feeding opportunities and and of course if organisms are too close they will compete but there may be some sweet sweet spot in between and we can get back to this I think uh the answer to this will really come from looking at the more advanced dynamical models will which uh analyze but I'm happy to go back to this question which is very interesting so um so we want to move to to these dynamical models and I want to show you how there was a different direction early on and now there is an explosion of different experiments of this kind but this is really these early days of trying to think of synthetic cooperation and now the idea of building synthetic microbial communities as a way of testing hypothesis and checking what is really happening when you put the organisms together can microbes cross feed and so on and so forth and there were these early attempts that were really interesting and a motivation for a lot of the things we did later on so this was a paper from um Winnin Shu and collaborators in 2007 and this was an engineered cooperation between two yeast strains one of which could not produce adenine and the other could not produce lysine and the idea was that only when grown together they could really uh would be able to survive and in fact this was indeed the case so this was an engineered cross feeding interaction that made each of these two strains uh completely dependent on the other it turns out and Winnin has continued working on this doing beautiful work showing for example that it's not clear um that the metabolites that you would expect being exchanged but it's the terminal portions of um of these pathways are the one being exchanged and there is a lot of interesting aspects of these dynamics and I'll mention more later on the other example um more focused on uh just trying to find different types of interactions as opposed to engineering them this was done with E. coli strains uh library of mutants and the idea was to put these mutants together first I mean if you grow them individually they grow fine in rich medium uh individually they would grow very poorly in minimal media because they're mutants that lack the capacity to synthesize I think these are only amino acids but occasionally when you put them together you could see synergistic growth so this was a way of trying to detect new interactions and there are a number of new interactions that were detected this was worked by Ed Wintermuth and Tom Silver so when we started thinking about this we thought it could be interesting to um try and mimic some of these ideas using stoichiometric models this was work that a former student in my lab Neil's Clipcord pioneered and we tried to do things in a slightly different way so rather than tweaking the internal circuits of the cell as done in these previous examples or you can do mutations and try to induce interactions based on changes in the circuits in particular auxotrophies or removal of genes that are essential for producing essential compounds we thought that perhaps it would be interesting to tweak the environment and so take two organisms that are natural occurring microbial strains and ask whether we can induce an interaction not by changing the circuits inside but changing the environment and the idea was that well first of all this is you know simpler to test potentially because if you want to test in particular a high throughput interactions of this kind it's much simpler to just provide different nutrients experimentally than having to do mutations to the strains and the other aspects of this which turned out to be really the beginning of a new line of research is that and a lot of people are obviously interested in this the environment clearly have a has a strong effect on modulating interactions so this is a way of starting to look at how the environment can really you know kind of environment environmental changes induce interactions and what is the role and how much variability there is in these interactions as a function of environmental composition you can change in principle the carbon the nitrogen the sulfur phosphorus source and so on so there is an endless combinations of different nutrients that can be used to try and induce these interactions of course one could use a rational approach and we'll see more of this but for now what we did was just simply use flux balance modeling to try and find in a large space of possible compounds some that would induce interactions and I need to tell you a little bit more about how this is done in practice because it's non-trivial right when you look at this we saw how to model an individual organism but how do you know how do you go from a stoichiometric model of an individual organism to a stoichiometric model of a community where you have two organisms together and the answer in the end is really something that existed already in the flux balance world but was used for different purposes and the idea is to use compartments so you can build a compartmentalized model and I'll illustrate this with this very simple example where you have two organisms one and two and they have a very minimal network organism one can produce b from a organism two can produce c from a but each of them has a biomass that depends on a b and c so each of them needs all all of these three compounds to survive and if a is the only compound provided in the environment the only possibility for these two organisms to survive is to exchange b and c so this is a you know minimal example of cross feeding if you wish but is all it also illustrates how you can build a model of a community using stoichiometry and flux balance modeling and the idea is that you can define you'll have multiple versions of each metabolite so you'll have an environmental metabolite a and you'll have an environmental sorry a metabolite a that is in organism one you can label it as a one and you have a metabolite a that is in organism two you can label it as a two and so on and so forth and you can write the system of reactions just labeling the organism the metabolites based on which organism they occur in and and you'll have essentially essentially a block diagonal matrix representing this system of two species interacting so this is um you know in a very superficial way that you know the way this stoichiometric models for communities can be built based on this multi compartment model this was first proposed by uh stoner and um david stahl in a very nice molecular systems biology paper in 2007 so we took this approach and used it to scan systematically systematically the space of possible environmental metabolites um and just to illustrate briefly the way this algorithm was designed you can first take two organisms and ask under what conditions can these two organisms grow both a growth rate that is above a minimal threshold and you can search all the possible carbon sources you have all the possible nitrogen sources and you can do the same for other elemental sources but you'll have you find those that provide growth to the pair of organisms together you'll have a set of putative media that support growth of the whole ecosystem and now what is interesting once you take this media you can ask for each of them whether it also supports growth of each organism on its own and all of this again to remind you you do you can do easily using flux balanced analysis so you have by definition right we've had many many different environments all these different combination of carbon and nitrogen that all support growth of species one and two together in this joint stoichiometric model but then you can take a given environment and ask will it that environment also support growth of organism one alone and organism two alone and if that the answer is yes then you found an environment right that supports the pair together but supports also each individual organism and this is a case where the organs are not really interacting they can grow on their own they can grow together nothing interesting about it but you can start finding then environments where for example the two can grow together that's again how this were originally found organisms one can grow but organisms two cannot grow and then what this means is that the two organisms grow together but two cannot grow by itself this means that one must be providing an essential component to organism two same situation here so if you find environments that satisfy these conditions these would be environments that would support a commensal interaction where one organism depend on the other and if none of the two organisms can grow on its own then again because the pair by definition was growing then what you find is a set of conditions that imposes induces a mutualistic two-way cross-feeding interaction between the two species so what is nice is that you can easily make this list of many many environments and for each of them you can test whether which of this is the case and ask how many times will you find environments that induce for example these mutualistic interactions and I should say when we started doing this we really didn't know what to expect how often would this happen and the idea was to start getting an idea of how frequent how prevalent how large is the space of this possible cross-feeding interactions in metabolism and what Niels found was actually oh yes sorry just a quick question maybe I missed this but in the joint FBA what's the objective function is it the sum of both great questions thank you yeah thanks for asking yeah so there are different flavor flavors of this joint FBA in this early so I'll we'll get back to this because that's actually this very question motivated a lot of other things I'm gonna talk about but compartment models you have to choose an objective function and the very first case was based on maximizing a linear combination of the two biomasses so you can create a new reaction that builds a linear combination of these two biomasses with a fixed proportion but then you determine in advance right what is the proportion of the two species you can do this by scanning many different proportions and seeing which one seems to be most growing faster but it's it's a little bit tricky so you can see already that this question of what is the objective function of a community turns out to be a really interesting question but also a tricky one and if you choose an objective function for these communities you know you this is a little bit like testing a hypothesis what we did in this case because all well all we wanted to know is whether we could find an environment that supports growth of each organism so what all we did was in this case ask that the growth rate of each organism has to be above a certain threshold so we asked that each of them grows at least a certain amount doesn't matter you know they don't have to grow optimally they have to grow above a certain threshold and then what we did we used mixed integer linear programming to minimize the number of exchange reactions so we asked what is the minimal way for these two organisms to potentially exchange something so that they can go grow both above a certain threshold which is why if the minimal number is zero that's totally fine maybe the two organisms grow together without you know having to exchange anything and in this case is there will be a non-zero number of exchange reaction but thanks for asking these questions because I forgot to mention it so that's clarified yes yes thanks okay great so so let's go back to the results here which again were quite interesting and this exemplifies some of what we found so these are seven species for which we run all these pairwise interactions and there is organism one and organism two here but these are the same organisms you can look for example at the interaction between E. coli and Salmonella and what you see in this pie chart the overall size of the chart pie chart represents the number of media of different combination of nutrients that we found that could support the two organisms together so for example if you look at E. coli and Salmonella there are millions of different nutrient combinations that can support growth of the pair whereas for example E. coli and H. pylori have a very small number of sets of nutrients that can support both of them together and then out of all these possible nutrients you can look how many of these are of this neutral kind that is environments that support also growth of each organism on its own and Salmonella and E. coli are very similar in their metabolic capability so as expected you find a large proportion of this green portion of the pie chart of nutrients nutrient combinations media that are essentially good minimal media both for E. coli and Salmonella and of course they support growth of them together but there is nothing interesting about this but there was also a lot of interactions that are commensal of one organism providing something for the other organism and what was most stunning and we really didn't expect that there are a lot of opportunities for this cross feeding all the yellow portions in this pie chart are cases where really this would be environment such that if you feed those nutrients to those two species they really need each other in order to survive and I should say that this is only based on stoichiometry right if we were to do exactly the experiments in the lab I wouldn't expect all of this interaction to occur because the fact that this stoichiometry have this property doesn't mean necessarily that the organism will have the right regulatory program to express the right genes to induce that interaction and so on so this is a purely theoretical flux balance stoichiometry based diagram but what it illustrates right is that there is out there in the microbial world there are millions of opportunities for cross feeding and that they're strongly dependent on the environments in which the organisms grow and also illustrates that in principle if you learn how to manage these possibilities there could be a lot of opportunities for engineering communities where you by designing the environment you could decide whether or not two organisms will depend on each other so this was promising but as I was hinting to and you know the question asked was hinting to and there are some underlying assumptions in this type of model that are that are somehow tricky and will so in particular right the require assumption on this ecosystem level objective of what you know how do you manage these two biomasses how do you know there there is some interesting hypothesis on the possibility that maybe you could use thermodynamic base objective function so I think this is a fascinating question but there are other limitations of this approach for example in the same way as flux balance will not allow you to predict intracellular metabolite concentrations because those are factor out when you assume the steady state for similar reasons with this type of compartmentalized based approaches you cannot predict the amount of each species in the community which is often one of the main main things you would like to be able to know so you cannot predict how much there is of each species at steady state and this is a major limitation of this approach another limitation is that it's very difficult to do spatial temporal dynamics you cannot really do dynamical models based on this because it's all steady state approximations and you can just compute one steady state and as we'll see later the solution to this is going to be what is called dynamic fba or dfba which is an extension of flux balance analysis which will simultaneously solve a lot of these issues and I think open up a lot of new opportunities now before we go there I want to pause for a second and think about this question of why would microbes exchange metabolites right and there are many different angles for this and we'll see this from different perspectives and this came up and we'll come up again I'm sure in other talks but you know metabolites are part of this strategy that microbes develop to you know grow and produce their own biomass why should they give out metabolites to someone else and as we saw last time right there are some pathways such as fermentative pathways that inherently give rise to secretions and these secretions might be helpful to other organisms but it's not clear how prevalent these secretions are and whether indeed these secretions are typically you know something that would be very costly for microbes to produce and then give rise to questions about stability cheaters and so on and we'll kind of get more into this so so one way we started thinking about this was first by quantifying really the cost of metabolic secretions and this is work by Gardison and the lab Alan Pacheco in the first observation we made that motivated this this was initiated by Niels Klickler before it was asking the following question if you take a flux balance model say for E. coli and you can grow it on different combination of nutrients and ask the following question if you impose a secretion flux will you induce a reduction in growth rate so how much how much will you have to pay in terms of the growth rate if you impose that that organ is secretes a certain metabolite and as you might expect right there are secretions in this case succinate such that if you ask the cell you know before maximizing growth you say okay there has to be this amount of flux of succinate going out of the cell and then you maximize growth and the growth rate you obtain in this case for example on glucose glucose and glycerol as carbon sources the growth rate you obtain is significantly smaller than the maximal growth rate when you don't impose a secretion flux so this would be a case of a metabolite by this costly and kind of corroborating what we're saying earlier whether or not metabolite secretion is costly depends strongly on what are the nutrients on a different set of nutrients succinate production is not not very costly until you produce a lot of it but then what is interesting and let's look at this first actually because that's an example we already illustrated before there are metabolites such as acetate if you're growing under carbon lean sorry oxygen limited conditions this is something that will spontaneously happen and in fact if you impose a secretion flux of acetate that is small or zero right the cell will not be able to grow optimally and in fact it will grow better when you impose that there is a high secretion of acetate possible okay so in this case the secretion is actually beneficial and there are some some cases in between such as format where apparently for the two environments explored here secretion doesn't change the objective function doesn't change the growth rate so these are kind of neutral secretions so we we will call both you know these two kinds of secretions for the purpose of this up we'll call them costless that is metabolic secretions that do not impose a cost or at least a reduction in the growth rate under this assumption and I want to remind you this is particularly relevant since we just heard about the cost of protein production and the capacity of embedding into flux balance model the or consumer resource model that is also possible in flux balance models the cost of protein production so this doesn't include any of that this was done with regular flux balance models but one could extend this kind of approach to models that also include protein production and the cost of protein production so let me show you what we found so what Alan did he wanted to I find how frequent are these costly costless secretions in the microbial world and again the idea was to scan many different flux balance models looking for how often would we find we find costless secretions so I designed the following experiment the idea was to have enough variability of environments to explore systematically a large space so again this was done only in silico and I'll mention later there is a follow-up work that is being done done experimentally now on this but the idea here was to give two different carbon sources chosen out of a pool of different carbons and also choose whether or not to provide oxygen so we could do this in silico experiments aerobically or anaerobically and as hinted to before of course this can have a big consequence on whether or not there are secretions or what secretions being produced and then we computed the possible the maximum growth rate of two organisms of a pair of chosen organism under these conditions and what we did was estimate if any of the organisms could grow what metabolite could be secreted in a costless way so we asked is there any metabolite that upon maximizing growth each of these organisms could produce and now we did iterations where these costless metabolites was added to the medium rather than medium and we repeated the experiment growing the same organism but now on the original medium plus the costlessly produced metabolites so this the idea of this analysis was that we could get inside both into what costless metabolites could be produced but also whether these costless metabolites were really useful for facilitating growth of a second organism so we did this for these two different conditions of oxygen availability 108 carbon sources 14 different species for a total of over a million unique simulations and I'm not going to show all the details there is a lot of data that came out of this that is available in this nature communications paper but I just gonna illustrate what you know summarize what we found which is that there are many different type of secretions I know much more than what we originally thought and not just the organic acid the organic acids are this light brown portion so there is a large portion of organic acids but there are a lot of other molecules carbohydrates and your organic compounds the metals are not necessarily so interesting because they're just coming in and out of the models but there are some peptides some phosphate that are being exchanged and for an overall total of about 60 metabolites and a little bit more when you have no oxygen and as expected somehow there is a little bit a larger number of secretions when oxygen is unavailable which is interesting in itself and and again it's interesting to think of this in terms of ecological niches and whether really you know that's something that would be testable whether indeed there is prevalence or additional metabolic interactions in anaerobic conditions and the other thing that was interesting is that these secretions could really induce interactions across different species so for example overall all the different species if you focus first or for example the oxygen dependent one you can count how many see how many simulations both organisms could initially grow and there is a certain number and there is this is what is interesting this is the number of combinations of microbes and environments where growth could occur after cross-feeding so after at least one round of costless lipidus metabolites being fed back into the medium so there is a large increase in the possible growth capabilities induced by this costless production and these are the proportions of which either zero or one of the two organisms grow so the interesting part is that you can almost double the the number of combination of nutrients in an organism in which there is growth of a pairs of organisms because of this cross-feeding interaction so again this was based only on flaxbanas modeling based only on stoichiometry but it's a different illustration and points out the fact that in this case right the cross-feeding can be induced by metabolites that are really not inducing not causing a decrease in the growth rate of each individual organism and the idea that right where there's a lot of interesting work and I'm sure there is many cases in which interactions are due to metabolites that are costly and this could be evolved traits and we're gonna talk about this soon but it would be evolved traits where an organism produces a metabolite that is costly because it leads to an advantageous interaction but there is the idea that is merging from this analysis that there are a lot of opportunities out there for costless interactions things that are induced just by organising what is best for themselves and at the same time in doing so throwing out there something that someone else can use I view this as a little bit like as recycling right in social community social you know human societies you know there are things you know when you are finished with with your milk you know we throw away the bottle and if you can actually recycle it instead it doesn't cost anything to you but it actually can be valuable for someone else and the idea is that this kind of interaction may be very abundant in the microbial world and I this is a map of the specific metabolites that can be exchanged I'm not gonna go into the details and to this but if anybody's interested you can actually look at what specific metabolites are being secreted under what conditions and there is a lot of interesting follow-up analysis one can do on this but I want to summarize this just by showing that the emerging network that we you know we we looked at by analyzing this costless interaction so this is a network of what organism could feed which other organism in this set of organisms we analyze based on costless secretions and the picture that emerged here is that there is a dense network of possible interactions that emerge spontaneously between different species that may not require organisms to give up anything anything valuable but are just emerging property of the system and somehow this was similar to the result that in parallel Alvaro Sanchez and when you hear about this found in this organ in this community grown from on simple carbon sources from plants and soil and and this is one of the illustration from that work which you'll hear I'm sure more but what is interesting again the same picture emerged that on each organism could grow on the spent medium of each other organism somehow suggesting again that there is really there are a lot of dense networks of exchange out there and and this is somehow I must have been very different from what I was expecting initially that is that it's not necessarily individual targeted interaction but there is probably a dense network of possibilities now um and as we started thinking about this with these models um yeah I think yeah 50 moments as we started thinking of whether in addition to looking at the natural interactions in communities we could use flux balance models to also purposely design cross feeding interaction between species um and the idea was that you know when we look at this natural interactions through stoichiometry we really look at many different organisms many environments and and we look at what are the possible outcomes but we wanted to do this in a more targeted way and also potentially get insight into what what I'd like to think of as deep symbiosis where the exchange metabolites are not necessarily byproducts end of end results of byproducts of specific pathways such as amino acids but more convoluted interactions that you may not be able to look at or find intuitively such as exchange of two amino acids again so let me show you you'll see in a second what I mean so the idea and this is sorry I worked by Megan Thomas another from a student in the lab with the annus plus the annus plus colitis and others and the idea here was the following we took E. coli and you can ask the following question you have a certain number of reactions in E. coli about a thousand but you can force E. coli to use a smaller number of either internal reactions or exchange reactions so imagine the way I think about this is you have a knob you can say okay instead of using a thousand internal reactions now you're allowed to only only use 900 and you can ask how well can you do and you can tweak also the number of exchange reaction how many metabolites you can transport from the external environment and at some point you can imagine let's say focus on the internal reaction if you turn this knob too much to the left right you you decrease the number of possible reactions too much at some point the organism will not be able to grow you might see at some point a decreasing growth if you limit the number of reactions and at some point there is no way for the organism to grow but then one can ask the same question for a pair of organisms so you can start with two E. coli that are initially exactly the same and you can impose these constraints on the internal reactions on both of these but now each of them could choose a different set of reactions to use and now you say okay I limit to let's say seven percent of the original number of reactions they can use but now this you know top organism could choose one set of reactions these other organisms could choose another set of reactions and what we were wondering was whether we could find a constraint that would not allow an individual organism to grow but would make it possible for the pair of organisms to grow together again in an obligate synergistic cross-feeding interaction that would be now induced by our arbitrarily tuning the number of possible reactions and again this was done using classical fba in this multi-compartment model and I'll show you a couple of outcomes of this analysis this was done using mixed integer programming linear programming where in addition to the variables representing each flux we had Boolean variables representing whether or not each reaction is present in each of the two organisms and I'm not going to go into the detail of the optimization it doesn't complete a trivial and it gets pretty time consuming as you go to full genome scale models so it's there is I think interesting computational work to be done in terms of trying to improve this kind of algorithms but let me show you first the outcome that we got from looking at the core metabolism of E. coli so this is a simplified model of E. coli metabolism and what was fascinating is that one of the solutions that the algorithm found was these two E. coli I would say subspecies right so these are species that are limited in their capabilities and you may recall here this is glycolysis this is the TCA cycle but here each of the two two organisms uses a different half of the TCA cycle this species uses one half this other species using this using this other portion and the exchange with each other multiple metabolites including pyruvate here but also some of the byproducts of this you know this intermediate byproducts of the TCA cycle and what is interesting here is you know there's there's a few things one is that it would have been very difficult to come up with such a scheme for a possible cross feeding without the algorithm this is again not the exchange of two amino acids this is what again they're called deep symbiosis where two organisms exchange interest uh metabolites that are uh are known to be uh transportable but are part really really of central carbon metabolism and there are multiple exchanges that are required for this to happen the other part that is interesting is that in nature there are indeed organisms that do half of the TCA cycle the incomplete TCA cycle as we hinted to um some of these are related to the capabilities of producing amino acid through these reactions but it's quite interesting that one of the solutions of this algorithm really resembles some of these half TCA cycle strategies that are found in submarine bacteria now when you go to uh genome scale models this is much more complicated than it's really impossible to visualize the whole network so this is another way of visualizing what happens so what you see here again is the number of exchange reaction the the limit on the exchange reactions and the limit on intracellular reactions so you can imagine right you start from this is where you have this corner top right all the reactions are possible and you gradually can decrease the number of allowed exchange reactions or internal reactions and these uh uh areas shaded areas in green and blue represent the feasibility and the growth rate that is possible um as you do this and as you can see if you start with one organism right one organism is feasible in this region so what this means is that right if you decrease the number of reactions to below about 250 one E. coli cannot grow anymore on its own uh same if you decrease uh through this it seems like a Pareto frontier this exchange reaction if you decrease them too much the single organisms cannot grow but this is where you can see again what we're hinting to before that if you have two organisms in each of the each of them has these limitations then there is a much larger space where the two organisms can grow if they grow together in the exchange metabolites even under constraints uh below this 250 thresholds up to 210 and so or so where uh yeah individual organisms would not be able to grow but the two organisms can grow together and then there is a lot of data again imagine for each of these cases you have this genome scale models of the E. coli networks and you can look at what is the structure of this networks what metabolites are exchange exchange and so on and this is just to exemplify the kind of insight you can get you can see that there are regions in this space as shown here where acetate is one of the key uh exchange metabolites which is not surprising again acetate comes back again but there are regions when you go to the extreme uh you know you push this pair of organisms to the limit then turns out succinate uh again one of the TCA cycle intermediate uh is one of the metabolites you would exchange you expect to be exchanged in order for these two organisms to coexist and there are areas where some um amino acids need to be exchanged so probably the two organisms will do will perform complementary metabolic functions in exchange amino acids and again there is much more one thing that I you can't really see here but I just want to um highlight is that there is an interesting very thin layer here between the one organism the two organism uh regions where one organism is still possible and two organisms of course are possible but there is a an area here where the two organisms growth rate is faster than the one organism single organism growth rate under those conditions so what this would imply is that if you were to put a chemostat experiment and force this organism to grow at a certain rate there would be a situation where the two organisms would outcompete the single organisms hinting to the possibility right that this could be a transition where even if a single organism could grow on its own but cross feeding could be evolutionary advantages and and give an advantage to a pair of synergistic organisms it's you know this is not not saying anything about the details of how this could happen in real life but it's showing that in principle there is this um overlap that would give an advantage to two organism solution rather than the single organism solution okay we have uh five more minutes and I will um tell you a little more about another aspect of this genome scale model so right we dealt so so far only with organisms that are very well characterized you know well-built models such as E. coli either organisms for which we had very good models but um as we hinted to right we want to start understanding complex microbiomes or more complex microbial communities and one of the limitations that we have to be aware of is that in many of these cases the knowledge about the metabolic capabilities of this organism is much more limited than what we have for E. coli or yeast and so on so the question is can we get around some of these limitations and um David Bernstein another former student in the lab did some really nice analysis using what we call metabolic percolation in order to address this question and this I think is a really interesting area because if you think about this when you have a metabolic network that is not working if you do flux balance analysis and you cannot produce a certain amino acid the network will just not grow and will will give you zero information of why it's not growing so it's a little bit like I think of it as a broken computer and if you don't know you know where to start you know that one element in the computer is missing but the computer is not working doing diagnostic could be very challenging and so on and and this is all with the situation we face when you build a flux balance model and it's not working it's very challenged to find out why and of course there are methods for doing this you can look at pathway by pathway but the idea that David started thinking about is that one could think of the problem of biomass production as a percolation problem where metabolites that are present in the environment could be present and chosen with a certain probability and then you can ask about the probability of producing a given metabolite and you can ask this for any metabolite that is part of the biomass of an organism and systematically choose many many different random environments based on these probabilities and ask what is the chance that by giving as inputs the different metabolites I'll be able to produce one of these biomass components and the advantage of this is that even if the network has some holes you'll occasionally add some of these metabolites by chance and you'll get an overall picture of how producible a metabolite is given all these different chances of different metabolites from the environment being available and this is illustrating for example how much more robust this algorithm is regular relative to regular FBA so for example if you remove randomly reactions from a network with the you know the fraction being let's say one in 100 or one in 10 FBA will soon not be able to give you really accurate prediction of the growth rate but the productivity will tell you whether or not an organism can grow even after you remove a pretty large number of reactions from the network so it doesn't give you accurate prediction of the growth rate but it will tell you what is the productivity of each metabolite in the network in a way that is very robust and I'll illustrate one of the applications of this this was done again circling back to the human oral microbiome and I'm showing here just a big heat map of 456 strains from the human oral microbiome and 88 metabolites that are part of their biomass and the darkness of the red shade here indicates how producible each metabolite was for each of these strains so this is basically a map of the metabolic capabilities of all these different organisms based on this percolation algorithm and I'll there is a lot of information here that could be compared then to for example co-currents of different organisms in microbiomes and we started doing this but I want to illustrate one specific example that was quite interesting one of these oral microbes is what is called TM7 you can really see it from here but there is a set of organisms that are not are uncultivated bacteria so these are organisms that are cannot be grown in the lab on a medium like E. coli and many other bacteria they depend on something else that is unknown and they can only be cultivated in cooperating together with another organism in this case octino bacteria and there is a lot of laborious work at the foresight and other places to find these partnerships and there is also very recent efforts that allow to sequence this TM7 organisms through metagenomic sequencing or single cell sequencing something that was unthinkable a few years ago but now we have the four genomes of this organism we could compute this productivity and what we found analyzing the reproducibility and the capabilities of this TM7 uncultivated bacteria and the host the octino bacteria we found putative metabolites that are complementary between these two so for example the host the octino bacteria could provide vitamins and amino acids to the TM7 and what is interesting is that there are cell wall components that could be exchanged potentially also from the TM7 to the host producing a two-way interactions that gives rise to a lot of testable predictions and that gave rise to some follow-up studies but this is just to illustrate how one can expand these ideas of flux balance modeling beyond just computing the detail growth rate and using it to start analyzing real complex microbiomes and their metabolic capabilities and I think it's time to stop and I'll just hint to the fact that what we'll talk about Monday is about how you extend you know go beyond this compartmentalized model and look at dynamic flux balance modeling where you can start thinking of really not just a more realistic way of how to model exchange with an organism but this will allow us to look at the dynamics and also the spatial structure of communities and a lot more so I'll stop here and see if there is any question. Thank you Daniel. For the last example I mean the crossfading between TM7 and the host have you tried to validate the finding here in Rachel? So I haven't this is I mean we're doing experiments but this is well beyond the kind of capabilities we have I think there are a couple of labs that can do this kind of experiments because and this is just really fascinating work that is being done for example the Forstner Institute but just being able to isolate this TM7 organs which are very small to do with the host is a whole lot of itself. So there is I think listed here there is some evidence from preliminary studies for example that there is gene expression studies where it's really it was found that some of this for example enacetylbicocosamine is really implicated in the expression of the host in co-pultivation with TM7 so there are now a lot of efforts to try and characterize some interactions and and I think it will be very interesting to see if you know this is valuable. So yeah we're not doing this but it is being done and that there is a lot of interest. These TM7 are a fascinating very big clade in the tree of life that has just starts being characterized. Yeah thank you yeah I think it's definitely very interesting to delineate which is the metabolite among all of this potential carcinometrolite is playing the key role of the interaction that would be really fascinating. Yeah thank you. That was a raised hand by Miguel Rodriguez please. Yes thank you. I have a question about well I don't know if it might be more relevant for the DFBA but I will ask anyway so you explicitly model you have been showing how to explicitly model syntrophy and sharing some of these metabolites but is there a way to also explicitly model the fact that competition has a very strong attractor for one dominating species? Yes yes and absolutely and as you hinted to the answer is going to be dynamic FBA in dynamic FBA this comes very naturally as we'll see because I can just hint to this right in dynamic FBA do this step-wise approximation of the growth curve where you solve flux balance time by time at each time point you predict the growth rate and you know how much nutrient is being depleted so if you have multiple organisms in the same environment this is a little bit like a consumer resource model where you keep track of the nutrient extracellularly and each organism will try to use it in its own best capabilities but they will all compete for the same nutrient and therefore competition will come as a very natural outcome of this simulation which is why now basically most of what we do and I think dynamic FBA is really the way to go for modeling community. If I can I don't see any raise hands so if I can ask another question are these is there a way to explicitly model also toxic by-products the accumulation I don't know yeah at which by-products become toxic? Yeah that's another very good question the short answer is no I mean the you can think of right reactions if you block internal reactions and there is no way for a metabolite to go this will give an infeasible solution in FBA so that is a little bit similar to toxicity but not really I mean toxicity if you think about this is really about a metabolite having a very high concentration and starting to do stuff that it shouldn't be doing inside the cell but by definition then FBA regular FBA does not have any notion of concentration inside the cell so it really cannot do that you know there are kind of you know just hint to the fact that if you look at things like shadow prices that give you sensitivity of the biomass production rate to changes in the constraints and including the mass conservation constraints you start having a notion of concentration inside the cell you can do this also with thermodynamic flux balanced models where you can put back the concentration so in principle I think it will be possible to do this but I think it's we're not there yet and I think it's a super important question but the other thing I'll say is that even if you have the concentrations you need to know somehow what compounds do to proteins why they're toxic so if you have knowledge in advance you can model the kind of rising of a toxicity in a certain situation but where we can predict the noble the toxicity I think that's even more challenging and I think that's a beautiful question but I don't think I think that's really beyond FBA it would require structural modeling or other approaches I don't seem to see any further question I am sorry I have a question but I cannot I cannot find the thing to raise my hand thank you very much for the very nice talk my question is to be related to what was asked before and I wanted to ask like if I understood well in this in the in the balance analysis like you use genomic data like so already sequenced organisms I was wondering just if this could be applied to non-cultivated organisms but let's say if this if it would be too much of a stretch to apply this to max or to differently sequenced organisms I'm thinking of the unculturable extreme environments my community yes yes beautiful question I think you know again the answer until a few years ago was no because all of these FBA models really are based on a knowledge of the genome and without that you know there there may be other ways to you know typically characterization but I don't think I think you're really in the genome but what is amazing now is that with the new technology of single cell sequencing right if you can see the sequence single cell and I think that starts being possible with with bacteria but the other thing that I think will really change this dramatically and there are some beautiful example already out there is that from a deep metagenomic sequencing now you can get enough resolution to close individual genomes so even you know when in a metagenome if you have an organism that is uncultivated you may have enough information to determine its its genome so I think that will open up huge possibilities in terms of looking at this unculturable two-flex planas yeah I was and I was thinking perhaps help guiding like the the culture ability of these organisms right right I think that would be absolutely fascinating again you know this is what we we're trying to do here the problem I just want to put a caveat there is that we know nothing about these unculturabilities right and there are many different hypotheses if metabolites is what determines those then yes fba can help but there could be you know signaling molecules like protein factors you know all sort of other things that are beyond fba and if those you know for those cases actually we will not be able to say much thank you very much yeah I have a really good question then you so you mentioned so there's a question from Martina okay raise that right and then it's okay I mean it's okay if it's related I can wait no problem okay please go ahead yeah so me can I ask or I yeah sure yeah so yeah my question is about reconstructing model I mean which is related to the previous question about the reconstruction model from metagenomics I mean even if the we can't identify the the genome the specific genome from the metagenomics is that possible or is there any work to you know reconstruct the model from the metagenome you know by using you know the conception of a super bus you know not compartmentalized you know we probably don't have the enough information about the genome of each bacteria in the metagenome but it's possible to construct a super bus like model you know solving the metagenomics is that possible yeah thank you very much so yeah very great question the it is possible and we'll talk a little bit about this next time we there are some approaches that essentially do a little bit of that but it's not clear to me that I think yeah it's not clear to me that they you really miss something and destroy something so I'll tell you very briefly one analysis really a few years ago was looking at yeast that has compartments and we compared FBA of yeast with compartments to an FBA of yeast where you destroy all the compartments so it's a little bit like a simulation of exactly what you said where you decompartmentalize the model and what we found is that the decompartmentalized yeast you make a lot of mistakes in the predictions basically because anything that depends on energy production across the membrane oxidative phosphorylation doesn't work anymore so I think it might be possible still to do ecosystem flux balance models I don't know that anybody has actually done this but it could be possible if you keep track maybe also of meta compartments but there are other approaches network expansion based which we'll talk about that actually do a little bit of that so the answer is that it's not clear it's a very interesting area and more to come on this yes that's very much yes Martina yeah hi um so my question is so for example uh I don't know pH and temperature can change uh the availability of the byproducts these kind of things is it possible to integrate these things in the flux balance analysis the dynamic one I don't know in general yeah uh it's a it's a very good question the the it's hard I mean it's not technically it's not impossible I think there's there's been a little bit of work out there on temperature um I mean the tricky part of temperature is that it changes so many things right you could in principle try to put the some constraints on um right the rate of reaction based on the Arrhenius equation and try to figure out some way of putting the the temperature in there but but you know if you think about this there is denaturation of proteins or you know suboptimal so there's so many things that could be happening and I think it will it would be very difficult it's it's a very painful thing for me because I and many others I think because there is a lot of important applications uh of how micro communities will change with temperature for example climate change and so on but unfortunately I don't think there is a very effective way of incorporating temperature and similar for pH uh we actually we had did some work which is unpublished but on trying to simulate exactly what are the molecules that are being secreted and how this will induce changes in pH in the medium so this is in principle possible but but it it's very complicated and uh one of the things is that let's say proton exchange and so on is very hard to keep track of um so I think for pH there is hope or temperature is much harder I would love you know if anybody had ideas of how to do this I think that would be hugely important but so far I think we you know it's not been possible okay thank you great I think we had a very lively and interactive uh lecture uh with Daniel uh more to come uh over the next phase um and so thanks thanks again Daniel and thank you thank you everyone and we'll basically move directly to the next lecture by Mercedes Pasquale Mercedes I think you're here already um yes I'm here good morning or good afternoon whatever yeah so I guess that we are I just want to check that you can hear me yeah can you hear you well and I can see your slides great great okay thank you see you in a short while welcome back everybody for the last lecture uh of today's session we welcome back again Daniel Fischer who will lecture on uh the college revolution in high dimensions okay thank you um thank you so again please um put questions in the chat and I will try to keep an eye on them and because Antonio or probably if I haven't seen uh I haven't seen one of them um so um yesterday talked about some and assembled communities which a lot of speakers have talked about and particularly talked about the effects of having many islands with migration um uh between them and what the effects of that are so what I'm going to talk about first today is how one gets to some of the results that I talked about the dynamical wind field theory um and unfortunately I'm not going to be able to go to that in in huge detail it's it's quite technical and difficult to actually solve the the dynamical wind field equations so I will give what they are and sort of motivate it and then sort of give some heuristics about those but if you're interested in that in in detail it's in this paper in um uh PNAS from Michael Pierce and Atisha Agaruala who'll be involved in all of this um all of this work the um some of the recent things that I'm gonna um uh mention or come to wars I won't really get to his current work which is also involves Aditya Mahadevan who where if we were all in Trieste together you would um meet since he's at the at the workshop um at the school okay so after talking about um that I'll then talk about some of the robustness of this phase that comes that I showed last time this spatiotemporal chaotic um chaotic phase and then I'll turn to things that are really um uh open and ongoing um asking about the question of whether communities like this can evolve or how they evolve in the future and then makes some brief comments about phenotype um um models um and where particularly in the focus in the context of bacteria and phage um um and phage interactions okay so just um first remind of what the um um of what the model is so we have k strains okay closely related strains labeled i equals one to k these all exist on i islands alpha equals one to the total amount of islands the islands are all identical um the populations of on island i sorry an island alpha of strain i is n and the total per island n alpha is is equal to big n which is roughly a um a constant kept by the overall limits and resources the frequencies which will use or the fractional abundances are the new i alphas which are just the ratio of the on that island to n so those are the basic dynamical variables that we'll um work with okay then there is a small migration rate between islands it's small compared to the typical growth rates on the islands which are of order one or actually order one of a root k but um um uh the m is small compared to um uh compared to those um and that's actually it's important to be relatively small for the um for the behavior for this phase okay then there um can be some selective differences between the types they just overall grow faster than others or slower than others the si and that's going to have some variance sigma s squared and mostly i'm going to ignore this but i'm going to come back and say some things about it towards the end okay the interactions they're only within the islands they don't depend on the island so there's matrix vij we also can have an interaction of a type um with itself a strain with itself which is much stronger potentially with a um which would be then minus a q but since we're interested in closely related strains where there's no particular reason for that to be stronger we're going to mostly set this to be equal to um zero okay well then the crucial parts with the v's instead the v's are random so i have to tell you the um statistics of the v's so the average of the v's is um zero as is the average of the s's and then the um the variance just v1 and this just sets the um uh sets the time scale um uh here so this basically just gives you the time um scale okay but then and then the v's are independent except for correlations across the um across the diagonal so in particular the correlation of the effect of what j does to i with the effect of what i does to j and that has this parameter gamma and we're going to particularly focused on gamma negative which is sort of motivated by particularly by the predator prey um um uh uh context but i'll say something about i'm thinking more generally so we're looking at these anti-symmetric correlations then gamma in this um um in this range and the canonical value for the simulations and things is gamma of minus 0.8 um a good value okay so what is the basic um um the basic dynamics then so here's my dynamics so there's the overall growth rate that'll depend on ghost minus death that'll depend on the selective differences it'll be a niche interaction if we include it but we're mostly not we mostly ignore that and then there's the interaction with all the others on the same um on the same island um and then there is this piece which is the Lagrange multiplier whose role is to keep the total n on that island being um being fixed at um n so it acts on each island separately and it's going to be in you know transient things at least it's going to be dependent on time okay and then there is the part from the migration and the crucial feature in the migration is that the migration comes from all islands to all other islands so the total migration in them will come from all of the other islands be the sum over the same strain or all of the other islands which is basically going to be the island average in the limit that i is very large okay we're going to take entirely deterministic equations with no stochasticity there is a possibility of local extinctions we'll sort of add that in if the news since they're the fraction of population become less than one over n so this is less than one um individual but they can get repopulated by migration from the um uh from the others and so we can understand the effects of this but we're initially not going to include that and take n to infinity okay so i showed last time from the simulations that what the system goes into is it goes into a spatio-temporally chaotic phase some fraction of the um all the strains go globally extinct it turns out to be a small fraction um usually a small fraction depends on parameters um they go globally extinct but the surviving ones that persist they form a they go into a chaotic steady state after some initial transients and the crucial part of that chaos is it's de-synchronized across the islands and the reason is if you have chaotic dynamics on two islands there's a positive gap on an exponent and the coupling between them from the migration is relatively small then they um um you will they will tend to um uh de-synchronize and that happens all except the migration is quite large so we've got this this phase and i'll just show the one of the figures that showed from last um um from last time so this is one type one strain across 10 of the um islands and this is plotting on a log scale and that's because the natural bouncing around is on the log scale because the growth rates um and death rates um vary so this is bouncing around they can bloom up to high abundances here so they all have these um um each of them has a bloom um they all bloom up to high abundance but then mostly go down and sort of hang out down here um down here here now what stops them going too low well is the input here from the migration there's a called a migration floor which is this curve here it's bouncing around because that's an average over all the islands and this is this new bar coming in so that stops them going to um too low they can go extinct but i put the extinction threshold down here so that's the condition that the population's migration is big enough the product of those so that's the extinction threshold then go they can go below and you know it's here actually um the the global population goes below that but some of them are surviving and they uh um repopulate this island that actually went um um went extinct there locally okay so the crucial part here as far as the you know the qualitative features is that the fluctuations of each island are on a log scale they're fluctuating over the log scale um um log scale here and that log scale is set by the size of the um the size of the migration so the range this goes over here these fluctuations is a range of log one over m will basically be the um the range of the fluctuations so if m is small they can go over a big um a big range okay um and the um oftentimes in fact if many of the strains most of the time they sort of hang out near the migration um near the migration floor um but they occasionally then bloom up to high um abundances now the blooms are high abundances that of course happens on a linear scale because what comes into this is on a linear scale so these blooms then will actually dominate the average if i look at this strain and i look at the average it's going to be dominated by the bits when it's way up here so at any given time only a small number of islands will uh um will dominate okay and those crucial then the blooms because of course it's when it blooms that it can give a migrants into the other islands when it's down here it's not going to give much migration doesn't matter much but when it's up here it's important when it's up here it of course also when it has the biggest effects on the other strains okay um so the um um this crucial bit is going to be understanding some of what these blooms blooms are and how they get there and they get there in a very irregular way as you can see from the wiggles um uh wiggles in here okay so these blooms then as they dominate the um average the average of islands they also dominate the average over the um over time so they also dominate the average of new i alpha on the one island of t average over time and i'm going to use this angular brackets to mean average over um average over over time so they'll dominate this and then of course into all the islands are equivalent something that overall islands will give you should give you this island um um uh this island average okay okay so how do we um um understand this behavior so the the goal is to try to understand this um um uh this behavior okay well the nice thing here is that we can do a systematic um a theory of this and the systematic theory is strictly living the limit that the number of strains goes to infinity and the number of islands goes to infinity initially we'll also take the population size to infinity but we can um handle that um um afterwards and this is coming under the general approach is make things as simple as possible and then add features and of course one of the features we want to be able to add is local um uh local extinctions okay even with the deterministic dynamics so an n is infinite i can still have global extinctions i can have all the strains on all the islands one strain on all the islands just keeping coming down and um and die out so i can still have the global um um global extinctions okay so how do we do this well what we do is i mentioned in the last time is we focus on one type one strain on one um island so that's um um uh alternating between strains and types um so one strain strain i on one island and since the statistics are independent of the island i'm just going to drop the um alpha index and there and call it new i okay so we have this the dynamics of this has several um um has several parts okay so first it's got the sort of obvious things in its growth rate so here's growth rate it's got the s i it's got this interaction with itself if we take that into account but then it has um and then it's got the Lagrange multiplier which keeps the population on the island um constant okay and then it's got the migration coming from all the other islands um and the migration out from there okay but then the effects of all of the interactions with all the others are coming from these two pieces uh here and i've shown this piece with a double minus sign because gamma is negative so this overall sign will be um will be negative negative there okay so what are um what are these okay so the way we turn you could try to understand this is we add a type and we add one type um a new type and i'm going to call that um a type um a type zero i'm just to distinguish it from the others so we put that in and we ask what are the effects of the others okay so the effects of the others um all the other ones on same islands of the others what will they do well the effects of the others will give it an effect in its um growth rate um which is going to be it's going to be where the zeta naught is going to come from okay and that's going to be the sum over all the other strains of v naught j times nu j of t so that's going to be the the sum of the others and this is going to be something which is going to be approximately Gaussian it's the sum of a large number of things that these are independent so there's a large number of things this is going to be approximately um um this is going to be approximately Gaussian okay and it's going to be Gaussian and it's going to have some correlations so it's going to have mean zero so the average is going to be zero um equals zero and then it's going to have some covariance so it's going to have a covariance which we're going to call c of t and t prime okay which is going to be the average of the um zeta of um of t um zeta of t prime okay now that zeta of course because we've got the um the this is coming in for the zeros that's going to have a um zero in it um but the statistics of it is going to be the same for um uh for all of them okay but we'll have to um think carefully about what the effects of the um the particular type um are okay so that's the one type okay so that's this part I'm here how it's interacting okay but then there is a really crucial part and the really crucial part is the feedback of this on the other types the other strains okay where is that feedback going to um uh uh going to come from okay so I now imagine that this new new is now growing so it has some time dependence and we're going to want to look at its effects on the presence of the future so we've got new um the new naught um which is going to um have some um time dependence and I'm going to look at this say in the past so I'm going to call that um t prime and what does that do well what this will give rise to this will give rise to an extra force on the other islands so this will give rise to on each of the other islands it will give some delta j to j zeta j of um at that time okay and what's that going to be well what that's going to be that's going to be equal to then the just the sum sorry the vj naught is coming from this island times new naught of uh of t prime okay so that's going to be the um the zeta okay but what's that going to do well that's going to change the new j only of the other stream so this is going to then result in a change this is going to result in a change okay delta new j okay and we're interested in what that change can be at later times so this of course can be at later times there with the t bigger than t prime so that's its effect okay well what will its effect be well roughly speaking the effect of each one on each of the others is small because they're a very large number and there's a total number the effect of each one is small so we can approximate this effect here of this delta new j is it going to be the extra force which is the zeta j times delta the derivative of new j at t with respect to zeta j t prime okay so this is like the response this is the response of um of j to the um changing the force on it changing the zeta on it right because the other ones this is the force um the other ones are feeling um we've now got the other ones that are feeling they are those and so they'll get this extra force okay so this is the the effects of there but what does this do this is now change the new j so the changing of the new j then that'll give us a change back of the an extra sort of force on on new nought but now this force on new nought is going to be at this later time well well how's it going to do that well that's just got the vj nought we've now just got to sum this up here okay so we've now got this is v nought j so it's the feedback back coming on this and then we've got the sum overall um all j okay so that's our extra force that we're going to um that we're going to get that's the extra force back on the um um on new nought okay so what is that term this term is going to have some average value why is it going to have an average value because we've got these two things here which are correlated this one is correlated with that that is correlated exactly with this parameter um um uh gamma okay so what is this going to this is going to be in the limit of large numbers then this is going to be just someone j okay of now we're going to get this parameter gamma coming from that averaging times this average of new j with respect to zeta j t p prime okay and this is then going to be times and this is going to be then um our quantity which is just going to be r of t t prime okay times the new nought at um sorry the new yorkity file i've got i'm already there okay so this is the quantity here what this is here is this is this whole bit coming in here so this is this whole part all of that is going to come down and give us the r okay and sorry i've got the gamma in the r um um the r there um so i've got the gamma times the um um the r okay so what is the r r is then going to have to be equal to this okay so we now have a self-consistency condition so the crucial part here is we have the self-consistency conditions self-consistency for this approximation and that's that statistically all the the strains are equivalent so there's nothing special about this one that i called um zero so this self-consistency has to be that this r so here i applied the r into this equation i applied the r i applied the zeta the zeta had some correlations the r was coming from a response right so this r here is a response so that's the response okay and the c here is the correlations okay so those have to be determined um um um uh self-consistently and those are the zeta and of course then this i have to determine the self-consistently so i have to find the statistics of each of the new j statistics for each of those i have to compute the correlations and then the self-consistency is that c of t and t prime is going to be the sum on j of the um average of new j of t um new j of t prime okay and this is then the average over all of the noise so this is really the averaged over all of the effects of the um of the zeta j okay so that's that's the correlation function and then i have similar for the response function which i've already um um written out in terms of this which is the sum on j of the um d new t prime and then obviously have to integrate those effects over all previous um over all previous times okay so the effect in here this is the effects of all the others it's a Gaussian random variable with with correlations in it i've got this response from the feedback of i on the other types and back again on the on type i this is of course time lagged there can be general time lag there so it's integral all the way up to um t and the coefficient of that coming from the correlations in the v's of the effect of i on j and the effect j on back on i has this coefficient here minus gamma so this is a negative effect so it's a feedback effect that stops new getting large okay so this effect here this effect here really is this kill the winner um kill the winner effect okay the effect on the others is the when the new i is big so when new i gets large that has the effects on the others they then do well and they give the feedback on this okay so that brings it back down again and that's responsible for the this dynamics up here that's crucially responsible for this part here which is the turnaround and stops them getting too big okay if i do have the queue as well the queue also of course stops them i'm getting too big but for most of the time at least we're just gonna ignore this term um and um um it doesn't matter unless it's particularly large okay it has to be larger than the other effects to matter and i say that's corresponding to assuming niches which we specifically don't want to uh don't want to do okay so this is the basic structure then of the um of the dynamical um mean field theory um just one more second um this is the basic structure of the of the dynamical field mill dynamical mean field theory and now our task is a simple task seemingly simple task that one has to figure out then self-consistently these two um uh these two functions and so we have to get these once we figure out the new we assume something and then we solve okay then we've got an additional self-consistency we then with the migration so if i then plus adding the migration okay so how do i do that i assume some new i um bar general will be dependent on t so i assume some new i bar then i compute i get the actual new i is coming um uh coming out from from that and then this has to be equal to um that so i compute the new um and then i have to make itself consistent so the time average of the um of the new let's sorry that's actually the average of the islands um the one over i times the sum on the islands of of new i has to be equal to new bar okay of course i need to adjust that so i assume new bar i get a new i and then i have to adjust until i get this so what's gonna happen well there's gonna be some news which go extinct so sometimes we will get that this will go extinct um and so i'll have some fraction here where it's um extinct so some strains there's no solution which is bigger bigger those and those go globally extinct sorry then it can make a question so yes uh so the new i are broadly distributed you said that uh at any given time there is one uh few few of them that will dominate the sun yes yes well is this approach uh uh does is there any problem with this approach where you take uh where you assume essentially that uh things are self averaging yes so the condition that one needs is that the number which are large at any time is big and the number which is large at any time is basically going to be the total number divided by this factor because they're roughly uniform on that scale so the condition we actually need is not that k be much bigger than one but k be much bigger than this parameter here this log okay but this is this is you know modestly a modestly large parameter in practice the with you know in modus k that one can see this a phase so like i've got here in practice it often is only dominated by a few of them okay so this is not in the regime where the mean field theory is strictly valid and associated with that also we have these fluctuations because the number of islands is not very large okay so this is showing from modus numbers one could do it for larger numbers and there's ways of trying to get some convergence numerical so that's a good a good point is that strictly speaking i did have many large at the same time however it turns out they turn around fast enough that only having a small number large at the same time the behavior is essentially the same but that's one of those things that we can put in afterwards and understand understand that yeah thank you that's a good important question that's a raised hand my arm oh yeah um i guess could you remind me what you took to infinity um in terms of so the things i took to infinity is k to infinity so that gets around this problem that Mateo just just raised i mean i always have a large number of i'm affecting all the others and that's coming in here the fact that i've got a sum over large number of roughly independent things okay um and then the i also took the number of islands to um um uh to infinity um and the reason i can do that is then i can treat the um self consistently of the number of islands i can treat this as something which doesn't depend much on the islands these are going to be roughly independent of each other because of the chaos the uncorrelated chaos and so i'll get something which is well-behaved average there but again one can i'll make something you can go by number of islands so then the Lagrange multiplier then what at what level is it keeping um the okay so i also have to adjust the um the epsilon okay i need to adjust that um that as well that i can do as i go along with the um with the dynamics it's part of the same um same thing so i better put that also in um in here um i'm i'm finding a new event because this is i get this and i also get um i also have to get the um epsilon of t sorry for not having said that i'm that part thank you the epsilon on t um will be because again the the large number of types and the statistics being the same the epsilon t can be the same on each island after transit so i'm now going to simplify um simplify things so some strains have gone globally extinct and then the assumption is that the rest go to a statistical steady state okay the statistical steady state and that means that for example the correlation function will just be a function of t minus t prime okay the response function will also just be response function of t minus t prime okay the um epsilon will be approximately constant so i'll have a those um those simplifications and each new bar it'll still depend on i okay um this will also go to a constant but depends on i depends on the strain capable to lose its time dependence so now i've got a time translation variant problem and i can try to solve that okay now i would just like to make a side note for people who've seen dynamic theory and spin glass context or other contexts usually in people the situation people do you can take these um a correlation response function you can go back in and work it out and you can directly get a self-consistent equation for the correlation response function and once you've done that you no longer need to do the stochastic dynamics here we don't have that behavior here you have to do the full stochastic dynamics you have to assume as data you have to do the stochastic dynamics understand the statistics of the new get the average of it get the correlations of it these quantities here both have long tails in time this has a long tail in time this has long tails in time and you have to work through those self-consistent and that's what's hard okay so the real challenge here is sort of the applied math problem now of going and trying to understand this um uh self-consistently say that's done in detail in this um um um the pna s paper okay so we have to assume we get a steady state and then we work out all of these things off consistency okay so what i want to just do is i want to do a bit of the um uh a bit of the heuristics um to give a flavor of that and since the distributions are on abroad the natural thing is to look at the log variables to look on a log scale okay so i'm going to define li is going to be um the log of a new i okay so l actually got i mean it was negative because the new is bounded by um the new is bound by one and then i can write that i've got the log i um uh dot here so this is now just going um uh just going up and down so it has this part coming from the um the zeta i um of t minus the um epsilon which is roughly a constant and then it has this feedback um effect of the um of the r times the um up to t this is now two minus t prime of the n well what is the n the n is going to be e to the now l at an earlier time right and there you see the exponential waiting if you like in terms of the natural variables which are the um the l's okay then there's going to be part the other part well there's just going to be the minus the migration out but then the important thing is that the migration in the migration in has this average coming in um here which is now just a constant but now of course that's divided by n so that means it has an e to the minus l here okay has an e to the minus l of t okay so these are all functions of t has an e to the minus l of t so what does this do okay this one here cuts it off at the top end right when l gets large it cuts it off this one here gives you a floor that you're not likely to go down much below this okay so this term here this term here keeps l usually being um uh bigger um than um log one over m times this new i bar okay so what is that that's this exactly in this picture um this picture here that's exactly what this floor is so that's this floor which is set by um which is set by this we have to adjust that floor itself consistently if this is a not very good type then i'll go up and come down oh and i should i let me put in the um leave in the si um um in here okay so what do i want to do well i want to divide this into two things i want to look at this and then it's going to have some average value okay so in the steady state that's going to have some average value but that's going to depend on i so i have a quantity then which i'm going to call psi i okay and that's going to have several parts it's going to be si plus the average over time this is going to be the average over time um the um minus the epsilon okay and then of course i've also got a part of the zeta which isn't average over time so i've got an extra part which doesn't average um so i can write zeta is the average zeta plus some eta where the average of eta equals zero okay so i have some other part there and the eta has the correlations associated with the remainder part of the um of this okay so now this will depend upon the i and this i'm going to call the bias okay what is that well in connection of things that Stefano Alacina talked about well this is this is just the invasion eigenvalue from small numbers why is that well if i look back up at my um equation here if i look when nu is small they've been small in the past this is what's going to determine how it invades okay so this is exactly the invasion of the invasion eigenvalue okay so this is the invasion eigenvalue now the surprising thing is well the not surprising thing is if if minus psi i if it's strongly negative so it's strongly negative they'll get we go extinct okay but psi i can be less than zero but bigger than some critical um critical value which is negative it can be in that range it can still persist okay so a crucial things here is even things that are biased downwards on average even things that are biased downwards on average can persist and in fact they can be biased down quite strongly on um um um on average and the reason that they um um uh that they persist sorry um the reason that they persist here is associated with what this um um uh form is here of what comes okay so even though on average they're going downward they're being pushed down towards there they've got um this effect here which stops them but in order for that to work in order for that to work they have to burst upwards so they have to even though they hang out down here if the psi is quite strongly negative they hang out down here but occasionally they burst upwards those blooms are crucial without those blooms you don't survive okay so you need them to keep the new i bar up you have to um bloom and in order for this to happen they need to have occasional blooms okay and those blooms then is what dominates the new i um uh the new i bar okay so you can work after work out the statistics of those blooms that's uh that that's subtle and the reason that subtle is that these r and c's have long-range um correlations in time so the r and the c here and the correlation function both have long-range correlations in time they decay with a two-thirds exponent over some range they have two other different regimes as well um and it gets and life gets very complicated but the crucial thing is when that's to understand the statistics of these blooms okay and that's a rare event um um uh calculation and they say this is coming from this is going to be the average of e to the li right i'm so it's an exponentially weighted average it's dominated by when the l's are almost close to zero i'm not enormously well okay so this is now um the um uh this is now the problem that one has to um uh the one has to solve okay so i'm not going to go into more of that this gives you a sort of qualitative picture of the uh behavior um it turns out that most of them actually persist with most of the parameter range unless the migration gets large you actually get most of the persisting only a few of them um uh go extinct most of them persist you can work out what this is how it depends on the parameters at least roughly so this is all doing asymptotics there's almost no things you can write down exactly you can write down sort of bounds on things which really give you useful results but the crucial thing is understanding the statistics of these blooms the crucial thing is seeing how they give rise to long-time correlations in the response and correlation functions okay so that's that that's the basic um um uh basic behavior okay so i just want to think at a um at a page here um okay okay so what do i want to just ask about is the robustness of this phase okay so um uh Roy Felix Roy and collaborators have looked at this for gamma equals zero so independent interactions and um q being uh um positive but not um uh too large um well it's order k root k but not um not too large okay and they find similar behavior there's some um differences associated with what happens how they turn around when they get large so if something comes up and when it gets large it's turned around by the q um rather than by the response but rather than by the feedback but the behavior is qualitatively similar okay so it seems as if this actually applies over very large parts of the sort of phase diagram in the basic um basic model okay so that's one part but another part of looking for the robustness is what happens if i have finite number of islands okay so if it's finite n finite population of each island okay then if um if m n so the total migration into each island is large okay then it's still okay you so some extra ones um uh go extinct but most of the strains still um persist okay a few extras go extinct if the new bar falls below one over n okay if i hadn't the figure okay finite number of islands i'm finite number of islands as well then what you do is you get to this the survival time um the uh survival time in this population goes as e to the um i um uh a number of islands exponentially long divided by some characteristic scale which depends upon the bias of that type it depends on log m and it depends on log n and again figuring out what this um this is takes quite a bit of um quite a bit of work okay and one can check that numerically 10 islands and you know 100 types um i've forgotten what it is 80 or something like that have survived very long times and some of them will survive i say exponentially long time as the number of islands get large so even though this is asymptotics and in fact it's asymptotics in log variables in some sense it's asymptotics in log variables it turns out that it's very um robust and so works over a large large range we could add other features we could add in um some extra environmental stochasticity we could add slight differences from island to island um uh Felix and company have looked a bit at um that and again the behavior seems to be assist so we really seem to have this robust phase for an assembled community with correlations in the interactions which are not strongly competitive right gamma being positive it's gone to the more competitive interactions this certainly is going to persist for some positive gamma we may persist in some sense all the way up to gamma just below one um but that we don't know yet okay so we don't know how persistent this will be if i put in say interactions via chemicals in the um um in the environment so that's still a lot of questions associated with that okay so i want to um um to talk in the last bit about um um whether this community can um um can evolve but let me just pause here if there are questions at this um um at this stage um and i say i certainly don't expect you to understand this in detail but to understand the spirit and how one does the calculations and sort of the heuristics so let's now ask the absolutely crucial question we assume it was an assembled community we let things go extinct it stayed but we want to know is this phase stable to evolution okay and also can evolution give um uh give rise to this so how do i want to um want to evolve it i want to take just choose some strain i um okay so i'm we're gonna first look at this we're gonna look at slow evolution that's the hardest case as far as the thing's persisting so we're gonna add one mutant at a time one mutant at a time and we're going to take type i and it's going to mutate to some type um um i tilde so this is the mutant it has a given parent and these will be correlated in some way their s's and the v's will be correlated in some way with a strength with a strength rho so root one minus rho squared is essentially like the difference between them between the parent and the mutant okay now one has to do these correlations in the right way to keep the sort of cystics and so on um in there this bit we have some analytic understanding of it mostly um uh uh most numerical at this stage okay so this is now looking at the um what to what happens so i start with some number of types here i start with number of some number of strains initially here those um some fraction go extinct so there's the rapid extinction here on the ecological time scales then we we let it equilibrate and then we add one more okay so we've add this mutant we let it then equilibrate um that'll drive some extinctions um possibly some new extinctions and of course a crucial thing if we're adding one sometimes it'll drive the parent out but much of the time the parent and the offspring um coexist and the mutant um uh coexist they're slightly different from um uh from each other okay and then i pick another random parent i do the same thing again so this is looking what um what happens and if you start with a small number of types you tend to get that it goes extinct if you start with an intermediate number it sort of fluctuates around and then starts um going up and if you start with larger numbers it just goes up so this suggests that there's sort of roughly a threshold at which you need a threshold in the complexity that you need before it starts to take off okay now where this is it will depend upon the migration rate it depends on much more details about the model we don't understand this quantitatively one understand qualitatively what's associated with it so when you go in here the number of types ends up plummet if you start with small numbers you basically it's exponentially rare um in a small of large exponential factor in order to get up and get going okay so this behavior one won't see unless one sort of gets the community going in some way but one could do that in various um um various ways we haven't explored in detail okay there's a very subtle thing and this is what a detail amount of fun is working on as to how the invasions actually occur so how does a new type come in and that turns out to be rather hard and if it's too close to its parent it's even it's even harder um for this we haven't explored the details of that um yeah but say that we're working on now okay so what happens is this something which is persistent well here's now looking on longer time scales so this is of course sometimes the it doesn't invade at all I should say um sometimes the mutant doesn't come in um so we collaborate often the parent mutant mutant some that sometimes um or even often the mutant fails and doesn't invade and sometimes it does um if does sometimes it replaces the parent can replace the parent okay so what happens then well depending on what the correlation is so this is now putting in independent ones that's assembling the community more this is very highly correlated ones tiny differences between the parent and the offspring um there you don't really see much happening yet um mostly happens here is that the the mutants replace the the parent but as soon as you get you decrease the correlations by a tiny bit you started getting if you add you lose a half um and so you just go up at a steady um right actually the these ones so um you go even faster than that okay so this this tends to come up fast you get the community which gets richer and richer as it goes along however this here we assume there were no generalist mutations what does that mean you can do better in general by having all of your v's larger or having all the v's against you being less negative okay but the bigger the population the number of the community is the less likely you are to have that um um uh to have that happen okay so this has no generalist mutations what do i mean we mean that the s i's there's no s i's the s i's rule zero okay and s i would just mean that you'd better in general if your s i's bigger than your parent you do better your than your parent in general okay so what happens now if we put in generalist mutations okay so we're going to do this but we're going to make a variance of the sigmas um uh so the variance of the sigmas is going to be much less than um uh one that's the assumption initially that things are already pretty well adapted they're pretty well adapted but what then happens is you can still go out into the tail so if i look at the distribution of the um of the s's so i look at the distribution of the uh um s's here distribution of the the s's so that they'll say it's some Gaussian um a Gaussian distribution initially okay well even very early on if i look though the ones that um that survive i'll tend to not most of these ones down here won't survive so they'll go um they'll go extinct and as i go up i will start to get um as i as i evolve the start types i'll start to get that this distribution will tend to concentrate more and more towards the tail as i go up further it'll concentrate even more towards the tail and it'll keep creeping up towards the tail so this i go with successive um uh successive invasions um i go along and i start pushing it out towards the tail okay what happens then is one can see that um and you can look at the um uh the correlations here if you look at the mean of the the um s's that come in so this is the generalist mutations then you that that goes gradually upwards you push further and further out into the um further and further out into the tail okay the potus also gets slower and slower because if you're an s here if you find a smaller s most the time your mutant will have a smaller s right so the one once you get up in this regime here okay the most of mutants um um in fact even the successful mutants so the s of the mutant will be less than the s of the parent nevertheless if it has good v's there's good interactions it can um it can invade okay but generally it's more likely to invade if it's s is s is larger because it's got an overall average high average growth rate or higher bias so what happens in this case is it continues to diversify it continues to diversify just gets slower and slower it's harder and harder to invade it slows down but we have a um analytic understanding of this and for at least for the Gaussian tails here it should keep on growing um um indefinitely um but um just getting more gradually slower and slower okay so even if you allow the generalist mutations this phase can still exist it evolves more slowly it gets harder and harder to invade that's of course a general property the things will evolve for a same in constant conditions will tend to get harder for new things to come in interestingly if you didn't put specifically these generalist mutations in that doesn't really matter this just keeps on going up and the statistics don't change substantially and the reason sort of is if there are so many ways to do well up here that um it doesn't really doesn't doesn't really gain to sort of do better overall you do better about whichever the current ones um current ones are so it keeps going up um in a steady rate a substantial fraction of all of the invaders can um uh can come in when I get down here most invaders fail so most mutants fail but you get um uh you get you get someone you this will continue to grow and this only grows logarithmically in um um logarithm of the entire could ask a question yes so if you so on some of these because the x-axis is successful invasions and some of them is time so is time the same as attempted invasions or yeah this is this is yeah thank you um so this time here this is proportional to attempted um attempted invasions and we haven't put in all the subtleties which you've thought about of the of that invasion process um we allow them to come in at substantial numbers to make it um um uh to make it easier to run things at a reasonable time okay so this is the number of attempted invasions sorry I should have said that we're not allowing reinvasion so if something goes extinct it stays extinct so if you drive extinctions then the um they'll stay they'll stay okay other other questions here so if um the the last thing I want to um talk about and again there's even less of just a tiny bit of a flavor on it is the question about the interactions okay so everything we've done so far everything we've done so far is that the phenotype of type i is defined by the interactions so by the whole set of vij with all of the others and the and the um similarly so this is the phenotype of type i so that's a very weird thing to do we can't just we shouldn't be defining our phenotype by interactions we should define it by some properties of the organism okay so we want to look at phenotype models where the interactions that are turned by properties of the organisms okay so for this I'm going to go explicitly back to the bacterium phage model so I have now bacterial strains a whole bunch of bacterial strains indexed by i phage strains indexed here by m l and the populations um of the bacteria in the phage and the bacteria dynamics um uh growth rate killing by the phages so the hs are all positive um competition with the other bacteria uniform competition known niche like interactions the phages will die without food they will then grow with the um the effects of bacteria and the specificity to the extent that there is is contained in these okay and these then will have some average value this is one species of phage one species of bacteria right so these are both of them are one um one species um uh of each and just strains so they're not specialists they course they could evolve to be specialists but we don't start off with them being um being specialists okay so what will the correlations look like will there be some average value the f and then say it's only slightly different from each other there'll be some small variations delta f and delta range okay and these will be those will be strongly correlated okay now where do we think these are coming from well this is now where i'm going to put in the phenotype okay so i'm going to put a d-dimensional phenotype okay and if you like crudely this is the this is the this is the a is labeling the amino acids a crude way in a receptor so the bacteria has a receptor the phage has a tail and the tail binds the receptor and depending on how well it binds that'll determine it so we're doing the absolutely simplest thing here there's only one phenotypic property of each the bacteria has a receptor the phage has a tail that's a d-dimensional thing so just a string of numbers the and then the binding strength is just the the i'm defining these in a way that they're energies so it's just the binding between so this is the binding strength of the um phage tail of type l to the bacteria of type i and then i'm going to assume that what the interactions do what this this interaction do they they will give rise through some function just some function the way that the bacteria harm the phage and the way that the phage feed on the bacteria okay so the simplest thing to do would be to assume that it's a linear um um a linear function so the simplest is if this is linear okay so if this is linear a linear then i get a low rank matrix so my rate matrix of interactions my matrix of interactions is going to end up being low rank okay and then you can't get much diversity it's going to be rank d um limits the diversity but what if i put in something which is somewhat more biologically um uh biologically motivated so if i look at what the effects of the of the phage are on the bacteria you know if they don't bind much it doesn't do much um if they bind um strongly it does something and if they bind more strongly it sort of saturates once they're being killed they're killed it doesn't matter okay for the phage i want to put in a bit different function again it's going to have this um um this behavior um it's going to have similar behavior um down here of course it only affects it when the bacteria does so it'll start coming up okay but one can imagine this will keep coming up harder how well it binds might more strongly affect the um the phage it can keep doing better even once the bacteria is already um died because it can get in more effectively and so on and maybe produce more so these functions are different from um each other but the fact that i've both got functions they both depend on the um on the same thing so they both depend on the g's right this implies that these are correlated of course it forces correlations in there okay but it's correlations just coming from this phenotype and i can look at it being in there okay so the only thing i know so far roughly is that if um uh if d is bigger than about six this is just numerically um uh d is bigger than around six so very low dimensional phenotypes and you and for some functions h and um of g and f of g at least for some functions you get diversity continues to grow so you get the similar behavior to um what i showed um um up here you get similar behavior to what um um goes on on here you actually don't get this you actually get something which is more like this you get something more like this case it's sort of bouncing around quite a lot sometimes it can have plunges but it keeps on growing up okay so this leads rise to a very interesting conjecture that even with a low dimensional phenotype okay so d is not large d is going to be some modest number even with a low dimensional phenotype and then deterministic um interactions that are determined just by those phenotypes that looks as if it give can give rise to a continuously um um increasing phase and it is of this this spatial-temporal-ecotic phase okay so it really is this um this phase and in fact they do tend to do somewhat better as generalists they the back phase particularly tend to do somewhat um push towards the upper um upper end but nevertheless they don't become so generalist that that limits the um diversity and they don't become particularly specialist either you can look at the specialist generalist correlation okay so what have i um um done i hope i've i've um got across some things okay so the first thing is the value of trying to look at really simple models to get an idea of what can happen and if something can happen in a simple model then i would say that's not so surprising that we might see it in in nature it doesn't mean we understand it but it means we get not so surprised okay there was one um um uh quant as i looked at um the first day um which i've already sort of summarized as far as looking at evolution in a sort of snowscape where you continually change it but the bit which ties much more into this um school generally is the last two days where i've looked at these random lotcavolterra models i'm motivated the randomness is coming from strains that were very closely um related so things were sums and differences of um the two effects they were not much different overall so the s is were small there were no niche interactions i did not assume anything special about the interactions of a strain with its siblings being any stronger on average than its interactions with its 23 cousins okay so that was that that was the basic model and then in those models we now have a solid analysis and a very good theoretical understanding of the spatially temporal chaotic phase that can exist in those when the correlations are in this anti-symmetric direction but it seems to persist more generally if i have a bit of niche interactions or interactions via chemicals which will give rise to that we'd be a resources which you're trying to consume okay so that's the part which which is a solid um and one can think about what its predictability is for um for nature one thing i forgot to um um say in talking about that um robustness um was that we would really like to add um um i'll put it in as a question mark um real spatial structure so that my um uh interaction my things can't move all over the place and this is particularly um relevant in the uh um uh in the ocean where things are getting moved around by um uh turbulence okay and so certainly if one wants to make contact with reality one has to think about um think about it okay the other bits of this question about it can whether it can evolve it is certainly again possible that these models can evolve higher and higher um diversity under what circumstances that tends to get slower and slower we don't um we don't know the specific assumptions about everything interact with everything that is responsible for some of that slowing down and if one goes away from that and looks at more sort of hierarchical interactions like i guess josh or wates particularly talked about then um maybe the diversity can increase more um um more easily and then the very last bit which is a even more speculative in connection with these phenotype models is that you do not need high-dimensional nanofenotype to get diversity okay so in my sense in which i talked about before what matters this is the dimension of the nanofenotype right so this was the nano um uh the nanofenotype um uh it was the only things that matter in this um uh in this model which is determine how they interact with each other and that's sufficient at least in principle to give rise to increasing um uh increasing diversity and you don't need to have high dimension um um high dimensions to do that okay this should have to be somewhat hard to find if you if you choose the wrong um the wrong functions or if you choose functions that may be nicer than the one more reasonable than the ones which which i used it's going to be harder to get it but um i think that's there's a lot to be still understood in this so there's a huge number of open questions a lot of interesting directions some of those we're trying to um um trying to pursue and there's still more needed on understanding the things that i have um i have talked about okay so i'll stop um stop there and apologies for going on um too long and for going too fast on um um much of it but i say i hope i at least got some of the flavor flavor of course thank you thank you thank you very much let's see whether there are any further questions from the audience so you mentioned if you change h of g and f of g that so that does that increase the um the dimension needed to have diversity well it does increase the dimension and it's not clear with at least some h and g which sort of are reasonable forms it's not clear you actually get this diversification at all what you do is you get things that look more like um more like this you start with a some number you know the numbers sort of hang out for a while maybe gets a bit more diverse and then it crashes and it's hard to get it back okay and even if you start with bigger numbers you get some more things coming down by the way i should mention if you take the perfectly anti-symmetric model on one island you get the same thing no matter how big a number you start with um there's an overall tendency for it to decrease that model is not stable to um to evolution even within the very perfect anti-symmetric model there okay it's not stable it tends to go down we don't have a general understanding of in what cases model will go up in what cases will go down that's like understanding you know when you take a microscopic model and you ask does it form a superconductor or an insulator or something else we have no idea how to do that still in physics but what we do know is if something happens then a whole bunch of other things happen as well and so here once it sort of starts going up we have some understanding of whether it'll continue when with the generalist mutations we have some understanding that once it goes starts going up then it'll tend to slow down in a particular way and we can sort of predict how this um um uh how this does uh um does that um and so the we have some um understanding of that with these phenotype models i don't have much understanding the subtle thing is you have to these can't be perfectly correlated these can't be equal to each other they have to be some different functions so you need sort of sufficient correlations but not too much i think if you put in a little bit of extra um kinds of uh um uh phenotypes coming in as well so it isn't just one quantities maybe two quantities two different proteins say are important then you can maybe be able to get it more more easily okay so this is now where i'm going to appeal to biology okay everything that we see is conditional upon evolution and looking back conditional upon evolutionary success over long times things are going to look special they're going to look at the special things that happen okay so my feeling is if one has a sort of you know choice of models or models in different regimes with one of which can give rise to continuing evolution and diversification and the other one can't then the longer term effects of the the evolution which are ones which are often the things that just happen to have happened and made the evolution keep going is going to mean that one is going to end up in sort of a phase where things wander around okay so the concrete thing i would say is in the random landscape models with just a single strain there i believe there's a family of models also a generic family where you don't have a continuing evolution with small feedback small ecological feedback however if you had such a system it's also much less responsive to environmental changes it's much more um be more likely to be destroyed by environmental changes if you have one which tends to wander around it'll wander around differently in different locations and it's much more likely to be robust to environmental changes so my sense is that the long-term evolution will drive the systems in a way that will tend to be the ones that have these kinds of these kind of properties that does not mean there's evolutionary pressures to do that what it means is that the ones which happen to be successful for very long times and by producing lots of an offspring in the sense of many types of bacteria or many types of insects that those ones are going to be ones that along the way somehow got these properties but it doesn't mean there's necessary evolutionary pressures for it to do that so i was getting more into the philosophical questions are there some some concrete questions on some of the the sort of analysis or the sort of ways of trying to trying to do things okay well i said if you have follow-up questions and a couple of you sent some really good follow-up questions previously i'm happy to answer them by email and they all mail some prompts and names that might come up in the in the discussions at the round table so thank you thank you very much for these and preceding lectures it's been a long day and thank you everybody who has followed all the lectures today and we'll meet again next Monday thank you have a good week