 is because of setup and feedback issues, there's two pictures of me. One of them is the one which has a picture of me actually talking, the other one, which is the one you'll see if you're on the speaker, which is of me just a snapshot. If you'd rather see my mouth moving, you can put that one on and you can pin that one. So I'm going to do a mixture of pre-prepared things and not pre-prepared, and so I hope that will go more slowly than if I just write things out. So this is very much a theoretical talk, except for some things I'll show at the beginning for motivation. And I'll explain what I mean by the subtitle of what should not be surprising. So the general focus is going to be thinking of biology in the real sense of being high dimensions and the role of randomness in modeling. Can anyone see my screen now? I think your audio cut out for the last few seconds. Yeah, can you hear me now? Can you hear me now? Just a few seconds. Sorry, you can hear me now. That's OK? Yes, it's OK. For some reason, my computer one seems to have cut out. OK, there, sorry. OK, sorry. So I'm going to add ecology. And then the second and third lecture, I'm going to come to something which has already been introduced by many speakers, particularly by Stefano Alicina, of talking about the ecology in high dimensions in the sense of lotteria models with large numbers of species. And then I'm going to add a crucial part to that, which mostly raising questions about the evolution on that. So first, some of the questions and motivation. So one of the key questions, why is there so much biological diversity? There's obviously a big general theme of this school. And the classic example of beetles, there are various factors that contribute a lot to being able to have that. There are many different niches. There are many different environments. And they geographically separated. What does that mean? That means they're not all competing with each other. Well, one of the things that's become clear in recent years and becoming more and more clear than more people dig into it is that the diversity really goes down to much smaller scales. And I'm going to show one of the most dramatic examples. So this is bacterial diversity within a species. And this is species of prochlorococcus. It's a cyanobacteria that dominates the synthesis in the tropical oceans. And this is showing a tree of a thing from 96 single cells in a very closely related subclade of this species. And there's tremendous amount of diversity here and more data with more and more cells that even from the same sample, and these are all taken from a few samples, even from the same sample, you see basically every cell is different genetically. And the time since the most common ancestor of this whole group is about a million years of the smaller subclades of this is about tens of 1,000 years. And these are all mixed together. They're free floating in the ocean. And in fact, they're mixed together over the whole ocean in about 100 years or globally in about 1,000 years. So this is extensive work of Pentatrism's group in this particular paper by Kashtan and collaborators. So that's really dramatic. And that's something which we'd like to try to understand. Surely there are not so many niches here that we can explain every possibility with a niche. So we'd like to ask about general questions about the diversity. So a question we'll want to ask is whether or not sufficient organismic and environmental complexity in a general sense implies that one should expect this very kind of extensive diversity, which is an enormous amount of diversity all the way down to within species, subspecies, and so on. And particularly in the most dramatic case where everything is mixing together and competing. Okay, so that part I'm really gonna talk about in the next two lectures. Today I'm gonna start off with the other part which is about evolution in simple environments. And so this is really intentional experiments done in the lab where they're trying to make a very simple environment where there's no ecology. Everything is well mixed together. It's E. coli in low glucose and the selective pressure from the low glucose. And this is a very long and beautiful experiment carried out by Rich Lansky over I guess is now more than 30 years. And what they find is that the E. coli evolved. There are many different genetic roots to higher fitness. Even in this very simple environment the E. coli find many ways to do better. Now, when this experiment was started there was essentially no DNA sequencing at all. And the kind of things that have become possible with this experiment in recent years from the enormous advantages of the sequencing have meant that one can go back and look at the fossil record and try to look at the fossil record and track the evolution that's gone on, okay? So the particular thing I'm showing them here is the dynamics of the frequencies of the mutations. So this is the fractions of the population that each mutation has and the mutations as they arise. And generally it's over here at this end the mutations, a lot of mutations arise quickly and take over the population. So this is one line, this is one flask. There's always shaken, so there's no geographical structure. He's done many different flasks in parallel and they all do somewhat differently and some aspects quite different. So these mutations come in quite fast and then not surprisingly, it's sitting in a fixed environment. Gradually, evolution starts slowing down. The evolution's come up more slowly. They tend to come up in groups. We have theoretical understanding of that, okay? But then after 10 years or so something sort of funny happens. You get things coming up and coming back down again and then some of those sort of swerve up and down over here. And there's a whole sense in here as to what's going on. And it's clear that there's something in there which is that they're interacting with each other. There's more than one type at a time. And so this is really indication of the development of ecology, even in this very simple conditions. And that's seen that the ecology that develops is different in the different runs of the experiment. But then, and this is the bit that I want to just focus on now, then something happens later on. It seems to go back to being relatively simple here with sweeps coming up. But the weird thing that notices is that the sort of rate at which mutations come up and take over here is not that much different than what it was much earlier on. It's slower than it was at the beginning, but it's not different much later on. To try to compete these with the original ancestor, they may be doing slightly better, but not much. And it's not clear if that can account for the rate at which they come up and how fast they come up, how come up they there. So this is a really question mark as to what's going on here. I'm not gonna claim to really understand it, but I want to use that as a motivation. Because this raises the possibility of whether even in a constant environment, so in a constant environment here, whether or not the evolution can continue forever. Okay, so the question I want to ask here is whether evolution in a constant environment can continue forever without slowing? And I really wanna put a can in here. Oh, sorry, I put a can at the beginning. Can evolution in a constant environment continue forever without slowing? Okay, so this is a thing which we're gonna address from theoretical considerations today and makes them wrong about whether or not we should expect this or not expect this. And therefore, is it something which is intrinsically surprising or something which is not so surprising? Okay, so I want to, but before going into that, I really want to sort of think about the sense in which I mean by high-dimensional biology. So if I think of within the cell, then we've got what we could call the nanofenotype. So the nanofenotype, which is the properties of all the proteins and how the proteins bind to DNA and so on, okay? And that's some very high dimension, a very high dimension, at least tens of thousands even for a simple cell and already substantial for a bacteria. So that's the sort of dimension of the phenotype that's all the different ways in which the organism can change. Now, of course that's driven by the changes in the genotype but the change in the genotype, there's something which doesn't directly couple to the environment or the properties. So we're gonna want to work entirely in terms of the phenotype in order to try to understand what goes on. And then of course, one can try to connect that to the genotype as well. So then this nanofenotype, this codes for the organismic phenotype which is still something very high-dimensional. It's all the ways in which it interacts with the environment, affects the environment, how it's affected by the environment, how it interacts with other organisms and so on. Okay, so we've got the organismic phenotype still very high of dimensions but the space in which the evolution is working is really of this nanofenotype of the microscopic level of the proteins. And then of course we've got the environmental and the simplest way to think about this would just be say the number of chemicals which affect the organism and the number of chemicals that with it can affect the organism. Now, Daniel Segre just said in the question at the end of the talk that if you took each cell and lice the cell and so all the chemicals in the cell came into the environment which others could use or could be affected by and be poisoned by and so on, that's something which already is very complicated. That surely is gonna give all kinds of roots for evolution, potential evolution and it's something which unfortunately no one seems to have really done that experiment carefully over time to evolve in those conditions. But the one thing we can be sure of is that the environment is very high dimensional, the very large number of chemicals and the phenotype is very high dimensional, okay? So what we want to, what I want to do is try to be motivated by this and think about the consequences of this high dimensional environment. Let's see how we make sure I'm good. Yeah, okay. So I want to think about the consequences of this high dimensional environment. So what we want to consider is organisms that are already well adapted. They've been around for a long time so we're not making artificial organisms usually, okay? And so then we have to think about the fact that everything is conditioned on the evolutionary history. Okay, so the evolutionary history of the organisms and of course that includes the history of the environment. What is that going to imply? Well, if they're really well adapted, that's going to imply any changes. So changes which can be genetic changes or changes in the environment. Okay, genetic and environmental changes, their effects will be a sum of positive and negative parts. Okay, well, if you have a large number of positive and negative parts, why do I say that? Well, supposing the genetic chains always had positive contributions, then it would already affect the population, something which is universally positive. So whether it's positive effects or negative effects will depend on the environment, it'll depend on the genetic background, the rest of the evolution is acquired before. So this is something which we can generally expect as a consequence of the evolutionary history. So then we can take just a sort of guess, well, if it's sort of sum of positive and negative parts, it makes it very unpredictable and this will give rise to something which we can try to approximate in a way, approximating by some kind of randomness and then look at the consequences of that. Okay, so this already, you've seen some of thinking about them, the interactions has been complicated, they can sort of be effectively random, but I'm going to really take that and try to run with it somewhat. So what is the hope? Well, the hope comes from the fact that we have high dimensions, okay? So this very general quote of Phil Anderson's more is different, that when you've got a large number of things interacting together, there's something in the general character of high dimensions, then things are different than you would expect from looking at small numbers interacting together. There are certain kinds of behaviors, some of the behavior is independent of details, so we want to know what the behaviors are that can occur, behaviors in a loose general sense, or the possible things that can occur. And one of the things that we've learned from physics is that not all the details are going to matter, so we can hope to get things from simple models. So not all the details matter. In particular, we're interested in behaviors that are sort of robust, they don't depend upon all of the details, and that's a sort of loose term, and I'll say a bit what it means in some particular contents, okay? So it would like to try to get a robust understanding, okay? So this really comes to the sort of goal of these talks. So what we're going to try to do is look at some very simple models where a key feature is that they're high dimensional and the interactions and things are complicated enough that we can approximate those as being random, and of course we then have to go beyond them, okay? So those simple models, we want to then what can exist, what things can occur, okay? Now this doesn't mean that it applies in biology at all, or that it implies in ecology in any particular system, but it's asking what can occur. If we see something that can occur in a simple model, that suggests that seeing it in nature should not be surprising, okay? Now we have to be a little careful there, and that's where this robust part comes in. If we can sort of argue that that can sort of robustly happen into the class of models, a range of models, then we can say some more about, well, okay, that wouldn't be surprising if we saw that. We don't fully understand it, but we get some general sense of it, okay? Now this is in a context which is going to be, loosely we can sort of think of phases, meaning phases like a solid or liquid or a superconductor, so the analogy with phases in physics. And we would like to ask what kind of phases can occur. And obviously we can't ask about that generally, but we're going to talk about particular, I want to talk about particular examples, okay? So that's the general scope and sort of what the goal is going to be. And now I'm going to go to some specifics. But before I do that, maybe I can pause if there are any questions on sort of where we can try to go. Okay, Antonio, can you just say something so I make sure that at least you can hear? Sure, I can hear, I don't think there was a question. So not questions at this point? No, not that. Okay. Oh, okay, there's one question just now, can you read that? Yes, this is just to clarify, when you say what can occur, do you mean behaviors or patterns are rising out of the nanofenotypes? So yes, I mean things that can come from the interaction between the nanofenotypes as manifested in the organismic phenotypes and the environment. And what can occur, like for example, do we expect extensive diversity with a very large number of closely related strains that can coexist, okay? And then also the question of things can evolution continue forever if the environment is constant? So that's the kind of thing that I want to be able to do, okay? So we're gonna now consider an idealized system, a system we're gonna consider evolution in the phenotype space, okay? So this is gonna be of all the properties of the organisms and I'm gonna say this is D dimensional, which like D was the dimension of phenotype, I'm gonna drop the phenotype for now. And my phenotype then is going to be given by some characterized by some vector, which is all the D properties of the organisms that will matter, okay? And then I'm gonna make some things incredibly simple. I'm gonna make, there's just at any given time, there's just one strain, okay? No spatial structure, no spatial structure or variations in the environment, everything is mixed together, everything can compete with each other. And then my fitness in that case, I can make a well-defined, it's just gonna be some function phi, which is gonna be a function of the phenotype and the environment. So this is a vector describing the environment, okay? It's gonna be a function of the phenotype environment and this is just gonna be the relative growth rate, okay? Now I personally, I hate the word fitness. It seems like it's a singular term, it's usually used as the fitness, this is fitter than that, okay? Well, it's even though it's two Ss, it really is a very parallel quantity and I've considered trying to get into the literature the idea of the fit known, I hate the word know things ending in own generally, but I think this is actually a illustrative of the complexities going on. And because at the very least, even when the environment is fixed, of course everything depends on the environment. If I change the environment, then things are going to change, okay? So that's the quantity of God, but we're gonna fix the environment at least for now, okay? So then we get a small mutation, a smaller fact mutation and that mutation will take X to X plus DX and I'll try to keep doing the vectors at least for a while, I'll get lazy, I'll get lazy shortly. And then if the Phi of X plus DX in this environment is bigger than Phi of X, okay? Then that mutation fixes, so the mutation takes over, takes over. So what does this mean? It means that I can treat my dynamics as being approximately deterministic if the DX is very small, so that the evolution is now approximately deterministic. Okay, and where will it go? Well, it would just go uphill. So it just goes simply uphill in this fitness landscape, okay? So my DXDT, the way in which it changes with evolution, okay? That's just gonna be the gradient with respect to X, so the gradient with respect to X of the Phi. I'm fixing the environment and so that's not changing. So I've got a very simple thing and then of course, if we're doing this for the fixing environment, then this Phi is just the landscape that I've got. There's a real evolution in a continuous landscape. This is the simplest possible model, okay? So then of course things are trivial or at least seem to be trivial, okay? So I can of course draw the behavior in the case where dimensions are two. Some people can draw things in three dimensions, I can't. And so what I've drawn here is contours of the landscape. So these are contours of Phi and then in those contours, I will get dynamics. So if I start at some point here, I'll come up. I'm here, I'm going up the gradient, so I'll bend around like this and I'll come to a fitness maximum. So I'm going to go up like that. Of course, if I started over here, then I'll come up like this, I'll come up here and go over to this maximum. If I came here, I would come up and go and by mistake erase my contours, which is cheating and I would go over in this way there. So here I've got two possible maxima and these are separated by a saddle, which is sort of the decision point of what goes on. And depending on whether the saddle, it comes to one side of that or to the other, then it will go to one of these maxima models. And of course I can have many maxima, but the thing which we know is that it's a long times, as we get the long times, the evolution will go to one of the maxima. Okay, there was a question when I say relative growth, what is it relative to? It's genetic drift important. So the first one, what I mean is that it's growing faster, it's higher fitness will mean it goes faster with something with lower fitness, okay? So the only things that really matter are the fitness differences. So it's really the difference between this fitness and this fitness that matters. That means this will grow faster and we'll take over. I'm going to ignore genetic drift. One can consider the effects of genetic drift if we're in bacterial populations, particularly if they live in the ocean, populations are very large, drift is not very important, even though possibly of extinction is important. Okay, so that I'll talk about next time, okay? So this is the behavior in two dimensions and the general assumption and the way in which most talk about evolution in simple conditions goes is that there are some number of fitness maxima and long times you go towards the maxima. So the question is that what really happens, okay? So we're now going to want to consider things in higher dimensions, but let me first show a bit of the behavior that the complex is gonna happen in two dimensions. So in two dimensions, I can look at all of the stationary points, all of the points in which there's no dynamics. So those are the points at which x dx dt is equal to zero. So those stationary points, they can be maxima as I've already shown, there's a maxima. They can be saddles like this one here, well, they can be saddles of index two and the index of these points is just the number of the unstable eigenvalues. So a maxima is zero and one which is a minimum, this is the minimum in two dimensions is two and the intermediate one is one, okay? Now if we go to high dimensions, so we now go to high to your general D, okay? So then we can get all possible indices, all possible indices of the saddles from i equals zero, which will be a maxima all the way up to D and of course, the expectation would be that one would go towards these maxima at long times, okay? Now what am I gonna consider? What kind of land escapes I'm gonna consider? I'm gonna consider landscapes where they have lots of maxima. It's gonna be the complexities of the biology, the complexity of the evolutionary history, evolutionary constraints and so on. I'm gonna assume that five is a complex, some complex landscape, okay? And I'm gonna approximate this as some random landscape and I'll be specific about it, example shortly, approximate this as random with some statistics, okay? So this is meaning that the number of extrema, this is the number of stationary points, so I was calling that number of stationary points goes exponentially with the dimension and that coefficient then will depend on i. If I look at the number with a given index, the exponential and dimension and all possible indices will occur. Generally, the maxima's will occur at higher phi and the minimum will occur at lower phi, but of course I can have local maxima and local minima which occurred in immediate effects, okay? So one of the general features which is known from some mechanics is you'll get exponentially many maxima and a minimum is one. Okay, well we'll still expect that the behavior would go towards one of the maxima, okay? So what actually happens? What happens in the limit of high D is if we look at a function of time, so we can look at function of time, if we look at function of time and we can look at the phi, so what happens is initially phi will go up fast, then phi will go along and it'll saturate. So it'll go very slowly, very slowly upwards. So that's phi, okay? I can look at the index. I can look at the index coming here. The index will start out being a water D over two and the index will gradually come down, okay? And what happens here is that the behavior is that it wanders around the saddles. So it wanders around the saddles or the saddles. It's generally getting to lower and lower index saddle, meaning they're getting closer to maximum, but they always have some unstable directions. And the crucial part here is that it never, never in the limit of infinite D commits to a maximum, okay? So there's no sense in the limit of very high dimensions that it approaches the maximum. It always keeps wandering around the saddles. It's as if in two dimensions, which of course I can't get this, it went, it came from this saddle, it wandered that saddle, it wandered to another saddle after that, wandered over here, went to some other saddle and so on, and just kept wandering around the saddles. And of course it can't do that in two dimensions, but in high dimensions it can do that. And that's the generic behavior. So this is the generic behavior that happens in high dimensional complex landscapes. This is the generic behavior. It's robust under a specific assumptions about what the dynamics, what the landscape looks like, okay? So that's already a surprise is that one's intuition about low dimensions is wrong when it comes to high dimensions, okay? So here the evolution continues forever. It doesn't commit to a saddle, but it still slows down, right? So it's definitely still slowing down here, right? The fitness is going up slowly, the index is going down slowly. If I think in terms of mutations, the mutations will fix less and less often. It's harder and harder to find the mutations. So evolution will slow down. So of course not surprisingly, the answer to my question of whether evolution can continue forever without slowing down is no in a constant environment. But now what we have to do is we have to add in the ecology. So we have to add in a little bit of ecology. What does that mean? Well, any organism changes its environment, okay? So if you have a phenotype X, so that's the whole population is just one strain, right? With phenotype X, what that will do it'll take the environment to being an environment will be equal to some function of that phenotype of the organism which is there and the external conditions, the things that you're holding fixed, okay? So these you're holding fixed in the experiment, but you can't hold fixed the effects of the organism because of course if the organism mutates, if X changes, then it will change, okay? So it's simple then to think in terms of the fitness of the organism when it's the fitness as a function of X, okay, so that's the function of phenotype, okay? And then it's gonna be the E, which is this E here. So it's now gonna be E, I'm given by Y, okay? And then this I'm gonna call a different quantum, I'm just gonna call it Psi of X and Y, okay? And this X here, this is the resident organism. It's the one that controls the environment, right? This is the one that controls the environment and this one can be any. So any other organism that comes in can have this fitness, if this is larger than the resident, right? So if this is larger than the resident, the fitness, then that can invade and come in. So again, I look at a mutant, okay? So I start off with X equal to Y, it's just there, but then I look at a mutant, which is X plus DX. And I ask if this is higher, so if Psi X plus DX and this fixed on Y, okay? And this fixed on Y, okay? Is greater than Psi at X and Y. And of course, this is at Y equal to X, right? This is at Y equal to X is the resident that's there, then it invades, then it comes in and this takes over, okay? And my environment changes and my fitness changes. So it takes over, the environment now goes to, I get now the environment becomes E of this X plus DX, okay? So what does that imply? That implies that the dynamics, the dynamics is now going to be driven by these changes. And so the dynamics is DX, DT, this is now the evolution, is DX, DT is going to be gradient with respect to X of Psi at X and Y, but I evaluate that at Y equal to X, right? Cause that's the thing that's already there, okay? So that's going to be my dynamics and this is my dynamics here, the, put that in red, that's my dynamics, okay? Now an important thing about this is this is not gradient flow, okay? So this dynamics here is not gradient flow, it doesn't just go up here, okay? If you like the physics thing, you can think of it as some curls to it, it doesn't intrinsically take Psi up. What it does is it takes Psi up at fixed value here, right? So it's increasing here, the fitness is going, the fitness is going up, but then that'll change it, so it's then going up a different, okay? So what this is going to look like as we move this, okay, damn, that's not working. Okay, so let me look at what happens here. I get flows coming around the saddle. So if I, for example, have my flow here, so now I'm sitting at fixed Phi, I now say get a flow which comes like this, so I'm starting getting a flow, it starts, gets near the saddle, it has to decide, right? So it bends over in this direction as that side of the saddle, okay? But now what can happen, of course, is the environment can change. As it's going along, the environment is changing, so if I do this correctly, if I do this correctly, my environment is now changing, so my environment changes here, my saddle has moved, okay? I'm doing it, of course, as a discrete step there, it isn't really a discrete step, it's continuous. So what does that mean? That means that this now goes a different direction, okay? So the environment changes can qualitatively change where it goes, okay? So I've got my environment change as it's evolving, driven by the evolution, okay? It gives you qualitative changes, you go in different directions. So this is a picture in two dimensions. Again, I can't do things, they don't get very interesting because I've only got a small number of a maximum that I can go to and so on. Okay, so this is a question. Did we assume separate timescales for the ecology evolution? Ah, thank you very much, okay? I should have certainly said that. We're assuming all the way through this, and I should have said it right at the beginning, is that the evolution is very slow, EVO is very slow, okay? And the environmental changes are fast with the evolution of a fast. This means evolution is slow in the sense of rare mutations, only one mutation is coming in at a time, okay? And the crucial bit here is this idealized, the fact that I said it's always one strain, it's one strain except in transient when I get a new mutation coming in. Yeah, thanks for that question. Okay, so what we want to ask is what will happen in high dimensions? We've now got something which is more complicated than this because as we're going up here in phi, then the whole function is changing so I'm going up somewhere different. Okay, so we want to ask what happens when we've got on this, okay? So what happens if we're in high dimension? In particular, we would ask, like to ask what happens with minimal ecology, okay? So what does the minimal ecology mean? I'm gonna put small changes in eco, small changes in the environment, in the environment, okay? And the specific thing that's gonna mean is that if I look at the sort of scale of the gradients with respect to X of psi, so this is the change with the phenotype, okay? This is going to be much, much larger than the typical scale of the gradients with respect to Y of the psi, okay? So this was the environment part, environmental change, and this was the phenotypic change. And in particular, I can make a ratio between these, so I'm gonna say this is much bigger. This is going to be, oops, this is going to be here, say, of order of some parameter here, which I'm gonna call delta, okay? So it's gonna be of order of that where I'm gonna be considering delta much less than Y, okay? So what I want to ask is I want to ask minimal effect, minimal effect of ecological changes. And I want to ask what happens with delta small, okay? Well, in this two-dimensional example, it's clear what happens when delta is small. If delta is small, then the amount that I'm gonna move the saddle by as this comes up is gonna be very small. So only if it was extremely close initially, if it was extremely close initially, and this changes a bit, will it go in a different direction, okay? So here, it can change it, it'll make little changes where it goes, but not much. Okay, so I think this is, again, that sort of conventional view, okay, of course they change the environment, particularly bacteria, they're gonna change it by a little bit, but it's not gonna be very interesting, okay? So what will happen if in high dimensions? So what happens is that the phase that comes in, I'm gonna call red queen phase, okay? And this red queen phase occurs for any arbitrarily small delta. This occurs for any delta, no matter how small, in a limit of d to infinity. In high dimensions, any delta will change it. Any delta will make it that I have the following behavior. If I, again, plot things versus time on these evolutionary time scales, okay? So if I look at the psi, which is the fitness, so I'm gonna look at the psi, how that behaves. This will come up, it'll come up, it'll come up, slow down, get slower and slower. Then I'll roughly saturate. Now if this saturates, you can say, wait a minute, that means the evolution stopped, but no it doesn't, because evolution is keeping going because the environment is changing, even though the overall fitness is staying the same, okay? So the fitness, of course, now is not very well defined since the environment is changing, but this quantity, which I've defined the psi, is staying the, roughly the same. Now we can ask how far has it got, okay? So we can draw a line up here, okay? And this is the line of the typical maxima, the fitness of which I would find the maxima as a function of psi, a function of x. Okay, so if I fix the environment, I would find the, I find some maxima, and it's not getting to that. There's some gap there. So there's some gap as to where it gets to, there's a little gap here, okay? There's a little gap there, and this gap is gonna be proportional to some power of delta, okay? The smaller delta raises the closer it gets, but it never quite gets to where the maxima are. So this is really going to be very much, is the wandering around the settles. Okay, how can I see that? How can I see that? Well, I could plot the index, so I can plot the index here. The index will come down, and saturate, so this is now the index. So this means I'm getting closer to the settles, if this was zero, every maxima, and this will saturate at a value here, which goes to some other power of delta. So it doesn't go to zero, so this is where I goes to infinity, okay? If I'm more precise as to what I meant by over here, I meant that the psi will go to, so psi at infinity will be approximately the psi at which the maxima, I first find maxima, first my maxima minus something which is delta vl. Okay, so I'm never seeing the maxima at all, I'm just keeping wandering around, and what this behavior is, is deterministic chaos. So this behavior is that I get deterministic chaos. It's deterministic, because I'm assuming no specificity, I'm just going uphill, but uphill in this weird way, about pulling this gradually changing environment. So of course the effects of small delta and the limit of small delta, this effects of small, it's going to get very close to this, it's going to slow down. If I look at the typical size of the DXDT, if I look at the DXDT, typical magnitude of it with the evolution, this typical magnitude is going to go to zero, and it'll go as T goes to infinity, as T goes to infinity, this will go as delta to some other power, some other parking lot, okay? And you can work out what those powers are in the concrete example, which I'm now going to show. So this is all very abstract. So this is, but however, it's the result of at least semi-honest calculations. And so I want to at least say what the model is, which will give this, okay? But the claim is that this is going to be very robust. There's a whole class of models that will give this behavior, okay? So I'm now going to write down and say a concrete model, okay? The convenience is convenient to put the X's on a sphere, a hypersphere with X squared, say equal to D, that gives you high symmetry and you can do various analytical things. And a lot is known about such landscapes, random landscapes on spheres, but where to fix landscape, okay? So a lot is known about that and all kinds of things about the statistics of the landscapes and the uphill dynamics on that landscape, the effects of fluctuations, temperature or fluctuations like drift are not good, okay? So I'm going to now write down a particular model. I'm going to write down my side. This is going to be a sum of all of the components. So my vector X has now components on XI with I equals one to D. And this is some J, I, J, K, XI, XJ, XK. So that's my landscape in the absence of the ecological feedback and the J are IID Gaussian, they're independent Gaussian variables with some variance, we mean zero, okay? So that's my math, my landscape. It's a simple form of a complex landscape and is known to be one of the generic classes of landscapes. But now I'm going to add a part to this. I'm going to add a part here, okay? Plus a part here, which is now going to have my parameter delta in front of it. It's going to be small on plus delta. And now I'm going to put again, the sum on the I, J, and K of some other set of random variables, I, J, K. And this is of course is going to depend on the X. That's the way it depends on the phenotype, okay? But then it's going to have, instead of the X coming in there, it's going to have Y, K, okay? So this is the way it depends on the environment. That's the way it depends on the environment, okay? And my W's are also IID Gaussian, okay? So I've got a Gaussian random potential on the sphere, the random landscape coming from this part, and then I'm putting a modification that comes from this, which depends on the environment, okay? And then what is my, what is my dynamics? My dynamics is that the X, I, D, T, we've got one component of it, and I, T is D by DXI, Psi, okay? Evaluated, it's evaluated at Y equals to X, right? That's the environment that's there. It's now changing coming from the mutations, right? So this is the evolutionary dynamics on this landscape. The devaluated there. And then I needed a piece which has a Lagrange multiplier to keep the, this bit keeps the, keeps it on the sphere. Technique two. Okay, so this is my dynamics, and then this is a model which one can analyze by methods I'm going to talk about tomorrow. So this is analyzed by methods for dynamical mean field theory. Damn. I'm sorry about that. Okay, dynamical mean field theory, and I'm going to talk about how to do that tomorrow. It's analogous to things done for spin glasses for those of you who are familiar with that. And I said, I'm going to explain it tomorrow, okay? So this is where the source of these predictions, the source of these understandings. And the one thing which we know from analysis of this is the kinds of behavior. So particularly the phase, this red queen phase is generic. You can meaning, you can change the form, you can make this having, you know, quartet terms, you can put other things in here, and the small other changes won't change the behavior. And this is really then in the sense of the physics is like a generic phase. I shouldn't mention, why am I calling it red queen? It's red queen because you have to keep going, running very fast just to stay still. You have to keep up with the changes in the environment by changing, and that's what driving the evolution. And with deltage small, you don't have to go very fast. It's not very red queen, but it's, I'm always have to keep racing and never really getting anywhere. So there's no sense in which is an overall improvement. So statistically, you can't tell the difference between whether you're at this time or whether you're at much longer times. Once you've got these initial transients, things will saturate, and it'll get to a statistical steady state out here at long, at a long time. Yes, so the question is, was the saturation of psi over long times on average? Well, this is one of the advantages of high dimensions. This is a bit like thermodynamics, is the quantities will fluctuate, but the quantities will fluctuate by something which is much less than the amount in which the average. So this will go to essentially a constant amount. It'll have fluctuations of relative magnitude, something like one over square root of D coming around this, that'll depend on delta as well. Okay, so it will saturate. Roughly, the index will come around, the index will also fluctuate around a little bit, but not by much. Okay, and this is the big advantage of the large dimensions or one of the big advantages for doing. The other is the ability to use these methods. Okay, so what I just want to, I'm gonna say I'm gonna explain how you do those kinds of problems a bit tomorrow, but what I want to do today is just ask some questions and extensions about this. So one question is, is there also a phase with different random potential without red queen? Meaning that phi slows down, everything slows down, it always just keeps slowing down, everything just keeps slowing down. So that's one possibility. I think I know the answer to this, but I'm not sure. That's one possibility. The other, another one is if I'm in a simple environment, to the extent that that exists, a simple environment. So there I would sort of think, well, in some sense, there's not gonna be very many peaks. So there's a modest number of peaks, or at least fewer peaks, but I have a lot of evolutionary constraints on how, why can change? In other words, it can only go in certain directions. Whether it can go in one direction or not will depend on where it's gone before. So that'll introduce what's like the settles and will introduce something which has a large number of possible places that it could get to, even though they're the phenotypically sort of the same, there's fewer peaks in the organismic phenotype. But in the level of the nanofenotype here, the changes are constrained, different changes could have similar effects. And so I could change in this way, or I could change in that way, that's similar effects, but a way, say in between, this way I'm changing like that, this would be forbidden by the constraints. Okay, so I'd have a large number of constraints. And so I would like to ask what happens if I have a large number of constraints. So there are many of these, many of these, and particularly actually I would like to ask about what happens if there are order D constraints. Okay, then we'll analyze some things as well. And again, I have some sense of what then happens, but not certainly not fully understand. Okay, and the last one and the most important one is about diversification. In particular, I can get what's called evolutionary branching. And when I should mention this whole framework, this whole framework is called adaptive dynamics. Framework of where the environment changes due to the evolution. And particularly Michael Dobly. Michael Dobly has a enormous amount of work on this, including finding chaos in some models of low dimensions. They're doing it merrily, primarily so can't do high dimensions. But one of the important phenomena is that happens is you can get this evolutionary branching. So say I've got some, again, I've got some saddle in the here, and I'm coming along, and so I'm coming along here, and I'm coming along, say, getting near the saddle. Okay, but then I get a mutant. My mutant can go over to here. My mutant is now not infinitesimal. Okay, so this mutant then can go off in another direction. It comes here, it'll go off, say, another direction. Another direction, this will go this way, and these can coexist. So these two then can coexist. So now I've got two types, okay? And the system will be, quite generically, will tend to be unstable to this branching. Meaning it'll be unstable to having coexisting types. Okay, so the question here is if we ask, what is the question? Can you get large number of types, strains coexisting? Now, of course, some will go extinct. Other ones will be driven extinct deterministically. Other ones will branch, and so you could get a continuing turnover from this. And this is, I think, the most interesting question as to whether or not this can occur, because that starts bringing together the questions about the evolution and the ecology. So I don't know what the answer to this is. Possibly you could get maybe order D, different strains. That would be one of the most interesting possibilities. This I don't know. I've somewhat some thoughts on it, but not very advanced, okay? This I'm gonna talk about now tomorrow. This I'm gonna talk about tomorrow in the general question of many strains coexisting, not in this framework, but in a lot of Otero one, and then come back to this sort of question at the end of Friday, okay? So the main message is here is that evolution in high dimensions is very different than in low dimensions. Okay, it's very different than one's intuition. In particular, that any small amount of ecological feedback, any small amount of that will make the evolution continue forever. So in some sense, if evolution does continue forever, even in a simple environment, we shouldn't be too surprised, okay? So I'm gonna say stop there and take questions and sorry, I suspect I've gone on a bit longer than I should have already. So questions waiting for questions. Yeah, if you want unmute and ask the questions directly. Can I make a comment? Hello Daniel, thank you very much. Hi Mercedes, hi. Very interesting to try to understand you. I think there is a fascinating question, which is when you get this interaction of ecology and evolution, how does the same evolutionary forces that have to do with coexistence of the nanismal phenotypes will determine the dimensions of the nanofenotype. So how big is the variation, the pull of variation from which you assemble the high coexistence at the higher level? I think those things are connected and the way they are connected may be super important for thresholds that we may cross when we set up conditions where we lose the capacity of keeping a diverse nanofenotype. So I think this is a fascinating question. I think where you have hyper diverse systems in biology, evolution over long times and large space has set up a very high dimensional nanofenotype. But that's not by chance. And I think understanding that connection is critical to maintaining diversity. So that's a great point or I guess many great points in there. So most of the interesting properties of environment that an organism feels are properties made by the rest of the biology. And so they said simple example when you take a cell which is complex, you lice it, you've got all of those chemicals in it, all kinds of potential. Of course with geographical structure, the biology will also set up gradients, set up different conditions and you'll get the development of all the further complexities which enable the more evolution and more diversification. So the way I want to sort of try to ask that is is that something which is sort of, we should expect that once the biology has got complicated enough, they're much less complicated than this now. Once it's got complicated enough, do we generally expect that one will get in this state or this phase of continuing diversification, the environment in some sense continue to get complex as well. But even if we sort of saturate how complex the environment is bacteria, not that much more complex now than they were a billion years ago, and we can still have everything continuing once we get to such a state. But there there's a question as you say about sort of threshold as some threshold will have to go over where this sort of picture might start to apply. In particular, there's in some of the parameters which I had here. Of course, D isn't infinite, it's large. For any finite D, there's a delta, but if it's not big enough, if you don't get enough feedback, then things will just slow down and you'll go to a maximum. So you sort of need the D to be big enough to get this going. And then the question is once it gets going, does it sort of continue? And should we be surprised that it continues? I'm gonna say a little bit tomorrow about thresholds in the diversification in a lot of Altaira models in absolute simplest situation. Again, I don't know whether any of this has anything directly to do with the biology. It's really the goal is to sort of make some conceptual things and give them some sort of mathematical footing in the sense of knowing this is something which could happen with sufficient complexity. But a really interesting question and to some extent, maybe it makes some comments on this Friday is that one can ask if you start with something which is not very complex and get the complexities added by the, as the evolution occurs, can you sort of get into some kind of high diverse state? Yeah, I'll give an example tomorrow, perhaps in the malaria case under very high transmission of the existence of such threshold. It may be of interest. Ah, great. Well, so I was gonna make a point, I listened to your lecture yesterday and really enjoyed it and I was gonna make a point of listening to it tomorrow. And one of the advantages of doing things on whiteboard rather than slides is I can modify in real time things based on the other. To know that there's a connection and I'd be interested in that connection. Yeah. So I think these are good questions I guess for the round table next week also. Are there some questions from students or postdocs? Yes. Okay, from Roberto. Is it physically reasonable to think about a rotating saddle environment like the electric field in a pole trap for ions so that X is trapped in periodic dynamics? Ah, okay. So that's interesting. So one of the things that Michael Dobly and collaborators find is that you can get things in these kinds of conditions where you actually do get, you get a limit cycle, you get some rotating around as it's going around, it changes the environment and it just goes on the cycle. You can also get where it branches, you can get where it branches and you have two of them going on, both going on cycles. So something like this, this would go on a cycle, this would go on a cycle and they'd go around and do that together. So you definitely can get cycles. Generally, and I'll say something about this tomorrow, generally in high dimensions, cycles tend to be sort of unstable. You generally tend to have either sort of static things or chaos, okay? And so in simple situations, you can get cycles, but you probably, if you go for a while, one of the organisms will find a way to sort of get out of that by doing something different and it'll go off in that direction. Other questions? So when you show the, you had these plots of the index saturating, the index of fixed point saturating and some of the value. So are these saturations usually, do they happen fast? Is there some time scale like exponential? Okay, so there's a time scale, so there's a time scale associated with this, okay? So there's a time scale associated with this coming down and this going up, depending on your initial conditions. So if you thought of going into a new environment, externally controlled environment, then initially it wouldn't be very fit, you would get, you would go up here. This would be pretty much independent of delta, small changes. You're just doing better in that environment. But these small ecological changes and they would only really start to matter when you'd sort of gotten near to what would be maximum in that environment that wasn't changing. So there's a time scale associated with this, which is just sort of basic time scales in the model. When you go out over here, you don't converge this exponentially, you converge this as a parallel. So you convert this parallel convergence towards these at this end. And so you get slow convergence. And that's the case also, if you look at the slowing down here, this is sort of a parallel slowing down of an exponential slowing down. Okay, but there are characteristic scales. I sort of set them all to one to make life simpler. So does that imply that if you could, so the rate at which new mutations fix would also go down as a parallel, I guess, is that? Yeah, so this is the rate of which is, this is, in fact, this is, I should have written it down, this is the rate of mutations fixing, okay? So that'll go down and how big the effects of those mutations are, so how fast they fix will also go down, but it won't continue to grow down, right? As you go to reward for infinite times, this will still, this won't slow down completely. It'll keep going at some rate depends on self. If we turned off the ecological feedback, it would just get slower and slower. Okay, thank you. So maybe I have a question. Hi, Daniel. So, I'm sorry, I lost the beginning of your lecture, so probably you already addressed this, but so the question is this is based on deterministic picture of the dynamics, if I understand. So are you thinking about, so the question is what is the effect of a noise on this? So are you thinking in your species, I mean, quasi-species picture where essentially your variables are robust with respect to noise or... So how does... So I'm not, for now, today I'm not talking about quasi-species because at any given time I have one strain and then a mutant comes in and the mutant can take over. So I have two strains temporarily. So I only have one strain at a time, okay? Certainly the fluctuations will be demographic fluctuations, there'll be fluctuations in the mutations. And particularly this thing I showed here about the evolutionary branching, this mutation actually has to occur some distance away. It can't be infinitesimal, otherwise you don't get this. So it occurs some small distance away. So that's a stochasticity associated with mutations. So any stochasticity will just add to it looking more chaotic, right? If it's deterministically chaotic, then a bit of stochasticity just sort of adds to that. If it's not chaotic, so if I'm in a situation like I get in two dimensions or low dimensions where I have a small number of, small numbers of maxima and I go, I'll just go to those and just talk about the saddles, then when I get near a saddle, of course the fluctuations will be important. And so which direction I go here will depend on the mutation, okay? So there's an analogy about evolution which I give when giving talks to physics, some audience or general ones, which is the analogy is of a qualifying exam in physics in which some departments, it's traditional to ask one general question or sometimes even a thesis defense. And the professor asked the student, how do you measure the height of a building with a barometer? So the student answered, I throw it off the roof and measure how long it takes to get to the ground. So he was failed on that part, but the professor realized that was kind of unfair. So he asked one of his colleagues to re-examine the student and she did and she wrote the story down. So it's actually based on a true story. I'm sure it's been embellished. So she asked him, what other way would you do it? She said, I would go up the stairs and use it as a ruler and measure the height of the building. She goes, what if you didn't have access to the building? She goes, well, then I would measure the shadow of the barometer and the shadow of the building and I would get the height from that. So she said, what about something that uses more interesting physics? She said, I would make a very good pendulum with the barometer and I would measure the period of the pendulum on the ground, the period of pendulum on the top of the building and being able to get its height from the change in gravity from the bottom to the top. She goes, what about something that uses its values of barometer? So he says, well, I would go to the superintendent of the building and I would say to him, if I give you this really nice barometer, I'll give you this very nice barometer if you tell me how tall the building is. So at that point, the professor said, you pass and as they were leaving the room, she goes, but surely you know what the right answer is. And he says, well, I know what answer you want, but I see no reason that I should give it. And I would say all of the lessons from evolution experiments in the lab is that you try to select on one thing and the organisms do something different. The very first chemist that experiments, as Leo's Lord, or early ones, where he was trying to select on faster growth rate, the bacteria would get all flushed out. What did they do? All of the stick to the walls instead. They just played a different game. So this enormous number of possibilities, and I think this relates, goes back to Mercedes' question, is if you think about the number of sort of possible ways in which organisms can do better in some conditions, that way gets more and more, the more complicated the biology and the ecology, meaning how many other species and things are around gets. So there's really more possibilities. The stochasticity will certainly be important or change things in detail, but in big populations, the stochasticity doesn't have to change things intrinsically. Okay, and I think that's related a bit to what I'll talk about tomorrow. Well, mostly talk about deterministic things, so we understand a little bit about putting in stochastic effects. So I'm a believer that for, you know, microbial populations except possibly pathogens, in some extent, drift doesn't matter. Extinctions matter. Drift matters when you have only one of you, but otherwise the effects of the drift are very important. The stochasticity as far as mutations is certainly important, but even that there are some aspects of it which don't have to be such as, say, the statement that I got once you're over here, you will tend to go there, you're not so likely to jump all the way over here, you know, small mutations can have big effects. I don't really answered your question, but that was just... Thanks. So I said, tomorrow I'm gonna be much more concrete and deal and work through things with concrete models and so on, and be more precise about what the kind of questions and things that I ended with here. Can we have just one more question? Yeah, some kind of traveling way of effects. So there's very interesting things in this, I will say something about as to what happens if you put in spatial structure. And so by spatial structure, I'm gonna mean the simplest possible, the external environment is the same everywhere, things can move around or get carried around and ask what the effects of that are, okay? And there are very big effects of that in some circumstances, they can sort of look like wave effects, but when you get lots of traveling waves, they tend to again get more chaotic like in the ocean. And so there are again, sort of chaotic things can dominate, but the spatial traveling waves is certainly important. Waves in phenotype space that's like the question that I've asked previously about sort of cycles in the phenotype space. So all kinds of things are possible, which things are sort of generic in the sense of not special but particular models, that one has to get at, and that's really sort of deep methods in developed in physics in the last 50 years enable one to sort of think about that, to talk about which things are more generic. And again, I'm gonna do example tomorrow with Lotka-Votera of some very interesting behavior that's completely non-generic, but I will use that to get some more understanding of some more generic, more robust behavior that shouldn't depend on all of the details. A question, I imagine evolutionary branching in two or three dimensions as a possible spherical wave. I'm not sure I understand that. I think here there's a discrete number of strains at any time. And so it isn't that there's a whole sphere of them branching out. It would be two points that would be separating. And then of course, one of them branch again and you can get more. A radiation, going into a new environment, completely new environment, you can get a lot of radiation into many different directions all happening simultaneously. And of course, the mutations can be fast enough and things that you'll get a lot going on at once. Okay, so it appears that we don't have any further question. Thanks very much, Daniel, for a beautiful... If you have further questions or you want clarifications of things that come up overnight, then please email me and I'll try to address those more generally tomorrow. Thank you. Thank you very much. And see you all tomorrow for a new session. Goodbye. Hello, Armon, ciao. Oh, pardon, I just saw that. I just saw it. We are still there, so I don't know why you are here in my screen. People are still there. I'm leaving right now. I'll see you sometime. Thank you. Thank you, ciao.