 Can you hear me? Yes. OK, perfect. So first of all, thank you very much to the organizer for giving me the opportunity to talk to you today. This talk is not going to be about quantum theory, but rather evolution and evolutionary biology. What I will show you more precisely is how approaching evolution from a stochastic thermodynamic perspective, we can gain some insight about the forces that shape evolution by natural selection. And I will show you how some of these forces gives us some surprise in the sense that some forces are, I will show you how robustness of reproduction, the variation, may originate out of one of these forces. But more in to begin with, I would like to start by highlighting some historical aspects related to how evolutionary dynamics have been compared to thermodynamics. And first of all, when I speak about an evolutionary dynamics, what I would like you to have in mind is what we have assisted with coronavirus. Maging a population of organism, it characterized by a genome, a DNA length. And they, in fact, reproduce as a consequence of reproduction. Some mutations happen. And those viruses that happen to reproduce faster, they make it to the next generation and in this way, they propagate. And now if you follow Jacques Monod's reasoning about evolution, here I report an excerpt from his book published in 1971, Chance and Necessity. You will immediately realize that the process of evolution bears striking similarity with the laws of thermodynamics, precisely the second one. Indeed, he observes as if you take a mutation, a point mutation, namely a substitution of one letter in the DNA for another, this process is reversible. But because of the huge amount of mutation that can happen, both because of the size of the genome and because of the size of the population, then this process becomes effectively irreversible. And indeed, he states, evolution in the biosphere is therefore a necessarily irreversible process defining a direction in time, a direction which is the same as that enjoyed by the law of increasing entropy. And that is to say, the second law of thermodynamics. And the reason why I report Monod, let's say, meditations about thermodynamics is that even though he was probably not the first recognizing this analogy, as a biology had an exceptional view of both evolution and thermodynamics. And indeed, he had a clear picture about the fact that the second law of thermodynamics is a statistical statement rather than a deterministic if you want a law. And indeed, it clarifies in the second part, the second law formulating only a statistical prediction, of course, does not deny any macroscopic system the possibility of phasing about and with a motion of very small amplitude and for a very brief space redescending the slope of entropy, thinking, as it were, a short step backward in time. What he had in mind of 30 years before the early development of stochastic thermodynamics was precisely fluctuation theorems. And indeed, with the modern development of equilibrium statistical mechanics and stochastic thermodynamics, this idea could be formulated and explored mathematically. But so far, it had been mainly remained in the domain of equilibrium statistical mechanics or detailed balance systems. Indeed, people have been mainly thinking of evolution in certain regime in which mutation rates are low. This is far from what happens for viruses, for instance. Because in this regime, the population becomes characterized by one or few organisms. And these organisms simply compete one another. The fitness, the fitness survive. And evolution can be described as a walk on a fitness landscape. But the fitness plays a little bit the same role as an energy in a system. In other words, evolution is described as a maximization of a free fitness in which one or few organisms compete one against the other. With the early development of stochastic thermodynamics, people have been thinking of the equivalent of driven detailed balance systems in which the environment changes over time, the fitness landscape then changes over time. And most student and co-workers derive a fluctuation relation for the gain of fitness, which they call fitness flux, written here, F. And more precisely, if you want, this is what Mono had in mind, or it's a reminiscent of Mono had in mind when he was speaking about the second law of thermodynamics. This is a statement about the fluctuations of the gain of fitness as a population evolves in a fitness landscape changes in time. And it tells us that despite the fact that evolution proceeds in a certain direction, on average, small fluctuation in which fitness decreases are possible. For the sake of clarity, N here is the effective population size, which is the analog of an effective temperature for the system, and this is a potential which described an entropic factor which described initial and final condition. But as I already mentioned, all these models describe regime of evolution, which is generally detailed balance. And so what happens beyond this model? What happens in general cases? Would one expect the dynamics to be non-equilibrium? And the fact that the dynamics is far from equilibrium and by taking a general model, what would one learn in this general picture? So what I will present to you today is a more generic approach which I will take generic model of evolution, not preliminary restricted to detailed balance condition. And although this model cannot be solved in general, by approaching this system as a stochastic thermodynamic process, and by looking precisely at the notion of forces driving this evolution, I will gain insight about how evolution proceeds. But before doing that, I need to clarify what I have in mind when I speak about thermodynamic forces. And I will recapitulate this notion in the next two slides. And I will do so with a very simple example, namely three level systems which I will later generalize. So what is depicted here is a system characterized by three states, couple to two thermal reservoir, one hot and one cold. And as you know, the dynamics can be described by a master equation describing the probability that the system hops from one state to the next. Thermodynamic consistency imposes local detailed balance, namely the log ratio of transition rates must be equal to the entropy change in the reservoir. And if we combine this expression with the conservation of energy, namely the fact that the energy changes in the system must be equal to the energy that the system gains from the exchange with the reservoirs, could it be the hot one, could it be the cold one? If we use this conservation law, as well as use the first one of those two reservoirs as a reference, let's take the cold one, we can recast the log ratio of rates into a conservative contribution and a non-conservative one. The conservative contribution describe a force associated who the fact that the system is driven down in free energy. The non-conservative force instead accounts for the gradient of temperature generated by the coupling between different reservoirs. Indeed, you see the difference of interest temperature here. And this is the force if you want that drives the system far from equilibrium. It prevents from reaching equilibrium. It's only when the temperature of the bath is the same that the non-conservative force vanish. And the system relaxes to equilibrium by maximizing this conservative potential. Now this result I derived it for a three level system. It can be generalized to an arbitrary stochastic dynamics described by a master equation. You could think of a generic system with an arbitrary internal topology coupled to arbitrary number of bath with different, with which it exchanges different quantities such as particle numbers, not only energy or work. The general result, which one can derive is to decompose the log ratio of rate into a conservative contribution, which contains all conservation laws which constrain the dynamics. Psi will be in generalized massiocontential. And the non-conservative force which account for a minimal, in the sense of non-redundant, independent set of contribution deriving from difference of chemical potentials, difference of temperatures, all those mechanisms that prevent the system from reaching equilibrium. And now it is this intuition that I will use in evolutionary dynamics, namely the fact that the log ratio of rate decomposing to a conservative and non-conservative contribution identify those mechanism that drive the dynamics in a certain direction. Now, more precisely, and here we'll summarize the main result, you could think of a generic evolutionary dynamics, which is characterized by a certain population of organism. Each organism is described by a genome or an epigenome, whatever variable describes how successful a certain organism is in a certain environment. And this variable, which I denote by gamma, is inherited from generation to generation. As a consequence of evolution, this population is subject to some variation, such as mutation or a combination or horizontal gene transfer. I would not be, the model is very generic. I will speak about mutation for the sake of clarity. And then after that, so the organism reproduce and then organism reproduce faster, have higher chances to be selected. Now these dynamics can be described as a mark of jump process in the space of population abundances. And despite the fact that there is no reservoirs here, there was no conservation laws, there was no entropy change, we can still exploit the fact that variations such as mutations are reversible to characterize the log ratio of forward and backward probability of transition. And here I need to clarify that the forces that arise out of this procedure, the conservative one and the non-conservative one, they tell us in which direction the dynamics is likely to go where this potential increases or where the non-conservative force is positive, but these forces do not arise because of external mechanism or because of coupling with different environment, but because of the internal structure of the evolutionary dynamics. Namely dynamics characterized by variation, reproduction and selection as ingredient. And what we find, okay, first of all, I will in a later, in the next slide, I will clarify what I mean, what is W, how to write that down. But what we find is first of all that the non-conservative force contribution does not banish in general, the dynamics is generally non-equilibrium. But even the conservative contribution do not go beyond in, we see that it does not describe a mere climbing in a fitness landscape. What I will show you is that effective interaction arise in these interactions, not only favors organism that are fitters whose reproduction rate is higher, but also organisms whose mutants are fitter or they have higher reproduction rate. And what I have in mind is illustrated in this picture, take two organisms, two dino types, two types, here in the middle, they will have a high reproduction or fitness and the mutants or the organisms that are obtained by upon mutation are here surrounding this type. Fit mutants have a higher characterized by this being square and low fitness mutant by this small square. Now you can see that this type is has surrounded in the space of mutation by fit genotype in contrast to this other one. You would say that the fitness of this organism is robust to mutation because most of the time the mutant will be fit as well. And what I will show you is that the conservative force contribution favors precisely this type of organisms. And clear to say this is not the outcome of specifically choosing, making a specific choice about fitness or variation. This is a general statement about evolutionary dynamics. So now I will go through the details of this result. First of all, the dynamics that I have described can be described through this kernel here. I've already mentioned what are the steps that characterize a genetic evolutionary dynamics variation, reproduction and selection. Here we made in a population starting from a certain abundance and prime. Pi describes here the rate, the probability rate in which the type gamma prime mutate into gamma. F is the rate of reproduction and the multinomial product surrounding this expression simply accounts for the fact that after the population has mutated and reproduced, I'm selecting for N, capital N new organisms that make to the next generation. It's clear at this point that I'm describing a process generation after generation. The time step is one generation and I'm keeping constant and large the overall abundance of organisms. These models can be of course generalized but for the sake of simplicity, I will use these assumptions. Now, if we investigate the log ratio of grades and look at the forces, what we find is the following. The non-conservative force contribution, as I've already mentioned, does not vanish in general. It's written here, it accounts for the asymmetries in the process of mutation and selection. It is in general difficult to interpret this. We can see in certain regimes, especially when the mutation rate is low, is that this force favors those organisms that are most abundant in the population. And in this way, it's responsible for decreasing the variation in the population because the organism which is more abundant gets selected and selected over time. For those of you familiar with the jargon of population genetics, the phenomenology behind this force contribution is called genetic drift. Regarding the potential instead, we see that it's composed of three contribution. The first is favors organism that reproduces faster. This is the typical picture of climbing a fitness landscape. There is an entropic contribution which favors diversity. And then there is a contribution which is higher. The higher the expected reproduction rate of the organism is after one generation. You see that this is an average of the production rate over the mutants of a given population. And this is precisely what describes the fact that a population whose mutants are fitter is favored. This is an effective interaction term and this goes beyond the trivial picture of evolution as climbing a fitness landscape. In the last slide of my presentation, I will clarify analytically how this term is connected to robustness and I will derive some prediction which can be tested against experiments more precisely. And to do that, I will need to restrict the model through certain assumption. I will assume that the mutation rate is constant and that's roughly the same for all organisms. Also we'll assume that selection is strong and to give you a flavor, this is a little bit what happens with viruses. They're characterized by certain genomes and this genome mutates with a certain constant probability and selection is strong. Under this condition, I can explicitly write the potential in terms of the sensitivity of the population reproduction to mutation. Think of the sensitivity as the inverse of robustness. This is the relative change of reproduction rate of the population upon mutation. When the sensitivity is zero, means that there is not that much change between a certain population and the population of mutants in terms of fitness or reproduction rate. And for this reason, the sensitivity is zero and the population is robust to mutation. Contrast when this is one, it means that upon mutation, the population loses a lot of reproduction rate. And if we now also neglect the non-conservative force contribution and approximate these dynamics to an equilibrium dynamics, we can precisely derive a fluctuation response relation for the sensitivity in which the field associated to the sensitivity a little bit like an energy in this context, the field associated to the sensitivity is the mutation rate mu. And the fluctuation response simply binds the response of the expected sensitivity or the expected robustness to changing this overall mutation rate mu. And it binds it to the size of the fluctuation themselves. Now, what is important about this fluctuation response relation is rather the qualitative prediction that it allows us to make. And more precisely, how the expected sensitivity changes upon increasing the mutation rate mu. And what it tells us is that, you've just passed the 20 minutes, so I'll just turn. Okay, this is my last slide and I'll jump to conclusion. There are two predictions that the mutation rate increases with the sensitivity, sorry, as the mutation rate increases, the sensitivity decreases and the fluctuations of sensitivity decrease. This is a little bit the analog of what happens to a thermal system when energy is reduced, the energy, sorry, the temperature is reduced means that the energy is reduced and the fluctuation reduced. Simply to mention that this statement can be tested against experiment at least defines qualitative agreement with experiment on viruses which can be evolved under lower high mutation rate and the sensitivity of the reproduction rate of the virus can be proved. And indeed, find people who have been found, the sensitivity and the fluctuations of the sensitivity decrease with increasing the mutation rate. So what I've showed you in absolute generalities, how one can build generic evolutionary dynamics. These generic evolutionary dynamics won't be a detail-balanced system in general. And we would not in general describe a trivial climbing in a fitness landscape. We could establish the forces that drive evolution in general, in absolute generality. And now some of these forces are responsible for engendering organism or robustness of reproduction to generic variation such as predation. The question which arises here, I've considered the constant environment. What happens in the environment? One can, what we're working on is this could be seen as changing as additional forces that shape evolution sort of another direction. With this I will conclude and thank you very much for your attention. Okay, thanks a lot for your very nice talk. We have a couple of raised hands for questions. So I think the first question came in this chat by Vladimir Villegas. Do you want to unmute yourself maybe and ask a question or? Hey, good evening from the Philippines. And I was just curious if this study, well, I saw in body environments, did your study already take into account an environment? If yes, what form? Yes, yes, but not, okay. There's the environment would in general, let's say the environment is what is behind the fitness or the reduction rate F here. I'm not making any specific statement about F if not the fact that it's constant. If you consider a changing environment mean you could imagine that this F changes and as I mentioned, you could see this is an additional force. It doesn't affect the statement here about the forces then the drive evolution in general. Simply this force would be written for a specific environment, for a specific F. A change of environment would be more what I described the year and the beginning in which the authors of this paper in which they derive a fluctuation relation imagine that the fitness landscape changes over time because of a changing environment. Do I answer your question? Yes, yes. Just a follow up on it. What is the environment being considered like having a strong coupling or weak coupling with the system concern? No, there is no coupling if not the fact that not really in the sense that imagine that the environment simply has a parameter and the parameter characterizes and I should have mentioned either the reproduction rate or also it could be equally described equally characterized the rate at which mutation happen. All right, thank you. There is no, I should have emphasized it more. This is not a thermodynamic system in the sense of coupling with reservoir. This is an evolutionary system in which forces arise because of the intrinsic dynamics. These forces arise internally from the mechanism of variation reproduction and selection. The environment simply represents the specific values of the rate of reproduction or the reproduction rate. So an example could be in a different environment a certain type, a certain organism may have a different reproduction rate and we could mutate with a different probability. That's it. Got it. Thank you. Yeah, I think the second question was by Samuel Jacob in the chat. Do you want to unmute yourself as well? If not, I will simply read it. Hello, can you hear me? Yes. Hey Ricardo. How's it going? Can you please re-explain on the meaning of the non-conservative force? You called it genetic drift. It's okay. If you could just... Yes, yes, yes. Now, it's difficult to see. If you imagine that the probability at which mutation happened is very low, you could rewrite this expression in a way that it highlights the following fact. Imagine that you have... Imagine these cases. You have three organisms of type one, the one organism of type two and two organisms of type three. What this force would do would favor this one only because it's three... Yeah, yeah, yeah. Because it's the largest in the population. I see. And so at the next population maybe there are gonna be four of this, none of this and two of this. And then this will be again favored because it's a larger. And this phenomenon in population genetics has been referred to genetic drift, which has nothing to do with the drift of physics. They call it... Simply the... No, sorry, thanks. Is it easy to see that? How can you see that's what type of... You need to work it out a little bit of the math. You need to work the math behind and then build specifically these conditions. Is more of a technical derivation than something that can be intuitively seen from this expression? Okay, thanks. Welcome. Yeah, all the best. I think we can have one very short last question. Maybe in the meantime, Obinna Avak would already try to share the screen. I think the first one to raise his hand was David Zivak. So please go ahead. Thanks, Ricardo. That was a very cool talk. My question is, I like this idea of the success of an individual being dependent on the success of its near mutants. And I guess, can you push that further and say, well, it depends on mutants of the mutants. And you sort of get this endless chain of dependence. Should I think of this as sort of a leading order effect of the nearest mutants? Or how do you sort of think about that? Eventually, would we see numerically or... Because again, the model in general, we cannot solve it exactly if not in the approximation, which the dynamics is non-equilibrium. What I observed numerically is that eventually interaction is not restricted to nearest neighbors in the sense that eventually the dynamics will make in such a way that it's not only the nearest neighbors that matter from one generation to the next, but this is more complicated to see than from the force. So the force is a deleting term, but the fact that the dynamics is non-equilibrium eventually will amplify this effect, and precisely in the limit in which mutations are very large when organism experience mutations and they become robust as a mutation. Okay, thanks a lot for this explanation. So unfortunately, we are out of time for now for the questions, but please everybody hold your questions for the discussion section after the next two talks. They will have more time to go into more detail. So now it's a pleasure to welcome Ubina.