 Thank you very much for introduction, and first of all, I'd like to thank the organizer for having this school and workshop. I'm very happy to be involved in this activity. So today I'm going to talk about the kind of theoretical attempt to understand self-replication from physical and chemical viewpoint. So as you know, self-replication is the unique property of the living system. Then actually, I think one ultimate goal of biophysics is to review the property that can differentiate living and also non-living systems. And actually, we already know that living systems should be non-equilibrium and open systems. This is why we are studying stochastic thermodynamics and non-equilibrium physics in this school. But this is not the sufficient condition, I think. A living system should, I think, have the ability to replicate itself. More specifically, so living systems should have a compartmentation that differentiates the system from the environment. So this is very important. And also, this compartment, I will call it volume or cell, should double the ingredient in it and also double itself in size to have the replication like this. So we here focus on how we can represent the self-replication from physical and chemical viewpoint. And actually, in history, we have, yes, we have long history to investigate necessary and sufficient condition for self-replication or self-publication. The most famous one is here, done by Von Neumann in 1940. He used the mathematical logic and also cellular automaton to consider this problem. Then actually, he proposed a set of the system that is sufficient logically to do self-replication. The important point is that this system contains the so-called universal construct that can work like the ribosome. So this is a system. So this is a system that can create other systems based on the so-called descriptors that correspond to the DNA in the actual biological system. So this was amazing that he proposed these kind of things ahead of the discovery of their biological counterparts, so ribosome DNA or other things, about 10 years. However, so this is, yeah, his argument is based on, as I said, cellular automaton. And so this is a very specific setting. So from a physical viewpoint and also when, if we consider, we want to understand the biological self-replication. So his argument has some factors, missing factors or implicit factors. And actually the kind of old catalytic idea of old catalytic cycle exists in this argument. But what are missing, what are missing is one of them is the kind of metabolism. In other words, energetics, he did not consider any energetics because cellular automaton does not require any physical energy in its salary. So it's a kind of logical thing. And also membrane or the kind of, the compartmentation were not so explicitly considered in his argument. So this is very important that we already know. So the cell should have certain kind of compartmentation by this kind of membrane. So he did not consider that explicitly because we have, in the cellular automaton, we have the two-dimensional space of the memory then so we can automatically assign the system on it without considering the membrane explicitly. And also therefore the expansion of the space, the volume, also was not considered. Actually we have other factors, missing factor from biological viewpoint about, so we focus on these three in this work specifically. And also we have other attempt to understand self-replication or kind of evolution from physical and chemical viewpoint. So this kind of attempt, as far as I know, can date back to the work by Alfred Lotoka. He proposed the physical biology and also investigated energetics of evolution. And after that, so we have the hyper cycle by Egan, also the auto catalytic set by Kaufmann. And very recently we have the new attention to this kind of topic due to the rediscovery of the so-called bacterial growth law in 2010 by Scott and Fuhr. Actually this bacterial growth law was originally reviewed in 1950, but they clarified it more by using more sophisticated way. And so motivated by that kind of experimental observation, so many physicists and sanitation now makes a kind of self-replicating reaction system looks like this. However, most of this work focus on the auto catalytic cycle mainly, so the problem of compartmentation or space is mainly ignored. So as long as I know we have almost no theory for the thermodynamics considering both the reaction and also the change of the size of the system. And if you know that, so please let me know. I'm very happy to know that kind of activity. So I don't know why, but one possible reason is that so a conventional theory of chemical reaction presumed that the volume, the reaction volume is constant by assuming the existence of non-reactive solvent or something like that. So if the system, the volume of system is constant, even if we have auto catalytic reaction and molecule and net, then first so this reaction proceed. But eventually so the volume will be occupied by this molecule then so diverse reaction should be dominated and therefore this reaction will stop at certain point by the diverse reaction or degradation of the molecule or something like that. And we can avoid this kind of situation by for example pumping out the new molecule from this system, but so this is the thermodynamics of chemistry. It's not the thermodynamics of self-replication in which the system should expand spontaneously in response to the change of the molecule in it. So therefore we have to have a theory that can address the thermodynamic coordination of reaction of the volume and also the growth of the volume itself. So this is what we tried in this recent paper we published. So in this theory, in this paper specifically we obtained the thermodynamic condition for steady growth realized. And also we clarified that so thermodynamic steady growth state is thermodynamically constrained a lot. So we have so any kind of steady growth state is not possible. So we have the actual constraint from thermodynamics. And also we derived representation of thermodynamic cost meaning that entropy production rate hit dissipation and the chemical work associated with this steady growth state. So this theory I am going to show you soon is based on the macroscopic thermodynamics. The usual thermodynamics you are taught in the under grass shade course. And therefore no stochastic thermodynamics is required and no kinetic detail assumed. And so this is based purely on the second row of some dynamics as I am going to show you. And actually this work was led by my two collaborator Yuki Sugiyama and Atsushi Kamimura. They are working as the assistant professor in my lab. So this is the excellent work by them. Okay let me move on to the setup we have. So we consider the system. So omega is the volume of the system that is valuable. So then so we consider the case there is a pressure external pressure to the system is constant. And so we have two types of molecule in the system. The blue one called constrained chemical that are constrained inside of the system. And the red ones so called open chemical that can diffuse across this membrane or barrier between the system and the environment. As I said volume is valuable the system of the volume is valuable. So an example this is just an example of the blue and the red molecule something like this. So A1 and B1 react to generate two A2. So therefore B1 is a kind of a resource to generate two A2 from A1. Then so A2 break down to one A1 and one B2. So B2 is a kind of waste. And this waste and resource can diffuse across this boundary. And so A1 and A2 form the autocatalytic cycle here. So we consider that so the intensive variable of the bars is the temperature of the temperature kept constant. The pressure is kept constant. And also the chemical potential of these open chemical and the environment are kept constant. So actually physically more correct picture of our model is something like that like this. So here we have rigid wall here and this is a system we have. And here we have semi-permeable and movable barrier. So the later one can diffuse across this barrier but blue one cannot. So this is a more physically feasible picture that we are working on. So from the sum of dynamics so we assume that there is a entropy function of the total system. And so this entropy function is the sum of the entropy of the system and also the bars. So what's the second law of some of the dynamics tell us is that the system can evolve to increase this total entropy. That is some dynamically allowed. Therefore what we have to do is to know when the total entropy increases as the number of blue molecule and the volume goes to infinity. So this kind of expansion forever in the number of molecule and also number size of the volume. Therefore we want to know that this kind of behavior is allowed thermodynamically. So this is so specifically this corresponds to this kind of picture. So this is a space of X and this is the total entropy. If entropy looks like this unbounded towards this direction towards X, capital X is infinity then the steady growth state can be possible some day now. So to derive that conditions first so we convert the total entropy. We introduce the entropy density. I just divide entropy of the system by the volume and here. And the small x is the concentration of molecule. Capital X is the number of molecule in the system. So they are different and also small n is the concentration of open chemicals. From the extensivity of the entropy function then so we can represent the entropy by using the entropy density times the volume like this. And here for simplicity we assume that this reaction inside it is the slowest in timescale. Therefore we consider that energy flux between the system and the environment is much faster. And also diffusion of open chemicals that run much faster than the reaction here. So therefore we can treat them as in treating them in the equilibrium state situation. So for such a case so it is convenient to introduce the Legendre transformation of this entropy function. This phi is the Legendre transformation of entropy density with respect to energy flux and also this open chemical. And this is a function of the concentration of blue run the molecule in the system. And also we extensively use each Legendre transformation that is phi star. So now this is a function of the chemical potential of blue molecule here. So the point is that so this function is can be interpreted as the internal pressure of the system. If these molecules has the chemical potential of mu under the condition that the volume is constant. This is very important so we are going to think about the volume is valuable but so this is the pressure when the volume is kept constant. Anyway and ok then I introduce a second assumption this is kind of equilibrium assumption. We assume that system this system converges to equilibrium state x e q if we keep the volume constant. So this is kind of virtual situation in our system because we consider the volume is valuable. But if we keep keep the volume constant then the system should go to the equilibrium state. So this is because we want to exclude the occurrence of the usual non-equilibrium steady state. That makes the theory much much more complicated and inaccessible. So this is for that case. But the point is that transient non-equilibrium state is allowed even under this condition. And also this equilibrium state is determined by the chemical potential and temperature in the environment. And also the chemical potential of open chemical and temperature. Ok and as I said the volume is valuable and so under that condition so the diffusion of the open chemical is very fast. Therefore if we fix the amount of the constrained molecule blue one then so we can obtain the volume in this state by solving this minimization problem. So this part is just entropy as a function of the volume and capital X. And so we now fix the pressure to this value. Therefore this is just a regenerative transformation and so the omega should be the minimum of this. And therefore omega volume is a function of the number of molecules. And also because of that so concentration of molecules is non-linear function of the number of molecules. So usually this is fixed, this is constant but now this is the function of capital X. Therefore the concentration of molecules will be non-linear function of capital X. By introducing this quantity so we can relate the total entropy as a function of capital X like this after some computation. So here we have the volume and here we have the bar star. This is the internal pressure of the system if the system and the chemical equilibrium. But this is the case when we consider the isobaric situation. So this is external pressure. Therefore this is a kind of difference between the internal pressure at equilibrium and also the external pressure. So here we have the Bregman divergence. Bregman divergence is a generalization of the carbacolibre divergence for convex function. Here we use the phi star for the convex function. And so this is the Bregman divergence between the chemical potential at equilibrium and also the chemical potential at current state. So I'm going to explain the physical intuition of this relation but so we can obtain this. So our first result is that by using this representation we could prove that the state of the system can be categorized into three cases. One is that so this part is positive. In this case the system eventually grows forever so meaning that the time derivative of volume is positive. If this term this two term is the same so this is equal then the system eventually equilibrate. So eventually the time derivative of volume becomes zero. And finally if this is negative then the system eventually shrinks. So for these three cases the entropy function the shape of entropy function looks like this as a function of capital X. So in this case so we have unbounded entropy function towards the X is infinity. So in the second case we have for this dashed line is the maximum of the entropy function. Therefore starting from certain point then system eventually goes to any point on this dashed line. So this is the equilibrium state. Then finally in the last case so entropy function looks like this it's maximum locate at the origin of X. Therefore the system evolve to shrink to the zero. So this is the very general consequence from this representation. So the next what is the meaning of this quantity? I will give you a physical intuition behind it. So let's first consider isochoric case such that volume is kept constant. In this case the pressure external pressure is variable to keep the volume constant. So in this case at certain state so we have chemical potential. Corresponding this chemical potential we have internal pressure that is obtained by the buster. So this system chemically tend to relax to the equilibrium state mu eq. At this point also we can evaluate the pressure. Usually these pressure two pressures are different but if the volume is kept constant this internal pressure automatically balances with the external pressure from the wall. Therefore there is no problem to relax to this state. However we consider the isochoric case where the volume is variable and the external pressure is constant. Therefore if we at certain point then we can evaluate the internal pressure. And this internal pressure should be balanced to the external pressure that was specified by the bath. If this equilibrium, chemical equilibrium state does not have the internal pressure. So this internal pressure is usually different from the external pressure from the bath. In that case the system cannot relax to this state. Eventually the system goes to infinity or goes to shrink to zero. So this is the mechanism when we consider the variable volume. So this is the second result we have. So let's consider the expanding case. So the system expands forever. So we can expect that in that case the concentration of the molecule will converge to a certain point. So actually the number of molecules converges to infinity. The volume also expands to infinity. But we can expect that their ratio converges to certain finite concentrations. So we call it a steady growth state concentration. And so this concentration should satisfy this inequality. So this term came from, so this part. So we just plugged XSG here and so we have this one. So this should be positive in the case of steady growth state. So this inequality can be decomposed into the two conditions. One is from the constant pressure. So external pressure should be always balanced with the internal pressure. So therefore this is this defined accessible state. This is this one. So this is an example. So accessible state under this external pressure. And also this Z determines the region where the entropy production is positive. This is from the second law of thermodynamics. And so this Z region is shown in red. Therefore a steady growth state, the chemical potential at this point should be constrained only on this line. So on this eye and also within this red region. So this is very strict constraint that can be accessible in the steady growth state. So this is our second result. If the system is in the equilibrium case, then so this intersection, the intersection of this and this is unique only one point where so we have the equilibrium state. So in the shrinking case they are not intersecting. So this is the second result. And then so from the representation of the total entropy we can obtain the thermodynamic cost for this steady growth state. So this is a representation of the steady entropy production rate. So that should look like this. And by using this we can evaluate growth efficiency like this. So this is the entropy production rate is divided by the rate of volume expansion. So the right hand side is always positive here. And the point is that so if this right hand side is close to there, the efficiency is maximum because we can expand the volume very much for unique entropy production. So this term becomes zero means that so the system is in equilibrium therefore maximum growth efficiency is achieved at the near equilibrium. And finally so we can obtain the heat dissipation representation. So this is a heat and also we can obtain the chemical and the mechanical work. So this is total work and this is chemical work part and this is mechanical work part. And so this is very reasonable because here this is external pressure and this is expansion of the volume therefore so this is a mechanical actually work that the system does to the environment. Okay that's all of we derive in our work. So let me discuss a bit what's missing in our story and what we have to do in the future. And actually so we assume that old catalytic reaction is the slowest. So this means that the old catalytic thing is the limiting step in the system. However it could happen that diffusion or volume expansion is the slowest. When the if we think about the real biological system so the real biological system we have transportation system is transportation between the system and the environment is very slow. Then so diffusion or volume expansion would be the late limiting. In that case we have to consider that situation by switching the role of reaction and also diffusion in our framework. And also in our theory we assume that the external pressure is constant. This means that the tension of the membrane is constant irrespective of the size of volume. This is sound unrealistic if we consider actual biological cell like this. And so yes tension is generally a function of volume and also the size of the volume of a cell usually changes by the synthesis reaction of the membrane. So in this case we have to consider coupled some dynamic system that involves self-replication and also the generation of the membrane and also that expansion or shrink or something like that. So these are what we have to do in the next case. And finally so in the second assumption we excluded the possibility of usual non-equilibrium steady state. Actually non-equilibrium steady growth state that origin is the extensivity of entropy function. So if the concentration is the same then so from the extensivity so we have so these two are the same. So this extensivity is the origin of the kind of eternal expansion. But the usual non-equilibrium steady state come from the kind of cyclic flux so we have to combine them. So to do that we need new types of non-equilibrium chemical reaction theory that can handle both usual non-equilibrium steady state and also this kind of volume change. Actually so we are very much interested in that kind of extension. To this end so what we are now thinking is that geometrically behind some dynamics is very important I think. I'm from applied mathematics so for me so this time so I learned a lot about some dynamics from my collaborator. My collaborator is a specialist of some dynamics and so to me the equilibrium and also non-equilibrium thermodynamics are largely related to the convex optimization problem with complex constraints mathematically. Actually in our case so we consider the dual space of the extensive variable concentration and also chemical potential. So this dual relation is induced by the general transformation of some dynamic function phi so this is convex function. And from that so difference of total entropy can be described very generally by the Bregman divergence. Actually so we are in another work here we are working on extending this kind of picture to non-equilibrium situation so in which we consider another dual relation between some dynamic force and some dynamic flux. So now so we have the dissipation the on-server's dissipation function. So this dissipation function induced the Legendre duality between F and F force and flux. And in non-equilibrium system so these two things so this is equilibrium some dynamics and this is non-equilibrium some dynamics. They are linked by very fundamental equation of continuity equation and also gradient equation. So in order to understand this kind of complicated structure it is fundamental to use the word of geometry. And then actually this is the geometry so what we need is a geometry of convex function and the Legendre duality. That is the information geometry some of you know about it. So this is very useful to understand this kind of structure. If you are interested in this kind of things so please take a look at our other paper we published in very recently. Okay I'd like to thank my collaborator and also funding agency. Thank you for your kind attention. The extensivity that you're assuming right? Yes. In thermodynamics and statistical physics we usually get that from assuming that boundary terms are being neglected. I could just say the last part again. The convexity that we get or the extensivity that we use in statistical physics is basically because of neglecting boundary terms. That's the reason why thermodynamic quantities are extensive. The crucial thing I always thought in self-replication and this has always been a conceptual problem for me as to how to actually mathematically model replication is that there's a discrete event when a cell closes off from its daughter. Alright. And there's an information content to this because there actually has to be information on both sides of this barrier being you know shutting. Yeah. Right? Yes. I've never been able to disentangle how one could talk about entropy and thermodynamics without disentangling this you know absolutely essential information partitioning. Okay. Right? Because there is information that is being put in that roughly half the contents especially the replicating part of the cell contents have to be roughly equally distributed over all the parts. But if the system is uniform sufficiently then. There's no living system that can be uniform because there is a specific quantity that is carrying the information that allows. Like DNA or something like what you mean. Okay. Yes. So there's a very specific how should I say obstruction to extensivity. You know what I'm saying. Yes. I just thought about this because this is kind of the first thing that hits you in the face if you try to model replication is that irreversible thing of the membrane contracting. Right. Yeah. Actually so and so our salary is not enough to obviously our salary ignore that kind of program. So we just focus on how the volume can expand. But that's it. That is necessary condition to realize the self-replications. Yes. I really very much interested in your point. So but that kind of thing. So we have to have the kind of stochastic some of them that we we need because the we have discrete number of molecules. So this is this work is based on the macroscopic some of them. And therefore we need to have. Yes. But very, very. Yes. Yes. There's a wild conjecture. There's a wild conjecture that in your steady state growth condition at some point you get to a metastable state. In other words your Legendre function is it's no longer convex. Okay. Yes. And that's when you have to the metastable state collapses to two separate replicates. Yes. That's the way I would envisage your beautiful growth dynamics of interfacing with replication. Yeah. I'm very curious about that kind of. Yes. Sorry. Maybe I think the good toy model is kind of self-replicating self-spontaneous dividing trot cell or membrane. So some experimentalist find that kind of reaction. Therefore it would be nice to play with that kind of very basic physical model to answer that. Yes. I'm very much interested in that. Okay. If not let's thank the speaker again.