 You can start. So in terms of announcement, there is anything I have to announce. No, I mean, if you have a look at the program, you know that this afternoon it's like free. So I encourage you to have a discussion. You can have the discussion wherever you want in the terrace, in the water, in the city. But please use it to chat with others. And then we have the first speaker of this morning, Roberto Mollet, who is going to talk about interacting cells from toy models to the industry. All right, good morning. All right, this is my name. I am a physicist. I'm sorry for that. I'm in the sense that I am a real physicist. Still half of my life is pure physics. So spin glasses, stuff like that. The other half arrived to biology like six or seven years ago. I was a bit bored about physics and about the fact that the Cuban community of physics is small. So I have a few friends working on the industry. And they actually, in this company, the Center for Molecular Immunology, they produce vaccine for cancer, COVID, a lot of stuff. And they are pretty good. And the community is large. And one of the head of the management board of the company is a good friend of mine. He's a physicist from formation. And he asked me to help him to solve a couple of problems. Now I am going to introduce you to them. And they are not problem from physics. They are not even actually problem from biology. They are problem from the industrial technology. And essentially, he introduced me to this problem. They have these big production processes. So they do basic research in biochemistry, but also they produce these vaccines. And they had, in practice, the problem that they have this continuous production process of monogonal antibodies, let's say. And it works perfectly. And sometimes it just fails. And that's it. It just fails. And that's it. Nobody knows why. They have to start everything from zero. And that's a good, big mess. And they didn't understand why. So I say, all right, but we are in Cuba. I mean, the condition goes up deep. And something changed. The air conditioner didn't work, whatever. But he convinced me that no, that something was implicitly wrong. And at the same time, they don't have, or they didn't have, a conceptual understanding on how to improve the production processes. In the sense that there is a lot of expedient from the engineers on the alternation on what's the proper dilution rate in the fermenters or whatever. But they don't know why. So they find different outputs using different parameters of the system. And the outputs are really different in the sense that they don't always get the same stationary state. So the problem he posed me was something like that. It's very simple. In this picture, you have a big vessel where your culture cells, the density of cells is x. So you have nutrients given to the cells. But not only nutrients, cells produce, I mean, the proteins that they are interested in. Si, i is an index for the product they are producing or the nutrients. So the nutrients are put inside, in a continuous mode, the vessel with rate d. And they are classified by ci, the concentration. And then with the same usual rate to make it simple, cells and byproducts are extracted with the same rate d. This is not necessarily exactly like this in an industry, but this is a good picture, right? So he explained me the problem. And he said, all right, this is working out sometimes. It gets down or sometimes changing a bit the parameters, the results are completely different. I don't understand why. So let's try to see. It's important to know that the cells that they use are mammalian cells. So it's a mess. So therefore, I will try to put as less biology as you can. You need to put as less biology as you can, because I mean, it's a mammalian cell. So what you know about biochemistry there is much less than what you know about biochemistry or biology in bacterial cells, right? So I went to my books, his books, actually. And he asked me this. And all right, this is the way in which you essentially describe this chemostata, just something very easy. So the rate at which the number of cells increased depends essentially on the growth rate, the death rate, and the dilution rate. This is a question everybody knows. And this is also an equation for how the concentration of products in the vitamin change on time. So you have on the right or on the left, I don't know. So you have this term that essentially accounts for how often you introduce nutrients and how often they go out. And this is a term that explains that gives you how often they are consumed by the cells. That's very easy from this point of view. Nothing to discover. Then all right, who are mu and who are all right? I can do that. That's physics. That's very easy. That's the differential equation. Then he tells me, all right, let's try to understand who is mu. Mu depends on the fluxes inside the cells. And a thing like the death rate depends on some chemical component, may depend on some chemical component that are outside in the system. That could be also a situation. But then things become hard. Because everything is connected. So you have the mu depends on the fluxes inside the cells. That's clear. And sigma, which is the death rate, depend on the concentration of the nutrients or everything. This is the connection between what is happening. And all right, I say, all right, let's try to see what's happening inside the cell. This is the metabolic alkali, all right, forget about it. I cannot do that. I cannot solve both problems at the same time. This is too complicated. I'm a poor physicist. And then what they say is, all right, let's try to simplify stuff, let's try to understand what the simplest model we can deal with. And then I found this. It's a model that proposed a friend of mine, I say, basques. And it's probably the simplest model in metabolic network. It's a network in which you consume something, let's say glucose, for simplicity. This transforms energy and pyruvate. The pyruvate then goes to energy again, through respiration cycle, or to lactate, all right? And there is one reaction that is bounded. And the idea of this paper was essentially that if you bound this reaction that essentially says that you cannot use too many enzymes, you can go to overflow and explain barburet hypothesis, whatever. That was more or less the idea I say at that time. So I say, all right, let's now try to understand what's happening with this model. If we combine with the chemical start, that was the first point. And essentially, the model, you can produce in these equations, one equation for the conservation of mass, one equation for the gold rate, and one upper bound for the reaction. And the intuition was, all right, let's try assumes, as usual, that we are in the situation which cells are trying to optimize or to maximize biomass. You can maximize, I mean, a more complex environment. You can make more complex cells. You can maximize many other things. But here, essentially, mu is a proxy for E. And that's what you are asking. So in practice, you get to this set of equations. So these are on the top. You have the general equation I wrote in the first slide. On the bottom, the same equation, but specify for these, no, I'm sorry. In the top, you have the equation for the density of cells. On the bottom, you have the equation for the nutrients and waste. And then if you put together these, you have to solve. This equation, given that mu is represented by the maximization of biomass given by this equation. And these are a couple of equations. Because how much glucose, for example, you can take will depend not on the substrates, but on the density of cells, the density of cells x. That's why, because you don't want S to be negative. So you can't guarantee that S will never be negative, only if you guarantee that the maximum of half of u you can take is bounded in this way. And this is a collective equation. So you have these equations here that receive information from the solution of the cell. And the cells that receive information at the same time from the density outside in the chemostat. That's more or less the way in which you close this simple equation. And in stationary state, that's a picture you get. So you get a picture with, when you move dilution, let's focus on the left figure, when you move dilution, you have one big branch. That is, I would say, engineers who like to have the maximum number of cells. But then you can have also second branch that depends on the death rate connected with the quantity of lactate. And what may happen, our intuition is that depending on where you start, so the number of cells to which you start, or the dilution at which you start, you can get in one branch or in the other branch. More generally, you can have jumps from one branch to the other if you have fluctuation. That was the picture we figured out from the system on the right-hand side. You can imagine, I mean, you find the same plot, the same stationary system in which you don't have this contribution of the death rate to the death rate given by the toxics by product of the cells. In this case, the lactate. Let's say this is the standard picture. The density of cell increased, but then if the dilution rate is too large, then at some point you are just washing out the chemostat and the dilution of cells goes down. That's the standard picture. What we say is that the picture may be more complicated if you assume that there should be some ways that you can get at the toxic system. That was the picture we found. And then at some point, you can play with that. And here, for example, what happens if you start a given dilution rate with a fixed number with a few cells. And what is happening is that, all right, this is x. That's what is signed there with the arrow. And then you have this production of lactate and glucose. If you start with a different protocol for the dilution rate, you have to find a different solution. And so on and so forth. This is just a picture of how the dynamics of the system may look like. They look like, right? And then, of course, Si, I have to remember you, is this relation between the density of cells and the dilution rates. And essentially defines different transition of the system from overflow to respiration and from competition to no competition. That's very clear. And what is important is that this side is a parameter that we will try to use further because it's the relevant one to characterize the stationary state of the system. So then we say, all right, now we have a picture of a toy model that's very beautiful that explains qualitatively what is happening with the complexity of this chemostat of you. And let's try now to see whether or not this phenomenology is reproduced in a larger system. Now we did the same work. Now we threw away the simple model that we understand because that's what we can understand. And now we put on the same methodological framework the genomes came in the network of this mice, I don't know is something like that. And those are now we move from three reaction to 6,600 reaction. And the picture is the same. So on the left, you will solve the equation for the chemostat. On the right, you will solve the FBA equation for this kind of stoichiometric matrix. And the picture, more or less, if you move size similar that you first have a lot of nutrients because the x is very small. So you are eating everything that you are giving the media to this cell. So for the media, we propose the same media they use in the company. So they have a media, they gave us the composition, we put this stuff there. And all these things you start to eat. And varying psi, which is equivalent to changing x or changing d, essentially. So what you find is this kind of transition. And if you look again for the x versus d curve, you find that the phenomenology is maintained. And that's nice. And there is no special feeding parameter. The only thing we are considering is that the death rate depends on the concentration of waste. In the system, in this case, acetate and ammonia. And we use number from the literature, essentially. That's what we find is essentially that the picture that we have for the toy model is a picture that is acceptable, at least in the genome scale metabolic network. And at that time, we're very happy. Then we started to speak with engineers. All right, now we need to try something. We need to see whether this is true or not. These experiments are very complicated in the sense that these are mammalian cells. So to see something, you really need to spend a lot of time, because the reproduction rate is like 24 or 48 hours a day. And so as you see, the whole experiment, this is the viable cell density in the system. And the whole experiment took like 90 days. So you can imagine in Cuba, so this is not the production system. So they are doing this in smaller chemostats, which they are not so careful attended as the big one that where they have to use to produce. So therefore, sometimes things go to the hell and they have to start from zero. So it looks like a year almost to have all this data organized. And all right, let's try to explain this experiment. So this gear, essentially, this I didn't do, that was a fantastic student engineer we have. All right, this is time. So they put the culture. They decide a given dilution rate, in this case 4.5 per day, I don't know in which scale, essentially. But the point is that during some time, they find what they call a stationary state. Stationary state one, right, at this dilution rate. Then they change the dilution rate. They decrease the dilution rate. And they found a new stationary state. And then they change again so they continue to decrease the stationary state and find a new dilution rate increased again, and now they started to increase again the stationary state. And the point is that they started to take the same numbers here. So they use 0.4, 0.4, 0.4, 0.4, 0.45, 0.45. And the intuition is that you want to see and to prove here that you really find, at the same values of the parameters, different stationary states that the history depends. It's important in the system. And that's what more or less at the end what they found. So that the density of variable cells is dependent on the history and that you have these two stationary states that more or less are consistent with the picture I showed you before. So at that time, we were super happy with the results. Then we did some more study about what now is fashion within the engineers about that analysis. And they found that the pathways that are activated are more or less the same path where we proposed. Sorry, I don't understand something on this picture. So the idea is that you do an experiment where you shift the dilution and then you move between these two. Right, that was more or less. At this point, our engineers were very happy. All right, now we want to optimize the production process. So to optimize the production process is harder. And the intuition is, on some point, either you have to dilute the disease or to optimize the dilution protocol. That's something we are working on. I'm not going to say anything now because it's work on progress. So we are not happy yet with the results. But there is another way in which you can optimize the production process, which is to reduce the cost of the nutrients that you're giving to the cell. So the intuition is, right, if you give no food to the cell, so if you don't waste money buying to some big company, the media to make the cell grow, but still the cell grows and produce the vaccines or the monochrome antibody that you want, then you're done. Because all right, for cheap, for free, you're getting produce, you're producing something. But that's, of course, not possible to get it. You have to feed the stuff. And the idea is whether you can decrease the cost of the production process. At that point, we say, all right, let's now try to assume that the curve we found is reasonable. And now you know that there are some costs associated with the production process. In particular, this is a fixed cost. This is the cost of processing the volume of the effluent. And this is the cost actually connected with the nutrients you are providing to the cells. This is the first step, and the question is whether we can optimize the first step. So the second and the third term, so the third term is the fixed cost, so I mean, higher less drivers, higher less people cleaning the rooms, higher less engineers, whatever. This is beta, gamma is associated with the cost of processing what is coming out, so the volume of the cells. Again, you cannot change this too much. The only thing you can actually work on besides fighting people working on the company is to nutrients you are giving to the fermentators. All right, that's what we try. And from this point of view, the idea is you have the concentration, which is CI, and then you have to optimize how much. I'm sorry, you have the cost of each nutrient, and you have CI, which is the concentration. The intuition is that if something is too costly within the media, you have to provide the less concentration of that in the media, but guaranteeing that the cell keeps growing. That's more or less the idea. So question is, can we fix CI in order to minimize C? So C is the cost, I'm sorry. And the answer is yes, if you look at this, this is a linear optimization problem. So in principle, it's very easy. The only thing you need to do is to pose the same problem. You say, all right, if the engineers tell me that I want to have this cell here, this density of cell, for example, this red point there, but the way these numbers are compatible with the number in the industry, this is the X they want to have. And this X was obtained with the given media that they gave me. This is the media we're using. Let's try to invent a different media that produce the same number of cells. That was the point. And to invent a different media, it means to change the concentration of the media, but minimizing the cost, but guaranteeing that the production of cells is the same. That's more or less what's the question. And these are more or less the results. If you look at it, so you will have. Psi, as I told you, is the parameter that matters here. It's D over X, or X over D. And each panel is a different nutrient. So you've seen proline, cysteine, arginine, aspargate. I'm sorry for the names in English. And the red curve, so what is red there, is the media they're using. So it's the concentration that they have in the media. And the green curve at the result of our simulation. And of course, one of the results that matters here is that, in principle, the amount of media that you have to give to the cell or to the culture depends on the situation of the culture. In particular, not only the situation, on the ratio between D and X. That's one important result. Then if you see this continuous line, it is just because this is an unstable phase that we described before between the high density phase and the low density phase. Of course, in most of the case, what we found is that you should decrease, as that is possible, in principle, to decrease the concentration of some nutrients in the media. You see that in most of the times, the green curve goes below the red curve. You can have a poorer media and still guarantee the same X. But of course, now you go to the engineer and you say, all right, we're done. Use a poorer media that will cost you less. And they tell me, no, you are crazy. First, because cells do not only use metabolism. So these products may be there for all the things. And they have been optimized by some big company somewhere in India, I guess, to produce this stuff. Second, because even if you manage to do that, you cannot call the company and say, all right, I need a specific media for myself because it will cost you a lot. You cannot say the company give me this media just for me. That's something they will not do. So when we say, all right, but now what you can do is to try to keep everything constant and try to introduce more nutrients of where we propose that there should be more concentration. Like, for example, this thing. So you see this thing here, you see that we propose that you should increase the amount of glycine in the system to get the same X. And now they are doing experiments on that. So in the next conference, I will explain if we succeed or not to improve the production of the production process, all right. But then at that point, I got bored. And that was in general, it was the problem were technically easy if you want. And then I told you, I am a physicist, so I have a bias on that direction. And then we start to ask, all right, but one of the big assumption in this problem there was that the chemostat, that all the cells in the chemostat were the same. So that was in fact, kind of mean what we call in physics and in field approximation in which we said, all right, there is not much difference. There is one cell or many cells, all the same, all the cells are the same. You can imagine the system like a supercell that is just growing and increasing the density. It makes not too much sense. So the idea is what's happening is the system is inhomogeneous. And from this point on now, let's try to remind a bit what does it mean in practice. So we have, we already, Marcelo explained a couple of years ago, a few days ago, and somebody gave a tutorial about that, but I will review everything. So the way in which we imagine the cell is a way in which what matters is the psychometric matrix. So how metabolites interacts to produce other metabolites. And this is easily encoded in this equation there. So this is the psychometric matrix. This is the velocity of the fluxes and this is the input and output on the system, all right? And what it tells you is that there is a whole space of solution of possible solution because differently from what happened in algebra. In algebra, when you start the first years of physics or mathematics or even biology, what you find is that you usually find the solution that solves a system of equation. And you are guaranteed to find a solution because you have the same number of equations than variables. Here you have a problem that is completely different because you have much more variables than equations. In this sense, all these solution in the gray area are possible solution of this system there, right? And the bounds are given because you know the bounds of the reaction. So in principle, a reaction cannot be as fast as the velocity of light. So the reaction has been expanded by biochemical processes. And if you're lucky, biology is biochemically state u, which is a bound from experiment or from intuition about which are the reaction that is happening there. And in principle, you find the solution. What FBA does, or let's say the restitution of FBA is to say, all right, this is true. But in principle, to figure out what's the real solution of the cell, what cell is happening, you have to assume that cells is doing something. Optimizing biomass, reducing the amounts of, minimizing the glucose, maximizing the gel, maximizing the production of ATP, that's the way in which you tweak the problem to find out one solution. That's more or less what FBA does. And most of the time, especially when you work in bacteria, but it's not necessary, what you say is all right, the cells what it's doing is maximizing biomass, which is more or less the first approximation reasonable. And this is translated in the fact that together with these restrictions there that defines your problem, you have to maximize some linear function of the parameters. Again, this is not necessarily true. Actually, this is not true in general. It is true only in some, it have been proved to be true in some particular case, but let's take with that, right? That's what we need in the first part of the talk to assume that the cells was maximizing biomass and let's, we assume that this is a general picture, right? You will try to maximize biomass or maximize something error, minimize something else. But the idea is if you say to me that it is not biomass, it is something else, that what is changing is, which are the reaction that you put in this part here, right? That's more or less the point. The intuition then is how could you assume that the system is not homogeneous in the sense that not all the cells are doing the same. But the easy or the easy way to achieve this tax because in one case you have all the cells are doing the same. In the other case, you'll say, all right, every cell is doing whatever she wants. It wants, I think. And all the possible space of solutions are at the end. Each cell is just a different point in this space. That's more or less the intuition. Well, I lost. Well, if you realize that a good way to describe this is to use in this Boltzmann factor, this exponential factor, then then we are going to justify. But the idea is that if you take, instead of assuming that P of B, so the space is flat, you will use this beta parameter that may help you to tune the system from all the cells are equivalent to, which is one point, to all the solutions are equivalent. And this is the way in which you explore this. So in principle, I mean, if beta is zero, so you don't take care of the exponential. Exponential is not there. So you have only the data vector that is guaranteeing you that you are here inside. I have to control this, all right? If beta is infinite, you, oh, wait, wait. It's strange, all right. If beta is infinite, you have to be careful. If beta is infinite, then you guarantee that the system will get a point. We'll have to be sensible with that. All right, that's more or less the way we take. And then we move on. In principle, it means that if you want to understand the problem, you want to understand how this property, what are the properties of this property? So beta is just a proxy to describe the homogeneity or heterogeneity of the culture. That's more or less the solution. Fortunately, it can be used to solve even big problem, large problem in the system. Now we have a smart algorithm called Expectation Propagation that was published in 2013. Now have more than 2,000 sites. Most of them are to archive work from. We were discussing yesterday of the importance of archive. And then there was a rewriting, actually the discovery of the algorithm, but Alfredo Brausten and Paola and Pagnani that they thought it was a new proposal. Then they realized that it was already used in the computer science community, or in particular to study this kind of problem in the metabolic space. But the idea is that the same way we could do very easily linear programming, we can solve this kind of problem, we can explore this kind of space now using this algorithm. It's a bit more complicated than linear programming, but it's also doable. You can find different implementation in Python. I think Julia, for sure, and C in GitHub, also has one of them. That's the picture. So what we are doing now is, now we are to the problem we had at the beginning, now we are introducing this idea of the heterogeneity. And this is the result, I'm not going to go to the detail, but in principle what you find is that this is the toy model, this is the since how, six thousandth reaction model. And what you find is that heterogeneity, of course, will change the kind of results, in particular if there is no optimization, you will lose the two branch structure. More importantly, if you go to the complex system, so the system with six thousandth reaction, what you find is that it's set in case, so you will find that here, you will find here that it goes above the red curve, although it's less heterogeneous, more heterogeneous. So you will find this kind of trade-off in which heterogeneity may help you to increase the density of cells, all right? But then they are- Sorry, Roberto, just so in the previous plot, the fact that the scale of this lambda M is very different is because you have a pre-factor which depends on the size of the network. Yes, and the value of the fluxes, and the value of the fluxes, because here you have a toy model in which the fluxes goes from zero to one, and here you have the fluxes that are bounded by three biological numbers, let's say, so the numbers are completely different. But yes, this is a good question, I'm taking your question. All right, then, all right, I get bored again because I say, all right, this is a very trivial way to introduce heterogeneity, and I know that there should be much more than that, or I had the intuition at that time, and then we move to the second question, and I want to remind you this plot that was presented by Andrea de Martino. He essentially showed these cells moving around a film, and he showed that some of the cells were producing acetate and changing the pH of the system, and some other cells were just consuming the acetate, I think was the system, and so you have the corals of the, on the top you have different pictures with corals saying which is consuming or which is producing something, and then you have this beautiful panel you find in the paper that I'm not going to describe. It's not my work. And then, of course, it's not the only case. Then I find a different example. So this is a culture in which they have gist. In principle, it's an isogenic culture. This is the colonies start to grow, and what they find is that cells in the border of the culture are eating something, sorry, eating sugar in particular, and cells that are inside the culture are essentially eating the byproducts produced by the cells in the border, all right? And then, of course, there is a lot of literature about this heterogeneity, even in a simple culture, and here we have discussed a bit about that, although in a more general context. Therefore, all right, I say, all right, now this is the picture I had for these cells that were there. They were not interacting. The only thing I said was they were just using different metabolic state. Now they are interacting. If you see this, and you are a physicist, there is a trivial way to assume that cells are interacting, right? And then it's assumed that this age is not anymore what you have here, but now that you have this kind of interaction between cells. This is the three, I mean, just come to my, to your mind in two seconds, and that's the way which you will assume that cells are interacting. Then we can discuss what's the meaning in this context, but the idea is that you have the first terms in which cells are maximizing biomass. This is tuned by beta, that is multiplying everything. And then now you have a second term in which cells are not only maximizing biomass, but depending on the value of J, they are exchanging given fluxes or they are competing for a given nutrient trying to both maximize the same, yes? All right, I will be. All right, there are two indices. I is an index for the cell. Now you have many cells in your system, which is this I and I and J. So cells I interact with cells J, right? And R is an index for the reaction. So inner reaction for the cells I, so how do people do? All right, in cell I here, you will have, you will try to optimize, for each cell I, for each element of the sum, you will try to optimize some combination of the reaction. This is what happened before. This is FBA, if you want, right? And then you have cell I interact with cell J through some flux, G, R, I, J. So R says, which is the flux? So the cell flux that cells I is, with which cells I is interacting with cell J. That's more or less the index. I'm not, no, it's clear, right? Then of course once you are there, now I mean the physicists come out, so this is your physics come out. What do you mean diagonal? So yes, flux three interact with flux three. Never within your flux. I mean only through the stoichiometric matrix directly, no. But in directly, yes, because then you have the stoichiometric matrix, all right, that if you are introducing something, everything should maybe, should conserve and be there, so this delta, the direct delta should be valid. And also because you are trying to optimize the stuff yourself independently. So interacting directly, right? If you consume something, still everything within the cell should be conserved. That's the way. So the idea of interacting flux three with flux three is that I produce lactate and you consume lactate. This is flux three. Flux three may be bi-directional in this case. So I produce lactate and you consume lactate. That's the intuition more or less. So that's something that we have seen here around in the community, but that's, let's say the easy way in which physicists can put this in a Hamiltonian form. If you, you, you can also imagine a lot cover terrex scenario in which this is valid if you have symmetric interaction. So you can imagine that you can write down dynamical equation that have a minimum in this kind of functional function. But that's, let's say for physicists, whatever. So this is the point. And then what we need to say, all right, now let's try to assume that J is a random number that come from a Gaussian distribution. And this is, and then we say, all right, again we start with the simplest possible model. This is the, this arrow there is coming in. So it's, the upper arrows should be coming in. And this is a glucose. And this is the simplest model you put there. And then you take, I mean you go to Paris's course, take all the machinery of the replica symmetry stuff. And this is the picture you get, all right. If there is no interaction. So this delta there is the randomness in the interaction. This, this is what you get. So this is for J equals zero. So there is no special pair. Then you have an uniform distribution at beta infinity. Then you have the things that to happen and move to the right. And then you get the usual solution of FBA, right. This is the picture that came out from physics. For this simple model, of course. That's what you can do on a little bit if you want. And then if there is a randomness, some cells that eat something that produce for dollars, the picture changes. And this is beautiful. Because what you take is first, an uniform distribution, which is trivial. So if beta is infinite, what is happening is that all the cells are doing whatever they want. So they don't care about their neighbor. Sorry. Then when beta start to increase, so the interactions start to be relevant. So you start to find different minima. And of course at beta zero, instead of having just one minima in the border of the polytope, but you find that you have different minima. And the different minima indicates you, so the intuition is that they indicate you, the possible state in which the conscious can be. So you will find cells with different, that's what in physics we call spin-glass state. So we have a lot of minima, and depending on the initial condition, or fluctuation, or whatever, you can figure it out different minima solution. That's the intuitive picture you find from this simple model. Then you can go further. And again, using expectation propagation, try to solve a similar problem, the same problem, but not for the toy model, but you can go to the network of Ecoli. And then you can ask yourself what would be the distribution of the fluxes in a culture made by Ecoli. And we, Ecoli, because it's a corner, was very easy to use. And you may assume that some Ecoli cells are producing lactate, and some of them are consuming lactate, they buy product. And what you find is that indeed the distribution of the fluxes is different. So you find in each line, Yes, I will. So in the x-axis, you have flux. In the y-axis, you have the probability of having this flux. So you can ask, what's the probability of in this system that if I take a cell at random, it will have a given flux. And then you take many cells and you make the probability. But here, this is an analytical computation. Modulo, you are using max n to make the computation. So what you find is that the picture changes enormously if you move from one system to the other. From one situation where you have different noise terms and interaction terms from the other. There you have the flux of consumer glucose, production of acetate, citrate, lactate, biomass, whatever. So you can see that the situation is completely different. And what I think is that this is a message to you if you want to experiment that is that one should be very careful to understand what is measuring, right? Because in the dependency, in principle, it depends a lot if you want to understand what is happening inside the cell or with the fluxes inside the cell, whether you are in situation in which cells are interacting or are not interacting. That's what I think is the main message we send there. And then, all right, now I am almost finished. I have five minutes, two minutes, how? Five minutes? Yeah, you are very relaxed, you are not like the bastard that was yesterday here organizing this stuff. All right, now I will do the problem the other way around. Because of course, as soon as I move a lot from, you know, to physics, so people from the, my friend from the university, all right, but try to go back and close again, and close again the gap, you know. And all right, the problem was, all right, now they have this chemostat and they would like to understand what is happening inside the system with the minimum of information. So they were in Cuba, we can not measure many things. Actually, we can measure only a few things, cause and alls. And therefore they say, right, don't give, don't ask, I mean, don't tell me that you will need all the proteomics of the system, whatever, because it's your don't, so it's no way that I will give you that to, don't disturb what is happening inside, all right. I say, all right, I will do my best. It's not easy. And then we say, all right, let's try to now imagine that we have a question in the chemostat. All right, I just, all right, we forget about it. I will go, we have a few times. We use something similar to what Marcelo explained the other day, which is maximum entropy. So there are many standard approaches in the literature. One of the, and the one we are going to compare is essentially FBA that given the data, maximize some linear function, essentially, and try to figure out what is inside the system. So the point is, you have some experimental data, which is this VI average X, this is experimental data. You know that if you are inside the polytope, it should fulfill this equation there, which is the constraint, that's a typical inference problem. And the point is, all right, which is the P of V that guarantee that you maximize the entropy of the system. But that's a typical inference problem. I mean, for those of you that still remind the talk of Marcelo, the difference here is that while Marcelo made an answer about the properties of the P of V, here the P of V are calculated exactly. The price you pay is that the computation is much more hard, that you need to go to expectation propagation to solve it. So you cannot do it fast in your computer, it's kind of painful. But you can do it for certain cells. And then we went back and let's say, remember our top model of the chemostats? This is the toy model, sorry. This is the toy model of the chemostats. Then we'll say, all right, now let's try to design a toy model in which you have heterogeneity, in which you have fluctuation in cells, and not only, cells may move from one metabolic state to the other. This is the standard way to do it. So if you look carefully, you will have the same term. Now forget about the epsilon. This is mu x. This is minus vx. And then the epsilon is, and you're telling you that if you are in one state, you can move back from this state, or you can go into this state. That's what this epsilon is giving you. If epsilon is zero, again, you recover the toy model. If epsilon is different from zero, you get a lot from the toy model. And then we started with that. But then my super student realized that this is too simple. If you take a model with just two reactions, then it's not too different what Max N can show you with respect to the approaches used by FBA. Because essentially, since you have just two reactions, one of them is already fixed by the constraint of the problem of the five-day journey in the chemostat. So that's not interesting. Sorry? No, the states are continuous and are, you know, the grown rate and, for example, the consumption of foods. Now the states are defined by the metabolic state. So what's the value of the flux on a given reaction? Then we move back. We move again. We'll say, all right, try me something simple and he told me, no, it's easy. Take the one we use in the picture, in the paper, I cannot try to explain this kind of reaction. Please make me a draw of the system. And then my biochemical student said, all right, this is the draw. This is simple. I said, no, please, because I know we didn't solve this problem. So we solved something much more easier. And he, now again, yesterday, finally sent me something like that. So it's the same problem we had before, just with one more reaction, essentially. That's more or less the way we deal with the problem. And therefore the questions move to this, in which you just introduce one more reaction, right? This is the dynamics of the chemostat in which you can imagine that cells are not always in the same status. All right, and more or less, this is a picture increasing epsilon. So increasing the disorder, you find a system that move from this kind of curve there in which there are a stationary state to this one here, which is visible. If the cells are not maximizing biomass, which is what we were assuming, and there is a lot of noise in the system. So of course the growth rate cannot be so large, so x will not be so large, and there is this change in x, and of course this change in the consumption of glucose. From the simulation, are they going to extinction or that they're stabilizing? They may go to extinction at some point. I'm not sure, but they'll say, eventually they will extinct. No, they will not extinct. The problem is that, of course, the density will be so low that in principle, if it were not continuous. Yeah, if you look at log x, is it that exponential? So if you look at log x. But there will be all these cells that are almost consuming very few of glucose, so they will be moving around the whole polytope. There will be, if epsilon is too large, they will essentially be moving around the polytope all the time, so some cells will be alive all the time. But the density will be slow because essentially, only a few of them, only a small fraction will be consuming a lot of glucose and producing a lot of biomass. But I realize here that perhaps there is a phase transition, but we didn't check. All right, and now, all right, let's pose the problem with this toy model. Now you have put in concrete terms what we had before about how a maximum entropy works. So you have the growth rate that is equivalent in this experiment to the dilution rate. So this is fixed. This is something the experimenter will tell you. You have this constraint, then you have a second constraint that is given by the concentration of the nutrient limitant. In this case, it's very easy because it's glucose. And then you know if you find out the maximum entropy principle that you have this exponential form for this stuff. And this is the plot. All right, now what we did. We took the dynamics of this system in which you have this heterogeneity. We assumed that this is an experiment. Somebody gave us the result of the experiment, right? The density of cells and the concentration of glucose at the beginning. And now we tried, these are different panels in which we tried different optimization techniques, right? And you see is that every techniques in which they tried to maximize one given flux or biomass or some kind of oxygen or glucose or whatever, they all differing. At some point, one of the flux is lost. So you see all the curve on the top, right? Some times they get lost. So in principle, you have in the x-axis, you have the prediction or the experimental value, the one that the simulation gave you. And on the y-axis, you have the prediction, right? And you would like this to be a straight line. And what you find is that it is only a straight line in the max entropy case. So in the case we tried. So otherwise, you lose always something. And just to finish, we did the same for genomes came in our network. So you take the same, similar data from Ecoli experiment in chemostat. If somebody knows there, somebody else that has more data, even in more complex network, it would be very welcome. But it's hard to find this. And again, you will find different columns are different FBA inference problem. The last one is maximum entropy and different lines and different dilution rates, right? To see the difference, it's better to plot it here. So you will find different symbols are different FBA approaches. And in blue, which I think is the only one is clear, is our approach. You see that the data is poor. So you cannot improve too much over standard FBA. But if you know this, the most stable one that is more or less always working is maximum entropy, right? Otherwise, they may take one of the views better than maximum entropy, but then you take more. And now it's done. So I would like to finish saying which are some of my old problems. The problem had that were interesting to me before I arrived here. Now I have to re-chaff everything after the conference. But one big problem for me was beta. So what's the physical or biological way in which you interpret beta? Because for us, beta can be interpreted very mathematically as a multiplier to fix stuff if you are doing an inverse problem. But clearly, it is very intuitively or a very easy trick to be used when you try to move from the FBA solution to the full space solution. But it's not clear from the point of view what is the actual meaning. People have been trying. But I think that nobody, no, our friend have tried to interpret beta are very happy with the answers we have about what's the interpretation, the proper interpretation, if there is a proper interpretation about it. Then an important thing that I think that I would like to answer is, how can you differentiate between different heterogenities? Because it is clear that there is an heterogeneity that is given by the fact that the cells are not the same. So let's say. But the cells are, let's say, prone to suffer from a lot of internal fluctuation. So regulation effects, a number of proteins produced inside the cell at this moment. So even if the cells are isogenic, there is a lot of internal fluctuation. But then there is a different heterogeneity in which cells interact. So even if we are the same, for some reason, I produce like this and you start to consume like that. That is something we have seen here a lot of time. And the idea is whether or not I can differentiate from outside just making macroscopic measurement, why are these heterogeneity arising? Right, that's a second. Then we have another problem, which is what's the best, which is again close to the industry. What is the best strategy to functionalize the psychometric matrices? So the point is that when you find a psychometric matrix in the web, this is a super psychometric matrix. This is the super E. coli, the super human network, the super G's network in the sense that this is the network that in principle does everything. So if you go to the human, you have a network that is the same network for the brain, for the neurons, and for the liver, and for the kidney. And it's clear that for most of the people, the cells in the brain are different from the cell of the kidney. Then we find that it's not absolutely true, but in principle it is. And then you would like to understand how to functionalize this network using, again, the less amount of information. All right, another thing I would like is to introduce a regulation in this kind of model, even in the toy model. This is not easy to put together metabolites and a regulation. Try to design a simple model that is solvable, that is comprehensible, and where you can control both processes, it's hard to define. And then, of course, can we define or define a minimum metabolic core? That's it. And these are my collaborations. On the left-hand side, you have half of the people are my PhD students. The other half have super engineers from the Center of Molecular Invalorgy that brought me to this problem. And on the right-hand side, you see my collaborations here in Europe on this subject. With that, I'm late. I'm sorry. I will be more gentle tomorrow. Thank you very much. OK, I suggest that we have time for a question. And while the question is asked, we can transition to the next speaker. So the next speaker is checking devices. So any question? So it looks very interesting to introduce this heterogeneity in the models. But a big problem with FBA is, of course, that if you make it somewhat more complicated, you can't do it for big models anymore. So do you have an estimate of how large the models can be that you can still solve them in this way? Let's say for FBA, in the sense of maximizing a growth rate or whatever, infinite. Sorry, this is in the programming. This is infinite. If you want to compute the space of solution, I think now it's feasible all to 1,000 of reactions. So we are close to really big stuff. There, what is hard is that there may be many details that are not compatible with the solution. We may be losing a lot of stuff because of wrong annotation in the system. But in principle, if it's painful, it's not like linear programming that you just work with 5,000 equations, 100,000, that's it. But I think that up to a few thousands, you can play with that. So it's already, so expectation propagation almost get to a genome-scale metabolic network. Of course, if you want to do something in which you have to explore a large range of parameters or condition, that's more complicated because it may take a much more time. But for a few stuff, you can do it easily in your laptop even. So it's not that you need something. Right? Yeah, just a comment to the beta. I imagine that as a Lagrange multiplier, it's related to the temperature in physical systems. And so my idea was that one interesting thing about the temperature is that it tells you whether two, if you take two systems, whether they can coexist in that state or not. So I was wondering if also in this case, if you maybe have two populations, it tells you something about the relation between them. I don't know. OK. As I told you, let's say, these beta have been around in the community already for a while, at least five or six years. We always give a general explanation, but we don't have really a derivation. So my intuition, if you want to go back to physics, is that the same way the temperature in physics, at least in statistical physics, like the degrees of freedom you are not controlling, you are not either in the phonons, for example, if you are looking for a spin system. So the temperature is a measure of how these phonons are acting. Here, I would like to have a similar picture in which, instead of having the metabolites or the fluxes, I will have the noise that's somehow tuning this stuff. But we don't have a model. We need piles to really frame this stuff. I think we haven't done that. I have a question here about the replica machinery you are employing. Because when you say, OK, I set myself into the replica symmetric phase of the theory, whatever. But you have order parameters. Do you have any comments about the equations for the order parameters and the interpretation? Let's say the problem is simple. You can solve everything for the order parameters. And you find different phases, as in the replica machinery, depending on the value of j and delta. If you realize it's not a complex problem in the sense that it's like a piece of spin in which the spins are, let's say, oh, I'm sorry for the biology. But it's like a piece of spin in which you take all the fluxes, but instead of having the sum of all the fluxes equal to 1 or 1 over n, here the sum may have some minus because it's defined by the stoichiometric matrix. So some fluxes are coming out or in. That's first. Second, you don't have three spin multiplying. So you have a kind of two-spin interaction with continuous variables. And then you have a standard field that is given by the growth rate of the cell. So from this point of view, it's quite easy. The problem is that what we did that was not easy at all was to how you solve this when you don't have three fluxes, but when you have many, like in E. coli. And once you have many, so all the machinery must go within the expectation propagation that is the one that allows you to explore the configuration space, which is equivalent to, it's a fast way if you want also to compute the partition function, the replicated partition function. But that's, you know, that's technically, sorry. Okay, so I suggest to thank Roberto again. And...