 in a given environment, arbitrary environment. So if you know an environment, meaning you know the concentration in this growth leader and you know exactly how it changes in time, can we even define, can we find or even exist optimal strategy in this environment? And the second question is can microbes actually implement this kind of optimal strategy or at least approximate that. Today I want to completely focus only on the first question, right? So I'm not talking about the second one. So to start with, we have to define some things. We have to define what would be the specific growth rate and it's just defined as the proportional change in the number of cells in a population. So we can define with this, sorry, derivative here. And in the very specific and special conditions of that we have a constant environment that you can achieve, for example, in a lab experiment. We can achieve a state of balance growth, meaning that all the concentrations and the growth rates itself of this population remains constant in time. So that's a very specific kind of scenario. And the specific growth rate in this particular kind of scenario can be a measure of fitness for this population. But in general, that's not what you have in nature, of course, right? So the problem that you want to address today is the dynamic of growth. Meaning if you have change environments, and so the concentration change in time and the specific growth rate change in time, we can talk about the absolute fitness, meaning that's I think a term from population genetics. So it's just defined as the final number of cells of our population after a period of time, T, divided by the initial one. So the proportional change in the population. So that we can achieve that just by taking the integral of this first equation, right? That's a simple procedure. So you have this exponential, that just to simplify things in the next slide, we are considering that our measure of fitness is just this simple one at the bottom. It's just we don't take the exponential, but in essence, they represent the same thing. So we are talking about dynamical systems and optimality and just to make the point that these two things go hand in hand from the beginning. So as you may guess, from the title of this talk also, we take a great inspiration from classical mechanics. So from a few that comes from physics, that would be very familiar to you. So almost for 300 years, so we have oil, agrangea and others that developed this theory, basic theory in physics, that they have optimality principles built in. And just to put this in a little perspective, we have this nice figure here that represents three kinds of hierarchical kind of problems that one can solve, right? So the simplest one is that we have a function and just plug in some values, x. Yeah, okay, so we can see that the first one has some values x and the output is some other values. But more complicated problems, we don't know this function, right? The function itself is something you want to find. So in general, then we have differential equations and the solution to the differential equations are these functions. So the input of boundary conditions and initial values. But even more fundamental than that, many times we don't know that these differential equations and these differential equations can be found themselves as a solution to another more fundamental kind of problem that is formalized by the calculus of variation. That's something that was developed by the same people that developed the classical mechanics and very fundamental for mathematics and physics. And in this particular case, the inputs of the problem are themselves, the sum objective function and some constraints. So we are getting close to the scenario that we're looking for. So what they do in physics, for some of you that are familiar with that in physics, they define what they call a functional with just, yeah, something in physics is the action, just a name here, it's just an integral of something. And this function inside the integral is the Lagrangian that defines the system of interest. And just important here is that the system is defining, you'll notice there's a vector q that defines the system state. The q dot is just the first derivatives in time and time itself. So if you have some kind of problem like this, there's these curves of variation, this mathematical machinery that you can use. And if you want to find the maximum point of the streamer point of this functional, we just have to solve what is known as the Euler Lagrange Equations. That's a very general thing. Okay, so we are close to what we want to define. They probably want to define. And to really define this, the problem, formally we are going to use what we call this growth balance analysis, which is general framework that we developed some years ago. And basically just a general or simplified framework for to develop general cell models. And just taking heavy inspiration of what was already existing, we just try to simplify that to the very minimum assumptions. So one of the things that we use mass units, there's something that simplifies a lot to the expressions and I think they understand it also. And we call this, we are interested in kind of what you can name as holistic model in the sense that we model, we are interested in modeling the whole self-replicated system, basically very close to the sense that Mola and I colleagues introduced in 2009. It's a little bit different, but essentially it's the same thing. And because we are talking about the most general possible kind of modeling, these need to be kinetic models in the sense that we want to be able, want to be able to be realistic and to simulate to find the properties that are non-linear to this kind of model. So this figure here, the schematics here on A is just a very simple representation of the kind of system we are interested in. So we have this rectangle represented by cell and the spirit of economics. So things are colored like bronze and silver and gold. So the bronze things are these compounds in the environment, right? That's transported in and outside of the cell by this, maybe you cannot see very well, but the green squares there, these are the transporters. Then you have blue squares that represent enzymes that are converting these compounds in a network of chemical reactions. And at the end, the products of this network are used by this special reaction, red here, which you just name it, ribosome, that produce this most valuable thing which are the proteins in gold color here. And these are, we assume, instantly distributed among all reactions in order to, for the protein that's catalyzed the reaction can be a transport, an enzyme, or the ribosome itself. And we see that these little arrows down here just represent that we assume is explicit that things are diluted by growth. So there's no biomass composition assumption. It's not an input, it's an output of this kind of model. And on the right we have what would be representing more or less the stoichiometric matrix of the kind of system. We can call M just because we're using different units. So we have S at this transport reactions, we have enzymatic reactions, and the ribosome is this special reaction here. That's the only one which produces this special compound you just copied for proteins or polypeptides that are instantly distributed to catalyze the reactions. So that's a very general scenario. So the system state is completely defined by the vector C or reactant concentrations can be, we can call these metabolites, but can be anything else in the system, for example RNA or DNA. And one entry is just the total protein concentration of this vector C and the fluxes. Okay, so now we are able to formally define the problem we want to solve. So there will be some equations as you noticed, but we don't want to, we are not getting too deep into equations, more the qualitative understanding. Okay, so the problem that you want to solve today is that we have this problem. So as inputs, we have a given model, meaning we have a stoichiometric matrix, a vector tau, here we call tau is just, essentially these are the kinetic rate laws, we assume that we know this information with all parameters and everything, and tau because these are the time that each reaction takes, depending on the rate law, and rho is the density of the cell. We assume to be constant. And a given, and let's assume we know the environment, how environment changes, the concentrations in the environment, the concentrations X, changing time. Assuming we know that. So you want to find the optimal fluxes and concentrations such that fitness is maximized and under these constraints here. So the first one is mass conservation. So these equations just saying that if you take the flux balance of everything, all reactions are counted for the dilution at each point in time by growth. The net production consumption, this difference gives the net production consumption of the reactants. So that's the dot here is just the time derivative of C. The second one is just saying that the sum of all proteins that you have in the system have to sum up to this total protein concentration, just a definition. And here, we have a vector here as transpose of the vector fluxes to the vector tau. It just means that we multiply each flux by the time this reaction takes, the time by the kinetics. This gives the amount of protein that has to be allocated to this reaction. And if you sum everything, this is the total protein concentration. And the last one is just that the total, the sum of mass concentrations is fixed is the density of the cell. So we have showed recently that if you assume balance growth, so in other words, if the growth rate itself and the concentrations are constant in time, so basically the difference is that this is zero and this is just a fixed number, we can solve this problem or in the sense that we can find the exact analytical conditions necessary for optimal state. But that's not what we want to solve today. We want to solve this more general problem. So the key is to do something very similar that we did before that we can describe as a non-dimensionalization process, just meaning we want to find the minimal set of variables that uniquely define the system and these variables are non-dimensional, are dimension. Okay. One question, yes. Am I understanding it right that normally the way people think about this is that once the concentration are given, then the kinetic parameters of the enzymes determine the fluxes, but you're gonna treat the fluxes as if they were three parameters themselves and you're optimizing both over the concentrations and the fluxes. Yes, at the same time, yes. The trick for doing that is exactly what you're presenting is that the core of this paper that you just published, but the key trick to do that, at least under some assumptions, is that especially the main assumption is that we assume that all proteins have the same composition. So that's generalization, but if you can assume that we can show that exactly, if you define these new variables, that not exactly, sorry, not exactly fluxes, but you just call the flux fractions just the flux is divided by the growth rate and total density, so that gets a dimensional. We can show that. So what we see here is just the equations, the constraint is solved before, but in terms of F, right? And just to note here, that if you have balanced growth, meaning that this C dot here is zero, we can already see that maybe I can just point that because that's easy to learn. So if this here is zero, the concentrations are uniquely defined by this vector F because the growth rates basically disappeared in this expression, so that's the main trick. And if the concentrations are uniquely defined by F, so these kinetic functions are already uniquely defined by F and we can then completely formulate the problem in terms of these variables F only. And that's what we have in this paper that was published recently in this particular case. And the result we get just to give a very quickly qualitative understand is that assuming this, that the protein concentrations are not negative, we can use this analytical method called the KKT conditions to find analytical conditions for that. And these are here, we find these expressions that we don't have to go into detail but what they're saying is that for each flux Fj here, each reaction to be active, so this flux to be non-zero, these things in parenthesis have to be zero, right? So it has to be optimum in this sense. In the other, in the opposite direction, if this thing in parenthesis is not zero, the flux has to be zero, so the flux has to be inactive. So this gives the exactly analytical property or conditions for each reaction to be active or not at the optimality. And here just to simplify the way to present that, we define this matrix E in this way and this gamma J is just the sum of each column in M. So just very quickly to go back to reality, just a little bit to show you guys that this not just only a theoretical exercise here, we had two previous papers that basically apply this kind of analytical principle in a very particular scenario, so more specific scenarios. So in this first paper here, we show that using these analytical conditions derived, we could predict the amount of active ribosome at different growth rates for E. coli and yeast. So that's the red line here in both. Without any 50-parameter, we just assume a previous model of translation that just which has metakinetics and see that it's very close to reality, right? And in this other paper, there's another simplification that one can make and we show that basically the message there is that enzyme substrates are all very close to optimal balance that is related to this automatic condition I mentioned here. Yes. Back in these equations in parentheses, that for example, the only solution that you can get is when only FJs are non-zero that... Yes, and yeah, that's something. I don't want to have the time to get too much into the mathematics, but thank you for the question, yes. And I think to me at least I'm biased, but I think there's a very direct way to see that here is just because this is a matrix that depend on kinetics and in order for the optimal state to not to be elementary flux mode or elementary growth mode, there would be a necessary linear dependency between these things and we can assume it's not. So essentially reproving do the same kind of proof again. Yes. Yeah, just to comment on that, that's exactly the kind of simplification we made in these previous papers here. So that solution is more general in this sense. Okay, so but today we want to solve this more general problem. So just to give the meat of the derivation that you want to make, or the main trick that we need to make is that we can take this flux as F, that we can think of as flux is in a funny unit that they can even later make the argument that I think there's a very actually natural units to use. We decompose that the main thing is that we want to decompose these fluxes into other vectors, Q and U, right? And this Q vector satisfy this equation is just saying that we can see this as fluxes that each point in time, they satisfy or they would be able to maintain balanced growth. So imagine that the cells change is composition state in time, but at each slice of time, these Qs would be the flux that would be able to sustain this balanced growth. But of course, in this general setting is not, so this U vector here would be exactly the correction for that because the concentrations are changed. And the trick here is that if you compare both equations, just take the time derivative of that, it's just a linear system of equation. We can show that this U vector is uniquely determined by the first derivative of Q divided by growth rate. And there are only a extra assumption here is exactly this one that we have a matrix that's full column rank, which means sounds very esoteric, but it's a very practical meaning, which is we assume that our network, our metabolic network has no alternative pathways. So if you can produce something, there's only one way to produce it. You cannot have alternatives. And there's still a limiting assumption here, but we still get very general results. So if you assume this, if you make these definitions, just keeping a few steps of derivation here, we can show that we can condense all these constraints or this equation had before into two equations. One is one equation for the specific growth rate at each point in time. And you see that magically now, all the problem depends on Q dot, which is the first derivatives in time of Q. The Q itself and X is a vector that's given as input and all functions of time. For those of you who are familiar with classical mechanics, you see that the choice of Q is also because it really resembles what would be the generalized coordinates for classical mechanics. So we can call generalized flux maybe here. So we have two equations. One equation is exactly the term, the specific growth rate. And so if you go back to the very initial problem that you want to solve, so what you want to maximize is this integral of the specific growth rate that's uniquely termed by these variables here. And the only constraint is a very simple linear constraint here. So that's why we go back to this thing of the Lagrangian or classical mechanics. So the way to solve the problem now is very straightforward. And if you're familiar with classical mechanics, we can just define this new function, the Lagrangian is just the objective function and you have now a constraint. And we want to maximize what can call a functional. So let's call this G. So instead of action as in physics, we have some other kind of quantity which is in this case, I would argue that's much more realistic or much less, much more practical. That's something you can measure that. And that's the integral of this function here. So we just reformulated whole problem in a way that may seem very strange. But it's just because you're formulating this way, you can use now exactly what I mentioned before is this Euler Lagrange equations. So we know that the maximum, if you have an optimal strategy, it has to satisfy these equations. So now the only thing to do is to calculate these derivatives. So it's a little work, I'm not showing to you. I think that's not very interesting. And if you do that, we find this system of differential equations, one for each reaction. So it's a very long, seems very long equation that I would say we don't even have to care exactly what's there. Maybe the only thing important to mention that it could be much worse, it could be much complicated. Because if you look at that, what we have here is these are ordinary differential equations that's not partial, so that's not as bad. And these are first order differential equations. So it's not so bad. So that's something that in principle it's not so difficult to deal with. Because of Lagrange derivation. Shouldn't it be a second order by definition? Yes, in physics, because kinetic function is defined as the quadratic of velocity, but... Ah, here it is not. Sorry? Here it is not defined as the quadratic of the variable. No, it's... Well, I would get back to that, maybe this is for a coffee break, but you see that what is velocity, he actually is linear, it's not quadratic. So that's why it doesn't be, yes. Actually, that's some very, you can think of kind of a growth potential and growth kinetic growth, maybe, here. Because that's something that depends on velocity and something that doesn't depend on velocity. But that's made for later. Okay, so that's it. We have an equation. And the only thing that I just want to mention is that we define this matrix epsilon here that just to make everything in matrix notation doesn't look, believe me, it can look much worse if you use summation here. So what we get here, so if you solve these equations, these equations would be the equivalent of the analogous what we have in physics would be the equations of motion. So in physics, if you find the equations of motion, you know exactly how a system evolves in time in the sense that how it changes in time. And this would be the equivalent of that. So if you solve that, what you have is, you find q, so with that, you just take the derivative, find q dot and following the previous equations, we find all the other variables that define the system state in each point in time. Yes. So, I'm just trying to understand what's happening here, right? It seemed to me that in your stationary solution, you already had that stationary solution as a function of the inputs x, no? Yes, x. So now you say x is going to vary in time in some way. Isn't the solution just again simply the solution as a function of x at every time point, almost like adiabatic sort of, it's not? It's not, and actually, we can see exactly here, because if it was, what we have here, so we can see that the stationary solution is here, right, the stationary solution would be this term, this term, this term, and this term here. So this, there are all the other... I can see that, but I cannot see that. Yeah, yeah, but I mean, I'm just saying that, yeah. For the first time, that's too much to see, but I'm just making the point that is different from, the equation is different from the original one. So there's still the stationary solution plus a correction because things are changing. If the environment is changing, it's impossible for the system to stay on the optimum as x is changing. Well, I'll show something about that in the next slide. So there's no yes or no answer to this, or? Yes, it's possible, but unstable. So no. Maybe I can follow just the next slide, it would be about that. Okay, so at least theoretically what we did here. So we define formally a mathematical problem that the inputs are given more than meaning the stoichiometric matrix in these different units, but stoichiometric matrix. The kinetics, we assume, you know, and the density of the cell of the system. And we know this x and how they change the time exactly, right? So you assume, you know this information. So we derived what would be the necessary conditions for any, if there is any optimal strategy in this environment, they have to satisfy these equations. And meaning they are optimal in the sense that they maximize this proportional growth under this constraint of mass conservation, kinetics, and cell density. So there was no extra assumption here in particular, but with the exception that the system has no alternative pathways. So today at least there's this extra assumption. So, okay, so just to answer the question about the relationship to stationary states. Just to show you guys a very, very rough numerical solution to show you what the solution would look like. I just consider this very simple model for reactions. So we have two transport reactions and that's something like metabolism here. I mean, everything here is just a time limit, right? There are labels, but they don't mean anything real. And there's a ribosome that produce the protein. There's KNs here. The AK gets forward and backwards. And there are this density here. And assume that the kinetics here, the kinetic rate laws are reversible Michales or hard-done rate laws. So there's a back and forth component. Okay, so to get to a point of the difference between stationary, the optimal balance growth for a stationary state state solution and this dynamical solution. So the first thing I compared was exactly that. And I just mentioned that to solve that numerically, it's something I started working on a few weeks ago. So the very rough solution is using, try to solve a problem that actually is a, what's called a system differential algebraic equations because the first order derivatives are not explicit but they are implicit in the system of equations. So there's some algebra that can be done and then I just apply the, use the Euler method which is the simplest method to solve this kind of problem which is particularly not ideal for stiff problems which that's apparently what we have here. But that's, the results I think are still meaningful. So the first example that I want to show is exactly this question of comparing what would be, maybe a sanity check, right? So would that dynamical, optimal dynamical strategy be better than just a nice state state optimal condition? And from the solution that we see here, the answer is yes. So what we have here is the specific growth rate against time. So that's the simulation. And the red line here is the growth rate that you find at each point in time, the specific growth rate. And you see that we started a point that maybe you cannot see but the growth rate at first, we are starting at the state that would be optimal for this environment if you constrain ourselves to be steady state, right? If you don't change the concentration. So you see that for some time we stayed there but after some time because of numerical fluctuations is unstable, so we start doing something else that seems to be a kind of quasi periodic behavior, right, so it seems like. And these two lines here are just to answer is actually this question that the dash, black dashed line is what would be the equivalent growth rate if you take the average here of this dynamical behavior. And the red dashed line is what would be the growth rate if you should just stay as a stationary condition. You see that there's an improvement here. So even if the environment is fixed, there's a self-imposed self-isolation behavior that seems to be optimal for the cell. So just to finish, just a second thing is that of course then I compared one example is just with some dynamical environment and in a very general sense I just put some cosine here for one of the external concentrations and what we see is just a different solution. It's a very small window of time so we just want to show as an illustration what would happen but yeah, that's just a very rough simulation. So just to wrap up what I had in these simulations from this very preliminary numerical exploration of different models what we have that even in fixed environments the optimal balance growth states are unstable fixed points so that's what I mentioned before. There's often a self-oscillatory behavior that emerges naturally even in these constant environments and one thing that's very important is that we need to use reversible rate laws to get more stable solutions. That's something that seems to be very important and just summarize that what we have here is actually a formal tool, a mathematical tool to study or to make sense or to rationalize what I think we have discussed especially last week is that the maximum growth rate of organism in environments is not always a good measure of its fitness. And with this kind of framework we have the formal way to study what fitness actually is which is this G here, this measure of the proportional growth in time. So with that I'd just like to conclude we have these exact analytical conditions for this mathematical problem defining what would be the absolute optimal strategy growth strategy in time and these conditions can be seen as fundamental principles right as we are interested in that. And the next question I think many of you have thinking is that, okay, that's all nice and cool but how close are that to reality and do microbes implement that to any extent and that's what we are trying to do next. We are using more realistic models and more realistic numerical solvers to compare that to actual data. And with that I'd like to acknowledge Martin Lashin his group in Dusodorf and Wolfram which is also part of this project and thank you very much. Thanks a lot. So we have time for questions. Thank you, this is promising progress I think. So you showed us these unstable fixed points and then you said that the optimal solution is not stable but it's clear that you get a higher growth rate with the unbalanced thing, right? So that doesn't, it seems to me that balanced optimal solution is just not optimal anymore, right? Yes, maybe the three questions here that what the conditions are fine but that's because I think that's a bias we have from physics because in physics the actual problem is convex, right? And I think that one can show that the traditional thing. Here the condition only say that the strategy is stationary, right? The fitness is stationary. So yeah, you're right. So the point of optimal balanced growth would be a stationary point but it's not optimal, it's not a maximal. So it could be that you have different fitness different points. I don't know if that's... Yeah, so I would say if you look at the solutions of the black line, do you see some kind of accumulation and then depletion of metabolite that was not possible in the balanced optimal state that now gives you an advantage? Yes, and that's... Here's the plot of this, the top one is the violet one is the total protein and the other ones are metabolites. And you see that's this oscillatory behavior. And you see there is even different phase change and I think this relates... Actually, Wolf has some previous work on exactly oscillatory behavior and that's the kind of thing you see. Like the first things are produced first. So let's say the first enzyme is more saturated and then you produce the second thing and so on. And you see that with the production of proteins also. There's this kind of... Yeah, the allocation goes from the first of the network and goes to the end. There's a kind of wave like that. That's what you see. And of course that's not possible in balanced growth so that's why it's a different one. Yeah, I was just wondering, it was something very close to the question that was made and because the Euler Lagrangian formalism it's always extremizing or functional actually. It's never giving you... There are some conditions that are some theorems that are gonna tell you if it's just a maximum or a minimum but I was just wondering if there is some like heuristic argument or something simple just for us to see if it's really a maximum in your fitness or... Okay, that's a very important question. The short answer is no. So not yet. So to be clear, what I showed, the equations I showed are necessary conditions but they're not sufficient for optimality. But that's a very important question that needs to be answered. We are working on that also, yes. Okay, so maybe you can help me gain some intuition. So this just plugging X as a functional time in your previous stationary optimum, that does worse or better than what you get here? Well, I didn't compare but it has to be worse, right? If this is correct. And the reason is that maybe at least to meet the intuitions that if you change things in time, there's maybe, I don't know to me that's too abstract but there's a notion of inertia, right? That's something that a cost of change in your state in time. So if you just follow what would be the optimal balance growth for each point in time for the environment, it's not accounting that you're trying to go from A to B in an optimal way. You're just following a trajectory that maybe is not optimal. And that's because in the previous equation you had the extra constraint that the sort of time derivative of all the concentrations has to vanish because it's balanced and now they're not vanishing, or? Exactly, yes. Any other question? Okay, let's thank. Okay, now we have the coffee break. Just.