 or also you can find it via the web. So, yeah. Okay. Okay, great. Thank you. So, doing my talk today, I'm gonna try to merge traffic models, non-equilibrium traffic models, and growth laws. So that is the starting point. So the question that this week, last week, we have seen several times. So, we have the famous growth laws. Which are relations between physiological states, like the growth rate, and the ribosome allocation for the ribosome protein fraction. And so, I'll try to make a model during this talk today that will be in between these more phenomenological models and the very refined models, mechanistic models like the one that we've seen presented by Andrea by last week. So, I think that was the fourth derivation of the growth laws, mathematical derivation. So we have, we can write the biomass production, like which should be proportioned to the number of active ribosomes times, well, a time scale that will be the codon elongation rate. So how fast the ribosome, an active ribosome can attach an amino acid to the growing peptide. So if you do the simple calculation, you will obtain these growth laws. So, I will start with a quite long introduction now. It's gonna be quite longer, but I think it will be useful for what comes afterwards. And to start with, I would like to start where I come from. I come from a community, like a community, not that well, a community that is doing models of messenger and translation. And what I've been doing was like to estimate the, what is called the translation efficiency. So let's say the ribosome flow along the messenger and chain, which in principle can depend on many things. The codon choice that you have, the initiation rates. And actually, in this community, initiation rate is thought to be the limiting, the very important limiting step. So the ribosome recruitment is a central point of translation efficiency. So now if we go back to the growth laws, my question when I first saw them, okay, I see that how much you produce is proportional to the amount of ribosomes, okay? So the ribosome recruitment might be important, but I don't see it explicit in the formulation of the growth laws. So there is no initiation rate. We can see only the elongation rate. So last part of the talk today, so the main objective of the talk was to include the initiation in this picture. And to start with, I start from the other community where I come from, so it's the non-equilibrium unidimensional traffic, which is called, well, I'll tell you what is called. So the exclusion process is the model that I'll be using today. It's a unidimensional traffic model in which you have particles. So you have to think that these particles are ribosomes, for instance, joining the lattice with a given rate and that will be the initiation. I will call the initiation alpha. So alpha is the tentative rate for initiating translation. Then a particle move along the lattice with the translation elongation rate, which I always call epsilon. But well, if there are two particles next to each other, they cannot move in the sense that one particle cannot overtake another particle. And that is what we call the traffic effect, ribosome interference, if you will. And then, of course, there is termination. At the end, you pop out the messenger RNA and you produce the protein. So this model has been introduced in the 68 by McDonough and coworkers. And it was actually for modeling messenger RNA translation. And it's called the totally asymmetric simple exclusion process. So totally asymmetric because, well, there is a preferred erection from here from the left to the right. The movement exclusion process because the only interaction that we have is the exclusion, the steric exclusion between the particles and simple because that is the only interaction that we have. So, totally asymmetric simple exclusion process for friends that doesn't, okay? And so if you have questions, so I don't hesitate to interrupt. So what I'm gonna show you is the simple model that was sketched before with particles covering just one side of the lattice, but actually want to do models of translations. So I'm considering bigger particles. So our ribosome covers more or less 10 codons, okay? And things, well, anyway, we'll show the equations and I will explain everything as if the particle will cover one side. The question will change, will be more complicated but qualitatively, the thing do not change. So, how does it work? So we can measure, if we know the parameters for the initiation, the elongation and the termination rate, we can evaluate the particle density. So how many ribosomes you have, okay? On your lattice and the particle current that will be linked to the protein synthesis rate and at the end that will be linked with growth. And the phenomenology is quite simple. So the current is the probability that you have a particle on one side that is a row times the probability that the next site is empty that is one minus row and then epsilon is the fixer of the timescales of the process. And so that is a parabola and if you keep increasing the density while the particle current goes up, you produce more and more but at some point, traffic effects become relevant. So you saturate, that is what is called maximal current phase and then if you keep increasing the density well, then you will have more and more traffic so you will even decrease the particle current for your protein synthesis rate. So, how do you fix the particle density? Well, the particle density is fixed by the boundaries. So the initiation and the termination. And that is the very well-known phase diagram of the process. So intuitively, you are in a low density phase with just a few particles on the lattice if the initiation is limiting and that is what is gonna be our physiological important regime. You are in a high density phase if termination is limiting. So you have lots of particles going up and the maximal current phase, which as a density, which is always one half is at the maximum of this parabola. So, what can we do with this model? So the model, as I told you, is being introduced at the end of the 60s, okay? And then it was kind of forgotten for a long time. And then physicists started taking over, they discovered it. And actually, the TASAP is the Prototypical Model of Non-Equilibrium Systems, which can be solved exactly today. So, but then in the last years, being used again as a model for translation, yes? Perhaps it's a very trivial question. So I don't understand why you say that people believe that we are in the, I mean, bacteria operates in the initiation limited phase. I didn't understand what is the evidence for, I mean. I'm gonna tell you now. I mean, there are some, well, I'm gonna tell you this slide. So what can we do with the model? If we know the rates, we can even solve it or we can simulate the model and get to the density and the current. So let's say for now that we know the elongation rate that can be estimated probably to the concentration of tRNAs, but roughly we can also measure order of magnitude is 10 amino acids per second. So the speed of the ribosome. And so there are some experimental measurement of ribosome densities. So if we know the experimental ribosome density, knowing the elongation rates, we can get the initiation rates. And that's what we did 10 years ago. So like 10 years ago, we did that for Easter. And for Easter, we found that the initiation rate, well, among different genes is quite broadly distributed, but so on average, there is a 0.12 per second while the elongation rate is 10 per second. So to order of magnitude difference between the two. That doesn't mean that codons are not important. So they still play a role, but the initiation rate is the most limited processing during elongation. And then there are also experimental measurement, the experimental measurement, other theoretical works and they can confirm these values in Easter. And I'm not sure if there are some estimates in Nikolai, but in principle it can be done if there are some ribosome densities. So it can be measured directly. Alpha might vary a lot across MRNAs of different genes, right? This alpha vary a lot, that's what the test is. It may, yeah, I'm assuming it might even be orders of magnitude different for different genes, right? Yes, I mean it could vary a lot. It could depend on sequences upstream of the star codons, for instance. So what we found in Easter, that's actually varies quite a lot, from the distribution of the initiation rate that we could find. Experimentally, I think you don't see orders of magnitude difference between different initiation rates. And as there is some paper by Teriwa's group, they could estimate, so they don't measure directly alpha, but they could estimate starting from ribosome densities. And so the distribution was quite big of the initiation rates. So but there are no extensive measurements of initiation rates. So that is what I just said. So there are simple equations that can relate, yes, no, no. So the initiation rate can also be limited by the density, right? It could just be that you have to wait until the initiation site is free, right? Okay, there is a big, I think in the literature, there is a big misunderstanding what initiation rate is. I completely agree. But so initiation rate is the tentative of initiating. So it's the probability per unit time that you bind here on the first site if there is nothing. Then if you have the first site that are occupied, okay, then the effective initiation rate will go down. And since you always assume steady state, I mean, I will always assume steady state that is equal to the current. So the initiation rates and the current are two different things because they're affected by the traffic, yeah. Okay, that's debated. There was these ramp hypotheses. And I think in the last months, there has been another paper discussing about these ramp hypotheses that are at the beginning of the coding sequence. You have a slow ramp of codons. But I don't know if that is still valid. But anyway, the first codons has been proved, experimented at really matters on the final thing. So if you change the first codons, and that's actually my next slide here, if you change the first codons, you're gonna strongly affect the current because you're gonna affect the recruitment of the ribosomes. So actually, that is quite important. So if the codons in principle can, the speed of the codon, the principle can change from codon to codon. But that's the big problem of codon users and codon buyers. And so in this case, we can forget about these simple solutions, okay? There is no exact solution. So it's still an open problem. In the last years, we made a developed power series approximation of the steady state solution of the TASAP when the initiation is limited. So in this case, we know we have an approximate solution and we can do it for any order of the approximation. And but the important thing, I think, is that we still find this solution that the ribosome density depends only not on the absolute values of elongation and initiation rates, but on the ratio of the two. I'm sorry, now the equation rate depends on the size. Exactly. In principle, elongation rate can depend on the size. And the sequential dissolving. Yes. And then I don't... You can give a sequence and then you compute the outcome. So then you can do different things. But it will strongly depend on the position of the actual sequence of the sequence. Yes, but no question. Once you choose position, you can do the same thing. Yes. And actually we now have a Python package if you're interested that can develop this TASAP approximation. But so far, okay, the important point is that no code on usage has been included so far in these drought laws frameworks. But okay, the take home message of this slide is that densities will depend only on the ratio between elongation and the duration between initiation and elongation rates. So I think that at the end there's always a balance between how well you recruit ribosome and how fast they go, so between initiation and elongation. So let's take this picture in which initiation is faster than elongation. Then you might be happy because you have a larger protein synthesis but at some point you might even saturate. But actually there are a lot of resources. When I talk about the resources, I mainly think about ribosomes that are used. On the other picture, you can have a stronger elongation, faster elongation and slower initiation. But in this case, you will have a lower protein synthesis and on the other way, less resources are used. So some time ago we analyzed some ribosome profiling. In the east, substantially in this experiment you have the ribosome footprint for each position. So it's a proxy for ribosome density on each codon. And then with our approximation, so by fitting the data into the model, we could estimate the initiation efficiency. So how well the ribosome will initiate considering this traffic effect and how well the ribosome will elongate so one doesn't find traffic. And so each point here is one gene in the east that we could analyze. And when initiation is efficient but elongation is low, so there is a lot of ribosome interference or traffic and so ribosome resources are wasted because when there is traffic, you don't contribute a lot to the final protein synthesis. And instead effective elongation and weak initiation will still finely tune production with codon choices, but you will not have a waste of ribosome resources. So okay, that was my big, larger motivation section. So now I try to go back to the growth loads and to the growth loads and I will try to implement what I said before. So the biomass production for me now will be not something that is proportional to the active ribosome, but more generally, I will say that is the ribosomal flow that was J here. And then you have, well it will depend on M which is here, how many messengers in the US. And then in principle you can have degradation. I will come back to that later. So far there is no degradation in the month. So what I'm assuming is that the initiation rate, the initiation rate will depend on the amount of free ribosomes that you have, the concentration of free ribosomes, times rate constant that I call alpha on. And with that I can compute the, how many ribosome are bound. So it will be just the density of the ribosome on one loptis times the length of the messenger RNA. But I don't have just one messenger. So anyway, I can also compute the current, okay, like the questions we saw before. I can, if I know there are bound ribosome, I can compute the free ribosomes. But there is not only one messenger, there are many messengers, so the bound ribosome will depend on the total amount of messenger that you write. So at the end I can write, if I put everything together, I can write how the initiation rate will depend on the free ribosome, now that the free ribosomes are a function of the total ribosome. But also something that reminds the ribosome density. Anyway, the elongation rate, but importantly also on the messenger RNA amount. So I can write these in at least two different ways. I quite like writing that in this way, because you see that if you increase the amount of messenger RNA that you have, your initiation rate will go down because there is a competition between the different messengers. So I will say that this way of writing thing is kind of highlighted the competition that you have among the different messengers. And that is potentially what Ludo presented last week. So there is a complex formation limitation, what we called, and so if you want to know more, you should see Ludo's talk of last week. So now I know the initiation, I can put everything back in the growth load. So I know that, okay, I know the current and I know the amount of messengers, so I can compute how lambda depend on FIAR, but importantly we depend also on the concentration of messenger RNA. So if we neglect the traffic, you get a complicated complex formation limitation that Ludo presented last week. And so there is a preprint out if you're interested here, I'm sure that you talked a little bit already. So if you consider traffic, you have to consider these factor here. And now I want to show you two applications of this framework. So one is what Philippe, a story class here, so I'll go very quickly about that because you probably remember the talk, but there are new people, so I just do a very quick presentation about that, and then something about the work that you want us to do. And the first work is also in collaboration with Ludo Vico and Marco. So in Philippe's model, so we can use exactly the same framework that I presented for translation, but coupling also transcription as well. So we have two different models, so I don't fix the amount of messenger RNA, but the amount of messenger RNA is given by the transcription model. So we have two different currents, the currents of transcription that will depend on the amount of genes that you have and the amount of polymerases that I call N here, and the current of translation that will depend on the amount of messenger RNAs that you have and the amount of ribosomers here. And then of course the output we are interested in is growth. So we can be with both systems in low density, but there are four options, low density, low density, or you can saturate like transcription and low density for translation, or you can have low density for transcription and high density or anyway, saturation for translation or everything is saturated. So we can make a phase diagram out of it. And so if we change the polymerase concentration of the ribosome concentration, you will get these four different regimes. And physiologically we are mainly, unless if you put some stress then you still don't know where we are. Physiologically we are on these, along this line here. So that is a isodensity line. So when we fix the density of the ribosome, that is what has been experimentally seen, at least in one of the last papers of Terry Wasgrove. So assuming that the density is constant, what can we say along this line? And we know the density is more or less of four ribosome per messenger RNA and the party doesn't change a lot at different conditions. So let's start by assuming that we move on this density, so that somehow ribosome density is regulated across different conditions. And what can we obtain? So if we move along this density, this isodensity line. So here there are, in red you see the estimated mRNA synthesis fluxes from the Balakrishnan et al. paper, last years, yes. Now a quick question. The four ribosomes per mRNA are like including the inactive ones? No, no, no. Is it only there? That is the actual density. Oh, density on, okay. It's not the total. They don't have a direct measurement of that, but they could estimate, measuring the elongation and the initiation, but they have this estimate. And they observe that it doesn't change across the condition. So assuming that we are on these isodensity lines, what can our model say? So these are the data from their group. And this is the theoretical prediction. So substantially there are no fitting parameters. So we just feed the model with the data. And then we obtain, we can predict the mRNA synthesis flux, which is gonna be, yeah. Can you clarify the unit on the y-axis per minute per micrometer? I think I plotted it wrong, it should be volume, right? Per unit volume? Yeah, yeah, yeah, yeah. Is it right? Yeah, okay. It should be per mRNA concentration or per mRNA, I'm a bit confused. Yeah, but concentration is a number per volume, so sure. But that might change with growth rate too, no? The concentration of the mRNAs. Yes, but that is the concentration of messenger and it changes with volume, with the growth rate, but that's in the model. So the concentration of the... Yeah, but that's exactly why I'm asking, right? Because if the y-axis is per concentration mRNA, then you've taken care of whatever change in concentration of mRNAs with growth rate. But if it's per unit volume, then it still also depends on how the concentration of mRNA changes with the growth rate, right? It changes with growth rate. Okay, we should go for it, okay. It still depends. Yeah, but that is all average, I mean, all the genes. So the assumption is that DNA concentration is constant. So the assumption is also that genes are the same. Sorry. And also on the same line, so they have measurements of ribosome concentration and RNA-P concentrations, and well, we can quite reproduce the experimental value without imposing any effect on the model. Okay, that is a lot of work in progress, right? So if you have ideas or questions, just come to me or Filippo or Budovico, who is also involved. And now I would like to go a bit more slowly on the second part. And here, so I was thinking that I would like to spoil the oneness presentation for last Saturday of Friday, but okay, I'm not gonna do that. I'm gonna more talk here about the role of initiation of translation. So I come back to my motivations, my first motivations. So that is our starting point, right? The question from the framework there that I showed you before. And so I told you that in my motivation, I wanted to highlight the role of initiation because the translation community is really important. So the initiation here is in these A, so these constant rate affinity, let's say, of messenger RNA ribosomes. So what can we take from the literature or the experimental data on the messenger RNA concentrations? The elongation rate has been measured. So we have seen the talk last week. And well, of course, the ribosome number fraction has been measured. So I have all the ingredients here to infer my initiation rate. And that's what I'm gonna do now. So but let me write maybe the initiation in this way. So I put all the regulation that can happen on the initiation on a factor F, okay? That will be something between zero and one. And it regulates the initiation of translation. It could be for many different reasons, I don't know why. It could be for initiation factors. It could be for hibernating factors as a question in the free ribosomes. So but I'm putting all the regulation into that. And now I would like to estimate F subsection. So when we do that with our model, we'll see that, so that is a function of the rate, the F parameter, which is going up and then it saturates. So it's going up means that here at the very slow growth, substantially you have no initiation, okay? Somehow you are regulating a way that you don't initiate the messenger name translation. And because of that, because of that you see it accumulates free ribosomes. It accumulates free ribosomes in your pool. So and F describes all kind of regulations. And I think the most accepted view is to consider that ribosomes are sequestered by some hibernating factors. So that is what the alpha will look like. So that is a function of the growth rate. Alpha will tend to zero and then will saturate some point. And that is similar to the order of magnitude of what I told you at the beginning for Easter. So for Easter, the average behavior was 0.12 per second, order of magnitude. And here we are the order of 0.3 per second of fast growth and remember that, so the elongation rate is order of tens per second, so it's the order of magnitude. So initiation is still meeting and should be important. So now let me do plot the estimated initiation as a function of the number traction, okay? Phi R. And so we see these, there is offset in Phi R and substantial that will be the amount of free ribosomes that you have at zero growth. Exactly like these, so the two things should match. But exactly, so that's the amount of, that will correspond to the amount of free ribosomes that you have here. But anyway, I think, and it was, I think, done questions in the first week. So what about initiation? I mean, what about inactive ribosome at zero growth? So you have not zero, you have zero active ribosomes, but you can measure elongation rates. So that is a bit inconsistent, right? So because you can still measure some elongation rates, but you have nothing that is actually initiating the translation. So that is something that is there in the simple model of having inactive ribosomes. So at zero growth, your extrapolate should have no biosynthesis, but it's a bit inconsistent with the fact that you can still measure elongation rates. So how do we want to, how can we fix it? Well, we thought that we fix it by including protein degradation. And so the next couple of slides that I'm going to present are actually the work in collaboration with Ludovico, Marco. For Marco, I found a very nice figure in the lab coat. And also with Jacopo, who's here. And so Ludovico went back to the literature, and there is actually not much literature about protein degradation, but we could find papers from the 70s, 80s. I didn't put the references here, but you can find them in our paper. And so we first try to see if, because everyone says that we can neglect the protein degradation. So it's still true that we can neglect the protein degradation at those low growths. So we can neglect it if the degradation and the dilution are very different, but it's low growth, so the two things might become comparable. And when we went back to the literature and analyzed the data, so we found that the degradation rate is actually increasing at those low growths. These are data from the 70s paper. And so in the next, so what I will show later, I just assume that this trend, the fitted curve here. And the same thing seems to happen in Easter as well. And we would like to do more accurate measurements of what happens here, because it's something that hasn't been explored so far that much. So anyway, let's go back to the contradiction in the active and active ribosome at those low growth. So we substantially, our thought is that there are some ribosomes that will be, are doing protein synthesis, but do not actually contribute to growth, but actually do maintenance to balance the degradation, the degradation flux. So if we do that, we can have a model with inactive ribosome, free ribosome, bound ribosome, and the bound ribosome can do maintenance or contribute to growth. And if you estimate the fraction of bound ribosome, including this term from the data, which are the crosses here, you see that actually the amount of active ribosome doesn't go to zero, but there is still an offset in the amount of ribosome that are translating, extrapolating the result of zero growth. So and that is the paper that we published last year. So now let's go back to the initiation problem that we had. So we have the problem that at zero growth, so here I plot in FIAR, so that's FIAR mean here. I have no initiation. That is without degradation, the model that I explained before. If I include degradation, I can actually, of course I can see an offset in the initiation rate at FIAR mean. So it means alpha zero here is the initiation rate, that we expect to have at zero growth. And substantially that is kind of consistent in the sense that you measure some elongation rates. Also initiation, some elongation rates that don't go to zero, you still have a protein expression, protein synthesis at zero growth. So the same thing should kind of happen with the initiation. And so now we, these are preliminary data again, so we'll try to do some more refined work on that. So I plot, so you're right. So that is not lambda, so that's FIAR mean, so it's the famous FIAR mean, there's the offset in the growth load. I could have plotted it to zero, but it's easier to see that FIAR mean when you're at zero growth, you have no, you still have some elongation rate and initiation rates that are different from zero. So it's just because it's easier. I could have plotted that as a function of alpha. So, and now let's go back to my motivation because remember the ribosome density should be a function of the ratio, right, of the initiation rate and the elongation rate. So now I can compute actually the ribosome density for different growth rate and compute the ribosome density for different growth rate. That is what we obtain. So that's the ribosome density as a function of the growth rate, assuming the epsilon, so the elongation rate and initiation rates that we estimated. And for fast growth, we are at a constant ribosome density and the gray line that is here is for more or less 4.5 ribosomes. The messenger and I, that is what it has been previously estimated. And actually the lowest value here is for more or less 3 ribosomes for messenger and I. So there is no big difference. So we don't have any mechanism yet on how initiation is actually regulated. You think that most of the thing could be hibernating ribosomes, but it seems that one of the outcome of having this both elongation and initiation are regulated is actually to keep the ribosome density more or less constant across conditions. Yes. Again, a super basic question. So do you use these as, I didn't understand when you were talking about the paper by Balakrishnan. I don't understand if you use the fact that the ribosome density is constant to infer this. So it's really a... That's an outcome of, so in that, sorry, it was a bit confusing. The first application, so Philippe's model that he presented last week, yes, we are assuming that. We're assuming that ribosome density is more or less constant. And in this part, so when we estimate initiation here, there is no assumption that ribosome density. So the only inputs that we have are those. So the messenger concentration, elongation rate, and the welfare. So these are the only input that we feed into the model. And then with these, we can estimate the initiation rate. And as an outcome, we have the density is more or less constant. Okay, and I think I can wrap up. Okay, so my question is, is there a certain input of a model or an output from the degradation that we have as an input? So okay, the question, if I understand is, if the fire mean here is an input of the model or an output, right? I mean, it's in the data, right? That we are giving to, it's in this data here, the offset. Yeah, I mean, yes, I mean, yes, in the data, yes, we are in the input of the model. But the degradation rate is necessary. So the degradation rate can give you the offset, okay? If you don't have this as an input, if you change the degradation rate, it will give you the offset, substantial. So now, if you analyze the data, so if you have an input with an offset and you don't have the degradation rate, you arrive to something that is inconsistent, like that you don't have initiation but you measure elongation, right? So if you, the degradation is not a fit, is actually taking the data from the literature. So also the degradation is an input. So if you take the model without considering degradation, you arrive on inconsistent results. If you add the degradation, data from the literature, you get to a value of initiation with an offset, with an offset. But is the offset observed also? This offset, well, it's not observed. In the sense that there are no measurements of initiation rates, as far as I'm aware, across different conditions. But I could estimate that starting from the data. So one thing to do will be probably to measure these. Which can be done. So regarding the degradation, that means that experimentally, if you were to manage to increase the degradation, you would see in these data from diet R that the offset of sort of the number, the fraction of ribosome at zero gross would go up. So if I understand the question, you're saying if you add more degradation, you should see that the data will go up. Well, no, I don't think so in the data of the data. So it must take this as an input, but it will probably change the interpretation that you give to the data. So if you consider no degradation, the offset is given by inactivated ribosomes. So how can you test these offset? Well, probably by changing the degradation, you should change the offset, that's for sure. I don't know what else can happen to this. One more question. So you said there are no measurements of initiation, right? But I'm just wondering, maybe I'm missing something, but isn't it just the number of ribosomes divided by the number of mRNA copies? I mean, aren't all the ribosomes on some mRNA? No, I wouldn't say so. So the question is, is the initiation rate just the number of ribosomes divided by the number of messenger RNA, right? In the cell? No, because that will give you the current. So the current is a combination, is a function of both initiation and elimination. And I want to decouple these two terms. But, no, but that's, so you're saying, I'm just saying, like, sorry, okay, so you also need to know how long it takes to finish the translation. Well, but even if you know how long it takes to finish, I mean, if you measure just put in synthesis, it will be like the time that you need to initiate, time that, plus the times that you need to translate. Right? Yeah, okay, so not everything is known. No, so the elongation rate, so they have been measured in the data that I show you, is actually the translation of the first ribosome. So the first thing, so it doesn't see any traffic. So that should be something that is very close to the microscopic elongation rate. If you want to do that at the steady state, let's say, you're actually measuring the current. But you're not measuring elongation or initiation. So I want to do a microscopic model starting from this to this. Okay, but I think I can wrap up. So just to make a quick summary. So what I've done and presented is that more general framework that can consider for complex formation limitation for the part that you have competition on the messengers for the ribosomes and which also includes traffic. So we can use, we can then build up on this framework. We can couple transcription and translation. So, and that being the choice that Ludovico and Philippe did last week. And we can investigate also the role of translation initiation. And also the message here is be careful because putting the gradation shouldn't be neglected at the slow growth when the dilution term and the gradation term becomes all the same order. Now, we can build and do a lot of things from now on. And the idea now is to perturb different steps of translation. So perturb not just elongation, like has been done in the literature. We can perturb initiation or we can perturb like with antibiotics that will affect on different ways than just programming. So with that, I would like to thank all these people, in particular, Alphilippe and Johannes who are here. And of course, thank you for listening. Yeah, thanks for that very interesting talk. I'm confused about something. You at some point you showed a graph of ribosomal fraction versus growth rate. The straight line which you mentioned was being done without any fitting any parameters to the data. Can you just go back to that? Okay, probably I went too fast on that part. Like this. Ribosome concentration, yes, versus that, right? Yeah. So essentially here, the thing that we did was to couple transcription and translation and then we can have a phase diagram according if you are in the saturated regimes or not, or in the low density regime as a function of RNAP concentration and ribosome concentration. And then we said, okay, now we know from data that the ribosome density is constant on the transcript. So we assume that, and there is one isodensity line that fixes the density at four ribosome per messenger. So what happens on this line? Now the question, what happens on this line? Well, we can, so I think that was the question. So we can see how as a functional growth rate being on this isodensity line, the ribosome concentration will change and the RNAP concentration will change. And that is the plot that I showed you. So what is confusing me to ask my question is that I thought that this particular straight line is a consequence of some regulation which is the balance, which is what the cell does to regulate the balance between ribosomal protein and metabolic protein. Now, where is that sort of conceptually entering your? Yeah, you're right. So the regulation is here by imposing that you want to live on an isodensity line, that you somehow the cells regulate the ribosome density to be constant across different conditions. So when we look at what happens around this line here, it's where we implicitly put all the regulation that this cell is doing. I see, so that's where, so, but why is that balance between ribosomal protein versus metabolic protein? I don't think we can answer with this one. But yeah, everything is implicit. Our questions? So with all these offsets and the degradation rate not being ignorable, is it more natural to look at growth rate plus degradation rate in all the plots, or would something else go wrong in your model? Sorry, I didn't understand because is it more reasonable to look at? So you always plot things versus growth rate, right? All these plots, sir. But then you say the degradation rate plays the same role at zero growth rate. So isn't it natural to just make all these plots with the growth rate and the degradation rate added? Or do they play a different role somewhere in the model? No, no, no, that will be, no, no. We were talking about the initial rate. Why did I choose to plot the initial rate as a function of fire, for instance? No, no, no, they, I mean, we have as an input with the fire, with the red as an option, we can get to the lambda, but they are, no. But it was just a choice, maybe not for the next one. One question? I have one. When you plot, if you go to this line that you showed before, all right, this one there. It seems that you have this limit there, this horizontal line, it's a sort of, I mean, a phase transition or is it just something that, just... You mean here? No, there is something on one, the black one. Ah, here. At the center one. There is something on one, yes. Okay, this one, yes, that's a phase transition. But you don't see it on the data. No, we don't see, I mean, by nutrient conditions, by changing the condition, we are in this region here. In the white one. It would be nice to see if it's changed when you, actually, but that would be... When you stress? Yes, because that would be a strong violation at the moment that Judo can actually see this situation. I agree. One more question. Right, we are on time.