 Thank you for allowing me to speak today. So I'm going to talk about my PhD work, one part of my PhD work. It's done in joint work with Vincent Frommion from INRA and Philippe Robert from INRIA. So I'm going to talk about proteins. So first of all, I need to make a small introduction about the biological context, the biological question we want to answer to. Then I will explain how we put that into mathematics, into probabilities language, let's just say. Then the main result, mathematical result, and here if I have time, I will explain some other aspects with some simulations. So first, the biological context. So you all know that the unit of life you sell in the particular case of bacteria, their goal is to grow and divide as fast as possible. And to do so, they are using functional molecules called proteins, which basically do everything in the cell. They are bringing the energy, they are composing the world of a cell, they are used as enzymes, et cetera. And they are produced from information based on the small part of the DNA, which are the genes. So it's a big deal, the proteins are a big deal into the cell. It represents 50% of the dry mass. There are counts of three million molecules of 2000 types and in each type you can have a few dozen of molecules up to 10 to the power of five. And everything has to divide to double in size and divide in a relative short amount of time, let's just say 30 minutes for the faster. And if we think of that, bring the biologists to say that 85% of the resources of the protein are dedicated, the resources of the cell are dedicated to the protein production. So it's the main mechanism into the cell. So how do we do the cell produce protein? So it's the most classical way of presenting it. It's the one I've learned in high school. So it's in three steps. First, you have the gene regulation. The gene can be either in active state or inactive. If it is active, then it can produce the transcription, an mRNA, an intermediate molecule called a messenger RNA, mRNA. And the mRNA has a left time of few minutes until it's degraded and each mRNA can be translated into a protein. And the protein is a very long lifetime, much longer than the cell division. So in all the model we present, I would just simply say that at some point the protein just go away in one of the two daughter cells. It's just simply dilution. So the problem is that this way of producing protein is subject to high variability. Why? Because the interior of the bacteria is non-organized medium. Everything goes with random diffusion and each mechanism to produce a protein is due to random collision between molecules. So everything of that make it a stochastic process with a large availability in the protein production. And the problem is that, as I said, 85% of the resources of the protein are impacted by a large availability. So it's a huge problem for the bacteria. And some biologists came up with the idea that there might be some mechanism in order to reduce the availability for some proteins, at least for the most important one, and so, for example, they have seen that there's other ways of producing proteins like this one with a feedback. So it's basically the same thing as I've presented before. The only difference is that each protein here has a tendency to bind on its own gene and hence deactivate it. So each protein here has a tendency to bind on its own genes and it's staying here for a certain amount of time until it's released by just by thermologic agitation. And the important thing to know here is that the more protein you have here, the more so this particular type of protein you have, the more the gene will be inactive and the other way around. If you have few proteins, then the gene will tend to be more active. So it's really a feedback process. And we came up with the idea that it might be a way of reducing the viability. It's the conjecture. There is less variability with this feedback mechanism than there is in the, I will call it, the classical model. So our goal would be to put that into equations and through probabilities equation if I into mathematical models and to compare the two distribution and to see if there with a feedback model we have less viability, lower variance than in the classical model. So how to put that into equation? So modeling protein production is not new. It did back to the 70s. And all the model have basically the same assumption. It says that every event, and the contributing molecules, the allegation of molecules that is to say the production of mRNA and the production of proteins and the lifetime of the molecules, everything of that follow exponentially distributed variables, which rate depends on the current state of the model. Of the current state of the model, yes. So for example, here you have the classical model. So it can be active or inactive and the transition is an exponential time of rate lambda in one minus. And when it's inactive, when I is equals to zero, it can active again at a rate lambda in one plus. If it's active, so if I is equals to one, it can produce an mRNA. So in that case, the big M, which is the number for mRNA would be increased by one with a rate lambda two. And each mRNA has a lifetime of rate mu two. So the global rate of mRNA degradation is mu two times M. And we have exactly the same with the protein. Here, each mRNA has a tendency to produce a protein at rate lambda three. So the global rate is lambda three times M. And P would be the number of protein would be increased by one in that case. And each protein has a lifetime of mu three until it's just simply diluted. So the global rate would be mu three times P. Okay. So we have that. It's a, this model has been studied since sometimes as I said, and you can have equilibrium flow, which is the mean and the variance. Everything is the terminated. So everything is good. So we try to do the same with our feedback model. So what will change here? The only thing is here, you have to see if you have each protein that does not seem to bind on its own gene. So I would say that a protein here will bind on its own gene at a rate lambda one minus hat. I will explain what there is a hat here. So the global rate of the activation will be lambda one minus hat times the number of protein. When I put a F, it means feedback. So each time you see a F, it will be simply a feedback. So why a hat is just simply, I will compare the two models and what you have to have in mind is that the average of lambda one minus hat times PF will be in the same order of managers as this lambda one minus in order to have on the average, the same deactivation of the gene. So it's just why I put a hat. It's not the same parameter. Okay. So we tried to do the same as before. We tried to make a little more flow and there is a little problem is that when you are trying to make, to search for example, the mean, you have to compute the average, sorry, it's not a C here, it should be a F, sorry. Each time you see a C, it's a F. So you want to produce, you do know the average protein gene activity, which is the mean of I and you can find it because there's a correlation when you try to make that. So we cannot, we have no way of knowing an expression for the mean and for the more for the variance. So in order to deal with that, we introduce a scanning. We try to simplify the two models in order to have a simplified enough model in order to be able to make some comparison between the two. So it's a clean result. So we introduce a scanning, which is to say that we are going to increase the speed of some parts of the model. So here I'm just saying that the activation and deactivation of the gene will be very fast and the creation and degradation of mRNA will be very fast also compared to the protein timescale. So how do we do that? So we introduce N, which is a scaling parameter will be increased to infinity and we multiply each parameter here by N here. So as N will go to infinity, we will have activation and deactivation of the gene very fast and the creation and degradation of the mRNA very fast compared to this part, which is the creation and degradation of protein. So what will happen? So you have a state with I, M and P. I and M will be on a quick timescale compared to P. In fact, we'll have any change in P where you have a creation of a protein and instantly you are I and M will reach their nucle embryo depending on the current amount of proteins. So I'll try to put that in a little bit more mathematical words. So if you have the toe one, which is the first jump of P and X will be the current amount of proteins, what you will have is that the average, the average of the gene activity will reach instantly the equilibrium with the scaling, which is depending on the current amount of protein X. And it will be exactly the same. The average amount of mRNA will reach before any change in P, an equilibrium, which is also depending on X, the current amount of protein. And so you will have a rate of production of protein, which will directly and instantly depends on the current amount of proteins. So that would be the theorem. The process P, which is the number of proteins, converts in distribution to a birth and death process with if X is the number of proteins, the birth rate will be this one. It depends on X, the current amount of proteins, and the death rate never changes. It's always new three times the number of proteins. Here it's depending on the amount of proteins. The more proteins you have, the less the birth rate will be high and the other way around. So that it is for the feedback model. For the classical model, we can do exactly the same and it's almost the same thing. The only difference is here it's not depending on X. So if X is high or low, is very high or very low, it does not change the amount of the birth rate. How much time, okay. Okay, five minutes, I'll be a bit quick. So you have equilibrium things. The first one for the classical model is it's easy, it's just a Poisson process. It's just a Poisson random distribution, sorry. And the other one is a little bit more complicated, but we can work with it and especially we can add these results. So here it's for the classical model. Here it's for the feedback model. What is interesting here is that you have two way of producing the same protein. One is the classical model, the other one is the feedback model. If you are producing the same amount on average of proteins in these two ways, in the classical model you have a variance will be the same as the mean. And the other way with the feedback model you have a variance which is lower than the mean. So it tends to go in favor of the conjecture thing that the feedback indeed reduce the variance. So since I don't have much time, I will skip a little bit that's just trying to explain the spirit of it. We try to know, okay, you have a lower variance in case of a feedback, but there is a low, is there a lower bound, lower bound? And we could find it with the asymptotic result. I'm not explaining why, but just share. With the asymptotic result, we have the ratio between variance and mean will reach one half. So that is to say that in the feedback model, yes, you have in the scaling with the scale version of the feedback model, you have a variance lower than the mean, but it cannot reach, it can reach below half of the variance, which is not very high. If you want to look at the distribution, just like a cloche, something like that, and it will be reduced only by one other square root of two because it's the standard deviation, so it's just the square root of the variance. So it is reduced, but not that much. So we came up, yeah, this is basically what I explained here. So the reduction of the variance is limited in feedback model. But the feedback model really exists in nature, so if it's not that useful, what it is useful for the cell. So we came up with another idea very quickly. We say, okay, everything will look until now just equilibrium results. Is there something different between the two models for dynamical reasons? So here we have done some simulation. Which one is going faster to reach the equilibrium? For example, you have a cell, we have antibiotics, and the cell needs to produce the counter-antibiotics very quickly, and to produce the counter-antibiotic in a very short amount of time. Which one, which way will go faster to reach the new equilibrium? So that's the biological example. So we make a simulation, we start at a low-production process, you have to obtain a high-production average production, and you see which one is going faster. So here's the other result. Here in blue you have the feedback model, and in red you have the classical model, and indeed it's going much faster for the feedback model to reach their new equilibrium. Then the classical one is going 20% faster for this set of parameters particularly. And it's quite in an equation with biological results, where they did the real experiments, and it can be 50% faster, but we are in the same range of things. So basically is everything I have to talk, sorry, at the end I'll go a little bit quick, but maybe you will have a question about that. If you're interested, there is an article about that. For now it's only on archive, but it will be published soon. Okay. Thank you for now. Questions? So on the last part of your talk, you concluded the analysis with the simulation. Is there any analytical results? No, we tried to have a look about if there are analytical results, we could not find a way to do it easily. So we just make some simulations for that part, because equilibrium is quite relatively easy to have analytical results, but then dynamically it's a little bit harder. Any other questions? If not, thanks for knowing again. Thank you.