 Okay, so good afternoon and thank you for this opportunity. In this talk, I want to present to you a project to say is to contribute to the mathematical foundation of adaptive immunity by building a mathematical environment, adapt to describe those mechanisms involved in antibody affinity maturation. This is a key process in adaptive immunity as it allows to produce antigen-specific antibodies against almost all kinds of pathogens. Besides the biological motivations, this is a general model which could also be relevant to model other evolutionary processes but also gossip or virus propagation as well. So our aim is to understand the interaction between division, mutation, and selection in elementary models which already brings interesting mathematical problems. And our method is based on the complementarity between probabilistic tools and numerical simulations. So my presentation will be divided into four main parts. In the first one, I will introduce the biological context. Then I will talk about the pure mutational models. And after that, I will introduce division and finally pass to the conflict model including division, mutation, and selection. I will always start by giving the definitions then some theoretical results. And finally, I conclude with some numerical simulations. So the biological background is the immune system. The immune system can be divided into innate immune system and adaptive immune system. The immune system defenses react very quickly but they are non-specific, meaning that they respond to pathogens in a generic way. On the other hand, the adaptive immune system takes a more long time to react but it can adapt against almost all pathogens that could penetrate our organism and infect diseases. In this context, immunity is conferred by special proteins called antibodies and the agents that causes their production are named antigens. The production of antibodies is assured by special lymphocytes called B cells which have to be trained in order to improve their ability to recognize the specific antigen presented. Activated B cells travel to the nearby lymph node and here they give rise together with other immunity cells, of course, to a special microenvironment called germinal centers in which they proliferate, mutate and differentiate. At the same time, they are also submitted to power selection mechanisms in order to improve their antigen affinity. On the outer surface of each B cells are localized 100,000 of identical transmembrane receptor proteins called B cell receptors, BCR for short, which allows the B cell to recognize the specific antigen and the binding between B cell and antigen is possible if the amino acids composing the BCR and presented on the surface of the antigen are distributed in such a way to create bonds. The BCR are Y-shaped molecules composed of two parts, a variable part and a constant region. The binding site is localized on the variable region of the BCR, which is the only one involved in mutation during the germinal center reaction. In this particular context, mutations are called somatic hypermutations with these terms, we mean mutation which arrives at a very high rate and which are essentially random mutations, so we clearly need selection in order to obtain specialized antibody in a reasonable time. So our aim is to build a mathematical framework in which we can pattern and study this kind of processes. The analysis we made of the problem suggests us to model these mutation processes, random walks on graphs with characteristic change depending on the rotational rule allowed. In particular, we suppose we are able to classify the amino acids which compose the BCR and characterize the chemical properties into two classes named zero or one. They may correspond to positively charged amino acids and negatively charged amino acids. Therefore, BCR and antigen are represented as N-length binary strings. We call HN the state space of all possible BCR. Therefore, we model the affinity using the amine distance between these two strings. In particular, we suppose that the optimal BCR is obtained when the amine distance between the BCR-representing string and the antigen-representing string is zero valued. And finally, to define a rotational rule means to define random walks on the graph. Well, I start with pure rotational models. We can, of course, define very different rotational rules in particular. We define the rule of simple point mutations in which at each time step, a randomly chosen amino acid switches the class it belongs to. Mathematically, this gives rise to a simple random work on the n-dimensional hypergroup. Of course, this kind of random words has been already studied in many different contexts. But here, it represents for us the basic rotational model, and we will use it to define more complex rotational rule. And I give the notation of the transition probability matrix, which is P, and we will use it later also. Of course, we can complexify this model in many different ways. For example, here we propose a model of class switch of one or two length strings, depending on the affinity. In particular, if the aming distance between BCR and antigen is greater or equal to two, then we have to switch two bits, one bit otherwise. And mathematically, what we obtain is a graph divided into two subgraphs. The one containing the antigen target cell is accessible from the other, but not conversely. We can understand maybe better from this little example how this rotational model works. At the beginning, the aming distance is equal to three, the blue bits. So we have to change for the first step two elements. Therefore, here, we have just one bit. The aming distance just equal to one. And so for the next step, we will have to change just one bit and so on. Finally, I propose here also another rotational rule, well, a class of rotational rules, which is the multiple point mutations. That means that a teach time step with given probability a, i, I am allowed to do i independent simple point mutations with i between one and k and k fixed. In this context, once we fix k, I propose two variants of the model of multiple point mutations. In the first case, sorry, the first case with equal probability one over k I'm allowed to do between one and k mutation. And in the second case, I always do exactly k star mutation at teach time step, where k star is the greater or the value smaller or equal to k. We were interested in understanding how the typical timescales of state space exploration change depending on the introduced rotational rule is for this reason that we get interested in understanding the eating time. The eating time is defined as a specter number of steps. We need to reach a specific position of the graph given the departure node. It has a clear biological interpretation. And the time needed to obtain the optimal BCR given the naive BCR trait. And for the models, rotational models I introduced, we were able to obtain explicit formulas to evaluate it or at least estimations for n big enough. And so we can compare these different rotational models and their ability of exploring the state space. Here, I summarize the main results we obtained for the three rotational models I introduced. While I was saying, I just collected the main results we obtained for the three rotational models I introduced. So we evaluated the mean eating time to cover an initial I mean distance d bar. And I conclude the first part with some numerical simulations. In this first simulation, I compared the eating time for the BC model and for the model of one or two, of switch of one or two line streams depending on the affinity. As demonstrated by mathematical analysis for n big enough, we obtained that the eating time for this second model is hard compared to the basic rotational model. And here I give the results over, we obtained for n equal to 10 over 5,000 simulations. So n is not so big actually. And in these last simulations, I compare the meeting time to cover an initial, I mean distance d for the two models I introduced of multiple point mutations. We can demonstrate by spectral analysis that the model represented by p power k star actually optimize this mean eating time for k strictly greater than k, for k strictly greater than two. While now I introduce the division process, mathematical we will talk about simple branching, simple two branching random work. The process starts at time zero with a randomly chosen VCR entering of the germinal centers and the library is active. So we have at the initial, a random node of our state space HN, library is active and at each time, each active node chooses two of its neighbor randomly and with replacement to become active at the time t plus one, and it becomes inactive, even if of course another active node chooses it. And in this context, we are not interested in counting how many times a node is chosen to become active. We also, I also work over this model but I don't want to show you here these results. So of course the addition of the division process imply a substantial speed up in the state space, in the state space explorations. So in this context, we were interested in observing the proportion of active nodes we can obtain after an order of N time steps and comparing the different rotational models I introduced. So using a process they introduced here in this paper to evaluate partial covert times, more general context and evaluating the expansion properties of the graph underlined respectively by metrics P and PK. We can obtain, we were able to obtain these two results for the simple two branching random work referring to P and PK respectively. And we observe that while using a simple point mutation model we obtain after grito N time steps, just a little portion of active node in the state space by using this mutational model almost a half of the state space will be activated after a time of the order of N and independently from K greater than two which is really interesting. I conclude the first part, the second part with another numerical simulations here I compare the evolution of the size of the active set comparing the three mutational models I already discussed. So in blue I represent the simple point mutations. In red the multiple point mutation corresponded to metrics P power K star and in green the other model of multiple point mutations. We observe that this last process is the faster one in spreading and also is the only one which allowed to cover all nodes simultaneously. This is due to a characteristics of the graph underlined by P and P power K star respectively which is the bipartitness. The only non-bepartic graph is the one corresponding to the red curve. In the horizontal blue and green lines represent the size of the active set as we estimated by the theorems I gave in the previous slide. Finally, I introduce this more complete model. This is we are working just right now on a model including mutation division and selection. More precisely the process starts this time with an initial B cell entering the germinal center reaction and with initial distance from the antigen target cell H0. At each time step each cell within the population can die with rate Rd. If not it can divide with rate Rd and after that when it divides it give rise to two newborn cells with a trait which is mutated referring to the mother cell and the mutations correspond it's the new trait is obtained depending on the mutational rule we allow. So we are still working on our state space of the trait Hn. In this context we define two different models of selection. We can say a positive negative selection for model A and just a positive selection for model B. So in particular in the first model when a B cell is submitted to selection we observe affinity to the target cell. If our having distance is greater than a certain threshold then we fix that so the cell will die by apoptosis. If it's affinity is good enough it exits the germinal center and enters the selected pool. The only thing that change in the second model is that if a negative selected cell just stay in the germinal center for the next time step and so eventually it can give rise to good trait in further time step by mutation. Okay, we were interested in and just in study the evolution of the selected pool. In order to do that we introduce N plus three N plus three type Galton Watson process. So that Ti is the vector containing the number of cells of each type starting from a time zero from a single I type cell. In particular for each J between zero and N that Ji comes the number of germinal centers B cells having I mean distance J from the target cell the type N plus one correspond to selected B cells and the type N plus two to death B cells. So by using all this formalism we are able to study many quantities of the process in particular the expected number of selected B cell till at time T or at time T their average affinity but also what happened in the germinal center. So the number the size of germinal centers and the affinity inside the germinal center. I conclude this last part with some numerical simulations. As I said we can evaluate many things by using the formalism of the multi type Galton Watson process. Here I represent the expected number of selected B cells after 15 time steps depending on the selection rate and comparing the two models of selection we can clearly observe that there is an optimal selection rate is for that reason that we evaluate numerically this optimal eras. We obtain two curves which are similar for the two selection models and as expected this optimal eras is decreasing in time and finally here I represent again the expected number of selected B cell after 15 time steps but depending this time from the threshold that I choose for positive selection this time I see two different behaviors between the two models in particular for the model of just positive selection I have again a phenomenon of maximum. So what we have done is to build a mathematical environment which is flexible enough in order to introduce and study different mutational rules and their interaction between division and selection for example. Also it allows us to fix our point of view and decide if you are looking to the mutation at the DNA lever or directly observing the effective mutation on the amino acid chain actually we obtain the models which are of the same kind we have just to be clear about the affinity and what we want to do is mathematical analyze the complete model including mutation, division and selection and that's what I'm working on just right now and also try to evaluate the other characteristics of the process in order to have a more global view of the process one. Thank you for the attention that's all I wanted to say today here. Thank you. Is there any question? When you look at the experimentally how the B cells mutate does it fit better one of your three mutations? Well actually experimentally during somatic hypermutation we have almost only single point mutations but also some other kind of mutation can happen for example what we call indels mutation which is insertion or deletions that I didn't take into account here because it makes other problems like the length of the chains can change for example and I don't take it into account but they are basically simple point mutations sometimes you can have two mutations I teach from one generation to the next one but not more actually. So you have assumed that your selection criteria is constant in time does that come from biology consideration? Not necessarily we just take this selection model because it seems to us easy to study and also we can understand it it can be possible but maybe you have a selection pressure that changes during the general center reaction and also you have to consider other effects. We also take a selection function that is just a step function which is zero after HHS and one before HHS for example this is also an assumption which is not necessarily the right one biologically. Okay, thank you. So let's thank you again.