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Branching random walks applied to antibody affinity maturation

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Title Branching random walks applied to antibody affinity maturation
Title of Series Les Probabilités de Demain
Part Number 10
Number of Parts 17
Author Balelli, Irene
License CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
DOI 10.5446/20266
Publisher Institut des Hautes Études Scientifiques (IHÉS)
Release Date 2016
Language English

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Subject Area Mathematics
Abstract Antibody affinity maturation is a key process in adaptive immunity: it’s a mechanism which allows B-cells to produce high affinity antibodies against a specific antigen. Besides the biological motivations, the analysis of this learning process brought us to build a mathematical model which could be relevant to model other evolutionary systems, but also gossip or virus propagation. Our aim is to propose and analyze a mathematical model of the division-mutation-selection process of B-cells during an immune response. In particular, we investigate how the combination of various mutation models influences the typical time-scales characterizing the efficiency of the state space exploration. Our method is based on the complementarity between probabilistic tools and numerical simulations.
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I was 1 of the world's most Back to what some good afternoon and thank you for this treaty and in this sort I Want to presented a project whose aim is to contribute to the mathematical foundation of adaptive immunity by beating a mathematical environment and adapted to describe those mechanisms involved in TV in unduly defeating moderation is key process that immunities as it allows to produce a in specific antibodies against almost all kind of pathogens and besides the biological motivations and this is a genetic model which could also be relevant to modern or the revolutionary process but also the gossip or virus propagation as well so our aim is to understand interaction between the Indonesian inspections in elementary models in which already being so interesting and mathematical problems and our method is based on the complementarity between brevity 6 tourists and numerical simulations so that my condition will be divided into 4 main parts in the first one I would introduce the biological context and then I will talk about the punitive action on Mondays and after that I will introduce division and finally passed to the conflict to modern included division imitation selection unbelievable I would always starts giving the definitions and son and theoretical results and finally I conclude this new mechanism so their biological as background is the insistence the new system can be divided into enacting assistance and battered in system in a year the system defenses react very quickly but they are specific meaning that they responding to pathogens in a minute and generic way a on the other hand the adaptive immune system takes a more time to react but it can adapt to the against almost all pathogens that could penetrate our old in and in the 2 diseases In this context in discomfort of special proteins called antibodies and the agents that causes their production are named antigens as the production of DVDs is assured that by especially emphasize called the which are which have to be trained in order to improve their ability to recognize the specific antigen presented activated the sent traveling near to the nearby that leaves no doubt and that year that gives rise together with other sense of course to especially from Byron and called a general centers in which they proliferated mutate and differentiate them and at the same time the so a and submitted to power so powerful selection the mechanisms in order to improve their Andean antigen Washington and on the outer surface of each of the sensors are localized 100 thousand of the identical membranous etc. proteins called the the service centers BCI for fraud which allowed to be sent to recognize the specific antigen and abandoning their between NBC and ABC 17 is possible if the enemy in composing abandon the VCR and presented on the surface of anti-Asian and with in such a way to create bonds and the BCR our shaking molecules and composed of 2 parts of pasta and a a constant region their binding site is localized on the variable region of Debussy on which is the only 1 involved in mutation during the DeGeneres interaction in this special industries in this particular context invitation called a synoptic I've imitation with this announcement with new means edition which arrives at a very high rates and there which I said essentially around imitation so if you will clearly need this selection in order to obtain a specialized antibody in a reasonable time so our aim is to minimize medical framework and in which we pattern and the study of this kind of process is the analysis we made of the program of suggests us to a model of this this limitation processes around the malls on grass was correct 6 change depending on the location of the rule allowed the impressively was supposed to be able to classify the amino acids which compose the VCR and the and directed by the chemical properties into 2 classes named or want corresponds to a positively charged amino acids and negatively charged amino acids therefore BCR and anti-Asian represented as an answer binary streams the legalization the stated this the status of all possible be stopped therefore we monitor their Fiji using the ending distances between these 2 extremes in particular was suppose that the optimal BCI is obtained when the it's them in descended into the besotted sentencing and into European industry is irrelevant and finally into the final competition ruling means to define around the world on on the ground was I suddenly pure additional models we can of course are a defined very different petitioner is impractical we as defined in the rule of single point mutations and we should teach time that there are randomly chosen eminence in the streets of the classic belongs to a mathematically this give rise to a in Brandamore upon interests Hydro-Quebec of course this kind of from the worlds has been already started many different context but yes it represents for us their designated and reviews the final accomplishment additional room and I'd even the imitation of the transition from the genetics which is speedy and we will use its at and of course we can complete survivors model in many different ways for example a year we propose a model of class which have 1 chewing strings depending on defeating in particular if Ugandan distance between this year and Andijan is greater or equal to show that we have to switch tributes 1 otherwise a mathematically what 15 years there have divided into 2 should grasp the 1 containing the contingent on it said it's accessible from the other but not conversely as we can understand better from this it is an example Our view is that the addition of the world's swept in at the beginning of the Anderson's is equal to 3 the so we have to change for the 1st step of a 2 elements that for years we have just 1 meter the end just equal to 1 and so for the next time step we have to changes when they beat and so on and finally I
propose here also another and additional when a class of medicinal which is the multiple point mutations that means that the teach stands stepped was given probability ATI I am I'm allowed to do I'm independent and simple point mutations with between 1 and carrying case seeks in this context was refutes Kate and I proposed to variants of the mother of my multiple point mutations in the 1st case of its preferred to this case With the Quetta ability 1 of OK and I to do between 1 intimidation and in the 2nd case that I always do accept the cat case sanitation and teach science that where Questar is there the Greater Orlando values more equipped to take but in that we were interested in understanding out to be no tanks caves offers state space inspiration change depending on the introduce an additional work is voters for this reason that we get interest in extending the eating time visiting time is the final suspect a number of steps so we needed to reach a specific position of the graph giving the patent on it as a clear biological interpretation and the time needed to obtain the optimal BC are given on the nite of the BCS trade and therefore that a moderate militiamen as I introduced were able to obtain a speech at former US tool evaluated or at least its initials for ending enough as so we can at least can compare these new friend additional modern and the ability of restoring the state's please yeah I summarize the main results we obtained for the theme additional models introduced what I was saying I just collected the main results we obtained for the 3 efficient models introduced Teleservices we evaluated the meeting time to govern the the Indonesian and decency in the back and and
with the 1st part of his some mechanism relations said in his 1st simulation by comparing the waiting time for them as a model for the moment for the mother of 1 of 2 of switches of well-entrenched things depending on their feeding as demonstrated by the mathematical and friendly enough that we taken the lead in time for this 2nd is they compared to their presentation on Monday and you're view the results overlooked in for an equal to 10 of thousand simulations so and it's not so the to and in his
last assimilation contract at the meeting time to and nation indecency afforded the tumult introduced multiple point mutations weekend a mistake by inspector analysis that the model the represented by keep our case start actually optimize this meeting demand for K Street the greeters indicate that and chill while the injured and introduce the revision process mathematical we would total about the simple branching single-issue mention on the walls the process starts at the time of the year with a randomly chosen a and B C are entering the dominant centers and libraries active so we have around them no under the cover of the space station that is active in time at each factor known chooses chew of its neighbor and randomly and the replacement of the 2 to become active at the time the press 1 and becomes inactive even if of course another actor known to the state and in this context that interest in counting down many times and on beaches into comments we also a tireless worker over this that this model but by but I don't want to show you hear this result as well as an off-course edition of the division process imply a substantial speedup in state space is in the suspicious operations so this context that we were interesting observing that and the proportion of active knows we can obtain after half time after an order of any kind steps In comparing them the different additional design introduced so using get other process so that they they introduced here in this paper to evaluate their partial good times more general context and evaluating the expansion properties of the graft underlined respectively then that's the end of the cake we can obtain the whereabouts of the industry results of for this simple to mention what referring to P & P K respectively and the observed that there there while using it as simple as that .period mutation model and we obtain after a great all in an offense that's just a little portion of the active known in the state space by using this additional and almost half of the state's space to be activated after after the time of the Order of and independently from K greater adventure which is really so interesting and
I can be the 1st part of the 2nd part with another notice relations Iraq compared a dilution of the size of the acted as a set of directives set comparing the 3 additional cited radio discusses the soul and in Peru I represented the single by mutations in the and that the motherboard .period mutation correspondents people talks ending greener on the other hand model of multiple point mutations that we observed that this last process the faster breeding and Boston is the only 1 which allowed to go over all notes simultaneously in this due to characteristics of the grass underlined by the anti-poverty start respectively which is the departed the only on departed Rafah is on the minor Chrisman into the red color in an DOD longtime both blue and green lines represent is the size of the active a sector as a whole we estimated instigated by the few instigating the previous slide finally a
introduced his confidence and this is what we are working just right now on a model including the addition lesion and selection of more precisely and move the process starts this time than the time with initially the set entering the generous into reaction and there with the initial coming at a distance from the indigenous targets set at age of the year at each time instead of each send in the police and the population with the rates are indeed if not it can divides with rates of odd-even ended after that day it's said and 120 devices that give rise to tune Newman says with a traitor which has mutated that referring to the mother said and limitations that charisma it said that the new traits of the ESA is obtained depending on the additional role we allowed so we are still working on our state spaces of the trait agent and in this context so we define should different models of selection as we can see positive negative selection from other AT and just a positive selection from others be so inviting and the 1st known that 1 of the Senate's submitted to selection we observe that are affiliated to the targets set in if are having standards is which than a certain threshold and we fix that so this will die by a part of this as if it's I think is enough issuing exits the seminal center and enters the selected for forward the undertaking of the claims in the 2nd 1 is is that the firm as a negative and selected the site just say in the German a center for the next time step and so invented can give rise to good traits In the 1st time set by sedimentation OK and we weren't interested in understanding the study of the dilution of the selected put it in order to do that to be introduced under strain of them and endlessly tied up in Watson process a still better day TI is better containing the number of officers of each type of starting up from time Europe from a single bite type said impractical for each Indira and that JI it counts a number of German centers be spending understands Jake from the targets set the time and was 1 of the correspond select the Senate and the time and the type of interest to the 2 deaths the so that by using all these formalities and we are able to study many in quantities of the process in particular the expected number of selected the safety at 20 or act until their average if did but also of what happened in the German centers so that the number of size of demand centers and their feeding inside the and I conclude that is not part of this amendment reconciliation as acidity in evaluating many things that using the deformities of the merger type person want to process a PRI representatives spent a number of selected the after 15 spent steps depending on honor as assimilation rates and and comparing the 2 models of selection weekend clearly observed that there is enough to mount selection rate is
ready and that we evaluated numerically this optimum as we obtain should Council which courses which have seen enough of the juice election monitors and as expected there is the ultimate air is decreasing in time and finally I represent that again despite number of 2nd instead of 15 times test but the attending this time from the threshold that I choose for an incentive for buyers is positive selection at this time I seat 2 different areas between the 2 Manderson party Ukraine for the mother of positive selection they began a phenomenal amount of maximum
when I saw it what we have done is to be the mathematical environment which is flexible enough in order to introduce and study the fragmentation of rules and their interaction between the region and patient and selection for example also it allows us to fix our point of view and decide if you are looking at today in addition and the DNA even or directly observing the effect imitation on the amino-acid chain are actively wilting models of which are of the same kind of just to be clear about the feeding and what we want to do it is a mathematical analyzes the complete Mallinckrodt imitation division selection and that's what I'm working on it just right now and also tied to the bottom of the characteristics of all the process in order to have a you all of the process was Thank you for a the attention that's what I wanted to say to the thank you forget that you moved the through the whole of the music is 1 of the woman who was actually an experimentally during semantic implementation we have almost a only a single point mutations but also some other kind of condition can actions in for example a what we call those mutation which insertion of nations that and then they didn't take into account here because it makes other problems like the lessons of this of the change can change for example and I don't think it will be found as so it there but they are basically simple point mutations sometimes you can have 2 mutations a teacher get it from the 1 generation to the next month but not more active the top not all of this you know that your selection criteria used Constantine time the user doesn't come from her approach has not necessary we just taken this selection demanded because it seems to us is to study and a also there while it can be it can understand it it can be possible but maybe you have a section presses the changes it during the general's interaction and also we have to consider all the other facts we also take selection function but this is just a step function which is the root of a after each had a shares 1 before agents for example this is also has an assumption which is not necessarily in the right 1 I thank you so the unanimity in thanks
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