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Anti-viral therapies are typically designed to target only the current strains of a virus. Game theoretically, this corresponds to a short-sighted, or myopic, response. However, therapy-induced selective pressures act on viruses to drive the emergence of mutated strains, against which initial therapies have reduced efficacy. Building on a computational model of binding between antibodies and viral antigens (the Absolut! framework), we design and implement a genetic simulation of viral evolutionary escape. Crucially, this allows our antibody optimisation algorithm to consider and influence the entire escape curve of the virus, i.e. to guide (or "shape") the viral evolution. This is inspired by opponent shaping which, in general-sum learning, accounts for the adaptation of the co-player rather than playing a myopic best response. Hence we call the optimised antibodies shapers. Within our simulations, we demonstrate that our shapers target both current and simulated future viral variants, outperforming the antibodies chosen in a myopic way. Furthermore, we show that shapers exert specific evolutionary pressure on the virus compared to myopic antibodies. Altogether, shapers modify the evolutionary trajectories of viral strains and minimise the viral escape compared to their myopic counterparts. While this is a simplified model, we hope that our proposed paradigm will facilitate the discovery of better long-lived vaccines and antibody therapies in the future, enabled by rapid advancements in the capabilities of simulation tools. Our code is available at https://github.com/olakalisz/antibody-shapers., Comment: Preprint

We investigate a class of stochastic chemical reaction networks with $n{\ge}1$ chemical species $S_1$, \ldots, $S_n$, and whose complexes are only of the form $k_iS_i$, $i{=}1$,\ldots, $n$, where $(k_i)$ are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter $N$. A natural hierarchy of fast processes, a subset of the coordinates of $(X_i(t))$, is determined by the values of the mapping $i{\mapsto}k_i$. We show that the scaled vector of coordinates $i$ such that $k_i{=}1$ and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as $N$ gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.

In this paper we study a class of stochastic chemical reaction networks (CRNs) for which chemical species are created by a sequence of chain reactions. We prove that under some convenient conditions on the initial state, some of these networks exhibit a discrete-induced transitions (DIT) property: isolated, random, events have a direct impact on the macroscopic state of the process. If this phenomenon has already been noticed in several CRNs, in auto-catalytic networks in the literature of physics in particular, there are up to now few rigorous studies in this domain. A scaling analysis of several cases of such CRNs with several classes of initial states is achieved. The DIT property is investigated for the case of a CRN with four nodes. We show that on the normal timescale and for a subset of (large) initial states and for convenient Skorohod topologies, the scaled process converges in distribution to a Markov process with jumps, an Additive Increase/Multiplicative Decrease (AIMD) process. This asymptotically discontinuous limiting behavior is a consequence of a DIT property due to random, local, blowups of jumps occurring during small time intervals. With an explicit representation of invariant measures of AIMD processes and time-change arguments, we show that, with a speed-up of the timescale, the scaled process is converging in distribution to a continuous deterministic function. The DIT analyzed in this paper is connected to a simple chain reaction between three chemical species and is therefore likely to be a quite generic phenomenon for a large class of CRNs.

Motivated by a general principle governing regulation mechanisms in biological cells, we investigate a general interaction scheme between different populations of particles and specific particles, referred to as agents. Assuming that each particle follows a random path in the medium, when a particle and an agent meet, they may bind and form a pair which has some specific functional properties. Such a pair is also subject to random events and it splits after some random amount of time. In a stochastic context, using a Markovian model for the vector of the number of paired particles, and by taking the total number of particles as a scaling parameter, we study the asymptotic behavior of the time evolution of the number of paired particles. Two scenarios are investigated: one with a large but fixed number of agents, and the other one, the dynamic case, when agents are created at a bounded rate and may die after some time when they are not paired. A first order limit theorem is established for the time evolution of the system in both cases. The proof of an averaging principle of the dynamic case is one of the main contributions of the paper. Limit theorems for fluctuations are obtained in the case of a fixed number agents. The impact of dynamical arrivals of agents on the level of pairing of the system is discussed.

We investigate the asymptotic properties of Markov processes associated to stochastic chemical reaction networks (CRNs) driven by the kinetics of the law of mass action. Their transition rates exhibit a polynomial dependence on the state variable, with possible discontinuities of the dynamics along the boundary of the state space. We investigate the scaling properties of these networks when the norm of the initial state is converging to infinity and the reaction rates are fixed. This scaling approach is used to have insight on the time evolution of these networks when they start from a ``large'' initial state. The main difference with the scalings of the literature is that it does not change neither the graph structure of the CRN, nor its reaction rates. Several simple and interesting examples of CRNs are investigated with this scaling approach, including the detailed analysis of a CRN with several unsual asymptotic properties in the last section. We also show that a stability criterion due to Filonov for positive recurrence of Markov processes may simplify significantly the stability analysis of these networks.

A Hawkes process on $\R$ is a point process whose intensity function at time $t$ is a functional of its past activity before time $t$. It is defined by its activation function $\Phi$ and its memory function $h$. In this paper, the Hawkes property is expressed as an operator on the sub-space of non-negative sequences associated to distances between its points. By using the classical correspondence between a stationary point process and its Palm measure, we establish a characterization of the corresponding Palm measure as an invariant distribution of a Markovian kernel. We prove that if $\Phi$ is continuous and its growth rate is at most linear with a rate below some constant, then there exists a stationary Hawkes point process. The classical Lipschitz condition of the literature for an unbounded function $\Phi$ is relaxed. Our proofs rely on a combination of coupling methods, monotonicity properties of linear Hawkes processes and classical results on Palm distributions. An investigation of the Hawkes process starting from the null measure, the empty state, on $\R_-$ plays also an important role. The linear case of Hawkes and Oakes is revisited at this occasion. If the memory function $h$ is an exponential function, under a weak condition it is shown that there exists a unique stationary Hawkes point process. In this case, its Palm measure is expressed in terms of the invariant distribution of a one-dimensional Harris ergodic Markov chain. When the activation function is a polynomial $\Phi$ with degree ${>}1$, there does not exist a stationary Hawkes process and if the Hawkes process starts from the empty state, a scaling result for the accumulation of its points is obtained.

Apparent parallels between natural language and biological sequence have led to a recent surge in the application of deep language models (LMs) to the analysis of antibody and other biological sequences. However, a lack of a rigorous linguistic formalization of biological sequence languages, which would define basic components, such as lexicon (i.e., the discrete units of the language) and grammar (i.e., the rules that link sequence well-formedness, structure, and meaning) has led to largely domain-unspecific applications of LMs, which do not take into account the underlying structure of the biological sequences studied. A linguistic formalization, on the other hand, establishes linguistically-informed and thus domain-adapted components for LM applications. It would facilitate a better understanding of how differences and similarities between natural language and biological sequences influence the quality of LMs, which is crucial for the design of interpretable models with extractable sequence-functions relationship rules, such as the ones underlying the antibody specificity prediction problem. Deciphering the rules of antibody specificity is crucial to accelerating rational and in silico biotherapeutic drug design. Here, we formalize the properties of the antibody language and thereby establish not only a foundation for the application of linguistic tools in adaptive immune receptor analysis but also for the systematic immunolinguistic studies of immune receptor specificity in general., Comment: 19 pages, 3 figures

In this paper we study an important global regulation mechanism of transcription of biological cells using specific macro-molecules, 6S RNAs. The functional property of 6S RNAs is of blocking the transcription of RNAs when the environment of the cell is not favorable. We investigate the efficiency of this mechanism with a scaling analysis of a stochastic model. The evolution equations of our model are driven by the law of mass action and the total number of polymerases is used as a scaling parameter. Two regimes are analyzed: exponential phase when the environment of the cell is favorable to its growth, and the stationary phase when resources are scarce. In both regimes, by defining properly occupation measures of the model, we prove an averaging principle for the associated multi-dimensional Markov process on a convenient timescale, as well as convergence results for fast variables of the system. An analytical expression of the asymptotic fraction of sequestrated polymerases in stationary phase is in particular obtained. The consequences of these results are discussed.

Deep neural-network-based language models (LMs) are increasingly applied to large-scale protein sequence data to predict protein function. However, being largely black-box models and thus challenging to interpret, current protein LM approaches do not contribute to a fundamental understanding of sequence-function mappings, hindering rule-based biotherapeutic drug development. We argue that guidance drawn from linguistics, a field specialized in analytical rule extraction from natural language data, can aid with building more interpretable protein LMs that are more likely to learn relevant domain-specific rules. Differences between protein sequence data and linguistic sequence data require the integration of more domain-specific knowledge in protein LMs compared to natural language LMs. Here, we provide a linguistics-based roadmap for protein LM pipeline choices with regard to training data, tokenization, token embedding, sequence embedding, and model interpretation. Incorporating linguistic ideas into protein LMs enables the development of next-generation interpretable machine-learning models with the potential of uncovering the biological mechanisms underlying sequence-function relationships., Comment: 27 pages, 4 figures

Antibodies are canonically Y-shaped multimeric proteins capable of highly specific molecular recognition. The CDRH3 region located at the tip of variable chains of an antibody dominates antigen-binding specificity. Therefore, it is a priority to design optimal antigen-specific CDRH3 regions to develop therapeutic antibodies. However, the combinatorial nature of CDRH3 sequence space makes it impossible to search for an optimal binding sequence exhaustively and efficiently using computational approaches. Here, we present \texttt{AntBO}: a combinatorial Bayesian optimisation framework enabling efficient \textit{in silico} design of the CDRH3 region. Ideally, antibodies are expected to have high target specificity and developability. We introduce a CDRH3 trust region that restricts the search to sequences with favourable developability scores to achieve this goal. For benchmarking, \texttt{AntBO} uses the \texttt{Absolut!} software suite as a black-box oracle to score the target specificity and affinity of designed antibodies \textit{in silico} in an unconstrained fashion~\citep{robert2021one}. The experiments performed for $159$ discretised antigens used in \texttt{Absolut!} demonstrate the benefit of \texttt{AntBO} in designing CDRH3 regions with diverse biophysical properties. In under $200$ calls to black-box oracle, \texttt{AntBO} can suggest antibody sequences that outperform the best binding sequence drawn from 6.9 million experimentally obtained CDRH3s and a commonly used genetic algorithm baseline. Additionally, \texttt{AntBO} finds very-high affinity CDRH3 sequences in only 38 protein designs whilst requiring no domain knowledge. We conclude \texttt{AntBO} brings automated antibody design methods closer to what is practically viable for in vitro experimentation.

We investigate spike-timing dependent plasticity (STPD) in the case of a synapse connecting two neural cells. We develop a theoretical analysis of several STDP rules using Markovian theory. In this context there are two different timescales, fast neural activity and slower synaptic weight updates. Exploiting this timescale separation, we derive the long-time limits of a single synaptic weight subject to STDP. We show that the pairing model of presynaptic and postsynaptic spikes controls the synaptic weight dynamics for small external input, on an excitatory synapse. This result implies in particular that mean-field analysis of plasticity may miss some important properties of STDP. Anti-Hebbian STDP seems to favor the emergence of a stable synaptic weight, but only for high external input. In the case of inhibitory synapse the pairing schemes matter less, and we observe convergence of the synaptic weight to a non-null value only for Hebbian STDP. We extensively study different asymptotic regimes for STDP rules, raising interesting questions for future works on adaptative neural networks and, more generally, on adaptive systems.

In neuroscience, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called synapses and represented by a scalar value, the synaptic weight. A Spike-Timing Dependent Plasticity (STDP) rule is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. A general mathematical framework has been introduced in~arXiv:2010.08195. In this paper we develop and investigate a scaling approach of these models based on several biological assumptions. Experiments show that long-term synaptic plasticity evolves on a much slower timescale than the cellular mechanisms driving the activity of neuronal cells, like their spiking activity or the concentration of various chemical components created/suppressed by this spiking activity. For this reason, a scaled version of the stochastic model of~arXiv:2010.08195 is introduced and a limit theorem, an averaging principle, is stated for a large class of plasticity kernels. A companion paper~arXiv:2010.08790 is entirely devoted to the tightness properties used to prove these convergence results. These averaging principles are used to study two important STDP models: pair-based rules and calcium-based rules. Our results are compared with the approximations of neuroscience STDP models. A class of discrete models of STDP rules is also investigated for the analytical tractability of its limiting dynamical system., Comment: arXiv admin note: substantial text overlap with arXiv:2010.08195

Mathematical models of biological neural networks are associated to a rich and complex class of stochastic processes. In this paper, we consider a simple {\em plastic} neural network whose {\em connectivity/synaptic strength} $(W(t))$ depends on a set of activity-dependent processes to model {\em synaptic plasticity}, a well-studied mechanism from neuroscience. A general class of stochastic models has been introduced in \cite{robert_mathematical_2020} to study the stochastic process $(W(t))$. It has been observed experimentally that its dynamics occur on much slower timescale than that of the main cellular processes. The purpose of this paper is to establish limit theorems for the distribution of $(W(t))$ with respect to the fast timescale of neuronal processes. The central result of the paper is an averaging principle for the stochastic process $(W(t))$. Mathematically, the key variable is the point process whose jumps occur at the instants of neuronal spikes. A thorough analysis of several of its unbounded additive functionals is achieved in the slow-fast limit. Additionally, technical results on interacting shot-noise processes are developed and used in the general proof of the averaging principle.

In neuroscience, learning and memory are usually associated to long-term changes of neuronal connectivity. In this context, synaptic plasticity refers to the set of mechanisms driving the dynamics of neuronal connections, called {\em synapses} and represented by a scalar value, the synaptic weight. Spike-Timing Dependent Plasticity (STDP) is a biologically-based model representing the time evolution of the synaptic weight as a functional of the past spiking activity of adjacent neurons. If numerous models of neuronal cells have been proposed in the mathematical literature, few of them include a variable for the time-varying strength of the connection. A new, general, mathematical framework is introduced to study synaptic plasticity associated to different STDP rules. The system composed of two neurons connected by a single synapse is investigated and a stochastic process describing its dynamical behavior is presented and analyzed. The notion of plasticity kernel is introduced as a key component of plastic neural networks models, generalizing a notion used for pair-based models. We show that a large number of STDP rules from neuroscience and physics can be represented by this formalism. Several aspects of these models are discussed and compared to canonical models of computational neuroscience. An important sub-class of plasticity kernels with a Markovian formulation is also defined and investigated. In these models, the time evolution of cellular processes such as the neuronal membrane potential and the concentrations of chemical components created/suppressed by spiking activity has the Markov property.

This article has been removed by arXiv administrators because the submitter did not have the rights to agree to the license at the time of submission, Comment: This article has been removed by arXiv administrators because the submitter did not have the rights to agree to the license at the time of submission

In this paper we analyze the equilibrium properties of a large class of stochastic processes describing the fundamental biological process within bacterial cells, {\em the production process of proteins}. Stochastic models classically used in this context to describe the time evolution of the numbers of mRNAs and proteins are presented and discussed. An extension of these models, which includes elongation phases of mRNAs and proteins, is introduced. A convergence result to equilibrium for the process associated to the number of proteins and mRNAs is proved and a representation of this equilibrium as a functional of a Poisson process in an extended state space is obtained. Explicit expressions for the first two moments of the number of mRNAs and proteins at equilibrium are derived, generalizing some classical formulas. Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results. Several convergence results for the distribution of the number of proteins at equilibrium are in particular obtained under different scaling assumptions.

We introduce a probabilistic generative model for disentangling spatio-temporal disease trajectories from series of high-dimensional brain images. The model is based on spatio-temporal matrix factorization, where inference on the sources is constrained by anatomically plausible statistical priors. To model realistic trajectories, the temporal sources are defined as monotonic and time-reparametrized Gaussian Processes. To account for the non-stationarity of brain images, we model the spatial sources as sparse codes convolved at multiple scales. The method was tested on synthetic data favourably comparing with standard blind source separation approaches. The application on large-scale imaging data from a clinical study allows to disentangle differential temporal progression patterns mapping brain regions key to neurodegeneration, while revealing a disease-specific time scale associated to the clinical diagnosis.

Illumination estimation is often used in mixed reality to re-render a scene from another point of view, to change the color/texture of an object, or to insert a virtual object consistently lit into a real video or photograph. Specifically, the estimation of a point light source is required for the shadows cast by the inserted object to be consistent with the real scene. We tackle the problem of illumination retrieval given an RGBD image of the scene as an inverse problem: we aim to find the illumination that minimizes the photometric error between the rendered image and the observation. In particular we propose a novel differentiable renderer based on the Blinn-Phong model with cast shadows. We compare our differentiable renderer to state-of-the-art methods and demonstrate its robustness to an incorrect reflectance estimation.

The equilibrium properties of allocation algorithms for networks with a large number of nodes with finite capacity are investigated. Every node is receiving a flow of requests and when a request arrives at a saturated node, i.e. a node whose capacity is fully utilized, an allocation algorithm may attempt to re-allocate the request to a non-saturated node. For the algorithms considered, the re-allocation comes at a price: either an extra-capacity is required in the system or the processing time of a re-allocated request is increased. The paper analyzes the properties of the equilibrium points of the asymptotic associated dynamical system when the number of nodes gets large. At this occasion the classical model of {\em Gibbens, Hunt and Kelly} (1990) in this domain is revisited. The absence of known Lyapunov functions for the corresponding dynamical system complicates significantly the analysis. Several techniques are used: Analytic and scaling methods to identify the equilibrium points. We identify the subset of parameters for which the limiting stochastic model of these networks has multiple equilibrium points. Probabilistic approaches, like coupling, are used to prove the stability of some of them. A criterion of exponential stability with the spectral gap of the associated linear operator of equilibrium points is also obtained.

Alzheimer's disease (AD) is characterized by complex and largely unknown progression dynamics affecting the brain's morphology. Although the disease evolution spans decades, to date we cannot rely on long-term data to model the pathological progression, since most of the available measures are on a short-term scale. It is therefore difficult to understand and quantify the temporal progression patterns affecting the brain regions across the AD evolution. In this work, we tackle this problem by presenting a generative model based on probabilistic matrix factorization across temporal and spatial sources. The proposed method addresses the problem of disease progression modelling by introducing clinically-inspired statistical priors. To promote smoothness in time and model plausible pathological evolutions, the temporal sources are defined as monotonic and independent Gaussian Processes. We also estimate an individual time-shift parameter for each patient to automatically position him/her along the sources time-axis. To encode the spatial continuity of the brain sub-structures, the spatial sources are modeled as Gaussian random fields. We test our algorithm on grey matter maps extracted from brain structural images. The experiments highlight differential temporal progression patterns mapping brain regions key to the AD pathology, and reveal a disease-specific time scale associated with the decline of volumetric biomarkers across clinical stages.

The joint analysis of biomedical data in Alzheimer's Disease (AD) is important for better clinical diagnosis and to understand the relationship between biomarkers. However, jointly accounting for heterogeneous measures poses important challenges related to the modeling of the variability and the interpretability of the results. These issues are here addressed by proposing a novel multi-channel stochastic generative model. We assume that a latent variable generates the data observed through different channels (e.g., clinical scores, imaging, ...) and describe an efficient way to estimate jointly the distribution of both latent variable and data generative process. Experiments on synthetic data show that the multi-channel formulation allows superior data reconstruction as opposed to the single channel one. Moreover, the derived lower bound of the model evidence represents a promising model selection criterion. Experiments on AD data show that the model parameters can be used for unsupervised patient stratification and for the joint interpretation of the heterogeneous observations. Because of its general and flexible formulation, we believe that the proposed method can find important applications as a general data fusion technique., Comment: accepted for presentation at MLCN 2018 workshop, in Conjunction with MICCAI 2018, September 20, Granada, Spain

The stochastic models investigated in this paper describe the evolution of a set of $F_N$ identical balls scattered into $N$ urns connected by an underlying symmetrical graph with constant degree $h_N$. After some random amount of time {\em all the balls} of any urn are redistributed locally, among the $h_N$ urns of its neighborhood. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The main original features of this context is that the cardinality $h_N$ of the range of interaction is not necessarily linear with respect to $N$ as in a classical mean-field context and, also, that the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. The approach relies on the analysis of the evolution of the local empirical distributions associated to the state of urns located in the neighborhood of a given urn. Under convenient conditions, by taking an appropriate Wasserstein distance and by establishing several technical estimates for local empirical distributions, we are able to prove mean-field convergence results. When the load per node goes to infinity, a convergence result for the invariant distribution of the associated McKean-Vlasov process is obtained for several allocation policies. For the class of power of $d$ choices policies, we show that the associated invariant measure has an asymptotic finite support property under this regime. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of $d$ choices policies.

In this paper, we investigate a stochastic model describing the time evolution of a polymerization process. A polymer is a macro-molecule resulting from the aggregation of several elementary sub-units called monomers. Polymers can grow by addition of monomers or can be split into several polymers. The initial state of the system consists mainly of monomers. We study the time evolution of the mass of polymers, in particular the asymptotic distribution of the first instant when the fraction of monomers used in polymers is above some positive threshold $\delta$. A scaling approach is used by taking the mass $N$ as a scaling parameter. The mathematical model used in this paper includes {\em a nucleation property}: If $n_c$ is defined as the size of the nucleus, polymers with a size less than $n_c$ are quickly fragmented into smaller polymers, at a rate proportional to $\Phi(N)$ for some non-decreasing and unbounded function $\Phi$. For polymers of size greater than $n_c$, fragmentation still occurs but at bounded rates. If $T^N$ is the instant of creation of the first polymer whose size is $n_c$, it is shown that, under appropriate conditions, the variable $T^N{/}[\Phi(N)^{n_c{-}2}{/}N]$ converges in distribution, and that the first instant $L_\delta^N$ when a fraction $\delta$ of monomers is polymerized has the same order of magnitude. An original feature proved for this model is the significant variability of the variable $T^N$. This is a well known phenomenon observed in biological experiments but few mathematical models of the literature have this property. The results are proved via a series of technical estimates for occupation measures of some functionals of the corresponding Markov processes on fast time scales and by using coupling techniques.

In this article, we quantitatively study, through stochastic models, the efects of several intracellular phenomena, such as cell volume growth, cell division, gene replication as well as fuctuations of available RNA polymerases and ribosomes. These phenomena are indeed rarely considered in classic models of protein production and no relative quantitative comparison among them has been performed. The parameters for a large and representative class of proteins are determined using experimental measures. The main important and surprising conclusion of our study is to show that despite the signifcant fuctuations of free RNA polymerases and free ribosomes, they bring little variability to protein production contrary to what has been previously proposed in the literature. After verifying the robustness of this quite counter-intuitive result, we discuss its possible origin from a theoretical view, and interpret it as the result of a mean-feld efect.

In this paper a stochastic model of a call center with a two-level architecture is analyzed. A first-level pool of operators answers calls, identifies, and handles non-urgent calls. A call classified as urgent has to be transferred to specialized operators at the second level. When the operators of the second level are all busy, the operator of first level handling the urgent call is blocked until an operator at the second level is available. Under a scaling assumption, the evolution of the number of urgent calls blocked at level~$1$ is investigated. It is shown that if the ratio of the number of operators at level $2$ and~$1$ is greater than some threshold, then, essentially, the system operates without congestion, with probability close to $1$, no urgent call is blocked after some finite time. Otherwise, we prove that a positive fraction of the operators of the first level are blocked due to the congestion of the second level. Stochastic calculus with Poisson processes, coupling arguments and formulations in terms of Skorokhod problems are the main mathematical tools to establish these convergence results.

Distributed storage systems such as Hadoop File System or Google File System (GFS) ensure data availability and durability using replication. This paper is focused on the analysis of the efficiency of replication mechanism that determines the location of the copies of a given file at some server. The variability of the loads of the nodes of the network is investigated for several policies. Three replication mechanisms are tested against simulations in the context of a real implementation of a such a system: Random, Least Loaded and Power of Choice. The simulations show that some of these policies may lead to quite unbalanced situations: if $\beta$ is the average number of copies per node it turns out that, at equilibrium, the load of the nodes may exhibit a high variability. It is shown in this paper that a simple variant of a power of choice type algorithm has a striking effect on the loads of the nodes: at equilibrium, the distribution of the load of a node has a bounded support, most of nodes have a load less than $2\beta$ which is an interesting property for the design of the storage space of these systems. Mathematical models are introduced and investigated to explain this interesting phenomenon. The analysis of these systems turns out to be quite complicated mainly because of the large dimensionality of the state spaces involved. Our study relies on probabilistic methods, mean-field analysis, to analyze the asymptotic behavior of an arbitrary node of the network when the total number of nodes gets large. An additional ingredient is the use of stochastic calculus with marked Poisson point processes to establish some of our results.

The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has $x$ requests to transmit, then it receives a fraction of the capacity proportional to $\log(1{+}L)$, the logarithm of its current load $L$. A stochastic model of such an algorithm is investigated in the case of the star network, in which $J$ nodes can transmit simultaneously, but interfere with a central node $0$ in such a way that node $0$ cannot transmit while one of the other nodes does. One studies the impact of the log policy on these $J+1$ interacting communication nodes. A fluid scaling analysis of the network is derived with the scaling parameter $N$ being the norm of the initial state. It is shown that the asymptotic fluid behaviour of the system is a consequence of the evolution of the state of the network on a specific time scale $(N^t,\, t{\in}(0,1))$. The main result is that, on this time scale and under appropriate conditions, the state of a node with index $j\geq 1$ is of the order of $N^{a_j(t)}$, with $0{\leq}a_j(t){<}1$, where $t\mapsto a_j(t)$ is a piecewise linear function. Convergence results on the fluid time scale and a stability property are derived as a consequence of this study.

The problem of reliability of a large distributed system is analyzed via a new mathematical model. A typical framework is a system where a set of files are duplicated on several data servers. When one of these servers breaks down, all copies of files stored on it are lost. In this way, repeated failures may lead to losses of files. The efficiency of such a network is directly related to the performances of the mechanism used to duplicate files on servers. In this paper we study the evolution of the network using a natural duplication policy giving priority to the files with the least number of copies. We investigate the asymptotic behavior of the network when the number $N$ of servers is large. The analysis is complicated by the large dimension of the state space of the empirical distribution of the state of the network. A stochastic model of the evolution of the network which has values in state space whose dimension does not depend on $N$ is introduced. Despite this description does not have the Markov property, it turns out that it is converging in distribution, when the number of nodes goes to infinity, to a nonlinear Markov process. The rate of decay of the network, which is the key characteristic of interest of these systems, can be expressed in terms of this asymptotic process. The corresponding mean-field convergence results are established. A lower bound on the exponential decay, with respect to time, of the fraction of the number of initial files with at least one copy is obtained., Comment: To appear in the "Annals of Applied Probability"

We consider a server with large capacity delivering video files encoded in various resolutions. We assume that the system is under saturation in the sense that the total demand exceeds the server capacity $C$. In such a case, requests may be rejected. For the policies considered in this paper, instead of rejecting a video request, it is downgraded. When the occupancy of the server is above some value $C_0{<}C$, the server delivers the video at a minimal bit rate. The quantity $C_0$ is the bit rate adaptation threshold. For these policies, request blocking is thus replaced by bit rate adaptation. Under the assumptions of Poisson request arrivals and exponential service times, we show that, by rescaling the system, a process associated to the occupancy of the server converges to some limiting process whose invariant distribution is computed by a Wiener-Hopf technique. This allows us to derive an asymptotic expression of the key performance measure of such a policy, the equilibrium probability that a request is transmitted without downgrading. Numerical applications of these results are presented.

Self-assembly of proteins is a biological phenomenon which gives rise to spontaneous formation of amyloid fibrils or polymers. The starting point of this phase, called nucleation exhibits an important variability among replicated experiments.To analyse the stochastic nature of this phenomenon, one of the simplest models considers two populations of chemical components: monomers and polymerised monomers. Initially there are only monomers. There are two reactions for the polymerization of a monomer: either two monomers collide to combine into two polymerised monomers or a monomer is polymerised after the encounter of a polymerised monomer. It turns out that this simple model does not explain completely the variability observed in the experiments. This paper investigates extensions of this model to take into account other mechanisms of the polymerization process that may have impact an impact on fluctuations.The first variant consists in introducing a preliminary conformation step to take into account the biological fact that, before being polymerised, a monomer has two states, regular or misfolded. Only misfolded monomers can be polymerised so that the fluctuations of the number of misfolded monomers can be also a source of variability of the number of polymerised monomers. The second variant represents the reaction rate $\alpha$ of spontaneous formation of a polymer as of the order of $N^{-\nu}$, with $\nu$ some positive constant. First and second order results for the starting instant of nucleation are derived from these limit theorems. The proofs of the results rely on a study of a stochastic averaging principle for a model related to an Ehrenfest urn model, and also on a scaling analysis of a population model.