Dynamical replica analysis of processes on finitely connected random graphs: II. Dynamics in the Griffiths phase of the diluted Ising ferromagnet

Mozeika, Alexander; Coolen, A. C. C.

doi:10.1088/1751-8113/42/19/195006

Cited by 20 publications

(42 citation statements)

References 50 publications

(174 reference statements)

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“…In order to model the process of affinity maturation, a further important ingredient of the adaptive immune system which has not been included yet, one can also assume that, in addition to B clones and T clones, also the interactions J µ evolve in time, and use the slow (or fast) variable assumptions to compute stationary distributions of the more complicated process. Furthermore, the assumption of separated time-scales can be relaxed, but to make progress in this purely dynamical scenario would require application of sophisticated analytical tools such as the (dynamical) path integral method [46] or the dynamical replica theory [47]. However, the most important next step would be to connect the theoretical framework developed in this article with concrete experimental data.…”

Section: Discussionmentioning

confidence: 99%

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Statistical mechanics of clonal expansion in lymphocyte networks modelled with slow and fast variables

Mozeika

Coolen²

2016

J. Phys. A: Math. Theor.

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We use statistical mechanical techniques to model the adaptive immune system, represented by lymphocyte networks in which B cells interact with T cells and antigen. We assume that B-and T-clones evolve in different thermal noise environments and on different timescales, and derive stationary distributions and study expansion of B clones for the case where these timescales are adiabatically separated. We compute characteristics of B-clone sizes, such as average concentrations, in parameter regimes where T-clone sizes are modelled as binary variables. This analysis is independent of the network topology, and its results are qualitatively consistent with experimental observations. To obtain the full distributions of B-clone sizes we assume further that the network topologies are random and locally equivalent to trees. This allows us to compete these distributions via the Bethe-Peierls approach. As an example we calculate B-clone distributions for immune models defined on random regular networks.Statistical mechanics of lymphocyte networks modelled with slow and fast variables 2 However, a BCR recognises only a specific part of the Ag, which is called epitope, so potentially a single BCR can recognise many different Ags [3,4]. An Ag that binds to the BCR is subsequently internalised and is broken by the cell into peptides (short proteins) which are then displayed on its surface. The T cell also has its T cell receptor (TCR) to recognise Ag, however, in contrast to B cells, a T cell binds to the peptides on the surface of antigen presenting cells (APCs), such as dendritic cells, B cells, etc. The adaptive immune response is triggered when T helper cells that are attached by their TCRs to B cells, activate these B cells (by cytokine-mediated signals) that recognise the same Ag. The strength of this interaction between the BCR (TCR) and the Ag is called affinity. Upon activation, the B cells with highest affinity begin to proliferate in a process known as clonal expansion. A group of B cells (or T cells) with the same BCR (or TCR) is called a clone. While B cell clones (B clones) are expanding the T cells, activated by Ag, they are also going through the process of clonal expansion. We note that the mode of B cell activation described here is called "T-dependent B cell activation", however for some "simple" Ags with repetitive patterns, such as polysaccharides, the B cell can also be activated directly by Ag and without the help from a T cell ("T-independent activation").Some proliferating B cells differentiate into plasma cells, which secrete large quantities of antibodies (BCRs in soluble form). The antibodies (Ab) protect the host from infection by binding to the pathogen and thus, by blocking its external parts, make the attack ineffective or "mark" the pathogen for ingestion by other host cells. However, the Abs produced in this initial clonal expansion are of very low affinity, especially if the host is infected with this pathogen for the first time. The other proliferating B cells are used to seed the germinal ce...

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Section: Discussionmentioning

confidence: 99%

“…(h)h, where P (h) is the RHS of the equation (51) with L = L + 1. The latter result is obtained by comparing the equations (B.10) and(47).Let us first consider the case of fast B-clone equilibration. This regime can be studied using equations (51), (52) and(29), with n = 1.…”

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confidence: 97%

Statistical mechanics of clonal expansion in lymphocyte networks modelled with slow and fast variables

Mozeika

Coolen²

2016

J. Phys. A: Math. Theor.

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“…We will show how the abundance of short loops can then be handled by a graph inflation technique such that the expectation values can be computed by ordinary BP on an auxiliary locally tree-like graph (the expanded graph). Details are and comparisons to the approach taken in [20,21,22,23] are given in Appendices A-C.…”

Section: Perturbative Large Deviations For Non-equilibrium Dynamicsmentioning

confidence: 99%

“…The paper is organized as follows: in Section 2 we introduce the general approach, and in Section 3 we show how it can in principle be used to set up a perturbation theory. In Sections 4 and 5 we analyze the model problem of a relaxing Ising chain, first on the level of approximation of a "two-parameter theory", and then on the level of the "joint spin-field theory" a higher-order approximation developed in [19,20,21,22,23]. We here show that the accuracy of each level of approximation can be assessed internally, where the largest inaccuracies of the "two-parameter theory" are taken care of in the "joint spin-field theory", but where higher-order inaccuracies would need a higher-order approximation.…”

Section: Introductionmentioning

confidence: 99%

Perturbative Large Deviation Analysis of Non-Equilibrium Dynamics

Ferraro

Aurell

2014

J. Phys. Soc. Jpn.

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Macroscopic fluctuation theory has shown that a wide class of non-equilibrium stochastic dynamical systems obey a large deviation principle, but except for a few one-dimensional examples these large deviation principles are in general not known in closed form. We consider the problem of constructing successive approximations to an (unknown) large deviation functional and show that the non-equilibrium probability distribution the takes a Gibbs-Boltzmann form with a set of auxiliary (non-physical) energy functions. The expectation values of these auxiliary energy functions and their conjugate quantities satisfy a closed system of equations which can imply a considerable reduction of dimensionality of the dynamics. We show that the accuracy of the approximations can be tested self-consistently without solving the full nonequilibrium equations. We test the general procedure on the simple model problem of a relaxing 1D Ising chain.

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“…Similarly, in dynamical studies based on generating functional analysis [20][21][22] we would be required to evaluate…”

Section: Tailored Random Graph Ensemblesmentioning

confidence: 99%

Random graph ensembles with many short loops

Roberts

Coolen

2014

ESAIM: Proc.

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Networks observed in the real world often have many short loops. This violates the tree-like assumption that underpins the majority of random graph models and most of the methods used for their analysis. In this paper we sketch possible research routes to be explored in order to make progress on networks with many short loops, involving old and new random graph models and ideas for novel mathematical methods. We do not present conclusive solutions of problems, but aim to encourage and stimulate new activity and in what we believe to be an important but under-exposed area of research. We discuss in more detail the Strauss model, which can be seen as the 'harmonic oscillator' of 'loopy' random graphs, and a recent exactly solvable immunological model that involves random graphs with extensively many cliques and short loops.Résumé. Les réseaux observés dans la Nature ont souvent des cycles courts. Ceci contredit le postulat de hiérarchie sur lequel se base la majorité des modèles de réseaux aléatoires et la plupart des méthodes utilisées pour leur analyse. Dans cet article, nous esquissons des directions de recherches possibles, afin de progresser sur les réseaux contenant beaucoup de cycles courts, faisant appel à des modèles de réseaux aléatoires éprouvés ou nouveaux, et des idées pour de nouvelles méthodes mathématiques. Nous ne présentons pas de solutions définitives, mais notre but est d'encourager et de stimuler de nouveaux travaux dans ce que nous croyons être une direction de recherche importante, bien que insuffisamment explorée. Nous discutons en détail le modèle de Strauss, qui peut être interprété comme 'l'oscillateur harmonique' des réseaux aléatoires 'Ãȃ boucles', ainsi qu'un modèle immunologique soluble exactement qui implique des réseaux aléatoires avec de nombreux cliques et cycles courts.

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Dynamical replica analysis of processes on finitely connected random graphs: II. Dynamics in the Griffiths phase of the diluted Ising ferromagnet

Cited by 20 publications

References 50 publications

Statistical mechanics of clonal expansion in lymphocyte networks modelled with slow and fast variables

Statistical mechanics of clonal expansion in lymphocyte networks modelled with slow and fast variables

Perturbative Large Deviation Analysis of Non-Equilibrium Dynamics

Random graph ensembles with many short loops

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