A Large Scale Dynamical System Immune Network Model with Finite Connectivity

Abstract

We study a model of an idiotypic immune network which was introduced by N. K. Jerne. It is known that in immune systems there generally exist several kinds of immune cells which can recognize any particular antigen. Taking this fact into account and assuming that each cell interacts with only a finite number of other cells, we analyze a large scale immune network via both numerical simulations and statistical mechanical methods, and show that the distribution of the concentrations of antibodies becomes non-trivial for a range of values of the strength of the interaction and the connectivity. §1. IntroductionThere are many numerical and theoretical studies of biological networks in which there is a global coupling between the constituent elements, e.g. Hopfield-type neural networks and networks of Kuramoto-type phase oscillators. 1), 2) In contrast, relatively few studies have been carried out of networks where each component interacts with only a small number of randomly selected other components. One such system is the immune network, introduced by Jerne 3) to explain the activation of immune cells in the absence of external stimulation.Let us briefly summarize the main mechanism of cell-interaction in the immune system. Its main constituents are B-lymphocytes (B-cells), T-lymphocytes (T-Cells) and antibodies produced by B-cells. B-cells and T-cells have receptors on their surfaces. The receptors of B-cells are antibodies, which recognize and connect to antigens in order to neutralize them; they have specific 3-dimensional structures which are called 'idiotypes'. A family of B-cells which are generated from a given B-cell is called a 'clone'. Hence, all members of a clone as well as the antibodies produced by this clone have the same idiotype. In general, each antibody could present several 3-dimensional structures which can be recognized by other B-cells. This then generates, indirectly, an effective interaction between antibodies. This paper is organized as follows. In §2 we formulate the model, followed by a summary of previous studies in §3. In §4, we present the results of our present study. Section 5 is devoted to a summary and a discussion.

Immune networks: multi-tasking capabilities at medium load

“…Given the assumed scaling of N B , we get This quantity can be interpreted as the average link probability in G. The average degreez over the whole set of nodes 10 can then be written as…”

“…We evaluate the typical environment for node i by averaging P ( |k i , k j , N B ) over P T (k j , c, N B ), as given by (10), and get…”

“…After a long period of dormancy since the pionering paper [1], we have in recent years seen a renewed interest in statistical mechanical models of the immune system [2,3,4,5,6,7,8,9,10]. These complement the standard approaches to immune system modelling, which are formulated in terms of dynamical systems [11,12,13,14].…”

Retrieving infinite numbers of patterns in a spin-glass model of immune networks

“…A consequence of its underlying topological sparsity is that the adaptive immune system exhibits only weak ergodicity breaking, so that also spontaneous switch-like effects as bi-stabilities are present: the latter may play a significant role in the maintenance of immune homeostasis.Beyond the so-far-classical approaches by Cohen, DeBoer, May, Nowak and Perelson (see e.g. [1,2,3,4,5]) that paved the main route for mathematical modelling in Immunology, and after a pioneering early paper by Parisi [6] followed by about two decades of dormancy, there is now increasing interest in statistical mechanical approaches to modeling the immune system [7,8,15,13,14,9,10,11,12,16]. This interest is stimulated in part by the potential of new quantitative methods for the study of systems with complex network topologies [18,19,20,21,17].…”

“…Beyond the so-far-classical approaches by Cohen, de Boer, May, Nowak and Perelson (see, e.g., [1][2][3][4][5]) that paved the main route for mathematical modelling in immunology, and after a pioneering early paper by Parisi [6] followed by about two decades of dormancy, there is now increasing interest in statistical mechanical approaches to model the immune system [7][8][9][10][11][12][13][14][15][16]. This interest is stimulated in part by the potential of new quantitative methods for the study of systems with complex network topologies [17][18][19][20][21].…”