Pattern formation in B-cell immune networks: Domains and dots in shape-space

Abstract

The immune system contains many types of B-cells, which can activate each other if the shapes and surface properties of their receptors (or antibodies) match well. The dynamics of the resulting network is analysed using a recently derived B-cell activation function which captures the effects of the binding and crosslinking of B-cell receptors. All receptor/antibody shapes are parametrised by a continuous 'shape-space', such that matching pairs of shapes interact locally.The model produces a variety of activation patterns across shape-space for a wide range of parameters. The spatio-temporal structures differ qualitatively from those seen with a previously used type of activation function. In either case, the pattern formation can largely be understood analytically, by first solving exactly for the various uniform fixed solutions, and then computing the evolution of spatially modulated perturbations. For the more realistic activation function, the following scenario is found.Most (random) initial conditions first lead to the formation of coarse domains, of three possible types: the 'virgin'-(V) state, the 'immune/suppressed' (I/S)-state, and its reverse (S/I). V-domains are stable, but the other two types are unstable to spatial perturbations with a wavelength which is of the order of the interaction range. In the second stage, this instability causes big I/S-and S/I-domains to split up into arrays of small 'dots' which preserve the I/S-asyn~netry of their parent domain. These dots are stable, even in isolation, which allows them to act as a 'memory' for previously encountered antigens.No stable dots are obtained when the model is made to emulate the simpler activation function which has been used widely in earlier models. With this less realistic choice, unstable waves propagate from the boundaries of coarse I/S-domains, eventually filling up most of shape-space. This instability was previously described as 'percolation'.

Clonal selection based Artificial Immune System for generalized pattern recognition

“…An expression has been derived for the population dynamics of the B-cells that incorporates all of these characteristics [34]. The population of the B-cells is governed by the differential equation:…”

“…which is a modification of the original dynamics equation [34] that restores a term for the direct affinity of the cell for the incoming antigen [35]. In this expression, b i is the number of cells of clone i, s is the rate of influx of newly generated cells, p is the maximal growth rate, θ is the growth clone-size threshold, f(h i , h i ') is the cell activation function, K iA is the affinity that the cell i has for the incoming pattern, and d is the death rate.…”

“…In all experiments, b i , the number of cells of clone i, was initially set to 100 for all i, the influx rate s was set to 0.0001, the maximal growth rate p was set to 8, the growth clone-size threshold θ was set to 75, the activation w was set to 0.1, the weight σ in the affinity function (5) was set to 5.0, the exponent η in the cross-linking term (4) was set to 1.0, and the death rate d was set to 1. These settings are essentially the same ones as those used in the original study [34] and based on numerous studies, the system does not seem to be sensitive to their exact values. All updates of (1) were done using a Euler rule with a time step of 1.0, the death decision for any population i was set to 50.0, and a maximal number of generations for each B-cell clone set to 5000.…”