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/suppresed’ (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-asymmetry 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’.