PATTERN-FORMATION IN ONE-DIMENSIONAL AND 2-DIMENSIONAL SHAPE-SPACE MODELS OF THE IMMUNE-SYSTEM

A large-scale model of the immune network is analyzed, using the shape-space formalism. In this formalism, it is assumed that the immunoglobulin receptors on B cells can be characterized by their unique portions, or idiotypes, that have shapes that can be represented in a space of a small finite dimension. Two receptors are assumed to interact to the extent that the shapes of their idiotypes are complementary. This is modeled by assuming that shapes interact maximally whenever their co-ordinates in the space-space are equal and opposite, and that the strength of interaction falls off for less complementary shapes in a manner described by a Gaussian function of the Euclidean "distance" between the pair of interacting shapes. The degree of stimulation of a cell when confronted with complementary idiotypes is modeled using a log bell-shaped interaction function. This leads to three possible equilibrium states for each clone: a virgin, an immune, and a suppressed state. The stability properties of the three possible homogeneous steady states of the network are examined. For the parameters chosen, the homogeneous virgin state is stable to both uniform and sinusoidal perturbations of small amplitude. A sufficiently large perturbation will, however, destabilize the virgin state and lead to an immune reaction. Thus, the virgin system is both stable and responsive to perturbations. The homogeneous immune state is unstable to both uniform and sinusoidal perturbations, whereas the homogeneous suppressed state is stable to uniform, but unstable to sinusoidal, perturbations. The non-uniform patterns that arise from perturbations of the homogeneous states are examined numerically. These patterns represent the actual immune repertoire of an animal, according to the present model. The effect of varying the standard deviation σ of the Gaussian is numerically analyzed in a one-dimensional model. If σ is large compared to the size of the shape-space, the system attains a fixed non-uniform equilibrium. Conversely if σ is small, the system attains one out of many possible non-uniform equilibria, with the final pattern depending on the initial conditions. This demonstrates the plasticity of the immune repertoire in this shape-space model. We describe how the repertoire organizes itself into large clusters of clones having similar behavior. These results are extended by analyzing pattern formation in a two-dimensional (2-D) shape-space. A lattice mapping is employed, whose rules are rigorously derived from a simplified version of the underlying differential equations via a logarithmic transformation of variables. A novel feature of the lattice model is that the neighborhood of cell (i, j) is centered around cell (-j, -j). Thus, interactions are non-local. The 2-D patterns that emerge are reminiscent of those found in reaction-diffusion systems, and contain many hills and valleys. (In contrast with most reaction-diffusion models, pattern formation in this model is not dependent on long-range inhibition and short-range activation.) The scale of the pattern depends on neighborhood size, with small neighborhoods generating fine scale patterns with narrow peaks, and large neighborhoods generating large scale patterns with wide peaks and valleys. Both one- and two-dimensional models support patterns in which a fraction of the clones are not stimulated by network interactions. The fraction of such "disconnected clones" increases with both dimensionality and σ. © 1992 Academic Press Limited.