Pattern formation in one- and two-dimensional shape-space models of the immune system

Abstract

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 coordinates 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 sigma of the Gaussian is numerically analyzed in a one-dimensional model. If sigma is large compared to the size of the shape-space, the system attains a fixed non-uniform equilibrium. Conversely if sigma 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.