Pattern Formation in One- and Two-Dimensional Shape Space Models of the Immune System

We analyze a large-scale model of the immune network 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, which have shapes that can be represented in a space of a small finite dimension. Two receptors are assumed to interact only when the shapes of their idiotypes are complementary. We model this by using a Euclidean metric and assuming that shapes interact whenever their coordinates in the space-space are of opposite sign. The range of the interaction is defined by a standard Gaussian function with variance sigma squared. 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. We study the stability properties of the three possible homogeneous steady states of the network. 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 responsible 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. Using numerical methods we study the non-uniform patterns that arise from perturbations of the homogeneous states. Identifying with the in vivo stiuation, these patterns represent the actual immune repertoire of an animal. In contrast with most reaction-diffusion models, pattern formation in this model is not dependent on long-range inhibition and short-range activation. Embedding our model in a one-dimensional shape-space, we numerically analyze the system upon varying the parameter sigma. 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 our 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. We extend these results by analyzing pattern formation in a two-dimensional (2D) shape-space. To do this we use a lattice mapping whose rules are regorously 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 call $(i, j)$ is centered around cell $(-i, -j)$. Thus interactions are nonlocal. The 2D patterns that we find are reminiscent of those found in reaction-diffusion systems with many hills and valleys. 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 inteactions. The fration of such "disconnected clones" increases with both dimensionality and sigma.