Shape space coverage with distance-dependent matching

The concept of a shape space was introduced by Perelson and Oster (1979). Since then, it has been used in numerous theoretical studies of the immune system, of which I will only mention a few (). In this framework, it is postulated that molecular interactions can be understood in terms of the "shape" of the molecules. The crucial assumption of this model is that the "shape" of a molecule can be represented by a vector of discrete values, from a finite, generally small, alphabet. Rules are specified for determining the "affinity" between two such "shapes". There have been attempts to relate this model to measurements that can be obtained in biological systems As (). I therefore decided to use this conceptualization for my study on the evolution of antibody gene libraries.

It is generally assumed that the number of pathogens in the environment of a species is very large. Indeed, if this number was small, the immune system would be able to distribute its resources, such as antibody molecules, among these pathogens. Each of the pathogens would raise an effective immune response. This is clearly not what we observe in reality. Therefore, I will assume that the pathogen universe is large. One has to keep in mind, though, that the failure of the immune system to cope with all the pathogenic challenges that it encounters may be due to other factors. Pathogen evolution sets a moving target for the immune system. As I will show in the following section, the rate at which the antibody library adapts to an evolving pathogenic environment might be too slow for the immune system to ever pin down even a small pathogen set.

Let us assume that the number of antibody shapes encoded in the genome is considerably smaller than the number of antigen shapes that the organism encounters during its life time. To understand the role of the antibody gene libraries in the generation of the immune repertoire, I will address the following questions:

• How does the survival probability of the organism scale with the size of its immune receptor repertoire?
• What structure do antibody libraries evolve in different types of pathogenic environments?
• Can an antibody repertoire that has been selected for interaction with pathogens perform equally well in the interaction with non-pathogenic antigens?