Size and connectivity as emergent properties of a developing immune network

Abstract

The development of the immune repertoire during neonatal life involves a strong selection process among different clones. The immune system is genetically capable of producing a much more diverse set of lymphocyte receptors than are expressed in the actual repertoire. By means of a model we investigate the hypothesis that repertoire selection is carried out during early life by the immune network. We develop a model network in which possibly hundreds of B cell clones proliferate and produce antibodies following stimulation. Stimulation is viewed as occurring through receptor crosslinking and is modeled via a log bell-shaped dose-response function. Through secretion of free antibody B cell clones can stimulate one another if their receptors have complementary shapes. Receptor shapes are modeled as binary strings and complementarity is evaluated by a string matching algorithm. The dynamic behavior of our model is typically oscillatory and for some parameters chaotic. In the case of two complementary B cell clones, the chaotic attractor has a number of features in common with the Lorenz attractor. The networks we model do not have a predetermined size or topology. Rather, we model the bone marrow as a source which generates novel clones. These novel clones can either be incorporated into the network or remain isolated, mimicking the non-network portion of the immune system. Clones are removed from the network if they fail to expand. We investigate the properties of the network as a function of P(match), the probability that two randomly selected immunoglobulins have complementary shapes. As the model networks evolve they develop a number of self-regulatory features. Most importantly, networks attain a specific equilibrium size and generate a characteristic amount of "natural" antibody. Because the network reaches an asymptotic size even though the bone marrow keeps supplying novel clones, clones must compete for presence in the network, i.e. repertoire selection takes place. Networks comprised of cells with multireactive receptors remain small, whereas networks consisting of cells with specific receptors become much larger. We find an inverse relationship between n, the number of clones in a network, and P(match), and a linear relationship between n and M, the rate at which novel clones are produced in the bone marrow. We present a simple phenomenological model for the number of clones in the network that accounts for the inverse relationship between n and P(match), and that can account for the relationship between n and M. Additionally, the phenomenological model suggests that there are two qualitatively different network equilibria.(ABSTRACT TRUNCATED AT 400 WORDS)

Immune network behavior—II. From oscillations to chaos and stationary states

“…2 before the global bifurcation involving the unstable manifold of the HH state at p = 3.7. Similar examples in the original coordinate system can be found in De Boer and Perelson (1991). In Fig.…”

“…Conversely, we observed in Part I that the interpretation of the nullclines of the SL coordinate system is not intuitive. Perelson, 1991). Varela (1989, 1990) have also searched for conditions for which oscillatory chaotic behavior could be sustained in the absence of a continuous bone marrow source.…”

Instability of Natural Antibody Repertoires in Systemic Lupus Erythematosus Patients, Revealed by Multiparametric Analysis of Serum Antibody Reactivities

“…The instability of SLE antibody repertoires, however, suggests some speculation. Previous evidence has indicated that circulating Igs contribute to the selection of the emerging B cell repertoires in the bone marrow [20][21][22], and to the level and repertoire of natural antibodies [23][24][25][26][27]. The surprising conservation of the size and repertoire of natural antibodies in normal individuals also suggests that they are recursively selected, the circulating Ig pool promoting its own dynamic maintenance.…”

A new bell-shaped function for idiotypic interactions based on cross-linking

“…Jerne's (1974) hypothesis that the immune system functions by means of a network of idiotypic interactions has been investigated extensively by theoretical biologists. Theoretical models that can account for memory, tolerance and repertoire selection (Hoffmann, 1980;De Boer and Hogeweg, 1989a;Weisbuch et al, 1990;De Boer and Perelson, 1991) have been developed and they have interesting self-organizing properties (De Boer and Perelson, 1991;De Boer et al, 1992aStewart and Varela, 1991).…”

“…Many recent idiotypic network models are based upon a phenomenological log bell-shaped function determining B cell activation as a function of the total concentration of ligand, or more precisely of the "field" (De Boer, 1988;De Boer and Hogeweg, 1989a, b;Weisbuch et al, 1990;De Boer and Perelson, 1991;De Boer et al, 1992a, b;De Boer et al, 1993a-c;Anderson et al, 1993). This log bell-shaped function is frequently interpreted as a phenomenological description of the fraction of B cell receptors crosslinked by ligand (Perelson, 1984).…”