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

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Abstract

Most recent models of the immune network are based upon a phenomenological log bell-shaped interaction function. This function depends on a single parameter, the “field”, which is the sum of all ligand concentrations weighted by their respective affinities. The typical behavior of these models is dominated by percolation, a phenomenon in which a local stimulus spreads globally throughout the network.

The usual reason for employing a log bell-shaped interaction function is that B cells are activated by cross-linking of their surface immunoglobulin receptors. Here we formally derive a new phenomenological log bell-shaped function from the chemistry of receptor cross-linking by bivalent ligand. Specifying how this new function depends on the ligand concentrations requires two fields: a binding field and a cross-linking field.

When we compare the activation functions for ligand-receptor pairs with different affinities, the one-field and the two-field functions differ markedly. In the case of the one-field activation function, its graph is shifted to increasingly higher concentration as the affinity decreases but keeps its width and height. In the case of the two-field activation function, the graph of a low-affinity interaction is nested within the graphs of all higher-affinity interactions.

We show that this difference in the relations among activation functions for different affinities radically changes the network behavior. In models that described B cell proliferation using the one-field activation function, network behavior was dominated by low-affinity interactions. Conversely, in our new model, the high-affinity interactions are the most significant. As a consequence, percolation is no longer the only typical network behavior.

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Authors and Affiliations

  1. Theoretical Biology, Utrecht University, Padualaan 8, 3584 CH, Utrecht, The Netherlands

    Rob J. de Boer

  2. Department of Zoology, Oxford University, S. Parks Road, OX1 3PS, Oxford, UK

    Maarten C. Boerlijst

  3. Theoretical Division, MS-K710, Los Alamos National Laboratory, 87545, Los Alamos, NM, USA

    Bernhard Sulzer & Alan S. Perelson

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  1. Rob J. de Boer
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  2. Maarten C. Boerlijst
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  3. Bernhard Sulzer
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  4. Alan S. Perelson
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de Boer, R.J., Boerlijst, M.C., Sulzer, B. et al. A new bell-shaped function for idiotypic interactions based on cross-linking. Bltn Mathcal Biology 58, 285–312 (1996). https://doi.org/10.1007/BF02458310

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