dorsal/arxiv
View SchemaThe Architecture of Idiotypic Networks: Percolation and Scaling Behaviour
| Authors | Markus Brede, Ulrich Behn |
|---|---|
| Categories | |
| ArXiv ID | physics/0006015 |
| URL | https://arxiv.org/abs/physics/0006015 |
| DOI | 10.1103/PhysRevE.64.011908 |
Abstract
We investigate a model where idiotypes (characterizing B-lymphocytes and antibodies of an immune system) and anti-idiotypes are represented by complementary bitstrings of a given length d allowing for a number of mismatches (matching rules). In this model, the vertices of the hypercube in dimension d represent the potential repertoire of idiotypes. A random set of (with probability p) occupied vertices corresponds to the expressed repertoire of idiotypes at a given moment. Vertices of this set linked by the above matching rules build random clusters. We give a structural and statistical characterisation of these clusters - or in other words - of the architecture of the idiotypic network. Increasing the probability p one finds at a critical p a percolation transition where for the first time a large connected graph occures with probability one. Increasing p further, there is a second transition above which the repertoire is complete in the sense that any newly introduced idiotype finds a complementary anti-idiotype. We introduce structural characteristics such as the mass distributions and the fragmentation rate for random clusters, and determine the scaling behaviour of the cluster size distribution near the percolation transition, including finite size corrections. We find that slightly above the percolation transition the large connected cluster (the central part of the idiotypic network) consists typically of one highly connected part and a number of weakly connected constituents and coexists with a number of small, isolated clusters. This is in accordance with the picture of a central and a peripheral part of the idiotypic network and gives some support to idealized architectures of the central part used in recent dynamical mean field models.
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"abstract": "We investigate a model where idiotypes (characterizing B-lymphocytes and\nantibodies of an immune system) and anti-idiotypes are represented by\ncomplementary bitstrings of a given length d allowing for a number of\nmismatches (matching rules). In this model, the vertices of the hypercube in\ndimension d represent the potential repertoire of idiotypes. A random set of\n(with probability p) occupied vertices corresponds to the expressed repertoire\nof idiotypes at a given moment. Vertices of this set linked by the above\nmatching rules build random clusters. We give a structural and statistical\ncharacterisation of these clusters - or in other words - of the architecture of\nthe idiotypic network. Increasing the probability p one finds at a critical p a\npercolation transition where for the first time a large connected graph occures\nwith probability one. Increasing p further, there is a second transition above\nwhich the repertoire is complete in the sense that any newly introduced\nidiotype finds a complementary anti-idiotype. We introduce structural\ncharacteristics such as the mass distributions and the fragmentation rate for\nrandom clusters, and determine the scaling behaviour of the cluster size\ndistribution near the percolation transition, including finite size\ncorrections. We find that slightly above the percolation transition the large\nconnected cluster (the central part of the idiotypic network) consists\ntypically of one highly connected part and a number of weakly connected\nconstituents and coexists with a number of small, isolated clusters. This is in\naccordance with the picture of a central and a peripheral part of the idiotypic\nnetwork and gives some support to idealized architectures of the central part\nused in recent dynamical mean field models.",
"arxiv_id": "physics/0006015",
"authors": [
"Markus Brede",
"Ulrich Behn"
],
"categories": [
"physics.bio-ph"
],
"doi": "10.1103/PhysRevE.64.011908",
"title": "The Architecture of Idiotypic Networks: Percolation and Scaling Behaviour",
"url": "https://arxiv.org/abs/physics/0006015"
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