Randomly evolving idiotypic networks: Modular mean field theory

Schmidtchen, Holger; Behn, Ulrich

doi:10.1103/physreve.86.011931

Cited by 3 publications

(13 citation statements)

References 29 publications

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“…Given these values, the influx probability p remains as main control parameter. In previous work (60, 61), we have studied a range for p from 0 to 0.1 and found that the architecture, which is of interest here evolves for p from 0.026 to 0.078. The results presented here explicitly are for p close to 0.078, where it is easier to initiate a reorganization of the pattern, but we have also studied a broader range of p .…”

Section: The Modelmentioning

confidence: 89%

“…The groups are built from clones of different idiotypes, which have an individual dynamics but share certain statistical properties. The building principles of these architectures can be described analytically (59, 60), and the statistical properties can be calculated within a mean-field theory in good agreement with simulations (61). …”

Section: Introductionmentioning

confidence: 87%

“…Whereas the preceding work by Brede and Behn (58), Schmidtchen and Behn (59), Schmidtchen et al (60), and Schmidtchen and Behn (61) considered the autonomous idiotypic network, i.e., the network of B-lymphocytes and their antibodies without foreign or self antigen, we investigate here the evolution of the idiotypic network, in the presence of self, toward an architecture where the expansion of autoreactive clones is controlled by idiotypic interactions. Self is modeled by permanently present idiotypes which influence the evolution of the network but are themselves not affected by idiotypic interactions.…”

Section: Introductionmentioning

confidence: 94%

“…They can be calculated (61) adopting the concept of mean-field theories, which was developed in statistical physics to describe phase transitions and has been transferred to many other problems in different fields. The main argument goes as follows.…”

Section: Mean-field Theorymentioning

confidence: 99%

“…The average occupation of the densely linked core groups is kept below 0.07, and the holes are almost empty, for details see Figure 8 in Ref. (61). Note that the singletons have no links to the connected part of the occupied network, which is built of the core and periphery groups.…”

Section: Idiotypic Network and Selfmentioning

confidence: 99%

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Self-Tolerance in a Minimal Model of the Idiotypic Network

2014

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We consider the problem of self-tolerance in the frame of a minimalistic model of the idiotypic network. A node of this network represents a population of B-lymphocytes of the same idiotype, which is encoded by a bit string. The links of the network connect nodes with (nearly) complementary strings. The population of a node survives if the number of occupied neighbors is not too small and not too large. There is an influx of lymphocytes with random idiotype from the bone marrow. Previous investigations have shown that this system evolves toward highly organized architectures, where the nodes can be classified into groups according to their statistical properties. The building principles of these architectures can be analytically described and the statistical results of simulations agree very well with results of a modular mean-field theory. In this paper, we present simulation results for the case that one or several nodes, playing the role of self, are permanently occupied. These self nodes influence their linked neighbors, the autoreactive clones, but are themselves not affected by idiotypic interactions. We observe that the group structure of the architecture is very similar to the case without self antigen, but organized such that the neighbors of the self are only weakly occupied, thus providing self-tolerance. We also treat this situation in mean-field theory, which give results in good agreement with data from simulation. The model supports the view that autoreactive clones, which naturally occur also in healthy organisms are controlled by anti-idiotypic interactions, and could be helpful to understand network aspects of autoimmune disorders.

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Section: The Modelmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 94%

Section: Mean-field Theorymentioning

confidence: 99%

Section: Idiotypic Network and Selfmentioning

confidence: 99%

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Self-Tolerance in a Minimal Model of the Idiotypic Network

2014

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Self-tolerance and autoimmunity in a minimal model of the idiotypic network

Landmann

Preuss

Behn

2017

Journal of Theoretical Biology

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We consider self-tolerance and its failure-autoimmunity-in a minimal mathematical model of the idiotypic network. A node in the network represents a clone of B-lymphocytes and its antibodies of the same idiotype which is encoded by a bitstring. The links between nodes represent possible interactions between clones of almost complementary idiotype. A clone survives only if the number of populated neighbored nodes is neither too small nor too large. The dynamics is driven by the influx of lymphocytes with randomly generated idiotype from the bone marrow. Previous work has revealed that the network evolves toward a highly organized modular architecture, characterized by groups of nodes which share statistical properties. The structural properties of the architecture can be described analytically, the statistical properties determined from simulations are confirmed by a modular mean-field theory. To model the presence of self we permanently occupy one or several nodes. These nodes influence their linked neighbors, the autoreactive clones, but are themselves not affected by idiotypic interactions. The architecture is very similar to the case without self, but organized such that the neighbors of self are only weakly occupied, thus providing self-tolerance. This supports the perspective that self-reactive clones, which regularly occur in healthy organisms, are controlled by anti-idiotypic clones. We discuss how perturbations, like an infection with foreign antigen, a change in the influx of new idiotypes, or the random removal of idiotypes, may lead to autoreactivity and devise protocols which cause a reconstitution of the self-tolerant state. The results could be helpful to understand network and probabilistic aspects of autoimmune disorders.

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Randomly evolving idiotypic networks: Structural properties and architecture

2012

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We consider a minimalistic dynamic model of the idiotypic network of B-lymphocytes. A network node represents a population of B-lymphocytes of the same specificity (idiotype), which is encoded by a bitstring. The links of the network connect nodes with complementary and nearly complementary bitstrings, allowing for a few mismatches. A node is occupied if a lymphocyte clone of the corresponding idiotype exists, otherwise it is empty. There is a continuous influx of new B-lymphocytes of random idiotype from the bone marrow. B-lymphocytes are stimulated by cross-linking their receptors with complementary structures. If there are too many complementary structures, steric hindrance prevents cross-linking. Stimulated cells proliferate and secrete antibodies of the same idiotype as their receptors, unstimulated lymphocytes die.Depending on few parameters, the autonomous system evolves randomly towards patterns of highly organized architecture, where the nodes can be classified into groups according to their statistical properties. We observe and describe analytically the building principles of these patterns, which allow to calculate number and size of the node groups and the number of links between them. The architecture of all patterns observed so far in simulations can be explained this way. A tool for real-time pattern identification is proposed.

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Randomly evolving idiotypic networks: Modular mean field theory

Cited by 3 publications

References 29 publications

Self-Tolerance in a Minimal Model of the Idiotypic Network

Self-Tolerance in a Minimal Model of the Idiotypic Network

Self-tolerance and autoimmunity in a minimal model of the idiotypic network

Randomly evolving idiotypic networks: Structural properties and architecture

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