Unreasonable implications of reasonable idiotypic network assumptions

and asp@tlO.lanl.goc) 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 higheraffinity 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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“…Following De Boer and Hogeweg (1989b) we define the connectivity C as the average number of non-zero interactions per clone. The classical results of Erdiis and Renyi (1960) proved that for C > 1 all clones tend to be connected to one another in one large graph (see also De Boer and Hogeweg, 1989b;Perelson, 1989). Here we study networks governed by an affinity matrix formed by assigning random values between 0 and 1, chosen from the uniform distribution, to randomly selected matrix elements ~~~ = ~~~ until an average connectivity of C is attained.…”

Section: F(handh')mentioning

confidence: 99%

“…There exists a large body of literature studying single-field immune network models having different affinities. This includes shape space models (Stewart and Varela, 1989;De Boer et al, 1992a;Detours et al, 1994), bit string models (De Boer and Perelson, 1991), Cayley tree models (Neumann and Weisbuch, 1992a, b), random networks (De Boer and Hogeweg, 1989b) and the models with autobodies (Stewart and Varela, 1990). The results of these previous publications are likely to be affected by choosing the more realistic two-field interaction function.…”

Section: Timementioning

confidence: 99%

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

Boer

¹

1996

Bulletin of Mathematical Biology

4

0

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and asp@tlO.lanl.goc) 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 higheraffinity 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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“…[4,5]) neglected the correlation between the two steps; the essence of this situation can be (5) This is formally the limiting case 0 ~ ~ of Eq. (1), so the simplification corresponds to a lack of clonal selflimitation.…”

Section: Introductionmentioning

confidence: 99%

Pattern formation in B-cell immune networks: Domains and dots in shape-space

Noest

¹

,

Takumi

²

,

Boer

³

1997

Physica D: Nonlinear Phenomena

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The immune system contains many types of B-cells, which can activate each other if the shapes and surface properties of their receptors (or antibodies) match well. The dynamics of the resulting network is analysed using a recently derived B-cell activation function which captures the effects of the binding and crosslinking of B-cell receptors. All receptor/antibody shapes are parametrised by a continuous 'shape-space', such that matching pairs of shapes interact locally.The model produces a variety of activation patterns across shape-space for a wide range of parameters. The spatio-temporal structures differ qualitatively from those seen with a previously used type of activation function. In either case, the pattern formation can largely be understood analytically, by first solving exactly for the various uniform fixed solutions, and then computing the evolution of spatially modulated perturbations. For the more realistic activation function, the following scenario is found.Most (random) initial conditions first lead to the formation of coarse domains, of three possible types: the 'virgin'-(V) state, the 'immune/suppressed' (I/S)-state, and its reverse (S/I). V-domains are stable, but the other two types are unstable to spatial perturbations with a wavelength which is of the order of the interaction range. In the second stage, this instability causes big I/S-and S/I-domains to split up into arrays of small 'dots' which preserve the I/S-asyn~netry of their parent domain. These dots are stable, even in isolation, which allows them to act as a 'memory' for previously encountered antigens.No stable dots are obtained when the model is made to emulate the simpler activation function which has been used widely in earlier models. With this less realistic choice, unstable waves propagate from the boundaries of coarse I/S-domains, eventually filling up most of shape-space. This instability was previously described as 'percolation'.

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Unreasonable implications of reasonable idiotypic network assumptions

Cited by 39 publications

References 43 publications

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

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

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

Pattern formation in B-cell immune networks: Domains and dots in shape-space

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