Proc R Soc Lond B Biol Sci  1993 Nov 22;254(1340):139-45 

Crosslinking of membrane immunoglobulins and B-cell activation: a simple model
based on percolation theory.

Faro J, Velasco S.

Departamento de Fisica Aplicada, Universidad de Salamanca, Spain.

In immune network models it is assumed that membrane immunoglobulin (mIg)
crosslinking leads to B-cell activation. To analyse further the implications of
this idea, a model of B-cell activation by ligand-induced mIg crosslinking in
the absence of cell-to-cell interactions is proposed. The present model, based
on a simple crosslinking mechanism previously proposed by other authors, assumes
that activation of B-cells is possible once crosslinks of mIgs percolate and
that percolation of crosslinks can only happen within a relatively short time
tau. Given a lattice (regular or not), a molecular cluster is said to percolate
or to become a percolating cluster if it spans the whole lattice (this is the
case, for instance, of a polymer in a gel phase). From this model of B-cell
activation we define the activation function fa (LK) as the fraction of B-cells
activated after tau minutes of interaction with a ligand at concentration L and
with affinity K. Numerical calculations show that, for current estimates of
kinetic constants involved in the interaction of a given ligand with a B-cell
clone, the activation function fa shifts when k-, the dissociation rate
constant, is varied below 10(-3) s-1, this shift being linearly proportional to
the variation of k-. This result contradicts and, therefore, challenges the
assumption in immune network models that the activation function is identical
for all ligands. This is important because the behaviour of at least some of
those immune network models is quite sensitive to the relative values of the
activation function thresholds.