Cross-linking reconsidered: binding and cross-linking fields and the cellular response

Abstract

We analyze a model for the reversible cross-linking of cell surface receptors by a collection of bivalent ligands with different affinities for the receptor as would be found in a polyclonal anti-receptor serum. We assume that the amount of cross-linking determines, via a monotonic function, the rate at which cells become activated and divide. In addition to the density of receptors on the cell surface, two quantities, the binding field and the cross-linking field, are needed to characterize the cross-linking curve, i.e., the equilibrium concentration of cross-linked receptors plotted as a function of the total ligand site concentration. The binding field is the sum of all ligand site concentrations weighted by their respective binding affinities, and the cross-linking field is the sum of all ligand site concentrations weighted by the product of their respective binding and cross-linking affinity and the total receptor density. Assuming that the cross-linking affinity decreases if the binding affinity decreases, we find that the height of the cross-linking curve decreases, its width narrows, and its center shifts to higher ligand site concentrations as the affinities decrease. Moreover, when we consider cross-linking-induced proliferation, we find that there is a minimum cross-linking affinity that must be surpassed before a clone can expand. We also show that under many circumstances a polyclonal antiserum would be more likely than a monoclonal antibody to lead to cross-linking-induced proliferation.

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

“…When xi is plotted versus the total ligand concentration L = Cj Lj on a logarithmic scale, the curve is bell-shaped. Since xi is a monotonic function of Si, the log bell-shaped nature of the cross-linking function is determined by Si only and, as we show in Sulzer et al (1995), Si itself is a log bell-shaped function of L.…”

“…In Sulzer et al (1995) we derived an expression for the total concentration of cross-links on a B cell that is stimulated by a collection of bivalent ligands. This analysis proceeded analogously to classical derivations for the case of one ligand (DeLisi and Perelson, 1976;Perelson and DeLisi, 1980;Perelson, 1981).…”

“…In deriving equation (3) we have used the excess ligand approximation (Sulzer et al, 1995), i.e. we have assumed that the concentration of ligand greatly exceeds the concentration of receptor sites, so that ligand is not noticeably depleted during binding.…”

“…Thus, we can redefine the fields in terms of non-dimensional quantities as Here lj = KL, is a non-dimensional ligand concentration and K, is the non-dimensional cross-linking constant already introduced above (cf. Sulzer et al, 1995).…”

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

“…Both cases are discussed in more detail in Sulzer et al (1995). ' Here K, is non-dimensional and is identical to K: R,, where Kk has units of area (e.g.…”