# Combinatorial protein-protein interactions on a polymerizing scaffold

## TL;DR

This study explores how polymerizing scaffold proteins enhance ligand interaction opportunities in cellular signaling, revealing unique concentration-dependent behaviors that surpass fixed-site scaffolds, with implications for understanding signalosome assembly.

## Contribution

It introduces a novel analysis of combinatorial ligand interactions on polymerizing scaffolds, highlighting their distinct concentration-dependent properties compared to multivalent scaffolds.

## Key findings

- Polymerizing scaffolds increase ligand interaction potential beyond fixed-site systems.
- The dependency of interaction potential on scaffold concentration switches from first to second order.
- Polymerizing systems mitigate the decline in interaction opportunities at high scaffold concentrations.

## Abstract

Scaffold proteins organize cellular processes by bringing signaling molecules into interaction, sometimes by forming large signalosomes. Several of these scaffolds are known to polymerize. Their assemblies should therefore not be understood as stoichiometric aggregates, but as combinatorial ensembles. We analyze the combinatorial interaction of ligands loaded on polymeric scaffolds, in both a continuum and discrete setting, and compare it with multivalent scaffolds with fixed number of binding sites. The quantity of interest is the abundance of ligand interaction possibilities---the catalytic potential $Q$---in a configurational mixture. Upon increasing scaffold abundance, scaffolding systems are known to first increase opportunities for ligand interaction and then to shut them down as ligands become isolated on distinct scaffolds. The polymerizing system stands out in that the dependency of $Q$ on protomer concentration switches from being dominated by a first order to a second order term within a range determined by the polymerization affinity. This behavior boosts $Q$ beyond that of any multivalent scaffold system. In addition, the subsequent drop-off is considerably mitigated in that $Q$ decreases with half the power in protomer concentration than for any multivalent scaffold. We explain this behavior in terms of how the concentration profile of the polymer length distribution adjusts to changes in protomer concentration and affinity. The discrete case turns out to be similar, but the behavior can be exaggerated at small protomer numbers because of a maximal polymer size, analogous to finite-size effects in bond percolation on a lattice.