  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.
