![]() Much attention has been given to the physics underlying sol–gel transitions and polymerization ( 13– 15). Interest in intracellular phase separation phenomena has increased since the discovery of P body dissolution/condensation in Caenorhabditis elegans ( 12). These observations suggest a possible link between the size distribution of scaffolding aggregates and disease. Deletion of Axin’s DIX domain abolishes degradation of β-catenin ( 5), and mutations in APC that drive familial adenomatous polyposis map to truncations reducing the number of SAMP repeats at which APC binds Axin ( 11). By virtue of their polymeric nature, such scaffold assemblies have no defined stoichiometry and may only exist as a heterogeneous combinatorial ensemble ( 8, 9)-also called “pleiomorphic ensemble” ( 10)-rather than a single well-defined complex. In addition, the DIX domain in Axin allows for oriented Axin polymers ( 5, 6), while APC (another scaffold) can bind multiple copies of Axin ( 7), yielding Axin–APC aggregates to which kinases and their substrates bind. β-Catenin is modified by CK1α and GSK3β without binding any of these kinases directly but interacting with them through an Axin scaffold ( 3, 4). A case in point is the so-called “destruction complex” that tags β-catenin for degradation in the canonical Wnt pathway. Scaffolds are involved in the formation of signalosomes, which are transient protein complexes that process and propagate signals. In particular, the scaffolding function of a protein can be conditional upon activation and serve to recruit further scaffolds, thus creating opportunities for network plasticity in real time. These devices facilitate the evolution and control of connectivity within and among pathways. Protein–protein interactions underlying cellular signaling systems are mediated by a variety of structural elements, such as docking regions, modular recognition domains, and scaffold or adapter proteins ( 1, 2). 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. We explain this behavior in terms of how the concentration profile of the polymer-length distribution adjusts to changes in protomer concentration and affinity. 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. This behavior boosts Q beyond that of any multivalent scaffold system. 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. 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 quantity of interest is the abundance of ligand interaction possibilities-the catalytic potential Q-in a configurational mixture. 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. Their assemblies should therefore not be understood as stoichiometric aggregates, but as combinatorial ensembles. Several of these scaffolds are known to polymerize. Scaffold proteins organize cellular processes by bringing signaling molecules into interaction, sometimes by forming large signalosomes.
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