Allosteric models for cooperative polymerization of linear polymers

In the cytoskeleton, unfavorable nucleation steps allow cells to regulate where, when, and how many polymers assemble. Nucleated polymerization is traditionally explained by a model in which multistranded polymers assemble cooperatively, whereas linear, single-stranded polymers do not. Recent data on the assembly of FtsZ, the bacterial homolog of tubulin, do not fit either category. FtsZ can polymerize into single-stranded protofilaments that are stable in the absence of lateral interactions, but that assemble cooperatively. We developed a model for cooperative polymerization that does not require polymers to be multistranded. Instead, a conformational change allows subunits in oligomers to associate with high affinity, whereas a lower-affinity conformation is favored in monomers. We derive equations for calculating polymer concentrations, subunit conformations, and the apparent affinity of subunits for polymer ends. Certain combinations of equilibrium constants produce the sharp critical concentrations characteristic of cooperative polymerization. In these cases, the low-affinity conformation predominates in monomers, whereas virtually all polymers are composed of high-affinity subunits. Our model predicts that the three routes to forming HH dimers all involve unstable intermediates, limiting nucleation. The mathematical framework developed here can represent allosteric assembly systems with a variety of biochemical interpretations, some of which can show cooperativity, and others of which cannot.