A Cooperative Model for Symmetric Ligand Binding to Protein Fibrils

A hallmark of neurodegenerative diseases like Alzheimer's Disease (AD) and chronic traumatic encephalopathy (CTE) is the presence of toxic protein aggregates in neurons. In AD and CTE specifically, the protein tau forms insoluble fibrils that are hundreds of nanometers in length. Intriguingly, recent experimental structures suggest that tau ligands like the disaggregator EGCG and positron emission tomography (PET) tracers like GTP-1 and MK-6240 bind to tau fibrils in long stacks reflecting the symmetry of the protein across many binding sites. In these stacks, each ligand makes more contact with its symmetry mates than it does with the protein. To interpret the binding of these molecules and new ligands, we must understand the effects of the cooperativity between sites and the entropy coming from the number of sites. Here, we investigate a nearest-neighbors model of cooperativity and use statistical mechanics to derive binding isotherms for saturation and competition experiments. This model allows us to relate measured EC₅₀ and IC₅₀ values to the intrinsic binding affinity to a single site and to cooperativity across sites in ways resembling the Cheng-Prusoff Equation. Depending on the degree of cooperativity between molecular species, this model permits solutions that lack the steep binding curves expected from cooperative systems and even solutions resembling 2-site systems. We finally consider conditions for a fibril's detection in a PET scan and practical matters of fitting this model's parameters to data.