Scaling and dimensionality in the chemical kinetics of protein filament formation

The formation of elongated supra-molecular structures from protein building blocks generates functional intracellular filaments, but this process is also at the heart of many neurodegenerative conditions including Alzheimer’s and Parkinson’s diseases, where it occurs in an uncontrolled manner. When observed at appropriate concentration and time scales, the chemical kinetics of filamentous protein self-assembly exhibits the remarkable property of self-similarity: the dynamics appears similar as the observation scale changes. We discuss here how this property leads to crucial simplifications of the fundamental laws governing protein filament formation and the emergence of scaling laws that provide the basis for connecting microscopic events with macroscopic realisations of such processes. In particular, we review recent developments in the modelling of linear protein self-assembly phenomena in the light of the concepts of dimensional analysis and physical self-similarity. We show how these tools and concepts can be used to elucidate the nature of the scaling laws for filamentous protein self-assembly, which illuminate the ultimately simple mathematical and physical principles underlying this seemingly highly complex phenomenon, and are expected to guide further developments in the field of linear self-assembly.