Kinetic theory of protein filament growth: Self-consistent methods and perturbative techniques

Filamentous protein structures are of high relevance for the normal functioning of the cell, where they provide the structural component for the cytoskeleton, but are also implicated in the pathogenesis of many disease states. The self-assembly of these supra-molecular structures from monomeric proteins has been studied extensively in the past 50 years and much interest has focused on elucidating the microscopic events that drive linear growth phenomena in a biological setting. Master equations have proven to be particularly fruitful in this context, allowing specific assembly mechanisms to be linked directly to experimental observations of filamentous growth. Recently, these approaches have increasingly been applied to aberrant protein polymerization, elucidating potential implications for controlling or combating the formation of pathological filamentous structures. This article reviews recent theoretical advances in the field of filamentous growth phenomena through the use of the master-equation formalism. We use perturbation and self-consistent methods for obtaining analytical solutions to the rate equations describing fibrillar growth and show how the resulting closed-form expressions can be used to shed light on the general physical laws underlying this complex phenomenon. We also present a connection between the underlying ideas of the self-consistent analysis of filamentous growth and the perturbative renormalization group.