Stochastic calculus of protein filament formation under spatial confinement

Abstract

The growth of filamentous aggregates from precursor proteins is a process of central importance to both normal and aberrant biology, for instance as the driver of devastating human disorders such as Alzheimer's and Parkinson's diseases. The conventional theoretical framework for describing this class of phenomena in bulk is based upon the mean-field limit of the law of mass action, which implicitly assumes deterministic dynamics. However, protein filament formation processes under spatial confinement, such as in microdroplets or in the cellular environment, show intrinsic variability due to the molecular noise associated with small-volume effects. To account for this effect, in this paper we introduce a stochastic differential equation approach for investigating protein filament formation processes under spatial confinement. Using this framework, we study the statistical properties of stochastic aggregation curves, as well as the distribution of reaction lag-times. Moreover, we establish the gradual breakdown of the correlation between lag-time and normalized growth rate under spatial confinement. Our results establish the key role of spatial confinement in determining the onset of stochasticity in protein filament formation and offer a formalism for studying protein aggregation kinetics in small volumes in terms of the kinetic parameters describing the aggregation dynamics in bulk.