A Kinetic Study of Amyloid Formation: Fibril Growth and Length Distributions

Abstract

We propose a kinetic model for the self-aggregation by amyloid proteins. By extending several well-known models for protein aggregation, the time evolution of aggregate concentrations containing r proteins, denoted c(r)(t), can be written in terms of generalized Smoluchowski kinetics. With this approach, we take into account all possible aggregation and fragmentation reactions involving clusters of any size. Correspondingly, an aggregate of size x + y could be formed by or break up into two smaller constituent aggregates of sizes x and y. The rates of each aggregation or fragmentation reaction, called kernels, are specified in terms of the aggregate size, and we solve c(r)(t) for large cluster sizes using numerical techniques. We show that by using Smoluchowski kinetics many pathways to fibrillation are possible and quantities, such as the aggregate length distribution at an arbitrary time, can be calculated. We show that the predicted results of the model are in agreement with the experimental observations.

Asymptotic solutions of the Oosawa model for the length distribution of biofilaments

“…We note that due to the alternating nature of the series in Eq. (34), truncation at odd k max yields unphysical negative numbers for small j in the long time limit. …”

“…The classical theory of nucleated polymerisation delineated by Eq. (1) is based on the simplest possible model and many variations of this model have been put forward, 29,34,[41][42][43][44][45][46][47][48] including the effect of heterogeneities in primary nucleation for polyglutamine aggregation. 48,49 In general, other microscopic processes, including filament fragmentation, association, or secondary nucleation processes, may play a role in determining the shape of the filament distribution.…”

“…(1) assumes that the rate constants for elongation and dissociation do not depend on the degree of polymerisation. This assumption may in general be not well-satisfied because features of the reaction, including the size of fibrils or the diffusion and conformation of proteins, influence the rate constants, 34,52 but it keeps the number of model parameters to an absolute minimum and renders the analytical treatment of the corresponding equations easier to achieve.…”

“…Due to absolute convergence of the series in Eq. (34), this error can be reduced by increasing k max . As it can be seen from the figure, the series converges in a small number of terms in the limit of large aggregation numbers or small times.…”

A Kinetic Model for Cell Damage Caused by Oligomer Formation

“…Based on our studies and other related works (11,12), most observations on amyloid formation can be well interpreted by this kind of model. The necessities of surface-catalyzed secondary nucleation (28), thickening (29,30), protein diffusion (31), and so on are left to future studies. In this study, for simplicity, oligomers are modeled as on-pathway intermediates; however, in many amyloid-related diseases cytotoxic oligomers are shown to be off-pathway aggregates, which means conformational transition (32,33) or off-pathway polymerization (34) should be considered as important generalizations in the next step.…”