An Efficient Kinetic Model for Assemblies of Amyloid Fibrils and Its Application to Polyglutamine Aggregation

Abstract

Protein polymerization consists in the aggregation of single monomers into polymers that may fragment. Fibrils assembly is a key process in amyloid diseases. Up to now, protein aggregation was commonly mathematically simulated by a polymer size-structured ordinary differential equations (ODE) system, which is infinite by definition and therefore leads to high computational costs. Moreover, this Ordinary Differential Equation-based modeling approach implies biological assumptions that may be difficult to justify in the general case. For example, whereas several ordinary differential equation models use the assumption that polymerization would occur at a constant rate independently of polymer size, it cannot be applied to certain protein aggregation mechanisms. Here, we propose a novel and efficient analytical method, capable of modelling and simulating amyloid aggregation processes. This alternative approach consists of an integro-Partial Differential Equation (PDE) model of coalescence-fragmentation type that was mathematically derived from the infinite differential system by asymptotic analysis. To illustrate the efficiency of our approach, we applied it to aggregation experiments on polyglutamine polymers that are involved in Huntington’s disease. Our model demonstrates the existence of a monomeric structural intermediate acting as a nucleus and deriving from a non polymerizing monomer (). Furthermore, we compared our model to previously published works carried out in different contexts and proved its accuracy to describe other amyloid aggregation processes.

The Impact of Mathematical Modeling in Understanding the Mechanisms Underlying Neurodegeneration: Evolving Dimensions and Future Directions

“…They tend to assemble as toxic fibrillar structures, which may in turn associate into mature amyloid inclusions. Several mathematical models have been proposed to explain the protein aggregation process in ND: Stochastic nucleation and fibril dynamics, for instance, have been modelled using a two‐step reaction mechanism 42, 43, 44, 45, 46, 47 and validated with empirical data 48.A micelle intermediate on the pathway of protein aggregation has been described using mathematical models 49, 50.Furthermore, models of inhibiting end‐blocking drugs have also been formulated for describing the protein aggregation process 51, 52, 53. …”

“…A micelle intermediate on the pathway of protein aggregation has been described using mathematical models 49, 50.…”

Size Distribution of Amyloid Fibrils. Mathematical Models and Experimental Data

“…As a conclusion, the final size distribution may have the same shape for protein polymerization that involves a nucleation as for synthetic polymerization of living polymers that starts by an initiation, but the kinetics induced by a nucleation may strongly differ from the one induced by an initiation. However in the cases of protein polymerization that are quickly activated, i.e., where nucleation does not exist (nucleus size considered as equal to 1) or is not a rate-limiting step [39], the kinetics might be similar to that of synthetic living polymers.…”

Insights into the variability of nucleated amyloid polymerization by a minimalistic model of stochastic protein assembly

“…T V for δ small -5 to 10% -represents an alternative definition for the lag-time of the reaction [17].…”

“…Such a simplification is also justified by the fact that current kinetics measurements of amyloid growth exhibit variability on the timecourse of the total polymerised mass, without giving any information on the size distribution of fibrils. Previous studies (see for instance [17], Supplemental material (S.M.) 2) have shown that the detail of the reactions of secondary pathways, such as a fragmentation kernel, may have a major impact on the size distribution of polymers, but comparably smaller effects on the timecourse of the polymerised mass.…”