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Self-assembly of proteins into amyloid aggregates is an important biological phenomenon associated with human diseases such as Alzheimer's disease. Amyloid fibrils also have potential applications in nano-engineering of biomaterials. The kinetics of amyloid assembly show an exponential growth phase preceded by a lag phase, variable in duration as seen in bulk experiments and experiments that mimic the small volumes of cells. Here, to investigate the origins and the properties of the observed variability in the lag phase of amyloid assembly currently not accounted for by deterministic nucleation dependent mechanisms, we formulate a new stochastic minimal model that is capable of describing the characteristics of amyloid growth curves despite its simplicity. We then solve the stochastic differential equations of our model and give mathematical proof of a central limit theorem for the sample growth trajectories of the nucleated aggregation process. These results give an asymptotic description for our simple model, from which closed form analytical results capable of describing and predicting the variability of nucleated amyloid assembly were derived. We also demonstrate the application of our results to inform experiments in a conceptually friendly and clear fashion. Our model offers a new perspective and paves the way for a new and efficient approach on extracting vital information regarding the key initial events of amyloid formation. [ABSTRACT FROM AUTHOR]

This study is devoted to the long-term behavior of nucleation, growth and fragmentation equations, modeling the spontaneous formation and kinetics of large polymers in a spatially homogeneous and closed environment. Such models are, for instance, commonly used in the biophysical community in order to model in vitro experiments of fibrillation. We investigate the interplay between four processes: nucleation, polymerization, depolymerization and fragmentation. We first revisit the well-known Lifshitz-Slyozov model, which takes into account only polymerization and depolymerization, and we show that, when nucleation is included, the system goes to a trivial equilibrium: all polymers fragmentize, going back to very small polymers. Taking into account only polymerization and fragmentation, modeled by the classical growth-fragmentation equation, also leads the system to the same trivial equilibrium, whether or not nucleation is considered. Finally, when taking into account a depolymerization reaction term, we prove the existence of a steady size-distribution of polymers, as soon as polymerization dominates depolymerization for large sizes whereas depolymerization dominates polymerization for smaller ones—a case which fits the classical assumptions for the Lifshitz-Slyozov equations, but complemented with fragmentation so that “Ostwald ripening” does not happen. [ABSTRACT FROM AUTHOR]

In the formation of large clusters out of small particles, the initializing step is called the nucleation, and consists in the spontaneous reaction of agents which aggregate into small and stable polymers called nuclei. After this early step, the polymers are involved in a number of reactions such as polymerization, fragmentation and coalescence. Since there may be several orders of magnitude between the size of a particle and the size of an aggregate, building efficient numerical schemes to capture accurately the kinetics of the reaction is a delicate step of key importance. In this article, we propose a conservative scheme, based on finite volume methods on an adaptive grid, which is capable of simulating well the early steps of the reaction as well as the later chain reactions. [ABSTRACT FROM AUTHOR]

Protein polymerization is involved in many important biological phenomena and gives rise to spontaneous formation of amyloid fibrils or, more generally, polymers. The starting point of this process, called nucleation, exhibits an important variability among replicated experiments. To analyze the stochastic nature of this phenomenon, one of the simplest models considers two populations of chemical components: free monomers and polymerized monomers. Initially there are only monomers. There are two reactions for the polymerization of a monomer: either two monomers collide to combine into two polymerized monomers or a monomer is polymerized after the encounter of a polymerized monomer. It turns out that this simple model does not explain completely the variability observed in the experiments. This paper investigates extensions of this model to take into account other mechanisms of the polymerization process that may have an impact on fluctuations. The first variant consists in introducing a preliminary conformation step to take into account the biological fact that, before being polymerized, a monomer has two states, regular or misfolded. Only misfolded monomers can be polymerized so that the fluctuations of the number of misfolded monomers can also be a source of variability of the number of polymerized monomers. For the second variant, based on numerical considerations, the reaction rate a of spontaneous formation of a polymer is of the order of N--v for some large scaling variable N representing the reaction volume and v some positive constant. Asymptotic results involving different time scales are obtained for the corresponding Markov processes. First and second order results for the starting instant of nucleation are derived from these limit theorems. The proofs of the results rely on a study of a stochastic averaging principle for a model related to an Ehrenfest urn model, and also on a scaling analysis of a population model. [ABSTRACT FROM AUTHOR]