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Prions are proteins capable of adopting misfolded conformations and transmitting these conformations to other normally folded proteins. Prions are most commonly known for causing fatal neurodegenerative diseases in mammals but are also associated with several harmless phenotypes in yeast. A distinct feature of prion propagation is the existence of different phenotypical variants, called strains. It is widely accepted that these strains correspond to different conformational states of the protein, but the mechanisms driving their interactions remain poorly understood. This study uses mathematical modeling to provide insight into this problem. We show that the classical model of prion dynamics allows at most one conformational strain to stably propagate. In order to conform to biological observations of strain coexistence and co-stability, we develop an extension of the classical model by introducing a novel prion species consistent with biological studies. Qualitative analysis of this model reveals a new variety of behavior. Indeed, it allows for stable coexistence of different strains in a wide parameter range, and it also introduces intricate initial condition dependency. These new behaviors are consistent with experimental observations of prions in both mammals and yeast. As such, our model provides a valuable tool for investigating the underlying mechanisms of prion propagation and the link between prion strains and strain specific phenotypes. The consideration of a novel prion species brings a change in perspective on prion biology and we use our model to generate hypotheses about prion infectivity.

Recent advances in the study of Alzheimer’s Disease and the role of Aβ amyloid formation have caused the focus of biologists to progressively shift towards the smaller protein assemblies, the oligomers. These appear very early on in the disease progression and they seem to be the most infectious species for the neurons. We suggest a model of spatial propagation of Aβ oligomers in the vicinity of a few neurons, without considering the formation of large fibrils or plaques. We also include a simple representation of the oligomers neurotoxic effect. A numerical study reveals that the oligomers spatial dynamics are very sensitive to the balance between their diffusion and their replication, and that the outcome in terms of the progression of AD strongly depends on it.

The use of yeast systems to study the propagation of prions and amyloids has emerged as a crucial aspect of the global endeavor to understand those mechanisms. Yeast prion systems are intrinsically multi-scale: the molecular chemical processes are indeed coupled to the cellular processes of cell growth and division to influence phenotypical traits, observable at the scale of colonies. We introduce a novel modeling framework to tackle this difficulty using impulsive differential equations. We apply this approach to the [PSI+] yeast prion, which is associated with the misconformation and aggregation of Sup35. We build a model that reproduces and unifies previously conflicting experimental observations on [PSI+] and thus sheds light onto characteristics of the intracellular molecular processes driving aggregate replication. In particular our model uncovers a kinetic barrier for aggregate replication at low densities, meaning the change between prion or prion-free phenotype is a bi-stable transition. This result is based on the study of prion curing experiments, as well as the phenomenon of colony sectoring, a phenotype which is often ignored in experimental assays and has never been modeled. Furthermore, our results provide further insight into the effect of guanidine hydrochloride (GdnHCl) on Sup35 aggregates. To qualitatively reproduce the GdnHCl curing experiment, aggregate replication must not be completely inhibited, which suggests the existence of a mechanism different than Hsp104-mediated fragmentation. Those results are promising for further development of the [PSI+] model, but also for extending the use of this novel framework to other yeast prion or amyloid systems. Author summary: In the study of yeast prions, mathematical modeling is a powerful tool, in particular when it comes to facing the difficulties of multi-scale systems. In this study, we introduce a mathematical framework for investigating this problem in a unifying way. We focus on the yeast prion [PSI+] and present a simple molecular scheme for prion replication and a model of yeast budding. In order to qualitatively reproduce experiments, we need to introduce a non-linear mechanism in the molecular rates. This transforms the intracellular system into a bi-stable switch and allows for curing to occur, which is a crucial phenomenon for the study of yeast prions. To the best of our knowledge, no model in the literature includes such a mechanism, at least not explicitly. We also describe the GdnHCl curing experiment, and the propagon counting procedure. Reproducing this result requires challenging hypotheses that are commonly accepted, and our interpretation gives a new perspective on the concept of propagon. This study may be considered as a good example of how mathematical modeling can bring valuable insight into biological concepts and observations. [ABSTRACT FROM AUTHOR]