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Prions are proteins most commonly associated with fatal neurodegenerative diseases in mammals but are also responsible for a number of harmless heritable phenotypes in yeast. These states arise when a misfolded form of a protein appears and, rather than be removed by cellular quality control mechanisms, persists. The misfolded prion protein forms aggregates and is capable of converting normally folded protein to the misfolded state through direct interaction between the two forms. The dominant mathematical model for prion aggregate dynamics has been the nucleated polymerization model (NPM) which considers the dynamics of only the normal protein and the aggregates. However, for yeast prions the molecular chaperone Hsp104 is essential for prion propagation. Further, although mammals do not express Hsp104, experimental assays have shown Hsp104 also interacts with mammalian prion aggregates. In this study, we generalize the NPM to account for molecular chaperones and develop what we call the enzyme-limited nucleated polymerization model (ELNPM). We discuss existence, uniqueness and stability of solutions to our model and demonstrate that the NPM represents a quasi-steady-state reduction of our model. We validate the ELNPM by demonstrating agreement with experimental results on the yeast prion PSI(+) that could not be supported by the NPM. Finally, we demonstrate that, in contrast to the NPM, the ELNPM permits the coexistence of multiple prion strains.

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]

Prions are a special class of proteins capable of adopting multiple (misfolded) conformations, some of which have been associated with fatal diseases in mammals such as bovine spongiform encephalopathy or Creutzfeldt–Jakob Disease. Prion diseases, like protein misfolding diseases in general, are caused by the formation and amplification of ordered aggregates of proteins called amyloids. While such diseases in mammals can take decades to form, yeast have a variety of prion phenotypes that occur over a few hours, making this system an ideal model for protein misfolding disease in general. Most experimental assays of colonies with yeast prions provide steady-state population observations which complicate the inference of biochemical parameters both by the inability to directly measure aggregate amplification and by obscuring heterogeneity between cells. We develop a mathematical and inverse problem formulation to determine the amplification rate with prion aggregates from single-cell measurements observed in propagon amplification experiments. We demonstrate the ability of our formulation to determine heterogeneous amplification rates on simulated and experimental data. Our results show that aggregate amplification rates for two prion variants are strongly bimodal, suggesting that the generational structure in the yeast population impacts the ability of prion aggregates to amplify. [ABSTRACT FROM PUBLISHER]

Prions adopt alternative, self-replicating protein conformations and thereby determine novel phenotypes that are often irreversible. Nevertheless, dominant-negative prion mutants can revert phenotypes associated with some conformations. These observations suggest that, while intervention is possible, distinct inhibitors must be developed to overcome the conformational plasticity of prions. To understand the basis of this specificity, we determined the impact of the G58D mutant of the Sup35 prion on three of its conformational variants, which form amyloids in S. cerevisiae. G58D had been previously proposed to have unique effects on these variants, but our studies suggest a common mechanism. All variants, including those reported to be resistant, are inhibited by G58D but at distinct doses. G58D lowers the kinetic stability of the associated amyloid, enhancing its fragmentation by molecular chaperones, promoting Sup35 resolubilization, and leading to amyloid clearance particularly in daughter cells. Reducing the availability or activity of the chaperone Hsp104, even transiently, reverses curing. Thus, the specificity of inhibition is determined by the sensitivity of variants to the mutant dosage rather than mode of action, challenging the view that a unique inhibitor must be developed to combat each variant. [ABSTRACT FROM AUTHOR]

The yeast Saccharomyces cerevisiae has emerged as an ideal model system to study the dynamics of prion proteins which are responsible for a number of fatal neurodegenerative diseases in humans. Within an infected cell, prion proteins aggregate in complexes which may increase in size or be fragmented and are transmitted upon cell division. Recent work in yeast suggests that only aggregates below a critical size are transmitted efficiently. We formulate a continuous-time branching process model of a yeast colony under conditions of prion curing. We generalize previous approaches by providing an explicit formula approximating prion loss as influenced by both aggregate growth and size-dependent transmission. [ABSTRACT FROM AUTHOR]

• Novel conservative finite volume framework on deforming geometry. • Key idea: use spatio-temporal control volumes. • Essential to ensure mass conservation and guarantee convergence. • Simulate prions aggregations in dividing yeast cell. • Investigate how heterogeneity can create large asymmetries in protein partitioning. Motivated by experimental observations of asymmetric protein aggregates distributions in dividing yeast cells, we present a conservative finite volume approach for reaction-diffusion systems defined over deforming geometries. The key idea of our approach is to use spatio-temporal control volumes instead of integrating the time-discretized equations in space, as it is common practice. Both our theoretical and computational results demonstrate the convergence of our method and highlight how traditional approaches can lead to inaccurate solutions. We employ this novel approach to investigate the partitioning of protein aggregates in dividing yeast cells, leveraging the flexibility of the level set method to construct realistic biological geometries. Using a simple reaction-diffusion model, we find that spatial heterogeneity in yeast cells during division can alone create asymmetries in the concentration of protein aggregates. Moreover, we find that obstructing intracellular entities, such as nuclei or insoluble protein compartments, amplify these asymmetries, suggesting that they may play an essential role in regulating molecular partitioning. Beyond these findings, our results illustrate the flexibility of our approach and its potential to design realistic predictive tools to explore intracellular bio-mechanisms. [ABSTRACT FROM AUTHOR]