A mathematical model of the dynamics of prion aggregates with chaperone-mediated fragmentation

Abstract

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.

Keywords: Nucleated polymerization; Prions; Protein aggregation; Protein misfolding; Yeast.

Fig. 1

Nucleated polymerization model: conversion and fragmentation ($n_0=2$). Conversion of healthy protein (circles) lengthens the aggregate (squares), which may in turn fragment. If a daughter fragment is smaller than the stable nucleus size ($n_0$), it is immediately disassociated into healthy protein monomers

Fig. 2

NPM is based on a random breaking of the prion aggregate (stars), while our model introduces the Hsp104 enzyme (hexagons) that mechanistically fragments the aggregate. Healthy Sup35 (squares) are converted by the ends of the prion aggregate in both models; proteins undergoing conversion are represented with pentagons to demonstrate the active change in conformation. a Nucleated polymerization model (stochastic fragmentation) b enzyme-limited polymerization model (mechanistic fragmentation via Hsp104 chaperone)

Fig. 3

Plot of steady-state $x_m=u_{m+n_0-1}/\mathit{\eta }$ with parameter values chosen from Tanaka et al. (2006), appropriately modified to match the steady-state, effective fragmentation rate with the paper’s constant rate

Fig. 4

Non-dimensionalized plots of aggregate density over time with varying $R_0=R_0(p_0)$. The system is initialized with a ${10}^{-9}$ perturbation of aggregated protein from an otherwise healthy initial state

Fig. 5

p(t) over time with varying $R_0=R_0(p_0)$; parameters are the same as in Fig. 4. The transient fragmentation efficiency may be higher (for small $R_0$) or lower (for large $R_0$) than the asymptotic efficiency

Fig. 6

Theoretical shifts in the steady-state concentration of aggregate size distributions $\{m{u}_m\}$ by increasing the synthesis of normal protein (${\mathit{\alpha }}_s$). Left $m{u}_m=\mathit{\eta }m{x}_{m-n_0+1}$ from the NPM. Though $x_m$ is invariant to ${\mathit{\alpha }}_s$, $\mathit{\eta }$ does have a dependency, resulting in the small changes in scaling. Right $m{u}_m$ from our ELNPM. Both the scaling and translation are affected by ${\mathit{\alpha }}_s$. Initial kinetic parameters are as in Fig. 3

Fig. 7

Hsp104 production is up-regulated or Hsp104 is deactivated after 3 h, both by a factor of ${10}^4$. The new system in either case is unable to stably support the presence of prion aggregates with our engineered parameters

Fig. 8

The reproductive number $R_0$ as a function of ${\mathit{\alpha }}_h$. Prion strains will only be driven to extinction by Hsp104 over-expression if ${lim}_{{\mathit{\alpha }}_h\to \infty }{R}_0({\mathit{\alpha }}_h)<1$

Fig. 9

Different parameter regimes exhibit fundamentally different behavior with respect to coendemic stability. The labeled regions denote the surviving strain, and gray regions denote areas of mutual coexistence. Refer to Table 1 for the parameter values. a No stable coexistence, b region of stable coexistence

Fig. 10

Plots of a specific parameter set admitting asymptotically stable, coendemic behavior. We note that the size densities and aggregated protein take on distinct values, despite very similar reproductive numbers. a Steady-state aggregate densities, b concentration of Sup35 in aggregates