A mathematical model of the dynamics of prion aggregates with chaperonemediated fragmentation
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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 enzymelimited nucleated polymerization model (ELNPM). We discuss existence, uniqueness and stability of solutions to our model and demonstrate that the NPM represents a quasisteadystate 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 YeastMathematics Subject Classification
92B05 92C451 Introduction
The central dogma of molecular biology stipulates that phenotypes, an organism’s expressed states, are determined by genotypes, the vertically transmitted DNA (Crick 1958). However, the link between genotype and phenotype is not always this direct. Today we understand that a number of phenotypes are determined epigenetically, without a change to the nucleotide sequence of DNA (Goldberg et al. 2007). In 1965, a number of yeast phenotypes were found to violate the laws of Mendelian inheritance and were thus inconsistent with DNAbased transmission (Cox et al. 1988). Further experimental studies demonstrated that the phenotypic states were not the function of the underlying DNA but were the function of a misfolded (prion) protein (Wickner 1994). As such, the phenotypes were transmitted by the proteins themselves. This phenomenon of “protein only inheritance,” also called the prion hypothesis, has over time gone from highly controversial to commonly accepted (Tuite and Serio 2010). Today, nearly a dozen proteins in yeast have been shown to be able to behave as prions (Liebman and Chernoff 2012). Of course, prions extend far beyond yeast. In mammals, prions are associated with a number of irreversible fatal neurological diseases such as Creutzfeldt–Jakob disease, fatal familial insomnia, chronic wasting disease and bovine spongiform encephalopathy. Mammalian prion diseases have varying modes of transmission and have been shown to be able to pass from one species to another. At present, all mammalian prion diseases are the result of a single protein, PrP (Aguzzi and Polymenidou 2004). In addition, prion diseases are closely related to other protein misfolding diseases such as Parkinson’s, Huntington’s, Alzheimer’s and diseases (Brundin et al. 2010; Knowles et al. 2014; Brettschneider et al. 2015).
Although humans are vastly different from yeast, the dynamics of prion proteins in both hosts is quite similar. Both mammals and yeast have cellular machinery dedicated to identifying and removing misfolded proteins (Nelson et al. 2008)—prion proteins are capable of evading such protective mechanisms and transmitting their misfolded (prion) state to other normally folded proteins. Prion proteins aggregate into complexes which act as templates for initiating further misfolding of normally folded protein. These aggregated complexes may also fragment into smaller units, each of which can template further misfolding (Sindi and Serio 2009; Tuite and Serio 2010). Finally, in order to spread the prion state to a colony or throughout a tissue, prion aggregates must be transmitted to other cells. In yeast colonies, prion aggregates are transmitted from mother to daughter cells during cell division (Tuite and Cox 2003; Derdowski et al. 2010). In mammalian prion diseases, PrP aggregates are thought to be transmitted extracellularly (Collinge 2001). For other mammalian neurodegenerative diseases, there is a growing body of evidence suggesting neuron to neuron propagation of the misfolded proteins (Brettschneider et al. 2015).
Many mathematical models have been developed to study the dynamics of prion aggregates primarily in the context of the mammalian host (Masel et al. 1999; Prüss et al. 2006; Greer et al. 2006; Calvez et al. 2010). Tanaka et al. (2006) applied these mathematical models to the \([\) PSI \({}^{+}]\) prion in yeast. However, experimental studies of \([\) PSI \({}^{+}]\) have shown that the molecular chaperone Hsp104 is essential for fragmentation, and recent studies have demonstrated that Hsp104 acts in a rate limiting fashion with respect to fragmentation (SatputeKrishnan et al. 2007; Derdowski et al. 2010). We also note that fragmentation is important not only for efficient conversion of normal protein by providing more templates, but also to ensure there are sufficiently many templates to allow efficient transmission; thus an accurate model of fragmentation is essential to understanding the in vivo dynamics of prion aggregates. As such, to accurately model prion aggregates in yeast, the dynamics of Hsp104 and its interaction with prion aggregates must be considered.
Further, modeling Hsp104 will lend insight to more general prion and protein misfolding disorders. Although no chaperones are known to be involved in mammalian prion dynamics, prion amplification in mammals necessarily requires aggregate fragmentation (Masel et al. 1999). In addition, while mammals do not express Hsp104, recent in vitro work demonstrates that engineered mutants of Hsp104 suppress the toxicity of misfolded protein aggregates associated with mammalian neurodegenerative disorders (Jackrel and Shorter 2014). At present, no mathematical model exists which considers the dynamics of protein aggregates in the presence of a chaperone mediating fragmentation.
In this study we develop a mathematical model of prion dynamics where fragmentation requires the interaction of Hsp104 with aggregated proteins. In Sects. 2 and 3 we provide the mathematical background and analysis of our model, which we call the enzymelimited nucleated polymerization model (ELNPM). In Sect. 4 we illustrate the necessity of including enzymelimited fragmentation by demonstrating important experimental properties of \([\) PSI \({}^{+}]\) that are not described by previously published mathematical models. We also demonstrate that in contrast to models which consider only the prion aggregates, interactions with the enzyme Hsp104 permit stable coexistence of multiple prion strains. In Sect. 5 we provide a summary and concluding remarks.
2 Mathematical models of prion aggregate fragmentation
We develop our model of enzymemediated fragmentation by considering the key biochemical processes involved in the dynamics of prions. We first discuss the dynamics included in previous mathematical formulations and then detail the additional features necessary to depict interactions between enzymes and aggregates. Finally, we demonstrate that through a series of assumptions, consistent with the yeast prion \([\) PSI \({}^{+}]\), our system of infinite ordinary differential equations can be analyzed with a 5dimensional system of differential equations which approximates the full system dynamics.
2.1 Prion aggregate dynamics
While the biochemical processes depicted differs between prion model formulations (Masel et al. 1999; Prüss et al. 2006; Greer et al. 2006; Calvez et al. 2010), in all cases aggregates change size through conversion and fragmentation. That is, a prion aggregate increases in length by actively converting and incorporating normal (healthy) protein monomers. Typically, aggregates are assumed to be linear fibrils and, as such, conversion of normal protein can only take place on one of the fibril ends. Aggregates may also fragment into two smaller aggregates, each of which now act as a template to convert additional protein. It is often assumed that any aggregates smaller than the minimum stable size, \(n_0\), immediately disassociate into healthy prion monomers (see Fig. 1).

\(\alpha _s\) the basal rate of transcription of Sup35,

\(\mu _s\) the decay (or dilution) rate of Sup35,

\(n_0\) the minimum stable aggregate size (aggregates of size smaller than \(n_0\) immediately disassociate into soluble Sup35),

\(\mu _0\) the decay (or dilution) rate of aggregated protein,

\(\beta \) the rate of conversion of healthy protein (from the end of a prion filament), and

\(\gamma (m1)\) the fragmentation rate of a prion aggregate of size m.
2.2 Enzymemediated fragmentation
We now draw attention to an underlying assumption in these equations that we will modify: for the mammalian prion PrP, it was assumed that the fragmentation rate is an intrinsic function of the aggregate size itself. However, with yeast prion systems, it has been demonstrated that fragmentation requires the additional presence of heatshock protein 104 (Hsp104) (SatputeKrishnan et al. 2007). Its under or overexpression can eliminate prion aggregates entirely (Chernoff et al. 1995). Additionally, overexpressing Sup35 results in a translational shift in the aggregatesize density (Derdowski et al. 2010)—the model equations of Masel et al. (1999) do not admit such behavior (see Sect. 4.3), suggesting the possibility of a ratelimited fragmentation mechanism rooted in the Hsp104 interactions.
 Translation and Dilution
 Aggregation (Polymerization/Coagulation)
 $$\begin{aligned} \text {Sup35}+X_{i,j} \xrightarrow [2\beta ]{} X_{i+1,j},\qquad \end{aligned}$$
 Enzyme Kinetics
 Fragmentation (with unspecified probability \(\displaystyle \kappa (m,n;i,j)\))
 $$\begin{aligned} X_{i,j} \xrightarrow [\gamma j]{} {\left\{ \begin{array}{ll} X_{m,n} + \text {Hsp104} + X_{im,jn1} &{}\quad m,im \ge n_0 \\ m\text {Sup35} + (n+1)\text {Hsp104} + X_{im,jn1} &{}\quad m < n_0, im \ge n_0 \\ X_{m,n} + (jn)\text {Hsp104} + (im)\text {Sup35} &{}\quad m \ge n_0, im < n_0 \\ i\text {Sup35} + j\text {Hsp104} &{}\quad m,im < n_0. \end{array}\right. } \end{aligned}$$
2.3 Enzymelimited nucleated polymerization model
We further observe that the systems \((s,\{u_m\})\) and \((s,\eta ,z)\) are identical to that of the original NPM (Masel et al. 1999), except the constant fragmentation rate \(\gamma \) has been replaced by the timevarying quantity \(\gamma p(t)\). This provides the interpretation of p(t) as the measure of how effectively the system is fragmenting at a given time. Furthermore, this formulation suggests a quasisteadystate interpretation of our approximation, since now every aggregate is always bound with Hsp104 proportionally as p(t); that is, the enzyme binding reaches equilibrium before any conversion or fragmentation events occur. We finally note that the first 2 terms in Eq. (12e) reflect a Michaelis–Menten simplification of the enzyme kinetics (Segel and Slemrod 1989); however, since Hsp104 offbinding results in a change in the amount of binding substrate with probability \(\gamma /(\gamma +k_{\text {off}})\), we may view the last term as the correction to preserve Hsp104 conservation.
Before detailed analysis of the ELNPM, we briefly examine the qualitative form of the aggregate size distribution. In Fig. 3 we plot a typical equilibrium solution to Eq. (13), where we have defined \(x_m=u_{n_01+m}/\eta \) to be the corresponding probability mass function over the natural numbers. As expected, given the asymptotic similarity of our system to (Masel et al. 1999) (in that \(\gamma p(t)\) presumably converges to a fixed \(\tilde{\gamma }\)), the equilibrium size density is of the same distribution.
3 Analysis of the ELNPM
We first prove a few results on existence and uniqueness for this system and then provide a nondimensionalized, transformed system we will use to study stability. We analytically demonstrate the stability of the diseasefree state and derive conditions which will ensure aggregate persistence.
3.1 Existence and uniqueness of solutions
Theorem 1
Trajectories of Eq. () remain invariant under a bounded, “feasible” subset of the nonnegative cone \({\mathbb {R}}^5_+\).
Proof
Let us define the feasible subset to be the set of all \((s,h,\eta ,z,p)\) where \(s,h,\eta ,z \ge 0\) and \(0 \le p \le 1\), with the further restriction that \(z \ge n_0\eta \), \(0 \le s+z \le \alpha _s/\mu _0\), and \(0 \le h+p(z\eta ) \le \alpha _h/\mu _0\). For analytical convenience, and in line with typical parameter regimes, we assume \(\mu _0 \le \mu _s, \mu _h\) and \(\frac{\gamma }{2\beta } < \frac{\alpha _s}{\mu _s}\).
Now consider the violation of any single constraint. It is straightforward to show \(s'\big _{s<0}, h'\big _{h<0}, \eta '\big _{\eta <0},p'\big _{p<0} \ge 0\). Similarly, \(p'\big _{p>1} \le 0\). Next, \((zn_0\eta )'\big _{z<n_0\eta }= 2\beta s \eta  \mu _0 (zn_0\eta )  n_0 \gamma p (zn_0\eta ) \ge 0\), which also demonstrates the nonnegativity of z since \(z \ge n_0\eta \ge 0\). Writing \(\mu _s/\mu _0=1+\epsilon /\mu _0\), we have \((s+z)'\big _{s+z>\alpha _s/\mu _0}=\alpha _s\mu _0(s+z)\epsilon s \le 0\). With a similar approach, we also find \((h+p(z\eta ))'\big _{h+p(z\eta )>\alpha _h/\mu _0} \le 0\). \(\square \)
Theorem 2
Solutions satisfy the existence and uniqueness criteria within the invariant, feasible region.
Proof
3.2 Nondimensionalized equations
3.3 Asymptotic behavior of ELNPM
3.3.1 Diseasefree steady state
Theorem 3
There is a unique solution to Eq. (19) in our feasible region of trajectories.
Proof
We now establish local stability criteria of this root; let \(p_0\) and \(\omega _0\) be the unique solution to Eq. (19).
Theorem 4
The unique diseasefree equilibrium is locally stable when \(R_0(p_0)<1\) and unstable when \(R_0(p_0)>1\).
Proof
A table of values used for the plots
Figures  \(\alpha _h\,(\upmu {\mathrm {M}}\,{\mathrm {min}}^{1})\)  \(\beta \,(\upmu {\mathrm {M}}^{1}{\mathrm {min}}^{1})\)  \(\gamma \,({\mathrm {min}}^{1})\)  \(k_{\text {on}}\,(\upmu {\mathrm {M}}^{1}{\mathrm {min}}^{1})\)  \(k_{\text {off}}\,({\mathrm {min}}^{1})\) 

0.002567  6.0  0.00294  0.20000  2.00000  
0.00142  –  0.00294  0.15964  2.93706  
\(4.16503\times 10^{6}\)  0.216456  0.0421667  81.8524  4.38533  
–  0.216456  0.0421667  81.8524  4.38533  
9a  0.002567  (6.0, 6.0)  (0.00294, 0.00294)  (0.2, –)  (2.0, –) 
9b  0.002567  (6.0, 8.0)  (0.00294, 0.002)  (0.2, –)  (2.0, –) 
0.002567  (6.0, 8.0)  (0.00294, 0.002)  (0.2, 0.21567)  (2.0, 2.0) 
3.3.2 Endemic steady state
4 Discussion
Formulating the ELNPM allows us to consider aspects of prion aggregate dynamics that cannot be explained by prior mathematical approaches that neglected the role of the Hsp104 chaperone in fragmentation. We first demonstrate that the NPM is a limiting case of the ELNPM and comment on implications of the ELNPM to the larger question of the appearance of prion strains in a population. We then demonstrate that the ELNPM is the first model capable of supporting two experimentally observed phenomena. First, the ELNPM is the first model capable of reproducing shifts in the aggregate densities associated with increases in synthesis of Sup35. Finally, we demonstrate that the binding kinetics of Hsp104 in the ELNPM allows the possibility of multiple coexisting prion strains. Intriguingly, the coexistence of multiple strains is thought to be crucial towards understanding the transmission of prion diseases between species (Chien et al. 2004).
4.1 The NPM is a limiting case of ELNPM
Our momentclosed ELNPM model, Eq. (), is nearly identical to the original NPM model but with a timevarying, effective fragmentation rate \(\gamma p(t)\) instead of the constant \(\gamma \). When \(p(t)\rightarrow \tilde{p}\), the dynamics of ELNPM will mirror that of the NPM with \(\gamma \) replaced by \(\gamma \tilde{p}\). As such, it is convenient to think of the NPM as a quasisteadystate approximation of the full enzyme kinetics we have considered in our model. We informally used this observation in Sect. 3.3 to motivate (but not prove) global stability based on known results of the NPM system.
We plot in Fig. 5 p(t) over the same parameters in Fig. 4 and note that—since yeast has a doubling time of roughly 90 min (Hartwell and Unger 1977) – p(t) will not reach its asymptotic value for a few celldivisions. As such, even though the NPM represents a quasisteadystate approximation it may not represent the aggregate dynamics during the early cell divisions following the introduction of prion aggregates.
4.2 Transient fragmentation efficiency may impact prion stability
Although the NPM may be viewed as an asymptotic simplification of the ELNPM, we remark that the differences in transient behavior may provide insight into the underlying stochastic dynamics that arise when a single prion seed is introduced into a healthy yeast colony. The ELNPM predicts an initial fragmentation rate that can be larger or smaller than the asymptotic rate—this is because the availability of enzyme (Hsp104) is much larger than the availability of substrate (binding sites) in this initial configuration (see Fig. 5).
4.3 Hsp104 acts as a rate limiter for fragmentation
4.4 Prion extinction and Hsp104 expression levels
Beyond translational shifts in aggregate size distribution, the \([\) PSI \({}^{+}]\) prion phenotype in yeast has been shown to be very sensitive to the amount of Hsp104 in the system (Higurashi et al. 2008; Doyle and Wickner 2009; Shorter and Lindquist 2008). Our mathematical formulation correctly captures the dependency of all prions to the underexpression of Hsp104. In contrast, and in agreement with recent experimental studies (Klaips et al. 2014), overexpression of Hsp104 does not necessarily drive prions to extinction.
First, sufficiently high concentrations of guanidine hydrochloride GdnHCl have been shown to severely disrupt the fragmentation process by inactivating Hsp104 (Ferreira et al. 2001; Byrne et al. 2007). Since fragmentation is halted the total number of aggregates will not change and aggregates will eventually be lost through dilution in the population due to cell division. Over time the population will be cured of the prion disease as the fraction of cells with aggregates approaches zero.
While earlier studies seemed to indicate overexpression of Hsp104 could cause loss of the prion state (Chernoff et al. 1995), more recent experimental evidence indicates that Hsp104 overexpression is not sufficient to drive prions to extinction (Klaips et al. 2014). In our formulation overexpressing Hsp104 drives \(p \rightarrow 1\). This is readily demonstrated by assuming \(k_1h = 1/\epsilon \gg 1\) and rescaling time by this quantity; then \(p' = 1p + O(\epsilon )\). However, as is experimentally, this alone is not mathematically sufficient to cure the prion state.
4.5 Stability of the coendemic prion strains
Up until this point we have considered the aggregates of a single prion species; however, a prion protein is capable of adopting a host of misfolded confirmations each of which is capable of the biochemical processes of conversion of normal protein and fragmentation (Sindi and Serio 2009; Tuite and Cox 2003; Tuite and Serio 2010). That is, \([\) PSI \({}^{+}]\) or PrP\(^\text {Sc}\) does not refer to a single prion phenotype, but many related ones, each characterized by different pathology (implying different kinetic parameter values). These distinct prion states are referred to as prion “strains.”
Biologists have observed multiple coexisting strains (Strbuncelj 2009; Polymenidou et al. 2005), but there has been limited mathematical modeling of multiple prion strains. Previously, Tanaka et al. (2006) considered the NPM, under the continuous relaxation of aggregate sizes, with \(n_0 = 1\) and demonstrated that if two strains were present then, asymptotically, one strain would dominate and drive the other to extinction. The outcome was determined by the strain which had maximized \(\beta \gamma \) (which is proportional to the reproductive number in the case of continuoussize, \(n_0=1\) NPM). Since level curves of \(\beta \gamma \) represent a set of measure 0 in parameter space, realistically this prevents asymptotic prion strain coexistence. By coexistence, we mean that there exists \(i \ne j\) such that \(\lim _{t\rightarrow \infty } \eta _i(t), \eta _j(t) > 0\) when \(\eta _i(0), \eta _j(0) > 0\) (where \(\eta _i\) is the concentration of aggregates of strain i).
Because our mathematical formulation also requires the molecular chaperone Hsp104, this opens up the possibility for an alternative pathway to prion strain coexistence—rather than outcompete solely on conversion \(\beta \) and fragmentation \(\gamma \), a second strain may be more efficient at sequestering Hsp104 (\(k_{\text {on}}/k_{\text {off}}\)). Increasing this ratio improves the strain’s own fragmentation efficiency, as well as decreases the other strain’s efficiency by decreasing the amount of available Hsp104.
Remarkably, Fig. 9b demonstrates that strain coexistence is possible for parameters lying in a nonzero area of parameter space. We note that, in contrast to prior models, this type of coexistence is biologically feasible because is it robust to small perturbations in parameter space. Thus, our numerical studies demonstrate that strains with different reproductive numbers (in isolation) can coexist. Further, at the coendemic state each strain’s “cooperative” reproductive number is different from their isolated value, but equal to each of the other strains’ cooperative numbers.
5 Conclusion
In this work we successfully developed a mathematical formulation of aggregate dynamics where fragmentation occurs through the molecular chaperone Hsp104. We demonstrate that, under certain restrictions, our model reduces to a numerically tractable form which we call the ELNPM. By including chaperonemediated fragmentation, this work represents an important step towards a more complete understanding of prion and protein misfolding in vivo.
We derived a unique diseasefree steadystate of the ELNPM and analyzed its stability. We demonstrated that the ELNPM supports experimentally observed results such as shifts in the aggregatesize distribution with increasing Sup35 synthesis and response to over and under expression of Hsp104. Additionally, it represents a first step towards quantifying prion strain coexistence.
While the ELNPM successfully describes the effects of varying amounts of Sup35 and Hsp104 in the system, we note that there are factors common to enzymesubstrate kinetics that were not included in our model. First, in many biochemical systems there is evidence of cooperation between binding sites (Nelson et al. 2008). Since there is no evidence of interaction between binding sites for Hsp104, we have modeled binding events as purely a function of the free enzyme and available binding sites.
Second, by assuming that \(k_{\text {off}}\) was large we were able to assume that for an aggregate consisting of i Sup35 monomers with j sites bound by Hsp104, all possible configurations of bound Hsp104 are equally likely (see the Supplemental Material). Since under normal expression Hsp104 is observed to be only minimally bound to \([\) PSI \({}^{+}]\) aggregates (Klaips et al. 2014), we interpret this to indicate that \(k_{\text {off}}\) must indeed be large relative to \(k_{\text {on}}\). Third, we considered a generalization of the uniform fragmentation kernel which corresponds to equality in fragmentation at all binding sites. Together, these three assumptions allowed the use of analytical approaches previously employed in the analysis of the NPM to demonstrate existence, uniqueness and asymptotic stability of the diseasefree steadystate.
Beyond Hsp104, other enzymes have been identified as important players in the dynamics of prion aggregate fragmentation (Inoue et al. 2004; Shorter and Lindquist 2008). As such, our mathematical formulation may be taken as the representing collective impact of enzymes on fragmentation. However, compared to the other enzymes, Hsp104 occurs in the lowest molecular number (Ghaemmaghami et al. 2003) and is thus likely to represent a rate limiting step. In addition, we have evaluated our model by comparison to experimental results on the \([\) PSI \({}^{+}]\) prion which has shown to have greatest sensitivity to Hsp104 (Higurashi et al. 2008). Lastly, again note that the form of Hsp104 is a hexamer (Doyle and Wickner 2009)—we have assumed the kinetics of hexamer formation are not relevant to the aggregate dynamics.
In addition to including additional biological complexities, in future studies we plan to investigate global asymptotic stability and explore the conditions underlying prion strain coexistence.
Notes
Acknowledgments
We acknowledge the financial support from NSF INSPIRE Grant #1344279 and the UC Merced School of Natural Sciences.
Supplementary material
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