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* Department of Chemical and Biological Engineering, Department of Biochemistry, Molecular Biology, and Cell Biology, and the Rice Institute for Biomedical Research, Northwestern University, Evanston, Illinois
Correspondence: Address reprint requests to Vassily Hatzimanikatis, Dept. of Chemical and Biological Engineering, Northwestern University, 2145 Sheridan Rd., Rm. E136, Evanston, IL 60208-3120. Tel.: 847-491-5357; Fax: 847-491-3728; E-mail: vassily{at}northwestern.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATHEMATICAL MODELSRESULTSDISCUSSIONAPPENDIX: CONDITION 2 IN...REFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATHEMATICAL MODELSRESULTSDISCUSSIONAPPENDIX: CONDITION 2 IN...REFERENCES |
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). Chaperones are essential for guiding folding and protecting against self-association of misfolded species into protein oligomers and aggregates. These aggregates are thought to interfere with the normal functioning of the cell and promote cell dysfunction and death (2,3). For neurons, this leads to a variety of neurodegenerative diseases including Parkinson's, Huntington's, Alzheimer's, ALS, Scrapie, and others (4).
The primary stress response chaperone (Hsp70) binds to motifs rich in hydrophobic residues, thus holding intermediates and guiding folding reactions. Consequently, the sequestering activity of chaperones serves to prevent misfolding and aggregation (5). Hsp70 together with its co-chaperones is also capable of escorting substrates to the proteasome for degradation (6). This process of chaperones targeting misfolded substrates either toward the refolding or degradation machinery is called protein triage (7).
Threshold phenomena, where changes in a control parameter result in a sharp change in an output, have been observed in a variety of biological systems including the Cdc2-cyclin B cell-cycle network, the activation of the JNK cascade, and the lysis/lysogeny switch in the 
-phage (8–12). The nonlinear phenomenon common to these different systems is the presence of a bistable switch that generates these thresholds. In a bistable system, two stable steady states (and a third, unstable steady state) co-exist for a certain range of system parameters; through perturbations in the system parameters or environmental conditions, the system can switch between the two states, in a threshold-dependent manner.
Observations of protein aggregation or refolding often reveal a similar sharp transition in the system, when changing the intrinsic properties of the aggregating species or the concentration of molecular chaperones. For example, in Caenorhabditis elegans, the shift between nontoxic and toxic protein aggregates occurs when expression of a protein containing a 35-mer polyglutamine repeat was increased to a 40-mer polyglutamine repeat (13). A threshold response was also observed for the reactivation of denatured G6PDH in vitro in the presence of DnaK, where a twofold increase in DnaK concentration resulted in more than a fourfold increase in G6PDH reactivation (14). The formation of yeast prion aggregates also displayed threshold-dependent formation of aggregate fibrils at different concentrations of Hsp104 (15).
We employed mathematical modeling to study the origin of threshold behavior in protein misfolding and aggregation in the presence of molecular chaperones. We demonstrate that the introduction of molecular chaperones into a model of protein aggregation creates a threshold in the model through bistability, and that the range of bistability is a function of chaperone concentration and the propensity for self-association of the misfolded protein species. We further use the models to study the potential of the system for stochastic switching from low to high concentrations of aggregates in individual cells. These models focus on the initial formation of soluble oligomers of an aggregation-prone protein, which recent experimental studies have implicated as the toxic precursor species in neurons (16,17).
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MATHEMATICAL MODELS |
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TOPABSTRACTINTRODUCTIONMATHEMATICAL MODELSRESULTSDISCUSSIONAPPENDIX: CONDITION 2 IN...REFERENCES |
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,19). Similar models of protein aggregation, which included explicit autocatalytic formation of the aggregates, have been used previously to describe the prion aggregation process (19,20). Second, we separated the growing aggregate into discrete aggregation states. In this formulation, we included new species for each additional monomer we allowed to incorporate into the aggregate nucleus. Common to both aggregate models was the inclusion of disaggregation and refolding steps, which we assumed were functions of the molecular chaperone concentration.
The lumped aggregate model
The lumped aggregate model (Fig. 1 A) simplified the in vitro experiments to three protein states: a folded state (F), an unfolded protein (U), and the lumped aggregate (A). Consistent with our model of an in vitro process, we assumed that the total protein concentration was constant,
 | (1) |
 | (2) |
 | (3) |
 | (4) |
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–24). Based on these studies, we included a growing aggregate nucleus in the model through the discrete addition of unfolded protein monomers to oligomers (Fig. 3; explanation of each species in Table 2). In contrast to the lumped aggregate model, the molecular chaperones were treated as individual, dynamic species in the discrete aggregate model. The total concentration of protein and molecular chaperones were assumed to be conserved in the discrete aggregate model,
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 | (6) |
 | (7) |
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 | (8) |
 | (9) |
 | (10) |
 | (11) |
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 | (14) |
 | (15) |
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATHEMATICAL MODELSRESULTSDISCUSSIONAPPENDIX: CONDITION 2 IN...REFERENCES |
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,26). The steady-state concentration of aggregates was determined from the condition that the dimensionless aggregation flux, va, equals the dimensionless disaggregation flux, vd:
 | (16) |
Condition 1
The maximal dimensionless rate of protein aggregation (vm,a) must be greater than the maximal dimensionless rate of disaggregation (vm,d). Otherwise, only one steady state will exist (Fig. 1 B, case I).
Condition 2
The slope of vd must be greater than the slope of va at [A] = 0, or only one or two steady states will be present. This condition can be satisfied through the combination of several parameters, including decreasing the relative affinities of the aggregates for self-association versus association with chaperones (Fig. 1 B, case III), or increasing the chaperone concentration or activity for disaggregation (see Appendix).
With the regime of bistability in the model established, we tested the effect of variations in the protein quality control machinery on the appearance of aggregation thresholds. Conditions 1 and 2 are functions of the maximal rates of disaggregation and refolding. Therefore, changes in the molecular chaperone concentration affected the number of steady states possible in the system. Varying these maximal rates in the model confirmed that above a critical threshold in molecular chaperone concentration there was only one steady state of the network, mostly folded proteins, and no aggregated protein (Fig. 2, A–C). However, once the chaperone concentration was lowered below this critical threshold, a new state emerged, which contained the proteins in aggregated form. Cells that operate at a chaperone concentration below this bifurcation point can effectively switch between the two states; in other words, cells with no aggregates could switch to the high aggregation state if an aggregate seed of sufficient size appeared in the system.
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,28). The analysis also provided the values of dimensionless parameters that gave rise to bistability, which resulted in physiological ratios for the rates and affinities of the different biophysical processes (Table 3). For example, we chose the dimensionless affinity of the chaperones for their substrates to be significantly higher than the dimensionless affinity of the unfolded proteins for self-association to reflect the use of ATP to drive tight associations of Hsp70 with its substrates (29). Similarly, we chose the dimensionless rate constant for spontaneous refolding (
) as several orders-of-magnitude slower than the dimensionless rate constant of chaperone-assisted refolding (
f) to reflect the established catalytic role of chaperone protein refolding. Finally, we used the common assumption of fast reversible rate constants by choosing the dimensionless reversible rate constants (
) to be several orders-of-magnitude greater than the catalytic rate constants (x). Interestingly, similar pathways or processes had comparable orders of magnitude for their rate and equilibrium constants. Also interesting, the two Conditions of the lumped aggregate model for bistability also held for the dimensionless parameters of the discrete aggregate model. This demonstrated the utility of simple models, such as the lumped aggregate model, for elucidating the features or design of more complicated biological systems. Using the parameters in Table 3, we simulated the discrete aggregate model with varying total molecular chaperone concentration. The discrete aggregate model displayed bistability similar to the lumped aggregate model (Fig. 4 A). However, this detailed mechanistic model provided better insight on the response of protein aggregation to changes on chaperone activity, and allowed us to investigate the impact of other physicochemical parameters on the bistable behavior of the system.
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). However, the ratio of molecular chaperones to total substrate where the threshold transition to low or high aggregate concentrations occurs is dependent on both the type of chaperone and the substrate. For example, the in vitro threshold for reactivation of G6PDH by DnaK occurred between a ratio of concentrations of 3–6 for DnaK to G6PDH (14). In general, we would not expect an exact correspondence between the values of the model parameters and the experimentally observed threshold values because of unaccounted-for processes in vivo, such as chemical modifications of protein substrates, and an unknown relationship between the concentration of molecular chaperones and their effective activity for protein quality control.
In the discrete aggregate model, after the initial formation of the A2 dimer, subsequent aggregation steps were more thermodynamically favorable, a fact that effectively introduced the feedback activation of aggregate formation found in the lumped aggregate model. To assess the dependence of bistability on the relative equilibrium constants between unfolded protein and dimer/oligomer concentration, we performed a two-parameter bifurcation analysis (Fig. 4 B). We specifically investigated the behavior of the system for the different chaperone concentrations when the initial aggregate formation was thermodynamically less favorable 
equally favorable 
and more favorable 
than the larger aggregate formation. In all three regimes, the model displayed bistability. However, the critical chaperone concentrations depended on the equilibrium constants of aggregate formation; the bistable switch occurred at higher chaperone concentrations if the formation of the first, small oligomers was more favorable that the larger ones (Fig. 4 C). This implied that the formation of small oligomers was the determining step for initiation of protein aggregation.
The molecular chaperones are necessary for bistability
The process of protein aggregation, without molecular chaperones, is a competition between the thermodynamic driving forces for aggregation versus spontaneous refolding. The driving force of aggregation is proportional to the equilibrium constant between the unfolded monomers and the small oligomers 
and spontaneous refolding is inversely proportional to the equilibrium constant of unfolding 
Therefore, we quantified the competition between refolding and protein aggregation by defining a new dimensionless parameter,
 | (17) |
fold, the presence of a bistable transition in the system was only observed for non-zero values of chaperone concentration (Fig. 5 A). In fact, using deficiency theory (27,28), we determined that the refolding and disaggregation activities of the molecular chaperones were necessary for a bistable threshold to exist in the discrete aggregate model. Below the bistable regime, the total aggregate concentration decreased monotonically as the fold was increased to favor refolding (Fig. 5 B). However, within the bistable regime, the appearance of a second stable steady state allowed the system to transition from relatively high aggregate concentration to low aggregate concentration, without any change in the thermodynamics of refolding and aggregation. These results suggested that even at low chaperone concentrations and high aggregation potential, the chaperone quality control mechanism is capable of preventing aggregation through the introduction of this bistable threshold.
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,31). The intrinsic noise in a bistable intracellular process has the potential for the fluctuations in the concentration of certain species to cross a critical threshold and switch between steady states. Such phenomena occur in several biological systems such as the Sonic-Hedgehog signaling and the lysis-lysogeny switch of the bacteriophage- (12,32,33).
We performed a stochastic simulation of the discrete aggregate model to investigate the effect of intrinsic noise in protein aggregation. We simulated the stochastic dynamics of our system using the Gibson algorithm, an efficient modification of the Gillespie algorithm (34–36), and we studied the effect of intrinsic noise on the formation of aggregates within the low-chaperone concentrations, high-aggregate regime (Fig. 6 A). The deterministic response reached 50% of its maximal value at time = 9.5 (all times are dimensionless); however, only 63% of the cells had reached their 50% maximal value at this time (Fig. 6 B). The most rapid aggregating cells achieved 50% maximal aggregation in approximately half the time of the deterministic simulation (time 
5), and the slowest aggregating cells did not reach their 50% maximal value until time 25. Additionally, the wide distribution of transition times from low to high aggregated states led to a distribution of aggregation states (Fig. 6 C); in other words, the initially homogeneous population underwent a transition through which the population was distributed between a low and high protein aggregate concentration.
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,15). Thus, a perturbation to the protein quality control machinery that results in the system crossing the bifurcation point to high aggregate concentration will lead to rapid formation of protein aggregates. However, additional timescales influence the time of onset for protein aggregates in vivo. These processes include the time for expression and accumulation of disease-causing proteins, and the slow time for changes to occur in the activity or concentration of the protein quality control machinery. |
DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATHEMATICAL MODELSRESULTSDISCUSSIONAPPENDIX: CONDITION 2 IN...REFERENCES |
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,15). Additionally, the expression of different length polyglutamine expansions in C. elegans showed a polyQ-dependent increase in the number of observed aggregates, and a dramatic decrease in the time of aggregation onset between a polyglutamine expansion of Q35 and Q40 (13). In vivo aggregation processes are far more complicated than the reactions in the model. For example, the model does not contain any explicit information about chemical modifications of polyQ proteins post-aggregation. However, the model does implicitly capture the effect of expanding the polyglutamine repeat, which in the model is equivalent to increasing the propensity of the proteins for self-association. Based on the analysis of our models, we proposed that a nonlinear phenomena known as bistability caused the threshold. The presence of bistability in the system has significant implications for the onset and potential prevention of protein aggregation. Potential therapies must account for hysteresis in bistable systems. For example, reduction of the concentration of molecular chaperones drives the system to the upper aggregate state. Restoration of the system to the low aggregate steady state requires an increase of the concentration of the molecular chaperones above the level at which bistability exists to overcome the hysteresis phenomena due to bistability.
The essential nature of the molecular chaperones machinery for generating the bistable switch revealed an interesting interplay between protein aggregation and molecular chaperone production and potential consequences for the failure of the heat-shock response. Our analysis showed that the molecular chaperones maintained the system in the low aggregate steady state, even as conditions increasingly favored the formation of protein aggregates. However, one of the unique aspects of many neuronal cell types is their inability to induce Hsp70 expression above basal levels under stress (37,38). Therefore, cellular stresses coupled with slow degradation of chaperones activity or concentration could cause the system to cross the threshold from the bistable region to the high-aggregate steady state.
In vivo, each individual biophysical process of the model is more complicated than simple self-association or chaperone binding; mechanistically, each process might require the binding of cofactors, hydrolysis of ATP, etc. However, the use of different models and the parametric analyses showed that the basic observation of bistability is robust to several mechanistic assumptions and parameter values. The finding that a simple mechanism generates the expected threshold is significant because there are many in vivo mechanisms for irreversible protein aggregation. For example, for the particular case of polyQ-associated disorders, the aggregates are cross-linked by transglutaminase (39). This process of cross-linking generates irreversible aggregates. Although this cross-linking contributes to the long-time formation of Huntingtin aggregates, the model demonstrates that this modification and other possible biochemical modifications of the protein aggregates are not necessary for the system to display a switch to high levels of protein aggregates. The origin of the bistable switch is in the feedback from the cellular protein quality control machinery, and its interactions (or failure to interact) with the misfolded substrates.
It is important to distinguish between the steady state of the protein aggregates and the timescale for the appearance and formation of protein aggregates. Steady-state analysis indicates whether aggregates will form, given certain intrinsic and extrinsic properties of the system. Many of these properties influence the timescale or dynamics for the appearance and formation of aggregates. The activity of the molecular chaperones is an extrinsic property of the system that may drift (become impaired) over years or decades, and this drift in activity may cause the system to cross the steady-state threshold to the high aggregate steady state and induce the formation of protein aggregates. In addition, we showed that intrinsic stochastic fluctuations influenced the timescale for formation of protein aggregates. These fluctuations led to the ensembles forming aggregates over a broad distribution of times. Thus, individuals with the same genotype can form aggregates at very different ages. However, the current state of the model does not address at what point protein aggregates become toxic to neurons and lead to the onset of neurodegenerative disorders, since, like most of the diseases, this depends on the function of many other cellular processes.
A whole genome RNAi screen in C. elegans identified 186 genes whose protein products act as modifiers of protein aggregation (40). This revealed that the prevention of protein aggregation is not the sole responsibility of the chaperone machinery, but involves the complex interaction of many essential intracellular processes. For example, of the 186 genes, 14 genes were associated with protein degradation pathways and the proteasome. Chaperones enhance the in vivo rate of dissociation and degradation of aggregates. Because enhanced clearance of aggregates also requires the degradative machinery, we can suggest that it is both the chaperone folding machinery and the degradative/clearance machinery of proteasomes, and that autophagy must be involved. Although chaperones are an essential component of these events, they are not the only components necessary for the proper dissociation of damaged proteins in the cell. Therefore, a full description of the in vivo process and identification of potential points of failure of protein homeostasis will require a more detailed description of this process and its interaction with the molecular chaperone machinery and protein aggregates (41). Mathematical models allow us to consider each essential piece of the complex biology individually and to begin to incorporate these components into a more complete theory for the design of protein quality control in the human cell.
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APPENDIX: CONDITION 2 IN THE LUMPED AGGREGATE MODEL |
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TOPABSTRACTINTRODUCTIONMATHEMATICAL MODELSRESULTSDISCUSSIONAPPENDIX: CONDITION 2 IN...REFERENCES |
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 | (18) |
 | (19) |
 | (20) |
Taking the derivative of vd and va with respect to a:
 | (21) |
 | (22) |
The second term in Eq. 22 was neglected since a = 0. However, to evaluate the first term, we solved for u as a function of a. We solved for u as a function of a using the conservation of total protein (Eq. 4) and the steady-state expression for f:
 | (23) |
 | (24) |
m,f, 
= 1 + 2(m,f – vm,f), and = a – 2(m,f – vm,f + 1). Combining Eq. 24 with Eq. 4,
 | (25) |
The resulting expression for Eq. 22, at a = 0:
 | (26) |
Therefore, the full Condition 2:
 | (27) |
Submitted on May 15, 2005; accepted for publication October 25, 2005.
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is the reverse or backward step for reversible reaction ‘i,’ and
is the equilibrium constant for a set of reversible reactions ‘i’. For example, 
is the equilibrium constant associated with the reversible binding of HSP to A2 (forming A2:HSP), 
is the rate constant for dissociation of the A2:HSP complex back to A2 and HSP, and kd2 is the rate constant of the catalytic disaggregation of A2:HSP to U:HSP and U. For clarity, only the association is shown explicitly for each reversible reaction.
and scaled HSP concentration. Horizontal lines indicate 
: 
(solid line), 
(dashed line), and 
(dash-dot line). (C) Steady-states of state-scaled aggregated protein versus scaled HSP concentration with the three ratios of 
from B. (Solid line) 
Thin-line segments denote the unstable steady state.
). (B) Time of crossing to high aggregate concentration. The time [A]tot/[P]tot = 0.50 for each cell is summarized in the histogram. The deterministic time is included for reference (dashed line). (C) The scaled aggregate concentration in each cell at time = 9.5. The deterministic value is included for reference (dashed line).