A mechanistic model to predict effects of cathepsin B and cystatin C on β-amyloid aggregation and degradation

β-Amyloid (Aβ) aggregation is thought to initiate a cascade of neurodegenerative events in Alzheimer's disease (AD). Much effort is underway to develop strategies to reduce Aβ concentration or inhibit aggregation. Cathepsin B (CatB) proteolytically degrades Aβ into non-aggregating fragments but is potently inhibited by cystatin C (CysC). It has been suggested that decreasing CysC would facilitate Aβ clearance by relieving CatB inhibition. However, CysC binds Aβ and inhibits Aβ aggregation, suggesting that an intervention that increases CysC would prevent Aβ aggregation. Both approaches have been tested in animal models, yielding contradictory results, possibly because of the opposing influences of CysC on Aβ degradation versus aggregation. Here, we sought to develop a model that quantitatively predicts the effects of CysC and CatB on Aβ aggregation. Aβ aggregation kinetics in the absence of CatB or CysC was measured. The rate constant for Aβ degradation by CatB and the equilibrium constant for binding of CysC to Aβ were determined. We derived a mathematical model that combines material balances and kinetic rate equations. The model accurately predicted Aβ aggregation kinetics at various CatB and CysC concentrations. We derived approximate expressions for the half-times of degradation and aggregation and show that their ratio can be used to estimate, at any given Aβ, CatB, or CysC concentration, whether Aβ aggregation or degradation will result. Our results may be useful for designing experiments and interpreting results from investigations of manipulation of CysC concentration as an AD therapy.

␤-Amyloid (A␤) aggregation is thought to initiate a cascade of neurodegenerative events in Alzheimer's disease (AD). Much effort is underway to develop strategies to reduce A␤ concentration or inhibit aggregation. Cathepsin B (CatB) proteolytically degrades A␤ into non-aggregating fragments but is potently inhibited by cystatin C (CysC). It has been suggested that decreasing CysC would facilitate A␤ clearance by relieving CatB inhibition. However, CysC binds A␤ and inhibits A␤ aggregation, suggesting that an intervention that increases CysC would prevent A␤ aggregation. Both approaches have been tested in animal models, yielding contradictory results, possibly because of the opposing influences of CysC on A␤ degradation versus aggregation. Here, we sought to develop a model that quantitatively predicts the effects of CysC and CatB on A␤ aggregation. A␤ aggregation kinetics in the absence of CatB or CysC was measured. The rate constant for A␤ degradation by CatB and the equilibrium constant for binding of CysC to A␤ were determined. We derived a mathematical model that combines material balances and kinetic rate equations. The model accurately predicted A␤ aggregation kinetics at various CatB and CysC concentrations. We derived approximate expressions for the half-times of degradation and aggregation and show that their ratio can be used to estimate, at any given A␤, CatB, or CysC concentration, whether A␤ aggregation or degradation will result. Our results may be useful for designing experiments and interpreting results from investigations of manipulation of CysC concentration as an AD therapy.
Alzheimer's disease (AD) 2 is the most common cause of dementia, estimated to affect over 44 million people worldwide. According to the amyloid cascade hypothesis, the initiating event in AD occurs when the membrane-bound amyloid precursor protein (APP) is sequentially cleaved into a small peptide fragment, ␤-amyloid (A␤), and secreted into extracellular fluid (1,2). Processing of APP by a ␤-site-cleaving enzyme (␤-secretase) initiates the amyloid secretory pathway, followed by liber-ation of the peptide from its transmembrane domain by a ␥-site-cleaving enzyme (␥-secretase). Specificity of the ␥-secretase is broad, and cleavage generates various lengths of A␤, of which the most common are A␤ 40 and A␤ 42 (3). After liberation (and possibly before secretion), A␤ self-associates into small, soluble oligomers that mature into larger fibrillar structures. Eventually, aggregates precipitate and deposit on neuronal tissue as amyloid plaque. Aggregates are thought to be neurotoxic (4), with smaller A␤ oligomeric intermediates believed to be more toxic than mature fibrils (5). For this reason, the kinetics of amyloid protein aggregation have been studied extensively (6 -10). Although A␤ 40 concentration is higher than A␤ 42 in human AD brain tissue, A␤ 42 is believed to be more aggregation-prone and more toxic (2).
Cystatin C (CysC) is a small (13.4 kDa) extracellular protein that is ubiquitously synthesized and is at relatively high concentrations in cerebrospinal fluid (CSF) (11). The link between CysC, A␤, and AD was identified years ago with the discovery that CysC and A␤ co-deposit in cerebrovascular and senile plaques of AD patients (12,13) and in amyloid deposits in the cerebrovasculature of patients with cerebral amyloid angiopathy (14 -16). More recently, soluble CysC-A␤ complexes were detected in CSF from AD patients (17). CysC has been reported to bind A␤ and prevent further aggregation in vitro (18 -20) and to protect cultured neurons from A␤ toxicity (21). Binding epitopes have been proposed based on molecular docking simulations and limited proteolysis experiments (22). CSF CysC levels were reported to be lower in AD than in healthy controls (23,24), whereas plasma CysC levels were reportedly higher in AD (25), possibly because high levels are associated with an anti-inflammatory response. CysC levels may be low in patients with mild cognitive impairment but later increase as a general response to cell damage (25).
CysC is an inhibitor of cysteine proteases, including those in the cathepsin family. Cathepsin B (CatB), a cysteine-type protease of the lysosomal/endosomal pathways, is specifically inhibited by CysC with sub-nanomolar affinity (20,26,27). Although CatB is mainly active intracellularly, it can be secreted. Should CatB escape the cell (in the case of apoptosis or secretory pathway activation), CysC has the important role of suppressing proteolysis in the extracellular space (28). CatB has been shown to degrade A␤, primarily through C-terminal truncation (29). Acidic pH favors rapid sequential conversion of A␤ 42 into A␤ 40 and then finally to the less-amyloidogenic A␤ 38 (30). Reaction rates are slower at neutral pH but not completely diminished (30). CatB secretion from neuronal cells following stimulation by A␤ aggregates has been documented (31,32), and secreted CatB is known to retain a significant fraction of its activity (33,34). There is conflicting data with regard to the relationship of CatB to AD; CatB was expressed at lower levels in AD patient monocytes compared with healthy controls in one study (35), but in another study, CatB levels were higher in plasma of AD patients compared with controls (36).
Because of their roles in preventing fibrillation and in degrading A␤, manipulation of CysC or CatB has been identified as a potential AD therapeutic. However, there is disagreement as to whether CysC or CatB levels should be modulated and whether the levels should be suppressed or enhanced. Some researchers have reported neuroprotective effects of CysC enhancement in animal models. Mi et al. (37) found that transgenic mice expressing human CysC at ϳ2-fold higher levels than endogenous cystatins showed a marked decrease in A␤ fibril deposition. Kaeser et al. (38) showed that CysC overexpression reduced plaque formation in human APP-transgenic mice. In another study using CatB-deficient mice, overexpression of CysC reduced the total amyloid plaque load (39).
Other researchers have found evidence that CatB proteolysis of A␤ is an important neuroprotective mechanism and that high amounts of CysC interfere with A␤ degradation. CatB enhancement in transgenic AD mouse models reduced both the total amount of soluble A␤ 42 and the ratio of A␤ 42 /A␤ x , indicating CatB may preferentially degrade A␤ 42 relative to other forms (30). Sun et al. (39) found mice overexpressing CatB lacked symptoms of cognitive impairment and premature mortality compared with CatB-free controls. These researchers also showed that expression of CysC could neutralize the neuroprotective effects of CatB. Similarly, Wang et al. (40) overexpressed neuronal CatB in transgenic mice and observed significantly lowered A␤ 42 levels, without affecting APP processing. Transduction of neurons to express CatB accelerated A␤ degradation and suppressed its production in vitro and led to a reduction of both A␤ 40 and A␤ 42 (41). Deletion of stefin B (a cysteine protease inhibitor) in an AD mouse model reduced A␤ accumulation and inhibited development of learning and memory deficits (42).
Others interpret the outcome of studies in light of putative ␤-secretase activity by CatB (3,29). Inhibition of CatB reduced plaque load and improved cognitive defects in transgenic mouse models; this was postulated to be due to reduced cleavage of APP into A␤ (43,44). Wang et al. (45) found that exogenously-applied CysC reduced soluble A␤ levels in animal studies, which was attributed to CysC-mediated reduction in ␤-secretase activity.
Differing results in transgenic animal models may be caused by differences in concentrations of A␤, CysC, and/or CatB. In particular, CysC plays two conflicting roles in the regulation of A␤ as follows: by binding to A␤ it can inhibit A␤ aggregation, but by binding to CatB it inhibits A␤ degradation. CysC and CatB interactions have been postulated to form key control mechanisms to regulate A␤ aggregation potential (39,43,47). The aim of this study was to quantify the effects of CysC and CatB on the fate of A␤. We modeled the protein interactions by considering binding, degradation, and aggregation reactions, and then we measured each relevant thermodynamic and kinetic parameter. We sought to understand the sensitivity of A␤ aggregation to perturbations in CysC/CatB balance and to be able to predict the outcome simply by knowledge of total CysC and CatB concentrations.

A␤ aggregation kinetics
We measured A␤ aggregation kinetics at three different concentrations using ThT, a dye that fluoresces when bound to amyloid fibrils. Briefly, A␤ 40 (8,28, and 46 M) or A␤ 42 (2.5, 10, and 20 M) was loaded into microwells and heated to 37°C. Sample fluorescence was continuously monitored for up to 72 h. Data for each of three replicates are shown in Fig. 1. Postaggregation, A␤ 40 at 28 M was centrifuged, and the peptide content of the supernatant was measured. No peptide was detected (assay sensitivity Ͻ500 nM), so we concluded that all A␤s had converted to aggregated species. Therefore, at long time the fibrillar A␤ concentration (in equivalent monomer molar units) equaled the total A␤ concentration [A␤] 0 . To convert ThT fluorescence intensity to concentration of fibrillar A␤, we plotted the maximum ThT signal (at 72 h) against [A␤] 0 and fit the data to the polynomial shown in Equation 1,

Modeling CysC/CatB effect on A␤
where ThT b is background signal, and a and b were determined by least-squares fitting (Table 1). Equation 1 was used to convert ThT intensity at all time points to [A␤ agg ], where [A␤ agg ] is the concentration of aggregated A␤ expressed in equivalent monomer molar concentration. Aggregation data were fit to several different simple kinetic models, including the monomer partitioning, Gompertz, and autocatalytic models (supplemental Fig. S1) (6,48). The best global fits were obtained by the autocatalytic model, a two-step model that includes a first-order irreversible conversion of monomer to aggregation-competent form and a secondorder reaction between monomer and aggregate to produce more aggregate (6) as shown in Model 1.

Model 1
This kinetic model has an analytical solution shown in Equation 2, where [A␤ agg ] is the monomer-equivalent concentration of A␤ aggregates at any time t. Although such a two-step model cannot account for oligomer or protofibril formation, protofibril association, fragmentation, or other events (9), it does capture the sigmoidal shape and concentration dependence of the data, and it can describe accumulation of ThT-positive aggregates. Aggregation data for six (A␤ 40 ) or nine (A␤ 42 ) independently prepared samples were globally fit to Equation 2 by leastsquares regression to recover k 1 and k 2 for both A␤ 40 and A␤ 42 ( Table 2). Best fit lines were generated and compared with each aggregation data set (Fig. 2). Although k 2 was similar for the two A␤ forms, k 1 was 5000-fold higher for A␤ 42 than for A␤ 40 , consistent with its higher aggregation propensity.

CysC-A␤-binding interaction
CysC co-localizes with A␤ in AD plaques (12,49). A few studies show that CysC inhibits A␤ fibril formation in vitro, possibly redirecting A␤ self-association toward amorphous non-fibrillar aggregates (12,50). It is unknown whether CysC predominantly binds monomeric or aggregated A␤. We designed a FRET assay to measure CysC-A␤ binding in solution (Fig. 3a). Data were normalized to the maximum observed FRET efficiency and expressed as a fraction of bound A␤ (Fig.  3b). Data from three independent series were fit globally to determine the binding constant, K D , and the cooperativity parameter n. n was found to be 0.94 Ϯ 0.05, so the data were re-fit with n ϭ 1, to obtain a final value for K D of 72 Ϯ 6 nM ( Table 2). Our estimate is slightly weaker than that derived from an ELISA method, where K D was estimated to be 10 nM (50). a Parameters a and b were determined by non-linear least-squares fitting of ThT fluorescence data at 72 h to ThT Ϫ ThT b ϭ a͓A␤͔ 0 ϩ b͓A␤͔ 0 2 (Equation 1), where ͓A␤͔ 0 is the total A␤ concentration in M and ͓A␤ agg ͔ ϭ ͓A␤͔ 0 at 72 h. ThT b was determined by averaging three sample wells containing ThT but no A␤ and was found to be 520 Ϯ 40. Error estimates are 95% confidence intervals.  These data were not included in the data sets for parameter estimation. Because essentially no aggregation was detected, the information content is less, and the data put an upper but not lower bound on k 1 and k 2 .

Modeling CysC/CatB effect on A␤ CatB degradation of A␤
At neutral pH, CatB proteolysis of A␤ proceeds via C-terminal truncation; products of A␤ 42 degradation were reported to be A␤ 40 , A␤ 38 , and A␤ 33 (30). Preliminary experiments in our laboratory were consistent with this finding (data not shown).
To determine degradation rate constants, fixed concentrations of CatB were incubated with various amounts of A␤ 40 or A␤ 42 in physiological conditions (pH 7.4, 37°C). Degradation over time was measured by dot-blot assay, where membranes were probed using antibodies specifically against the C terminus of A␤ 40 or A␤ 42 (Figs. 4a and 6, a and b). Measurement linearity was determined by the concentration standard curves (Fig. 4b), and only those samples in the linear concentration region were reported. Sample intensity was converted to A␤ concentration using the standard curves. A␤ 40 concentration could reliably be determined as low as 50 nM, whereas A␤ 42 could only be determined down to 200 nM. With A␤ 40 , concentration decreased over time as expected (Fig. 4a). Concentrations over short time periods (1 h) were plotted to determine the initial reaction rate (Fig. 5a), and initial rates were plotted against initial A␤ concentrations to determine the degradation rate constant k deg ( Fig. 5b and Table 2). A␤ 42 degradation kinetics were measured in a similar manner (Fig. 6a), except A␤ 42 blots were stripped and re-probed with anti-A␤ 40 to show the generation of A␤ 40 directly from A␤ 42 as a result of CatB activity (Fig. 6b). Quantitative densitometry analyses at 600 nM A␤ 42 are shown (Fig. 6, c and d), demonstrating the close temporal correspondence between loss of antibody binding to the C terminus of A␤ 42 and gain of anti-A␤ 40 -specific antibody binding. Values of k deg for A␤ 40 and A␤ 42 were not statistically different.

CatB degradation of CysC
Optimal CatB proteolytic activity occurs in acidic conditions (pH Ͻ6), which is relevant for CatB contained within endocytic and lysosomal vesicles (51). Upon secretion to neutral CSF, CatB activity is expected to decrease, but not deactivate entirely (33,34). Others have reported that secreted cathepsin D specif-

Modeling CysC/CatB effect on A␤
ically degrades extracellular CysC in acidic to near-neutral tumor environments (52). We speculated that CatB could similarly degrade CysC, reducing CysC inhibitory capacity and promoting uncontrolled proteolysis. When considering the interaction between CatB and CysC, a complication arises due to coupling of the inhibitory and degradation reactions. To measure true degradation kinetics, it is necessary to first "turn off" inhibitory binding modes between CysC and CatB. Using site-directed mutagenesis, we produced and purified W106G CysC, a variant which has been shown to be a poor inhibitor of CatB (53). We confirmed by CatB activity assay that W106G inhibition constant is far weaker than WT (supplemental Fig.  S3). The CD spectrum of W106G was identical to WT, confirming that loss of activity is due to specific loss of interaction, rather than loss of protein fold (supplemental Fig. S2).
We used an immunoblot assay to measure the kinetics of W106G CysC degradation by CatB at 37°C for up to 2 h. There was no detectable CysC degradation at pH 7.4 (supplemental Fig. S4). We stripped and re-probed each membrane with a CysC N-terminal antibody (C-27) and a third antibody that does not overlap with the N or C terminus (Cyst24); neither showed significant degradation (data not shown). At pH 6.0, W106G degradation was noticeable (supplemental Fig. S4). WT was degraded much more slowly due to coupling of the inhibition reaction to degradation.

Model development
We developed a kinetic model to predict the effect of CysC and CatB on A␤ aggregation and degradation. CysC plays two distinct roles in this model: by binding to A␤ it inhibits A␤ aggregation, but by binding to CatB it inhibits A␤ degradation. We assumed all binding reactions are reversible and sufficiently fast to be assumed always at equilibrium, with equilibrium dissociation constants of K D (binding of CysC to A␤ 40 or A␤ 42 ) and K I (binding of CysC to CatB). We assumed that CysC binds only non-aggregated A␤. A␤ is enzymatically degraded by CatB, by the rate constant k deg . We assumed that CatB degrades A␤ monomers but does not degrade A␤ aggregates. Degradation of CysC by CatB was not included because it was negligible at pH 7.4.
The mathematical model includes coupled algebraic and differential equations. Material balance Equations 3 and 4 are derived assuming that the concentration of CatB bound to the substrate is negligibly small (K m Ͼ Ͼ [S], see the supplemental text for justification). Free CatB or CysC concentration [CatB] or [CysC], respectively, at any time is calculated as the initial concentration ([CysC] 0 or [CatB] 0 , respectively) minus the protein that is bound in the CysC-CatB complex or (for CysC) to A␤, where K D and K I are given in Table 2.
A␤ monomer material balances account for decreases due to binding to CysC, enzymatic degradation, or aggregation. Additionally, A␤ 40 can be generated from A␤ 42 degradation as shown in Equations 5 and 6. The rate of production of degraded or aggregated A␤ is described by differential Equations 7-10, (Eq. 8)  with rate constants determined experimentally and summarized in Table 2. We assumed no cross-aggregation of A␤ 40 and A␤ 42 . The set of equations is initialized by first simultaneous solution of the coupled material balance equations (Equations 3-6) given the initial concentrations of CysC, CatB, and A␤. Then the differential equations (Equations 7-10) were solved numerically, with iterative solutions of the material balances as intact monomeric A␤ is lost. We solved the model equations to calculate A␤ aggregation kinetic profiles at several different CysC and CatB concentrations. Specific conditions tested are shown in Table 3. We then measured A␤ aggregation kinetics at these conditions, using ThT fluorescence, and we compared experimental data to model simulations. There were no additional adjustments made to the model. Model output and data are shown superimposed in Fig. 7. Model simulations accurately captured the experimental data. At 2.5-fold excess A␤, CysC was able to significantly delay aggregation of A␤ 40 but had only a minor impact on A␤ 42 aggregation (cases ii and vii). In the absence of CysC, CatB degradation of A␤ was sufficiently fast to completely suppress A␤ aggregation (cases v and ix). When both CatB and CysC were added (at up to 15-fold excess CysC), the lag time for onset of aggregation was delayed, and the steady-state amount of aggregated A␤ was lowered due to degradation (cases iii, iv, and viii).

Comparison of time constants for degradation versus aggregation
We next sought to develop a simple parameter to predict whether A␤ aggregation or degradation would be the dominant outcome of A␤-CysC-CatB interactions. We defined a parameter , as shown in Equation 11, where t 50, deg is the time required for degradation of 50% of A␤ in the absence of any aggregation, and t 50, agg is the time required for aggregation of 50% of A␤ in the absence of any degradation. ¡ 0 if degradation is the primary outcome, and 3 ∞ if aggregation is the primary outcome. Aggregation time is calculated from Equation 12.  Table 3 Model validation sample set A␤, CysC, and CatB were mixed to the final concentrations given and incubated at 37°C with 10 M ThT. A␤ aggregation over time is shown in Fig. 7 and is compared with model predictions.

Modeling CysC/CatB effect on A␤
To a first approximation, degradation time is shown in Equation 13, (see supplemental text for derivation).
We solved the full-scale numerical model for several different A␤, CatB, and CysC concentrations and plotted the fraction of aggregated A␤ at long time from the simulations versus the calculated (Fig. 8a). The calculations collapse onto a set of overlapping curves. This plot falls into two clear regimes: aggregation-dominant ( Ͼ Ͼ 1) and degradation-dominant ( Ͻ Ͻ 1), with a sharp transition at ϭ 1. This behavior was identical for A␤ 40 and A␤ 42 , meaning that degradation and aggregation rates used to calculate are properly scaled. From this result, we can quantitatively predict whether A␤ will be completely degraded or will end up in toxic aggregated forms, simply from knowledge of initial concentrations of A␤, CatB, and CysC.
For illustration purposes, we calculated at 10 nM A␤ 40 over a range of CatB and CysC concentrations (Fig. 8b). This figures shows, for example, that switching from aggregation ( ϳ5) to degradation (ϳ0.5) at constant CatB requires an ϳ10-fold decrease in CysC.

Discussion
We developed a simple kinetic model to predict the outcome of interactions between CatB, CysC, and A␤ on A␤ aggregation and degradation. Several approximations and assumptions were made in developing the model that deserve brief comment.
ThT fluorescence is commonly used to measure aggregation of A␤ and other amyloidogenic proteins. We found that the maximum ThT intensity was not linear with total fibril mass, in agreement with other work (10, 54, 55). Non-linearity could arise due to differences in ThT-binding modes at high fibril

Modeling CysC/CatB effect on A␤
concentration and accompanying changes in absorbance (56). Additionally, self-quenching effects have been demonstrated at high binding site occupancy (54), which is expected when total fibril mass is small compared with ThT concentration. To account for non-linearity, we fit a polynomial function to a plot of the maximum signal versus aggregate mass and then used this fit to correct all other aggregation samples. Because we used a single experimental output, ThT fluorescence, as a measure of aggregation, we chose to describe the aggregation kinetics using a simple two-parameter model that captures the conversion of A␤ from ThT-negative to ThT-positive aggregates. The autocatalytic model was able to capture the shape and concentration dependence of the experimental data for both A␤ 40 and A␤ 42 and to return reasonable values for the two parameters. That said, the fact that the equations fit the data should not be interpreted as indicating that the mechanism is a correct description of the underlying self-association process. In particular, this simple model neglects A␤ oligomer formation, fibril elongation, fibril fragmentation, and other steps in the complex A␤ self-assembly process.
We assumed that CysC binds only the monomeric (unaggregated) form of A␤. This reduces the complexity of the model, as it eliminates the need to account for different aggregate sizes and to determine the binding affinity as a function of aggregation state. There is some experimental support for this assumption. Mi et al. (17) found stable CysC-A␤ complexes in brain homogenates corresponding only to monomer-monomer interactions and only in non-demented control subjects. No such complexes were found in AD patients. The researchers speculated that A␤-CysC binding was competing with A␤ selfassociation, with the latter dominating in AD patients. Other protein binders of A␤ are thought to modulate aggregation through more complex mechanisms. For example, transthyretin has been shown to preferentially bind small aggregates of A␤ rather than A␤ monomers (57), resulting in suppression of fibril elongation and lateral association but not initial nucleus formation (58). Although it is possible that CysC can bind A␤ aggregates, it was not necessary to invoke extra binding modes for the model to accurately simulate aggregation data. Our model is consistent with a mechanism whereby CysC sequesters monomeric A␤, which is slowly released to maintain equilibrium as free A␤ is converted to aggregates or is degraded. It would be of interest to repeat FRET measurements of CysC binding to oligomeric A␤, if technical difficulties, such as the heterogeneity and instability of the oligomeric A␤ preparation and the effect of aggregation state on donor emission, could be properly managed.
This model assumes that CatB does not degrade A␤ aggregates. Although A␤ fibrils are known to gain extensive resistance to proteolysis (59), CatB degradation of A␤ 42 fibrils in vitro has been reported at pH 7.0 and 37°C (30). However, we observed no degradation of ThT-positive A␤ aggregates. In A␤/CysC/CatB co-incubation studies, sample fluorescence reached a plateau and did not decline (Fig. 7, traces iii, iv, and  viii). Some samples maintained a plateau phase for nearly 48 h with no evidence of degradation. Lack of fibril degradation is likely due to a combination of experimental conditions favoring a very small effective concentration of CatB (due to CysC bind-ing) and very low k deg rate constant for fibril proteolysis (due to neutral pH).
Despite these approximations, we were able to accurately predict the aggregation profiles of A␤ 40 or A␤ 42 at several different concentrations of CatB and/or CysC (Fig. 7). Moreover, we developed a simplified parameter, , which can be quickly used to estimate whether specific concentration regimes will lead to predominantly A␤ aggregation or A␤ degradation. Knowing CysC and CatB levels in an AD animal model, for example, a researcher could find the corresponding value of and then determine whether conditions would support A␤ aggregation. CSF biomarker levels in human AD have been determined in multiple studies (23,24,36). Typical A␤ 40 and A␤ 42 CSF concentrations are 2 and 0.2 nM, respectively (36). CysC concentrations in CSF are reported to be from 200 to 5000 nM in CSF and 100 -1000 nM in plasma (23,24,36). Total CatB (as determined by ELISA and not residual activity) is ϳ0.3 nM in CSF and ϳ6 nM in plasma (36). Using A␤ 40 ϭ 2 nM, A␤ 42 ϭ 0.2 nM, CysC ϭ 500 nM, and CatB ϭ 0.5 nM, we obtain ϳ10 for A␤ 40 and ϳ15,000 for A␤ 42 , showing a strong propensity for A␤ aggregation under typical CSF conditions. If CysC and CatB concentrations were at typical plasma levels (100 and 6 nM, respectively), ϳ0.2 for A␤ 40 , the system moves to the degradation regime. It should be pointed out that the model was developed from experimental data obtained at concentrations higher than physiological for practical reasons. Analyses of at typical CSF and plasma conditions require extrapolation of the model to much lower concentrations of A␤ and CatB and should therefore be interpreted cautiously.
The model derived in this study could be expanded to accommodate more complex binding interactions, different cellular environments, and generation of A␤ from APP. For example, CysC is well-known to form domain-swapped dimers (18, 60 -63), which lose the ability to inhibit CatB (64). CysC reversibly forms domain-swapped dimers during normal cellular trafficking, but only monomers are secreted (65). The disease-causing L68Q CysC mutation may domain-swap extracellularly, resulting in a pool of predominantly inactive CysC (66 -68). If the domain-swapped dimer binds A␤ with the same affinity as monomeric CysC but does not inhibit CatB, CysC and CatB would no longer undermine the neuroprotective effects of one another. Additionally, we have observed oligomeric aggregates of CysC in vitro, which are not domain-swapped (18,20). Nonswapped oligomers remain fully active against CatB, but they inhibit A␤ aggregation much more potently than CysC monomers. It is not clear whether these oligomers exist naturally; if so, oligomer formation could allow CysC to potently control both CatB activity and A␤ aggregation. Engineered CysC mutants also provide a method for testing the model. For example, W106G binds CatB but is a poor inhibitor of its enzymatic activity; we anticipate we would observe rapid degradation of A␤ by CatB if this mutant were used in place of wild-type CysC. We plan to carry out these experiments in the near future.
We considered only the protein interactions at neutral pH typical of extracellular fluids. Most CatB is normally restricted to intracellular compartments. The mildly acidic conditions of the endosomal/lysosomal system could alter considerably the outcome. In acidic cellular compartments, CatB activity is expected to increase, leading to fast k deg rate constants for both A␤ and CysC substrates (33,34). Additionally, the assumption of protease-resistant fibrils may be less valid at pH 6 than pH 7.4. However, A␤ aggregation is also accelerated at pH 6 (30). The kinetic model could easily be adapted to predict outcomes at acidic pH once experimental data are collected, allowing determination of k 1 , k 2 , k deg , K I , and K D . Finally, the model could be adapted to consider A␤ generation. A␤ generation from APP is a complex process resulting from many enzyme interactions (3,69). It has been suggested that CatB may act as a ␤-secretase, truncating and priming APP for eventual liberation by a ␥-secretase (43)(44)(45). This action would be a countervailing effect to CatB's degradation of A␤.

Cystatin C preparation
Recombinant human CysC was produced and purified as described previously (20). Monomeric CysC was fractionated from any dimers or aggregates by membrane filtration and stored at 40 M in PBSA-E buffer (10 mM phosphate, 150 mM NaCl, 0.02% w/v NaN 3 , 1 mM EDTA, pH 7.4) at 4°C. W106G CysC was generated by site-directed mutagenesis (Agilent QuikChange II kit). W106G plasmids were transformed into BL21(DE3) cells (New England Biolabs). W106G was produced and purified as described for WT. Protein concentration was determined by BCA assay (Pierce), using known concentrations of WT as standards. After purification, W106G was stored at 28 M in PBSA-E at 4°C.

A␤ aggregation kinetics
Lyophilized A␤ 40 (Bachem, HCl salt) and A␤ 42 (Bachem, HCl salt) were reconstituted in denaturing buffer (8 M urea, 100 mM glycine, pH 10) to 2.8 mM, as described previously (9). Small volumes were aliquoted, snap-frozen, and stored at Ϫ80°C. Immediately prior to use, aliquots were thawed and gently diluted to 200 M in PBS (10 mM phosphate, 150 mM NaCl). ThT powder (Sigma) was dissolved to 100 M in PBS and filtered through a 0.45-m syringe filter. Freshly-diluted A␤ and ThT were combined in black 96-well plates to final concentrations of 8, 28, and 46 M A␤ 40 or 2.5, 10, and 20 M A␤ 42 and 10 M ThT. The maximum fluorescence intensity is limited by ThT concentration, rather than A␤ (70). For this reason, we chose to hold ThT concentration constant to allow direct comparison of maximum ThT intensity. A clear plastic adhesive strip was used to seal each well and prevent evaporative loss. Fluorescence of ThT was continuously monitored for up to 72 h in a Tecan Infinite M200 plate reader (excitation 440 nm and emission 480 nm), with temperature controlled to 37°C. Postaggregation, three reaction mixtures (28 M A␤ 40 ) were removed and centrifuged at 14,000 rpm for 30 min at 4°C. Residual soluble peptide concentration in the supernatant was determined by BCA assay, using freshly-dissolved A␤ monomers for concentration standards. Fibrillar A␤ concentration (in equivalent monomer molar concentration) was calculated from ThT fluorescence intensity, using maximum ThT fluorescence at 72 h for calibration.

FRET measurements
Hilyte Fluor 488-labeled A␤ 40 (HF488-A␤ 40 , Anaspec, AS-60491-01) was reconstituted in denaturing buffer (8 M urea, 100 mM glycine, pH 10) to 2.8 mM. Small volumes were aliquoted, snap-frozen, and stored at Ϫ80°C. HF488-A␤ 40 was thawed and diluted to 10 M with PBS. Concentration of HF-A␤ 40 was confirmed by absorbance (⑀ 497 ϭ 70,000 M Ϫ1 cm Ϫ1 , Anaspec) and then adjusted to 800 nM by addition of PBS. TF4-CysC was diluted from stock to 2 M. TF4-CysC and HF488-A␤ 40 were mixed to final concentrations of 400 nM HF488-A␤ 40 and 0 -600 nM TF4-CysC. Samples were incubated at 37°C for 0 -60 min and then transferred to a 130-l cuvette, loaded into the temperature-controlled sample pedestal of a QM-40 fluorometer (PTI, Inc.) and maintained at 37°C. Fluorescence spectra of all samples were measured using excitation at 488 nm and emission scan of 510 -630 nm. Spectra of the same sample taken at various times after mixing showed minor changes up to 10 min and then no difference thereafter (data not shown). Thus, all reported spectra were recorded after at least 10 min of co-incubation. The average of three spectra per sample was recorded. Emission of HF488-A␤ 40 (FRET donor) at 530 nm was used to determine overall fraction of bound HF488-A␤ 40

Cathepsin B-mediated degradation kinetics
CatB purified from human liver (16.5 M, Enzo Life Science, BML-SE198-0025) was aliquoted and stored at Ϫ80°C. CatB was activated by mixing equal volumes of CatB stock with activation buffer (10 mM phosphate, 150 mM NaCl, 0.005% v/v Tween 20, 10 mM DTT, pH 7.4) and incubating at room temperature for 15 min. CatB active-site concentration was determined by titration with the irreversible inhibitor, E-64. Stock E-64 (Sigma) was diluted to 1 M in PBS-T (10 mM phosphate, 150 mM NaCl, 0.005% v/v Tween 20, pH 7.4). Activated CatB (200 nM) was combined with E-64 (0 -1000 nM) and incubated at 37°C for 10 min. Samples were diluted 10-fold at 37°C PBS-T and then combined in equal volume with 200 M Z-FR-AMC fluorogenic substrate (R&D Systems). Final concentrations were 10 nM CatB, 0 -25 nM E-64, and 100 M Z-FR-AMC. Fluorescence was monitored over time on a BioTek FL X 800 plate reader (excitation 360 nm and emission 460 nm), maintaining samples at 37°C. Initial rates were plotted against E-64 concentration, and CatB active-site concentration was determined from the x-intercept of the plot. The active-site concentration was found to be ϳ95% of the total CatB concentration.
Kinetics of degradation of A␤ 40 , A␤ 42 , and CysC (WT and W106G) by CatB were determined by immunoblot assays. Stock A␤ 40  Mixtures were incubated at 37°C for up to 2 h. Every 15 min, 2 l was spotted onto a nitrocellulose membrane and allowed to dry completely. Membranes were washed in TBS-T (20 mM Tris base, 150 mM NaCl, 0.05% v/v Tween 20, pH 8.0) and incubated overnight in blocking buffer (5% w/v nonfat dry milk in TBS-T). Membranes were incubated with antibody against the C terminus of either A␤ 40 (Abcam, ab76317), A␤ 42 (Abcam, ab10148), or CysC (Novus, Cyst13) at 2 g/ml in TBS-T for 1 h at room temperature. The membranes were then washed in TBS-T and incubated with 2 g/ml goat anti-rabbit (Novus, NBP1-75,325, A␤ blots) or goat anti-mouse (Novus, NB7539, CysC blots) HRP-conjugated secondary antibody in TBS-T for 1 h at room temperature. The membranes were then washed in TBS-T, and antigen was imaged by addition of chemiluminescent substrate (GE Healthcare). Images were taken on a Chemi-Doc XRSϩ (Bio-Rad). Blot images were background-subtracted and quantified using ImageJ. Initial degradation rates were plotted against the substrate concentration and data were fit to Equation  to determine the degradation rate constant k deg .

Model building
All model equations were numerically integrated using ode15s solver package in MATLAB. Equations were input to the solver by a mass matrix to discriminate between material balance and ordinary differential equations. Initial conditions were determined by implicit solution to CysC, A␤, and CatB material balance equations, given the total amount of each protein. Integration time steps were determined automatically by the solver package. Model equations were solved until steady state was reached.