NOMENCLATURE

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ctot	=	Total mass concentration of peptide;
[D]	=	Molar concentration (in equivalent monomers) of stable dimer;
d	=	Filament or fibril diameter;
df	=	Filament diameter;
dF	=	Fibril diameter;
dsph	=	Hydrodynamic diameter of a sphere with equivalent diffusivity;
Dz	=	Z-average translational diffusion coefficient;
[fi]	=	Molar concentration of filaments;
[Fi]	=	Molar concentration of fibrils;
[I]	=	Molar concentration (in equivalent monomers) of amyloidogenic intermediate;
Ib(90°)	=	Average scattered intensity of buffer at 90° scattering angle;
Is(90°)	=	Average scattered intensity of sample at 90° scattering angle;
Itol	=	Average scattered intensity of toluene at 90° scattering angle;
K'	=	Instrument-dependent constant for light scattering analysis (Eq. 2);
kB	=	Boltzmann's constant;
kD	=	Rate constant, formation of stable dimer from unfolded monomer;
kI	=	Rate constant, formation of amyloidogenic intermediate from unfolded monomer;
kij	=	Rate constant, fibril elongation by end-to-end association;
k'ij	=	Rate constant, filament elongation by end-to-end association;
kla	=	Rate constant, lateral aggregation of filaments to fibrils;
kM	=	Rate constant, formation of stable monomer from unfolded monomer;
KMD	=	Monomer-dimer equilibrium constant;
kn	=	Forward rate constant, cooperative association of intermediate into nucleus;
kn	=	Reverse rate constant, disassembly of nucleus into intermediate;
kp	=	Forward rate constant, addition of intermediate to nucleus or filament;
kp	=	Reverse rate constant, dissociation of intermediate from nucleus or filament;
Lc	=	Contour length (includes filaments and fibrils);
Lf	=	Filament length;
LF	=	Fibril length;
lk	=	Kuhn statistical segment length;
[M]	=	Molar concentration of stable monomer;
[Mu]	=	Molar concentration of urea-unfolded monomer;
M1	=	Molecular weight of A monomer;
Magg	=	Wt-averaged molecular weight of all large aggregates;
Mf	=	Wt-averaged molecular weight of filaments;
MF	=	Wt-averaged molecular weight of fibrils;
Mi	=	Molecular weight of species i;
Mw	=	Wt-averaged molecular weight (includes all species);
[N]	=	Molar concentration of amyloidogenic nucleus;
n	=	Number of intermediates in nucleus;
nb	=	Refractive index of buffer;
ntol	=	Refractive index of toluene;
NA	=	Avogadro's number;
p	=	Number of filaments in fibril;
P(90°)	=	Particle scattering factor at 90° scattering angle;
Pf(90°)	=	Particle scattering factor at 90° scattering angle for filaments;
PF(90°)	=	Particle scattering factor at 90° scattering angle for fibrils;
q	=	Reaction order, filament to fibril association;
Rtol	=	Rayleigh ratio for toluene;
T	=	Temperature;
wagg	=	Weight fraction peptide in aggregated form;
wf	=	Weight fraction peptide present as filaments;
wF	=	Weight fraction peptide present as fibrils;
fib	=	Maximum allowable distance X angle between two associating fibrils;
fil	=	Maximum allowable distance X angle between two associating filaments;
0	=	Wavelength of incident light in vacuo;
f	=	Moment of the filament distribution;
F	=	Moment of the fibril distribution;
lin	=	Average linear density of filaments and fibrils;
vh	=	Partial specific volume of hydrated peptide.

	INTRODUCTION

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-Amyloid peptide (A) is the major protein component of senile plaques and cerebrovascular amyloid deposits from Alzheimer's disease (AD) patients (Glenner and Wong, 1984; Masters et al., 1985). A is a 39- to 43-residue proteolytic product of a membrane-associated precursor protein, APP, containing sequences from both extracellular and transmembrane regions of the parent protein (Kang et al., 1987; Masters et al., 1985). The spontaneous conversion of monomeric A into fibrillar aggregates is associated with the development of Alzheimer's disease (Joachim and Selkoe, 1992). The "amyloid hypothesis," that A amyloid deposition is a major causative factor in the onset of AD, is supported by biochemical, genetic, and transgenic animal studies (e.g., Yankner et al., 1990; Mattson et al., 1992; Games et al., 1995; Hsiao et al., 1996; Holcomb et al., 1998). Similar such conversions of soluble proteins or protein fragments into fibrillar polymers occur in diseases as diverse as Huntington's disease, senile systemic amyloidosis, transmissible spongiform encephalitis, and type II diabetes (Koo et al., 1999). Proteins unrelated to known disease states can be induced to form amyloid fibrils by reducing the conformational stability of the folded globular protein (Chiti et al., 2000). Indeed, it is possible to generate libraries of synthetic peptides with the tendency to self-associate into amyloid; these peptides do not share specific residue homology, but rather an alternating pattern of stretches of polar and nonpolar side chains (West et al., 1999).

Several studies suggest that A is toxic only when it is aggregated (Pike et al., 1993; Simmons et al., 1994; Lorenzo and Yankner, 1994; Seilheimer et al., 1997; Hartley et al., 1999; Ward et al., 2000). This apparent link between the physical state of A and its biological activity has motivated efforts to understand the kinetics and pathway of A self-association. Using turbidity to measure aggregation, Jarrett et al. (1993) proposed a qualitative kinetic model for A self-association. In this model, monomer is very slowly converted to an n-mer nucleus (lag phase), followed by rapid addition of monomer to the nucleus to form a fibril (linear phase), until equilibrium is reached and fibril mass concentration no longer changes (plateau phase). Tomski and Murphy (1992) used dynamic light scattering to investigate self-association kinetics of A(1-40) in phosphate-buffered saline. They hypothesized that A monomers spontaneously and completely converted to octamers, that the octamers stacked to form fibrils, and that longer fibrils grew by diffusion-limited irreversible end-to-end association of shorter rodlike fibrils. A quantitative mathematical model was derived to explain the data. This model accounts for changes in fibril length with time. However, it assumes complete conversion of monomer to an oligomer and therefore does not provide a mechanism whereby monomeric A co-exists with fibrils at equilibrium. Furthermore, it neglects monomer addition to the fibril tip as a mechanism of growth. Naiki and Nakakuki (1996) used thioflavin T, a dye that fluoresces upon binding to amyloid fibrils, to measure fibril growth of A(1-40), and proposed a simple mathematical model to explain their data. Briefly, fibril elongation was postulated to occur by reversible addition of monomer to preexisting fibrils. This model is appealing in its simplicity, but does not provide a mechanism for generation of new fibrils, nor does it simulate fibril length. Lomakin et al. (1996, 1997) used dynamic light scattering to study fibril growth from A(1-40) in 0.1 M HCl and proposed a detailed kinetic model based on these data. Briefly, rapid reversible equilibration between monomers and micelles was postulated to occur, followed by spontaneous and irreversible generation of nuclei from micelles. Fibrils then grew by addition of monomer to the nucleus or fibril tip. This work represents the most detailed mathematical model of A association kinetics published to date. The model accounts for the presence of both monomer and fibrillar forms, and can predict both the mass concentration of fibrils and fibril length as a function of time. However, the experiments upon which the model was based were conducted at non-physiological conditions (pH ~ 1).

More recent studies have revealed that linear assemblies of A are not homogeneous in structure or diameter. In electron microscopy (EM) and atomic force microscopy (AFM) studies, two types are commonly observed: 3-4-nm diameter "filaments" (also called protofilaments or protofibrils) and 8-10 nm diameter "fibrils" (Stine et al., 1996; Harper et al., 1997, 1999; Kowalewski and Holtzman, 1999; Ward et al., 2000). Some investigators have observed small globular structures that may be the building blocks for filaments and fibrils (Stine et al., 1996; Harper et al., 1999). Malinchik et al. (1998) suggested that fibers are made of three to five laterally associated filaments, each ~3 nm in diameter. Fraser et al. (1991) observed five to six globular units with diameters of 2.5-3 nm in EM cross-sections of amyloid fibers. None of the extant kinetic models specifically includes both filament and fibril formation and growth.

In this paper we propose a detailed quantitative model for the kinetics of conversion of unfolded A into fibrils. Briefly, A(1-40) was denatured in 8 M urea, then rapidly diluted into phosphate-buffered saline (PBS) to initiate "refolding." Size exclusion chromatography, dynamic light scattering, and static light scattering were used to follow the monomer/oligomer/aggregate distribution and the average length, diameter, and molecular weight of aggregates as a function of time. Experiments were repeated at three different concentrations, covering a range of kinetic behavior regimes. The experimental data, together with prior published information, were used to develop a detailed kinetic model that quantitatively describes A self-association kinetics from the unfolded state. Parameters were determined by nonlinear regression fitting of the model to the experimental data. The model incorporated information about both mass distribution changes and length changes, included co-existence of monomer, dimer, and aggregated species, provided mechanisms for both generation and elongation of fibrils, and explicitly accounted for filaments and fibrils. The model was able to capture all the essential features of the experimental data and represents, to our knowledge, the most detailed and complete quantitative description of A kinetics at physiological conditions published to date. Besides providing a clearer mechanistic understanding of amyloid fibril growth, such models may improve our ability to design compounds that modulate fibril formation, and therefore possess therapeutic potential.

	EXPERIMENTAL METHODS

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Sample preparation

Urea (electrophysiology/molecular biology grade) was purchased from Boehringer-Mannheim (Indianapolis, IN). All other chemicals were purchased from Sigma-Aldrich (St. Louis, MO) unless otherwise stated. Phosphate-buffered saline with azide (PBSA; 0.01 M K₂HPO₄/KH₂PO₄, 0.14 M NaCl, 0.02% (w/v) NaN₃, pH 7.4) was double-filtered through 0.22-µm filters (Millex); 8 M urea was prepared in 10 mM glycine-NaOH buffer, pH 10, then filtered through 0.22-µm filters. Lyophilized A(1-40) (Anaspec, Inc., San Jose, CA) was solubilized using pre-filtered 8 M urea, pH 10, at a concentration of 2.8 mM (70 and 140 µM final concentration) or 5.6 mM (280 µM final concentration) for 10 min. Samples were then rapidly diluted into filtered PBSA to 70, 140, or 280 µM A (equivalent monomer concentration). All final solutions were at pH 7.4 and contained 0.4 M urea. (Results were not affected by increasing the incubation time in 8 M urea to 1 h, data not shown.) Samples were then filtered through 0.45 µM filters directly into light scattering cuvettes or glass vials for further analysis. MALDI-mass spectroscopy analysis confirmed that A was not chemically modified by this procedure.

Size exclusion chromatography

Samples were analyzed with size exclusion chromatography (SEC) using a Superdex 75 column (Pharmacia, Piscataway, NJ) on a Pharmacia FPLC system. The mobile phase (PBSA, pH 7.4, containing 0.4 M urea) flow rate was set at 0.05 ml/min and elution peaks were detected by UV absorbance at 280 nm. The column was calibrated using the following proteins as molecular weight standards: insulin chain B (3500), ubiquitin (8500), ribonuclease A (13,700), ovalbumin (43,000), and bovine serum albumin (BSA) (67,000). To determine the distribution between small species that could be resolved on the column (MW 3-70), and larger species that could not be resolved, samples were injected without the column in place; this peak area was used to calculate the total A concentration of each sample.

The column was also calibrated with 8 M urea, pH 10, (with 150 mM NaCl added to the running buffer to prevent nonspecific interactions with the column), using urea-denatured insulin chain B and ubiquitin as molecular weight markers. Mobile phase flow rate was varied from 0.05 to 0.1 ml/min. Samples of A in 8 M urea, pH 10, were injected with and without the column. Apparent molecular weight was determined by comparison to the calibration data collected in the appropriate buffer, and total recovery was calculated by comparing peak areas of samples injected with and without the column in place.

Light scattering

Static and dynamic light scattering data were collected and analyzed as described previously (Shen et al., 1994). Briefly, samples in light scattering cuvettes (Hellma, NY) were placed in a temperature-controlled vat containing decahydronaphthalene. A Lexel (Fremont, CA) model 95 ion laser operated at 488 nm was focused on the cuvette and data were collected using a Malvern 4700c system (Southborough, MA). Dynamic light scattering measurements were collected at 90° scattering angle hourly or more frequently. Data were analyzed using the method of cumulants to yield a z-average translational diffusion coefficient D_z. For the purpose of reporting the data, D_z was converted to the hydrodynamic diameter of a sphere with equivalent translational diffusion coefficient, d_sph, using the Stokes-Einstein relationship. Average scattered intensity data at 90° scattering angle, I_s(90°), were collected at the same time intervals. For each data point intensity was measured for 10 s, and then averaged. Toluene was used as a standard reference (R_tol = 39.6 × 10⁴ m¹), and the buffer intensity was also measured.

Bis-ANS fluorescence

A samples were prepared as described above. Aliquots were removed at specific time intervals, then diluted into PBS, pH 7.4, such that final samples contained 2 µM A, 20 µM 1,1-bis(anilino)naphthaline-5,5-disulfonic acid (bis-ANS) (Molecular Probes, Eugene, OR), and 1 mM urea. Samples were incubated at room temperature in the dark for 4 min, then analyzed for bis-ANS fluorescence intensity. Samples were excited at 360 nm, and emission spectra were taken at 450-550 nm using a PTI spectrofluorometer (South Brunswick, NJ). A background spectrum of control samples containing 20 µM bis-ANS and 1 mM urea was subtracted from each sample emission spectrum.

Cellular toxicity

Toxicity of A was assessed as described in detail previously (Pallitto et al., 1999). Briefly, PC-12 cells were plated in 96-well polylysine-coated plates with ~15,000 cells/100 µL medium/well. Lyophilized A was dissolved in pre-filtered 8 M urea, pH 10, at 12 mg/ml for 10 min, then diluted to 70, 140, or 280 µM with sterile-filtered PBS. The samples were allowed to aggregate for 1 or 3 days at 25°C, then diluted to 35 µM with fresh media and added to plated cells. Plates were incubated for 24 h at 37°C, then toxicity was assessed using the 3-(4-,5 dimethylthiazol-2-yl)-2,5 diphenyltetrazolium bromide (MTT) assay. All final solutions (including controls) contained 0.4 M urea.

	EXPERIMENTAL RESULTS

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Unfolding of A in urea

To create a mathematical model of the kinetics of A aggregation, we needed a well-characterized and reproducible initial condition. The initial state of synthetic A is poorly defined; the method of synthesis, the lyophilization conditions, and the solvent used to dissolve the peptide all influence the starting conformation and aggregation state (Barrow et al., 1992; Shen and Murphy, 1995; Thunecke et al., 1998). For our studies, the preferred state was completely monomeric and unfolded. We chose 8 M urea, pH 10, as a starting solvent because 8 M urea usually unfolds polypeptides to the random coil state (Creighton, 1994), and because A aggregation is hindered at pH > 9 (Burdick et al., 1992). Furthermore, refolding of proteins from the urea-denatured state is a widely used technique in protein-refolding kinetic studies (Goldberg et al., 1991; Fink, 1998).

A(1-40) in 8 M urea, pH 10, eluted as a single sharp symmetric peak on a calibrated size exclusion column with a residence time equivalent to 5.3-6.6 kDa (compared to its known molecular weight of 4.3). The apparent molecular weight was sensitive to the mobile phase flow rate, decreasing with increasing flow rate. This sensitivity to flow rate is likely due to the presence of NaCl in the mobile phase, but not in the sample; NaCl facilitates aggregation of A (data not shown) but reduces nonspecific interactions with the column. Recovery of injected A in the monomer peak was complete within experimental error: at a concentration of 120 µM, 99 ± 4% (SEM) was recovered, and at 2.8 mM, 97 ± 4% was recovered. The intensity of scattered light from A(1-40) in 8 M urea, pH 10, was not greater than solvent alone, and there was no change in scattered intensity over a 24-h period, further indicating the absence of aggregate. Circular dichroic spectra contained no strong bands, and in particular lacked any trace of a minima at 218 nm or 222 nm, consistent with a lack of -sheet or -helix; however, the strong absorbance of urea precluded collection of reliable spectral data below 210 nm, and therefore more definitive secondary structure assignments could not be made. Together, these data indicate that 8 M urea, pH 10, is an effective solvent, and renders A completely monomeric and unfolded.

A monomer/oligomer size distribution in PBSA

Refolding was initiated by rapid dilution of urea-denatured A(1-40) into PBSA. The resulting solution was analyzed with size exclusion liquid chromatography (SEC). Representative chromatograms are shown in Fig. 1. Invariably, two peaks were observed in the inclusion volume of the column, eluting with retention times corresponding to molecular masses of 4.1 ± 0.2 and 9.5 ± 0.2 kDa. These peaks will be referred to as monomer and dimer, respectively. A peak that eluted at the void volume was observed, but not consistently. No difference in chromatograms was observed when the sample was centrifuged before injection (data not shown).

View larger version (16K): [in this window] [in a new window]   FIGURE 1   Representative chromatograms of A in PBSA. Lyophilized A was dissolved in 8 M urea, pH 10, diluted at least 20-fold into PBSA, then injected onto a Superdex 75 column. Running buffer was PBSA with 0.4 M urea; flow rate was 0.05 ml/min. Peak detection was by absorbance at 280 nm. Arrows indicate predicted retention times for monomer and dimer, based on column calibration.

The identical samples were injected using the same sample loop and detector, but without the column in place, to measure the total concentration (total peak area) of A. To calculate the fraction of A in monomer and dimer populations, the individual peak areas (obtained with the column in place) were divided by the peak area without the column. The fraction of aggregates (>70 kDa) was calculated by difference. A summary of the concentrations of monomer, dimer, and aggregates is given in Table 1. These values did not change appreciably over time, up until precipitates were visible (data not shown). A weighted nonlinear regression fit to the data yields a relationship between monomer and dimer of [D] = 0.6 ± 0.3[M]^2±0.2. The second-order dependence on concentration is consistent with assignment of the two peaks as monomer and dimer.

View this table: [in this window] [in a new window]   TABLE 1   Size distribution as function of total A concentration

A aggregate growth kinetics

Dilution of urea-denatured A into PBSA produced aggregates possessing a linear stiff or semiflexible morphology (Murphy and Pallitto, 2000). We examined changes in size of A aggregates as a function of time and concentration. Aggregate size was characterized by measuring D_z and converting this to the average hydrodynamic diameter d_sph, which is sensitive to the average length of the aggregates. (Scattering from monomer/dimer populations is too weak to be detected in the presence of aggregates.) These data were previously reported (Murphy and Pallitto, 2000) and are shown in Fig. 2. Within a few minutes of dilution of urea-denatured A into PBSA, large aggregates were already present. Interestingly, aggregates were initially largest at the lowest concentration (70 µM A). Patterns of growth were strongly concentration-dependent. At 70 µM, d_sph was nearly constant. At 140 and 280 µM, initial sizes were about the same (d_sph ~ 25 nm); both increased, but the rate of increase was much faster at 280 µM.

View larger version (23K): [in this window] [in a new window]   FIGURE 2   Average hydrodynamic diameter (dsph) as a function of time and concentration for 70 µM (), 140 µM (), and 280 µM () A (nominal concentration, equivalent monomers). A was prepared as described above; autocorrelation functions were collected at 90° scattering angle and analyzed as described in the text.

Average scattering intensity at 90° scattering angle, I_s(90°), was measured at the same time intervals. I_s(90°) is sensitive to the average molecular weight of the aggregates. Both the absolute intensity and the rate of change were dramatically dependent on A concentration (Fig. 3). At 70 µM, I_s(90°) was relatively constant with time. At 140 µM, I_s(90°) slowly doubled over the course of 24 hours. At 280 µM, I_s(90°) increased ~6-fold over ~10 h, then leveled off. I_s(90°) is related to the size and shape of the particles in solution as (Shen et al., 1994):

(1)

where c_tot = total peptide concentration, M_w is the weight-averaged molecular weight of particles in solution, P(90°) is the particle scattering factor at 90°, I_b(90°) is the scattered intensity of the buffer, and K' is an instrument-dependent constant,

(2)

where n_b and n_tol are the refractive indices of buffer and toluene, respectively, dn/dc is the refractive index increment (0.145 ml/g), N_A is Avogadro's number, ₀ is the laser wavelength in vacuo, I_tol is the scattered intensity from a toluene sample, and R_tol is the Rayleigh ratio for toluene.

View larger version (21K): [in this window] [in a new window]   FIGURE 3   Average scattering intensity Is(90°) for 70 µM (), 140 µM (), and 280 µM () A. Samples are identical to those described in Fig. 2.

Results from SEC and light scattering measurements were combined to evaluate the average linear density _lin (mass per unit length, M_agg/L_c) of the linear aggregates as follows. If only monomer, dimer, and large aggregates are present, then M_w = w_iM_i ~ w_aggM_agg, with M_w related to I_s(90°) per Eq. 1 and w_agg given in Table 1. The fibril contour length L_c was calculated from the measured d_sph along with an experimental estimate of fibril flexibility (Kuhn statistical length l_k = 180 nm, Murphy and Pallitto, 2000), using relations derived by Yamakawa and Fujii (1973). P(90°) was calculated from L_c and l_k, using the theory derived by Koyama (1973) and described in Shen et al. (1994). Then,

(3)

_lin, a measure of the thickness of the chains, is shown in Fig. 4. At 70 µM, _lin was nearly constant over time with an initial value of ~4 kDa/nm. At 140 µM, _lin was higher than at 70 µM (13 kDa/nm) initially, and increased modestly over time. The most dramatic changes were observed at the highest concentration, 280 µM; _lin was greater than the other concentrations initially (20 kDa/nm), and increased rapidly over the first ~8 h, then leveled out after ~20 hours at ~60 kDa/nm.

View larger version (23K): [in this window] [in a new window]   FIGURE 4   Linear density lin as a function of time and concentration for 70 µM (), 140 µM (), and 280 µM (). lin was calculated from dsph and Is(90°) as described in the text.

Aggregate hydrophobicity

The fluorescent dye bis-ANS was used as a qualitative probe for exposed hydrophobic surfaces on A aggregates. The dye binds to exposed hydrophobic patches on partially folded proteins, leading to an increase in fluorescence intensity and blue-shifting of the emission maximum (Gibbons and Horowitz, 1995). Freshly-diluted monomeric A does not cause bis-ANS fluorescence (Kremer et al., 2000). A was aggregated at 70, 140, or 280 µM, then diluted to 2 µM into a solution containing bis-ANS. In Fig. 5 A Bis-ANS fluorescence is shown as a function of the concentration at which A was aggregated. At 70 µM a fluorescence peak was observed, which did not change appreciably with time (data not shown). At 140 µM, the peak fluorescence is ~2-3-fold higher, and at 280 µM, the fluorescence increased another 3-4-fold and is slightly blue-shifted. At 280 µM (Fig. 5 B) and, to a lesser extent, at 140 µM (not shown), fluorescence intensity increased over the first few hours, then stabilized. These data suggest that there are distinct structural differences between the aggregates formed at these different concentrations.

View larger version (19K): [in this window] [in a new window]   FIGURE 5   Bis-ANS fluorescence spectra in the presence of 2 µM A. A samples were prepared at 70, 140, or 280 µM as described above, then diluted at the indicated time into a solution containing 20 µM bis-ANS. Excitation wavelength was 360 nm and a background spectrum was subtracted. Binding of bis-ANS to hydrophobic sites produces an increase in fluorescence intensity and a blue-shifting of the emission maximum to ~480-490 nm. (A) Sample aggregated at 70 µM (), 140 µM (), and 280 µM (), then diluted to 2 µM, taken 4 h after sample preparation. (B) Sample aggregated at 280 µM (), then diluted to 2 µM, taken 1 h (short dashed line), 4 h (solid line), or 2 days (long dashed line) after sample preparation.

	MATHEMATICAL MODEL

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Experimental results described above lay the foundation for developing a mathematical model of A self-association kinetics. The key observations that the model must capture are:

1.	A substantial amount of material remained in a nonfibrillar state, as monomers and dimers, after renaturation. Dimer-monomer concentrations were related by a second-order function;
2.	High-molecular-weight species formed very rapidly upon dilution;
3.	The initial aggregate size was greatest at the lowest test concentration;
4.	The rate of growth increased with concentration;
5.	The linear density of aggregates increased at higher concentrations with time, eventually reaching a steady-state value;
6.	Aggregates formed at different concentrations are structurally distinct, specifically in terms of exposed hydrophobic regions.

The development of the structure of the model will be discussed first, followed by a description of the detailed equations derived to solve the model.

Monomer, dimer, and aggregate mass distribution

We observed that 1) monomer and dimer concentrations depended on total A concentration; 2) monomer, dimer, and aggregate distribution was relatively constant with time; and 3) large aggregates appeared very quickly upon dilution into PBSA. We used these data to postulate that "refolding" of A from the urea-denatured state occurs extremely rapidly, and that refolded species become quickly and irreversibly committed to either monomer/dimer (nonamyloid) or aggregate (amyloid) status. A division between amyloidogenic and nonamyloidogenic populations of A has previously been postulated by Soto and Castano (1996), and is conceptually similar to the division between aggregated and correctly folded proteins observed in other protein refolding studies (Goldberg et al., 1991). We also observed that monomer and dimer concentrations were related by a simple second-order equation, and therefore propose that monomer and dimer are in rapid reversible equilibrium. Schematically this is shown in Fig. 6 under the "refolding" step, where M_unfold represents monomeric A in its urea-denatured state, M and D are monomer and dimer "native" conformations that are stable in PBSA, respectively, and I is an unstable intermediate that can form larger aggregates. In the refolding literature M and D formation would be considered "on-pathway" and I aggregate would be "off-pathway." The split between amyloidogenic and nonamyloidogenic populations was assumed to be irreversible.

View larger version (18K): [in this window] [in a new window]   FIGURE 6   Schematic of model describing kinetics of A fibril growth.

We considered, and discarded, several alternative schemes in which monomers (and/or dimers) were assumed to be all "off-pathway", e.g., M  D  I aggregates. For example, conversion to aggregates could occur via cooperative association of monomer (or dimer) to reactive intermediate. This model is similar to that proposed by Jarrett et al. (1993) and Lomakin et al. (1996, 1997). Under this model, [I] remains very small until the "critical concentration" of [M] is reached, then [M] remains constant (and equal to this critical concentration) as total A concentration is increased. We instead observed an increase in [M] and [D] as the total concentration increased, directly contradicting this model. Two other models, used commonly to describe linear polymerization processes (Billmeyer, 1971; Schmidt, 1998), were explored. A association was postulated to occur either via a stepwise (condensation) polymerization scheme, in which any two oligomeric species react to form longer polymers, or via addition (chain) polymerization, in which a highly reactive initiator is formed and monomers quickly add to the growing chain. With stepwise polymerization, the molecular weight distribution is very broad, and very few monomers are present for even modest degrees of aggregation. This model is inconsistent with experimental data. Chain polymerization produces a distribution that includes only monomer and high-molecular-weight polymer, which is closer to the observed distribution. However, the monomer concentration tends toward a constant value, independent of total concentration, in contrast to observation. Several variants of these ideas were evaluated; none were fully consistent with experimental data. Therefore, we conclude that formation of nonamyloid and amyloid A species occurs in parallel, not serial, fashion.

Aggregate size and shape

Two key structural features that have been consistently observed in AFM studies are "filaments" of 3-4-nm thickness and "fibrils" of 8-12-nm thickness (Stine et al., 1996; Harper et al., 1997, 1999). Cross-section EM images and x-ray diffraction studies indicate that fibrils contain three to six laterally associated filaments (Fraser et al., 1991; Malinchik et al., 1998). Analysis of the light scattering data revealed increases in _lin with concentration and with time, and substantially greater (~3-fold) steady-state _lin at 280 µM relative to initial values. Given these data and the aforementioned EM, AFM, and XRD studies, we incorporated into our model two possible long linear aggregated structures: filaments (thin) and fibrils (thick).

Development of the detailed kinetic model for filament and fibril formation and elongation relied on several key experimental observations. First, we observed that even at the earliest time points high-molecular-weight aggregates were present, and that the initial size (d_sph) was greatest at the lowest concentration. We postulated that this could occur due to a nucleation-dependent process akin to crystallization, where fewer longer crystals are observed at lower concentrations and more, shorter crystals are observed at higher concentrations (Jarrett et al., 1993). Thus, a nucleation mechanism was applied not to the initial partitioning between M/D and I, but to further self-association of I into larger species. A high-order, cooperative reaction is necessary to capture the inverse relationship between initial size and concentration. We modeled filament initiation as reversible self-association of I to form a nucleus, N, containing n I, characterized by forward and reverse rate constants k_N and k_N, respectively. This step is illustrated in Fig. 6 as filament initiation. We further assumed that nuclei N could elongate into filaments f by addition of I. This is illustrated in Figure 6 as filament elongation by I addition.

The second key observation was the significant increase in _lin at 280 µM with time, whereas at 70 µM _lin was constant and small. We postulated that the increase in _lin at 280 µM was due to lateral aggregation of several filaments into fibrils. This would be consistent with observations indicating that fibrils are close-packed filaments (Fraser et al., 1991; Malinchik et al., 1998) and would have a high reaction order, to explain the strong concentration dependence. Therefore filaments f were assumed to laterally associate into fibrils F, as shown in Fig. 6. Lateral association was assumed to be irreversible, characterized by a rate constant k_la.

Third, we observed an increase in d_sph without a corresponding increase in _lin at 280 µM and longer times (t > ~20 h, see Figs. 2 and 4). The increase in d_sph indicates an increase in fibril length. No change in _lin implies a constant fibril diameter. We postulated that the increase in d_sph at t > ~20 h was due to axial elongation by end-to-end association of shorter fibrils. Additional experimental evidence of this was presented in Murphy and Pallitto (2000) and Harper et al. (1999). End-to-end association was assumed to be slow and diffusion-limited and was modeled after the classic Smoluchowski equation. The rate constant associated with end-to-end association, k_ij, was assumed to depend on length as (Hill, 1983, Tomski and Murphy, 1992):

(4)

where i and j indicate the size of associating fibrils, k_B is the Boltzmann constant, T is the absolute temperature,  is the solvent viscosity,  is the maximum allowable distance between two fibril ends,  is the maximum allowable angle between two fibrils, L_i is the fibril (or filament) length, and d is the fibril (or filament) diameter. For completeness, a similar mechanism for end-to-end association of filaments was included. End-to-end association was assumed to be an irreversible process.

Model equations

The schematic depicted in Fig. 6 and justified in detail in the previous sections was used to derive a set of equations describing the kinetics of A aggregation. Refolding of denatured peptide to produce M, D, and I was described by simple irreversible kinetic expressions:

(5)

(6)

(7)

where [M_u] is the concentration of monomer in the urea-unfolded state. The initial distribution among M, D, and I was assumed to occur very rapidly, relative to further I association. The data were best modeled if I was assumed to be dimeric, hence the form of Eq. 7. Equations 5-7 were solved to give a pseudo-steady-state expression for [I], i.e., the value that would be obtained at long times if there were no further aggregation of I:

(8)

where [M_u]₀ is the initial A concentration in the unfolded state. The amount of M and D formed is simply:

(9)

We assumed that "native" monomer M and dimer D were always in equilibrium:

(10)

Equations 8-10 give the initial concentrations of M, D, and I just after dilution into PBSA. For convenience, M, D, and I concentrations are given in equivalent monomer molar concentrations.

I is consumed by initiation and elongation steps (Fig. 6) and associates to form N, or adds to N to form filaments, or adds to filaments to increase their length. The shortest filament is produced by addition of I to N and is referred to as f_n+1. Filaments can be any size i from (n + 1) to , where n is the number of I per N. (There are 2i monomers per filament f_i.) Summing over all species gives:

(11)

(12)

A filament of length i, f_i, is formed by addition of I to f_i1 or by end-to-end association of two shorter filaments (f_j and f_k, where j + k = i); f_i is lost by similar reactions: addition of I to form f_i+1, or end-to-end association with any filament f_j to form filaments of length i + j. In addition, filaments are lost by (irreversible) lateral aggregation to fibrils. Mathematically, this is expressed as:

(13)

where p is the number of filaments per fibril, q is the order of the lateral association reaction, k_la is the lateral association rate constant (assumed independent of length), and k'_ij and k'_jk are end-to-end association rate constants for filaments (defined in Eq. 5). Since i varies from n + 1 to , there are an infinite number of equations like Eq. 13. To make this system of equations finite, we defined moments of the filament size distribution _f:

(14)

Fibrils form by lateral association of filaments and grow in length by end-to-end association of shorter fibrils. We defined moments of the fibril size distribution _F:

(15)

and assumed that the length of fibrils formed by lateral association was equal to the number-average length of filaments at that time. Substituting in the definition of moments (Eqs. 14 and 15) into Eqs. 12 and 13, and similarly deriving equations for fibril formation and growth, produces a finite set of coupled differential equations:

(16)

(17)

(18)

(19)

(20)

(21)

(22)

where are were calculated from Eq. 4 using the number average length of filaments or fibrils, respectively.

Parameter estimation procedure

Equations 16-22 together with Eq. 11 were solved numerically using the program DDASAC (Caracotsios and Stewart, 1985). Model parameters were derived by fitting experimental data to model equations, using the parameter estimation package GREG (Stewart, 1987) and the following procedure.

First, we determined the "refolding" parameters. Estimates of K_md = 0.64 ± 0.08 µM¹, k_M/k_I = 80 ± 30 µM, and k_D/k_I = 0.65 ± 0.15 were obtained by nonlinear regression fit of the data in Table 1 to Eqs. 8-10. Because these are rapid reactions, only the ratios of rate constants, rather than the absolute values, could be evaluated. As shown in Fig. 7, the model calculations accurately reflect the observed population distribution at all three concentrations tested. We considered and discarded mechanisms in which k_M or k_D was set equal to zero, because these were not able to capture the observed trends satisfactorily (not shown). Addition of the extra parameter is statistically justified at the 99.5% level, using the F-statistic (Davies, 1954).

View larger version (24K): [in this window] [in a new window]   FIGURE 7   Observed (dark shading) and simulated (lightly shaded) weight fraction of (A) monomer, (B) dimer, and (C) aggregate. Samples were prepared at the indicated total A concentration; distributions were evaluated from size exclusion chromatograms. Concentrations are given as equivalent monomer concentrations.

Second, we determined the parameters involved in filament initiation and elongation. To compare experimental data to model simulations, a method was developed that relates the experimental observations d_sph and I_s(90) to the size distributions calculated from the model. The weight-average-molecular-weight of filaments M_f and fibrils M_F were related to model-generated distributions as: