Protein Model, Structure of Aβ_16–22 Oligomer, and Fibril.

The Aβ_16–22 aggregates are modeled using a coarse-grained protein model, called PRIME20 (30, 31), in which each amino acid residue is represented by three backbone spheres, one each for N-H, C-H, and C = O, and a single side-chain sphere R (Fig. 1A). The nondirectional interaction between side-chain spheres and the directional hydrogen bond between the N-H and C = O spheres are modeled as square-well potentials (Fig. 1 B and C). While the strength of the directional backbone hydrogen bonding interaction is the same for all residues, the strength of the nondirectional side-chain–side-chain interaction is side-chain–specific. Details are provided in Methods.

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Fig. 1.

(A) Representation of the Aβ_16–22 peptide in the PRIME20 model. (B and C) Schematic representations of the square-well interaction between two side chains and the directional square-well interaction (hydrogen bonding) between backbone NH and C = O spheres. (The spheres are not drawn to scale for ease of viewing.) (D–H) Simulation snapshots of non-HB oligomer; HB oligomer; and two, three, and four β-sheet fibrils, respectively, generated by Visual Molecular Dynamics (VMD) software. The peptides in the five aggregates and in solution are shown in green and magenta, respectively.

Here, we focus on five types of aggregates formed by Aβ_16–22: non–hydrogen-bonded (non-HB) oligomers (Fig. 1D); hydrogen-bonded (HB) oligomers (Fig. 1E); and fibrils composed of two, three, and four β-sheet layers (Fig. 1 F–H). The non-HB oligomer is defined to be an aggregate formed solely by interpeptide hydrophobic (HP) side-chain–side-chain interactions with no interpeptide hydrogen bonds; however, intrapeptide hydrogen bonds are allowed. The HB oligomer is defined to be an aggregate that has, at most, one interpeptide hydrogen bond between any two neighboring peptides and any number of intrapeptide hydrogen bonds. We consider the non-HB and HB oligomers to be two distinct states because the peptides in these two states have different numbers of hydrogen bonds and HP side-chain contacts with neighboring peptides (SI Appendix, Fig. S1). The β-sheet aggregates are stabilized by having side-chain HP contacts between tightly packed antiparallel β-sheets and a maximum number of backbone HBs between fully extended peptides within each β-sheet, consistent with X-ray diffraction and solid-state NMR measurements (2, 6). Our simulations show that the probability distributions for the number of HB and HP side-chain contacts for the five aggregates do not change significantly with temperature over the studied temperature range.

Measurement of Solubility for Aβ_16–22 Oligomers and Fibrils.

To determine the solubility of aggregates (oligomers and fibrils) formed by the Aβ_16–22 peptides, we perform MD simulations in the canonical ensemble. The solubility is defined to be the peptide concentration above which the peptide will join with others to form an aggregate and below which the peptide will remain in solution. In our simulations, the solubilities for the different Aβ_16–22 aggregates (oligomers and fibrils) at a given temperature are measured as the equilibrium monomer concentration at which the aggregate neither grows nor shrinks (details are provided in Methods). This approach was used by Bai and Li (32) to measure solid–liquid phase equilibrium for the Lennard–Jones fluid. Their calculation of the solid–liquid interfacial energy (surface tension) using classical nucleation theory was in qualitative agreement with results calculated using other theoretical approaches [e.g., interface fluctuation method (33), reversible work integration (34)]. Later, Auer and Kashchiev (28) and Auer (29) used this method, along with Monte Carlo simulation, to calculate a phase diagram for a simple homopolypeptide model, where each amino acid is represented by one Cα bead. The predicted values for the surface tension were in agreement with theoretical predictions for protein crystals using a simple lattice model (35).

We first measured the solubility for the non-HB and HB oligomers. Fig. 2A shows that the solubilities for the non-HB and HB oligomers increase with increasing temperature. This is expected, as at higher temperatures, the thermal energy makes it easier for peptides to detach from the oligomer so that higher concentrations are required to stabilize these aggregates. The latent heat of aggregation for monomeric peptides into oligomers can be obtained by fitting the van’t Hoff equation,Ce=Crexp(−LkBT),[1]to the simulation data (Fig. 2B), where Ce is the equilibrium fibril solubility, Cr is a temperature-independent reference concentration, k_B is the Boltzmann constant, T is the simulation temperature, and L is the latent heat of monomer peptide aggregation into the oligomer or fibril. From the simulation, we also calculated the average peptide binding energy (the sum of the HP and HB energies per peptide) within the non-HB and HB oligomers, including all of the peptides on the surface and inside the aggregate, and found that the latent heat is about half of the average peptide binding energy, E_B, (Fig. 2C) consistent with earlier work (29).

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Fig. 2.

(A) Plot of the solubility, Ce, for non-HB oligomer (olig.); HB oligomer; and two, three, and four β-sheet fibrils as a function of the temperature. (B) Solubility data for non-HB oligomer; HB oligomer; and two, three, and four β-sheet fibrils at four to six different temperatures. Lines are fitted to Eq. 1. The color scheme is the same as used in the plot in A. Ln(C_e), natural logarithm of C_e. (C) Comparison of the average binding energy per peptide within the aggregate and the corresponding latent heat of peptide aggregation into the oligomer and fibril. (D) Dependence of the fibril solubility on thickness (i = 2, i = 3, and i = 4) for two, three, and four β-sheet fibrils, respectively. Lines are fitted to Eq. 2.

Although the solubility of the oligomers should depend on the oligomer size as described by the Ostwald formula (28, 29, 36) (which predicts that solubility decreases with increasing oligomer size), we do not consider the effect of oligomer size here. This is because large oligomers do not form for this Aβ_16–22 system. The simulations show that the non-HB and HB Aβ_16–22 oligomers are relatively small (containing around 10 peptides) and easily break apart into smaller aggregates when they contain more than 15 peptides.

Next, we determined the solubilities for two, three, four, and infinite β-sheet Aβ_16–22 fibrils. The simulation results (Fig. 2A) show that at a given temperature, the solubilities for the two, three, and four β-sheets decrease with increasing thickness (number of β-sheet layers) of the fibril. As the number of β-sheets in the fibril increases, the average number of HP contacts per peptide increases, as shown in SI Appendix, Fig. S1C, leading to increased stability of the fibril. A quantitative description of the dependence of the solubility of a fibril with i layers (i = 1, 2, 3, etc.) of a β-sheet, Ciβ, on thickness at fixed temperature can be derived from classical nucleation theory and is given by (37)Ciβ=C∞βexp(2ψhkBT⋅1i) ,[2]where C∞β is the solubility of an infinite thick fibril, ψh=ahσh is the fibril surface energy parallel to the fibril thickening axis, σh is the specific surface energy of the face perpendicular to the fibril axis, and ah is the lateral surface area occupied by each peptide within one β-sheet as defined by Kashchiev and Auer (37). A linear fit of our simulation data (lnCiβ vs. 1/i) to Eq. 2 (Fig. 2D) yields an estimate for the solubility C∞β as a function of temperature for the infinitely thick fibril. As before, the latent heats of peptide aggregation into fibril phases from solution can be obtained by a fit of our data to Eq. 1 (Fig. 2B), and the obtained values are shown in Fig. 2C. From Fig. 2C, we find that for two, three, and four β-sheet fibrils, the values of the latent heat are comparable to the average peptide binding energy. The relation between latent heat and protein–protein interaction energy for a protein crystal was first derived by Haas and Drenth using a simple lattice model (35). They found that the latent heat should be half of the average protein–protein interaction energy; this was derived by using the zero-temperature approximation, which does not consider the entropic effects. We speculate that it might be the neglect of the entropic effect that causes the relation between latent heat and peptide binding energy to hold for oligomers but not for two, three, and four β-sheet fibrils in this work. In addition, the fit of our simulation data to Eq. 2 yields values for ψh=ahσh, and knowledge of ah enables us to estimate the average surface tension, σh, of the Aβ_16–22 fibril (SI Appendix, Fig. S2). The lateral surface area per peptide, ah=106.1Å, is the product of the length of a peptide within a β-sheet (22.1 Å) and the interpeptide distance in one β-sheet (4.8 Å). Hence, depending on temperature, the average surface tension, σh, for the Aβ_16–22 fibril ranges from 24 to 30 mJ/m² (SI Appendix, Fig. S2), which is in the range of 0.1–30 mJ/m² reported for protein crystals in aqueous solution (35, 38⇓–40).

Aβ_16–22 Phase Diagram.

The major finding of this study is our quantitative calculation of the thermodynamic phase diagram for the Aβ_16–22 peptide fragment (Fig. 3A). The peptide solution phase (Fig. 3A, cyan region) and the macroscopic fibrillar phase (Fig. 3A, yellow region) are the only two thermodynamically stable phases. The thermodynamic boundary between these phases is taken to be the solubility line for a fibril with an infinite number of β-sheets. [Error bars are not shown in Figs. 3 and 4A because the R² value for the linear fit of C∞β vs. T data to obtain C∞β(T) is close to 1.] Within the fibril phase region of the phase diagram, there exists a series of metastable aggregate phases separated by the metastable solubility lines for the non-HB oligomer; HB oligomer; and two, three, and four β-sheet fibrils. A hierarchy of metastable phases, with the ultimate stable boundary being between solution and infinitely thick fibril, was first observed by Auer and Kashchiev (28).

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Fig. 3.

(A) Phase diagram for Aβ_16–22 peptide. Solubility, C_e, versus temperature data are presented for the non-HB oligomer (orange dashed line), HB oligomer (purple dashed line), two β-sheet (2β) fibril (red dashed line), 3β fibril (green dashed line), 4β fibril (blue dashed line), and infinite layer β-sheet (∞β) fibril (black line). The fibril and solution phases are colored yellow and cyan, respectively. (B) DMD/PRIME20 simulation snapshots at the concentration and temperature of phase points A and B are taken at different time points, respectively.

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Fig. 4.

(A) Summary of both the simulation-predicted temperature-dependent solubility line, Ce(T), for Aβ_16–22 peptide (black curve) and the fibrillation experiments performed under given conditions. Red dots and red circles indicate conditions at which fibrils have been found to form and not to form, respectively, via TEM (at T = 277 K, T = 296 K, T = 310 K, and T = 330 K). Blue dots indicate that fibrils have been reported in the literature to form under these conditions (SI Appendix, Fig. S5). (B) Six selected TEM images (i–vi) showing that Aβ_16–22 forms fibrils under the conditions that correspond to the six red dots labeled i–vi in A. The buffer is 100 mM ammonium bicarbonate (pH 7), with a final concentration of 1% DMSO (vol/vol). (Scale bar: 200 nm.)

To justify the infinite thickness approximation in calculating fibril solubility, we calculated Aβ_16–22 fibril solubility versus fibril thickness (SI Appendix, Fig. S3) and found that fibril solubility remains unchanged after the thickness exceeds 30 layers. We think that an actual Aβ_16–22 fibril may contain more than 30 layers of β-sheets at neutral pH, because Aβ_1–42 (41), Aβ_10–35 (42), and Aβ_18–28 (43) form fibrils in vitro with 2, 6, and 24 layers of β-sheets, respectively, at neutral pH. This is consistent with a hypothesis by Lu et al. (44) that shorter amyloidogenic peptides tend to adopt more planar β-sheets and maintain more layers in the fibril than longer peptides.

To better illustrate why the solubility line for the fibril is regarded as the coexistence line, and why the oligomer and various multisheet fibril phases are metastable with respect to the fibril phase, we put our calculations in the context of more familiar thermodynamic arguments in which the compositions of two phases in equilibrium are found by minimizing their free energies. SI Appendix, Fig. S4 plots the Helmholtz free energies of the solution, oligomer, and fibril phases as a function of the peptide mole fraction, xp (which is simply related to the peptide concentration for a dilute solution), in a peptide and water mixture at fixed temperature and volume. A double-tangent (also called convex-envelop) construction on the free energies of the solution and fibril phases determines the peptide mole fractions, xpsoln(1) and xpfib, at which the solution and fibril phases are in equilibrium. The construction locates the peptide mole fractions at which the peptide chemical potentials (the xp = 1 intercepts of lines tangent to the free energy curves) are equal [i.e., μpsoln(1)(xpsoln(1))=μpfib(xpfib)]. The composition, xpsoln(1), is equivalent to the peptide concentration above which the monomer goes through a phase transition to another phase, the solubility line determined in our simulations. Similarly a double-tangent construction on the solution and oligomer phases yields peptide mole fractions xpsoln(2) and xpolig, with equal peptide chemical potentials, μpsoln(2)(xpsoln(2))=μpolig(xpolig), signifying equilibrium between solution and oligomer phases in this concentration range. This is, however, a metastable equilibrium compared with the equilibrium between solution at xpsoln(1) and fibril at xpfib, because the peptide chemical potential in solution is higher than that when the solution is in equilibrium with the fibril. Note also that having μpsoln(2)(xpsoln(2))>μpsoln(1)(xpsoln(1)) means that xpsoln(2)>xpsoln(1), consistent with what is measured in simulations. Notice also that in SI Appendix, Fig. S4, there are no metastable phases when xpsoln(1)<xp<xpsoln(2); the oligomer phase only becomes metastable with respect to the fibril phase when xp>xpsoln(2). This is also reflected in the solubility versus temperature phase diagram (Fig. 3A), which shows a region of concentrations where the HB oligomer is unstable (below the dashed purple line and above the black solubility line) and a region where the oligomer is metastable (above the dashed purple line). Similar arguments can be made for the other metastable phases, such as the two, three, and four β-sheet fibril phases. We emphasize that we did not calculate free energies as we were able to calculate the solubilities directly; the discussion here is mainly presented for pedagogical purposes.

Knowledge of the thermodynamically stable and metastable phases in the phase diagram allows us to determine the conditions under which the oligomer or fibril can form. A previous discontinuous molecular dynamics (DMD) study of Aβ_16–22 kinetic aggregation by Cheon et al. (6) found that at 20 mM and T* = 0.2, corresponding to 342 K, a system of peptides in a random configuration first forms a fibril nucleus, which then grows into fibrils. At the same concentration but at a lower temperature T* = 0.17, corresponding to T = 273 K, the peptides first form HB oligomers that later merge and rearrange to form fibrils. Snapshots from new simulations at T = 342 K and T = 273 K are shown in Fig. 3B. Our thermodynamic phase diagram helps explain why Aβ_16–22 aggregates via a “one-step” pathway at high temperature and a “two-step” pathway at low temperature. From Fig. 3A, at point A (20 mM, T = 342 K), the peptide concentration is greater than the solubility of fibril, C > C_e,fibril (T = 342 K) but smaller than the solubility of the HB oligomer, C < C_{e,HB oligomer} (T = 342 K). As point A satisfies the thermodynamic criteria for stable fibril formation but not for HB oligomer formation, it is not surprising that the aggregation kinetics are characterized by nucleus formation and subsequent fibril growth. In comparison, point B (20 mM, T = 273 K) satisfies the thermodynamic condition for formation of a stable fibril and a metastable HB oligomer C > C_e,fibril (T = 273 K) and C > C_{e,HB oligomer} (T = 273 K); this is conducive to two-step aggregation kinetics in which several oligomers form and later merge and rearrange to form fibrils. These arguments about the connection between the fibril formation kinetics and aggregate stability and metastability are meant here only to be suggestive. The best way to predict Aβ_16–22 aggregation pathways at different temperatures and concentrations would be to construct aggregation free energy landscapes along the appropriate reaction coordinates.

To validate the in silico prediction of Aβ_16–22 solubility at biophysically relevant temperatures, TEM was used to determine whether fibrils had formed after a predetermined time (2-wk incubation). Fig. 4 is a summary of our experimental results near the predicted solubility boundary line at temperatures ranging from 277 to 330 K. At T = 330 K, fibrils form at both 200 μM and 300 μM, but no fibrils are observed when the peptide concentration is at or below 100 μM, which is consistent with the predicted solubility (177 μM) at 330 K. Furthermore, at T = 277 K, T = 296 K, and T = 310 K, fibrils form when the concentration is equal to or above 20 μM, which also agrees with our predicted solubility at T = 277 K, T = 296 K, and T = 310 K of 0.2 μM, 2.6 μM, and 16.5 μM, respectively. It is important to note that even though fibrils were not observed in the TEM images at 10 μM at all temperatures tested, this does not rule out the presence of low populations of fibrils under these conditions that are below the detection limits of this method (i.e., low possible surface adsorption of the fibrils). Thus, we conclude that the boundaries predicted by the phase diagram quantitatively agree with the experimental findings, demonstrating the power of this approach to understand the thermodynamic stability of peptide assemblies.

The concentration and temperature conditions at which Aβ_16–22 forms fibrils reported from a number of other in vitro studies also agree with our simulation prediction (2, 3, 8, 45) (Fig. 4 and SI Appendix, Fig. S5). It should be noted that the only studies included in this comparison use Aβ_16–22 with capped N and C termini, since uncapping the termini can have a substantial twisting effect on the fibril supramolecular structure (46). Most of these studies were performed at high concentrations (>200 μM), placing them in regions of the phase diagram that would predict fibril formation, even at the highest temperature studied (55 °C, 328 K) (7). Only one study by Senguen et al. (3) demonstrated that fibrils are formed at a concentration of 55 μM at 37 °C (310 K), again in agreement with our phase diagram.

Knowledge of the peptide solution/fibril phase boundary could allow us to modulate the quality of nanofiber structures formed. This can be accomplished by carefully controlling the driving force for fibril formation (e.g., highly ordered fibrils can be obtained at solution conditions close to the phase boundary, while less-ordered entangled fibrils can be obtained at conditions away from the phase boundary). This suggests that our computation-based approach could inform future fabrication of amyloid nanofiber substrates/scaffolds with desired microstructure properties for various tissue engineering and biomedical applications (47, 48). In addition, controlling protein aggregation is a key problem for the purification and storage of therapeutic proteins (49, 50). Thus, knowledge of the solubility of specific peptides and proteins over a wide range of temperatures could be used to effectively prevent their unwanted agglomeration.

Our approach can be applied to investigate the effect of mutations on peptide solubility. For example, in preliminary simulations, we show that the F19A mutant of Aβ_16–22 remains soluble at C = 20 mM and T = 342 K in contrast to the fibril-forming behavior of the wild type at the same conditions. In fact, our predicted solubility of the F19A mutant is about 600 mM at 307 K (SI Appendix, Fig. S6). This result is consistent with experimental findings by Senguen et al. (3) that the F19A mutant of Aβ_16–22 is much more soluble (at least 350 μM) than its wild type at body temperature. One of our next steps will be to calculate the solubility of Aβ_16–22 mutants at positions 19 and 20 and to compare our results with experimental measurements.