Frustrated peptide chains at the fibril tip control the kinetics of growth of amyloid-β fibrils

To examine the growth of Aβ40 fibrils, we use time-resolved in situ AFM (Fig. 1 and SI Appendix, Fig. S1) (50). We deposit fibril seeds on mica and monitor the growth of both fibril ends toward fixed reference points (Fig. 1A) in solutions of Aβ40 monomer (SI Appendix, Fig. S2) with known concentrations. We evaluate the fibril growth rate as the slope of the time correlation of the fibril tip displacement (Fig. 1E) (50). Our previous work employing this method (50) revealed that the fibril growth rates and solubility measured using in situ AFM are close to those determined from time-dependent bulk growth of fibrils in solution (31). This comparison certifies that interactions with the substrate that may strain the fibrils or assist the supply of monomers to the fibril tip do not modify the growth rates, in contradistinction to results with Aβ42, the short amyloid peptide Aβ (12–28), and amylin (43, 51, 52). We also established that the opposite ends of individual fibrils grow at similar rates, and the growth was relatively steady (50). Time-resolved in situ AFM monitoring of fibril growth revealed that the fibrils readily dissolved in quiescent peptide-free solutions (Fig. 1 B and F).

The addition of urea led to significant increase of the fibril solubility: the fibrils dissolved at concentrations at which they otherwise grew in urea-free solutions (Fig. 1C). Urea-induced thermodynamic fibril destabilization, manifested as higher solubility, is consistent with urea’s known activity as a universal protein denaturant (53) owing to its favorable interaction with the amide groups of the peptide backbones (54). This interaction impairs not only the formation of contacts between segments of a single chain that support folded protein structures but also the formation of contacts between distinct chains within amyloid fibrils (55). Increasing the peptide concentration in the presence of urea, however, leads to growth rates significantly faster than the values recorded at the same peptide concentration in the absence of urea (Fig. 1D). The acceleration of fibril growth in the presence of urea would appear contrary to the destabilization of the contacts that support the fibril structure (56, 57).

The correlation between the fibril growth rate R and the Aβ40 concentration C is linear (Fig. 2A). The observed linearity implies that the fibrils grow by association of the dominant solution Aβ40 species, whether it be monomer, dimer, or a heavier oligomer (58). Whereas oligomers of varying compositions are present in Aβ solutions, they reside in equilibrium with the monomers, which capture the majority of the peptide mass in the solution (59, 60). Growth by association of oligomers in equilibrium with a majority of monomers would manifest as a superlinear (e.g., quadratic, for growth by dimer association) $R\left(C\right)$ correlation (58). We tentatively conclude that Ab40 fibrils grow by association of monomers.

The $R\left(C\right)$ correlation crosses the interpolated line of zero growth at C_e = 0.44 ± 0.07 μM (50). For C below C_e, the negative values of R correspond to fibril dissolution (Figs. 1B and 2A). A solution with concentration C_e is in equilibrium with the fibrils (i.e., C_e is the Aβ40 solubility with respect to the fibrils). The equilibrium $F_n\hspace{0.17em}+\hspace{0.17em}M\hspace{0.17em}\rightleftarrows F_{n+1}$, where $F_n$ and $F_{n+1}$ denote fibrils that differ in length by one monomer $M$, is characterized by a constant $K={\left[M\right]}_e^{-1}$, since the addition of a monomer does not modify the fibril concentration and $\left[F_n\right]=\left[F_{n+1}\right]$. Considering the dominance of monomers in the solution, we approximate the equilibrium monomer concentration ${\left[M\right]}_e$ with the total peptide concentration $C_e$ at equilibrium with the fibrils and arrive at $K=C_e^{-1}$.

For insight into the mechanisms that guide faster growth in the presence of urea despite the fibril destabilization that this denaturant enforces, we measured the R(C) correlations at two concentrations of urea, 1 and 1.5 M, and compared them to R(C) data in urea-free solutions (Fig. 2A). At the three tested compositions, fibrillization was reversible. The growth and dissolution dynamics revealed by AFM images (Fig. 1 C and D) and the R(C) correlations (Fig. 2A) demonstrate that urea acts as an apparent catalyst for growth and dissolution. It leads to both faster fibril growth and faster dissolution (Fig. 2A), while increasing the Aβ40 solubility with respect to the fibrils.

The solubility boosts with added urea define gains of standard free energy of fibrillization, $\mathrm{\Delta }{G}^o=-RT\ln K=RT\hspace{0.17em}\text{ln}{C}_e$, from −36.5 ± 0.4 kJ ⋅ mol⁻¹ in the absence of urea to −32.7 ± 0.3 kJ ⋅ mol⁻¹ at 1 M urea and −31.9 ± 0.2 kJ ⋅ mol⁻¹ at 1.5 M urea [i.e., about 3 kJ ⋅ mol⁻¹(mole urea)^-1 (Fig. 2E)]. The increasing $\mathrm{\Delta }{G}^o$ announces the expected urea-enforced destabilization of the fibrils relative to the solute monomers. We use fibril seeds that were generated without urea, and previous work has established that the structure of a fibril persists after the growth conditions deviate from those during fibril nucleation (11, 29, 61). The uniformity of the fibril structure during growth in the presence and absence of urea ascribes the destabilization of the fibrils relative to the solution to urea-imposed lower free energy of the solute peptide chains (Fig. 2 F and G).

We model the linear R(C) correlation as $R=a{k}_a\left(C-C_e\right)$, where a = 0.47 nm is the contribution of an incorporated monomer to the protofilament length (29), and k_a is the bimolecular rate constant for the reaction between monomers and fibril tips. We define protofilament as a single stack of peptide chains (SI Appendix, Fig. S3) (62, 63); we discuss alternative definitions (29, 61, 64) in SI Appendix. The expression for $R$ is akin to the result of a model, which assumes a two-step reaction of monomer association to the fibril tip, followed by incorporation into the fibril, under conditions where the first step is rate limiting (31). The constant k_a assumes values between 1.8 × 10⁴ and 2.8 × 10⁵ M⁻¹ ⋅ s⁻¹, depending on the urea concentration (Fig. 2C). These values are significantly slower than the expected diffusion limit for association of about 10¹⁰ M⁻¹ ⋅ s⁻¹ (65). The rate constant can be written as $k_a=k_0\text{exp}(-\mathrm{\Delta }{G}^{‡}/k_{B}T)$, where $k_B$ is the Boltzmann constant and T is temperature. We assume $k_0=$ 10¹⁰ M⁻¹ ⋅ s⁻¹ as the diffusion limit (65). It has been argued that $k_0$ for Aβ40 fibril growth should be closer to 10⁹ M⁻¹ ⋅ s⁻¹ (31); the exact value of $k_0$, however, does not modify the arguments presented in A Frustrated Complex at the Fibril Tip. We assume that the rotational and orientational entropy contributions to $k_0$ are not substantially affected by urea. We note that urea increases the solution viscosity by 4% at 1 M and 6% at 1.5 M (66). Accounting for this increase would depress the values of $G^{‡}$ in the presence of urea by about 0.1 kJ ⋅ mol⁻¹, which is within the experimental uncertainty of this variable (Fig. 2E). With this, the free energy barrier $\mathrm{\Delta }{G}^{‡}$ decreases from 33.0 ± 0.1 kJ ⋅ mol⁻¹ in the absence of urea to 28.1 ± 0.2 kJ ⋅ mol⁻¹ at 1.0 M urea and to 26.2 ± 0.2 kJ ⋅ mol⁻¹ at 1.5 M urea; that is, the barrier reduces by about 4 kJ ⋅ mol⁻¹(mole urea)⁻¹ (Fig. 2 E and G). Importantly, the correlation between $\mathrm{\Delta }{G}^{‡}$ and the urea concentration is linear (Fig. 2E). In analyses of protein folding kinetics, such linearity is taken as evidence that urea does not greatly modify the conformation of the transition state. In folding, observed kinetics nonlinearities have been shown to correlate to flatter free energy profiles with “malleable” transition states (67). The linearity here, while based on only three concentrations, suggests that lack of urea-enforced structure change in the transition state is a good first approximation.

The opposing activation free energy $\mathrm{\Delta }{G}^{‡}$ and fibrillization free energy $\mathrm{\Delta }{G}^o$ responses to urea eliminate certain mechanisms of amyloid fibril growth that have previously been suggested based on simulations. One such proposal assumes that monomeric peptides undergo a conformational transformation into an aggregation-prone state, whose lifetime is longer than the time required for collision with a fibril tip (24). In this scenario, the depression of the free energy of the peptides in the solution enforced by urea would boost the free energy barrier for incorporation (Fig. 2F) and impose slower fibril growth, contrary to actual observations. From a broader perspective, the opposing $\mathrm{\Delta }{G}^{‡}$ and $\mathrm{\Delta }{G}^o$ trends defy any mechanism that constrains the urea impact to the peptide chains in the solution.

The essential identity of the bulk fibril structure in the presence and absence of urea eliminates urea-driven bulk fibril structure modifications as the mechanism that regulates the high sensitivity of $\mathrm{\Delta }{G}^{‡}$ to urea. The exclusion of peptides in the solution and fibril structure as targets for urea attack implies an incoming chain passes through an intermediate state, in which it attaches, but only partially, to the fibril tip. The substantial magnitude of $\mathrm{\Delta }{G}^{‡}$ and its sensitivity to urea together imply that the intermediate complex is bound by strong hydrophobic contacts, distinct from the native contacts that support the fibril structure and dictate $\mathrm{\Delta }{G}^o$. The unraveling of the initial nonnative contacts between the incoming and terminal fibril monomers, in search of the native conformation typical of the bulk fibril, becomes the rate-limiting step for the attachment of the monomer to the fibril tip and contributes to $\mathrm{\Delta }{G}^{‡}$. The extension of the linear $R\left(C\right)$ correlation to dissolution in undersaturated solutions (Fig. 2A) indicates that disorder at the tip is an equilibrium feature of the fibril structure. Urea, by interacting with the backbones of monomers at the fibril tip (68, 69), weakens the nonnative contacts and thereby destabilizes the intermediate state, which lowers $\mathrm{\Delta }{G}^{‡}$ (Fig. 2G). Importantly, the linear correlation of $\mathrm{\Delta }{G}^{‡}$ with the urea concentration (Fig. 2E) indicates that the urea-induced weakening of the nonnative contacts likely stops short of modifying the structure of the intermediate and transition states. In protein folding, such energy-rich nonnative contacts have been called frustrated (70–72). Whereas simulations have foreseen frustrated states as amyloid peptides fold to incorporate into fibrils (25, 73, 74), the opposing $\mathrm{\Delta }{G}^{‡}$ and $\mathrm{\Delta }{G}^o$ responses to urea, based on the divergent effects of urea on fibril solubility and growth rate, provide direct experimental evidence for the role of frustration in fibrillization.

The pathway of association of peptides with the fibril tips suggested by the rates of fibril growth (Fig. 2) shares certain features with a mechanism put forth by simulations. In this mechanism, the incorporation of a monomer into a fibril divides in two steps. First, the association of an unstructured monomer to the fibril tip (often called docking), followed by conformational rearrangement toward the peptide structure in the fibril bulk (locking) (24, 65, 75, 76). In the simplest lock-and-dock scenario, the conformational transformation of each captured peptide chain is templated by the previously arrived peptide. The magnitude of $\mathrm{\Delta }{G}^{‡}$ and the opposing $\mathrm{\Delta }{G}^{‡}$ and $\mathrm{\Delta }{G}^o$ trends that we observe advocate a more complex picture whereby the docked state evolves to a frustrated complex, in which a monomer makes many nonnative contacts in the fibril tip.

Additional characteristics of the frustrated complex emerge when we quantitatively compare the urea-induced weakening of the frustrated contacts to what is seen for protein unfolding (77). Statistics over 45 proteins have revealed that urea-induced lowering of the free energy of unfolding scales with the protein surface area exposed to solvent upon unfolding (77). This proportionality indicates that the thermodynamic effect of urea of about 3 kJ ⋅ mol⁻¹ (mole of urea)⁻¹ is associated with the loss of about 3,200 Ų of solvent-accessible surface area (SASA) upon monomer incorporation into the fibril; we designate this loss as ΔSASA⁰ = −3,200 Ų. The urea-induced activation free energy drop of about 4 kJ ⋅ mol⁻¹ (mole of urea)⁻¹ corresponds to the exposure of ΔSASA^‡ = 5,300 Ų. The evaluations of ΔSASA⁰ and ΔSASA^‡ afford the opportunity to compare the results of the kinetics experiments to those of simulations and thus attain additional insights in the incorporation pathway.

To achieve greater molecular-level understanding of how an incoming peptide chain acquires the structure typical of the fibril bulk as it incorporates into a fibril, we carried out simulations using the associative memory, water-mediated structure, and energy model for molecular dynamics (AWSEM-MD) (78). The coarse-grained nature of the AWSEM simulation Hamiltonian leads to much more rapid sampling than models with explicit solvent molecules. Owing to the lack of solvent, 1 ps of simulation time compares to longer than a nanosecond of laboratory time. We used the Protein Database entry 2LMN, the polymorph examined with AFM (50), to construct the fibril structure. We evaluated the free energy profile for an incoming monomer to associate with a fibril composed of five chains arranged in a single stack (Fig. 3A). We characterized the conformational ensembles in terms of the distance between the center of mass of the peptide chain and the fibril end and by the similarity of the structure of the incoming monomer to that of a chain in the fibril bulk (Fig. 3B). We explored about 30 select configurations, divided into six groups, along the reaction pathway (Fig. 3 A and B). We then computed the SASA of fully atomic models generated from the coarse-grained structures (Fig. 3C).

The computed change, ΔSASA_eq = SASA₆ – SASA₁ (Fig. 3C), defines the sensitivity of the equilibrium $\mathrm{\Delta }{G}^o$ to the addition of denaturant, and the model result ΔSASA_eq = 3,400 Ų agrees well with the estimate based on the measured urea dependence of the fibril solubility. The simulations reveal that the SASA passes through a minimum at position 3 (Fig. 3 A and C). In this group of structures, an incoming monomer partially binds to itself forming nonnative inter- and intrachain contacts rather than forming native contacts with the fibril tip (Fig. 3 A and B). These frustrated contacts must unravel as the peptide reconfigures to fully integrate into the fibril in the cluster 6 structures, labeled in Fig. 3B. Remarkably, the finding of an intermediate frustrated complex at the fibril tip agrees with the conclusions implied by the responses of experimentally measured $\mathrm{\Delta }{G}^{‡}$ and $\mathrm{\Delta }{G}^o$ to added urea.

The positive ΔSASA between positions 3 and 6 represents only a rough estimate of the effects of urea on the activation free energy $\mathrm{\Delta }{G}^{‡}$. All-atom simulations of the conformations of the peptides at the fibril tip yield ΔSASA^‡ = 1,800 Ų. The results for monomer association to the opposite, odd, fibril end (Fig. 3D and SI Appendix, Fig. S5) are similar. Importantly, both values of ΔSASA^‡ are only about one-third of the value inferred from the response to urea of the growth kinetics of individual fibrils, measured by AFM. The discrepancy between the simulated and measured ΔSASA^‡s suggests that the structure of the frustrated complex at the fibril tip is probably more elaborate than what has emerged from these initial simulations.

We envision two possible models of a more elaborate frustrated intermediate. First, the frustrated complex may involve more than one peptide chain within a single protofilament; the ratio between the experimental and simulated ΔSASA^‡ suggests that three or four monomers would need to be involved. Simulations of a peptide chain association with a disordered fibril tip would involve characterizing a multidimensional free energy landscape and require significant additional efforts. Alternatively, the intermediate frustrated state may involve binding of the incoming monomers to both protofilaments in a filament. The experimentally measured thicknesses of the fibrils monitored by AFM indicate that the majority of the filaments are built of two parallel protofilaments, represented by the 2LMN structure (50).

We explore the possibility where the incoming peptide chain associates to both protofilaments in a filament. We note that the addition of monomers to a double filament must ultimately preserve the structure of the twinned protofilaments, and this involves at least two alternating conformations of the fibril tip: one where the two protofilaments exactly match in length and one where one of the protofilaments is ahead by one monomer. AWSEM simulations of monomer attachment to the tip of a two-protofilament structure reveal that, instead of folding on top of one of the two monomers at the fibril tip, an incoming peptide chain spans both stacks and enforces a stabilized frustrated conformation that may be incapable of further growth (Fig. 3 E and F and SI Appendix, Fig. S6). This state was predicted for both the matched and the mismatched protofilaments (Fig. 3 E and F). ΔSASA^‡ from such frustrated states to the final conformation are greater than 12,000 Ų (Fig. 3 E and F).

To discriminate experimentally whether the complex frustrated state at the fibril tip recruits more than one disordered peptide chain that crown a single protofilament or structures as a single chain that spans both protofilaments in a filament, we monitored the growth rates of freshly cut fibrils (Fig. 4A). In freshly cut fibrils, the peptides at the tip initially should carry the structure of monomers in the fibril bulk, and an intermediate state composed of more than one frustrated chains may not have had time to evolve. Thus, if the slow growth rate observed in the AFM experiments is due to a frustrated complex composed of several chains, freshly cut fibrils will grow faster than fibrils with normal tips. By contrast, if the high $\mathrm{\Delta }{G}^{‡}$ deduced from AFM experiments is due to interactions of a single incoming chain with two adjacent protofilaments, the frustrated complex that crowns the freshly cut fibril tips will be identical to the one at equilibrated tips, and the freshly cut tips will grow with rates similar to those of normal tips.

The AFM measurements reveal that freshly cut fibrils initially grow about twice as fast as fibrils with equilibrated tips (Fig. 4 B and C). In five of the cut fibrils, the growth rate transitioned to its “normal” value after 6 to 8 min, during which time the fibrils grew about 10 nm (Fig. 4B). The observed faster growth rates indicate that the freshly cut fibrils carry a simpler frustrated complex than equilibrated fibril tips. For a second test of the distinction between the kinetics of growth of freshly cut and mature fibrils, we measured the effects of 1 M urea on the growth of freshly cut fibrils and compare them to the growth rate of mature fibrils (Fig. 4C). The results reveal that urea accelerates the growth rate constant of freshly cut fibrils by about threefold, significantly weaker than its ca. sixfold effect on normal tips (Fig. 4C).

Collectively, the outcomes of the two tests with freshly cut fibrils indicate that the frustrated complex at equilibrated fibril tips comprises more than one peptide chain but is probably constrained to a single protofilament.

We have expressed Aβ(M1-40) (MDAEFRHDS GYEVHHQKLVFFAEDVGSNKGAIIGLMVGGVVIA) in E. coli and purified it by size exclusion chromatography (50) following published procedures (79). This method produces Aβ peptide with a methionine attached to the N terminus (SI Appendix, Fig. S2), which significantly simplifies purification and contributes to several-fold–greater yields (79). The fibrillization kinetics of the methionine-initiated peptide quantitatively matches that of the methionine-free peptide (79), justifying the wide use of Aβ(M1-40) and Aβ(M1-42) in amyloid-β fibrillization studies (31, 35, 80).

Fibril seeds grew at 37 °C while stirred at 300 rpm on a table Inkubator 1000 (Heidolph) for 24 h. Seed generation in continuously stirred solutions enforces a twofold symmetric fibril structure, 2LMN (31, 50). The generation of fibril seeds was performed without any exposure to urea even for growth rate determinations in the presence of urea. In those latter runs, urea was added to the solution immediately before fibril growth observations with AFM.

We used a multimode atomic force microscope (Nanoscope VIII or IV, Bruker) for all AFM experiments. To prepare samples for growth rate measurements, aliquots of 2 μL second-generation fibril seeds was added to the incubation buffer; urea was added based on experiment needs. AFM images were collected in tapping mode.

Existing fibrils deposited on the AFM substrate were monitored under AFM for several minutes. To cut a fibril, we chose a relatively long one, which spanned about 50% of the image width. To ensure a good cut, we rotated the scan angle so that the angle between the fibril and fast scan direction angle was at least 45°. To increase the force of interaction between the tip and the fibril, we lowered the amplitude set point to 50 or 60% of the original value and set the scan frequency to 1 Hz to better control the length of the cut out. After the cut was complete, we reset the set point and the scan frequency to their original values to monitor the growth of the freshly cut ends.

The simulations were carried out using the AWSEM force field (78). AWSEM represents proteins as coarse-grained structures, with every amino acid residue defined by the interactions of three beads: C_α, C_β, and O. The AWSEM Hamiltonian ensures an ideal backbone geometry between the explicit beads and includes terms accounting for secondary and tertiary interactions, implicit solvent effects, and residue burial preferences. To simulate the growth of an Aβ40 fibril, we used the structure of a twofold symmetric polymorph consisting of U-shaped protofilaments with a positive stagger (2LMN) (81). We chose this structure because it likely represents the polymorph that forms under the conditions employed in the AFM experiments (50).

To study the free energy associated with monomer addition, we introduce the order parameter Q-interface in the following form:

where r_ij is the interatomic distance between the C-α atoms of residues i and j, and r_ij^N is the distance between the same C-α atoms in the fully aligned fibril n + 1 structure. The term σ_ij defines the distribution width, which is minimally influenced by the sequence distance between the i and j C-α atoms. The value of Q-interface ranges from 0 (completely dissimilar contacts) to 1 (native contacts), as enforced by the normalization by N_p, the number of unique residue pairs that satisfy the |i − j| > 2 conditions.

Umbrella sampling simulations were performed with a biasing force between the center-of-mass of the free monomer and the center-of-mass of the chain sitting at the tip of the protofilament (65).

The set of simulations performed for modeling monomer additions at either end of the protofilament—to compare the dynamics of a positive and negative stagger—and across the full range of distance biasing and temperatures, resulted in trajectories totaling 0.46 µs. A timestep of 2 fs was used for all simulations, and each simulation ran for a total of 2,500,000 steps (5 ns), with the first 0.042 ns of each simulation disregarded to allow for equilibration. Potential of mean force plots were generated using the multistate Bennet acceptance ratio (82) to remove the biasing term contribution to the energy calculations from umbrella sampling (Fig. 3B).

To compare the simulations results with results of the fibril growth kinetics experiments, we evaluate degree of burial of sex distinct conformations along the reaction pathway in terms of the total SASA. Even though AWSEM represents each amino acid with three beads, which correspond to the C_α, C_β, and O atoms, it allows for the reconstruction of all heavy atoms of the backbone under the assumption of backbone planarity and ideal geometry. To perform the SASA calculations, the CG conformations were first back-mapped to all-atom representations using Modeler (83), with the lowest-energy predicted structure being chosen from 10 possible conformations. The positions of the rest of the heavy atoms of the side chains, except C_β, which is explicitly represented in AWSEM, were computed using SCWRL4 (84). The structures generated from this process were then visualized using PyMOL (85). The SASA values were estimated using a native function in PyMOL, in which a rolling a sphere with radius 1.4 Å, representing a water molecule, is used to race over the exposed atoms of the peptides and the fibril.