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Alloform-specific differences in structural dynamics between Aβ40 and Aβ42 appear to underlie the pathogenesis of Alzheimer’s disease. To elucidate these differences, we performed µs time-scale replica exchange molecular dynamics simulations to sample the conformational space of the Aβ monomer and constructed its free energy surface. We find that neither peptide monomer is unstructured, but rather that each may be described as a unique statistical coil in which five relatively independent folding units exist, comprising residues 1–5, 10–13, 17–22, 28–37, and 39–42, which are connected by four turn structures. The free energy surfaces of both peptides are characterized by two large basins, comprising conformers either with substantial α-helix or β-sheet content. Conformational transitions within and between these basins are rapid. The two additional hydrophobic residues at the Aβ42 C-terminus, Ile41 and Ala42, significantly increase contacts within the C-terminus and between the C-terminus and the central hydrophobic cluster (Leu17-Ala21). As a result, the β-structure of Aβ42 is more stable than that of Aβ40 and the conformational equilibrium in Aβ42 shifts towards β-structure. These results suggest that drugs stabilizing α-helical Aβ conformers (or destabilizing the β-sheet state) would block formation of neurotoxic oligomers. The atomic-resolution conformer structures determined in our simulations may serve as useful targets for this purpose. The conformers also provide starting points for simulations of Aβ oligomerization, a process postulated to be the key pathogenetic event in AD.
Alzheimer’s disease (AD) is the most common cause of late-life dementia1. The pathognomonic neuropathologic features of AD are extracellular amyloid deposits comprising primarily fibrils of the amyloid β-protein (Aβ) and intracellular neurofibrillary tangles formed by tau protein2. Compelling evidence supports a seminal role of Aβ in AD. Fibrils originally were thought to be central to AD pathogenesis3, but recent studies support the hypothesis that the proximate neurotoxic agents in AD are Aβ oligomers4–6. In fact, recent experiments have shown that some pathways of Aβ oligomerization and fibril formation are independent7,8 and that fibrillization may be protective9.
Aβ is produced naturally and ubiquitously in vivo as an ~4 kDa peptide10. It exists predominately in two forms, Aβ40 and Aβ42, that contain 40 or 42 amino acids, respectively (Fig. 1). Despite the small structural difference between Aβ40 and Aβ42 (the C-terminal Ile-Ala sequence), the peptides display significantly different behaviors in vitro and in vivo. Aβ42 is the principal component in parenchymal plaques11–13. An increase in the Aβ42/Aβ40 concentration ratio is associated with familial forms of early onset AD14,15. Treatments that reduce Aβ42 levels have been shown to correlate with decreased risk for AD16. In addition, Aβ42 displays enhanced neurotoxicity relative to Aβ4017–19. In vitro studies have shown that Aβ42 displays fibril nucleation and elongation rates that are significantly higher than those of Aβ4020–22 and that Aβ42 forms larger oligomers than does Aβ4023,24. These results support the conclusion that development of efficacious therapeutic agents for AD would be facilitated by knowledge in at least two areas: (1) the structural dynamics of Aβ monomer folding and oligomerization; and (2) differences in the dynamics between Aβ40 and Aβ42.
Experimental studies of Aβ monomer structure and dynamics are complicated by the lack of existence of a stable fold and the propensity of the peptide to aggregate into amorphous assemblies or multiple fibrillar forms 25,26. NMR experiments on Aβ fragments or full-length Aβ40 and Aβ42 done in the absence of solvent additives consistently reveal little regular structure27–35. A small increase in C-terminal rigidity has been observed in Aβ42 versus Aβ4034. Consistent with these data, studies of region-specific endoprotease sensitivity showed increased resistance of the Aβ42 C-terminus29. These studies have provided relatively coarse insights into local Aβ structure, but they were not capable of elucidating the Aβ conformational ensemble in atomic detail. Substantial helical structure was revealed in Aβ studied in mixtures of fluorinated alcohols36–40 or SDS41,42 with water. However, the relevance of these systems for understanding extra-membranous assembly is unclear.
Molecular dynamics (MD) simulations complement experimental studies through their ability to define the conformational space and dynamics of a macromolecule(s)43. This approach is being applied actively in the Aβ field (for recent reviews, see Teplow et al.44 and Urbanc et al.45). Recently, we studied Aβ42 dynamics computationally, integrating these data with experimental results obtained using ion mobility spectroscopy–mass spectrometry46. We found that Aβ42 conformational space is dominated by loops and turns. Comparative studies with Aβ40 were not done. Sgourakis et al. performed Aβ simulations using both Aβ40 and Aβ42 in an explicit water environment47. Structured regions were observed, one of which was a β-hairpin within the C-terminal peptide segment Ile31–Ala42. The simulation employed a virtual cubic space that was designed to contain a collapsed peptide that then was solvated by explicit waters. This system size may not accommodate extended conformers and thus not completely sample conformational space, a result that would produce a biased view of Aβ structure and dynamics. Here, using replica-exchange MD (REMD) with an all-atom protein model, we sample and compare the conformational spaces and the corresponding free energy surfaces of Aβ40 and Aβ42. We assess the relevance of the data by comparison with experimental information extant. Through comparison of the structural dynamics of Aβ40 and Aβ42, we establish their shared and distinct features. Finally, we discuss the implications of these findings for understanding and potentially controlling neurotoxic Aβ assembly.
Before performing extensive analyses of our simulation data, we sought to establish their physical relevance by comparison with experimental data produced using NMR, the experimental method that has provided the highest resolution structural information on Aβ. To do so, we used chemical shift values predicted from our simulations (δsim) and determined experimentally (δexp). The δsim values were predicted with the SHIFTS program48, based on the conformational ensemble collected at 278 K. We used δexp values of monomeric Aβ that had been measured at the same temperature32 and pH. As shown in Fig. 2, the small standard deviations of the chemical shift values indicate the convergence of our simulation. The δsim values of Cα and Hα atoms of both Aβ40 and Aβ42 were highly correlated with those observed experimentally. Strong correlation (r = 0.942) was also observed for the Nα atoms of Aβ40. The Nα chemical shifts of Aβ42 were less correlated (r = 0.883) than those of Aβ40, but remained well correlated with δexp. It is possible that the modestly weaker correlations of Nα atoms are illusory because the SHIFTS program is known not to consider all the factors contributing to N chemical shifts in proteins49. It is also possible that the decreased correlation of the Aβ42 δsim(Nα) data results from intermolecular interactions among monomers in aqueous solution in the NMR experiment (personal communication, Dr. Michael Zagorski). In conclusion, the high correlations observed between our in silico data and those produced experimentally indicate that our simulations reproduce the Aβ structural ensemble well.
We first calculated and compared the secondary structures of Aβ40 and Aβ42. The secondary structure was assigned according to criteria defined by STRIDE50 (Fig. 3) or by DSSP51 (Fig. S1). Both programs yielded qualitatively similar secondary structure assignments (see below). STRIDE defines a higher percentage of “turn” structure. This is consistent with the fact that distance criteria are used by STRIDE to assign turn structure, whereas DSSP uses H-bonding as a criterion. Turns defined by DSSP thus comprise a subset of all turns defined by STRIDE. Because Aβ is very dynamic conformationally, it does not display high levels of classical α-helix, β-strand, or β-turn. Therefore, to represent both classical and non-classical turn structures, we discuss here the results obtained using STRIDE. The term “turn” thus refers to short peptide segments in which the peptide chain reverses direction, regardless of the presence or absence of H-bonds.
Residues 6–40, except those at the N- or C-termini, exist as turn structures with ~30–80% probability. Residues more likely to exist as β-strand or α-helix are those at or adjacent to the central hydrophobic cluster (CHC) region (residues 17–21) and C-terminus (residues 30–36). In the CHC, significant (≈30–40%) secondary structure comprising these types was observed. Residues 16–27 and 30–36 of Aβ42 display β-structure with frequencies of ~6–8% and ~8–14%, respectively. These data are consistent with prior experimental studies that have suggested that Aβ belongs to the class of “natively disordered” proteins52. However, Aβ is not entirely a “random coil.” Two highly populated (>60%) turn structures were observed, centered at residues 6–9 and 23–27. The positions of these turns correspond precisely with turn or bend-like structures observed by NMR to occur at Asp7–Glu11 and Phe20–Ser2632. We found that residues 14–16 and 31–35 also had high probabilities of existing as turns.
One experimental approach that has been used successfully to identify structured and unstructured regions within Aβ has been limited proteolysis. This approach revealed a protease-resistant decapeptide segment, Ala21–Ala30, that was found in NMR studies to form a turn-like structure29. This region was postulated to nucleate folding of the Aβ monomer. Interestingly, and consistent with this interpretation, pathogenic (familial AD- or cerebral amyloid angiopathy-linked) amino acid substitutions within this region alter the stability of this turn53. Based on these experimental findings, simulations of turn dynamics have been done on the decapeptide54–56. These studies all revealed that a majority of the peptide conformers possess a turn-like structure that is stabilized by hydrophobic interactions between Val24 and Lys28 and electrostatic interactions between Glu22 or Asp23 and Lys28. Baumketner et al.46 have suggested that hydrogen bonds involving the side chain of Asp23 and the amide hydrogens of adjacent residues are important in stabilizing this turn. These hydrogen bond interactions also recently were reported by Fawzi et al., although their data suggested a lower occurrence frequency (~40%) for this type of conformer57. A stable turn formed by residues Val24–Asn27, and an intramolecular Asp23–Lys28 salt bridge, also were observed in computational studies of Aβ(10–35) structural dynamics58.
Experiments on the Ala21–Ala30 decapeptide, or on Aβ(10–35), cannot reveal effects of adjacent (missing) regions of the peptide that may occur in the biologically and clinically relevant full-length peptide. To determine whether such effects exist, we studied the structure and dynamics of the decapeptide turn element within the native Aβ monomer (Fig. 4). We found that the structure of residues 21–30 is highly similar (RMSD≈1.0 Å) to that obtained using the decapeptide alone55. The structure is stabilized by a H-bond network involving the side-chain carboxyl oxygens of Asp23 and the NαH atoms of the adjacent residues Gly25, Ser26, Asn27, and Lys28 (Table 1), an H-bond between the Asp23 Oγ and the Ser26 Hγ atoms, and by a salt bridge formed between the Lys28 Nζ and the Glu22 carboxylate anion. Similar structural features and stabilizing factors also were observed in simulations by two other groups in explicit solvent, one using the OPLS-AA force field47 and the other using the Gromos force field (Dr. Yundong Wu, personal communication).
Our simulations suggest that Aβ(21–30) adopts similar structures in its isolated decapeptide form and in the context of the full-length Aβ monomer. In practice, this result suggests that the decapeptide is a relevant proxy for the cognate segment within the Aβ monomer. In theory, our observation would be predicted for a structural domain that serves to nucleate protein folding. It should fold in isolation and this folding should not be affected significantly by adjacent regions. In the full-length peptide, decapeptide folding would facilitate contacts between the CHC and C-terminal regions of Aβ, contacts that comprise key stabilizing elements of fibril structure. However, this fold forms infrequently, as Aβ exists in many conformations with comparable free energies and does not display a dominant global minimum on its free energy surface (Fig. 8).
In our simulation, the C-terminus of Aβ42 is significantly more structured than that of Aβ40, as reflected by a higher percentage of turn structure within peptide segment 34–41 and β-structure within segment 30–42. It is noteworthy that residues 30–36 in Aβ42 have almost twice the β-structure that they do in Aβ40 and that residues 39 and 40 have no β-structure in Aβ40 whereas they can exist in β-structure in Aβ42 with a likelihood of ~5%. The more frequent turn and β-structure at the Aβ42 C-terminus may be due to increased intramolecular contacts stabilized by the two hydrophobic residues, Ile41 and Ala42. In fact, such stabilizing interactions may account for the increased C-terminal rigidity observed in NMR experiments32–34. Higher β-structure frequency is observed also among residues 20–27 of Aβ42, the basis of which is a β-hairpin centered at residues 28 and 29 (Fig. 8b, cluster 8). Cluster 8 does not exist with Aβ40. Interestingly, residues 3–6, 11–14, and 17–18 have higher β-content in Aβ40 than in Aβ42 and residues 17–28 are more likely to exist in α-helices in Aβ40. For both peptides, residues 14–28 are more likely to form α-helical structure than the C-terminus, consistent with a number of experiments in which central region α-helices were more stable than C-terminal helices40.
We assess tertiary structure by analyzing contacts among amino acids (Fig. 5). Two amino acids were considered in contact if any heavy atom (C, N, O, S) of one residue was ≤ 5.5 Å from any heavy atom of the other residue. Globally, the contact maps of Aβ40 and Aβ42 are similar and are characterized by relatively frequent contacts of residues within peptide segments 3–13, 10–20, and 17–36 (Fig. 5, red boxes). These segments correspond to populated turn structures centered at residues 6–9 (T1), 14–16 (T2), and 23–27 (T3), respectively. Weaker but noticeble contacts also were observed between two discontinuous segments 6–10 and 23–27. We note that the CHC contacts residues 30–36 at the C-terminus with 20–30% probability. However, with the two extra residues Ile41-Ala42 at the C-terminus, residues 30–42 of Aβ42 are more likely to contact each other, presumably in the context of a turn (Fig. 5b).
Despite the overall global similarity in contacts between Aβ40 and Aβ42, differences are discernible (Fig. 6, red boxes). Residues 24–30 of Aβ40 are more compact than those of Aβ42, as shown by the 2.5–30% more contacts in this region (Fig. 6a). This datum is consistent with the higher helical content of this region (Fig. 3). The Aβ40 CHC is more likely (5–17.5%) to contact the N-terminus. In contrast, the Aβ42 CHC is 2.5–10% more likely to contact the C-terminus (residues 34–41) (Fig. 6b). Residues 3–8 contact residues 10–14 more frequently, which corresponds to higher turn formation of residues 6–9 and higher β-content of residues 2–14. A peculiar residue is Phe19, which is more buried in Aβ40, as evidenced by its more frequent contacts with residues 3–36.
The extra two residues at the C-terminus significantly increase the contact frequency within peptide segment 31–42 and also increase modestly the contact frequency between the C-terminus and the CHC. It is interesting to note that Tyr10 in Aβ42 contacts residues 13–32 more frequently (Fig. 6b, red box), suggesting that it is more shielded from the solvent. In fact, recent chemical cross-linking experiments suggest that Tyr10 is more difficult to cross-link in Aβ42 than it is in Aβ40 (unpublished observation).
Contact maps (Fig. 5) suggest that the N-terminus of Aβ40 is more compact than that of Aβ42, whereas the opposite is true of the C-terminus. DMD simulations of Aβ oligomerization performed by Urbanc et al. showed that the N-termini of Aβ42 monomers comprising oligomers were significantly more extended and unstructured than those in the corresponding Aβ40 assemblies59. Our observation here suggests that this feature exists at the monomer level. We postulate that the increased hydrophobicity of the Aβ42 C-terminus, due to the presence of Ile41 and Ala42, increases the contact frequency between residues in the C-terminus and CHC. As a result, the N-terminus of Aβ42 has less contact with the CHC and exists most frequently as an extended or coil structure. This postulation is consistent with results of studies of the assembly state-dependence of the intrinsic fluorescence of a Tyr residue substituted for Phe2060. These studies showed that a significant increase of fluoresence intensity occurred in Aβ40, but not Aβ42, during the initial oligomerization of the peptides. This change would result from a disruption of N-terminus–CHC interaction in the Aβ40 monomer and the subsequent formation of the more apolar CHC–C-terminus complex, in which quenching of the fluorescence would be substantially lower. It is intriguing that recent scanning cross-linking studies have suggested that such competition between the N-terminus and C-terminus for interaction with the CHC is a fundamental feature of Aβ conformational dynamics61. In Aβ42, relative to Aβ40, the C-terminus “wins” because the presence of Ile41 and Ala42 facilitates formation of a C-terminal bend that creates a larger and more stable hydrophobic region, one that can interact especially strongly with the CHC. The biological consequence of this interaction is the formation of folded Aβ monomers with the well-known increased propensity to form neurotoxic higher order assemblies.
Secondary and tertiary structure analyses suggest that Aβ is relatively disordered and thus populates a large conformational space. However, like proteins in general62,63, and from the computational and experimental results discussed above, the Aβ monomer does not appear to be a random coil, if we define “random coil” as an entity that can populate every point in monomer conformational space with equal probability. The implication of this fact is that valuable knowledge may be obtained if non-random features of conformational space can be identified. One method to achieve this goal is to identify segments of Aβ whose motions are correlated. To do so, we quantify the cross-correlations of atomic positional fluctuations64. This approach has shown that many regions in proteins, especially those involving secondary structure elements, display correlated motions.
We represent each residue with its corresponding Cα atom. In Fig. 7, each grid square indicates the correlation (rij) between a residue pair. rij = 1 indicates two residues are completely correlated, whereas rij = −1 indicates completely anti-correlated motions. Complete correlation means that two residues always move together with the same phase and period. For simplicity, we arbitrarily define meaningful correlations to be those with |rij| > 0.5.
For both Aβ40 and Aβ42, positive correlations are only observed along the diagonal line, i.e., residues have motions that are correlated only with their immediate neighbors. All non-neighbor motions are anti-correlated. Aβ40 and Aβ42 each display four sets of peptide segments whose motions are correlated significantly (Fig. 7, red boxes 1–4). The location of each set is similar, but the sizes of the sets and the magnitudes of the correlations within them vary (Fig. 7, Table 2). In Aβ40, the largest negative correlations exist between residues 1–3 and 6–10 (box 1; T1), 9–11 and 15–18 (box 2; T2), 21–23 and 29–31 (box 3; T3), and 30–35 and 38–40 (box 4; T4). The correlations in boxes 1 and 2 of Aβ42 occur over a similar segment range as do those in Aβ40 and have similar total magnitudes. The largest difference occurs for boxes 3, in which substantially greater total correlation exists for Aβ42. The segment range in box 4 is translated two residues towards the C-terminus in Aβ42 relative to Aβ40 and the total correlation magnitude is greater. These differences result from the presence of Ile41 and Ala42, which likely stabilize a C-terminal turn centered at Gly38. This postulation is consistent with our secondary structure analyses (Fig. 3b) that show that residues adjacent to the Gly37–Gly38 dipeptide in Aβ42 have particularly high probabilities of forming β-structures.
If monomer tertiary structure is maintained during peptide oligomerization and fibril formation, our data allow certain structure-activity predictions. For example, models of Aβ fibril structure produced using NMR data suggest that the fibril is formed by parallel, in-register packing of β-strand-turn-β-strand motifs65,66 in which D23–K28 salt bridges stabilize the turn. The formation of the T3 turn identified in our simulation would increase contacts between peptide segments immediately adjacent to it, facilitating fibril formation. In contrast, formation of the T4 turn (centered at Gly38) would disrupt the extended conformation that exists in the C-terminal β-strand element of the fibril. In considering these predictions, however, it is important that Aβ conformational dynamics in the monomer state be distinguished from that in the fibril state. The two are related, but the most stable conformers in one state are not entirely isomorphous with those in the other. Experimental and computational studies have shown that conformational conversions may occur both in incoming monomers attaching to fibril ends and to the monomers comprising the fibril ends themselves67–69.
Our observation that all the negatively correlated peptide segments reside on the opposite sides of turns identified at residues 6–9, 14–16, and 23–27 (Fig. 3) suggests that segmental motion within Aβ is dominated by “zipping” and “unzipping” motions around these turn structures. It is intriguing to consider Aβ monomer folding as a process involving five relatively independent folding units, residues 1–5, 10–13, 17–22, 28–37, and 39–42, that are connected by four turn structures. This simple consideration would have a highly significant implication—the degrees of freedom* for Aβ folding would decrease from ~2000 to ~10 and this would enable ab initio determination of the structures of higher order assemblies such as Aβ oligomers, now thought to be the proximate neurotoxins in AD10.
To facilitate our understanding of the conformational equilibria of Aβ, we used principal coordinate analysis (PCoorA)70 to project its conformations onto a two-dimensional space comprising the first two principal coordinates. By defining a third coordinate as conformational free energy, we constructed the free energy surface for Aβ (Fig. 8). This method has been shown to effectively preserve the essential features of the conformational space. Surfaces of Aβ40 and Aβ42 share two global features: (1) two large basins, one dominated by extended or β-sheet conformations (“β-basin”) and the other dominated by α-helical structures (“α-basin;” except position 8 in Fig. 8b); and (2) a number (~20) of minima that have comparable free energies and are separated by shallow barriers. This latter feature predicts that conformational conversions (movement among these minima) occur frequently.
The first principal coordinate of Aβ40 dominates the conformational conversion within each basin and the α-helix→β-sheet conversion occurs along the second principal coordinate (Fig. 8a, white double-headed arrow). In contrast, the α-helix→β-sheet conversion of Aβ42 occurs along its first principal coordinate and conformational conversions within the α- or β-basins occur along the second principal coordinate (Fig. 8b, white double-headed arrow). This suggests that α-helix→β-sheet transitions occur more readily in Aβ42 than in Aβ40.
We also see that the local minima in the Aβ40 α-basin have lower free energies, on average, than those in the β-basin (Fig. 8a), whereas for Aβ42, the magnitudes of the free energies of local minima in both basins, on average, are comparable (Fig. 8b). Previous experimental studies have shown that α-helix-containing conformers are obligatory, but transitory, intermediates in Aβ fibril assembly, both for Aβ40 and for Aβ4271. During fibril assembly, in which a statistical coil→α-helix→β-sheet conformational conversion path is observed, the maximum α-helix content of Aβ40 and Aβ42 was 32% and 19%, respectively, and Aβ42 reached its maximal β-helix content significantly sooner than did Aβ4071. Consistent with this finding, solvent conditions that facilitate the initial coil→α-helix transition significantly accelerate fibril assembly, whereas increased α-helix stabilization blocks β-sheet formation72. The thermodynamics of these systems revealed in the respective energy surfaces provide a mechanistic explanation for this observation. First, the deeper α-helix basin in Aβ40 means that Aβ40 conformers will populate these regions more frequently. Second, the lower barriers for α-helix→β-sheet transitions in the Aβ42 surface mean that the equilibrium amount of α-helix-containing Aβ42 conformers will be lower. The differential stability of α-helix-containing conformers of Aβ40 and Aβ42 explains the well-known difference between the two peptides in assembly rate, a difference that appears to be related directly to the increased neurotoxicity of the longer Aβ alloform10.
The impetus for our work was the importance of elucidating alloform-specific differences in structural dynamics between Aβ40 and Aβ42, differences that appear to underlie the pathogenesis of AD. REMD simulations at µs time-scale reproduced aspects of Aβ conformational dynamics revealed previously using NMR, supporting the biological relevance of the simulations, and importantly, providing insights not obtainable experimentally. We find that neither peptide monomer is unstructured, but rather that each may be described as a unique statistical coil in which five relatively independent folding units exist, comprising residues 1–5, 10–13, 17–22, 28–37, and 39–42, which are connected by four turn structures. Incorporating this modular organization into folding algorithms could simplify analyses of Aβ assembly and facilitate ab initio studies of Aβ oligomerization. Our determination of the free energy surfaces of Aβ40 and Aβ42 revealed that both peptides may possess significant amounts of α- or β-structure and that conformers within and between each structural class are in rapid equilibrium. The two additional hydrophobic residues at the Aβ42 C-terminus, Ile41 and Ala42, significantly increase contacts within the C-terminus and between the C-terminus and the CHC. As a result, the β-structure of Aβ42 is more stable than that of Aβ40 and the conformational equilibrium in Aβ42 shifts towards β-structure. Considered together with our previous studies of the role of α-helix formation in Aβ assembly71,72, these results suggest that drugs stabilizing α-helical Aβ conformers (or destabilizing the β-sheet state73), would block formation of neurotoxic oligomers. The atomic-resolution conformer structures determined in our simulations may serve as useful targets for this purpose.
Our first consideration in selecting a method for simulating Aβ monomer conformational dynamics was how to model a highly flexible peptide chain that in its extended conformation is ~156 Å in length. A simulation space fully enclosing such a conformer would contain >100,000 water molecules. Simulations of systems of this size over meaningful timescales currently are computationally impractical. Solvating a more compact peptide conformer would produce a much smaller system. For example, a 50–60 Å simulation space would contain only ~5,000 water molecules. However, natively disordered peptides sample extended states, and when this occurs during a simulation, inter-image interactions can occur that would affect the results. Even in situations in which a pronounced bias toward compact states exists, interconversion among different states likely would proceed through extended intermediates. For these reasons, we used the Generalized Born (GB) implicit solvation model74 to mimic the aqueous environment and did not represent water explicitly.
By eliminating the degrees of freedom and the viscosity associated with solvent water, simulations using the GB model can converge rapidly and explore conformational space efficiently. However, as with any simulation system, limitations exist. In the absence of explicit water, the simulated events take place much faster due to the lack of frictional effects. This complicates studies of kinetics. Our primary focus here is the thermodynamics of the systems, not their kinetics, thus a more rapid attainment of equilibrium is useful. Another potential limitation of the GB model is that it does not allow the simulation of water bridges, but for a highly flexible molecule like Aβ, there is no evidence that water bridges contribute significantly to its stability. In practice, the GB approach has been implemented frequently and has yielded reliable results75–78, including in ab initio protein folding experiments in which structures of unprecedented accuracy were produced79,80.
We used the REMD technique to further enhance comformational sampling. The simulations were performed with the Sander module of the Amber simulation package (version 9)81. The proteins were modeled by PARM99SB, a recently improved Amber force field82. We implemented an effective 0.2 M salt concentration in the GB solvent model. Non-polar solvation effects were represented using a surface tension coefficient of 0.005 kcal/mol·Å2. Starting from extended Aβ monomer, sixteen replicas were created that exponentially spanned the temperature range 276–400K. The temperature of the system was regulated using the Berendsen coupling algorithm83 with a coupling constant of 1.0 ps. Hydrogen atoms were constrained using SHAKE84. The integration time step was 2 fs. Exchange between replicas was attempted every 2 ps. We used the default values of the Sander module for other relevant parameters81. This system mimicks a very dilute aqueous Aβ solution at neutral pH. For each replica, the simulation length was 110 ns and 110,000 conformations were collected. The first 10 ns was treated as equilibrium and the last 100ns was used for data analysis. The total simulation time was 3.52 µs.
Isotropically distributed ensemble (IDE) analysis85 was used to characterize the correlated molecular motions of Aβ. We represent each residue with its corresponding Cα atom. For a protein containing n amino acids, we first constructed an n × n matrix P with elements , where i (or j), the position vector of residue i (or j), originates from the center of mass of the protein and ends at each Cα atom. Each element Pij of the matrix is the vector product averaged over the entire conformational ensemble. The cross-correlation coefficient rij = Pij/(PiiPjj)1/2, with the overall rotational modes eliminated. If rij = 1, the motions of the two residues i and j are highly positively correlated. If rij = −1, the motions are strongly negatively correlated. For rij = 0, no correlations exists.
Energy surfaces were constructed by principal coordinate analysis (PCoorA)70. To do so, the collected conformations were clustered with a threshold RMSD distance of 3.0 Å. The cluster centers were used to build a distance matrix A with elements , where dij is the power distance86 between conformation i and j. The matrix A was “centered” by Eq. 2, where k is the mean over all specific indices k = i, j or ij.
This centering process guarantees a zero root of matrix A*. Matrix A* was diagonalized and the resulting eigenvectors were sorted in descending order according to their corresponding eigenvalues. Finally, we constructed the free energy surface by projecting all the collected conformations onto the two-dimensional space defined by the first two eigenvectors. This procedure is similar to that described by Yang et al.75, with the exception that the “minimum energy envelope” was not adopted. The conformations in our analysis were not minimized and all collected conformations were projected onto the surface to retain the free energy information.
We gratefully acknowledge the support of grants from the State of California Alzheimer’s Disease Research Fund (#07-65806), the UCLA Faculty Research Grant program, the NIH (AG027818), and the Jim Easton Consortium for Alzheimer’s Drug Discovery and Biomarkers. Computational resources were provided by UCLA Academic Technology Services and the National Center for Supercomputing Applications. Use of the Pymol and Gnuplot software packages (GPL licensed) is gratefully acknowledged.
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*We calculate degrees of freedom df = 3N – 6, where N is the number of elements in the system. For Aβ42, N = 630, the number of atoms in the peptide. For a system with five folding units, if the structures of the units are known, N = 5.