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Harvard-MIT Division of Health Science and Technology, MIT Department of Electrical Engineering and Computer Science, and MIT Computer Science and Artificial Intelligence Laboratory, Cambridge, Massachusetts
Correspondence: Address reprint requests to Collin M. Stultz, E-mail: cmstultz{at}mit.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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,2,3). The most straightforward approach for modeling solvent is to explicitly include solvent molecules in molecular dynamics (MD) simulations. However, molecular simulations with explicit solvent increase the degrees of freedom in the system and therefore can incur a significant computational cost. Consequently, a number of implicit solvent models have been developed to reduce the computational complexity associated with solvated simulations. Such models modify the potential energy function to reproduce the effects of solvation without explicitly representing solvent atoms (1,2). As simulations with implicit solvent models have led to important insights, these models have gained widespread acceptance in the field of biomolecular simulations (4). As evidence of this, the literature is replete with studies that make conclusions based solely on data obtained from such models (4). Recent studies, however, suggest that implicit solvent models can sometimes lead to results that are at odds with data obtained from explicit solvent simulations and experimental observations (5,6,7). Therefore, it is likely that not all implicit solvent models are appropriate for every application. Moreover, the correct choice of solvent model to use for any given problem likely depends on the system to be studied, whether qualitative or quantitative results are desired, and the degree of accuracy required.
In the study presented here, we explore whether conformational sampling with implicit solvent models can yield results similar to that obtained with explicit solvent simulations. The solvent models that form the basis of this work include: i), an early implementation of the generalized Born (GB) model as described by Brooks and co-workers (8); ii), an alternate implementation of the generalized Born formalism that is based on an integral equation approach and that employs a simple smooth switching function (GBSW) (9); iii), the effective energy function-1 (EEF1) implicit solvent model (10); and iv), the TIP3P model of explicit solvent (11).
The GB model uses a linearized form of Still's equation to estimate the electrostatic component of the solvation free energy (8,12). The equation itself contains six independent parameters that are varied to optimize agreement between GB solvation energies and solvation energies calculated with a finite-difference-Poisson-Boltzmann (FDPB) algorithm (8). As the Born radius is inversely related to the atomic polarization energy, Born radii can be calculated from the GB energies after parameter fitting (8). The model has been widely applied and its utility has been demonstrated in a number of applications (13,14).
The GBSW model, like the GB model, is based on Still's equation; however, GBSW employs a more rigorous integral equation approach to calculate the Born radii. In this method, the electrostatic solvation energy of a given atom is expressed as a sum of two terms—the self-solvation energy in the Coulombic approximation plus a term that accounts for the reaction field (9). Each term is calculated using a surface/volume integration that employs a smooth switching function at the dielectric boundary to ensure numerical stability during molecular simulations (9). Unlike the GB method, the GBSW model contains two adjustable parameters that dictate the relative importance of the Coulombic field term and the reaction field term (9). As before, the values of these parameters were obtained by minimizing the least-square error between GBSW energies and those calculated with a FDPB approach (9). Once the optimal values of the adjustable parameters are known, the Born radii can be calculated in a straightforward manner. The current implementation of the GBSW algorithm also incorporates a nonpolar contribution to the solvation free energy using the solvent-exposed surface area of the protein of interest, and a user-defined surface tension coefficient. The GBSW model has been used to refine model structures of the C-terminal domain of Hsp33 protein, obtained from sparse NMR data, into native-like folds that matched solved structures (15). In addition, GBSW has been used to examine intermolecular interactions between actin and myosin, leading to new observations regarding a mutation associated with familial hypertrophic cardiomyopathy (16). Overall, the model appears to be applicable to a broad range of problems.
EEF1 estimates the solvation free energy using a Gaussian solvent-exclusion model (10). EEF1 expresses the solvation free energy of a protein as a sum of group contributions, where each contribution is equal to a reference solvation energy (i.e., the solvation energy of the group alone) minus an integral over a solvation free-energy density function. The underlying assumption is that the integral over the free-energy density is well approximated by a sum of Gaussian functions (10). Important aspects of the model are that charged side chains are neutralized and a distance-dependent dielectric is used to further attenuate electrostatic interactions. The model has been used in a number of applications, and interesting results have been obtained. Most notably, EEF1 has been used to calculate unfolding trajectories of proteins (17), discriminate correctly folded from unfolded structures (18), and probe the interactions between regions of 
-lytic protease, leading to a better understanding of the relative importance of different interactions in stabilizing the native state (19).
In the study presented here, we address a specific, well-defined problem. We determine whether each of these solvent models can reproduce the set of local energy minima obtained from quenched MD (QMD) simulations with explicit solvent. To this end we perform QMD simulations with each of the aforementioned implicit solvent models and compare these results to those obtained with a TIP3P model of solvent. We note that QMD is a widely used method for locating local energy minima on a given potential surface. The procedure consists of high-temperature MD simulations (typically at 1000 K), followed by minimization of the resulting structures (20). High-temperature simulations ensure that a wide region of conformational space is sampled and the subsequent minimizations assure that only local energy minima are analyzed. Minimization can be performed by coupling the system to a heat bath at 0 K (21,21,22), or by using standard energy minimization algorithms such as steepest descent or conjugate gradients (23). QMD has been used to determine optimal positions and orientations of small functional groups in the binding site of an enzyme (21), estimate the density of states for proteins (24), and study the conformational landscape of peptides and peptide analogs (22,23).
Our studies focus on a six-residue peptide commonly referred to as paired-helical filament 6 (PHF6), which corresponds to the sequence found at the N-terminus of the third microtubule-binding repeat domain of tau protein (306VQIVYK311). Tau protein forms intracellular aggregates (also known as neurofibrillary tangles) in patients with Alzheimer's disease (AD), and PHF6 corresponds to the minimal region of tau needed for aggregation to occur in vitro (25,26,27). As the formation of intracellular aggregates may be responsible, in part, for neuronal death in patients with AD, the predominant low energy states of PHF6 are of particular interest (28,29). In performing an analysis of PHF6, the goals of this work are not only to evaluate the ability of several implicit solvent models to reproduce energy minima on a potential surface that explicitly models solvent, but also to determine the most stable conformations of this peptide.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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). Both the N- and C-termini of the peptide were patched using NTERM and CTERM patches, as is commonly done, resulting in charged termini (i.e., –NH3+ and –COO–). The resulting structure was solvated with an equilibrated set of TIP3P water molecules, and waters that overlapped with the peptide or that were outside of a 19 Å radius were removed. A total of 823 water molecules was added to the system. A stochastic boundary setup with a solvent sphere of radius 19 Å was used for these simulations (31). The system was minimized, then heated and equilibrated for 1 ns at 1000 K. Production dynamics were performed for an additional 10 ns at 1000 K. Sampling at this temperature facilitates a broad exploration of the conformational space. The temperature was maintained by weakly coupling (tcoup = 5 ps) the system to a heat bath using the Berendsen method (32). All explicit solvent simulations employed an electrostatic nonbond interaction cutoff of 17 Å, shifted between 14 Å and 16 Å. Switching was used to cut off van der Waals interactions at 16 Å. SHAKE was used to hold hydrogen bond distances close to their equilibrium values and a 2 fs time step was used (33). Structures were chosen from the trajectory every 10 ps and subsequently minimized, resulting in 1000 distinct minimum energy structures. Minimizations were performed on the entire system consisting of the peptide and all explicit water molecules. In addition, minimizations used the nonbond specifications outlined above and consisted of 2500 steps of steepest descent followed by 2500 steps of conjugate gradient minimization. A root mean-square gradient cutoff of 0.01 kcal/mol/Å was set, such that if the system achieved a root mean-square gradient below this value during the minimization protocol, the minimization was terminated. The procedure for heating, equilibration, sampling, and minimization was identical for all of the solvent models investigated in this study.
Quenched molecular dynamics in vacuum
Quenched molecular dynamics simulations were performed in vacuum (
= 1). Comparing the vacuum minima with minima obtained with the different solvent models enabled us to assess the effects of the solvent models on the structure of the peptide. The nonbond cutoffs and the minimization protocol were identical to those used in the explicit solvent simulations.
Quenched molecular dynamics simulations with implicit solvent
We performed a similar procedure for finding local energy minima on the potential energy surface of each implicit solvent model described above. One issue that needs to be resolved is the correct choice of simulation conditions for each implicit solvent model. In general, we rely on prior data to choose simulation conditions that optimize the chance that each implicit solvent simulation would reproduce minima obtained from the explicit solvent simulations. In this regard, we note that some temperature-coupling algorithms may not be appropriate for all implicit solvent simulations (34). In explicit solvent simulations with a Berendsen heat bath, the entire system, consisting of both the solute and the solvent, are coupled to an external heat bath. Implicit solvent simulations that utilize similar thermostats only couple the peptide to an external bath as a continuum model is used for solvent. It has been noted that some thermostats that couple the solute alone to a heat bath may lead to diminished atomic fluctuations, especially when the peptide itself is tightly coupled (34). Diminished root mean-square (rms) fluctuations would clearly be disadvantageous for an approach that attempts to map local energy minima on a large potential surface.
To determine whether a Berendsen thermostat with a coupling constant of 5 ps would be appropriate for our studies, we conducted MD simulations of PHF6 with each implicit solvent model outlined above and compared these data to simulations conducted with explicit solvent (when both the peptide and solvent are coupled to an external bath). The resulting rms fluctuations were then compared to rms fluctuations arising from the explicit solvent simulations. For PHF6, the rms fluctuations arising from all of the implicit solvent simulations are in reasonable agreement with the rms fluctuations from the explicit solvent simulations (Fig. 1). As we are primarily interested in mapping the local energy minima on the different potential energy surfaces, and not the dynamical properties of PHF6 in different models of solvent, these data suggest that simulations with a Berendsen thermostat would be appropriate for our studies.
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). As in our previous study (7), no truncation of nonbond terms was used as this approach yields better results relative to an approach that employs finite nonbond cutoffs (35). ). Nonbond cutoffs were set to 16 Å using a switching function for both van der Waals and electrostatic interactions. Of the 1000 structures, the minimization protocol described above failed for a single structure, which was excluded from the analysis, yielding 999 distinct structures. The singular failed structure was in a nonphysical conformation corresponding to an energy of 1.3 x 1011 kcal/mol, whereas all other structures had energies that were <–400 kcal/mol. ,30). As the nonbond cutoff parameters are integral to the model, the previously described nonbond cutoffs were used here.
Generation of Ramachandran plots
Ramachandran density surfaces were created from the minima generated from each of the quenched dynamics simulations. The 
/
values for residues Gln2-Tyr5 were calculated for each of the 1000 minima (999 for GBSW), and a density function was computed using the SCATTERCLOUD function (written by Steve Simon) obtained from the MATLAB central code repository (http://www.mathworks.com/matlabcentral/). The densities were normalized by their maximum values and rendered as surface plots using MATLAB (The MathWorks, Natick, MA). Approximate secondary structure regions as defined in Hovmöller et al. (36) corresponding to -helical and ß-structure are colored.
Generation of minimum pairwise distance plots
Histograms of minimum pairwise backbone rms deviations between minima from different models (a reference model and a comparison model) were computed. These histograms were used to determine whether each minimum in the reference model was adequately represented by a structurally similar minimum in the comparison model. For example, suppose explicit solvent is the reference model and data arising from the EEF1 simulations are the comparison model. The minimum pairwise distance (MPD) plot is used to determine if each explicit solvent minimum is represented in the set of EEF1 minima. For each TIP3P minimum, we find the EEF1 minimum with a backbone conformation closest to the TIP3P minimum in question. This set of rms deviations provides an objective assessment of how well the EEF1 minima reproduce the structures corresponding to the explicit solvent minima. It is also of interest to determine the converse; i.e., whether each EEF1 minimum is well represented by an explicit solvent minimum. The converse is computed by setting EEF1 as the reference model and the explicit solvent results as the comparison model. If EEF1 generated many spurious minima that did not correspond to explicit solvent minima, then the resulting histogram of rms deviations would contain many large values. Therefore, two sets of MPD plots were computed for each of the implicit solvent models. In one set of calculations, the explicit solvent minima formed the reference set, and in the other set of calculations, the implicit solvent model served as the reference. Histograms were computed using MATLAB and plots of aligned structures were constructed with Visual Molecular Dynamics (37).
Potential of mean-force calculations for PHF6
Free-energy profiles for PHF6 were computed for each solvent model. The reaction coordinate for these simulations was the radius of gyration of the peptide main-chain atoms. The simulations began by restraining the backbone to adopt an extended conformation with a radius of gyration of 5.5 Å using a harmonic constraining potential with a force constant of 25 kcal/mol/Å2. The system was then equilibrated at 300 K for 1 ns. The potential of mean force (pmf) for a given solvent model was calculated by running a series of simulations (windows), where the peptide is restrained to a different radius of gyration using a harmonic force constant of 25 kcal/mol/Å2. The first window was centered at 5.5 Å and subsequent windows began with the final state from the preceding window. The radius of gyration was decreased by 0.1 Å for each new window. Restrained molecular dynamics for each window involved 20 ps of equilibration followed by 80 ps of production dynamics. Additional dynamics were performed to extend the pmf boundaries and improve sampling for regions of the pmf that exhibited discontinuities. Specifically, windows for extended states of the peptide were run at 0.1 Å intervals for radius of gyration (rgyr) constraints ranging between 5.6 Å and 6.6 Å to extend the boundaries of the pmf.
To compute the potential of mean force, the radius of gyration was computed every 20 fs for each window of dynamics. From these data, a biased probability density, 
i* is computed and the potential of mean force, Wi(
), is computed using the relation (31)
 | (1) |
). To determine that our pmf had converged, we performed additional simulations for new windows constrained at rgyr that were offset 0.05 Å from the original window constraints and determined that this convergence criterion was satisfied. Our metric for convergence of the pmf was based on the location of the pmf minimum, since this is the primary quantity of interest for this study. Specifically, we required that the location of the pmf minimum changed by <0.25 Å as the window step size was halved.
Representative structures from the global energy minimum in each pmf were generated by first averaging the structures sampled at the window corresponding to the global energy minimum followed by minimization to the nearest local energy minimum. All molecular figures were constructed with Visual Molecular Dynamics (37).
Calculating vibrational entropies
Vibrational entropies were calculated from the 1000 distinct minima obtained from EEF1 simulations. To ensure that only nonnegative eigenvalues would be generated from the normal mode calculations, each minimum was further minimized using 1000 steps of steepest descent minimization followed by 2500 steps of adopted-basis Newton Rhapson minimization. The corresponding Hessian matrix was then diagonalized to yield the normal modes and their corresponding frequencies. The vibrational entropy for a given minimum was computed as follows (39,40):
 | (2) |
are the normal mode frequencies. CHARMM was used to create the Hessian matrix from minimized structures and MATLAB was used to calculate vibrational entropies from the Hessian matrix, yielding 1000 vibrational entropy measures; i.e., one for each minimum (30). We note that a harmonic analysis could only be performed on minima arising from the EEF1 simulations, as second derivative calculations with GB are not supported in CHARMMv32a2, and despite the extensive additional minimization, Hessian matrices for GBSW structures had negative eigenvalues, thereby preventing a normal mode analysis.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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A conformational analysis of the TIP3P minima suggests that the four residues in PHF6 with defined / angles (residues 2–5) preferentially sample regions of conformational space corresponding to ß-structure (Fig. 6). Gln2, in particular, is most likely to adopt / angles belonging to the ß-strand region of conformational space. The / densities of the GB, GBSW, and EEF1 minima are similar to that obtained from the TIP3P simulations in that ß-strand configurations are also favored (Fig. 6). By contrast, the vacuum simulations yield minima where residues 2, 4, and 5 adopt / angles that belong to the -helical region of conformational space (Fig. 6).
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5.2 Å, corresponding to a relatively extended conformation of the peptide (Fig. 7)—a finding consistent with the / densities of explicit solvent minima.
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/ angles that fall within a region of conformational space consistent with ß-structure. The GBSW average structure, however, is least similar to the average structure from the explicit solvent pmf minimum (Fig. 8). The backbone rms deviation between the GBSW pmf minimum and the TIP3P pmf minimum is 2.7 Å, whereas the GB and EEF1 structures are within 1 Å of the TIP3P pmf minimum structure (Fig. 8). Hence, whereas all of the implicit solvent models show qualitative agreement with the explicit solvent pmf, the average structure arising from EEF1 simulations at the global free-energy minimum is most similar to the average structure obtained from corresponding simulations with explicit solvent.
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,41,42,43). Moreover, a number of these studies suggest that the most stable state can be deduced from static energy calculations on energy-minimized structures (18,41,43). Given these observations, we explored whether static energy calculations on the different implicit solvent minima could provide enough information for identifying the most stable conformation. A comparison of the relative energies of the different minima is shown in Fig. 9. Both the GB and GBSW minima have a number of low energy states that are within 2 kcal/mol of the lowest energy structure, and all of these conformations are relatively compact with radii of gyration near 3.5 Å (Fig. 9). By contrast, the set of EEF1 minima contains a prominent minimum with a radius of gyration of 5.08 Å, a value close to the global free energy minimum in the EEF1 and TIP3P free energy profiles (Fig. 9). Hence the most stable conformation of PHF6 can be identified from an analysis of the EEF1 energies alone.
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,44,45). Such approximations may be valid for a number of proteins, but it is not clear whether such a premise is valid for small peptides like PHF6 (44,45). Although static energy calculations with EEF1 lead to results that agree with calculated free energy profiles, this does not necessarily imply that the solute entropy is the same at each minimum. Therefore, to explore the role that the solute entropy has in determining the relative stability of the PHF6 minima, we computed the vibrational entropy of each EEF1 minimum within the context of a harmonic approximation (40). The relative free energy of each minimum was then estimated using the sum of the internal energy (i.e., the EEF1 energy) and the vibrational entropy (Table 2). Ranking the EEF1 minima using this new measure leads to conclusions that are identical to what was obtained from an analysis of the EEF1 energies alone. In particular, the lowest energy conformations are extended, and the lowest energy structure is the same (Table 2). However, as is clear from Table 2, the vibrational entropy spans a range of more than 10 kcal/mol, a somewhat larger range than was noted in prior studies on proteins (44,45). Including the vibrational entropy also leads to a change in the ranking of the PHF6 minima. Consequently, even though our results are similar to those seen when the vibrational entropy is explicitly included, it is clear that it can play a role in determining the relative ordering of different minima.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONREFERENCES |
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–19). However, not all implicit solvent models are created equal, and some may be more appropriate for particular problems. As such, studies, such as the work presented here, which aim to delineate the limitations as well as the advantages of different implicit solvent models, may help to decide which model to use for any given application. This study was designed to address a specific question–namely, could selected implicit solvent models adequately reproduce the set of local energy minima found on a potential surface that explicitly includes solvent. Toward this end, we mapped local energy minima on different potential surfaces and compared these minima to minima obtained from simulations with explicit solvent. We found that GB, GBSW, and EEF1 performed quite admirably in that they were able to successfully reproduce the set of minima obtained from explicit solvent simulations. Ramachandran plots of the resulting structures confirm that all solvent models sampled similar regions of conformational space. Furthermore, free-energy profiles obtained from all three implicit solvent models were in good agreement with free-energy profiles obtained with explicit solvent. However, visual inspection of the structures suggests that EEF1 provides a slightly more accurate representation of the most favored conformations on the peptide's free energy surface.
All of the implicit solvent simulations generate pmfs that are in good agreement with the explicit solvent simulations in a fraction of the central processing unit (CPU) time required for the explicit solvent simulations (Fig. 10 A). Of the different implicit solvent simulations, EEF1 required the least CPU time (Fig. 10 B). This is due, in part, to the different nonbond cutoffs in each model. As the nonbond specifications in EEF1 are part of the model, all EEF1 simulations employ a relatively short cutoff of 9 Å (10). The nonbond cutoffs for the GB and GBSW models were considerably larger. The GB simulations employed an infinite cutoff because it has been shown that this cutoff scheme yields results that are in good agreement with explicit solvent for some systems (35). The GBSW simulations used a finite nonbond cutoff of 16 Å because this value leads to reasonable computation times with relatively small errors in the calculated forces (9). Nevertheless, a 16 Å cutoff for a small peptide like PHF6 leads to almost no truncation of the nonbond terms. As a result, the nonbond lists for the GB and GBSW simulations are quite similar. The longer simulation time for GBSW is due to the fact that, unlike GB, GBSW employs a relatively expensive surface/volume integration to calculate the electrostatic contribution to the solvation energy (8,9).
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It should be noted that this conclusion may not be generally applicable. Ranking EEF1 minima based solely on static EEF1 energies assumes that the solute entropy at each minimum can be safely ignored. However, estimates of the vibrational entropy reveal that the solute entropy can vary significantly at each minimum. Although our conclusions are the same when the vibrational entropy of each minimum is explicitly calculated, the ranking of the EEF1 minima is somewhat altered when this is done. Therefore, we cannot rule out that estimates of the solute entropy are needed to accurately identify the most stable conformation of other peptides. In this regard, we note that static energy evaluations of GB and GBSW minima lead to conclusions that differ from that obtained from the pmf calculations in explicit solvent. As normal mode analyses could not be performed on GB and GBSW minima, it may be that more accurate results could be obtained if a vibrational analysis was performed on these minima.
In our previous study, we found that both EEF1 and GB were unable to reproduce the free-energy profile obtained from simulations with explicit solvent using a different peptide system (7). In that work, we used umbrella sampling calculations with explicit solvent to calculate a peptide's potential of mean force as a function of its radius of gyration (7). The FRET efficiency for this peptide, which was calculated from the pmf, was in excellent agreement with experiment. Central to the success of the explicit solvent simulations was the formation of a stable salt bridge between glutamate 5 and arginine 11. By contrast, in both the GB and EEF1 simulations, the formation of a glutamate-arginine salt bridge was unfavorable, and consequently simulations with these implicit solvent models lead to calculated FRET efficiencies that disagreed with the explicit solvent results (7). Although the solvation energy of individual side chains is likely well modeled by these implicit solvation models, it is not clear that energetics of salt-bridge formation is appropriately modeled by these approaches (7,46). This may be particularly true for salt-bridges that involve arginine residues (46). As such, the absence of multiple charged side chains in the sequence of PHF6 likely explains the difference between the results presented in this work and those of our prior work. For PHF6, representative structures from the lowest energy state within the explicit solvent pmf contain one salt-bridge between the side chain of lysine 6 and the C-terminal carboxyl of the same residue (Fig. 8). Therefore, the explicit solvent pmf suggests that the lowest energy state is extended without any salt bridges or hydrogen bonds between moieties that are separated in the sequence. This simple extended state that lacks salt bridges or hydrogen bonds between distant residues is well modeled by the implicit solvent models investigated in this work.
All of the solvent models predict that PHF6 preferentially adopts extended structures in solution, and a conformational analysis of amino-acids in PHF6 argues that residues 2–5 adopt / values corresponding to the ß-strands. These findings have important implications for the pathogenesis of neurofibrillary tangle formation in patients with AD. In particular, there is growing consensus that the ability of amyloidogenic proteins like tau to aggregate stems from properties of the protein backbone. In many instances, protein aggregation requires the formation of intermolecular backbone hydrogen bonds yielding a cross ß-structure (i.e., the ß-strands are perpendicular to the axis of the fibril), and for tau this process is likely important for the initiation of neurofibrillary tangle formation (47,48,49).
Our findings imply that PHF6 exhibits a strong preference for extended ß-structures in solution—a finding that suggests that PHF6 promotes neurofibrillary tangle formation by facilitating the formation of cross ß-structure between tau monomers. This premise is consistent with recent data suggesting that the sequence of PHF6 is the minimal region of tau required for tau aggregation into cross ß-filaments and hence neurofibrillary tangles (25). As neurofibrillary tangle formation may play a role in neurodegeneration (28), therapies directed at modifying the structural preference for PHF6 may lead to new treatments for dementias like AD and the tauopathies (50).
Submitted on June 12, 2006; accepted for publication September 19, 2006.
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