Protein folding by zipping and assembly

Keywords: protein structure prediction, replica-exchange molecular dynamics

Control Simulations Starting from Experimental Structures.

Before we attempt to predict a native structure, we must first verify that the force field is an adequate model, i.e., that computer simulations do not drift away from the known experimental native structure under native conditions under extensive conformational sampling. The force field was tested on the proteins listed in Table 1. We conducted a number of simulations initiated from the experimental native structures, running REMD (33) for 10 ns per replica (details described in Methods). Although stability over the finite duration of such simulations does not prove that there is no more stable region of phase space, it is a minimal requirement that the force field must pass. To prove that a force field is not adequate, it would suffice to show that a protein that is known to be stable from experiments would unfold under the force field under native conditions.

The main results are as follows. Seven of the nine proteins considered here did not drift >3 Å C^α rmsd away from the experimental starting structures over the course of the simulation, suggesting that this force field is adequate for these proteins. In the case of α3D, the molecule deviated from the Protein Data Bank (PDB) structure by up to 4 Å rmsd, but the mobility was mainly in the loop regions; the rmsd over the helical regions, excluding these loops, remained within 2.6 Å.

Comparison with the Experimental PDB Structures.

We performed two kinds of tests. First, Fig. 1(ZA vs. PDB) compares the structures predicted by ZAM with the corresponding experimental native structures from the PDB. ZAM produces an ensemble of structures. We use the centroid of the dominant cluster as representative. The comparison of ZA vs. PDB is a combined test of both the force field and the search method. Second, a more direct test of the search method alone is to compare ZA with the force field's best (FFB) structure (ZA vs. FFB). The FFB structure is a representative conformation taken from the most populous cluster that was reached by REMD simulations that were started from the experimental structure.

In general, the differences between ZA and FFB structures average only ≈1.6 Å rmsd, indicating that for those seven proteins, the search method has converged to the native basin of the force field (Table 1). For src-Src homology 3 (SH3), ZA fails to find any structure better than 5 Å from the experimental structure (ZA vs. PDB). In that case, the problem appears to be the GB/SA implicit solvation model. Similar problems have been observed before of ion pairs that are too stable in proteins having charged side chains (20).

Some Predicted Folding Routes.

In our computational process, ZAM generates one or more folding routes, depending on the protein. In our simulations of protein A, helices 2 and 3 form first, then pack together, followed by the addition of the C-terminal helix 1. This result is consistent with the relative stabilities of the helices and intermediates that were found by Garcia and Onuchic in their explicit-solvent REMD simulations (29) and with experiments (41, 42).

ZAM finds that the albumin binding domain folds by first forming the C-terminal helix (helix 1), which then extends this helix. Helices 2 and 3 form independently, then assemble onto helix 1. The three helices of α3D form independently, and then assemble to form the full helix bundle.

The C-terminal 16-residue fragment of protein G is known from experiments to be stable by itself and has been studied by physics-based computational modeling (27, 43, 44). However, the full 56-residue protein has previously been beyond the range of high-accuracy predictions by molecular mechanics force fields. In our simulations, the folding of protein G (described in detail in Methods) begins with hydrophobic contacts forming in the N- and C-terminal β-turns, followed by hairpin formation. The C-terminal hairpin recruits the hydrophobic residues of the helical region to its core, causing the helix to form and pack against it. Finally, the N-terminal hairpin and the helix-C-terminal hairpin folded units assemble, completing the core.

We also studied the 35-residue N-terminal fragment of ubiquitin. In its ZA folding route, the 10-residue inner hairpin (Val-5–Thr-14) containing the β-turn is the first to form, followed by growth of the hairpin to its full length of 17 residues. The α-helix forms independently, then assembles onto the β-hairpin.

For both of the WW domains we studied (1E0N residues 7–31 and 1E0L residues 6–33), the two hairpins (β₁–β₂ and β₂–β₃) form independently at first, but the only successful folding route not ending up in a nonproductive trap involves adding the third strand (β₃) onto the β₁–β₂ hairpin.

The α-spectrin SH3 domain contains five antiparallel β-strands packed to form two perpendicular β-sheets. Folding begins by zipping the three-stranded β₂–β₃–β₄ sheet, followed by the addition of the seven-residue diverging turn (DT) and four more residues to the end of the DT, before a favorable hydrophobic contact between the RT loop and β4. Then the chain zips from the N terminus to form hydrophobic contact Tyr-15–Met-25 within the RT loop. The rest of the chain is zipped up in the last step, completing the structure. Folding steps for three additional proteins (α3D, protein A, and α-spectrin) can be found in supporting information (SI) Fig. 4.

Kinetic Impact vs. Experimental Φ Values.

To determine whether our ZAM folding routes are consistent with experiments, we computed Thomas Weikl's (45) kinetic impact quantity for each of the secondary structural elements based on stability and order of emergence of the units. According to his quantity, secondary structures are assigned a high kinetic impact value if those structures form early in the folding process and lead to the formation of other structural units, a low kinetic impact value if they form late, and an intermediate value if they form at some stage of folding in between. The relative stability of a structure is also taken into account: when two structures form in parallel in a kinetic model, a medium kinetic impact value is assigned to the one that is less stable. Following Weikl, we do not attempt finer discrimination here than high, medium, and low. To compare kinetic impacts with averaged experimental Φ values, we use the following values for the kinetic impact factor: high (0.6–0.8), medium (0.5–0.3), and low (0.2–0). We compared the computed kinetic impact with average experimental Φ values for their secondary structural elements (see Fig. 2). We applied the method to protein A, the Pin (YJQ8) WW domain, α-spectrin SH3, and protein G.

Based on these simple rules, the kinetic impact values that we computed from our ZA simulations for α-spectrin SH3 are as follows: the three-stranded β-sheet β₂–β₃–β₄ rapidly forms first by zipping, and, because β3 and β4 are more stable than β₂, β₃ and β₄ are assigned a high kinetic impact value and β₂ a moderate value. The formation of β₂–β₃–β₄ leads to the diverging turn and RT turns, which have moderate experimental Φ values. The strands β₁ and β₅ form last in our simulations, so they have low kinetic impact values, also consistent with experimental Φ values.

The kinetic impact values computed from the ZA folding routes correlate well with average Φ values measured for the proteins tested (see Fig. 2). The exceptions are that the β₂ strands of protein G and α-spectrin are estimated to have moderate kinetic impact, whereas the experimental average Φ values are negative, although negative values also imply kinetic importance in the folding pathway (46). The good general correlation between kinetic impact factors and experimental Φ values suggests that the ZAM routes in our simulations are consistent with experiments. However, as noted above, the experiments are, by their nature, much too highly ensemble-averaged to prove or disprove that the ZAM routes are physically correct.

Comparison with Other Protein Structure Prediction Methods.

David Baker and colleagues (7), using their Rosetta algorithm, have recently achieved an important milestone in protein structure prediction. Their predictions are “high resolution,” which we define to mean: (i) a backbone rsmd from the experimental structure over the entire protein (rather than just selected parts) of <3 Å, (ii) achieving this level of experimental agreement routinely (i.e., for a significant fraction of proteins tested), rather than rarely, and (iii) consistent performance over different classes of folds. For 10 of 16 proteins <85 aa in length, Baker's group (7) recently reported the prediction of native structures to 3 Å or better, averaging 4.7 Å over the entire set of 16. The 3-Å threshold is important because at this level computer-based models of proteins may be as good as experimental x-ray or NMR structures for initiating drug discovery (47). The Rosetta high-resolution method requires ≈0.5 CPU years to predict each protein structure. It appears to represent the state of the art in protein structure prediction for modeling that incorporates database-derived insights and does not require a template having high sequence identity.

For comparison, straightforward molecular dynamics simulations combined with molecular mechanics force fields have folded small proteins without using database-derived information. For example, the Pande group (26–28) has used the distributed computing platform Folding@Home to fold the 16-residue C-terminal β-hairpin from protein G and the 36-residue villin headpiece to within 3 Å backbone rsmd of the experimental NMR structures; Pitera and Swope (25) have used REMD to fold the 20-residue Trp-cage miniproteins; and the Simmerling group (30) has simulated the folding of a designed three-stranded β-sheet by using multiple independent trajectories. The aim of those studies using explicit water was not rapid prediction of protein structures. We have shown here that the ZAM algorithm predicts native structures for eight of nine proteins tested, up to 74 aa in length, to an average of 2.2 Å from their experimental structures, using just a physics-based force field without database-derived preferences.

Molecular Mechanics Model and Simulation Protocol.

REMD (33) was used to sample the conformation space during the growth and assembly stages of the ZA algorithm. REMD periodically attempts to exchange conformations between independent molecular dynamics simulations running in parallel at different temperatures, based on a Metropolis-like criterion. This allows individual replicas to heat up to overcome barriers and then cool back down to temperatures of interest. It has two main advantages: (i) REMD explores more conformational space than conventional molecular dynamics techniques (48) and (ii) REMD samples from the canonical ensemble at each temperature, giving estimates of free energies, not just energies.

Proteins and fragments were modeled with the AMBER96 force field (31) with a GB implicit solvent model (32) and a SA penalty term of 5 cal·mol⁻¹·Å⁻². All fragments were capped at the N and C termini with acetyl and N-methylamine blocking groups, respectively, to avoid undue influence from the zwitterionic termini. All simulations were conducted by using a custom Perl script wrapper around the sander program from the AMBER7 molecular dynamics package. Replica temperatures were exponentially distributed over the range 270 to 690 K, with the number of replicas chosen to give average exchange acceptance probabilities of ≈50%. Exchanges were attempted every picosecond, between which energy-conserving molecular dynamics was used with a 2-fs time step. Velocities were randomized from a Maxwell-Boltzmann distribution after each exchange attempt to ensure sampling from the canonical ensemble at the appropriate temperature.

The ZAM Algorithm.

We illustrate ZAM by describing how it folds protein G (shown in Fig. 3). According to the ZA model, on the shortest time scales, the chain does not have time to explore more than a few local degrees of freedom in any chain segment. Thus, in the ZA strategy, small peptide fragments of the chain first search for metastable structures, independently of other segments, i.e., in the absence of the rest of the chain. The computer first determines the locations within the protein chain at which structure might preferentially begin to form.

Because of limitations in our computer resources, we have done this first step of parsing into peptides in two different ways. For a few of the proteins (i.e., protein G, protein A, and α-spectrin SH3 domain), the process has been random, systematic, and not guided by knowledge of the native structure. In those cases, the chain is chopped into overlapping fragments 8–12 residues in length (e.g., residues 1–12, 5–16, 9–20, etc.). The fragments are chosen to be eight residues in length unless it is necessary to extend the chain up to 12 residues in length to include two hydrophobic residues. We then determine whether each peptide finds a metastable structure, by the method described in ref. 49. We believe that this strategy may work in general, because 133 different peptides from six different proteins are found to be structured by using our same force-field approach (49). For other proteins, we accelerate the first stage by simply choosing sites that we guessed would be nucleation points, based on knowing the native structure. The ZA algorithm then found routes to the folded states from those initiation sites. The latter tests prove that there are routes to the native structure that we would have found from the systematic random chopping process described above, but it does not show whether there might have been alternative routes, and it does not show what dead-ends or kinetic traps or misfolded structures might have hindered the search, if we had allowed a broader set of nucleation sites.

After discarding the first 2.5 ns per replica to equilibration, most such peptide fragments are found to populate a broad ensemble of structures. Some fragments, however, have strong conformational preferences based on a coarse-grained φ–ψ angles of backbone representation called mesostrings (49). One can calculate mesostring entropy by using the Boltzmann formula S = −k Σ_i p_i ln p_i, where p_i is the probability that the peptide is in the mesostring i. The fragments having low mesostring entropy can be considered as early folding nuclei to initiate the zipping (49).

Stable hydrophobic contacts within the fragments where the zipping is initiated are located by computing the PMF as a function of C^α distance between each possible pair of hydrophobic residues using weighted histogram analysis (38, 39). We identify all hydrophobic contacts that exhibit a substantially deep minimum in the distance range 0 to 8 Å in their PMF plots (see SI Fig. 5 for details). At this stage, evaluation of the PMF plots and the decision about which hydrophobic contacts needs to be restrained are not yet automated. The first tests show that automation can be done in two ways: (i) we compute the probability of a contact (i.e., integrating the PMF plots based on the contact distance ≥7 Å), hydrophobic residue pairs with contact probabilities >50% are considered as the contacts to be restrained; and (ii) we divide the PMF plot in two regions, the contact-forming region of distance separation (the region between 0 and 8 Å of distance separation) and contact breaking region (the region between 10 and 14 Å of the distance separation between two hydrophobic residues). We locate the minima in these two regions and compute the difference in free energy of these two minima. All contacts with contact free energies more stable than 2.0 kcal/mol are considered stable and become proposals for restraints.

To speed equilibration without affecting the final structure, we have found that seeding the REMD simulations with configurations taken from the Baker I-sites library to be useful in accelerating the convergence of computed PMFs if the fragments appear in the library with >80% confidence (50). For protein G shown here, we did not use this acceleration process.

For protein G, two fragments from this initial stage, Tyr-45–Phe-52 and Ile-6–Gly-15, contain stable hydrophobic contacts. To explore whether these transient contacts are able to recruit additional contacts, we conduct a growth phase whereby the fragment is extended to incorporate additional hydrophobic residues. More simulation is conducted to compute a conditional PMF to determine whether additional contacts may be stable in the presence of the first ones. The contact identified in the previous step is restrained by using a potential applied to the distance between C_β atoms that is zero over the distance range [0,6.0) Å, harmonic with a force constant of 0.5 kcal·mol⁻¹·Å⁻² over [6.0,6.5) Å and linear thereafter with continuous slope at 6.5 Å. In a small fraction of cases, such as α spectrin SH3, examination of all interresidue PMFs indicated that a hydrogen bond may have a smaller variance than a hydrophobic contact; in these cases, hydrogen bond distances are restrained instead. At each growth step, four additional residues are added in the extended configuration to each end of the fragment. Another set of REMD simulations is performed on all fragments that have been grown in this way, and a new set of conditional PMFs is computed. Such growth attempts are then repeated to identify additional stable contacts until no further such contacts can be found, at which point the algorithm switches to fragment assembly, described below.

In the case of protein G, the first fragment identified for growth initially spans residues 45–52 and contains contact Tyr-45–Phe-52. We then enforce this contact with a restraint as described above, and four additional residues are added to each end, so that the new fragment contains residues 41–56, the entire C-terminal β-hairpin. Similarly, the other fragment that has metastable structure, which includes the contact Ile-6–Gly-15, is subjected to the same treatment. Iterating this procedure leads to a highly structured segment (residues 28–56) containing a helix packing onto a strand and a β-hairpin in the N-terminal segment (residues 1–20).

In some cases, for example, protein A and α-spectrin SH3, this zipping procedure alone is sufficient to reach the native state. In other cases, the fragments grow only to a point at which the addition of new residues to the segment forms neither additional structure nor favorable hydrophobic contacts. In those cases, the ZA algorithm then attempts to assemble the existing structures. To assemble, two nonoverlapping fragments are selected from the growth stage, the intervening residues are incorporated, and an ensemble is generated of different relative conformations of those pieces. A new REMD simulation is then initiated from this ensemble of configurations, and more simulations are conducted, retaining all of the previously imposed restraints, until new contacts and additional structure are formed, as determined by the PMF criterion.

Either of two different methods has been found satisfactory for generating an ensemble of relative orientations of these fragments: (i) the anisotropic network model (ANM) or (ii) uniformly sampling the backbone torsion angles of the intervening chain before a new round of REMD sampling. ANM begins with coordinates of a structure, connects adjacent residues, based on a cut-off distance, with a spring, and computes elastic models by using a Hessian connectivity matrix (40). We diagonalize this matrix, decompose it into its eigenvectors, and take the two slowest (i.e., most global) eigenmodes. We add this (fluctuation) vector to the current atomic coordinates of the chain, up to the inverse force constant with a scaling factor of 10, to generate an ensemble of relative positions of the two pieces. Or, one residue near the center of the intervening chain in the assembly unit can be chosen randomly, from which we generate nine conformers by letting the φ and ψ angles each take on three values, −180 or ±60°.

Protein G reaches a stage where growth terminates when the fragments span residues 1–20 and 28–56, so the algorithm then attempts to assemble these fragments by using an anisotropic network model. To speed up assembly, we add additional spring constraints either when PMFs of hydrophobic interactions indicate a contact, as described above, or when the side-chain centroid distance between a pair of hydrophobic residues is <10 Å and closing to half of its initial distance in 600 ps. We then restart the REMD simulation in the presence of these springs until convergence. For protein G, PMFs for possible hydrophobic contacts obtained by weighted histogram analysis indicate further pairing of nonlocal hydrophobic contacts between Leu-5–Phe-30, Tyr-3–Ala-26, Leu-5–Phe-52, Leu-5–Trp-43, Tyr-3–Phe-52, Leu-7–Val-54, and Ala-20–Ala-26. At the end of the process, all restraints are removed and an additional REMD simulation is conducted to ensure the resulting structure is stable on its own, irrespective of the ZA folding pathway.

Supporting Figures

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Supplementary Materials