Protein folding under confinement: A role for solvent

The P_fold method was used to assess the effect of confinement on folding kinetics and thermodynamics (see Methods for an explanation of P_fold). We can determine the effect of confinement on folding by correlating the P_fold values of our confined (perturbed) and unconfined (unperturbed) systems. These correlations allow us to compare two systems and determine which is more likely to fold (Fig. 2A). We can also infer changes in native-state stability (ΔΔG_F; Fig. 2A) and changes in mechanism such as a shift or broadening of the transition state (Δx_TS, Δσ; Table 1).

When we correlate the P_fold of the bulk-like system (Fig. 1A) with the protein-only confinement system (Fig. 1B), we find that P_fold increases upon confinement (Fig. 2B). Therefore, confining the protein increases the probability of folding before unfolding. From our estimate of ΔΔG_F (Table 1) we can see that the native state is stabilized (−1.01 ± 0.19 k_BT). Using the method of Zhou and Dill (3), one would predict analytically that a system of this size would be stabilized by −0.35 k_BT. As a positive control, these simulations were repeated at larger confining radii. The estimated ΔΔG_F agrees very well with the values predicted from analytical methods. Thus our simulations agree qualitatively with predictions from polymer theory [see supporting information (SI) Text for details concerning the analytical calculation of ΔΔG_F for this system].

We can also see from our estimation of barrier width (Δσ; Table 1) and transition-state shift (Δx_TS, Table 1) that polymeric confinement is mechanistically different from bulk folding. By the Hammond postulate, these data also support a highly destabilized unfolded state. Thus, we observe that protein-only confinement increases folding by destabilizing the unfolded state, which is consistent with predictions based purely on polymer entropy.

However, when protein and solvent were both confined (Fig. 1C), the probability of folding before unfolding decreased (Fig. 2C). Our estimate of native-state stability shows that the folded state is destabilized under these conditions (ΔΔG_F = 1.05 k_BT; Table 1). We also observed that the transition state is shifted toward the folded state and the barrier does not broaden (Table 1). This finding implies a different mechanism from bulk folding and folding with only protein confined. Also, in terms of the Hammond postulate, these data agree with our prediction of decreased native-state stability. Overall, these observations imply that confined solvent has a negative effect on protein folding.

The errors reported in Table 1 arise from the statistical uncertainty in our P_fold correlations. Although there is some spread in the data, it still remains that the correlations are significantly nonlinear (which is further demonstrated by Fig. 2D, which directly compares protein-only confinement to protein and solvent confinement). The scatter in the data implies that while the values reported in Table 1 may not be sufficiently precise to be truly quantitatively useful these data can still yield a reasonable qualitative and potentially semiquantitative assessment of kinetic and thermodynamic perturbations to folding resulting from confinement. Despite this scatter it should be noted that the estimates of native-state stability are self-consistent in that the ΔΔG_F from Fig. 2D is approximately equal to the difference of the ΔΔG_F from Fig. 2 B and C.

Together, our results indicate that there is a polymeric confinement effect (Fig. 2B) in which the native state is stabilized and a solvent-confinement effect (Fig. 2C) in which the unfolded state is stabilized. It is important to note that the system depicted in Fig. 1C confines both protein and solvent. Thus it must also intrinsically contain any polymeric-confinement effects observed in the system shown by Fig. 1B. This finding implies that the solvent-confinement effect is greater in magnitude than the polymer-confinement effect for our system. It is also consistent with Zhou and Dill's model (3) in that large (5–10 k_BT) stabilizations occur only for larger systems that are extremely confined (such that the confining volume is only slightly larger than the volume of the native state).

When considering how these effects arise, it is convenient to look at a distribution of states according to various metrics. Fig. 3 shows the 2D histogram of distance rmsd of the protein α carbons from their native positions (dRMSD_Cα) and helicity (as these plots were constructed from nonequilibrium data, contours should not be taken to imply free energy, rather these plots are included to show the different regions of conformation space explored under confinement). We can see that in the bulk-like system, there are a number of states with high dRMSD_Cα (>7 Å) and moderate amounts of secondary structure (helicity) (Fig. 3A). This result is consistent with experimental observations of unfolded villin (21). These unfolded states have greatly reduced population in the polymeric confinement system (Fig. 3B), which is in agreement with the P_fold correlation shown in Fig. 2B. The system with protein and solvent confined is also missing this population of states but contains a new population with low dRMSD_Cα and low helicity (Fig. 3C). The same trend is observed in Fig. 3 D–F with radius of gyration. Thus in the absence of confinement, the unfolded state is populated mostly by more expanded structures (R_g ≈ 60% of what would be expected from an ideal polymer chain of the same length) that contain moderate amounts of secondary structure. In contrast, with protein and solvent confined the unfolded state is populated by compact structures with very little secondary structure. Direct observation of trajectories from these simulations has revealed that this compact unfolded state occurs when the protein migrates to the interface between the solvent and the confining wall. This loss of secondary structure is consistent with observations from Sorin and Pande (17) in which molecular dynamics simulations of an α-helix confined to a single-walled carbon nanotube with explicit solvent showed unfolding. Sorin and Pande's argument states that this loss of secondary structure results from the reduced entropy of confined solvent, which is completely consistent with our observations, as the free-energy difference between a protein–protein hydrogen bond and a protein–water hydrogen bond would be decreased at the solvent interface. These results seem to support a collapsed globule type mechanism (22) when protein and solvent are confined.

We have observed both a “folding effect” arising from polymeric confinement and an “unfolding effect” arising from solvent confinement. The folding effect we observe is qualitatively consistent with the polymer entropy model proposed by Zhou and Dill (3). The observed agreement with previous simpler models and analytic theory serves as an excellent validation for our simulation methodology. Likewise, the fact that we observe an unfolding effect when water is treated explicitly implies that polymer entropy models alone may be insufficient to describe confined folding for all systems. In addition, because our confining potential is completely repulsive, we can conclude that this unfolding action does not arise from interactions between the protein and the confining wall, but rather arises from interactions between the confined solvent and the confined protein.

For our system, the solvent-mediated effect is larger in magnitude than that of polymeric confinement. This effect is believed to arise as a result of the free-energy landscape created by confined water. Because the repulsive confining potential is effectively hydrophobic, water at the surface cannot have a full complement of hydrogen bonds. The protein is effectively pushed to this region, where there is a minimal free-energy penalty for disturbing the water's hydrogen-bond network. As the confining potential is purely repulsive, this effect must be occurring because of water–protein interactions rather than interactions between the protein and the confining wall. In previous works involving simple implicit models of solvent (23, 24), the net attraction between the protein and the wall must be included by hand. Our work using explicit solvent sheds light on the nature of this interaction and shows that for hydrophobic surfaces there can be an induced effective attraction, because of the nature of confined water. However, this may not be the case for all systems. Ideally, one would investigate how changing the hydrophobicity of the confining surface mediates this effect. Also, as chaperonins are water-permeable (25) it would be advantageous to perform these simulations at constant chemical potential for water (this would eliminate any pressure volume effects that cannot be treated with the current methodology). These modifications, however, involve a substantial increase in the complexity of our model and therefore must be treated in future work. Still, these results provide an interesting complement to the existing work that has focused on purely polymeric confinement and suggest that the solvent may not be merely a passive component for folding in chaperonins, but rather may play an important role in mechanism by which chaperonins aid protein folding.

Currently, a precise mechanism by which chaperonins help their substrates to fold is unknown, but several possible mechanisms have been proposed (26–28). Also, considering the fact that chaperonins act on many different substrates of various sizes (27, 29) it is feasible that more than one mechanism may be needed to describe chaperonin activity. Chaperonins are believed to rescue kinetically trapped non-native peptides by unfolding them. It has been proposed that chaperonins may do this entropically by binding nonnative peptide (30) or mechanically by exerting force on bound peptide during the chaperonin's own conformational changes (31). The results from our simulations indicate a third possibility. Confined solvent may create an energy landscape that is conducive to unfolding. If for some substrates chaperonins cause unfolding to a different unfolded state, this unfolded state may not be kinetically trapped nor aggregation prone. This idea is qualitatively consistent with both the fact that many iterations of chaperonin activity are frequently needed to fold substrates (26, 28), and bulk FRET experiments show that substrate is compacted (but not necessarily native-like) inside the cavity (30).

One may argue that the hydrophobic surface used in these simulations may not be relevant for modeling the behavior of proteins inside the closed chaperonin cavity (which is mostly hydrophilic). For this it is important to note that the prokaryotic chaperonin GroEL has long hydrophobic tails at its C terminus that are not resolved in the crystal structure (32). This hydrophobic surface may interact with solvent in such a way as to affect folding in a way similar to that described above. It is also conceivable that upon confinement larger substrates (where the volume of the chaperonin cavity is only slightly larger than that of the native protein) may be more affected by polymer entropy effects, whereas smaller substrates may be more sensitive to solvent-mediated effects. Even if the solvent effect described herein is not the driving force in chaperonin mechanism, solvent effects may still alter the energy landscape seen by the confined protein and most likely also affect the overall mechanism.

We would like to compare the folding kinetics of our confined and unconfined systems. One approach is to project the simulation data to one or a few reaction coordinates. However, choosing a proper 1D reaction coordinate for high-dimensional protein systems has long been a challenge in theoretical chemistry (33). Here, we use P_fold calculation to circumvent the choice of reaction coordinate. P_fold is defined as the probability of folding before unfolding for a given protein conformation (34). A structure with a P_fold of 1 is folded (committed to the folded ensemble). Likewise a structure with a P_fold of 0 is unfolded (committed to the unfolded ensemble), and structure with a P_fold of 0.5 is a transition-state structure (equally likely to commit to the folded or unfolded state). In this method, we choose a definition of the folded and unfolded states rather than choosing a single structural metric to use as a reaction coordinate. For our system we used the following definitions for the folded and unfolded states (see SI Text for justification of these definitions): Here RMSD_cα(9–32) is the coordinate rmsd of the α carbons of residues 9–32 from their positions in the native structure [the N- and C-terminal residues of the villin headpiece are less structured so this metric is limited to the central residues (35)]. N_helices is the number of stretches of consecutive α-helical residues as determined by DSSP, and helicity is the number of helical residues as determined by DSSP (36).

We selected structures from previous simulations of the villin headpiece (without confinement) with a full range of P_fold values (0 to 1). For each of these structures we performed a large number of concurrent simulations by using the Folding@Home distributing computing network (37). From these simulations we simply counted the number of trajectories that committed to the folded or unfolded state (according to our definition). We then calculated the probability of folding before unfolding (P_fold) for each structure by calculating the fractions of simulations that committed to either state.

For our set of starting structures, we have calculated the P_fold for each under bulk-like conditions (no confinement; Fig. 1A), protein-only confinement (Fig. 1B), and protein and solvent confinement (Fig. 1C). We examined the correlation between P_fold under these different confinement regimes. From these correlations we can infer how confinement affects the kinetics (mechanism) and thermodynamics of folding. We estimated changes in mechanism [in terms of shifts in barrier width (Δσ) and position (Δx_TS)] and changes in thermodynamics (ΔΔG_F) as a result of confinement. Changes in barrier width and position were calculated by using the technique described in ref. 13. ΔΔG_F was calculated by fitting the following relation from Berezhkovskii and Szabo (38) (see ref. 38 and SI Text for a derivation): Here P′_fold is the perturbed value (i.e., protein-only confined, protein and solvent confined), and P_fold is the original value (no confinement).

These estimates of kinetics and thermodynamics rely on the approximation that perturbations to the transition state are small. Although this is often the case for point mutations (in the case of φ-value analysis) this assumption may not be as valid in the case of confinement. For this reason, these values are meant as qualitative measure of kinetic and thermodynamic changes resulting from confinement.

For more detailed materials and methods on the nature of the simulation methodology, as well as analysis techniques, see SI Text and SI Figs. 4–8.

We thank Prof. Ken Dill, Relly Brandman, Jeremy England, and Chris Snow for advice in preparing this manuscript, Holly Duddy for assistance with figure preparation, and the Folding@Home participants for contribution of central processing unit time. This work was supported by National Institutes of Health Roadmap on Nanomedicine Grant EF-0623664.