Multiple stepwise refolding of immunoglobulin domain I27 upon force quench depends on initial conditions

Methods

Coarse-Grained Model for I27. The conformation of a polypeptide chain in a coarse-grained representation is specified by a set of coordinates of the C_α-carbon atoms, where N is the number of amino acids. The energy of a conformation for the C_α-Go model is (14)

[1]

where r_i, θ_i, and φ_i are the bond length between i and i – 1 residues, the bond angle between (i – 1, i) and (i, i + 1) bonds, and the dihedral angle around the ith bond, respectively. The distance between residues i and j is r_ij. Subscript 0 and superscripts NC and NNC refer to native conformation and native and non-native contacts, respectively. The values of the parameters are K_r = 250 kcal/(mol·Å²), K_θ = 50 kcal/(mol·rad²), , , ε = 1.25 kcal/mol, and C = 4 Å. The unit of f in our model is ≈23 pN. Beads i and j form a contact in the native state, if r_ij ≤ d_c = 0.6 nm. A contact between i and j in an arbitrary conformation exists if r_ij ≤ 1.2 d_c. The native conformation of I27 (Protein Data Bank ID code 1TIT) has 86 native C_α – C_α contacts.

Simulations. The folding dynamics is obtained by integrating the Langevin equations of motion by using the corrector-predictor algorithm (17). In the presence of a constant f and assuming a rapid alignment of the end-to-end vector along the force direction, the total energy of the polypeptide is E_f = E – fR.We calculated f-dependent thermodynamic quantities by using a version of the multiple histogram method, in which both temperature T and f are varied (18).

Results

Force-Quench Refolding Occurs in Steps. The equilibrium f – T phase diagram shows that when f exceeds f_c = 43 pN at T_s = 0.85 T_F (T_F is a folding transition temperature at f = 0), I27 unfolds (M.S.L., unpublished work). The calculated value of f_c is in approximate agreement with the estimate (f_c ≈ 30 pN) by Li et al. (19) based on nonequilibrium atomic force microscopy experiments. Given the simplicity of the model and the crude methods used to estimate f_c in the atomic force microscopy experiments (19), the semiquantitative agreement validates the use of the Go model. To follow the dynamics of approach to the NBA we initially prepared stretched structures by applying a constant f = 87 pN (Δf_I ≈ 2). A molecule is stretched, if R ≥ 0.85 L, where L = 30.4 nm is the contour length of I27. Starting from force-denatured ensemble (FDE) conformations, we quenched the force to f_q < f_c. The refolding dynamics is monitored for 100 molecules by following the time dependence of the number of native contacts Q(t), the end-to-end distance R(t), and the radius of gyration R_g(t).

Refolding trajectories upon force quench show considerable heterogeneity (Fig. 1). Different molecules traverse distinct routes in the FDE → NBA transition. The reduction in R for the f_q = 0 case occurs in multiple steps (Fig. 1 a and b). At very short times (t ≲ 30 ns) R decreases continuously from 25 nm to ≈10 nm (stage 1, Fig. 1a). R fluctuates ≈10 nm (stage 2) until a transition to R ≈ 5 nm occurs around t ≈ 50 ns (stage 3). In this refolding trajectory the molecule fluctuates in the compact state (stage 4, R ≃ 5 nm, R_g ≃ 2.5 nm, Q ≃ 30 nm) until the NBA is reached around t ≃ 200 ns. At t > 200 ns the fluctuations in R are considerably reduced (Fig. 1a). Surprisingly, the observed multistep reduction in R is masked in the time course of R_g, which changes continuously from the initial value of 7.5 nm to 1.3 nm in the NBA. The acquisition of the native structure, as measured by Q, also occurs in steps from the initial value of Q ≈ 0 until the NBA is reached. Although another trajectory for f_q = 0 quench (Fig. 1b) can be similarly interpreted, it is clear that there are significant variations in the lifetimes of the transient intermediates and in the time scales of transition between them. There is no “typical” trajectory, which implies that refolding pathways are heterogeneous.

The pathway diversity is also evident when f_q > 0 (Fig. 1 c and d). Although the decrease in R occurs in steps as for f_q = 0, the nature of intermediates sampled en route to the NBA is different. In Fig. 1c, there is a rapid reduction in R to ≈10 nm (stage 1) followed by a further decrease to R ≈ 5 nm on longer times (stage 3). Another trajectory (Fig. 1d) shows that the residence time of the R ≈ 10 nm intermediate is ≈300 ns (≈2/3 of the first passage time to the NBA). The number of native contacts reveals these differences even more dramatically. In both cases, there are three distinct metastable intermediates that are sampled. Their Q values are different (Fig. 1 c and d), which shows that molecules sample distinct regions of the energy landscape even though the starting conformations are similar. It is interesting that even though R(t) and Q(t) reflect the stepwise acquisition of structure, R_g(t) in all cases decreases more continuously, thus masking the presence of any barriers to folding.

Collapse and Folding Occur on Distinct Time Scales upon Force Quench. The time dependence of 〈R(t) 〉, 〈R_g(t) 〉, and 〈Q(t) 〉 that monitors the relaxation to the NBA from the FDE shows a spectrum of time scales (Fig. 2). Upon force quench to f_q the initial event is the entropically driven transition from the stretched FDE state to a RC-like state (RC_f). The time dependencies of probes of collapse and folding are well described by using a sum of two exponentials. When f_q = 0 (Fig. 2), the FDE → RC_f transition takes place on a time scale of and corresponds to a drastic contraction of the dimensions of the polypeptide chain. The amplitudes in the two exponential fits indicate that on 〈R(t) 〉 decreases by about two-thirds of the total variation during refolding, whereas 〈R_g(t) 〉 decreases by ≈75%. The second kinetic stage results in a further reduction of 〈R(t) 〉 by ≈30% on a time scale , whereas collapse, monitored by 〈R_g(t) 〉, takes place on a significantly longer time scale (). This difference shows that the overall chain collapse is more complex than the reduction of the end-to-end distance. The decrease in 〈R_g 〉 is followed by the acquisition of the native interactions. The formation of native interactions, 〈Q(t) 〉, also occurs in two kinetic stages with the time scales and . Only 20% of native contacts are formed during the first fast kinetic stage, whereas the majority (80%) are formed after the final chain collapse.

The plots of various time scales, describing the transition from FDE to NBA, as a function of f_q (Fig. 3) show no apparent dependence of time scales and on f_q. We rationalize this surprising finding by noting that the entropic force (f_e) needed to stretch the protein is much greater than the maximum value of the quench force (). From the worm-like chain model we find that f_e ≈ 100 pN for stretching Ig27 to 0.85 L. Because the initial compaction of the polypeptide chain that is driven by the magnitude of f_e is not sensitive to the small f_q values. In contrast, the collapse time scales, and , and the folding time scale τ_F increase exponentially with f. For example, , at f_q ≈ 0.13 f_c ≈ 6 pN, increases 3-fold as compared with f_q = 0. For all values of f_q, and are smaller than τ_F, which implies that chain collapse occurs before the acquisition of the native state.

Ensemble of Initial Unfolded Structures Determines the Folding Pathways. A RC-like temperature-denatured ensemble (TDE) is generated by increasing temperature. Upon temperature quench the folding transition starts from a high-entropy TDE state to a low-entropy folded (NBA) state. In contrast, in FDE refolding, the low-entropy NBA is reached from a low-entropy stretched state with the initial rapid kinetic stage being the FDE → RC_f transition. If the relaxation of tension along the chain is rapid compared with protein conformational changes, then the ensemble of transiently populated coil conformations upon force-quench RC_f must be distinct from the TDE. The transiently populated metastable RC state RC_f at T < T_F is under tension when f_q > 0. These physical arguments suggest that FDE refolding pathways and time scales must be different from those observed for TDE refolding (Fig. 4).

When refolding is initiated by a temperature quench, 〈R(t) 〉 decreases exponentially with a single characteristic time scale τ ^R = 8 ns (Fig. 2). In contrast, 〈R_g(t) 〉 and 〈Q(t) 〉 show two-stage kinetics. Their double exponential fits indicate that the first kinetic (“burst”) stage coincides with the contraction of the end-to-end distance ( and ). During this initial stage, the chain dimensions are reduced by ≈50%, and ≈25% of native contacts are formed. The final collapse and folding take place on a much longer time scale (). This stage is associated with the formation of the bulk of the native interactions (75%) and further compaction of the chain by ≈50%. The data show that collapse and folding starting with the TDE conformations are effectively synchronous.

In contrast to TDE folding, 〈R(t) 〉 in FDE refolding shows a two-stage relaxation (Fig. 2). The radius of gyration and the number of native contacts also follow a two-stage kinetics as for TDE. However, FDE time scales, and , increase by 30% and 50%, respectively (Fig. 2). The folding time τ_F shows a similar increase (from to ). More importantly, collapse and folding are no longer synchronous (Figs. 2 and 3) when folding is initiated from FDE.

Physics of Refolding Is Not Model Dependent. To ascertain that the qualitative results for the I27 Go model are not model dependent, we performed the force-quench refolding simulations by using the four-strand β-barrel model sequence S1 (20). This model differs from I27 in two important aspects. First, the energy function of S1 includes non-native interactions that are neglected in the I27 model. Second, the native topology of S1 is a β-barrel, which is simpler than the β-sandwich structure of I27. The S1 simulations (for details see Supporting Text and Figs. 6 and 7, which are published as supporting information on the PNAS web site) were performed under the conditions similar to those used for I27. All structural probes, which monitor TDE refolding, show a concerted collapse and folding on a single time scale of ≈85 ns. When folding is initiated with FDE conformations (f_q = 0), we observe an additional rapid relaxation, corresponding to contraction of the initially fully stretched conformations to partially stretched RC_f, whose average dimensions are about double that of TDE. In addition, the time scale for reaching the NBA increases by a factor of two. Surprisingly, the final collapse time scales in TDE and FDE refolding are nearly the same. Therefore, folding and collapse occur on distinct time scales upon the change in initial conformations. Hence, the findings for S1 and I27 are in qualitative agreement.

From the results for S1 and I27, we surmise that the qualitative differences between folding initiated by temperature or force quenches are primarily related to the variations in the nature of the metastable RC_f and the thermally equilibrated TDE (Fig. 4). It is the memory of the initially stretched conformation that makes the RC_f ensemble sampled along FDE refolding pathways different from the TDE. By shifting the FDE plots by Δt to the left, we find that the distribution of conformations in the FDE pathway approaches TDE if Δt ≈ 50 ns (Fig. 5). The striking agreement between FDE and TDE refolding plots upon Δt shift suggests that folding begins only after RC_f conformations start to “resemble” the TDE RC conformations. Subsequently, both kinetic processes are similar. The delay in FDE folding compared with collapse is primarily caused by the time required for RC_f conformations to equilibrate among the ensemble of coil-like structures. The time scale for local equilibration of RC_f is surprisingly long (). The long time needed to reach local equilibrium in RC_f, after which folding begins, explains the increase in τ_F for FDE folding as compared with that for TDE. This finding also shows that folding strongly depends on the starting point on the energy landscape.

To further probe the dependence of force quench τ_F on the initial conditions, we prepared distinct FDEs by stretching S1 to a different values of the initial end-to-end distance, R. This procedure is equivalent to varying f_I albeit in a nonlinear manner. In accord with the physical picture in Fig. 4. we find that, at f_q = 0, τ_F changes dramatically with R (Fig. 5 Inset). Thus, initial conditions, which can be readily manipulated in laser optical tweezer or atomic force microscopy experiments by altering f_I, determine the refolding mechanism and time scales.

Collapse and Refolding Times for I27 Depend Exponentially on f_q. We obtained the f_q-dependent refolding time τ_F from the distributions of first passage times computed by using 50–100 trajectories for each value of f_q. When f_q = 0, we find that , which is ≈60% more than the TDE refolding time at T_s = 0.85 T_F. When f_q > 0, the refolding times obey the relation

[2]

where Δx_f is the average location of the folding transition-state ensemble (Fig. 3 Inset). From the linear fit of log(τ_F) as a function of f_q by using Eq. 2 we get Δx_f ≈ 0.6 nm for I27 at T_s = 0.85 T_F. In the force-quench experiments R can be treated as a suitable reaction coordinate in the FDE → NBA transition (6). The dependence of τ_F on f_q can be written as τ_F ≈ exp(yΔG_UFΔx_f/X_UFk_BT), where ΔG_UF is the equilibrium free energy of stability of the NBA with respect to the unfolded states, y = f_q/f_c, and X_UF is the distance between the NBA and the unfolded states. In force-quench experiments, the search for the transition-state ensemble occurs among the ensemble of unfolded states. With this interpretation, a Tanford-like parameter can be written as β ≈Δx_f/ΔX_UF ≈ 0.2 for I27, where ΔX_UF ≈ a(N^3/5 – N^1/3). Fernandez and Li (6) found that the mean relaxation time for poly Ub also obeys Eq. 2 with Δx_f ≃ 0.8 nm, which implies that β for Ub is also likely to be close to the NBA. Given the simplicity of our model and the differences in the topology of I27 (β-sandwich protein) and Ub (α/β protein), the agreement in Δx_f suggests that the native states of these proteins might be “brittle,” i.e., once the extension exceeds a small value the structure unravels. A similar conclusion follows from the steered molecular dynamics simulations of mechanical β-sandwich proteins (12).

Discussion and Conclusions

Force-quench experiments offer a potential of initiating biomolecular refolding starting from arbitrary initial values of the end-to-end distance, R (6). Fernandez and Li (6) generated mechanical folding trajectories from the regions of free energy landscape that cannot be accessed in conventional experiments. In this work, we have used coarse-grained models for the β-sandwich protein I27 and β-barrel to probe refolding from the ensemble of stretched structures. The implications of our findings are discussed below.

The refolding pathways are heterogeneous, with the approach to NBA occurring in distinct stages. The longer refolding times in FDE simulations compared with TDE refolding are caused by the reduction in the rate of formation of low-energy compact structures. Similar to poly Ub, several long-lived intermediates with varying values of R and Q are sampled en route to the NBA. These results for a monomeric I27 are in qualitative accord with the conclusions drawn from mechanical folding trajectories for poly Ub (6).

The nature of the ensemble of initial states determines the collapse and folding pathways. When the protein is fully extended, the entropy of the stretched state is dramatically lower than that of TDE conformations. Refolding from FDE effectively starts from extended conformations (R_g ≈ N), whereas TDE refolding begins from the ensemble of conformations with a mean R_g ≈ N^0.6. The folding initiated with FDE differs from TDE folding in two aspects. First, the stretched polypeptide chain must initially contract to the ensemble of conformations with the dimensions approximately similar to TDE RCs. Second, FDE leads to the delay in the formation of native interactions, because chain must not only contract, but also “equilibrate” within a RC ensemble. Therefore, when folding begins from stretched state, the collapse and folding occur on distinct time scales. In contrast, when TDE is used, collapse and folding often occur in concert for two-state folders.

Force-quench experiments on poly Ub (6) and the present study on monomeric proteins show that the f_q-dependent refolding time obeys Eq. 2. However, the value of τ_F extrapolated to f_q = 0 is larger than that obtained by conventional experiments. Fernandez and Li (6) obtained for ubiquitin, whereas the folding time from ensemble experiments (21). Our results show that the difference between the two experimental values is caused by the vastly different initial conditions, from which refolding is initiated. In the transition from the stretched state to the NBA, a large free-energy cost is involved in burying substantial surface area . In contrast, the NBA is accessed from TDE by much smaller conformational fluctuations. The discrepancy between and should decrease, if f_I does not fully stretch the polypeptide chain (Fig. 5 Inset).

In the Fernandez and Li experiments (6), f_I is fixed (Δf_I ≈ 3) and f_q is varied, which affects the free energies of the folded and unfolded states. The linear increase in the free-energy barrier to folding, which is consistent with the Bell-type model (22), is caused by the stabilization of the RC_f states with increasing f_q. The presence of free-energy barrier to folding is consistent with theoretical (23, 24) predictions and simulation results (25) that show that equilibrium-force unfolding is a first-order transition. On the other hand, if f_q is fixed and f_I is varied, then we expect , where γ is an effective surface tension. This finding suggests that the shape of the dependence of τ_F on f_I should approximately resemble the equilibrium force-extension curve (Fig. 5 Inset). These arguments show that force-quench refolding with fixed f_I and varying f_q is qualitatively different from folding with f_q fixed and differing f_I values. This inherent asymmetry can be exploited to map the free-energy landscape of proteins.

It is not straightforward to infer the nature of monomeric protein folding from mechanical folding trajectories of polyprotein constructs. The folding of the ith domain may be influenced by the neighboring domains that act as “flexible” linkers. Resolution of folding trajectories of a single domain requires that the spring constant of a linker k_L > k_P, where k_P is a measure of the “softness” of a protein. This requirement may not be satisfied in polypeptide constructs. Therefore, it is preferable to use stiff linkers that sandwich the monomeric protein just as was done in the mechanical unfolding of RNAs (5). Computational studies of force-quench refolding of RNA hairpins with linkers show that relaxation dynamics of the hairpin and the linkers is additive (C. Hyeon and D.T., unpublished work). This finding implies that by varying separately the mechanical properties of the linker and the linker-biomolecule-linker constructs the refolding dynamics of the monomeric protein can be directly monitored.

The major limitation of the present study is the use of the coarse-grained models. For more realistic models, the difference between the time scales of TDE and FDE folding is likely to grow (6). Because the physics of the transition from stretched states to the native conformation is generic (Fig. 4), we propose that the qualitative aspects of force-quench refolding explored here should be valid for other single-domain proteins.

Supplementary Material