Protein folding guides disulfide bond formation

Significance

Anfinsen inferred the principles of protein folding by studying a protein containing four disulfide bonds in the native state. However, how protein folding drives disulfide bond formation is poorly understood despite the role such proteins play in variety of extracellular and intracellular functions. We developed a method to mimic the complex chemistry of disulfide bond formation in molecular simulations, which is used to decipher the mechanism of folding of bovine pancreatic trypsin inhibitor. The flux to the native state is through a complex network of native-like states. Crucially, we establish that it is the conformational folding of the protein that directs disulfide bond formation. Testable predictions and implications for enzyme-catalyzed folding of disulfide proteins in vivo are outlined.

Abstract

The Anfinsen principle that the protein sequence uniquely determines its structure is based on experiments on oxidative refolding of a protein with disulfide bonds. The problem of how protein folding drives disulfide bond formation is poorly understood. Here, we have solved this long-standing problem by creating a general method for implementing the chemistry of disulfide bond formation and rupture in coarse-grained molecular simulations. As a case study, we investigate the oxidative folding of bovine pancreatic trypsin inhibitor (BPTI). After confirming the experimental findings that the multiple routes to the folded state contain a network of states dominated by native disulfides, we show that the entropically unfavorable native single disulfide [14–38] between Cys₁₄ and Cys₃₈ forms only after polypeptide chain collapse and complete structuring of the central core of the protein containing an antiparallel β-sheet. Subsequent assembly, resulting in native two-disulfide bonds and the folded state, involves substantial unfolding of the protein and transient population of nonnative structures. The rate of [14–38] formation increases as the β-sheet stability increases. The flux to the native state, through a network of kinetically connected native-like intermediates, changes dramatically by altering the redox conditions. Disulfide bond formation between Cys residues not present in the native state are relevant only on the time scale of collapse of BPTI. The finding that formation of specific collapsed native-like structures guides efficient folding is applicable to a broad class of single-domain proteins, including enzyme-catalyzed disulfide proteins.

The landmark discovery that the information to fold a protein is fully contained in the primary amino acid sequence was based on oxidative refolding experiments on disulfide bond formation in ribonuclease A (RNase A) (1, 2). Anfinsen showed that the initially unfolded protein, generated by reducing the disulfide (S–S) bonds in the native state of RNase A, folds reversibly under oxidizing conditions by correctly reforming the four native S–S bonds (among 105 possibilities) between the eight cysteine (Cys) residues. Besides being central to the enunciation of the principles of protein folding, many secretary proteins, whose misfolding is linked to a number of diseases, contain S–S bonds (3). Although biophysical aspects of such proteins are not as well studied as those without S–S bonds, understanding the link between conformational folding coupled to disulfide bond formation (4–7) is important and challenging both from a chemical and biophysical perspective (8).

The formation of S–S bonds and their identities during folding can be monitored by quenching the oxidative process at various stages of the folding reaction (9). By arresting the reaction, it is possible to characterize the accumulated intermediates in terms of already formed S–S bonds (10). However, the relationship between protein folding and disulfide bond formation is nontrivial to establish because this requires separate reporters for disulfide bond formation and organization of the rest of the polypeptide chains. Even if the reaction can be arrested rapidly, the conformations of the intermediates are difficult to determine using experiments alone, although single molecule pulling experiments hold exceptional promise (7). Thus, well-calibrated computations are needed to decipher the precise relationship between conformational folding and S–S bond formation (11–14).

Here, we investigate the coupling between conformational folding and disulfide bond formation by creating a novel way to mimic the effect of disulfide bond formation and rupture in coarse-grained (CG) molecular simulations, which have proven useful in a number of applications (15–18). As a case study, we use the 58-residue bovine pancreatic trypsin inhibitor (BPTI) with three S–S bonds in the native state to illustrate the key structural changes that occur during the folding reaction. The pioneering experiments of Creighton (9) seemed to indicate that nonnative disulfide species (19–22) are obligatory for productive folding to occur (for a thoughtful analysis, see ref. 23). Subsequently, using acid quench technique (by lowering pH, resulting in slowing down of the thiol disulfide exchange reaction) and a superior way of separating the intermediates Weissman and Kim (24) found that only native single and multiple disulfide bonds accumulate during the folding process. A plausible resolution of these contradictory findings was provided using theoretical studies (5) and simulations using lattice models (12) showing that nonnative intermediates are formed only on the time scale of the global collapse of the polypeptide chain. On longer times, only native species (S–S bonds found only in the folded state) dominate, as surmised by Weissman and Kim (24).

The experimental studies could not resolve whether disulfide bond formation drives protein folding or vice versa, and has remained a major unsolved problem in protein folding. To solve this problem, we created a novel computational method to mimic disulfide bond formation and rupture within the context of a C_α representation of polypeptide chain by building on the demonstration by Scheraga and coworkers (6) that the formation or disruption of S–S bonds in these proteins can occur only if a few structurally important criteria (proximity of Cys residues, orientation, and accessibility of thiol groups to oxidative agents, see Figs. S1–S3) are met. We incorporated this physical insight in our model and simulated the oxidative folding of BPTI. Our results quantitatively capture the relative importance of all single and two disulfide intermediates that direct folding of BPTI. The initial rapid formation of single disulfide intermediates (in particular [14–38], an intermediate with disulfide bond between Cys₁₄ and Cys₃₈), occurs only after substantial compaction of BPTI and complete structuring of the central antiparallel β-sheet shown in Fig. 1A. Formation of two-disulfide intermediates and the species ${\mathrm{N}}_{SH}^{SH}$ ([5–55, 30–51]) that is poised to fold rapidly to the folded state N requires substantial unfolding of BPTI. Loop formation dictated by entropic considerations and forces that drive chain compaction place the Cys residues in proximity to enable S–S bond formation, thus directing BPTI to the folded state (5). Our work also provides a general framework to simulate oxidative folding of disulfide-containing proteins, and firmly establishes that early formation of specifically collapsed structures results in efficient folding of single domain proteins.

Fig. S1.

The mean values of the three structural parameters $\langle d_{\alpha }\rangle$, $\langle {\theta }_{\alpha }^k\rangle$, and $\langle n_{\alpha }\rangle$ obtained from equilibrium simulations at $k_B$T = $0.9\mathit{ϵ}$ for BPTI containing with different combination of S–S bonds. Here, $d_{\alpha }$ is the distance between the two cysteines in native state, ${\theta }_{\alpha }^k$ ($k=\mathrm{1,2}$) is the orientation angle defined in Fig. S2, and $n_{\alpha }$ is the number of residues within a spherical shell with radius R (Fig. S3). The label α = 1, 2, and 3 refer to the three disulfide bonds [5–55], [14–38], and [30–51], respectively. For example, for [5–55] (Left), these values are obtained from the simulations of BPTI mutants containing [5–55], N′, $N_{SH}^{SH}$, and N.

Fig. S3.

The definition of the number of residues ($n_{\alpha }$) within a spherical shell with radius R, drawn from O, the center of the α^th S–S bond.

Fig. 1.

(A) Ribbon diagram of the native structure of the 58 residue BPTI containing three disulfide bonds (marked in yellow) between residues Cys₅ and Cys₅₅ [5–55], Cys₁₄ and Cys₃₈ [14–38], and Cys₃₀ and Cys₅₁ [30–51], respectively. The antiparallel β-sheet is in red. (B) Simplified representation of the secondary tructure of BPTI and the three native disulfide bonds in BPTI. (C) Variables ${\beta }_O$ and ${\beta }_R$ mimicking the redox conditions. Small ${\beta }_O$ (${\beta }_R$) represent strongly oxidizing (reducing) condition. The star with ${\beta }_O$ = 1.0 and ${\beta }_R$ = 1.5 is used in most of the simulations. These values are a mixture of mildly oxidizing and reducing condition. (D) Distribution of fraction of native contacts obtained from high-temperature simulations.

Fig. S2.

Definition of the orientation angle ${\theta }_1$ and ${\theta }_2$. The indices i − 1, i, i + 1 and j − 1, j, i + 1 are the two different peptide segments such that i and j are the two cysteines that can form a disulfide bond in the native state; ${\theta }_1$ is the angle between the vector connecting the cysteine residues i and j (i < j) and the covalent bond connecting $Cy{s}^i$ and the neighboring $C_{\alpha }^{i-1}$. A similar definition holds for ${\theta }_2$.

Results

The native state of BPTI has three disulfide bonds between Cys₅ and Cys₅₅, Cys₁₄ and Cys₃₈, and Cys₃₀ and Cys₅₁ (Fig. 1A). The 5–55 disulfide bond is located between the two terminal helices [helix α₁ (from Asp3 to Glu7) near the N terminus and α₂ (Ala48 to Gly56) in C terminus]. The maximally solvent-exposed [14–38] disulfide bond is near the terminus of the β-hairpin (formed between the β₁-strand from Ile18 to Asn24, and β₂-strand from Leu29 to Tyr35). The 30–51 disulfide bond bridges the β₁-strand and α₂ (Fig. 1B). When the three native disulfide bonds are reduced BPTI unfolds, resulting in the R state. The two-disulfide species [5–55, 14–38] is designated as N*, [14–38, 30–51] is termed N′, and [5–55, 30–51] is ${\mathbf{N}}_{SH}^{SH}$. The native BPTI [5–55, 14–38, 30–51] is labeled N. In this standard description, the states of BPTI are described solely in terms of the [S–S] bonds without consideration of the conformations of the rest of the polypeptide chain.

Folding Network in Terms of Intermediates.

To validate our simulations by direct comparisons with experiments, we map the folding pathways of BPTI in terms of the accumulated S–S bond intermediates. The flux map in Fig. 2, obtained using 2,000 folding trajectories, quantifies the flow toward the native state from any given intermediate in the folding network. In the earliest stage of folding, nearly 84% of the reduced BPTI rapidly forms the [14–38] intermediate (Fig. 2). The populations of [30–51] and [5–55] in the initial stages of BPTI folding are considerably less under the simulated redox conditions (Fig. 1C). The finding that the metastable [14–38] is the first native disulfide to form rapidly before further rearrangement agrees with theoretical predictions (5) and subsequent experimental validation (25, 26). As folding progresses, the kinetically unstable [14–38] rearranges to form the more stable [30–51] and [5–55] (Fig. 2). In our simulations, which mimic mildly oxidizing conditions, the transition from [14–38] to [5–55] and [30–51] involves transient population of compact R′ ensemble [(not the same as R ensemble) structures devoid of S–S bonds but with persistent secondary structures (see below)].

Fig. 2.

The folding pathway, represented as a network of native-like states, connecting the fully unfolded state (R) to the folded (N) state with three disulfide bonds. For ${\beta }_O$ = 1.0 and ${\beta }_R$ = 1.5, the early event produces predominantly [14–38]. Subsequently, there is a bifurcation in the pathway with this intermediate rearranging to [5–55] and [30–51]. The percentages indicate the dominant route to N from R. Representative structures of all of the relevant states are shown. The numbers in parentheses were obtained from simulations with ${\beta }_O$ = 2.0 and ${\beta }_R$ = 4.0, which mimics the redox conditions used by Weissman and Kim (24). In red are the fluxes through the various native intermediates to the native state obtained from simulations that consider native and nonnative disulfide bond formation. The qualitative agreement between the two simulations is striking.

Formation of Two-Disulfide Species and the Native State.

The single disulfide species rearrange with varying probabilities to form intermediates containing two native S–S bonds. Based on the cumulative analysis of all of the oxidizing events, the predominant species are [14–38, 30–51] (N′) and [5–55, 14–38] (N*) rather than the productive [5–55, 30–51] (${\mathbf{N}}_{SH}^{SH}$). The relative equilibrium fluxes show that the likelihood of populating N* from [5–55] is nearly twice as large as the formation of ${\mathbf{N}}_{SH}^{SH}$. Similarly, N′ formation from [30–51] is nearly six times more likely than ${\mathbf{N}}_{SH}^{SH}$. Because among the single native disulfide species [30–51] forms with the highest probability after the population of [14–38] decays, it follows that formation of intermediates with two S–S bonds occurs by conformational rearrangement involving [30–51] (25).

The final step is the rearrangement of the two-disulfide species to N. The flux map in Fig. 2 shows that only a very small fraction of N* and N′ states is kinetically connected to N. Because the population of N* is less than N′, we conclude that N* is a kinetic trap. Our simulations show that these dead-end species must undergo rearrangement by reduction of [14–38], and subsequent formation of the productive two disulfide species, ${\mathbf{N}}_{SH}^{SH}$. In this process nonnative species are transiently populated in some of the folding trajectories (see below). The simulated folding pathway is in very good agreement with experiments, setting the stage for us to elucidate how protein folding controls disulfide bond formation.

Flux Map Depends on Redox Conditions.

The range of redox conditions is approximately mimicked by varying the [${\beta }_O,{\beta }_R$] values. One way in which redox conditions are altered in in vitro experiments is by changing the buffer concentrations of glutathione disulfide (GSSH) and glutathione (GSH) (27). To assess the effect of changing redox conditions on BPTI folding we performed simulations with ${\beta }_O$ = 2.0 and ${\beta }_R$ = 4.0, which mimics the conditions used by Weissman and Kim (24). The large value of ${\beta }_R$, corresponding to weak reducing conditions, means that the probability of an already formed [S–S] bond has very small probability of undergoing reduction (see SI Text for details). It is in this sense that the $\left[{\beta }_O,{\beta }_R\right]$ values used in these simulations qualitatively mimic the Weissman and Kim (24) conditions. The percentage of molecules that reach N is now considerably less, and the transition from N* is so slow that only a very small (the values are given in parenthesis in Fig. 2) fraction of N* reaches the ${\mathbf{N}}_{SH}^{SH}$. Thus, N* is effectively a dead-end kinetic trap. The overall findings are in excellent agreement with experiments (24).

Interestingly, the rate of formation and the fraction of R ensemble that reaches [14–38] do not depend on the redox conditions. To establish the robustness of this finding, we varied ${\beta }_O$ with ${\beta }_R$ fixed at 1.5 (Fig. 1C). Neither the yield nor the time (τ_[14-38]) for forming [14–38] changes appreciably as ${\beta }_O$ changes (Fig. 3A). Thus, [14–38] formation depends predominantly on the statistics of loop formation and compaction of BPTI resulting in the near complete ordering of the β-hairpin that brings Cys₁₄ and Cys₃₈ in proximity (5).

Fig. 3.

(A) Dependence of the mean first passage time (black line) for forming [14–38] as the oxidizing condition is changed from being strong (small ${\beta }_O$) to weak (large ${\beta }_O$). The red line gives the yield of [14–38]. (B) Time-dependent decay, $P_u^{\alpha }\left(t\right)$, of the three native single disulfide species. (Inset) $ln{P}_u^{\alpha }\left(t\right)$; the lines are linear fits.

Kinetics of Single Disulfide Bond Formation.

We quantify the kinetics of formation of single disulfide species using the time-dependent changes in the population, $P_{\alpha }\left(t\right)={\displaystyle {\int }_0^t{P}_{fp}^{\alpha }\left(s\right)ds}$, where α refers to [14–38], [30–51], or [5–55], and $P_{fp}^{\alpha }\left(s\right)$ is the distribution of first passage time for the formation of the α^th species. We calculated $P_{fp}^{\alpha }\left(s\right)$ by determining the first time the α^th species is reached in each folding trajectory. The results in Fig. 3B for $P_u^{\alpha }\left(t\right)=1-P_{\alpha }\left(t\right)$ show that although [14–38] forms rapidly with substantial probability (Fig. 2) (relative to [5–55] or [30–51]), it decays also rapidly rearranging to [5–55] and [30–51]. Closer inspection of $P_u^{\alpha }\left(t\right)$ for [5–55] and [30–51] shows a delay in the decay that is absent in $P_{\left[14-38\right]}\left(t\right)$, implying that only after [14–38] forms does one observe population of the other two native single disulfide intermediates. Because [14–38] is exposed to solvent, it is vulnerable to further oxidation or reduction. In contrast, both [5–55] and [30–51] are buried (Fig. 1A), and hence once they are formed they are not as vulnerable to further rearrangement as [14–38]. Consequently, [14–38] is unstable and rearranges to the more stable [30–51] and [5–55] involving substantial unfolding (Fig. S4) through structures that are more compact than reduced BPTI (Fig. S5).

Fig. S4.

(A) A trajectory showing the time-dependent changes during the [14–38] → [30–51] transition. The distance $d_{14-38}$ increases (red) abruptly around $t\approx$ 9$\times$ 10³τ and is accompanied by a decrease in $d_{30-51}$ (blue). During the transition, Q remains low (scale is on the right), which is also illustrated by the snapshot below. (B) Same as A except the trajectories and conformations are for the [14–38] → [5–55] transition. Note that the scales for the distance as well as Q are different in A and B. (C) Distribution of the radius of gyration for the ensemble of R′ structures sampled during the [14–38] → [30–51] transition, an example of which is shown in A. (D) Same as C except the distribution is for the [14–38] → [5–55] transition.

Fig. S5.

Distribution of $R_g$ using an ensemble of unfolded conformations generated at high temperature. The set of initial conformations with $R_g>$ 20 Å and fraction of native contacts $Q<$ 0.1 (Fig. 1D) is used to initiate oxidative refolding. The initial conformations are devoid of any disulfide bonds.

Collapse and Complete Ordering of β-Hairpin Precede Formation of Disulfide Bond.

The connection between S–S bond formation and folding is best captured by analyzing a typical folding trajectory shown in Fig. 4A (Movie S1). Another example is given in Fig. S6. A number of inferences can be drawn from Fig. 4A: (i) Fig. 4A shows that before the formation of a single S–S bond, BPTI is compact with $R_g$ decreasing from 24 Å in R state to less than 15 Å; (ii) upon compaction, Cys₁₄ and Cys₃₈ are in proximity with correct orientation resulting in the formation of the S–S bond. As reflected in Fig. 4A, [14–38] forms ahead of [5–55] and [30–51]. Most importantly, before formation of [14–38], the antiparallel β-hairpin is fully structured and the terminal α-helices adopt native-like structures. The rest of BPTI is disordered. We substantiate this finding quantitatively by a plotting $d_{\left[14-38\right]}$ as a function of the fraction of the native contacts, $Q_{\beta }$, between the two β-strands. The near perfect anticorrelation between these quantities (Fig. 4B) shows that as $Q_{\beta }$ increases, the distance between residues 14 and 38 decreases, thus facilitating the formation of [14–38]. Even at $d_{\left[14-38\right]}\approx$12 Å, a value that is too large for [14–38] to form, $Q_{\beta }$ adopts near native value (Fig. 4B), implying that the core hairpin formation is complete before [14–38] formation starts. (iii) When $d_{\left[14-38\right]}\approx$5.7 Å, which is close enough to form a disulfide bond, the distribution of $Q_{\beta }$ is peaked at a value that exceeds 0.8 (Fig. 4C), implying complete ordering of the hairpin. (iv) As time progresses, [14–38] undergoes oxidation and reduction multiple times under the conditions of simulations (Fig. 4A and Fig. S6) because it is exposed to the solvent. Although kinetically favored, the low stability of [14–38] results in its disruption as folding proceeds. Thus, collapse of the polypeptide chain and complete structuring of the β-hairpin followed by [14–38] formation are key events that direct protein folding.

Fig. 4.

(A) An example of one of 2,000 oxidative folding trajectories showing the route to the folded state from a fully reduced starting conformation with low Q. The blue curve shows the decrease in the radius of gyration, $R_g$ (scale is on the right). The gray lines show formation of various disulfide species labeled on the left. Snapshots (a–h) show some of the conformations sampled in the trajectory. (B) Plot of the distance between the distance between residues 14 and 38, $d_{14-38}$, as a function of the fraction, $Q_{\beta }$, of contacts in the β-hairpin shows that the hairpin (formed between β-strands from Ile18 to Asn24, and from Leu29 to Tyr35) is fully structured before [14–38] formation. (C) Distribution of $Q_{\beta }$ when $d_{14-38}$ = 5.7 Å (the distance at which [S–S] bond forms) for the first time in a folding trajectory.

Fig. S6.

A second folding trajectory is shown to illustrate that except for changes in the value of the first passage time, the behavior is similar to the one shown in Fig. 4A. Some of the sampled conformations (a–h) are explicitly shown. This figure, together with the one in Fig. 4A, shows that structures similar to the native state of BPTI are populated in the folding process relatively early.

Rearrangement of N* and N′ to ${\mathrm{N}}_{SH}^{SH}$ Involves Nonnative Species.

Although predominantly native-like interactions drive disulfide bond formation, experiments (24) showed that rearrangement of N′ ([30–51, 14–38]) to ${\mathbf{N}}_{SH}^{SH}$ involves transient population of nonnative species. In several of the folding trajectories, such as the one in Fig. 5A, we find that in the conversion process between two native-like disulfide bond intermediates nonnative species, [30–51, 5–38] and [30–51, 5–14], are transiently populated. These were precisely the ones identified in experiments (see figure 6A in ref. 24). Interestingly, Fig. 5A also shows that the protein unfolds substantially because establishing the disulfide bond between 5 and 55 requires exposure to oxidation agents (28). Similar conclusions can be drawn from Fig. 5B, which shows the dynamics of conformational changes in the N* →  ${\mathbf{N}}_{SH}^{SH}$ transition. Note that in this case BPTI unfolds to a much greater degree (compare the scales in Fig. 5 A and B) than in the transition from N′ →  ${\mathbf{N}}_{SH}^{SH}$. The snapshots of the structures show that the key secondary structural elements are intact, which means that the transiently populated nonnative species are compact (Fig. S7).

Fig. 5.

(A) Illustration of the dynamics of rearrangement from N′ to ${\mathrm{N}}_{SH}^{SH}$ using distance between the different Cys residues. The colors are illustrated in the figure. In this transition, BPTI samples both the nonnative species [30–51, 5–14] (structure on the left) and [30–51, 5–38] (conformation on the right). (B) Same as A except this trajectory describes the N* $\to$ ${\mathbf{N}}_{SH}^{SH}$ transition.

Fig. S7.

Distributions of the radius of gyration ($R_g$) for the three native two-disulfide intermediates calculated from simulations emphasizing only native-like interactions. The distributions are narrow and are roughly peaked at the value of $R_g$ corresponding to the native state, thus confirming the near native structures in these intermediates.

Stability of β-Hairpin Correlates with τ_[14–38].

Our simulations reveal that β-hairpin is fully formed long before the distance between residues 14 and 38 is close enough to establish S–S bond (Fig. 4B). It stands to reason that if interactions favoring β-hairpin formation are strengthened, then τ_[14–38] should decrease. To test this prediction, we stabilized the β-hairpin by increasing λ (Eqs. S1 and S2), involving only the residues in the β-hairpin. We find that τ_[14–38] decreases as λ increases (Fig. 6), thus demonstrating the correlation between τ_[14–38] and the β-hairpin stability. The yield of [14–38] (red line in Fig. 6) also increases as the hairpin stability increases. We repeated the simulations by creating a pseudo mutant in which only [14–38] can form, which is realized by setting $K_s$ = 0 (Eq. 1) for all other Cys residues. The changes in the rate of [14–38] folding essentially mirrors that for the WT. These simulations predict a twofold increase in rate if the hairpin is stabilized.

Fig. 6.

Time for forming [14–38] relative to the wild-type as function of λ (defined in Eqs. S1 and S2). The stability of the β-hairpin increases (decreases) as λ increases (decreases). The relative times are given for two variants. One of them (black line) is for the WT (λ = 1.0) and the other blue is for a pseudo mutant, in which disulfide bonds other than [14–38] cannot form. The red curves show the yield of [14–38] for the WT.

The finding that an increase in the stability of β-hairpin (Fig. 1A) results in a decrease in the folding time of [14–38] is in quantitative accord with experiments (29). Mutation of a number of bulky hydrophobic residues in the β-strands (residues 18–24 and 29–35), several of them being distant from 14 or 38, by Ala in a pseudo mutant (residues 5, 30, 51, and 55 were replace by Ala) showed retardation of [14–38] formation (figure 4 in ref. 29). The maximum decrease in the rate is about a factor of 2, with most substitutions showing a more modest decrease relative to the pseudo WT. Remarkably, our simulations capture the extent of rate changes found in experiments quantitatively, thus establishing the crucial role the β-hairpin plays in enabling [14–38] to form. More recently, experiments have further illustrated the link between hydrophobic interactions and disulfide bond formation in controlling fibrillar and globular aggregate formation in egg white lysozyme (30, 31).

Factors Determining Protein Collapse and [14–38] Formation.

To understand the importance of the initial formation of [14–38], which subsequently leads to native-like states N* and N′, neither of which contains [14–38], we performed a series of simulations. Consider a variant of BPTI in which S–S bond formation between Cys residues 30, 51, and 55 and interactions between nonbonded residues are all repulsive (obtained by setting ${\epsilon }_1$ = 0 in Eq. S2). This model, referred to as type I, is a polymer mutant of BPTI serves as a reference system for assessing the role of entropy in forming the single disulfide species. In this limit, we expect that the relative probability of forming the single disulfide intermediates should be proportional to the probability (see figure 1 in ref. 5) of loop formation, $P\left(l\right)\approx \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\theta }_3$}\right.\left(1-exp\left(-l/l_0\right)\right)$, where l is the number of bonds separating two residues, $l_0$ (roughly, 2-3) is the persistence length of the polypeptide chain, and ${\theta }_3\approx 2.2$ (32). The theory predicts that the ratio of the probability formation of [5–14] (l = 9) to [14–38] (l = 24) should be roughly (24/9)${}^{{\theta }_3}\approx$ 9. Based on the theory for loop formation kinetics (33–35), we predict a similar ratio for time scales for forming such contacts.

The type II model is the same as type I except that only the interactions between residues in the β-hairpin are purely repulsive. All other nonbonded interactions retain the values as in the WT. The type III model is a pseudo mutant of the WT created by preventing the S–S bond formation between Cys residues 30, 51, and 55. We achieve this by setting $K_s$ to zero in Eq. 1 for these three residues. Such mutants can be created in experiments by Ala substitutions (29). In all these simulations, ${\beta }_O$ = 1.0 and ${\beta }_R$ = 1.5.

From 3,000 trajectories for the type I model, the mean time, τ_[5–14], for forming the most probable [5–14] is $\approx$ 6,000 τ, whereas [14–38] forms in τ_[14–38] $\approx$ 8.7 × 10⁴τ (Fig. S8A). The ratio is ∼15, which agrees well with predictions based on polymer theory. The least probable [5–38] forms about 26 times slower than [5–14], which is also in rough accord with theory.

Fig. S8.

Kinetics of formation of three single-disulfide bonds for type I model (A), type II model (B), and type III model (C). The three models are described in detail in the main text. The mean first-passage time values, quoted in the main text, are obtained using single exponential fits shown as solid lines in A–C.

We generated 4,218 trajectories for the type II model in which the formation of the central β-hairpin is prevented. In this case, the three single disulfide bonds form in comparable times (Fig. S8B) with τ_[5–14] $\approx$ 2.5 × 10⁴τ, τ_[14–38] $\approx$ 1.9 × 10⁴τ, and τ_[5–38] $\approx$ 4.8 $\times$ 10⁴τ, suggesting that the presence of helices (Fig. 1A) and favorable tertiary interactions results in BPTI collapse enabling the three one-disulfide bonds to form albeit without discrimination.

For the type III pseudo WT model (only disulfide bonds involving 5, 14, and 38 can form), we created 2,249 trajectories. In all these trajectories, [14–38] forms, whereas in only a minority of trajectories the formation of [5–14] and [5–38] is observed; this is surprising because the mean folding time of the WT is less than the maximum time for which the simulations are run for this model. Thus, the calculated τ_[5–14] $\approx$ 0.8 × 10⁴τ is a lower bound. In sharp contrast, we find that [14–38] forms with unit probability in all of the trajectories with τ_[14–38] $\approx$ 5,400 τ (Fig. S8C). With the caveat that the estimates for τ_[5–14] and τ_[5–38] are lower bounds, our simulations show that [14–38] forms considerably faster (with substantial yield) than the nonnative one disulfide intermediates (Fig. S8C).

The results for the WT and the three mutant simulations show that the rapid formation requires not only collapse of the protein but complete formation of the central β-hairpin and the helices. The very slow kinetics associated with [5–14] formation in the type III model also rules out any significant role such nonnative species play in the folding of BPTI. In addition, the present work shows in no uncertain terms the inevitability of protein collapse and [14–38] formation in guiding the folding of BPTI. We should emphasize that despite the important kinetic role that [14–38] plays in the wild-type BPTI folding, mutants (Cys to Ser) lacking these two cysteines can also reach the native state with altered kinetics (36). We performed simulations with $E_{SS}=0$ (Eq. 1) for only [14–38]. In accord with experiments, we find that this in silico mutant does reach the native by different pathways (compared with the WT) through a simpler network of states (Fig. S9).

Fig. S9.

Folding pathways for an in silico mutant created by setting $E_{SS}=0$ (Eq. S1) for only the [14–38] disulfide bond. Although the mutant does fold clearly, the resulting pathways bear no resemblance to the ones found for the wild type (Fig. 2).

Effect of Nonnative Disulfide Bond Formation.

To investigate how nonnative (NN) disulfide bonds affect the folding pathways, we performed simulations allowing for [S–S] bond formation between all of the Cys residues. In these simulations the criteria for rupture and formation of [S–S] bonds are identical regardless of whether native or nonnative disulfide bonds are involved. We obtained the flux diagram from 2,000 folding trajectories, and the results are shown in Fig. 2; additional consequences for the initial steps in the folding are shown in Fig. S10. There are two important lessons that come from these simulations. (i) The differences between these simulations and the ones based on the Go model arises only in the early stages of BPTI folding. In simulations with NN disulfide bond formation ∼43% of molecules form a mixture of various single NN disulfide species, such as [5–14] and [5–38]. Nearly 50% reach the native [14–38] state slightly ahead of all other NN single disulfide species (Fig. 2). All of the single disulfide species (native and nonnative) are present only after substantial collapse of BPTI (Fig. S10). These findings are in excellent agreement with theoretical predictions (5). Interestingly, the percentages of molecules that form various single disulfide species (Fig. 2) are in close agreement with experiments (26), which is remarkable given the simplicity of the model. (ii) There is virtually no difference between results obtained using the Go model and the one with NN disulfide bond formation in the flux through two disulfide species (Fig. 2). Thus, the major conclusions reached based on the Go model simulations remain valid when NN disulfide bond formation is allowed.

Fig. S10.

Kinetics of formation of [14–38] (black) and the entropically favored nonnative [5–14] (red). We also show the decay of $R_g$ in blue (scale on the right). The results show that both the native [14–38] and the nonnative [5–14] only form after considerable compaction of the polypeptide chain. In addition, even with nonnative interactions [14–38] forms slightly ahead of [5–14] with substantially greater yield. The simulations were performed using ${\beta }_O=1.0$ and ${\beta }_R=1.5$.

Discussion

Nonnative Interactions.

Our earlier works (5) showed that on times exceeding ${\tau }_c$, the population of nonnative species decrease, whereas those of native intermediates increase. The present simulations (using models with and without nonnative disulfide bonds) show that, before the first native disulfide intermediate ([14–38]) forms, BPTI adopts compact conformation, which allows us to focus on the network of connected states involving native-like native intermediates. Because the initial collapse is nonspecific, it follows that nonnative intermediates are likely formed stochastically (5, 12) (SI Text), limited only by topological restriction due to chain connectivity and stability, and do not direct folding. Beyond the collapse stage the dominant native interactions facilitate the formation of the folded state. In the process of rearrangement from N* and N′, compact intermediates containing nonnative species (Fig. 5) (24) are transiently populated. Lattice model simulations show that one or two nonnative contacts are likely to be part of the TSE even in the folding of proteins without disulfide bonds (37, 38). Recent analyses of atomically detailed simulations and theoretical arguments also suggest that in the folding of small proteins only native-like interactions dominate (39, 40).

Predictions.

We have made testable predictions. (i) If the stability of the β-hairpin is compromised by suitable mutations, then the formation of the crucial kinetic intermediate [14–38] is impeded. As a result nonnative intermediates (in particular, [5–14]) compete with the formation of [14–38], thus derailing efficient folding. (ii) For the wild-type BPTI, we predict that the yields and the relative rates of native single disulfide intermediates do not depend on the redox conditions (concentrations of GSSH and GSH, e.g.). However, the flux through other states can be dramatically altered, as shown in Fig. 2. (iii) If the central β-hairpin is destabilized, then the nonnative intermediates [5–14] and [5–38] and the native [14–38] form equally efficiently even though the probability of forming [5–14] is greater than [14–38]. Only upon destabilizing the two peripheral helices and the β-hairpin, the population of [5–14] greatly exceeds that of [14–38], as expected from polymer theory (5, 33).

Concluding Remarks

To provide a theoretical description of how protein folding drives disulfide bond formation, we developed a novel method to mimic the chemistry of disulfide bond formation and rupture in simulations using coarse-grained models. The simulations reproduce the experimentally inferred parallel pathways, involving flux through a network of connected native-like states. Our work explains all of the key features inferred from experiments. (i) The global folding pathways found in simulations are in near quantitative agreement with experiments. The flux through the network of native intermediates, including the rate-limiting step involving the formation of ${\mathbf{N}}_{SH}^{SH}$, is in excellent agreement with experiments. (ii) In the process of conversion of native-like N′ and N* to ${\mathbf{N}}_{SH}^{SH}$, compact nonnative species [30–51, 5–38] and [30–51, 5–14] are populated (24). (iii) The decrease in the rate of [14–38] formation, as the antiparallel β-sheet is destabilized (Fig. 6), supports mutation experiments reporting similar retardation in the rates (29). We also predict that the rate should increase upon stabilizing the β-hairpin. (iv) Inclusion of nonnative disulfide bonds improves quantitative agreement in the flux through native single disulfide species. However, qualitatively, all of the features are captured using a native-centric model.

The finding that polypeptide collapse should occur before disulfide bond formation is supported by single molecule pulling experiments (7), investigating the role protein disulfide isomerase (PDI) plays in catalyzing oxidative folding (41) of a broad class of disulfide proteins. Although derived in the context of disulfide-containing proteins, the link between collapse and folding is also applicable to the folding of globular proteins (42, 43).

Methods

Coarse-Grained Model.

In our CG Go-like model (44), each residue is represented as a single reaction center located at the C_α position. We use a harmonic potential for the covalent disulfide bonds (14, 45, 46) given by

$$E_{SS}=K_s{\left(r_{\alpha }-r_{\alpha 0}\right)}^2,$$ [1]

where $\alpha =\mathrm{1,2,3}$ labels the three disulfide bonds, $r_{\alpha }$ is the distance in the ${\alpha }^{th}$ [S–S] bond, and $r_{\alpha 0}$ is the equilibrium distance in the folded structure. The values of $r_{\alpha 0}$ ($\alpha =\mathrm{1,2,3}$) are 5.61, 5.67, and 6.26 (Å) for 5–55, 14–38, and 30–51, respectively, and $K_s$ = 20 ε${\AA }^{-2}$, where ε is the unit of energy. The energy function and the details of formation and rupture of [S–S] bonds are described in SI Text.

SI Text

Energy Function of Coarse-Grained Model for Proteins with Disulfide Bonds.

We use a C_α representation of a polypeptide chain. The energy of a given conformation, $\Gamma$ [specified by the coordinates $r_i$ (i = 1, 2, …, 58)], with 0 denoting the corresponding coordinates in the folded structure, is given by

$$\begin{array}{l}{E}_P\left(\mathrm{\Gamma },{\mathrm{\Gamma }}_0\right)={\displaystyle \sum _{i=1}^{N-1}{K}_r{\left(r_{i,i+1}-r_0\right)}^2}+\lambda {\displaystyle \sum _{i=1}^{N-2}{K}_{\theta }{\left({\theta }_i-{\theta }_{0i}\right)}^2}\\ +\lambda {\displaystyle \sum _{i=1}^{N-3}\left[K_1\left(1-\text{cos}\left({\varphi }_i-{\varphi }_{0i}\right)\right)+K_2\left(1-\text{cos}\left(3\left({\varphi }_i-{\varphi }_{0i}\right)\right)\right)\right]}\\ +E_{NON}+E_{SS},\end{array}$$ [S1]

where $r_{i,i+1}$ is the distance between two consecutive residues, ${\theta }_i$ and ${\varphi }_i$ represent the bond angle formed by three consecutive residues, and dihedral angle formed by four consecutive residues, respectively. The corresponding quantities in the native state ${\mathrm{\Gamma }}_0$ are represented as $r_{0i}$, ${\theta }_{0i}$, and ${\varphi }_{0i}$. The values of parameters $K_r=100$ $\epsilon /{\mathit{\AA }}^2$, $K_{\theta }=20\epsilon /ra{d}^2$, $K_1=\epsilon$ and $K_2=0.5$ ε, with ε being the unit of energy.

The term $E_{NON}$ in Eq. S1 includes both the native and nonnative contact interactions, and has the form of Lennard–Jones (LJ) potential

$$\begin{array}{c}{E}_{NON}=\lambda {\displaystyle \sum _{i<j-3}\{{\epsilon }_1\left(i,j\right)\left[5{\left({\sigma }_{ij}/r_{ij}\right)}^{12}-{\left({\sigma }_{ij}/r_{ij}\right)}^{10}\right]}\\ +{\epsilon }_2\left(i,j\right){\left({\sigma }_{ij}^{nn}/r_{ij}\right)}^{12}\}\end{array}$$. [S2]

If the distance between any pair of heavy atoms between residues i and j is within 5.0 Å in the native structure, then i and j are considered to form a native contact. For native contacts, we set ${\epsilon }_1\left(i,j\right)=\epsilon$ and ${\epsilon }_2\left(i,j\right)=0$, and for nonnative contacts (interactions between residues that are not in the native structure), ${\epsilon }_1\left(i,j\right)=0$ and ${\epsilon }_2\left(i,j\right)=\epsilon$. In Eq. S2, the parameter ${\sigma }_{ij}$ is the distance between residues i and j in the native state, and ${\sigma }_{ij}^{nn}=4.0$ Å corresponds to the range for nonnative repulsive interactions.

The dimensionless parameter λ (= 1.0 for WT) in Eqs. S1 and S2 controls the stability of secondary and tertiary interactions. Increase (decrease) in λ results in an increase (decrease) in the stability of the central antiparallel β-sheet (Fig. 1A).

Simulation Details.

We generated the folding trajectories of BPTI by integrating the Langevin equation under overdamped conditions. The equations of motion include a damping term with friction coefficient γ, a random force $F_r$ term and the conformation force $F_c=-{\nabla }_r{E}_P$ is $m\ddot{\overrightarrow{r}}=-\gamma \dot{\overrightarrow{r}}+F_c+F_r$, where $\dot{\overrightarrow{r}}$ is velocity, and m is the mass. The random force, $F_r$, is obtained from a Gaussian distribution with a white noise spectrum. The standard variance is related to temperature T as $\langle F_r\left(0\right)F_r\left(t\right)\rangle =2\gamma k_{B}T\delta \left(t\right)$, where $k_B$ is the Boltzmann constant, t is the time, T (measured in unit of ε) is the temperature, and $\delta \left(t\right)$ is the Dirac delta function. The equations of motion are integrated using the velocity form of the Verlet algorithm. The integration time step is taken to be $\delta t=0.005\tau$, where the unit of time $\tau =\sqrt{m{a}^2/\epsilon }$ ($a=5$ Å). The friction coefficient is taken to be $\gamma =50$, which describes overdamped motion.

A folding trajectory is generated for a maximum time ${\tau }_{max}=5\times {10}^4\tau$. If the three native disulfide bonds are formed and BPTI reaches the native state within ${\tau }_{max}$, then the simulation is terminated. The information of all of the intermediates ([5–55], [14–38], [30–51], N′, N*, $N_{SH}^{SH}$) can be obtained from the generated oxidative folding trajectories. To obtain statistically meaningful results, we generated 2,000 folding trajectories.

Because of the CG nature of the model, the conversion to real time (47) typically yields values that are less than measured in experiments. In the simulations of [S–S] bond this problem is exacerbated. The formation of [S–S] bond is a two-step process. In the first step, a mixed disulfide between glutathione disulfide and the thiol group forms, followed by an intramolecular rearrangement in which glutathione is displaced after [S–S] bond formation between two Cys residues. Capturing the time scales associated with this process and including the details of the reaction are beyond the scope of any simulation method. As a result, only relative times are meaningful.

Formation and Rupture of Disulfide Bonds.

To overcome the difficult problem of simulating the formation and rupture of disulfide bonds in a buffer containing GSSH and GSH, we use three criteria to assess if disulfide bonds can form. Before we describe our method of simulating disulfide bond formation and rupture we should point out that these processes could be simulated within the context of CG model by creating a double-well potential with the one minimum corresponding to [S–S] bond and the other to the ruptured state. It is likely that had we adopted this strategy the results would be qualitatively similar because in both methods disulfide bond formation is mimicked using short-ranged potentials. (i) Proximity: for an S–S bond to form, the two reactive thiol groups have to be in proximity. To ascertain an appropriate value for the proximal distance, we calculated the distribution of distances between the Cys residues that form S–S bonds in the native state by generating folding trajectories at $k_{B}T=0.9\epsilon$, where the folded state is stable. If the distance between a pair of native S–S bonds is less than the mean value (Fig. S1) plus the associated dispersions (e.g., $d_{5-55}<{\overline{d}}_{5-55}$ + ${\delta }_{5-55}$ for [5–55]), then the proximity criterion is satisfied. (ii) Orientation: disulfide bonds can form only if the Cys residues also have correct orientation. The details of the orientation criteria are given in SI Text. (iii) Solvent exposure: besides the purely geometric criteria described above, S–S bond formation also requires that the thiol groups be accessible to oxidizing agents for which a measure is needed to assess if they are sufficiently exposed to solvent. To estimate the solvent accessible surface area (SASA), we calculated the number density, $\rho =3n/\left(4\pi R_s^3\right)$, where n is the number of residues within a spherical shell with radius, $R_s$, drawn from the center of a given disulfide bond (Fig. S3). For $R_s=R_0=6.12$ Å, all three disulfide bonds in the native state have $n=n_0=8$, and $\rho ={\rho }_0=0.21$ Å⁻³ for oxidation. From the distribution of $n_{\alpha }$ values ($\alpha =\mathrm{1,2,3}$ corresponding to the 5–55, 14–38, and 30–51 S–S bond, respectively), we computed the mean values ${\overline{n}}_{\alpha }$ ($\alpha =\mathrm{1,2,3}$) and the dispersions ${\delta }_{\alpha }$ ($\alpha =\mathrm{1,2,3}$). We found that ${\overline{n}}_{\alpha }$ − ${\delta }_{\alpha }$ is 5.27, 6.84, and 6.56 for the 5–55, 14–38, and 30–51 S–S bond, respectively (Fig. S1). If i^th conformation satisfies both the proximity and the orientation criteria and if the surrogate for SASA, $n_{\alpha }^i<\hspace{0.17em}{\overline{n}}_{\alpha }$ − ${\delta }_{\alpha }$, then we assume that S–S bond can form provided the Cys residues are in the thiol state. If the conformation already has an S–S bond, then it is reduced.

If the three criteria above are satisfied, then the disulfide bond forms, stabilized by the harmonic potential given in Eq. 1. As the polypeptide chain evolves in time, as determined by Brownian dynamics (SI Text), disulfide bonds and other nonbonded interactions can form or rupture without prejudice until BPTI reaches the folded state.

Mimicking Redox Conditions.

If $n_{\alpha }^i$ exceeds ${\overline{n}}_{\alpha }$ − ${\delta }_{\alpha }$, then the region surrounding the i^th S–S bond is more compact than in the folded state. In this case, neither the oxidizing nor reducing agents can access the S–S bond or the thiol groups for chemistry to occur. The decision to allow S–S bond to form or be reduced (if one already exists) is made by comparing $n_{\alpha }^i$ and ${\overline{n}}_{\alpha }$ − ${\delta }_{\alpha }$. We mimic differing oxidizing conditions by comparing $n_{\alpha }^i$ and ${\overline{n}}_{\alpha }$-${\beta }_O{\delta }_{\alpha }$, where small ${\beta }_O$ corresponds to a strongly oxidizing condition and large ${\beta }_O$ to a weakly oxidizing condition. Similarly, reducing conditions are mimicked using the threshold values ${\overline{n}}_{\alpha }$-${\beta }_R{\delta }_{\alpha }$ where small (large) ${\beta }_R$ corresponds to strong (weak) reducing conditions. By varying ${\beta }_O$ and ${\beta }_R$ we can simulate different redox conditions, which affect the rates and yields of intermediates (27). By varying the range of [${\beta }_O,{\beta }_R$], we mimic the ability of oxidizing or reducing agents to access the thiol groups (Fig. 1C). A large value of ${\beta }_O$ (${\beta }_R$) implies the effective SASA associated with the S–S bond (thiol group) is small, which gives rise to ease of reduction (oxidation). For example, if the i^th conformation has [S–S] bond, and if $n_{\alpha }^i>{\overline{n}}_{\alpha }-{\beta }_R{\delta }_{\alpha }$, then this bond cannot undergo reduction. A similar criterion holds for oxidation. In bulk of the present work, we set ${\beta }_O=1$ and ${\beta }_R=1.5$, corresponding to strongly oxidizing conditions, approximately mimics the conditions in the endoplasmic reticulum (48). Note that the method of mimicking the chemistry of disulfide bond formation can also be used in atomically detailed simulations.

Generation of Unfolded Conformations.

The ensemble of initial conformations, R, in which all of the S–S bonds are reduced, is generated at a high temperature. These structures, with fraction of native contacts $Q<0.1$ (Fig. 1D) and average radius of gyration $R_g>20.0$ Å (Fig. S5), have low similarity to the native BPTI ($Q=1$) and contain no disulfide bonds. To obtain statistically meaningful results we used Brownian dynamics (47) and generated 2,000 folding trajectories each for a time ${\tau }_{max}=5\times {10}^4\tau$ (described in detail in SI Text).

Implementation of Oxidation and Reduction of Disulfide Bonds.

The main text describes how formation and rupture of disulfide bonds is mimicked for a given redox conditions specified by ${\beta }_O$ and ${\beta }_R$ values (Fig. 1C). In practice, we calculate $n_{\alpha }^i$ in trajectory i at various times by a time average over a window $\mathrm{\Delta }$t = τ using

$$n_{\alpha }^i\left(t\right)=\frac{1}{\mathrm{\Delta }t}{\displaystyle {\int }_t^{t+\mathrm{\Delta }t}{n}_{\alpha }^i\hspace{0.17em}\text{d}s}.$$ [S3]

To assess if the oxidizing or reducing agents can access the S–S or the thiol groups that are in proximity, we calculated the distance between all of the C_α atoms within the sphere $R_0$ using

$$R_{\alpha }^i\left(t\right)=\frac{1}{\mathrm{\Delta }t}{\displaystyle {\int }_t^{t+\mathrm{\Delta }t}\frac{2}{n\left(n-1\right)}}{\displaystyle \sum \left|r_{ij}\right|\left(s\right)\text{d}s},$$ [S4]

where $\left|r_{ij}\right|$ is the distance between beads i and j, and n is the number of beads within $R_0$. If $R_{\alpha }^i<R_0$, then a preformed disulfide bond is reduced. The condition in Eq. S4 is applied only if $n_{\alpha }^i\left(t\right)>{\overline{n}}_{\alpha }$ − ${\beta }_R{\delta }_{\alpha }$.

Rearrangement of [14–38].

Fig. S4A shows that during the rearrangement process the distance between [14–38] increases, and simultaneously residues 30 and 51 come into proximity, enabling them to establish the S–S bond. Qualitatively similar results are found for the [14–38] → [5–55] transition (Fig. S4B). Fig. S4 A and B shows that during rearrangement an ensemble (R′) of structured conformations with intact antiparallel β-sheet as well as the peripheral helices (α₁ and α₂ in Fig. 1A) are sampled. The R′ ensemble in Fig. S4 A and B), devoid of any S–S bond, are compact. The distributions of radius of gyration, $R_g$, of the R′ ensemble are peaked around 15 Å (Fig. S4C for [14–38] → [30–51] and Fig. S4D for [14–38] → [5–55]), whereas $R_g$ for the fully unfolded BPTI from which folding is commenced exceeds 20 Å (Fig. S5).

Structures Sampled During Oxidative Folding.

The set of conformations of various disulfide species (Fig. 2) show that even the compact single disulfide species have considerable similarity to N. At the earliest stages of folding, secondary structures and even tertiary interactions are established. Comparison of [5–55] and the two-disulfide [14–38, 30–51] with N shows great similarity (Fig. S7). This conclusion is supported by NMR experiments, which showed that both [5–55] (49) and N′ (50) bear great resemblance to N. The good comparison between simulations and experiments allows us to assert that the conformations sampled in folding kinetics and in equilibrium are very similar.