Flexibly varying folding mechanism of a nearly symmetrical protein: B domain of protein A

Abstract

The folding pathway of the B domain of protein A is the pathway most intensively studied by computer simulations. Recent systematic measurement of Φ values by Sato et al. [Sato, S., Religa, T. L., Daggett, V., & Fersht, A. R. (2004) Proc. Natl. Acad. Sci. USA 101, 6952–6956], however, has shown that none of the published computational predictions is consistent with the detailed features of the experimentally observed folding mechanism. In this article we use a statistical mechanical model of folding to show that sensitive dependence of multiple transition state ensembles on temperature and the denaturant concentration is the key to resolving the inconsistency among simulations and the experiment. Such sensitivity in multiple transition state ensembles is a natural consequence of symmetry-breaking in a nearly symmetrical protein.

In the last decade we have witnessed a significant advance in protein folding study. Combined efforts of theoretical, computational, and experimental researches have led to a new picture that regards protein folding as diffusive motions on the energy surface of conformational change (1–3). From this picture folding is a convergent process of a variety of different stochastic trajectories to the unique native conformation, and the experimentally observed folding pathway should correspond to an ensemble of trajectories that have a relatively high probability of realization.

Computer simulations have played a key role to identify such folding pathways of various proteins (4–6). Among those proteins the B domain of Staphylococcal protein A (BdpA) has been most intensively studied. As shown in Fig. 1, BdpA is a 60-residue, three-helical protein, with Helix1 (H1, residues 10–19), Turn1 (T1, residues 20–24), Helix2 (H2, residues 25–37), Turn2 (T2, residues 38–41), and Helix3 (H3, residues 42–56). BdpA folds according to simple two-state kinetics (7) with a half-life time of microseconds (8). Because of its fast kinetics, BdpA has been a target of many simulation studies (9–22), which has made BdpA a unique system: Comparing accumulated simulation data with experiments, the folding mechanism of the protein should be analyzed in depth.

Fig. 1.

Native structure and contact map of the B domain of protein A (Protein Data Bank entry 1BDD).

Recently, Sato et al. (23) experimentally measured Φ values of BdpA in a systematic way and assessed predictions of pathway made by simulations. The assessment was also summarized from the theoretical point of view in the commentary paper of Wolynes (24). Simulations have given good predictions of the two-state kinetics, and of the concomitant formation of secondary and tertiary structures, but differed from the experiment in detailed points. The experimental data showed that in the transition state ensemble (TSE), H2 is fully formed whereas H1 is partially formed and H3 is only weakly formed. T1 is unstructured whereas T2 has some structure in TSE. Predicted pathways, on the other hand, are rather diverse. Boczko and Brooks (9) concluded that starting around the T1 region, interactions between H1 and H2 are formed at the early stage, and structures of H3 and the tertiary interaction between H2 and H3 are formed later. Garcia and Onuchic (12) derived similar results: T1 and H2 are formed in TSE. Although H3 is formed at the earlier stage, the H2–H3 interaction and the H1–H3 interaction are formed after passing the transition state. Other groups (13–22) put more emphasis on the role of H3: they predicted that H3 is formed earlier and H1 and H2 follow. The early formation of H3 looks consistent with the experimental results that the isolated fragment of H3 forms a more stable helix than the isolated fragment of H1 or that of H2 (7, 25, 26), but it disagrees with the observed Φ values of BdpA. Thus, none of the published predictions of pathway seems fully consistent with the experimental observation (23).

We should ask how such discrepancy between simulations and the experiment is remedied to have a consistent picture of folding of BdpA. Do we have to use a more realistic simulation model with more accurate atomistic interactions to get reliable results? In this article we take a different approach from such detailed modeling to show that a simple coarse-grained model can reveal the reason of discrepancy. With the simplified model, the free energy surface of the conformational change can be readily drawn in many different situations, which should facilitate the capture of the folding mechanism in a transparent way.

Emphases will be laid on two points. First, multidimensionality of the energy landscape has to be explicitly considered. Folding is originally a process in the extremely large dimensional space of conformational change, but often it has been useful to project the process onto some one-dimensional reaction coordinate. Here, we should point out that the different physics is shed light on by departing from the picture of one-dimensional coordinate. Second, care should be taken in the definition of the reaction coordinate. Different variables such as the native likeness in the backbone configuration, the number of native contacts between residues, and Φ values should describe different aspects of the structure formation process, respectively, so that pathway descriptions with different variables cannot be compared immediately. We show that by keeping these two points in mind, a simplified statistical model based on the minimum frustration principle (1) can describe the folding mechanism of BdpA in a consistent way. This consistent picture obtained from a simplified model should provide a strategy to describe BdpA or other more complex proteins with realistic simulations or from the experimental data.

As a coarse-grained variable, it is convenient to use m_i = 1 or 0, which represents the configuration at the ith residue: m_i takes unity when two dihedral angles of the backbone at the ith residue are within some narrow range around values in the native state conformation, and zero otherwise. Using a variable thus defined, Wako and Saito (27, 28) described folding pathways of proteins with the following Hamiltonian:

where the summation is taken over native pairs, [i, j] with i < j. The pair of ith and jth residues is defined to be the native pair when they are close in space in the native conformation. See Methods for the precise definition of the native pair. When the backbone of the segment from i to j takes the native configuration as m_ij ≡ m_i m_i+1 … m_j-1m_j = 1, then the native pair i and j should have a large chance to come close to each other to form the native contact. ε > 0 is the energy gain of forming a native contact. Thus, the native segment of m_ij = 1 is sufficiently stabilized in energy compared with the nonnative segment of m_ij = 0, so that H({m_k}) represents the energy bias of the conformational change to the native conformation. Nonnative contacts between residues are not counted in energy, so that Eq. 1 is a Go-type Hamiltonian (29, 30) with minimal frustration. Much later the same Hamiltonian was used by Muñoz and Eaton (31, 32) to analyze free energy landscapes and rates of folding of many proteins. Good agreement of their results with observed data convinced us that at least in the first approximation proteins are minimally frustrated.

The partition function is given by Z = Σ_conf A({m_i})e^−βH with A({m_i}) = exp[(S₀ − Σ_i=1^Nσ_im_i)/k_B], where Σ_conf denotes the summation over the set of configurations, {m_i}, N = 60 is the total number of residues in BdpA, and β = 1/k_BT with temperature T. S₀ is the entropy of the fully unfolded chain but does not appear in the results. σ_i > 0 is the entropic cost arisen from the ith residue that takes the native configuration. The partition function can be exactly calculated by using the transfer-matrix method without introducing any further approximation (33). The relevant parameters of the model are λ ≡ ε/k_BT and σ_i. Small λ implies weakening of the contact due to the addition of denaturant or increase in temperature and will be discussed in detail in the next section. Values of σ_i are given in Methods.

Results and Discussion

Two-State Transition.

Two-state behavior of folding is evident when we plot the average fraction of native contacts, 〈Q〉 = (1/n_c)∂(lnZ)/∂λ, as a function of λ as in Fig. 2A, where n_c = Σ_[i,j]1 is the total number of native pairs. 〈Q〉 shows a sigmoidal curve from the denatured state (D) at small λ to the native state (N) at large λ. The midpoint of the D–N transition is at λ = λ_c = 0.472. This two-state behavior can be more clearly seen by drawing free energy profiles, F(M) and F(Q), as in Fig. 2 B and C. Here, M is the number of residues that take the native configuration, M = Σ_im_i, and Q is the fraction of native contacts, Q = 1/n_c Σ_[i,j]m_ij. F(M) was calculated from the constrained partition function, Z_M = Σμ_(M)Ae^−βH, as F(M) = −k_BTlnZ_M, where Σμ_(M) is the sum over the constrained configurations, μ(M) = {m_i|Σ_im_i = M}. F(Q) was calculated in the same way as F(Q) = −k_BTlnZ_Q with Z_Q = Σμ_(Q)Ae^−βH, and μ(Q) = {m_i|1/n_cΣ_[i,j]m_ij = Q}. There are free energy minima at M_N ≈ 55 and M_D ≈ 18 in Fig. 2B, and at Q_N ≈ 1 and Q_D ≈ 0 in Fig. 2C, which confirms the two-state feature of the transition. Locations of the free energy maxima, M^‡ and Q^‡, shown in the figure are (M^‡ − M_D)/(M_N − M_D) ≈ 0.6 and (Q^‡ − Q_D)/(Q_N − Q_D) ≈ 0.3–0.5.

Fig. 2.

Two-state transition behavior. (A) Average fraction of native contacts 〈Q〉 as a function of λ ≡ ε/k_BT). (B) Free energy surfaces, F(M), for several values of λ (real lines) and the average fraction of native contacts 〈Q〉_M (dashed line) as a function of the reaction coordinate, M. (C) Free energy surfaces, F(Q), for several values of λ (real lines) and the average number of residues having the native backbone conformation 〈M〉_Q (dashed line) as a function of the reaction coordinate Q. In B and C, free energy surfaces are drawn for λ = 0.536 (a), 0.500 (b), 0.472 (c), and 0.445 (d).

The equilibrium populations of native and denatured states should be given by Z_N/Z and Z_D/Z, respectively, where Z_N = Σ_M<M^‡Z_M and Z_D = Z − Z_N. Then, difference in free energy between the native and denatured states, ΔF_DN, is calculated as ΔF_DN = k_BTln(Z_N/Z_D). At the midpoint of the D–N transition, λ = λ_c = 0.472, ΔF_DN/k_BT ≈ 0 giving equal native and denatured populations. Quantitative interpretation of λ can be made by comparing the value of calculated ΔF_DN with the experimentally observed ΔF_DN at T = 298 K (estimated from figure 3b of ref. 23): λ = 0.536 in 0 M guanidinium chloride (GdmCl) (ΔF_DN ≈ 8.4 k_BT), λ = 0.500 in 2 M GdmCl (ΔF_DN ≈ 3.7 k_BT), and λ = 0.445 in 5 M GdmCl (ΔF_DN ≈ −3.5 k_BT). The activation free energy of folding, F(M^‡) − F(M_D), is about a few k_BT for λ = 0.536 (0 M GdmCl), which is consistent with the experimentally observed rates of folding (23).

Multiple TSEs and Multiple Pathways.

The complex folding process can be analyzed by drawing the free energy surface in two-dimensional space. We define the two-dimensional coordinates, M_N and M_C, as M_N = Σ_i=1^N/2m_i and M_C = Σ_i=N/2+1^Nm_i with 0 ≤ M_N ≤ N/2 and 0 ≤ M_C ≤ N/2. Free energy is given by F(M_N, M_C) = −k_BT lnZ_MM,MC with the constrained partition function Z_MN,MC = Σ_μ(MN,MC)Ae^−βH, where the summation is taken over the conformations with constraints, μ(M_N, M_C) = {m_i|Σ_i=1^N/2m_i = M_N, Σ_i=N/2+1^Nm_i = M_C}. Free energy surface is shown in Fig. 3A–D for several values of λ. In F(M_N, M_C) a long ridge that separates basins of the folded and unfolded states has two saddle points. A straightforward assumption is that conformations designated by these two saddle points constitute two TSEs. In Fig. 3A two TSEs, TS1 at (M_N, M_C) = (24, 14) and TS2 at (M_N, M_C) = (12, 23), and dominant pathways passing them are shown for λ = 0.536 (0 M GdmCl). TS1 has the lower free energy than TS2, so that the pathway passing TS1 should be more important. In Fig. 3E the probability of the native contact formation between the ith and jth residues, 〈m_ij〉_MN,MC, at each TSE for λ = 0.536 is shown, where 〈···〉_MN,MC is the average calculated from Z_MN,MC and m_ij = m_ji. As λ changes, landscape of F(M_N, M_C) varies and the relative weights of TSEs are altered: For λ = 0.500 (2 M GdmCl), TS1 at (M_N, M_C) = (25, 15) and TS2 at (M_N, M_C) = (15, 25) have almost the same free energy, so that pathways passing each transition state should be nearly equally populated (Fig. 3B). For λ = λ_c and λ = 0.445 (5 M GdmCl), TS2 has lower free energy and should be more important than TS1 (Fig. 3 C and D).

Fig. 3.

Free energy surfaces and structures at saddle points. (A–D) Free energy surfaces in the two-dimensional reaction coordinate (M_N, M_C) for λ = 0.536 (A), 0.500 (B), 0.472 (C), and 0.445 (D). Gray and white arrows for λ = 0.536 denote the dominant folding pathways passing through the saddle points (TSEs), TS1 and TS2, where TS1 is at (M_N, M_C) = (24, 14), and TS2 is at (M_N, M_C) = (12, 23). A black arrow in D is a dominant unfolding pathway passing through TS2 for λ = 0.445. (E) The probabilities of contact formation 〈m_ij〉_MN,MC at saddle points TS1 (lower triangle) and TS2 (upper triangle) for λ = 0.536 (0 M GdmCl).

It should be noted that many pathways are possible on the two-dimensional surface and the statistical weight may be broadly distributed among them. Relatively highly populated pathways among them, however, should pass through TS1 and TS2. Examples of such pathways passing TSEs are “Pathway-F” shown by the gray line in Fig. 3A for λ = 0.536 (0 M GdmCl) and “Pathway-U” shown by the black line in Fig. 3D for λ = 0.445 (5 M GdmCl). Along those exemplified pathways, the structural formation/destruction process is described in Fig. 4A and B in terms of the average, 〈m_i〉_MN,MC, and the response to the perturbation of interactions involving the ith residue, q_i(M_N, M_C). See Methods for the detailed definition of q_i(M_N, M_C). The coordinate along each pathway is defined by ξ ≡ {(M_N, M_C) ∈ Pathway}, where ξ is 0 ≤ ξ ≤ 1 with ξ = 1 for the structured state and ξ = 0 for the fully unfolded state, and 〈m_i〉_MN,MC and q_i(M_N, M_C) are hereafter denoted by 〈m_i〉_ξ and q_i(ξ), respectively. 〈m_i〉_ξ and q_i(ξ) show the same sequence of the secondary-structure formation or destruction. Along Pathway-F the order of the structure formation is H2 → H1 → H3. Along Pathway-U the order of the structure destruction is H1 → H2 → H3. TS1 is the free energy maximum on Pathway-F, where the secondary structure of H1 and H2 and contacts between these helices are formed. TS2 is the free energy maximum on Pathway-U, where H2 and H3 are folded and the interhelix contacts are weakly formed at TS2. For the structure formation at turns, however, 〈m_i〉_ξ and q_i(ξ) give different results: 〈m_i〉_ξ shows that T1 folds concurrently with H1 and T2 folds concurrently with H3, whereas q_i(ξ) shows that contacts of T1 in Pathway-F or contacts of T2 in Pathway-U are not constructed at each transition state. q_i(ξ) shows that T1 forms when the C-terminal part of H3 is structured, and T2 forms when the N-terminal part of H1 is structured. These are because 〈m_i〉_ξ represents the backbone structure, whereas q_i(ξ) focuses on the native contacts: if partner residues in contacts are unstructured, q_i(ξ) is reduced. The contact map of Fig. 1 shows that T1 has many contacts with the C-terminal part of H3 and T2 has contacts with the N-terminal part of H1, which explains behaviors of q_i(ξ) on turns.

Fig. 4.

Comparison between structure changes along dominant pathways. (A) 0 M GdmCl (λ = 0.536). (B) High denaturant concentration 5 M GdmCl (λ = 0.445). q_i(ξ) is the response to the perturbation of interactions and 〈m_j〉_ξ is the probability that backbone dihedral angles are close to the native values for each residue j as a function of the reaction coordinate ξ, where ξ is defined along the pathways drawn with a gray line in Fig. 3A and black line in Fig. 3D, ξ = 1 for the structured state and ξ = 0 for the fully unfolded state. The free energy profile along the corresponding pathway F(ξ)/k_BT is shown on the left.

Sensitivity of the statistical weight of each pathway and TSE to temperature or the denaturant concentration should lead to the flexible alteration of the folding/unfolding mechanism. Under the condition of high temperature or high denaturant concentration, pathways passing TS2 are more important, which leads to the unfolding scheme that H1 is firstly unstructured, H2 is secondly unfolded loosing the H2–H3 interactions, and H3 remains weakly formed in the denatured state. This result is consistent with the all-atom simulation of unfolding at high temperature (15). In the folding process at the temperature of the folding transition or under the critical denaturant concentration, H2 and H3 and its interhelix contacts are formed at first and the formation of H1 follows. Under the physiological condition without denaturant, H2 forms early, H1 follows, H3 folds later, and the H2–H3 interactions are formed at the last stage. This sensitivity of the mechanism to temperature and denaturants implies that much care should be taken so that comparison between simulation and experiment is made on the same condition.

Comparison with Experimental Results.

The above discussion on the sensitivity of the folding mechanism is based on the assumption that most populated pathways should pass through the saddle points located on the ridge of free energy. Under this assumption, kinetics of folding depends sensitively on the relative free-energy height of saddle points. To verify this hypothesis, we estimate Φ values based on the free energy height of saddle points and compare them with the observed ones. See Methods for the details of the approximation used to calculate Φ values.

The calculated Φ values are compared with the observed values for 0 M GdmCl in Fig. 5A and for 2 M GdmCl in Fig. 5B. For 0 M GdmCl the calculated Φ values are high (0.5–0.9) in H2, middle (0.3–0.5) in H1, and <0.3 in H3. For 2 M GdmCl the calculated Φ values are lower in H1 and higher in H3 than those for 0 M GdmCl: 0.2–0.4 in H1, 0.5–0.9 in H2, and 0.2–0.5 in H3. In the T1 region the theoretical Φ values are zero or very low for both 0 and 2 M GdmCl conditions. Small Φ values at T1 show that the folding mechanism inferred from Φ values is close to the one inferred from q_i(ξ) along the representative pathway but is different from the one obtained from 〈m_i〉_ξ. Such disagreement between the Φ values and 〈m_i〉_ξ gives caution in choosing variables to monitor the folding mechanism in simulations.

Fig. 5.

Comparison between observed and theoretical Φ values. (A) 0 M GdmCl (λ = 0.536). (B) 2 M GdmCl (λ = 0.500). Lines represent values calculated with the approximate relations explained in Methods, and green filled squares are observed values. Observed values include both for mutations to glycine to scan tertiary structure and turns (tables 2 and 3 of ref. 23) and for the Ala → Gly scanning at helix surface (table 1 of ref. 23). (C) Predicted Φ values in the unfolding process for 5 M GdmCl (λ = 0.445). The gray line is φ_U = 1 − φ.

The calculated Φ values in Fig. 5 A and B give globally qualitative and partially quantitative agreement with the observed Φ values. Agreement between the calculated and observed values is better in the case of 2 M GdmCl, for which ΔΔG_DN were most precisely measured, so that the experimental data for 2 M GdmCl are more reliable than those for 0 M GdmCl (23). This agreement between the calculated and observed Φ values confirms the validity of the assumption of the kinetic importance of saddle points.

Although Sato et al. (23) wrote that there are no appreciable difference between the data for 0 M GdmCl and those for 2 M GdmCl, a slight rise in observed Φ values in H3 for 2 M GdmCl can be recognized from the data. This increase of Φ values in H3 should be due to the shift in weight of pathways, which is reproduced by our calculation. This tendency of increase in Φ values becomes evident when the system is in the unfolding condition of 5 M GdmCl. Fig. 5C shows the prediction of Φ values in the unfolding process for 5 M GdmCl with high Φ values in H3 and low in H1.

Mechanism of Pathway Change.

Position and height of saddle points of the free energy surface are determined by the balance between energy and entropy. Fig. 6A shows energy, E(M_N, M_C)/k_BT and Fig. 6B shows the relative entropy to the fully unfolded state, ΔS(M_N, M_C)/k_B, which takes account of both the entropy decrease at residues taking the native configuration and the entropy of mixing of native and nonnative residues along the chain. Free energy is F(M_N, M_C)/k_BT = E(M_N, M_C)/k_BT − (S₀ + ΔS(M_N, M_C))/k_B. Saddle points, TS1 and TS2, are the positions where the slope of E(M_N, M_C)/k_BT is equal to the slope of ΔS(M_N, M_C)/k_B, so that their positions are determined by the subtle functional form of E(M_N, M_C)/k_BT and ΔS(M_N, M_C)/k_B. ΔS(M_N, M_C)/k_B is almost perfectly symmetric in the two-dimensional space of M_N and M_C by definition in the model. E(M_N, M_C)/k_BT is also nearly symmetric due to the nearly symmetrical topology of BdpA. There is, however, a small asymmetry in E(M_N, M_C)/k_BT due to the difference between the H1–H2 interactions and the H2–H3 interactions. As shown in Fig. 1 the number of native pairs is larger in the H1–H2 interactions than in the H2–H3 interactions, and the relative weakness in interactions between H2 and H3 is compensated for by the T1–H3 interactions. This asymmetry gives rise to the shift of the location of TS1 to the lower energy side than TS2. Fig. 6C shows the average fraction of native contacts, 〈Q〉_MN,MC for λ = 0.536. The shift of the location of TS1 to the lower energy side results in the larger 〈Q〉_MN,MC at TS1 than at TS2. Also for 0.445 ≤ λ < 0.536, 〈Q〉_MN,MC at TS1 is larger than at TS2 for the same reason. Since free energy at λ + Δλ is F(M_N, M_C)/k_BT|_λ+Δλ = F(M_N, M_C)/k_BT|_λ − n_c〈Q〉_MN,MC Δλ + O((Δλ)²), free energy of TS1 depends more sensitively on λ than free energy of TS2. Thus, a small asymmetry in interactions results in different dependence of TSEs on temperature or the denaturant concentration, leading to the flexible variation of the folding mechanism, which explains the diversity among predictions made by simulations and the seeming discrepancy between simulations and the experiment.

Fig. 6.

Decomposition of the free energy to energy and entropy. Energy E(M_N, M_C)/k_BT for λ = 0.536 (A), relative entropy to the fully unfolded state ΔS(M_N, M_C)/k_B (B), and average fraction of native contacts 〈Q〉_MN,MC for λ = 0.536 (C) are shown as functions of M_N and M_C.

Symmetry-Breaking: A Guideline to Describe Folding Process in the Multidimensional Space.

There are an abundance of proteins having symmetrical native structures. Importance of symmetry in designing the energy landscape of folding was discussed (34, 35). The mechanism of flexible variation of folding kinetics discussed in here should not be a specific feature of BdpA but might be applicable to generic proteins which have symmetrical or nearly symmetrical topologies (36, 37). When there are K local regions within which residues are strongly interacting with each other, cooperativity within each region should favor the two-state-like local structure formation and destruction. With such local cooperativity there should be less chance of the concurrent structural formation of all of K regions, but independent structure formation at individual regions should be expected. Therefore, even in a symmetrical protein a structure on the folding pathway should have an asymmetrical structure. In this way the original symmetry of protein topology is broken in the realization of folding process. Thus, in the nucleation-condensation mechanism of folding (3), the symmetry-breaking should result in multiple ways of nucleation. BdpA is the case of K = 2 with the H1–H2 interaction region and the H2–H3 interaction region. In this case, drawing the two-dimensional free energy surface is suitable to distinguish two ways of nucleation, which lead to two pathways to the native state. For the case of K > 2, in general, the higher dimensional representation should be necessary to analyze multiple ways of nucleation and resultant multiple pathways.

Folding along each of the broken-symmetry pathways can be observed in single-molecule measurements. It would be interesting to see how the broken-symmetry pathways and TSEs are observed when the starting initial condition is controlled with the single-molecular manipulation. In the macroscopic measurement, on the other hand, only the average of many pathways is observed. When the protein has a perfectly symmetrical structure, broken-symmetry pathways should be equally realizable, and the symmetry is recovered in macroscopic observation. A slight difference among K regions, however, should enhance the statistical weight of specific pathways, which are observable in the macroscopic measurement. The way of enhancement should depend on temperature, solvation, and the sequence design, so that a perturbation of the folding condition or small-scale mutations should flexibly alter the enhanced pathway from one to the other. Thus, symmetry-breaking in a nearly symmetrical protein is the origin of the flexibly varying folding mechanism. This symmetry-breaking and the resultant multiple pathways should become more evident in proteins having multiple domains, and we can regard the multiple pathways in modular repeat proteins (38, 39) as its extreme examples.

Studies on BdpA have shown that the free energy surface contains rich structures, and those multidimensional structures in the folding process of single or multiple domain proteins should be further explored by comparing simulations and experiments.

Methods

Native Pairs and Parameters in the Wako–Saito–Muñoz–Eaton Model.

We define the pair of ith and jth residues as the native pair, [i, j] when an atom in the ith residue and an atom in the jth residue are closer than 4 Å and j > i + 2. Fig. 1 shows the contact map of the native structure of BdpA (Protein Data Bank entry 1BDD) in which [i, j] thus defined are designated by black squares. We assume that σ_i = 0.5 k_B for coil regions and σ_i = 1.0 k_B in other regions, where the coil regions are the N- and C-terminal parts, 1 ≤ i ≤ 9 and 58 ≤ i ≤ 60. Results are insensitive to changing values of σ_i when λ is appropriately adjusted.

Response to the Perturbation of Interactions.

Response to the perturbation of interactions with respect to the lth residue, q_l(M_N, M_C), is defined as follows. Energy is modulated as ΔH(l) = −ΔεΣ_[i,j]δ(i, l)m_ij by changing the strength of interactions that involve the lth residue from ε to ε + Δε, where δ(i, l) is the Kronecker delta with δ(i, l) = 1 when i = l and δ(i, l) = 0, otherwise. Then, q_l(M_N, M_C) is defined by q_l(M_N, M_C) = ΔF_l(M_N, M_C)/[−ΔεΣ_[i,j]δ(i, l)] with ΔF_l(M_N, M_C) = −k_BTln〈exp[−βΔH(l)]〉_MN,MC. We use βΔε = −0.1 to draw Fig. 4. For β|Δε| ≪ 1, q_l(M_N, M_C) can be written as q_l(M_N, M_C) = Σ_[i,j]〈m_ij〉_MN,MCδ(i, l)/Σ_[i,j]δ(i, l), which is a fraction of native contacts involving the lth residue.

Estimation of Φ Values.

Φ value is defined by φ = k_BTln[k_f(wt)/k_f(mu)]/ΔΔG_DN, where k_f(wt) and k_f(mu) are the rate constant of folding of the wild-type protein and that of a mutant protein, respectively, and ΔΔG_DN is the difference between the free energy to stabilize the native state of the wild-type protein and that of a mutant, ΔΔG_DN = ΔF_DN(mu) − ΔF_DN(wt). For the mutation at the lth residue, energy is modulated as ΔH(l), so that one can approximate ΔΔG_DN ≈ |Δε|Σ_[i,j]δ(i, l). We assume that the ratio of the rate constants is determined by the free energy height of saddle points as

where “TSEs” denotes a set of saddle points that should correspond to TSEs. βΔε = −0.1 was used to calculate Φ values of Fig. 5. For β|Δε| ≪ 1, Φ value can be reduced to φ(l) = Σ_{(MN,MC)∈TSEs}α(M_N, M_C)q_l(M_N, M_C) with α(M_N, M_C) = Z_MN,MC/Σ_{(MN,MC)∈TSEs} Z_MN,MC.

Abbreviations

BdpA

B domain of Staphylococcal protein A

GdmCl

guanidinium chloride

TSE

transition state ensemble

Note Added in Proof.

Two papers on the simulation of protein A have appeared recently (40, 41). These works also show diversity among simulation results.