Aborted double bicycle-pedal isomerization with hydrogen bond breaking is the primary event of bacteriorhodopsin proton pumping

Abstract

Quantum mechanics/molecular mechanics calculations based on ab initio multiconfigurational second order perturbation theory are employed to construct a computer model of Bacteriorhodopsin that reproduces the observed static and transient electronic spectra, the dipole moment changes, and the energy stored in the photocycle intermediate K. The computed reaction coordinate indicates that the isomerization of the retinal chromophore occurs via a complex motion accounting for three distinct regimes: (i) production of the excited state intermediate I, (ii) evolution of I toward a conical intersection between the excited state and the ground state, and (iii) formation of K. We show that, during stage ii, a space-saving mechanism dominated by an asynchronous double bicycle-pedal deformation of the C10═C11─C12═C13─C14═N moiety of the chromophore dominates the isomerization. On this same stage a N─H/water hydrogen bond is weakened and initiates a breaking process that is completed during stage iii.

The light-activated proton pump bacteriorhodopsin (bR) is an Archaea receptor contained in the purple membrane of Halobacterium salinarium. As shown in Scheme 1, upon photoexcitation, the all-trans retinal chromophore (PSBT) bounded to Lys216 via a protonated Schiff base linkage is converted to the 13-cis isomer (PSB13). This event triggers a series of conformational changes that ultimately result in a proton translocation from the cytoplasmic to the extracellular domain (1).

Scheme 1.

The bR protein environment affects both the spectroscopic and photochemical properties of PSBT (2–4). First, the absorption maximum (568 nm) is red-shifted with respect to the one observed in solution (440 nm in methanol). Second, time-resolved spectroscopy reveals a dominant excited state lifetime component of 450 fs (5, 6) almost matching the 500-fs time scale for formation of the vibrationally hot primary photoproduct J (7, 8). In contrast, PSBT in solution features a biexponential decay dynamics with a dominant (ca. 10-fold longer) 2-ps component (9–11). Finally, although bR photoisomerizes stereoselectively (leading exclusively to PSB13) with high (ca. 67%) quantum yield (9, 12), irradiation of PSBT in solution leads to a mixture of different stereoisomers with a smaller (ca. 25%) total quantum yield (9, 13).

Low-temperature spectroscopic studies provided evidence for the existence of a tiny (≤ 1 kcal mol^-1) energy barrier on the excited state (S₁) potential energy surface of bR (14, 15). Such a barrier would explain the 450-fs short-lived quasistationary state observed by Ruhman et al. (5) that is assigned to the fluorescent state I (16–18) and precedes decay to the ground state (S₀). Recent experiments by Léonard et al. (19) have also revealed that the photoinduced changes in the permanent dipole moment of PSBT are mirrored by the absorption of Trp86 up to the first stable (i.e., cryogenically trapped) photoproduct K (7). Relative to bR, K bears 11.7–15.9 kcal mol^-1 stored photon energy (20).

The investigation of the photoisomerization mechanism of bR is of great interest because this receptor is the archetypal ion pump machine. Its comprehension may open previously undescribed perspectives for the use of bR in nanotechnology or for the design of unique molecular devices (21, 22). In recent years numerous theoretical works, involving various hybrid quantum mechanics/molecular mechanics (QM/MM) approaches, have investigated the tuning of the spectroscopic properties of the retinal chromophore (23–28), the mechanism, stereoselectivity, and dynamics of its photoisomerization (29–32), and the energy storage (33). However, these studies have not yet delivered a coherent mechanistic scenario. For instance, the counterion effects on the spectroscopy of PSBT, as well as the mechanism of the PSBT space-saving photoisomerization, are not resolved.

Our recent work on the retinal chromophore in vacuo (34, 35) shows that the CASPT2//CASSCF/6-31G* protocol yields balanced values for geometrical parameters, energy differences, spectroscopic properties, and change in dipole moment. Most important, it delivers an accurate description of the photochemically relevant potential energy surfaces (PESs). The same protocol has been successfully coupled with the AMBER molecular mechanics force field to investigate the photoisomerization of the visual pigment rhodopsin (Rh) (4, 36), returning energies that agree with the experiments. These studies support a two-state model for the retinal chromophore isomerization: the S₁ path develops entirely along a charge transfer (i.e., hole-pair 1B_u-like) state and ends into a S₁/S₀ conical intersection (CI) where the reacting double bond features a ca. -90° twisted structure. Because of its geometrical and electronic structure, this point is consistent with that of a twisted intramolecular charge transfer (TICT) state (34, 37). The analysis of the S₁ isomerization coordinate along the minimum energy path reveals that this is sequentially dominated by different, almost uncoupled, modes. First, a stretching mode leading to single-bond/double-bond order inversion occurs. Second, a one-bond-flip (38, 39) twisting of the reacting double bond is activated in vacuo, whereas in the visual pigment Rh we have documented (4, 36) a space-saving mechanism that includes the bicycle-pedal motion proposed by Warshel (40, 41). This mechanism has been recently demonstrated using trajectory computations in vacuo (42) and in Rh (43).

Below the CASPT2//CASSCF/AMBER protocol recently applied to Rh (4, 36, 43) is used to construct the S₁ isomerization path of bR. Although structural (i.e., nondynamical), the data yield a mechanism of the initial stage of the photocycle [from the Franck–Condon (FC) point to the first isolable intermediate K] involving an asynchronous double bicyclo-pedal or folding-table (44) deformation. Indeed, it is shown that the ─C11═C12─ and ─C15═NH─ double bonds undergo a limited counterclockwise twisting, while the ─C13═C14─ bond is isomerizing in the clockwise direction. Remarkably, this photoinitiated motion induces a simultaneous hydrogen bond breaking.

1. Results and Discussion

1.1. Chromophore Structure and Absorption Maximum.

The optimized S₀ bR structure (see Fig. 1) displays a -160° value of the PBST C12─C13─C14─C15 dihedral angle corresponding to a 20° clockwise twisted C13═C14 double bond. In contrast, the PSBT structure optimized in vacuo is planar (45). The bR C13═C14 pretwist is larger than the one observed for the reactive bond of Rh (4, 36) and bias the photoisomerization in the clockwise direction (as detailed by the computed path; see below). This suggests that the stereoselectivity observed in bR (see, e.g., ref. 31) is decided by the asymmetry of the protein cavity that tilts the S₁ PES in order to favor a unidirectional C13═C14 photoisomerization.

Fig. 1.

Optimized structure for the first three states—bR, I, and K—of the proton–pump photocycle: The geometrical parameters (top, middle, and bottom values for bR, I, and K, respectively) are reported in Å and degrees (in parentheses).

The computed 53.6 kcal mol^-1 (534 nm) required for the vertical π → π^∗ spectroscopically allowed (f = 0.411) S₀ → S₁ transition in bR compare well with the experimental value of 50.3 kcal mol^-1 (568 nm); see Fig. 2. This value is blue-shifted (+6.9 kcal mol^-1) as compared to that computed in vacuo (see SI Text for details). This effect agrees with previously reported values by Rajamani and Gao (∼4 kcal mol^-1) (23), Vreven and Morokuma (2–6 kcal mol^-1) (30), and Houjou et al. (∼7 kcal mol^-1), (24, 25) and is due to electrostatic destabilization of the S₁ state by the protein (2, 46). The blue-shift effect due to the complex counterion (Asp85, Asp212, Arg82) alone is much higher (+20.1 kcal mol^-1 for the bare chromophore/counterion pair with respect to vacuo; see SI Text), revealing that the protein has a high shielding effect (ca. 65%). This conclusion is consistent with previous computational studies on Rh (4). A consequence of the above results is that the opsin-shift observed when going from solvent (440 nm) to bR (570 nm) (ca. -13 kcal mol^-1) is mainly due to (i) the binding pocket structure that forces the β-ionone ring into a planar 6s-trans conformation (a skewed 6s-cis conformation is expected in solution), effectively increasing the length of the conjugated chain, and (ii) a dumping of the chromophore/counterion electrostatic interaction due to the protein environment and leading to a red-shifted absorption as compared to the unshielded counterion.

Fig. 2.

CASPT2 relative energies (in kcal mol^-1) for the first three states—bR, I, and K—of the photocycle. Vertical electronic transitions (with their computed oscillator strength f) are also reported with the corresponding experimental values (in italics).

1.2. Excited State Intermediates and Transient Spectroscopy.

The PSBT displacement occurring after photoexcitation leads from the FC region to a shallow S₁ minimum displaying bond order inversion and a negligible twisting of the reactive C13═C14 double bond (see Fig. 1). This structure is assigned to the fluorescent state I (5, 6). The corresponding computed and observed transition energies agree within 1–2 kcal mol^-1 (see Fig. 2): (i) The ∼740 nm (38.6 kcal mol^-1) fluorescence maximum observed by Schenkl et al. (16) is matched by the 37.6 kcal mol^-1 (761 nm) energy gap computed for the S₁ → S₀ transition at this point. (ii) Hasson et al. (14) described a transient near-IR absorption peaking at 850 nm (i.e., 33.6 kcal mol^-1). This band can be attributed to the significantly allowed S₁ → S₃ transition, which is computed to be 35.9 kcal mol^-1 (797 nm). (iii) Finally, the transient absorption in the visible seen by Logunov et al. (47) and peaking at 490 nm (i.e., 58.3 kcal mol^-1) can be assigned to the S₁ → S₄ transition, which shows an energy gap of 57.2 kcal mol^-1 (500 nm) and a not negligible f. Notice that these properties are also consistent with the results of transient spectroscopic studies of artificial rhodopsins hosting a conformationally locked retinal analog (16–18).

1.3. Photoisomerization, Photoproduct Formation, and Retinal Dipole Moment Changes.

The < 1 kcal mol^-1 barrier located along the flat region of the S₁ path starting at I (see the difference between I and the transition state (TS) in Fig. 3) does not allow for a quantitative analysis. However, the existence of a barrier is in agreement with low-temperature experiments (14, 15) and with a quasistationary nature of I (5). The S₂-S₁ energy gap is initially decreasing [15.7 kcal mol^-1, 5.3 kcal mol^-1, and 3.3 kcal mol^-1 for FC, I, and TS, respectively (see Fig. 3 and SI Text)], leading to mixed ionic/covalent S₂ and S₁ wave functions. Then, it rises again on the way to the twisted and peaked S₁/S₀ CI (which prompts radiationless decay to S₀) and the path, after the TS, develops exclusively on a charge transfer S₁ state [whereas S₂ is a dark (π → π^∗)² covalent bound state. See charge analysis in Fig. 3B]. Consequently, the dark state shall not be involved in the photoisomerization and decay of bR, as found for Rh (4, 36) and for PSBT in vacuo. (35). This questions the S₂ assignment from two-photon absorption experiments by Birge and Zhang (48) and supports the recent reassignment by Andersen and coworkers (49).

Fig. 3.

The photoisomerization channel mapped on S₁. (A) CASPT2 corrected S₀, S₁, and S₂ energy profiles (molecular parameters are reported in Å and degrees). (B) Corresponding evolution of the fractional positive (Mulliken) charge resident on the right chromophore moiety and ionic character (ionicity %). (Inset) See dotted line: Ionic/covalent states are characterized by a positive charge that stays preferentially on the left/right moiety.

The located CI features a -120° C13═C14 dihedral angle. This CI belongs to a S₁/S₀ crossing seam that develops along the isomerization coordinate as indicated by the fact that the S₁ and S₀ energy profiles remain parallel and energetically close up to a -90° twisting (less than 10 kcal mol^-1 separation at the CASPT2 level and substantially degenerate at the CASSCF level; see Fig. 3). This situation parallels the one documented for Rh (50). Although in order to simulate/understand the reaction quantum yield, one should carry out trajectory calculations (that go beyond the scope of the present work), our results show that an extended low-lying S₁/S₀ intersection seam is energetically accessible upon relaxation from the Franck–Condon point. Therefore, trajectories that even slightly escape the computed path may access structurally different points of the seam contributing to the observed high photoisomerization efficiency.

The S₁ reaction coordinate displays three consecutive steps characterized by different structural changes (see Fig. 4A). The first (first step, bR^∗ → I, < 50 fs) takes place immediately after light absorption and is characterized by the bond order inversion leading to the formation of the fluorescent state I (16, 51). The following step (second step, I → CI, ∼450 fs) occurs when the initial C13═C14 isomerization motion is accompanied by small rotations about the nearby bonds allowing the system to cross the barrier and reach the CI at 120° (in Fig. 4A the 100° twisted CI—CI100—is used to produce a clearer picture of the deformation needed to reach K). This step initiates the isomerization and is characterized by a concerted double bicycle-pedal mechanism (40) (illustrated in Fig. 4B) with a minor contribution of hula twist (52) involving tiny pyramidalizations at C12 and C13 [this pyramidalization was also reported by Hayashi et al. (31), but at a different location, C14, perhaps due to reduced QM moiety employed in the simulations]. As shown in Fig. 4C, this makes the isomerization facile as a limited change in the chromophore steric hindrance occurs from bR to K. The simultaneous three bond twist has been observed recently in crystals of all-cis-1,6-diphenyl-1,3,5-hexatriene (53) and also described in dynamics simulation of mechanically constrained protonated Schiff bases, named folding table (44).

Fig. 4.

Photoisomerization mechanism. Schematic representation of (A) the photoinduced three-steps (bR^∗ → I → CI100 → K) motion [bonds in Å and angles in degrees (in brackets)], (B) the detailed double bicycle-pedal or folding-table motion occurring along the bR^∗ → CI100 excited state stage (the variation in the twisting angle—Δ°—for C11═C12, C13═C14, and C15═N_ζ is also reported), and (C) the space-saving character of the overall motion (the chromophore ends are at rest during the isomerization process). Structural parameters optimized for the mobile W402 setup are reported in italics when differing from those pertaining to the fixed W402 setup (see Computational Methods and SI Text).

In bR the concurrent twisting deformation of three adjacent double bonds C11═C12 (+27°), C13═C14 (-60°), and C15═N_ζ (+21°) occurs in alternate directions (counterclockwise, clockwise, and counterclockwise, respectively). This parallels/extends the structural evolution during the isomerization of the PSB11 retinal chromophore in Rh where the C11═C12 reactive bond isomerizes in the counterclockwise direction while the adjacent C9═C10 bond twists in the clockwise direction. As in Rh, the mechanism can be described as asynchronous (the extent of twistings of the ─C11═C12─ and ─C15═N_ζ─ bonds are smaller than the one experienced by the ─C13═C14─ bond). However, note that the combined +27° (C11═C12) plus +21° (C15═N_ζ) angle variations almost fully compensate for the -60° twisting of C13═C14 at the CI. More remarkably, progressive lengthening of the N_ζ─H/water-402 (W402) hydrogen bond occurs along the excited state path from bR to the CI (+0.14 and +0.2 Å for the fixed and mobile W402 set up, respectively; see Fig. 4A and SI Text), suggesting that photoexcitation prompts its breaking, and pointing to a W402 that stays always tightly bound to the complex counterion (see also Fig. 5). This process would be energetically demanding on S₀. However, the chromophore N─H positive charge translocation occurring on S₁ leads to a weakening of this hydrogen bond, which accounts for its lengthening. In fact (see SI Text for details), an instantaneous ∼3.0 kcal mol^-1 destabilization accompanies S₀ → S₁ excitation that can be mainly due to hydrogen bond weakening. This effect is predicted to increase steadily as the system evolves along the S₁ isomerization path. Indeed, positive charge translocation from the nitrogen head to the β-ionone ring progressively increases (till a net 100% transfer at the TICT point; see Fig. 3B).

Fig. 5.

Structures of the optimized binding site (hydrogen bonds are highlighted and reported in Å; structural parameters pertaining the mobile W402 setup are reported in parentheses). (A) bR (modeled from the available X-ray crystallographic structure) (62). (B) The primary photoproduct [for comparison, the K structure from X-ray crystallography (56) is also shown in transparence]. bR and K crystal structures display a substantially unmodified binding pocket pointing to a W402 that is tightly bound to the complex counterion.

To the best of our knowledge, a double bicycle-pedal mode followed by hydrogen bond breaking has never been proposed or detected in dynamical studies of bR (31, 32). It is also different from the single bicycle-pedal mechanism recently found in Rh (4, 36) and described by the rotation of the plane hosting the ─C10─C11─ moiety. Indeed, in bR there are two such adjacent rotating units: the ═C14─C15═ unit (plane α) and the ═C12─C13═ unit (plane β). The direction of twisting during the isomerization is given in Fig. 4B. As anticipated above, besides bond order inversion and triple isomerization (i.e., about the ─C11═C12─, ─C13═C14─, and ─C15═N─ bonds), hydrogen bond weakening is activated on S₁ and helps to release the electrostatic hindrances on retinal (as recently suggested by Schenkl et al.) (51).

The twisting of the reactive C13═C14 double bond initiated on S₁ is finalized after decay to S₀ through the CI, whereas the two adjacent bond revert to the original configuration (third step, CI → K). Indeed, S₀ optimization leads to a stable PSB13 chromophore (even if still 20° twisted about the C13═C14 bond. See Figs. 1 and 4A). The aborted bicycle-pedal motion seems to be compensated by a 60 ° N_ζ─C_ε conformational change (see Fig. 4A) allowing minimum volume C13═C14 isomerization. Finally, during this third stage the breaking of the hydrogen bond between W402 and the N_ζ─H retinal is completed. An animation of the simulated isomerization motion is included in SI Text.

The structural evolution described above seems to be consistent with the photoinduced changes in permanent dipole moment of the chromophore that have been experimentally monitored by Léonard et al. (19). In particular, the recorded transient response (i.e., modifications in UV-absorption spectrum) of the nearby Trp86 residue has been used as a probe of the ultrafast changes in the local PSBT charge distribution (i.e., the dipole moment variation—Δμ—with respect to the one recorded in ground state bR). The instantaneous rise (i.e., within the 50-fs time resolution of the experiments) of the Trp86 signal reveals the underlying retinal charge translocation upon photon absorption and the corresponding increase in μ (i.e., Δμ > 0). These experimental observations are consistent with our computations (see Table 1), because the latter show a significant increase in μ when going from bR to FC and I (Δμ ≈ 10) and even more (Δμ≥20) from I to the CI. Afterward, Δμ is shown to decrease with time constants that match S₀ recovery and photoproduct K formation, although the initial bR value is not fully recovered (19), calling for a change in the local electrostatic environment that remains long after isomerization. This trend is consistent with the small value of Δμ calculated at K consistently with a charge distribution similar to that found in bR.

Table 1.

Changes of the difference dipole moment—Δμ (in debye)—along the photoisomerization channel

Structure	FC	I	150°	TS (140°)	CI	K
Δμ	12.3	9.0	22.1	20.3	20.0	2.6

Δμ is calculated as the difference of the initial ground state bR dipole and the current transient dipole at each reported structure (this is the dipole computed on S₁ for all the structures but K, where it is referred back to the ground state).

1.4. K and Energy Storage.

The results above indicate that the computed 13-cis photoproduct is K. The J intermediate is assigned to either a vibrationally hot state on S₀ or S₁ (e.g., I with activation of hydrogen-out-of-plane modes) (54) and cannot be characterized through static computations. This conclusion is consistent with the calculation of the energy difference between K and bR (15.1 kcal mol^-1), which agrees with the experimental estimate of the energy storage in K (11.7–15.9 kcal mol^-1) (20) and also with previous computational studies (16 kcal mol^-1) (55). We have also compared the computed K geometrical parameters with those reported by Lanyi and Schobert (56) on the basis of X-ray diffraction data (see Fig. 5B). The two structures are in good agreement (the largest differences being located on the C13═C14 and C15═N_ζ bonds, which correspond to the less-resolved regions in the experimental structure). Finally, the structure and energy analysis reported here for K (including the predicted N_ζ─H/W402 hydrogen bond breaking) matches the results reported by Hayashi et al. (55) using a different QM/MM protocol. Thus, it is apparent that, despite a slightly overestimated simulated absorption maximum [the computed S₀-S₁ vertical energy gap is 53.0 kcal mol^-1 (540 nm), i.e., 5 kcal mol^-1 higher than the 48.0 kcal mol^-1 (596 nm) experimental absorption maximum for K; see Fig. 1] (57), our data support the assignment of the optimized photoproduct to K.

The energy stored in K brings the proton pumping to the end. In the available literature its origin is explained with two different hypotheses (58). The first assigns it to the augmented steric strain of the PSB13 chromophore that is transferred to the protein in later steps (20). The second emphasizes the weakening of the interactions between the chromophore and the nearby polar groups (59). To disentangle these contributions, we have analyzed the energy difference between Rh and K (see SI Text for details). The van der Waals energy difference is only ∼0.7 kcal mol^-1, revealing a negligible opsin/chromophore steric component. However, the intrinsic chromophore strain accounts for only 6.6 kcal mol^-1. The weakening of the interactions between retinal and its complex counterion (most of which is due to the breaking of the N_ζ─H/W402 hydrogen bond; see Fig. 5) contributes ca. twice that value (13.8 kcal mol^-1). This value, which agrees with previous computations (33, 55) is, in part, compensated by the electrostatic effects due to the remaining protein environment leading to a total of 7.8 kcal mol^-1 for the electrostatic component. It is thus concluded that, as also found for Rh (36), changes in both the intrinsic retinal strain and electrostatic interactions contribute almost equally to the energy stored at K, in line with the results of the pioneering work of Birge and Cooper (58) and more recent theoretical estimates (33, 55).

2. Conclusions

We have been able to construct a CASPT2//CASSCF/AMBER computational model that reproduces the observed spectral features of the initial part of the bR photocycle (i.e., bR → I → K) and the energy stored at K with a less than 5 kcal mol^-1 error. It has been shown that bR blue-shifts the absorption maximum with respect to PSBT in vacuo, but not as much as the counterion alone would do: In fact, this is significantly quenched (> 60%) by the protein. The bR opsin-shift is due to an increased double-bond conjugation (in going from the skewed 6s-cis conformation in solvent to the fully planar 6s-trans in bR) and to an opsin-induced dumping of the counterion electrostatic effect. The bR photoisomerization mechanism involves three stages: (i) a backbone rearrangement leading to an inverted bond order intermediate I on S₁ (first stage, ≤ 50 fs); (ii) the C13═C14 torsion accompanied by concurrent twistings of the adjacent bonds allowing the system to proceed along a space-saving isomerization coordinate reaching the deactivation region; a chromophore-N_ζH/W402 hydrogen bond breaking is also initiated during this stage (second stage, to ∼450 fs); (iii) the S₀ relaxation leading to double-bond reconstitution and N_ζ─C_ε single bond twisting that result in an aborted bicycle-pedal progression and lead to the K state (third stage, till ∼3 ps). This final step is responsible for completing N_ζ─H/W402 hydrogen bond breaking that is one of the main contributions to the energy storage.

3. Computational Methods

Consistently with previous works (4, 36, 50), the photoisomerization path is computed in terms of a relaxed scan (with the C13═C14 twisting angle fixed at specified values) carried out using our recently developed QM/MM potential [the GAUSSIAN98 (60) and TINKER (61) suite of programs are employed]. The retinal, Schiff base nitrogen and attached ε-carbon atom (see Scheme 1) constitute the QM part of the system, described at the CASSCF/6-31G* level (with an active space of 12 π-electrons in 12 π-orbitals), whereas AMBER is used for all MM atoms. All QM atoms and MM atoms comprising the Lys-216 side chain (C_γH₂, C_δH₂, and C_βH₂) are free to relax, whereas the protein framework [PDB ID code 1C3W (62) is used] is kept fixed: We assumed that this has no time to change (i.e., full thermal equilibration is prevented) during the subpicosecond time scale of the bR primary photoinduced event investigated here, a view that is supported by the similarity recognized between the bR and K crystal structures (see Fig. 5). W402 may be an exception: Although it belongs (and is tightly bound via hydrogen bonding) to the complex counterion (see Fig. 5), possibly limiting its motion, nevertheless it also directly/strongly interacts (via hydrogen bond) with the chromophore and can respond to its changes. Therefore, two sets of computations were employed to compute the path with a locked/unlocked W402, respectively, which eventually delivered the same mechanistic outcome (see Fig. 4A and SI Text). This validates the fixed W402 model, which the results reported in this work are generally referred to. The employed crystallographic structure is considered as a mean representation of the experimental position of the atoms (although at low temperature), and the computed path portrays a mean static view of the process. This approach to ultrafast photoinduced reactions has been already employed successfully to study the Rh primary event (4, 36, 43, 50). Multiconfigurational second-order perturbation theory computations [carried out with the CASPT2 (63) method using the MOLCAS5 (63) package] are then used to increase the accuracy of the energy profiles. Whenever possible, a single state approach is employed. Otherwise, when root flipping occurs, a state average procedure (by equally weighting the first five singlet roots) is used, together with a multistate approach (MS-CASPT2) (64) when required. Oscillator strengths (f) are computed using correlated (CASPT2) energies within the RASSI approach (65). See SI Text for further details on computations and protein setup.