A. Evaluation of computational model and proceduresThe results from the calculations comparing ground-state geometric parameters and wavelengths of maximum absorption for PΦB1, PΦB2, PΦB3, and PΦB4 are presented in Table 1. From the dihedral angles, we first and foremost note that the dipyrrolic moiety comprising rings C and D regardless of computational model and configuration is non-planar. Even though the Z,syn, E,syn, Z,anti, and E,anti designations strictly speaking refer to configurations with the C14−C15C16−C17 and C13C14−C15C16 dihedrals equal to 180 and 0, 0 and 0, 180 and −180, and 0 and −180°, respectively, these designations will for the sake of simplicity refer to geometry-optimized configurations in this section. Moreover, we observe (cf. PΦB1 and PΦB2) that none of the bond lengths, all of which one may expect to be of relevance for the isomerization, are sensitive to the exclusion of the propionate side chains of pyrroles B and C. This is also reflected by PΦB1 and PΦB2 having very similar λmax (0.01 eV difference for both Z,syn and E,anti). As for the influence of the methyl groups (cf. PΦB1 and PΦB3), a somewhat larger effect is observed. Yet, the absolute mean-deviation between the two sets of bond lengths is fairly small (<0.01 Å), as are the differences in λmax between PΦB1 and PΦB3 (0.05 eV for both Z,syn and E,anti). Even though an absolute comparison of calculated λmax, representing the absorption by the chromophore in the gas phase, with experimental spectra, representing the absorption of phytochrome in vitro, for obvious reasons cannot be made, it is encouraging to note that the maximum absorption of the E,anti configuration throughout is red-shifted relative to that of Z,syn, which is in accord with the maximum absorption of Pfr being bathochromically shifted relative to that of Pr.[5] We furthermore note that the calculated red-shifts essentially remain unaltered (0.07, 0.07, 0.07, and 0.10 eV for PΦB1, PΦB2, PΦB3, and PΦB4, respectively) upon enlargement of the computational model, and that these are in reasonable agreement with the experimental value of 0.18 eV.
Table 1 also lists HF ground-state and CIS excited-state bond lengths for PΦB1. The observed shortening and lengthening (relative B3LYP) of nominal double and single bonds, respectively, is a well-known deficiency of HF when applied to conjugated systems. By comparing HF and CIS bond lengths, we see that the bond alternation is less pronounced in the S1 state. This is a consequence of this state originating from a one-electron HOMO → LUMO π−π* transition. The most notable geometric feature of the S2 state (in comparison with HF S0) is the inversion of the C13C14 and C14–C15 bonds lengths, which is a result of this state including contributions from two π−π* transitions. Since CIS due to the inclusion of only singly excited configurations in general seriously overestimates excitation energies for states having pronounced double-excitation character (e.g., the S2 state of PΦB1), it should be stressed that the S2 geometries likely are less accurate than the S1 ones. Even though the fact that TD-DFT in general offers a significant improvement over CIS in reproducing transition energies for low-lying valence-excited states[40,42] – including to some extent also those having appreciable double-excitation character[43] – motivates the construction of excited-state PES using CIS geometries and TD-DFT energies, it is nevertheless important to emphasize that the S2 PES presented below should be regarded as preliminary. The observation that no inversion of the C13C14 and C14–C15 bonds is displayed in the S2 state of the E,anti configuration can probably be attributed to the aforementioned shortcoming associated with the CIS methodology.
By subjecting the optimized S1 geometries of PΦB1 to TD-B3LYP/6-31G(d) single point calculations, wavelengths of maximum emission were obtained as well (Table 1). Depending on phytochrome size, the maximum fluorescence of Pr in vitro is located in the 672–692 nm (1.79–1.85 eV) region,[12]i.e., 0.03–0.09 eV below the absorption maximum at 660 nm (1.88 eV).[5] Encouragingly enough, the calculated 0.09 eV Stokes shift (difference between vertical excitation and emission energies) for Z,syn PΦB1 accounts rather satisfactorily for this feature of the photochemistry of phytochrome.
The results from the calculations investigating basis set effects on electronic energies of PΦB1 are given in Table 2 together with calculated ZPVE and thermal corrections. It is seen that using a larger basis set than 6-31G(d) has only a minor (≤0.4 kcal mol−1) effect on relative ground-state electronic energies. Increasing the basis set does not significantly alter vertical excitation energies either, which is in accordance with recent TD-DFT studies on the absorption of the highly conjugated astaxanthin chromophore.[44,45] At the 6-311+G(d,p) basis set level, the S1 and S2 excitation energies are lowered by 0.04 and 0.02–0.03 eV, respectively, with respect to the 6-31G(d) values. The relative insensitivity of vertical excitation energies to the inclusion of diffuse s and p-functions on heavy atoms reflects the ‘valence-excited state’ nature of S1 and S2. Considering differences in vertical excitation energies between different configurations (of particular relevance for the present work), basis set effects on both states throughout do not exceed 0.01 eV. Turning to the results from the frequency calculations, we note that ZPVE corrections and Gcorr for the optimized ground-state structures differ within 0.3 and 0.2 kcal mol−1, respectively, between the different configurations. As for the optimized excited-state structures, the dependence of ZPVE correction (within 0.5 (S1) and 0.2 (S2) kcal mol−1) and Gcorr (within 1.2 (S1) and 0.6 (S2) kcal mol−1) on conformation is not pronounced either. These results indicate that the energetics of the PΦB1 isomerization can be assessed on the basis of electronic energies only. Even though the effects of neglecting ZPVE corrections and Gcorr for the individual states of course may add up when the states are considered simultaneously, the conclusions to be drawn from the present work are based on electronic energy differences that, we believe, are sufficiently large to marginalize such effects.
Vertical S1 and S2 excitation energies for PΦB1 in Z,syn, E,syn, Z,anti, and E,anti configurations were in addition computed at the CIS/6-31G(d)//HF/6-31G(d) level of theory. For both states, which respectively have HOMO → LUMO and {HOMO − 1 → LUMO/HOMO → LUMO + 1} character within the framework of CIS as well, this yielded considerably higher transition energies than did the TD-B3LYP/6-31G(d)//B3LYP/6-31G(d) calculations (0.8 and 2.3 eV higher for S1 and S2, respectively). This finding is in accord with previously reported calculations on the electronic spectra of hexamethylpyrromethene indicating that CIS grossly overestimates excitation energies for pyrrole compounds.[33]
In summary, the results reported in Table 1 suggest that the approximation imposed by not including the thioether linkage, the propionate side chains, and the methyl groups in the computational model is sound, and also that PΦB1 is a reasonable model in that the calculated absorption red-shift induced upon Z,syn → E,anti isomerization, as well as the calculated Stokes shift for the Z,syn configuration, agree rather well with the corresponding experimental shifts as obtained for phytochrome. The results given in Table 2, in turn, suggest that using a larger basis set than 6-31G(d) for calculating electronic energies is superfluous, and that the accounting for ZPVE and thermal corrections can be omitted. Finally, it should be noted that our conclusion that the computational procedure is suitable for calculating the relevant isomerization PES of course is directly dependent on the validity of the benchmark calculations also for distorted phytochromobilins along the isomerization path.
B. Potential energy surfacesIn the first part of this section, we will report S0 and S1 PES for PΦB1, and thereby propose a mechanism for the Z,syn → E,anti isomerization involving these two states only. In the second part, we will present different cuts of the S2 PES constituting further support for such a two-state model.
The S0 and S1 PES, and their difference, are shown in Fig. 3. Considering first the S0 PES, we note that thermal Z → E isomerization of both the Z,syn and the Z,anti configuration is prevented by a high-energy barrier (>35 and >32 kcal mol−1, peaking at β = 75°), as expected. The barriers for E → Z isomerization are of a similar magnitude (>34 and >32 kcal mol−1 for E,syn and E,anti). We furthermore observe that light does not seem to be an absolute requirement for syn → anti isomerization as such, as indicated by a barrier of ∼8 and ∼9 kcal mol−1 (peaking at α = −90°) for thermal isomerization of the E,syn and Z,syn configuration, respectively. The barriers for anti → syn isomerization are likewise low (∼7 kcal mol−1 for both E,anti and Z,anti).
Turning to the S1 PES, we recall that, as outlined in Section I, the primary photochemical reaction in phytochrome in all likelihood involves a Z → E isomerization of the chromophore. Moreover, it has been argued that also the syn → anti isomerization is part of the primary photochemical reaction and that the two isomerizations occur simultaneously in a concerted fashion,[5,22] whereas others have suggested that syn → anti is initiated thermally upon completion of Z → E,[19] thus favouring a stepwise process. Hence, it is interesting to note that – while photochemical Z → E isomerization of the Z,syn configuration is associated with a flat energy profile (maximum at β = 135° lying ∼4 kcal mol−1 above Z,syn) and a subsequent energy minimum region (at β = 90° lying ∼7 kcal mol−1 below Z,syn) enabling the system to decay to the S0 PES and evolve to E,syn – the reaction path corresponding to a concerted Z,syn → E,anti photoisomerization has an unfavourable energy profile. For example, the central {α = −90°, β = 90°} structure, the 9 structures covering the {−105 ≤ α ≤ −75°, 75 ≤ β ≤ 105°} region, and the 25 structures covering the {−120 ≤ α ≤ −60°, 60 ≤ β ≤ 120°} region lie ∼30, ∼17–32, and ∼11–32 kcal mol−1 above Z,syn, respectively. Furthermore, no S1 → S0 decay channel seems to exist along this path (cf. Fig. 3, S1−S0 PES). These findings strongly suggest that the photoactivation of phytochrome is less likely mediated by a concerted Z,syn → E,anti photoisomerization than by a stepwise Z → E, syn → anti mechanism involving a photochemical Z → E isomerization (∼4 kcal mol−1 barrier) followed by a thermal syn → anti isomerization (∼8 kcal mol−1 barrier).
The S1 PES also suggests that an alternative mechanism (that has not been discussed in the experimental literature) involving consecutive syn → anti and Z → E photoisomerizations could be energetically feasible. The estimated barriers for these isomerizations are ∼6 and ∼3 kcal mol−1, respectively, which effectively means that the calculations do not unambiguously show that photochemical Z → E followed by thermal syn → anti is the most probable route to the activation of phytochrome. Nevertheless, of the scenarios actually discussed in the experimental literature,[5,19,22] the calculations clearly show that photochemical Z → E followed by thermal syn → anti is energetically preferable over a concerted Z,syn → E,anti photoisomerization.
The decay process by which the excited system returns to the ground state is a key mechanistic element of photochemical reactions. By means of CASSCF calculations, it has been shown that the decay channel in a number of systems corresponds to a conical intersection of the ground and excited-state potential energy surfaces.[25,46,47] Given that single-reference DFT methods in general fail to explicitly locate conical intersections, the possibility that the decay along the photochemical Z,syn → E,syn path is governed by an actual S0/S1 crossing cannot be excluded. This uncertainty is however irrelevant for distinguishing Z,syn → E,syn from Z,syn → E,anti as the most probable photoisomerization pathway. The S1−S0 PES shows that the two states are the closest in energy at {α = 0°, β = 75°} (∼11 kcal mol−1 apart) and {α = −180°, β = 75°} (∼12 kcal mol−1 apart).
It has been suggested that the primary photochemical reaction in phytochrome takes place within picoseconds.[10,48] This is slower than the photoisomerizations underlying the biological activity of the rhodopsin family of proteins, which take place in the sub-picosecond regime and hence are essentially barrierless. Still, the barrier for PΦB photoisomerization should be very low. Even though the prediction that ∼4 kcal mol−1 are needed for excited Z,syn to reach the decay channel is compatible with an isomerization occurring within a photochemically reasonable period of time, it is clear that the calculations, due to, e.g., neglected interactions with the surrounding protein and limited computational accuracy, overestimate the barrier for photoisomerization. Furthermore, a more elaborate quantum chemical treatment would require the explicit localization of excited-state stationary points and minimum energy path calculations.[25–27] In order to, in turn, put the prediction that ∼8 kcal mol−1 are required for thermal E,syn → E,anti isomerization into an experimental perspective, it can be noted that an overall energy barrier of 17–20 kJ/mol (∼4–5 kcal mol−1) for Pr → Pfr conversion has been derived through the monitoring of changes in Pr fluorescence in vitro.[49] Even though the calculations once again appear to provide an overestimation, we believe that – considering the various approximations employed – the fair agreement between theory and experiment is rather encouraging.
Finally, as for the S2 PES, we note that while an equal number (99) of S1 and S2 geometry optimizations were initiated, roughly one third of the latter were not successfully completed. However, we are here primarily interested in the shape of the S2 PES along the Z,syn → E,syn isomerization coordinate. Rather than presenting a S2 PES covering a lesser range than S1, we therefore focus our attention on the Z,syn → E,syn cut of S2. This particular cut, as well as that along the Z,anti → E,anti coordinate, is shown in Fig. 4 together with the corresponding cuts of S1. We note that there are no signs of an S2/S1 avoided crossing along the Z,syn → E,syn isomerization coordinate, as evidenced by the facts that the S2–S1 energy gap increases (from ∼11 to ∼36 kcal mol−1 at β = 90°) as the system evolves towards the decay channel, and that the S2 state throughout retains its character. The same features are observed along the Z,anti → E,anti coordinate. These results indicate that the two-electronic state model involving only S0 and S1 previously assigned to, e.g., the retinal photoisomerization in rhodopsins[25–27] is applicable to the PΦB photoisomerization in phytochrome as well. As emphasized in Section IIIA, the S2 PES may be qualitatively incorrect due to the inadequacy of CIS in treating the PΦB S2 state. Therefore, the proposed two-state model for PΦB photoisomerization should be regarded as a tentative model, rather than as a rigorously derived one.