Applicability of Transition State Theory to the (Proton-Coupled) Electron Transfer in Photosynthetic Water Oxidation with Emphasis on the Entropy of Activation

1. Introduction

In oxygenic photosynthesis, water is oxidized at a Mn₄Ca(µ-O)₆(OH_x)₄ complex (denoted here as the Mn₄Ca-cluster) bound to the proteins of photosystem II (PSII)[1,2]. Light drives water oxidation at the Mn₄Ca-cluster indirectly[3,4], with a primary charge separation occurring at the electron donor side of PSII resulting in the formation of a delocalized chlorophyll cation radical (P680⁺). This is followed by the oxidation of a specific tyrosine residue, denoted as Tyr_Z or Y_Z . The subsequent electron transfer (ET) steps from the Mn₄Ca cluster to Tyr_Z^ox are closely related to the water oxidation chemistry coupled with proton movements over a large area on various length scales, see Figure 1. Herein we will discuss, in a broad sense, whether the Mn₄Ca cluster – Tyr_Z^ox ET can be described favorably by transition state (TS) theory in the framework of the Eyring-Polanyi equation[5,6,7] or whether elements of ET theories, e.g. the well-known semi-classical Marcus theory[8,9], are applicable or even essential for a correct description. We note that the following Results and Discussion section corresponds in parts to Sections SII.8 and SII.9 of the Supplementary Material of Greife et al.[10]. Our discussion addresses specifically the water oxidation reaction (or oxygen evolution reaction) in PSII, but it may, in parts, also be of relevance for redox reactions of other proteins, molecules, and redox-active electrodes in electrocatalysis.

2. Results and Discussion

2.1. TS theory applied to oxygen-evolution step of PSII

In Eyring’s TS theory, the reaction rate (k_TST) is governed by the Eyring-Polanyi equation[5,6] involving the entropy (ΔS^‡) and enthalpy (ΔH^‡) of activation:

$$k_{TST}=\frac{k_{B}T}{h}\kappa e^{\frac{{\mathsf{\Delta }S}^{‡}}{k_B}}{e}^{\frac{-{\mathsf{\Delta }H}^{‡}}{k_{B}T}}$$

(1)

with k_B being the Boltzmann constant, h the Planck constant, and T the absolute temperature in Kelvin. The κ parameter describes the transmission probability, i.e., the probability that the reaction will occur once the TS is reached; typically κ-values of 0.5 or 1.0 are assumed to be reasonable[5,6]. In Equ. 1, the activation enthalpy, ΔH^‡, describes the temperature dependence of the rate constant and is closely related the activation energy, E_a, as determined from a conventional Arrhenius analysis (ΔH^‡ ≈ E_a - k_BT₀, the difference ΔH^‡ and E_a stems from the weak temperature dependence of $\frac{k_{B}T}{h}$ in Equ. 1, see ref.[12]). The activation entropy, ΔS^‡, is usually a negative number that corresponds to a temperature-independent reduction of the k_TST.

In research on PSII water oxidation, Equ. 1 has been applied to the rate constant of the oxygen-evolution transition[10,13,14], which is initiated by the third flash of saturating light applied to dark-adapted PSII[15]. The oxygen evolution transition is referred to as the S₃->S₀ transition and involves an ET from the Mn₄Ca-cluster to Tyr_Z^ox. This ET step precedes O-O bond formation and O₂-release. It is generally believed that the rate-determining step of the overall reaction is the Mn₄Ca-cluster—Tyr_Z^ox ET step, which we have suggested to be coupled to the movement of four protons[10] see Figure 2.

Based on experimental determination of the rate constants for the O₂-formation at temperatures ranging from -5°C to 40°C step, we have calculated (by application of Equation 1) the enthalpy and entropy of activation of the rate-determining (proton-coupled) ET step for PSII of the thermophilic cyanobacterium Thermosynechococcus elongatus (T. elongatus), see Extended Data Figure 5 and elsewhere in Greife et al. We obtained values for the activation enthalpy of 310 meV (7.15 kcal/mol) and activation entropy of 284 meV (6.55 kcal/mol, value of -T₀ΔS^‡ for T₀ = 293 K), for κ = 1 [10]. A variation of the κ-value from 1.0 to 0.5 would leave the activation enthalpy unchanged and affects the numerical value of the activation entropy only marginally.

For the PSII of T. elongatus and a closely related thermophilic cyanobacterium, Thermosynechococcus vulcanus (T. vulcanus), detailed crystallographic data is available, not only for the dark-stable state, but also for intermediates of the reaction cycle[11,16,17,18]. Due to the availability of atomic-resolution structures, computational studies on the mechanism of PSII water oxidation mostly have been based on coordinates obtained for PSII of T. elongatus or T. vulcanus[19,20,21,22,23,24,25]. Therefore, herein we focus on the enthalpy and entropy of activation experimentally determined for the O₂-formation step of T. elongatus.

The application of Eq. 1 to experimental data is straightforward and has resulted in values for enthalpy and entropy of activation for the rate-determining ET transfer step in the oxygen evolution transition. The question arises as to how the application of TS theory relates to ET theories.

2.2. Relation to ET theory

In the framework of the Eyring-Polanyi equation, essentially instantaneous ET would have to occur whenever the appropriate nuclear coordinates are reached, corresponding to a κ-value near unity in Equ. 1. In terms of non-adiabatic ET theory, however, the relocation of electron density (and thus spin density) does not always occur once a suitable nuclear geometry has been reached, but with a finite probability that is determined by the electron-tunneling distance. A tunneling probability well below the frequency factor in Equ. 1 might mimic an entropic contribution to the activation energy. Therefore, the possibility of whether non-adiabaticity of the ET step could contribute significantly to the experimentally determined entropy of activation is explored in the following.

Following Moser et al.[26], we estimate the tunneling distance as the molecular edge-to-edge distance between donor and acceptor moiety, where herein the acceptor moiety is the tyrosine radical and the donor being the Mn₄Ca cluster. The ‘edges’ or rather the extension of the relevant orbital systems are not obvious and cannot reliably be identified without using computational approaches. We calculated several potentially relevant interatomic (internuclear) distances, see Figure 3 and Table 1. An internuclear distance close to 5.7 Å is obtained between Tyr_Z-O and O6. This distance could be especially relevant because in the rate-limiting ET step of the S₃->S₀ transition, formation of the O6^• radical is facilitated by ET to the oxidized Tyr_Z, with high spin density expected on the phenolic oxygen. This corresponds to a Van-der-Waals tunneling distance of only about 2.5 Å. Other potentially relevant internuclear distances in Table 1 are even shorter, predicting a particularly high tunneling probability. According to Moser et al., the tunneling distance of 2.5 Å corresponds to a free-energy optimized rate constant for electron tunneling on the order of 10¹² s^-1, or a time constant of 1 ps. The frequency that describes nuclear movements in the TS regime is about 6 times greater (k_BT/h in Equ. 1, about 6∙10¹² s^-1 at 20°C) suggesting that for oxygen-evolution steps in PSII a minor slow-down resulting from limiting ET probability cannot be excluded. Staying in the framework of TS theory, such a slowdown could be described by a reduced transmission probability (reduced κ-value), which would result in only a small reduction of the activation entropy that we calculated from experimental data using the Eyring-Polanyi equation (Equation 1).

The above considerations involve several approximations and estimates. Presumably the tunneling probability exceeds the above estimate of 10¹² s^-1 because the ‘medium’ between the donor and acceptor moieties comprises atom groups (specifically W3 and W4) that could lower the tunneling barrier significantly. In addition, multiple parallel tunneling paths could also increase the overall tunneling probability. Thus, it is well conceivable that the adiabatic limit, where electron tunneling probabilities become irrelevant, is reached.

In summary, uncertainties remain regarding the exact numerical value of the activation entropy, because of possible limitations due to electron tunnelling. However, it is likely that the limiting case of adiabatic ET is almost, or even fully reached. If the adiabatic ET limit were not reached, the activation entropy would be moderately lower than the value formally determined by application of Equ. 1 to experimental data. However, this does not affect the central conclusion that there is a pronounced entropic slowdown of the oxygen-evolution transition.

Based on minimum energy path (MEP) calculations, we have proposed that the ET from the Mn₄Ca cluster to Tyr_Z^ox is coupled to the movement of four specific protons[10]. However, the used computational approach cannot clarify whether concerted ET and proton transfer (PT) is involved, that is, the simultaneous tunneling of the electron and one or more protons at the TS. This question is of relevance when assessing the significance of activation entropy formally calculated by application of Equ. 1 to experimental data, because in the case of a truly concerted electron-proton transfer (CEPT), the proton tunneling probability could influence the reaction rate and mimic an entropic slowdown. However, the low value of the experimentally observed H/D kinetic isotope effect (about 1.2[27,28]) clearly disfavors a limitation of the reaction rate by the proton-tunneling probability.

Further evidence supporting the entropic slowdown hypothesis comes from experimental indications for entropy-enthalpy compensation in the total free energy of activation of the oxygen-evolution step. In PSII of higher plants (spinach) at room temperature, the rate constant is very similar to that of the thermophilic cyanobacterium (T. elongatus). However, the Arrhenius activation energy and consequently the activation enthalpy are significantly lower for the higher-plant PSII[14]. For the cyanobacterium Synechocystis sp. PCC 6803, again another combination of action enthalpy and entropy has been reported[13]. We propose that compensation of the decreased activation enthalpy by an increased activation entropy explains identical rate constant values at isokinetic temperatures in the physiological temperature range. This entropy-enthalpy compensation in the activation energy of the oxygen-evolution step is still insufficiently understood and requires further investigation (work in progress). In general, entropy-enthalpy compensation is well known for the free energy of activation relating to various catalytic processes in biological and non-biological systems [29,30,31,32]. Finding entropy-enthalpy compensation in PSII oxygen evolution supports the existence of a major entropic contribution to the total free energy of activation, and it disfavors alternative explanations like a dominating role of electron or proton tunneling probabilities.

2.3. Origin of activation entropy and its relation to Marcus theory

In PSII water oxidation, the environment of the two key protagonists, Tyr_Z and the Mn₄Ca cluster, is characterized by an extended network of H-bonded water molecules and amino-acid residues. See Figure 4 for a two-dimensional mapping of the H-bond network (HBN) within 20 Å of the Mn₄Ca cluster. There are over 40 resolved waters to be found, grouped into several different water channels. Some of these groups are likely involved in proton and water transport and have been shown to be variable during the S3->S0 transition[10,11]. We propose that multiple conformations, that is, multiple H-bond patterns with multiple orientations of water molecules and residues can explain the experimentally found activation entropy. The interplay between evolutionary optimized order (stabilization of favorable conformations) and thermally driven dynamics needs to be considered when discussing the functional relevance of the extended H-bonded protein-water network. Following Greife et al (Supplementary Information, section SII.9), we point out:

(1) It is expected and has been verified by extensive molecular dynamics simulations on PSII that the protein internal H-bond networks (HBN) are highly dynamic, especially regarding water positions and H-bonding directionality, with a multitude of roughly isoenergetic conformations reached within nanoseconds at room temperature (see, e.g., ref.[35]). Even though these HBN dynamics maybe in part conducive regarding PT reactions, they are largely ‘inevitable’ thermodynamic fluctuations.

(2) The arrangement of the water molecules in the extended H-bonded protein-water network surrounding the Mn₄CaO₅ cluster and the Tyr_Z is well resolved in the crystallographic structures determined at both cryogenic and room temperature. This implies that the HBN dynamics evolve around the mean positions of the individual nuclei that are detected by protein crystallography. The HBN and the related mean-value atom coordinates are likely evolutionarily optimized for efficient (fast) water oxidation.

In light of (1) and (2), the qualitative explanation for the energy of activation in Greife et al. appears plausible: ‘Although well-defined coordinates of individual oxygen atoms are resolved in crystal structures, the presence of an HBN that is at the same time and in every detail perfectly arranged for the here discussed proton-coupled ET, will still be a rare event. The limited probability to reach this perfect conformation of all atoms of the HBN explains the entropic contributions to the activation energy.’

We emphasize that even if the optimal HBN conformation perfectly matches the atomic coordinates resolved by protein crystallography, it will still be rare that all atoms are simultaneously at their evolutionarily optimized positions at physiological temperatures.

3. Conclusions

Marcus and related theories on ET or CEPT reactions are powerful tools because they can predict rate constants based on a small number of clearly defined parameters, which are often experimentally accessible. The extended H-bonded protein-water network in PSII might be seen as a special case of the solvent environment that in Marcus theory is described involving a quadratic energy dependence with the outer-sphere reorganization energy being the (only) decisive parameter [9]. However, such an approximative treatment does not consider individual solvent molecules. Instead, Marcus theory assumes a homogeneous solvent environment which responds to charges at the redox factors by a linear dielectric response; the total energy of charge solvation is decisive for the contribution of the outer-sphere reorganization to the activation energy.

In contrast to the consideration on the ET in PSII of the Results and Discussion section, in Marcus theory there is no special conformation of the solvent environment foreseen that uniquely supports the ET step; it is the total electrostatic solvation energy that matters, realized by a multitude of conformations of the solvent environment. Therefore, the underlying approximations of Marcus theory render its applicability to redox reactions in a protein environment with multiple conformations of a specific, evolutionary optimized protein-water network problematic, inter alia because it cannot easily capture the here discussed entropic contributions to the free energy of activation. Whether the predictive power of ET theory can be fruitfully used regarding the contribution of inner-sphere processes of contributing redox factors (in PSII and other redox proteins), remains to be answered. In conclusion, the common assumption (in ET theories) of nuclear movements in approximately quadratic potentials needs to be scrutinized for redox factors surrounded by a fluctuating protein-internal HBN.

In summary, our considerations support the appropriateness of discussing the rate-limiting proton-coupled ET step in photosynthetic water oxidation in terms of Eyring's TS theory. We point out the important but apparently poorly understood role of activation entropy and hope that future work can shed more light on the interplay between order and statistical dynamics in PSII water oxidation, biological redox processes in general, homogeneous and heterogeneous catalysis.