Theory of Triplet Excitation Transfer in the Donor-Oxygen-Acceptor System: Application to Cytochrome b₆f

Abstract

Theoretical consideration is presented of the triplet excitation dynamics in donor-acceptor systems in conditions where the transfer is mediated by an oxygen molecule. It is demonstrated that oxygen may be involved in both real and virtual intramolecular triplet-singlet conversions in the course of the process under consideration. Expressions describing a superexchange donor-acceptor coupling owing to a participation of the bridging twofold degenerate oxygen’s virtual singlet state are derived and the transfer kinetics including the sequential (hopping) and coherent (distant) routes are analyzed. Applicability of this theoretical description to the pigment-protein complex cytochrome b₆f, by considering the triplet excitation transfer from the chlorophyll a molecule to distant β-carotene, is discussed.

Introduction

Photosynthesis starts with the absorption of a solar photon by specific pigments, namely (bacterio)chlorophylls and carotenoids, bound to the so-called light-harvesting proteins. After a rapid equilibration the resulting excitation energy migrates between light-harvesting complexes, finally reaching the photosystems or reaction centers, where its transfer triggers a fast charge separation, through which it is transduced into the chemical potential. Although the whole process, from the initial photon absorption to the charge separation, occurs with a quantum yield close to unity, the chlorophyll molecules being in their excited singlet state may experience, with a low but significant yield, the transition into the triplet state through intersystem crossing (1). Chlorophyll triplet states are able to stimulate the production of the oxygen singlet state, one of the most harmful reactive oxygen species for living organisms. Carotenoid molecules are playing an important photoprotective role in the photosynthetic membrane by quenching the chlorophyll excitation in the triplet states through the chlorophyll to carotenoid triplet-triplet transfer (2, 3). Although the position of the energy level of the carotenoid triplet state is not precisely known, it is estimated to be at <7882 cm⁻¹ to ensure it does not sensitize the oxygen singlet (4).

As of this writing, triplet-triplet transfer from chlorophyll to carotenoid has been known for almost five decades (5). However, it is still not completely understood in many situations. In complexes from anoxygenic bacteria, it occurs in the nanosecond time range (6), while it was reported to be ultrafast in complexes from oxygenic organisms (7). This was proposed to be an adaptation of the photosynthetic apparatus of these organisms to an oxygen-rich environment (7), but the molecular mechanisms underlying this ultrafast triplet-triplet transfer are not fully characterized. It was proposed to involve a partial delocalization of the spin density of the triplet state on both chlorophyll and carotenoid molecules, but the details of this mechanism are still unclear.

In cytochrome b₆f, chlorophyll, the role of which is still largely unknown, is present (8). Although this chlorophyll is largely quenched, it produces, although with a low yield, a triplet state that is quenched by a carotenoid molecule situated nearby (9). Kinetics of such triplet-triplet transfer from chlorophyll to carotenoid is relatively rapid (<8 ns) (10). However, in the structure of this protein, the distance between the chlorophyll and carotenoid molecules is ∼14 Å, apparently incompatible with the observed kinetics (10). It was tentatively proposed that this excitation transfer was mediated by molecular oxygen O₂. In this article, we analyze in detail the bridging capacity of this molecule for the triplet-triplet transfer between a donor (D) and an acceptor (A) molecule.

Materials and Methods

Theory

To analytically describe the kinetics of triplet transfer in a DO₂A system, i.e., an oxygen-bridged triplet donor and acceptor, we use a model where each unit n (the donor, the acceptor, and molecular oxygen) are characterized by only two working electronic states, which are defined next.

Model formulation

For the D and A molecules, the working states refer to the excited triplet (³D^∗(m) and ³A^∗(m)) and the ground singlet (¹D₀ and ¹A₀) states, respectively (where m = 0, ±1 holds for spin projections). As to the bridging O₂ molecule, its ground state is the triplet state ³Σ⁻_g whereas the lowest singlet states are ¹Δ_g and ¹Σ⁺_g. These three states are formed by two electrons occupying π^∗_x and π^∗_y molecular orbitals (MOs) (11). From calculations and experimental data (11, 12), it follows that the state ¹Σ⁺_g lies at 5200 cm⁻¹ ≈ 0.65 eV above the ¹Δ_g state so that the singlet state ¹Δ_g is the closest to the ground state ³Σ⁻_g. If the energy difference corresponding to the ³D^∗→₁D₀ transition exceeds that of the ¹Σ⁺_g→³Σ⁻_g transition, the contribution of the oxygen state ¹Σ⁺_g is likely not to play any important role in the D-A triplet transfer in comparison to the contribution from the ¹Δ_g state. Thus, the twofold degenerated singlet state ¹Δ_g can be taken into consideration for this excitation transfer. For further consideration, the working states of molecular oxygen will be denoted as ³O₂(m) and ¹O₂(l), where the symbol l = π^∗_x,π^∗_y indicates the doubly occupied MO.

Physically, transfer of the triplet excitation occurs through the transition from the initial (I), bridging (B), and final (F) states of the whole DO₂A system. To specify these states, let note that at weak interaction between D(A) and O₂ molecules, the energies of the singlet (S = 0), triplet (S = 1), and quintet (S = 2) states of the ³D^∗(³A^∗)-³O₂ pair differs insignificantly. This means that instead of a complete set of nine states |S,M> with S and M being the total spin and spin projection of the pair, one can use a much more suitable complete set of nine states |³D^∗(³A^∗)(m)>|³O₂(m′)>, where m, m′(= 0, +1, −1) are the spin projections of a molecular triplet state. Thus, the proper states of the DO₂A system (including the ground state G) involved in the triplet-triplet energy transfer can be described as the product of separate molecular states:

$$\begin{array}{l}|I\rangle ={|}^3{D}^{\ast }\left(-m\right)\rangle {|}^3{\text{O}}_2\left(m\right)\rangle {|}^1{A}_0\rangle ,\\ |B\rangle ={|}^1{D}_0\rangle {|}^1{\text{O}}_2^{\ast }\left(l\right)\rangle {|}^1{A}_0\rangle ,\\ |F\rangle ={|}^1{D}_0\rangle {|}^3{\text{O}}_2\left(m^{\prime }\right)\rangle {|}^3{A}^{\ast }\left(-m^{\prime }\right)\rangle ,\\ |G\rangle ={|}^1{D}_0\rangle {|}^3{\text{O}}_2\left(m^{\prime }\right)\rangle {|}^1{A}_0\rangle .\end{array}$$ (1)

The fact that, during the excitation transfer, the total spin projection is conserved, is taken into account in Eq. 1. Because the bridging state B has a zero spin projection, in the I(F) state, the spin projections of the O₂ and D(A) molecules must be of the opposite sign. As to conservation of the total (zero) spin during the same excitation transfer, this circumstance is taken properly into account in the calculation of transition matrix elements with use of Klebsch-Gordon coefficients (see Section A in the Supporting Material).

In the absence of the magnetic field, a triplet state is threefold degenerated, and, thus, the molecular energy is independent of spin projection $m$, i.e., E(³j^∗(m)) ≡ E(³j^∗), and E(¹O₂^∗(π^∗_x) = E(¹O₂^∗(π^∗_y) ≡ E(¹O₂^∗)). Thus, the energies ε_a that correspond to the proper states under consideration can be defined as the sum of the relevant molecular energies:

$$\begin{array}{l}{\epsilon }_I=E\left({}^{3}D^{\ast }\right)+E\left({}^3\text{O}_2\right)+E\left({}^{1}A_0\right)\mathrm{,}\hfill \\ {\epsilon }_B=E\left({}^{1}D_0\right)+E\left({}^1\text{O}_2^{\ast }\right)+E\left({}^{1}A_0\right)\mathrm{,}\hfill \\ {\epsilon }_F=E\left({}^{1}D_0\right)+E\left({}^3\text{O}_2\right)+E\left({}^{3}A^{\ast }\right)\mathrm{,}\hfill \\ {\epsilon }_G=E\left({}^{1}D_0\right)+E\left({}^3\text{O}_2\right)+E\left({}^{1}A_0\right).\hfill \end{array}$$ (2)

Kinetics

Weak interaction of the bridging O₂ with the D and A molecules results in a nonadiabatic excitation transfer. As discussed above, a triplet excitation transfer from D could cause a transition into a singlet state of O₂, with a subsequent transformation of this singlet excitation into a triplet excitation of A. Such a sequential process includes the two transitions from I to B, and B to F. At the same time, an oxygen molecule mediates direct (coherent) I → F transition, during the course of which the singlet-oxygen states participate only virtually in the triplet-triplet excitation transfer. Fig. 1 summarizes these transfer routes.

Figure 1.

Oxygen-mediated sequential (hopping) $I\rightleftarrows B\rightleftarrows F$ and direct (coherent) $I\rightleftarrows F$ routes for triplet excitation transfer between D and A molecules (thin and thick arrows, respectively). (Wave lines) Decay of excitation within each separate molecule. Δε_BI(F) = ε_B − ε_I(F) is the energy gap among the bridging (B), initial (I), and final (F) states of the DO₂A system.

This model contains two routes for the excitation transfer, namely sequential (hopping) and coherent (direct), which are similar to the bridge-mediated process used to describe the nonadiabatic D-A electron transfer, developed in the framework of the coarse-grained approximation (for a detailed theoretical description, see Petrov and co-workers (13, 14, 15). However, in the case of the excitation transfer, the process occurs without any change of the energy states of the D and A (due to the presence/absence of charges in the case of the electron transfer). Here radiative and radiationless processes have to be additionally taken into account.

According to the model formulated above, the excitation energy transfer is described by the following set of kinetic equations (see Section A in the Supporting Material):

$$\begin{array}{l}{\dot{P}}_D\left(t\right)=-\left(r_{-i}+r_{if}+k_D\right)P_D\left(t\right)+r_i{P}_{{\text{O}}_2}\left(t\right)+r_{fi}{P}_A\left(t\right)\mathrm{,}\\ {\dot{P}}_{{\text{O}}_2}\left(t\right)=-\left(r_i+r_f+k_{{\text{O}}_2}\right)P_{{\text{O}}_2}\left(t\right)+r_{-i}{P}_D\left(t\right)+r_{-f}{P}_A\left(t\right)\mathrm{,}\\ {\dot{P}}_A\left(t\right)=-\left(r_{-f}+r_{fi}+k_A\right)P_A\left(t\right)+r_f{P}_{{\text{O}}_2}\left(t\right)+r_{if}{P}_D\left(t\right)\mathrm{,}\\ {\dot{P}}_G\left(t\right)=k_D{P}_D\left(t\right)+k_{{\text{O}}_2}{P}_{{\text{O}}_2}\left(t\right)+k_A{P}_A\left(t\right).\end{array}$$ (3)

Here, particular integral state populations P_j(t) are defined by Eq. A19 in the Supporting Material. In addition, the populations should also satisfy the normalization condition

$$P\left(t\right)+P_G\left(t\right)=1,$$ (4)

where

$$P\left(t\right)=\sum _{j=D\mathrm{,}{\text{O}}_2\mathrm{,}A}{P}_j\left(t\right)$$ (5)

is the probability to find one of the D(A) and O₂ molecules in the excited (triplet or singlet) states, respectively. Quantity P_G(t) is the probability to find the DO₂A system in the ground (unexcited) state. Transfer rates

$$\begin{array}{l}{r}_{i\left(f\right)}=3K_{{\text{O}}_2\to \text{D}\left(\text{A}\right)}\mathrm{,}\\ r_{if}=3K_{D\to A}\mathrm{,}\\ r_{-i\left(-f\right)}=2K_{D\left(A\right)\to {\text{O}}_2}\mathrm{,}\\ r_{fi}=3K_{A\to D},\end{array}$$ (6)

characterize the efficiency of transitions between the excited states of the DO₂A system (see Fig. 1). These rates are expressed via rate constants K_j→j′ and K_j′→j, which should satisfy the detailed balance relationship

$$\begin{array}{l}{K}_{D\left(A\right)\to {\text{O}}_2}=\text{exp}\left(-\Delta {\epsilon }_{D\left(A\right)}/k_{\text{B}}T\right)K_{{\text{O}}_2\to \text{D}\left(\text{A}\right)}\mathrm{,}\hfill \\ K_{A\to D}=\text{exp}\left(-\Delta \epsilon /k_{\text{B}}T\right)K_{D\to A},\hfill \end{array}$$ (7)

with ε_D(A) $\simeq$ ε_BI(F) (compare to Fig. 1) and Δε $\simeq$ ε_I – ε_F being the corresponding transition gaps. The latter reduces to the differences

$$\Delta {\epsilon }_{D\left(A\right)}=\Delta E_{{\text{O}}_2}\left({}^1\Delta _g{\to }^3{\Sigma }_g^{-}\right)-\Delta E_{D\left(A\right)}\left(S_0\to T\right),$$ (8)

and

$$\Delta \epsilon =\Delta E_D\left(S_0\to T\right)-\Delta E_A\left(S_0\to T\right)$$ (9)

between energies of the ¹Δ_g → ³Σ⁻_g and S₀ → T transitions in respective molecules. Rate constants K_j′→j are given in Eq. A2 in the Supporting Material with couplings defined by Eq. B5 in the Supporting Material. Rates $k_{D\left(A\right)}={\left[{\tau }_{\text{dec}}^{\left(D\left(A\right)\right)}\right]}^{-1}$ and $k_{{\text{O}}_2}={\left[{\tau }_{\text{dec}}^{\left({\text{O}}_2\right)}\right]}^{-1}$ determine the characteristic decay times τ_dec associated with the combined (radiative and radiationless) transitions from the excited states to the ground states of the intervening molecules.

Let us consider a transfer process by choosing the following initial conditions:

$$\begin{array}{l}{P}_D\left(0\right)=1\mathrm{,}\hfill \\ P_{{\text{O}}_2}\left(0\right)=0\mathrm{,}\hfill \\ P_A\left(0\right)=0\mathrm{,}\hfill \\ P_G\left(0\right)=0.\hfill \end{array}$$ (10)

At t = 0, the donor molecule is in its triplet state while the oxygen molecule and the acceptor molecule are in their ground triplet and ground singlet states, respectively. Excitation transfer within the DO₂A system takes place if the following inequality is satisfied,

$${\tau }_{tr}\ll {\tau }_{\text{dec}},$$ (11)

where τ_tr and τ_dec are, respectively, the characteristic time of an excitation transfer and the above noted decay time. The condition defined by Eq. 11 suggests a validity of the inequalities

$$\begin{array}{l}{r}_{-i}+r_{if}\gg k_D\mathrm{,}\hfill \\ r_i+r_f\gg k_{{\text{O}}_2}\mathrm{,}\hfill \\ r_{-f}+r_{fi}\gg k_A,\hfill \end{array}$$ (12)

which indicates that a total kinetics rate given by the set of expressions in Eq. 3 is decomposed into two principally different stages. The first stage includes an excitation transfer within the D, O₂, and A units. Time evolution of the probabilities P_D(t), $P_{{\text{O}}_2}\left(t\right)$, and P_A(t) occurs on a timescale on the order of Δt ∼τ_tr. At this timescale, a complete probability, Eq. 5, is nearly conserved so that

$$\begin{array}{l}P\left(t\right)\simeq \mathrm{1,}\hfill \\ P_G\left(t\right)\simeq 0.\hfill \end{array}$$ (13)

In line with inequalities given by Eq. 12, the solution of the expressions in Eq. 3 can be found for zero rate constants k_D, $k_{{\text{O}}_2}$, and k_A. This yields

$$\begin{array}{l}{P}_D\left(t\right)=P_D^{\left(\text{qst}\right)}\left(1-\alpha \left(t\right)\right)+\left[\left(c_1+r_i\right)/c\right]\beta \left(t\right)+\gamma \left(t\right)\mathrm{,}\\ P_{{\text{O}}_2}\left(t\right)=P_{{\text{O}}_2}^{\left(\text{qst}\right)}\left(1-\alpha \left(t\right)\right)+\left(r_{-i}/c\right)\beta \left(t\right)\mathrm{,}\\ P_A\left(t\right)=P_A^{\left(\text{qst}\right)}\left(1-\alpha \left(t\right)\right)+\left(r_{if}/c\right)\beta \left(t\right),\end{array}$$ (14)

where the time behavior is determined by the values

$$\begin{array}{l}\alpha \left(t\right)=\frac{1}{K_1-K_2}\left[K_1\text{exp}\left(-K_{2}t\right)-K_2\text{exp}\left(-K_{1}t\right)\right]\mathrm{,}\\ \beta \left(t\right)=\left[\text{exp}\left(-K_{2}t\right)-\text{exp}\left(-K_{1}t\right)\right]\mathrm{,}\\ \gamma \left(t\right)=\frac{1}{K_1-K_2}\left[K_1\text{exp}\left(-K_{1}t\right)-K_2\text{exp}\left(-K_{2}t\right)\right],\end{array}$$ (15)

with c = K₁ − K₂ and

$$\begin{array}{l}{K}_{\mathrm{1,2}}=\frac{1}{2}\left[c_1+d_1\pm \sqrt{{\left(c_1-d_1\right)}^2+4c_2{d}_2}\right]\end{array}$$ (16)

being the overall excitation transfer rates. Note that the following definitions are used:

$$\begin{array}{l}{c}_1=r_f+r_{-f}+r_{fi}\mathrm{,}\\ c_2=r_{if}-r_f\mathrm{,}\\ d_1=r_i+r_{-i}+r_{if}\mathrm{,}\\ d_2=r_{fi}-r_i.\end{array}$$ (17)

Overall transfer rates determine the characteristic times

$$\begin{array}{l}{\tau }_{tr}^{\left(j\right)}=K_j^{-1}\mathrm{,}\hfill \\ \left(j=\mathrm{1,2}\right)\mathrm{,}\hfill \end{array}$$ (18)

of a two-exponential kinetics describing the D-A triplet excitation transfer at t ≪ τ_dec. This transfer is accompanied by formation of an intermediate singlet excited state of an oxygen molecule.

If K₁ ≫ K₂, then the characteristic time τ_tr of an excitation transfer is associated with quantity τ_tr⁽²⁾. At large times, populations defined by the expressions in Eq. 14 reduce to their quasi-steady-state values:

$$\begin{array}{l}{P}_D^{\left(\text{qst}\right)}=\frac{1}{K_1{K}_2}\left(r_i{r}_{fi}+r_f{r}_{fi}+r_i{r}_{-f}\right)\mathrm{,}\\ P_{{\text{O}}_2}^{\left(\text{qst}\right)}=\frac{1}{K_1{K}_2}\left(r_{-i}{r}_{fi}+r_{-i}{r}_{-f}+r_{if}{r}_{-f}\right)\mathrm{,}\\ P_A^{\left(\text{qst}\right)}=\frac{1}{K_1{K}_2}\left(r_i{r}_{if}+r_f{r}_{if}+r_f{r}_{-i}\right).\end{array}$$ (19)

The second stage reflects a much slower kinetic process describing the transition of the excited DO₂A system into its ground state G. The process occurs on a timescale the order of Δt ∼ τ_dec, and already exhibits itself at t ≥ 5τ_tr. Due to the strong inequality defined by Eq. 11, the timescale Δt ∼ τ_dec is much larger than the timescale that is characteristic for the excitation transfer. Thus, the decay of the excited molecular states occurs on the background of a quasi-equilibrium distribution among the excited states. This means that a population of the jth excited state can be represented in the following form, as

$$\begin{array}{l}{P}_j\left(t\right)=P_j^{\left(\text{qst}\right)}P\left(t\right)\mathrm{,}\hfill \\ \left(t\ge 5{\tau }_{tr}\right)\mathrm{,}\hfill \end{array}$$ (20)

where quantities P_j^(qst) are given by Eq. 19 and satisfy the normalization condition

$$P_D^{\left(\text{qst}\right)}+P_{{\text{O}}_2}^{\left(\text{qst}\right)}+P_A^{\left(\text{qst}\right)}=1.$$ (21)

At the same time, a total normalization condition is given by Eqs. 4 and 5, and Eq. A19 in the Supporting Material. Taking into account Eqs. 20 and 21, the set of expressions in Eq. 3 is reduced to ${\dot{P}}_G\left(t\right)=k_{\text{dec}}P\left(t\right)$ and $\dot{P}\left(t\right)=-k_{\text{dec}}P\left(t\right)$, and thus

$$\begin{array}{l}{P}_G\left(t\right)=1-\text{exp}\left(-k_{\text{dec}}t\right)\mathrm{,}\\ P\left(t\right)=\text{exp}\left(-k_{\text{dec}}t\right)\mathrm{,}\\ \left(t\ge 5{\tau }_{tr}\right).\end{array}$$ (22)

Here, the decay rate

$$k_{\text{dec}}={\tau }_{\text{dec}}^{-1}=P_D^{\left(\text{qst}\right)}{k}_D+P_{{\text{O}}_2}^{\left(\text{qst}\right)}{k}_{{\text{O}}_2}+P_A^{\left(\text{qst}\right)}{k}_A$$ (23)

specifies the overall characteristic decay time τ_dec. The decay process is performed along the triplet$\to$singlet and singlet$\to$triplet channels associated with the D(A) and O₂ molecules, respectively. The characteristic channel decay times, τ_dec^(j) = k_j⁻¹, are determined by the partial decay rates k_j (j = D, A, O₂). Contribution of each channel in the total decay rate defined by Eq. 23 strongly depends on quasi-steady-state populations P_j^(qst) formed during the first (transfer) stage.

Results and Discussion

Analytical expressions given by Eq. 14 demonstrate the possibility to fulfill an excitation transfer within the DO₂A system as long as t ≪ τ_dec. In the absence of specific quenchers, the characteristic decay times ${\tau }_{\text{dec}}^{\left(D\left(A\right)\right)}$ for spin forbidden triplet$\to$ singlet transitions in the optically active organic molecules are ∼(1–100) μs. Because for a singlet excited oxygen, ¹O^∗₂, the lifetime is ∼20 μs (16), the expressions defined by Eq. 14 are applicable in a wide (0.1–100) ns time domain.

Let us consider two important cases when the energy difference, Δε_D, defined by Eq. 8 is positive or negative, while assuming the energy difference Δε_A to be positive and fixed. The condition Δε_A > 0 can guarantee a negligible generation of singlet oxygen at t ∼ τ_tr, i.e., after finishing the excitation transfer within the DO₂A system. The condition Δε_D < 0 reflects a downhill excitation transfer associated with conversion of the D and O₂ molecules from their triplet to their singlet states. The process occurs if the following relationship between corresponding rates is satisfied: r_−i ≫ r_i, r_if and r_f ≫ r_−f, r_fi. In this case, the two-exponential kinetics corresponding to quasi-steady-state populations as follows from the expressions in Eq. 19 is characterized by the overall transfer rates of

$$\begin{array}{l}{K}_1={\left[{\tau }_{tr}^{\left(1\right)}\right]}^{-1}\simeq r_{-i}\mathrm{,}\hfill \\ K_2={\left[{\tau }_{tr}^{\left(2\right)}\right]}^{-1}\simeq r_f.\hfill \end{array}$$ (24)

When the difference between these rates is large, the excitation transfer kinetics can be discriminated between fast and slow phases.

Fig. 2 a shows a temporary behavior of populations when r_−i ≫ r_f, thus, reflecting the situation when coupling between D and O₂ is stronger than that between O₂ and A. During the fast phase, the conversion of the triplet excitation located on D into the singlet state of O₂ is expected to occur with high probability. The slow phase is evidently associated with backward conversion of the singlet excitation (located at the O₂) into a triplet excitation on A. In this case, the characteristic time τ_tr coincides with τ_tr⁽²⁾.

Figure 2.

Kinetics of the D-A triplet excitation transfer on a timescale Δt ∼ τ_tr(≪τ_dec) at Δε_D$\simeq$ε_IB > 0. The major contribution to the transfer process arises from the sequential route. If r_−i ≫ r_f, then probability of the formation of intermediate state I (with the ¹O^∗₂) can be rather large (a), whereas at r_−i ≪ r_f the noted probability is small (b). The calculations are based on Eqs. 6–9, and Eqs. A2, B5, and A17 (see the Supporting Material) (at G⁰_FI = Δε_IF$\simeq$ −Δε) with T = 0.025 eV(≈290 K), Δε_D = −0.05 eV, Δε_A = −0.15 eV, Δε = 0.2 eV, λ_D$\simeq$λ_BI = 0.05 eV, λ_A$\simeq$λ_BF = 0.05 eV, λ_DA$\simeq$λ_FI = 0.06 eV, $V_{{\text{O}}_2,D}$$\simeq$V_B,I = 5 × 10⁻⁴ eV, and $V_{A\mathrm{,}{\text{O}}_2}$$\simeq$V_F,B = 2 × 10⁻⁴ eV (a); and $V_{{\text{O}}_2,D}$$\simeq$V_B,I = 1.5 × 10⁻⁴ eV and $V_{A\mathrm{,}{\text{O}}_2}$$\simeq$V_F,B = 5 × 10⁻³ eV (b).

The kinetics depicted in Fig. 2 b corresponds to the opposite situation, when r_−i ≪ r_f is satisfied. In this case, the excitation transfer when ³D^∗ + ³O₂ → ¹D + ¹O₂^∗ is much slower than the excitation decay due to interconversion when ¹O₂^∗ + ¹A → ³O₂ + ³A^∗. It is worthwhile to mention that the probability of the singlet oxygen formation is small during both fast and slow kinetic phases. Thus, during the course of the excitation transfer in the DO₂A system, the decay of the triplet excitation of D proceeds jointly with the appearance of the triplet excitation on A.

When Δε_D > 0, the hopping rate constants r_i and r_f should strongly exceed other rate constants. As a result, the fast and slow kinetic phases become very different as is evident from the definition of the characteristic times ${\tau }_{tr}^{\left(\text{fast}\right)}$ and ${\tau }_{tr}^{\left(\text{slow}\right)}$ of the different phases, which determine the overall transfer rates

$$\begin{array}{l}{K}_1={\left[{\tau }_{tr}^{\left(\text{fast}\right)}\right]}^{-1}\simeq r_i+r_f\mathrm{,}\\ K_2={\left[{\tau }_{tr}^{\left(\text{slow}\right)}\right]}^{-1}\simeq k_{\text{fwd}}+k_{\text{bwd}},\end{array}$$ (25)

where the rates

$$\begin{array}{l}{k}_{\text{fwd}}=r_{if}^{\left(\text{seq}\right)}+r_{if}\mathrm{,}\\ k_{\text{bwd}}=r_{fi}^{\left(\text{seq}\right)}+r_{fi}\end{array}$$ (26)

characterize forward (k_fwd) and backward (k_bwd) triplet transfer between D and A molecules. The terms

$$\begin{array}{c}{r}_{if}^{\left(\text{seq}\right)}=\frac{r_{-i}{r}_f}{r_i+r_f}\mathrm{,}\\ r_{fi}^{\left(\text{seq}\right)}=\frac{r_{-f}{r}_i}{r_i+r_f},\end{array}$$ (27)

define the contributions associated with the sequential route, ${}^{3}D^{\ast }{}^3{\text{O}}_2{}^1{A}_0\rightleftarrows {}^1{D}_0{}^1{\text{O}}_2^{\ast }{}^1{A}_0\rightleftarrows {}^1{D}_0{}^3{\text{O}}_2{}^3{A}^{\ast }$, of the excitation transfer. Contributions r_if and r_fi arise from the coherent (direct) route, ${}^{3}D^{\ast }{}^3{\text{O}}_2{}^1{A}_0\rightleftarrows {}^1{D}_0{}^3{\text{O}}_2{}^3{A}^{\ast }$. Along the coherent route, an intermediate bridging state ¹D₀¹O₂^∗1A₀ is virtually involved in the transfer process.

During the fast phase (covering, approximately, the time domain $0\le t\le 5{\tau }_{tr}^{\left(\text{fast}\right)}$), the populations of the excited molecular states exhibit only minor alteration and achieve their intermediate values $P_D^{\left(\text{int}\right)}\simeq 1$, $P_{{\text{O}}_2}^{\left(\mathrm{int}\right)}\simeq r_{-i}/\left(r_i+r_f\right)\ll 1$, and $P_D^{\left(\text{int}\right)}\simeq r_{if}/\left(r_i+r_f\right)\ll 1$ (Fig. 3). Main changes occur when $\Delta t\sim {\tau }_{tr}^{\left(\text{slow}\right)}$ so that the characteristic time of excitation transfer is defined by this value, i.e., ${\tau }_{tr}\simeq {\tau }_{tr}^{\left(\text{slow}\right)}$. A negligible population of the bridging state is associated with the excited oxygen ¹O₂^∗. This is explained by the fact that population of the bridging state requires thermal activation. Therefore, if the condition $r_{-i}/r_i=exp\left(-\Delta {\mathit{\epsilon }}_D/k_{\text{B}}T\right)\ll 1$ is satisfied, then the forward excitation transfer process ${}^{3}D^{\ast }+{}^3{\text{O}}_2\to {}^1{D}_0+{}^1{\text{O}}_2^{\ast }$ is less effective than the backward process ${}^{1}D_0+{}^1{\text{O}}_2^{\ast }\to {}^3{D}^{\ast }+{}^3{\text{O}}_2$. These properties of the sequential route come from the fact that each hopping step implies a transformation of two triplet states into two singlet states or vice versa. For instance, hopping rate r_−i characterizes the ${}^{3}D^{\ast }{}^3{\text{O}}_2{}^1{A}_0\to {}^1{D}_0{}^1{\text{O}}_2^{\ast }{}^1{A}_0$ transition corresponding to the reaction ${}^{3}D^{\ast }+{}^3{\text{O}}_2\to {}^1{D}_0+{}^1{\text{O}}_2^{\ast }$. At the same time, a coherent route appears as the direct distant transfer of the triplet excitation so that rate r_if characterizes the efficiency of reaction ${}^{3}D^{\ast }+{}^1{A}_0\to {}^1{D}_0+{}^3{A}^{\ast }$.

Figure 3.

Kinetics of the D-A triplet excitation transfer on the timescale Δt ∼ τ_tr (≪τ_dec) for Δε_D < 0. The major contribution to the transfer process is associated with the coherent route. Population of intermediate state I (and, thus, the ¹O^∗₂ state) is negligible. The calculations are the same as in Fig. 2 except Δε_D = 0.12 eV, Δε_A = 0.24 eV, Δε = 0.12 eV, $V_{{\text{O}}_2,D}$ = 1.5 × 10⁻⁴ eV, and $V_{A\mathrm{,}{\text{O}}_2}$ = 5 × 10⁻² eV.

Because realization of the sequential route depends strongly on thermal activation of the bridging state, whereas a coherent route is less sensitive to the temperature, a contribution of the coherent route may dominate the input given by the sequential pathway. Relative contribution of each of them is evidently estimated by the ratio

$$\eta =r_{if}^{\left(\text{seq}\right)}/r_{if}.$$ (28)

The analytical form for this ratio follows from expressions given by Eqs. 6, 7, and 27, and Eqs. A2 and B5 in the Supporting Material. For instance, if r_i ≪ r_f, then

$$\eta \simeq \frac{1}{4}{\left|\frac{\Delta {\mathit{\epsilon }}_D\Delta {\mathit{\epsilon }}_A}{\left(\Delta {\mathit{\epsilon }}_D+\Delta {\mathit{\epsilon }}_A\right)V_{A\mathrm{,}{\text{O}}_2}}\right|}^2\frac{{\left(FC\right)}_{{\text{O}}_2\to D}}{{\left(FC\right)}_{D\to A}}{e}^{-\Delta {\mathit{\epsilon }}_D/k_{\text{B}}T}.$$ (29)

In the opposite case, when r_i ≫ r_f, the substitution of $V_{A\mathrm{,}{\text{O}}_2}$ and ${\left(FC\right)}_{{\text{O}}_2\to D}$ by $V_{D\mathrm{,}{\text{O}}_2}$ and ${\left(FC\right)}_{{\text{O}}_2\to A}$, respectively, has to be performed in Eq. 29. Let us estimate the ratio given by Eq. 29 at room temperature (T $\simeq$ 300 K) using a simple Marcus form given by Eq. A17 in the Supporting Material for the Franck-Condon factors. As a rule, the reorganization energy λ for an electron transfer within and between proteins is ∼1 eV (17). In the case under consideration, a similar energy has to be reduced by more than one order of magnitude. Therefore, for estimations we can safely use λ_ab ≈ (0.05–0.15) eV. Setting Δε_D = 1000 cm⁻¹, Δε_A = 2000 cm⁻¹, and λ_IB $\simeq$ λ_FB $\simeq$ λ_IF ≡ λ, the expression given by Eq. 29 is satisfied when $\left|V_{A\mathrm{,}{\text{O}}_2}\right|/\left|V_{D\mathrm{,}{\text{O}}_2}\right|>10$. Moreover, if $\left|V_{A\mathrm{,}{\text{O}}_2}\right|{>10}^{-2}$ eV, the transfer along the coherent route is dominating (η < 1). Fig. 4 demonstrates that relative efficiency of the coherent route increases with couplings between O₂-D and O₂-A by decreasing the temperature. Indeed, the kinetics of the triplet excitation transfer is mainly associated with the coherent route of the triplet excitation transfer in the DO₂A system, as evidently demonstrated by curve 4 in Fig. 3.

Figure 4.

Relative contributions of sequential and coherent routes in the D-A triplet transfer as the function of reorganization energy λ ≡ λ_D = λ_A = λ_DA. At the fixed $V_{{\text{O}}_2,D}$, even small alteration in the coupling $V_{A\mathrm{,}{\text{O}}_2}$ or temperature leads to noticeable change in the contributions associated with sequential and coherent routes. Calculations with Eq. 28 are at the same parameters as in Fig. 3 except $V_{A\mathrm{,}{\text{O}}_2}$ = 5 × 10⁻³ eV, T = 0.025 eV (curve 1); $V_{A\mathrm{,}{\text{O}}_2}$ = 5 × 10⁻³ eV, T = 0.023 eV (curve 2); $V_{A\mathrm{,}{\text{O}}_2}$ = 6 × 10⁻³ eV, T = 0.025 eV (curve 3); and $V_{A\mathrm{,}{\text{O}}_2}$ = 5 × 10⁻³ eV, T = 0.025 eV (curve 4).

Additive contribution of the sequential and coherent routes to the forward and backward D-A transfer rates as defined by the expressions in Eq. 26 appear only at negligible population of the bridging state B (see Fig. 1 for definitions). A similar conclusion has also been made by considering the electron transfer in a D-A system (13, 14, 18). If the bridge population is not small, then both routes are strongly mixed. As a result, the kinetics cannot be characterized by the simple expressions in Eqs. 25–27, and a correct description of the transfer process requires the use of more general forms defined by the expressions in Eqs. 16 and 17.

Triplet energy transfer in pigment-protein complexes

It is known that chlorophyll (Chl) molecules are able to generate the singlet oxygen via the triplet-singlet conversion: ³Chl^∗ + ³O₂ → ¹Chl + ¹O₂^∗. Car molecules can prevent the singlet oxygen formation, though only at very short distances (in various photosynthetic proteins, Chl and Car molecules are often in close contact). If that is not the case, the triplet-triplet energy transfer (TTET) has to be mediated by a bridging unit. This surprising phenomenon was attributed to the presence of an oxygen molecule entering the protein via a specific intraprotein channel, which could play the role of a bridging unit (9, 10). Below, we will apply our theoretical results to describe this phenomenon.

Characteristic excitation decay times τ_dec for the ³Chl^∗a, ³Car^∗, and ¹O₂^∗ molecules are (100–400) μs (19, 20), (1–5) μs (21, 22), and 20 μs (16, 21), respectively. Because these times strongly exceed the characteristic time τ_tr (<8 ns) of the excitation transfer in cytochrome b₆f, the inequality described by Eq. 11 is then largely satisfied. Therefore, by describing the oxygen-mediated ³Chl^∗a→³Car^∗ excitation transfer, the general expressions from Eqs. 14–19, 22, and 23, can be used, setting D ≡ Chl a and A ≡ Car. Experimental data describing the kinetics in cytochrome b₆f at timescales of ∼(0.1–1) ns have not yet been measured. Thus, we can only suppose the possible types of routes responsible for a TTET in this complex.

Mobility of the molecular oxygen within cytochrome b₆f is very high. At room temperature, an oxygen shuttles through the intraprotein oxygen channel within tens of picoseconds (23). Because ¹O₂^∗ is not observed during the TTET, we can suppose that only the ³O₂ molecule shuttles in the oxygen channel. To avoid the presence of the ¹O₂^∗ molecule in the channel, it is necessary to assume a negligible probability for the formation of the bridging state B(¹Chl a¹O₂^∗1Car) during the ³Chl^∗a → ³Car^∗ TTET. Physically, a negligible probability becomes very possible if the major transfer process is associated with the coherent route. The latter can be active even at zero population of the bridging state (the relevant kinetics rate is depicted in Fig. 3). Note that for chosen parameters, the timescale of slow phase may correspond to the characteristic time τ_tr of the triplet-triplet transfer in cytochrome b₆f (3 ns).

From the kinetic description presented above, it follows that contribution of each route in the TTET is controlled by the gap, given by Eq. 8, which coincides with the difference of the gaps associated with singlet-triplet transition in Chl a (Car) and triplet-singlet transition in O₂. Estimation of the gap $\Delta {\text{E}}_{{\text{O}}_2}\left({}^1\Delta _g{\to }^3{\Sigma }_g^{-}\right)$ for isolated O₂ itself is problematic (24): the simple system has limitations and difficulties because both O₂ and O₂-complexes are open-shell systems. Methods such as MP2, CCSD, and MRCI predict the singlet-triplet energy gap of the isolated oxygen to be from 0.9 eV to 1.16 eV (24, 25) while the experimental estimations give 0.99 eV of this value (26). Electron paramagnetic resonance data show that in the b₆f complex, the heme b_n can bind the O₂ molecule via one of the pyrrol rings (27) fixing, thus, the O₂ molecule in the hydrophobic region between hemes b_n and c_n (28, 29). Because the influence of the nonpolar environment on the singlet-triplet gap of O₂ is insignificant, one can then take $\Delta {\text{E}}_{{\text{O}}_2}\left({}^1\Delta _g{\to }^3{\Sigma }_g^{-}\right)$ as (0.95–1.15) eV for estimations.

It is known that the B3LYP functional does not correctly account for charge transfer states, which are inherent for chlorophylls (30). This is due to the self-interaction error in the orbital energies obtained in the ground-state DFT calculations (31). If the transitions are limited to those that include only localized transitions, the triplet state of Chl a can be predicted as 1.28 eV using TD-DFT (B3LYP/TPZV). Calculations (that take into account different equilibrium position of nuclei in ground, S₀, and excited, T₁, states) give 1.19 eV for the vertical excitation of the triplet-singlet transition (32). Bearing in mind the fact that protein environment effects the S₀-T₁ gap, the qualitative estimations of ΔE_{Chl a}(S₀ → T) can be performed giving (1.05–1.25) eV.

The β-carotene can be also calculated using the DFT approach. The ground state can be predicted in terms of DFT (B3LYP/6-31G(d)) while the triplet state energies must be investigated using the TD-DFT formalism (33). The triplet energy of β-carotenes, ΔE_Car(S₀ → T), as follows from these estimations, is close to the experimental value 0.8 eV (the experimental value is equal to 0.88 eV (34)). Thus, in Eq. 8, the gap ΔE_A(Car) is essentially positive whereas the gap ΔE_{D(Chl a)} can be positive or negative. This means that in the b₆f complex, the TTET can be realized along both routes.

To understand the formation of transfer matrix elements $V_{D\left(\text{Chl}a\right),{\text{O}}_2}$ and $V_{{\text{O}}_2,A\left(\text{Car}\right)}$, we refer to Section B in the Supporting Material where possible physical mechanisms of hopping $T+T\rightleftarrows S_0+S$ intermolecular transitions are considered. Generally, these transitions could be associated with single-step two-electron and synchronous two-step single-electron exchange pathways in the framework of the HOMO-LUMO model for D(Chl a) and A(Car) (see Fig. 5). Similar pathways are well known in the literature (see, for instance, Closs et al. (35), Harcourt et al. (36), Scholes et al. (37), You et al. (38), Scholes and Fleming (39), Curutchet and Voityuk (40), and Voityuk (41)). An important result of the literature studies is that the attenuation factor that characterizes the decrease of the rate of the D-A TTET is approximately twice as large as typical values for electron transfer between the same D and A sites (35, 36, 38, 39, 40, 41). (Detailed analysis of mechanisms responsible for formation of bridge-assisted electron/hole D-A superexchange coupling in biological and chemical systems can be found in numerous articles (42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57).) We have only to pay attention that if D and A units are spaced by the bridging sites, then the Dexter’s mechanism (58) of single-step two-electron exchange is much less effective than those of the bridge-assisted two-step single-electron exchange. This result has been clearly supported by direct calculations of a distant dependence of TTET in water and organic solvents (40).

Figure 5.

Two-electron (a) and single-electron (b) exchange mechanisms responsible for T + T → S + S₀ transitions between the jth and O₂ molecules. Squares of respective transition matrix elements are given by Eqs. B8 and B12 (see the Supporting Material). If the j-O₂ coupling is performed via superexchange pathways, then the respective matrix element is given by Eq. B14 (see the Supporting Material).

To clarify the mechanism of rather fast (<8 ns) triplet excitation transfer in cytochrome b₆f, let us refer to the structure of this complex (9, 10, 59, 60). The complex contains the heme b_n situated between the Chl a and Car molecules, with the center-to-center distance from heme b_n to Chl a being ∼16 Å. As it has been noted above, the heme b_n is able to bind the oxygen (28) and this oxygen is kept in the region between the edge of the PR of the heme b_n and just opposite the plane of the PR of the heme c_n (28, 29). Based on this fact, we can assume that the immobilized oxygen can mediate the excitation transfer between the Chl a and Car molecules according to a two-stage sequential process, namely ³Chl a + ³O₂ → ¹Chl a + ¹O^∗₂ and ¹O₂ + ¹Car → ³O₂ + ³Car, as well as to a direct one-stage process, ³Chl a → ³Car. Because the heme b_n is situated closer to Car, one can expect stronger coupling of O₂ to Car than to Chl a. Therefore, if Δε_{D(Chl a)} < 0 (compare to the inset in Fig. 2 b) and if the Franck-Condon factors do not differ significantly in value for the transitions ³Chl a + ³O₂ → ¹Chl a + ¹O₂^∗ and ¹O₂ + ¹Car → ³O₂ + ³Car, then owing to a strong inequality between the couplings, $\left|V_{D\left(\text{Chl}a\right),{\text{O}}_2}\right|{}^2<<\left|V_{{\text{O}}_2,A\left(\text{Car}\right)}\right|{}^2$, the condition r_−i ≪ r_f has to be strongly satisfied during the TTET process. In line with our kinetic description, this means that the D-A TTET reflects a slow kinetic phase that is characterized by respective transfer rate $k_{D\to A}\approx k_{{\text{O}}_2\to \text{Car}}=r_{-i}$. At this regime, no distinct generation of a toxic oxygen is expected. The same conclusion is obtained in the case of a distant (coherent) regime when the rate of the D-A TTET is given by Eq. A2 in the Supporting Material with |V_ba|² = |V_{A(Car),D(Chl a)}|² as follows from Eqs. B5 and B14 in the Supporting Material. As to the transfer rate, it is given by the expression k_D→A ≈ k_{Chl a→Car} = (1/3) r_if (see Fig. 1 and the inset in Fig. 3).

Before estimating the Chl a-O₂ and O₂-Car couplings let us note that, generally, the most comprehensive investigation of any energy transfer process in a molecular system includes combined experimental and theoretical studies. One can find excellent examples of these in Scholes and Fleming (39) and Di Valentin et al. (61). However, studies of the oxygen-mediated D-A TTET in the b₆f complex are in the initial stages of development. Therefore, concerning the Chl a-O₂ and O₂-Car couplings, let us note that despite the protein structure of b₆f being available (29), the functional role of separate molecular groups responsible for a formation of TTET pathways, including the precise structure of an intraprotein oxygen channel, has not been completely established. We propose, thus, a probable, shortest pathway shown in Fig. 6, which does not contradict the physical point of view. More detailed studies of D-A TTET (including rigorous numerical estimations of respective couplings) can be done only after reliable specification of the spatial molecular groups responsible for the formation of Chl a-Car TTET pathways and after receiving detailed experimental data concerning the kinetics at a timescale of ∼(0.1–1) ns. Below, bearing in mind this circumstance, we give only qualitative estimations for the Chl a-O₂ and O₂-Car couplings.

Figure 6.

A possible route for the oxygen-mediated triplet transfer from Chl a to β-carotene in the b₆f complex (bold lines). It requires that the O₂ molecule be interacting with the b_n heme. Coupling to the A(Car) is performed through the participation of PR (O₂-1-2(A) pathway) whereas the coupling of O₂ with D(Chl a) includes two mediators, the PR and the Trp-molecule (O₂-1-2(Trp)-3(D) pathway.

If the molecular oxygen interacts with heme b_n, the direct overlap between wave functions of oxygen and β-carotene will be very small, and even smaller between oxygen and Chl a. Therefore, we associate the TTET mechanism with the superexchange electron/hole transfer pathways defined by couplings between Chl a and O₂, and O₂ and Car. These pathways are represented in Fig. 7. Respective matrix elements are given by Eqs. B3, B12, and B24 (see the Supporting Material). They have been derived by taking into account the fact that the D and A sites are not identical. Thus, the McConnel’s form for a distant D-A electron/hole coupling (40, 62) has been modified with basic expressions used in a quantum mechanics context for the transition matrix elements (63). Such a modification is reflected in the electron/hole coupling for the irregular expressions of Eqs. B26 and B27 (see the Supporting Material) and the regular expressions of Eqs. B28 and B29 (see the Supporting Material) for the bridging sites.

Figure 7.

Formation of superexchange $t_{{\pi }_x^{\ast },L^{\left(j\right)}}^{\left(\sup \right)}$ and $t_{H^{\left(j\right)},{\pi }_y^{\ast }}^{\left(\sup \right)}$ couplings at transitions (a)${}^3\text{O}_2\left(+1\right),{}^3{j}^{\ast }\left(-1\right)\to {}^2{\text{O}}_2^{-}\left({\pi }_y^{\ast },+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right),{}^2{j}^{+}\left(-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$ and (b) ${}^2\text{O}_2^{-}\left({\pi }_y^{\ast },+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right),{}^2{j}^{+}\left(-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\to {}^1{\text{O}}_2^{\ast }\left({\pi }_x^{\ast }\right),{}^1{j}_0$. The transitions are associated with the HOMO and LUMO pathways (left and right columns, respectively).

To estimate the transfer matrix elements, we employ Eq. B24 (see the Supporting Material) and write

$$\left|V_{{\text{O}}_2\left({\pi }_x^{\ast }\right)\mathrm{,}j}\right|=\left|V_{H^{\left(j\right)};{\pi }_y^{\ast }}\right|V_{{\pi }_x^{\ast };L^{\left(j\right)}}\Vert F_{{\text{O}}_2\mathrm{,}j}|e^{-\left(\beta /2\right)R}.$$ (30)

The quantity $\left|V_{{\text{O}}_2\left({\pi }_x^{\ast }\right)\mathrm{,}j}\right|$ in the case of j = Car may be estimated from the b₆f structure units (9,10) where an excitation transfer from the bound O₂ to β-carotene can include the heme’s PR (via its pyrrol ring 1) and the carotene (via its cyclohexene ring 2). In this O₂-1(PR)-2(Car) pathway (compare to Fig. 6), a PR exhibits itself as a single bridging unit. Therefore, one has to set R = 0 in Eq. 30. Concerning $\left|V_{H^{\left(\text{Car}\right)};{\pi }_y^{\ast }}\right|$ and $\left|V_{{\pi }_x^{\ast };L^{\left(\text{Car}\right)}}\right|$, where H^(Car) and L^(Car) are carotene MOs, respectively, it is worth noting that electronic coupling for π-electrons of organic and biological molecules lies in a wide range from 1 meV to 1 eV (26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46). In the polar environment, the oxidation potentials of organic molecules with conjugated bonds (like Chl, Car, Tyr, and Trp) are arranged in the domain of [−2,+2] eV dependent upon their electronic states (see, for example, Dashdorj et al. (9)). This means that the gaps obtained in Eqs. B22 and B23 (see the Supporting Material) are associated with respective electron and hole superexchange pathways (via the bridging PR), and are ∼(1,2) eV whereas $F_{{\text{O}}_2},\text{Car}=\left[1/\left(I\left({}^3\text{Car}^{\ast }\right)-A\left({}^3\text{O}_2\right)\right)+1\left(I\left({}^1\text{Car}\right)-A\left({}^1\text{O}_2^{\ast }\right)\right)\right]\sim \left(0.5-1\right)$ (eV⁻¹). Therefore, if one takes t_νν−1 ∼ (0.3–0.5) eV, then $V_{{\text{O}}_2\left({\pi }_x^{\ast }\right),A\left(\text{Car}\right)}|\sim \left({10}^{-2}-{10}^{-3}\right)$ eV.

Analogous estimations can be obtained for the absolute value of the matrix element $V_{{\text{O}}_2}\left({\pi }_x^{\ast }\right),D\left(\text{Chl}a\right)$, thus, giving $\left|V_{{\text{O}}_2}\left({\pi }_x^{\ast }\right),D\left(\text{Chl}a\right)\right|\sim \left({10}^{-2}-{10}^{-3}\right)\text{exp}\left[-\left(\beta /2\right)R\right]$ (in eV). Factor exp[−(β/2)R] can be evaluated considering a superexchange pathway from the center of heme b_n to the edge of Chl a including not only the bridging PR (1) and the Trp molecule (2) but also the space domain between the edge of heme b_n and the Trp edge nearest to this heme (Fig. 6). It yields R ≈ (8–10)Å. Experimental dependence of the long-range D-A triplet excitation transfer gives a value ranged from 0.3 to 0.5 Å⁻¹ for the attenuation factor β, depending on the conformation state of planar bridging units (64, 65). Therefore, one can suppose that in nonpolar biological structures containing the planar molecules, the effective attenuation factor β can be ranged between (0.5 and 0.8) Å⁻¹. Bearing in mind the fact that the planar Trp molecule covers more than half of distance R, we can take β = 0.6 Å⁻¹. At R = (8–10)Å, giving exp[−(β/2)R] ∼ (10⁻¹–10⁻²), it means that $\left|V_{{\text{O}}_2}\left({\pi }_x^{\ast }x\right),D\left(\text{Chl}a\right)\right|$ can be in the range of (10⁻³–10⁻⁴) eV. Thus, one can see that qualitative estimations of the couplings $V_{{\text{O}}_{2}D}=V_{{\text{O}}_2\left({\pi }_x^{\ast }\right)},D\left(\text{Chl}a\right)$ and $V_{{\text{O}}_{2}A}=V_{{\text{O}}_2\left({\pi }_x^{\ast }\right)},A\left(\text{Car}\right)$ cover the values that have been used to explain the kinetics of the D-A TTET in the model D-O₂-A system. The latter mimics a similar kinetics in the Chla-O₂-Car system on the timescale τ_tr < 8 ns.

Conclusions

We have considered the kinetic aspects of triplet excitation energy transfer from molecule D to molecule A in conditions when the transfer is mediated by an oxygen molecule. Weak interaction of the oxygen molecule with D and A molecules results in the nonadiabatic regime of the excitation transfer. At this regime, the transfer rates characterizing the transitions between states of the system actually coincide with transition rates between adiabatic terms of the molecules. When the excitation transitions in the DO₂A system are much slower than the deexcitation processes (see Eq. 11), a detailed analysis of the time evolution of populations during the transfer becomes possible. We show that the regime of the D-A TTET is strongly controlled by the transfer rates defined by either sequential or coherent transfer routes. The sequential route includes a hopping process, and thus is mainly responsible for singlet oxygen formation.

Decay of singlet oxygen is controlled by the subsequent triplet excitation transfer to the acceptor molecule. If the energy of the initial state I is above the energy of the bridging state B, then the probability for the occupation of the bridging state with singlet oxygen is defined by the relation between rates characterizing the appearance of the singlet oxygen molecule and its transformation into the triplet states (Fig. 2). When the rate of the former process is a dominant one, the portion of the singlet oxygen population is rather large. However, if the departure of the singlet excitation from O₂ to the A is much faster than the formation of such excitation, this portion is small.

The probability for the oxygen molecule to be excited becomes negligible when the population of the bridging state B requires a thermal activation. In this case, the efficiency of the sequential route drops down according to the importance of the activation energy. The coherent route then becomes a dominant one, and, thus, the state involving the bridging molecule becomes only virtually involved in the process. Because the virtual involvement does not implicate thermal activation, this route can give a major contribution to the D-A triplet excitation transfer even at room temperature. Thus, either the specific sequential or the coherent route could be responsible for the oxygen-mediated triplet transfer between Chl a and β-carotene in the cytochrome b₆f complex. The answer can be obtained by considering a temperature dependence of the D-A TTET with the process of penetration of the O₂ into the oxygen channel taken into account. From a physical point of view, the minimization of the excitation transfer along the sequential route is the best way to avoid the formation of toxic singlet oxygen. According to the model of nonadiabatic excitation transfer, such a minimization is achieved if energy of the ³Chl^∗ a → ¹Chl a conversion is less than the energy corresponding to the ¹O₂^∗ → ³O₂ transition. Such a situation can likely be realized, with a large probability, in the system BChl-O₂-Car (66). As to the b₆f complex, we show that the mechanism of D-A TTET in this complex is associated with the distant superexchange coupling of a molecular oxygen with Chl a and Car.

Author Contributions

E.G.P. and L.V. conceived and designed the contribution; E.G.P. and S.-H.L. performed the theoretical description of the model and relevant calculations; B.R. analyzed data from the point of view of their applicability to a real system under consideration; and E.G.P., B.R., and L.V. wrote the article.

Editor: Daniel Beard.

Supporting Citations

References (67, 68, 69, 70, 71, 72) appear in the Supporting Material.