БИОЛОГИЧЕСКИЕ МЕМБРАНЫ, 2012, том 29, № 5, с. 367-373

УДК 577.34

Combined Förster—Redfield Theory for Modeling Energy Transfer in Plant Photosynthetic Antenna Complexes © 2012 V. I. Novoderezhkin

Belozersky Institute of Physico-Chemical Biology, Moscow Lomonosov State University, Leninskie Gory, Moscow, 119991 Russia; e-mail: novoder@belozersky.msu.ru Received 25.01.2012

Methodological aspects of modeling of the energy transfers in plant antenna complexes are considered using the combined Förster—Redfield theory. We analyze the advantages of this approach as well as some questionable points, including the choice of a critical value of the exciton coupling (corresponding to the boundary between the Redfield and Förster limits), a spread of the transfer rates induced by energetic disorder, and the accuracy of the theory for the couplings near the critical value.

Keywords: energy transfer, excitons, photosynthetic antenna.

The photosynthetic antenna complexes of higher plants consist of ordered arrays of light harvesting chlorophylls (Chl) as main light-harvesting pigments. The distance between nearest-neighbour pigments can be as short as 9—10 A, thus giving rise to strong pigment—pigment interactions. As a result, the whole antenna is generally characterized by a complicated manifold of excited states, including collective electronic excitations (excitons) with a high degree of de-localization in combination with more localized excitations [1, 2].

In native antenna complexes, excitations are coupled to nuclear motions, i.e. low-frequency phonons of pigment—protein matrix and high-frequency intra-and inter-pigment vibrations. Such coupling gives rise to energy transfers within the excited state manifold, including fast (fs) relaxation between exciton states within strongly coupled clusters and slower (ps) energy migration between clusters or monomeric sites. This energy transfer is the basic mechanism of photosyn-thetic light-harvesting, providing ultrafast cascading from higher- to lower-energy states, an effective energy migration in the antenna and the delivery of excitation energy to the reaction center. In this paper we consider a possibility of application of the combined Forster—Redfield theory for modeling energy transfer in plant antennae, highlighting the advantages as well as some specific features of such approach.

RkkkT = 2 Re \dtAk( t)F* (t) Vkk( t),


Ak( t) = exp {- i&kt - gkkkk( t)}, Fk(t) = exp{- iakt-g*kkk(t) + 2i^k'k'k'k'(t)},


where F(t) and A(t) are line-shape functions corresponding to fluorescence of the donor state and absorption of the acceptor, respectively, while V describes the electrostatic interaction between donor and acceptor. The function gkkkk(t) determines the line-broadening of the k-th exciton state (with the energy œk) due to exciton—phonon coupling, Kkkkk is the corresponding reorganization energy. These quantities are [3, 5]:

gkkkk(0 = ^ (ckn)4gn(0 ; Kkkkk = ^ (ckn)4K ;

(t) = - ГJ-ÇLсп(ш) X J 2пш

- œ

( cos ot - 1 ) - i( sin ot - ш t) d ш

coth -


2 kBT


К =

г сn (ш),

J 2пш


The rate of energy transfer from the state k' to state k is given by [3, 4]:

where gn(t) and Kn correspond to the line-broadening function and reorganization energy of the n-th molecule, which are related to the spectral density Cn(œ) of the exciton—phonon coupling, T is the temperature, kB is the Boltzmann constant. The wavefunction am-








680 720 760

Wavelength, nm


Fig. 1. The 7K fluorescence spectrum measured [14] (dots) and calculated (line) for LHCII using Eq.(1) with experimental spectral density. Vibrational wing increased tenfold (x10) is also shown.

plitude cn corresponds to participation of the «-th molecule in the k-th exciton state.

If the donor and acceptor states are localized at the m-th and «-th sites (i.e. ckm = 1 and ckn = 1) then V is

given by [4]:

Vkk( t) — \M„,


Vkk( t) —


k A/T k


function overlap) are given by their interaction with phonons. The modified Redfield approach gives [3, 4]:

Vkkit) = exp(2gk.k.kk{t) + 2iXkkkkt) x [gkkkk(t) -

- {gkkkk(t) -gkkkk(t) + 2ikkkk} x

x {gkkkk(t) - gkkkk(t) + 2iK'kkk} L


Skkk'k'■(t) - Z c"c"c" c" gn(t) '


n — 1 N

k k k k

^kkk 'k " — Z c„c„c„c„ k„.

n — 1


where Mnm is the interaction energy corresponding to a weak coupling between the localized sites n and m. Switching to the Fourier transforms of F(t) and A(t) we can rewrite the integral in Eq. (1) in a form of donor-acceptor spectral overlap [4]. Thus, Eq. (1) is reduced to the well known Förster formula [6].

The standard Förster formula can be generalized to the case of energy transfer between two weakly connected clusters [7—11]. The rate of energy transfer from the k'-th exciton state of one cluster to the k-th state of the other cluster is

where n and m designate molecules belonging to different clusters. In this generalized Förster formula, the donor and acceptor states k' and k can have an arbitrary degree of delocalization (corresponding to arbitrarily strong excitonic interactions within each cluster). But it is important to note that the inter-cluster interactions Mnm are weak, thus producing only a small spatial overlap between the k' and k wavefunctions.

In the case of significant spatial overlap of the wavefunctions ckm and ck„ the transfer rate cannot be calculated by treating Mnm as a perturbation. In this case the energy transfer should be calculated in the basis of the exciton states of the whole system. The rates of relaxation between these states (with arbitrary wave-


As an example we calculate the energy transfer rates within a dimer as a function of the coupling M12 and energy gap (E1 — E2) between the two molecules. We use the Förster (Eq. (3)) and modified Redfield (Eq. (5)) expressions. Energies ®k and wavefunctions

ckn of the two exciton states can be calculated analytically or evaluated numerical by diagonalization of the free-exciton Hamiltonian of the dimer.

To be more specific we consider the two chlorophyll a (Chl a) molecules with the spectral density Cn(®) determined experimentally for the Chl a from the plant light-harvesting complex LHCII [12]. This spectral density can be approximated by low-frequency overdamped Brownian oscillator and 48 under-damped high-frequency modes [13]. Using this spectral density we can calculate the spectral line shapes and transfer rates (using Eq. (1)) for our model Chl dimer as well as for more complicated systems. In Fig. 1 we show the low-temperature fluorescence profile measured [14] and calculated for LHCII using Eq. (1).

In Fig. 2 we compare the transfer rates between two Chl molecules (as a function of the energy gap E1 — E2 between them) calculated using the Förster (Eq. (3)) and modified Redfield (Eq. (5)) expressions assuming the same spectral density as in Fig. 1. We show only the downhill rate, i.e., the rate of transfer from the molecule with higher energy (in the Förster approach) or the rate of relaxation from the higher exciton level (in the Redfield picture). The difference between energy transfer rates as predicted by the two theories is compared with the delocalization length calculated as the inverse participation ratio (PR) of the exciton wave-

functions, i.e. Ndel = PR-1, where PR = (c[ )4 + (ck )4. For a dimer the inverse participation ratio varies from 1 (in the localized limit) to 2, where the excitation is uniformly delocalized over the two molecules.

Figure 2 shows that in case the energy difference is large (E1 — E2 > 5M12) the excitation is almost localized (1 < Ndel < 1.1). The energy transfer has the form of



Energy gap, cm 1

Fig. 2. The energy transfer rate as a function of the energy gap between two Chl molecules calculated with the Förster and modified Redfield theories (only downhill rates are shown). The delocalization length Nde[ is calculated as the inverse participation ratio of the exciton wavefunctions. The Ndel = 1 line is shown to highlight the deviations from the localized limit. The interaction energy is M12 = 100 cm-1. The transfer rates have been calculated with the same spectral density as in Fig. 1 but for 77K. The specific non-monotonous dependence of the rates on the energy gap is determined by the shape of the vibrational wing.

hopping between the two molecules. In this case the Förster and Redfield theories give approximately the same rate.

Decreasing the energy gap (E1 — E2 < 5M12) results in formation of delocalized states (with Ndel increasing from 1.1 up to 2). In this case the excitonic interactions create a coherent mixing of the two sites. Instead of an excitation jumping from one site to the other, we now have a relaxation between two delocalized states. Increase of the spatial overlap between the delocalized wavefunctions corresponds to having much faster transfer. The deviation of the Redfield rates compared with those predicted by the Förster equation increases in proportion to the deviation of delocalization length from the localized limit.

Both for strong (M12 > 100 cm—1) and moderate coupling (M12 > 30 cm-1) the Redfield theory predicts 2—5 times faster transfer in the isoenergetic case (E1 — E2 = 0) due to delocalization. But this deviation of the Redfield rates from the Förster limit become anomalously (and unrealistically) high in the weak coupling limit (M12 < 20—30 cm-1) as shown in Fig. 3.

Thus, for M12 = 10 cm-1 and large energy gaps (E1 - E2 > 30 cm-1) we got localized excitations and transfer rates near 0.25 ps-1 (time constants around 4 ps) predicted both by the Förster and Redfield theories. On the other hand, for small gaps the Redfield theory predicts an abrupt increase in the transfer rate, i.e. up to 23 ps-1 (time constant of40 fs) in the E1 - E2 = 0 limit. Thus, we got about 2 orders of magnitude increase in transfer rate! Formally

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