PhotosynthesisResearch 42: 9-15, 1994. © 1994KluwerAcademicPublishers.Printedin the Netherlands. Regular paper

Exciton states of the antenna and energy trapping by the reaction center V l a d i m i r I. N o v o d e r e z h k i n 2'3 & A n d r e i P. R a z j i v i n 1'2 ~A. N. Belozersky Institute of Physico-Chemical Biology, Moscow State University, Moscow 119899, Russia (for correspondence and~or reprints), 21nternational Laser Center of Moscow State University and 3Scientific Research Center on Technological Lasers, Russian Academy of Science, Troitsk, Moscow Region, 142092, Russia Received9 December1993;acceptedin revisedform6 May 1994

Key words: energy migration, exciton dynamics, light-harvesting antenna, primary processes, trapping

Abstract Forward and back energy transfer between antenna and RC in the photosynthetic apparatus of purple bacteria was studied taking into account the exciton states of the antenna. The exciton states were calculated for core antenna configuration in the form of a circular aggregate of N identical BChl molecules with the CN-symmetry. The influence of pigment inhomogeneity on the proposed exciton description of the antenna and its interaction with RC was investigated. The ratio between the rate constants of forward and back energy transfer between the exciton levels of the antenna and RC was obtained as a function of the temperature, the number of antenna BChls and the antenna exciton level position with respect to BChl special pair level of RC. A versatile analytical expression for this ratio which is independent of the BChl special pair level position and its dipole orientation was derived. The proposed model results in an irreversible excitation trapping by RC even at room temperature.

Abbreviations: RC - reaction center; BChl - bacteriochlorophyll; ps - picosecond Introduction Several experimental facts obtained with the antenna of purple photosynthetic bacteria such as the shape of picosecond absorbance difference spectra (Borisov et al. 1982; Razjivin et al. 1982; Nuijs et al. 1985; Danielius et al. 1989), the absorption anisotropy kinetics (Sundstrom et al. 1986; Bergstrom et al. 1988), the anomalously high value of absorption changes per absorbed quantum (Novoderezhkin and Razjivin, 1993b) are difficult to interpret using traditional models based on the assumption of localized exciton migration (Borisov et al. 1982; Razjivin et al. 1982; Van Grondelle et al. 1988; Hunter et al. 1990; Sundstrom et al. 1986; Pullerits and Freiberg 1991; Timpmann et al. 1991; Visschers et al. 1993; Van Mourik et al. 1992; Pullerits et al. 1994). An alternative explanation starts from the models which draw on the notion of exciton interactions between BChl molecules in the light-harvesting

antenna. In (Novoderezhkin and Razjivin, 1993a, Novoderezhkin and Razjivin, 1993b) a model of the antenna as a circular aggregate of strongly interacting BChl molecules was proposed for interpretation of the above-listed effects. Such a model is consistent with circular structures formed by association of a,~pairs of antenna proteins (Zuber, 1985, 1987). If N pigment molecules of the associate are arranged in a closed ring with the CN-symmetry, the dipole transition moment of each molecule oriented at an angle to the ring plane, only two exciton levels (nondegenerative and two-fold degenerative) with different absorption anisotropy are dipole allowed. It would appear reasonable that the long-wavelength 'minor' band in the absorption spectra of B875 complexes corresponds to transition from the ground to the lowest exciton level of the circular aggregate whereas the 'major' band corresponds to transition to the higher (two-fold degenerative) exciton level (Novoderezhkin and Razjivin, 1993a; Novoderezhkin and Razjivin, 1993b). The

10 model of the light-harvesting complex in the form of a circular aggregate allows one to explain the shape of the picosecond absorbance difference spectra for low and high excitation pulse energy, the anomalously high value of the photoinduced absorption changes at a bleaching peak of the difference spectrum for low pulse energy as well as the kinetics of the induced absorption anisotropy. It should be noted that exciton interactions in circular aggregates were treated in (Pearlstein and Zuber, 1985) for calculating optical spectra of purple bacteria antennae and in (Reddy et al. 1992) for interpretation of the hole-burning data obtained for Rhodobacter sphaeroides at 4.2 K. The special case of a circular aggregate with dipole moments arranged in the plane of the aggregate was analyzed in both papers. This spatial model fails to explain essential optical properties, like circular dichroism (for coplanar dipoles CD is equal to zero). Also it might be well to point out that pigment inhomogeneity (Pullerits and Freiberg 1991; Timpmann et al. 1991; Van Grondelle et al. 1992; Visschers et al. 1993; Van Mourik et al. 1993) must influence the excitonic states of the antenna. The degree (power) of its influence was estimated in (Reddy et al. 1992). It was shown that the value of the inhomogeneous broadening for the B875 complex is equal to 80 cm - l . This value is much less than the energy gap between the exciton levels (i.e. the energy gap between the BChI880 and BCh1896 levels) which is equal to 200 cm -1. That is why the diagonal disorder due to pigment inhomogeneity will not destroy coherent exciton states. In this case the influence of the diagonal energy disorder Hamiltonian on the exciton eigenstates can be treated perturbatively as in (Reddy et al. 1992) for a molecular system with planar degenerated dipole moments. Similar calculations for a circular aggregate with dipole moments perpendicular to the circle plane was carried out in (Knapp, 1984). It is evident that the antenna model based on the exciton concepts must result in a new view of excitation trapping dynamics by RC. The processes of forward energy transfer from antenna exciton levels to RC as well as the processes of back energy transfer to the antenna differs essentially from those processes of localized excitations. In particular, the rates of forward and back energy transfer are nearly equal in the case of delocalized excitations which results in an irreversible energy trapping. The aim of this paper is to treat theoretically excitation dynamics in the core light-harvesting anten-

y

Fig. 1. The model of a photosyntheticunit. The opened circles correspondto light-harvestingBChl molecules;the closedcircleis the BChl dimerof RC; x, y, z are the Cartesiancoordinates;dR and dn are dipole momentsof the RC BChl dimerand the n-th antenna molecule, respectively;ARy is BChl dimerdisplacementfromthe geometricalcenterof the circle,formedby antennamolecules.

na and RC of purple bacteria in the context of the antenna model proposed before (Novoderezhkin and Razjivin, 1993a; Novoderezhkin and Razjivin, 1993b) with regard to pigment inhomogeneity.

Antenna model

A core light-harvesting complex of purple bacteria will be considered to be a circular aggregate with the CNsymmetry where N is the total number of identical BChl molecules (Fig. 1). In the general case a dipole transition moment of the n-th molecule makes some angle with the circle plane and is equal to d n = dlli'Cos0n -4- dl127sin0n + d ± Z

(1)

where K, ~ are the unit length vectors in the circle plane; 2' is the unit vector perpendicular to the circle plane. If the interaction of antenna molecules with RC (assuming it is weak) and the exciton-phonon interaction are ignored the Hamiltonian for the antenna would be expressible as (Davydov, 1971): H = E ( A E + An) P+Pn + M ~ P+Pm n n=m-4-1

(2)

where P+ and Pn are the Pauli creation and destruction operators; AE is the excitation energy of an isolated

11 (monomer) molecule; An describes the energy disorder due to pigment inhomogeneity; M is the matrix element of excitation transfer between the nearest neighbor molecules (let everywhere below M < 0). In the case of a homogeneous antenna (An = 0) exciton interactions of antenna BChls result in the splitting of the electronic level into N exciton levels with energies:

Ek = AE + 2McosOk,

0 = 27r/N

Since the x axis can be chosen arbitrarily in the circle plane the above mentioned absorption lines have different anisotropy. If antenna homogeneity is disturbed at the m-th site (An = Aan,m, i.e. the m-th molecule transition energy differs from the same parameter of other antenna molecules) then the dipole moment values are: '

(3)

and eigenfunctions:

do -- d o t

e-ik~n I n >

(4)

(

dl+ = 21/2 d l c o s 0 m + d o

n

where k takes N integer values 0, + 1,:~2,...4-(N/2-1), N/2 for even N ; In> is the state when the n-th molecule is excited. For An 5~ 0 in the zero order approximation of the perturbation theory each two-fold degenerative level [+k> with the energy Ek will split to two sublevels Ik>+ with the energies E~, where k=l, 2 .... (N/2-1):

= 2 -1/2 (lk > -¢-e-i~'l - k >)

gk~ = gk --b Vk,k 4- IVk,-kl; Vk,_ k = ]Vk,_k[ei40 Vkl.k2 -~- < kllAnP+Pnlk2 >

(5)

In the first order approximation of the perturbation theory one can obtain for the levels k=l, 2 .... (N/2-1): N/2- I

,

4-

Vk,, k

[k >+ = [k >4- + Z gk -- Ek, [k' >4k'=l +

Vk~,,k Z 2- U 2 - ] k ' > Ek -- Ek, k~=0,N/2

4- ~--- Vk,,k -I- e-teVkt,_k gk,,k

(6)

Vk"k Ik' >

I k > ' = Ik> + $

Ek_Ek-------~

[ %

E~ = Ek + Vk,k

(7)

(in all sums the term with k'= k is absent). For homogeneous antenna (An= 0) the dipole moment of transition from the ground state, Ig>, to the one-exciton state, Ik>, is non-zero only for the two lowest exciton levels, k=0 and k=+ 1. In the case of linearly polarized light wave I~ = (ezz'+ exg)-E, exp(iwt) the corresponding dipole moment values are equal: dt = NU2exdN/2

d'k- = 0 ; dk+ = 21/2 2dl

cos0m+

OEk_Eo J

E'o = Eo + A/N;Ek~ = Ek + A/N 4- IA/NI;

(9)

The presented equations show that in the case of a diagonal disorder transitions to higher exciton levels are optically allowed. The corresponding transition dipole moments are linear combinations of do and dl values which results in some smoothing of the wavelength anisotropy dependence within the line. The polarization at the long-wavelength side becomes lower and at the short-wavelength side some non-polarized absorption in the form of a broad wing can be seen. The level shift averaged over all antenna aggregates gives a broadening of each exciton component.

Interaction between antenna and reaction center

For the non-degenerative levels k=0, N/2 it will be:

do = N1/2ezd± ;

A/N l~2cos0m

d l_ = 21/2dlsin0m;

[k > = N-1/2 Z

Ik > e

d

(8)

Interaction of the antenna and RC involves the processes of energy transfer from the k-th exciton level to the BChl dimer of RC as well as those of back energy transfer. The BChl dimer of RC will be considered to be arranged near the center of the circle (Fig. 1). The interaction of the RC BChl dimer with the antenna BChl molecules is assumed to be weak. What this means is the structure of antenna exciton levels is not distorted by RC. The energy diagram for the antenna and RC is shown in Fig. 2. The kinetic equations for this system take the form: d ~Pk

= --FkPk + CtkPRc

+ f0 t (~k,k+qPk+q -- q0k+q,kPk) dt'

12

Ik>

IRC>

I0>

%

molecule; Mk is the matrix element of the interaction of RC with the k-th exciton level; f~k is the spectral overlap integral for transitions between k states and RC (f+/f~- = exp(fl(Ek - ER)) where/3 = 1/kBT is a temperature; ER is the energy of the IRC> state). As an example, let us consider the case where the dipole moment of the BChl special pair of RC, dR, is equal to (see Fig. 1) :

IRE+->

d~ = d~f÷ d.~

cxk c~Rc .... Ig> Fig. 2. The energy scheme of the antenna and RC. Ig> is the ground state of the antenna; IRC>is the state with excitation localization at RC (at BChl dimer); IRC+-> is the state with charge separation; Ik> is the delocalized exciton state of the antenna; ak and a aRC are the total rates of linear losses of the Ik> level in the antenna and RC, respectively (note that the total rates include fluorescencelosses for the dipole allowed levels); ac is the rate of charge separationin RC; a + is the transition rate from the antenna k state to the state with energy localized at RC; a~- is the rate of back transition.

d p --r'RcPRc R + cE

M° + Rn-3 where an is the unit vector aligned from the BChl dimer of RC toward the n-th antenna molecule; Rn is the distance between the BChl dimer of RC and the n-th molecule. Assuming A=0 and 6 + exciton states are denoted as Ik> for brevity. According to (Kenkre, 1978) a + and a~- are equal to:

MnMm < nlk > t . < ml k >,

II ~170

= f~ IMkl2;

the BChl dimer can be displaced from the central point of the circular aggregate in the y axis direction by the ARy value (relative displacement is 6 = ATy/R where R is the radius of the circle). In the dipole-dipole approximation

OLk+ Pk,"

=

a ~ = f~ E

(12)

(11)

where Mn is the matrix element of the interaction of RC (of the special BChl pair of RC) with the n-th antenna

36d dll

,

k = +2 (14)

The total rate of energy transfer to RC is equal to W + = + a k Pk; the back transfer rate is W - = ~ a k . If the BChl dimer of RC is placed at the center of the circular aggregate of antenna BChls then energy transfer occurs only from the lowest dipole allowed levels with the rate proportional to its populations W + = const • (X2po + P1 + P - l ) where X 2 = f+/f+(2dRdj_/d~dll) 2. For a little gap between the exciton levels or at a high temperature Po=Pt=P_I=I/N. In this case an excitation is localized at a single molecule. For a large gap or at a low temperature only low-lying levels will be populated. This situation corresponds to an excitation delocalization over the circular aggregate of antenna BChl molecules. This results in the transfer rate increase approximately N-fold. For purple bacteria at room temperature it may be assumed/3 (E1-Eo)=I which gives Po=0,576; PI=P_ t=0,212. The excitation transfer rate will be equal to 0,576.N for X=0 and to 0,212.N for X=c~.

13 Now let us discuss the ratio between forward and back transfer rates, W + / W - . This ratio is described by a versatile equation which is independent of the RC dimer position and of the mutual orientation of RC and antenna BChl molecules (parameters 6 and X). This statement is valid for the Bolzmann distribution of exciton levels population Pk = const • exp(--3EK) and fk+/f~- = exp(/~(Ek -- ER)): -1

W +/w-

= ( ~ k e x p ( - - 3 ( E k - ER) )

(15)

In the case of localized excitations when levels Ek are close together or at high temperatures it may be assumed that fl(Ek -- ER)=flEAR where EAR is the energy gap between the antenna and RC. Then: W + / W - = 1 / N . exp(C3EAR)

(16)

If the energy gap, EAR, is small enough the rate of back energy transfer from RC to the antenna will be N times that of forward transfer from the antenna to RC. Assuming the time constant of forward transfer is equal to 60 ps, one can obtain the time constant of back transfer equal to 2.5 ps for N=24. If the time constant of charge separation in RC is 3 ps then the probabilities of energy trapping and back transfer will be 45% and 55%, respectively. It means that RC is a non-effective (reversible) trap. In the case of delocalized excitations when the levels are well separated one may assume I/3(Ek --ER)I >> 1 for all but one or two levels. If the RC level is in resonance with the lowest exciton level k=0 one can assume 3(Eo - ER) = 0, ~(E-¢-I - ER) = 1, and fl(E+2 - ER) >> 1 at r o o m temperature. Then according to Eq. 15 W + / W - = 0.576. For the time constant 60 ps, the time constant of back transfer will be 34.6 ps. If the time constant of charge separation in RC is 3 ps then the probabilities of energy trapping and back transfer will be 92% and 8%, respectively, i.e. RC is an effective (quasi-irreversible) trap. If the RC level is in resonance with the other exciton level k = + l one can assume 3(Eo - ER) = --I, ;3(Ea:l --ER) = 0, and 3 ( E ± 2 - E R ) > 1. Then W + / W - = 0.212. For the time constant 60 ps, the time constant of back transfer will be 12.7 ps and the probabilities of energy trapping and back transfer will be 81% and 19%, respectively. In this case RC is an effective (quasi-irreversible) trap as well.

Discussion The most pronounced manifestation of exciton interactions of antenna molecules is an Nl/2-fold increase of dipole moment of the allowed exciton levels of the molecular aggregate compared to the monomer dipole moment. This involves an N-fold (within a factor of 3-4) increase of the photoinduced absorption changes, the rate of fluorescence emission and the rate of energy transfer to RC. The structure of exciton levels determines the shape of the temperature dependence of these parameters as well. The interaction of RC with the antenna exciton state may be interpreted as interaction with one supermolecule. For 6=0 the matrix element of interaction is quite the same as for interaction with one discrete molecule (see eq. (13)), except that the dipole moment of the molecule is replaced by the dipole moment of transition to the exciton state determined by equation (8) or (9). In this case the rates of energy transfer from the supermolecule to RC and back are equal within the Bolzmann factor. This kinetic scheme suggests irreversible trapping of excitation in RC and fits the experimental data (see for example (Abdourakhmanov et al. 1989, Otte et al. 1993). Within the framework of the proposed model kinetic dependencies must be biexponential. The fast component, reflecting energy redistribution between the exciton levels, will have the time constant within the pico- or subpicosecond range. The slow component will reflect energy quenching by opened (,=60 ps) or closed (r=200 ps) RCs or linear losses in RC-less antenna (r=600 ps) respectively. For describing kinetic dependencies adequately one should take into account that each exciton level of the molecular aggregate has a set of sublevels corresponding to librational modes with relaxation times of 20-30 ps (de Boer et al. 1987). The similar time constants correspond to vibronic relaxation of a photoexcited chromophore embedded in a protein matrix (Li and Champion, 1994). Generalization of equations (10) with regard to the processes of excitation and subsequent cooling of librational modes will allow one to describe the dynamic red shift with a time constant of 10-30 ps observed in fluorescence (Timpmann et al. 1991) and photoinduced absorption spectra (Sundstrom et al. 1986; Bergstrom et al. 1988; Hunter et al. 1990). The properly accounted excitation of librational modes must introduce corrections to the selection rules for light absorption and to the kinetics of energy transfer from the antenna to RC.

14 It should be noticed that the proposed theory is an alternative view of more traditional energy transfer and trapping theories in photosynthesis, based on localized exciton approach (Timpmann et al. 1993; Pullerits et al. 1994).

Acknowledgements The research described in this publication was made possible in part by Grant No. MKG000 from the International Science Foundation and in part by Grant No. 94-04-12779-a from the Russian Foundation for Basic Research.

References Abdourakhmanov IA, Danielius R and Razjivin AP (1989) Efficiency of excitation trapping by reaction centres of complex B890 from Chromatium minutissimum. FEBS Lett 245: 47-50. Bergstrom H, Westerhuis WHJ, Sundstrom V, Van Grondelle R, Niederman RA and Gilibro T (1988) Energy transfer within the isolated B875 light-harvesting pigment-protein complex of Rhodobacter sphaeroides: at 77K studied by picosecond absorption spectroscopy. FEBS Lett 233: 12-16. de Boer S, Vink KJ, and Wiersma DA (1987) Optical dynamics of condensed molecular aggregates: accumulated photon-echo and hole-burning study of the J-acrgregate. Chem Phys Lett 137: 99-106. Borisov A Yu, Gadonas RA, Danielius RV, Piskarskas AS and Razjivin AP (1982) Minor component B-905 of light-harvesting antenna in Rhodospirillum rubrum chromatophores and the mechanism of singlet - singlet annihilation as studied by difference selective picosecond spectroscopy. FEBS Lett 138: 25-28. Danielius RV, Mineyev AP and Razjivin AP (1989) The cooperativity phenomena in a pigment-protein complex of light-harvesting antenna revealed by picosecond absorbance difference spectroscopy. FEBS Lett 250: 183-186. Davydov AS (1971) Theory of Molecular Excitons, 262 pp, New York: Plenum Press. Hunter CN, Bergstrom H, Van Grondelle R and Sundstrom V (1990) Energy transfer dynamics in three light-harvesting mutants of Rhodobacter sphaeroides: a picosecond spectroscopy study. Biochemistry 29: 3203-3207. Kenkre VM (1978) Model for trapping rate for sensitized fluorescence in molecular crystals. Phys Stat Sol (b) 89:651-654. Knapp EW (1984) Lineshapes of molecular aggregates. Exchange narrowing and intersite correlation. Chem Phys 85: 73-82. Li P, Champion PM (1994) Investigations of the thermal response of laser excited biomolecnles. Biophys. J. 66: 430--436. Novoderezhkin VI and Razjivin AP (1993a) Exciton interactions and their influence on picosecond absorbance difference spectra of light-harvesting antenna of purple bacteria. In: KorppiTommola J (ed) Laser Spectroscopy of Biomolecules (Proc. SPIE Vol. 1921), pp 102-106. Novoderezhkin VI and Razjivin AP (1993b) Excitonic interactions in the light-harvesting antenna or photosynthetic purple bacteria

and their influence on picosecond absorbance difference spectra. FEBS Lett 330: 5-7. Nuijs AM, Van Grondelle R, Joppe HLP, Van Bochove AC and Duysens LNM (1985) Singlet and triplet excited carotenoid and antenna bacteriochlorophyll of the photosynthetic purple bacterium Rhodospirillum rubrum as studied by picosecond absorbance difference spectroscopy. Biochim Biophys Acta 810: 94-105. Otte SCM, Kleinherenbrink FAM and Amesz J (1993) Energy transfer between the reaction center and the antenna in purple bacteria. Biochim Biophys Acta 1143: 84-90. Pearlstein RM and Zuber H (1985) Exciton states and energy transfer in bacterial membranes: The rule of pigment-protein cyclic unit structures. In: Michel-Beyerle ME (ed) Antennas and Reaction Centers of Photosynthetic Bacteria, pp 53~1, Berlin: SpringerVerlag. Pullerits T and Freiberg A (1991) Picosecond fluorescence of simple photosynthetic membranes: evidence of spectral inhomogeneity, and directed energy transfer. Chem Phys 149: 409--418. Pullerits T, Vischer KJ, Hess S, Sundstrom V, Freiberg A, Timpmann K and Van Grondelle R (1994) Energy transfer in the inhomogeneously broadened core antenna of purple bacteria: A simultaneous fit of low-intensity picosecond absorption and fluorescence kinetics. Biophys J 66: 236-248. Razjivin AP, Danielius RV, Gadonas RA, Borisov A Yu and Piskarskas AS (1982) The study of excitation transfer between light-harvesting antenna and reaction center in chromatophores from purple bacterium Rhodospirillum rubrum by selective picosecond spectroscopy. FEBS Lett 143: 40-44. Reddy NRS, Picorel R and Small GJ (1992) B896 and B870 components of the Rhodobacter sphaeroides antenna: A hole burning study. J Phys Chem 96:6458-6464. Spano FC, Kuklinski JR and Mukamel S (1990) Temperaturedependent superradiant decay of excitons in small aggregates. Phys Rev Lett 65: 211-214. Sundstrom V, Van Grondelle R, Bergstrom H, Akesson E and Gillbro T (1986) Excitation energy transport in the bacteriochlorophyll antenna systems of Rhodospirillum rubrum and Rhodobacter sphaeroides, studied by low-intensity picosecond absorption spectroscopy. Biochim Biophys Acta 851: 431-446. Timpmann K, Freiberg A and Godik VI (1991) Picosecond kinetics of light excitations in photosynthetic purple bacteria in the temperature range of 300-4K. Chem Phys Lett 182: 617-622. Timpmann K, Zhang FG, Freiberg A and Sundstrom V (1993) Detrapping of excitation energy from the reaction center in the photosynthetic purple bacterium Rhodospirillum rubrum. Biochim Biophys Acta 1183: 185-193. Van Grondelle R, Bergstrom H, Sundstrorn V, Van Dorssen RJ, Vos M and Hunter CN (1988) Excitation energy transfer in the lightharvesting antenna of photosynthetic purple bacteria: the rule of the long-wavelength absorbing pigment B896. In: Scheer H and Schneider S (eds) Photosynthetic Light-Harvesting Systems, pp 519-530, Berlin, New York: Walter de Gruyter & Co. Van Grondelle R, Van Mourik E Visschers RW, Somsen OJG and Valkunas L (1992) The bacterial photosynthetic light-harvesting antenna aggregation state, spectroscopy and excitation energy transfer. In: Research in Photosynthesis, Vol 1, pp 9-16. Kluwer Academic Publishers, Dordrecht, The Netherlands. Van Mourik E Visscher RW and Van Grondelle R (1993) Spectralinhomogeneity of the light-harvesting antenna of Rhodospirillum rubrum probed by triplet-minus-singlet spectroscopy and singlettriplet annihilation at low temperatures. Photochem Photobio157: 19-23.

15 Visschers RW, Van Mourik E Monshouwer R, Van Grondelle R (1993) Inhomogeneotts spectral broadening of the B820 subanit from LHI. Biochim Biophys Acta 1141: 238-244~ Zuber H (1985) Structure and function of light-harvesting complexes and their polypeptides. Photochem Photobiol 42: 821-844.

Zuber H (1987) The structure of light-harvesting pigment-protein complexes. In: Barber J (ed) The Light Reactions, pp 197-259, Amsterdam: Elsevier Science Publishers