Refinement of a Structural Model of a Pigment−Protein Complex by

J. Phys. Chem. B 2007, 111, 10487-10501

10487

Refinement of a Structural Model of a Pigment-Protein Complex by Accurate Optical Line Shape Theory and Experiments T. Renger,*,† I. Trostmann,‡ C. Theiss,§ M. E. Madjet,† M. Richter,| H. Paulsen,‡ H. J. Eichler,§ A. Knorr,| and G. Renger⊥ Institute of Chemistry and Biochemistry, Free UniVersity Berlin, Takustrasse 6, D-14195 Berlin, Germany, Institute of General Botany, Johannes-Gutenberg-UniVersity, Mu¨llerweg 6, D-55099 Mainz, Germany, Institute of Optics, Technical UniVersity Berlin, Strasse des 17. Juni 135, D-10623 Berlin, Germany, Institute of Theoretical Physics, Nonlinear Optics and Quantum Electronics, Technical UniVersity Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany, and Max Volmer Laboratory for Biophysical Chemistry, Technical UniVersity Berlin, Strasse des 17. Juni 135, D-10623 Berlin, Germany ReceiVed: March 2, 2007; In Final Form: June 7, 2007

Time-local and time-nonlocal theories are used in combination with optical spectroscopy to characterize the water-soluble chlorophyll binding protein complex (WSCP) from cauliflower. The recombinant cauliflower WSCP complexes reconstituted with either chlorophyll b (Chl b) or Chl a/Chl b mixtures are characterized by absorption spectroscopy at 77 and 298 K and circular dichroism at 298 K. On the basis of the analysis of these spectra and spectra reported for recombinant WSCP reconstituted with Chl a only (Hughes, J. L.; Razeghifard, R.; Logue, M.; Oakley, A.; Wydrzynski, T.; Krausz, E. J. Am. Chem. Soc. U.S.A. 2006, 128, 3649), the “open-sandwich” model proposed for the structure of the pigment dimer is refined. Our calculations show that, for a reasonable description of the data, a reduction of the angle between pigment planes from 60° of the original model to about 30° is required when exciton relaxation-induced lifetime broadening is included in the analysis of optical spectra. The temperature dependence of the absorption spectrum is found to provide a unique test for the two non-Markovian theories of optical spectra. Based on our data and the 1.7 K spectra of Hughes et al. (2006), the time-local partial ordering prescription theory is shown to describe the experimental results over the whole temperature range between 1.7 K and room temperature, whereas the alternative timenonlocal chronological ordering prescription theory fails at high temperatures. Modified-Redfield theory predicts sub-100 fs exciton relaxation times for the homodimers and a 450 fs time constant in the heterodimers. Whereas the simpler Redfield theory gives a similar time constant for the homodimers, the one for the heterodimers deviates strongly in the two theories. The difference is explained by multivibrational quanta transitions in the protein which are neglected in Redfield theory.

1. Introduction Pigment-protein complexes (PPCs) are the essential constituents of the photosynthetic apparatus that transforms solar radiation into Gibbs energy (for a review see ref 1). These units are rather complex and a challenge for thorough theoretical analyses. Small model proteins provide an invaluable material to test the potential of theoretical models for optical spectra and excitation energy transfer in PPCs. Two types of interactions need to be taken into account: (i) the electronic coupling between pigments giving rise to excitation energy transfer and delocalized excited states and (ii) the exciton-vibrational coupling between the pigments and the protein that leads to dissipation of excess energy and to lifetime broadening and vibrational sidebands in the optical spectra. An analytical solution for linear optical spectra of a single pigment coupling * To whom correspondence should be addressed. E-mail: rth@ chemie.fu-berlin.de. † Institute of Chemistry and Biochemistry, Free University Berlin. ‡ Institute of General Botany, Johannes-Gutenberg-University. § Institute of Optics, Technical University Berlin. | Institute of Theoretical Physics, Nonlinear Optics and Quantum Electronics, Technical University Berlin. ⊥ Max Volmer Laboratory for Biophysical Chemistry, Technical University Berlin.

to a manifold of protein vibrations in the harmonic approximation can be obtained from the work of Lax2 and Kubo and Toyozawa.3 However no exact solution exists if, in addition to the pigment-protein coupling, pigment-pigment coupling occurs, as in larger complexes. Different approximation schemes have been developed4-10 that can be applied to take into account both types of interactions. The different approximations are best compared by Kubo’s generalized cumulant expansion.4 Partial ordering prescription (POP) for the cumulants leads to a time-local equation of motion for the reduced statistical operator5-9,11-14 whereas the chronological ordering prescription (COP) is equivalent to a time-nonlocal equation of motion.10,13,15,16 It is much debated, but still an open question, which of the two approximations offers a better description of reality.12,13,17-22 For simple model systems, the approximations have been compared with an exact solution.17-20,23 Mukamel investigated the optical line shape of a two-level system with a stochastically fluctuating energy gap. He found that in the case of a dichotomously fluctuating energy gap, a second-order COP theory is exact, whereas for a modulation of the energy gap by a Gaussian random process POP yields the exact solution.17 Palenberg et al.19 investigated population transfer in a symmetric dimer with dichotomously fluctuating

10.1021/jp0717241 CCC: $37.00 © 2007 American Chemical Society Published on Web 08/14/2007

10488 J. Phys. Chem. B, Vol. 111, No. 35, 2007 energies and found that in the critical parameter range the POP theory gives results that are closer to the exact solution. The spin boson system has been used as a simple model case as well.18,23,24 Using path integral functionals Xu et al.23 obtained a hierarchical set of equations of motion for the reduced statistical operator. Different truncation schemes for this hierarchy were tested, and the one that corresponds to a POP treatment was reported to yield the most accurate results.23 A similar hierarchical set was used by Schro¨der and Kleinekatho¨fer recently to investigate the population decay of a damped harmonic oscillator, using a Drude spectral density for the system-bath coupling.25 Whereas POP in a certain parameter range was found to be converged already in second-order, second-order COP showed artificial oscillations in the populations that vanished only in higher orders.25 Alternatively, the line shape functions predicted by the different theories can be compared directly with experimental data, if a suitable model pigment-protein complex is available.12 The smallest possible model protein to test theoretical predictions should contain an excitonically coupled pigment dimer. We have recently investigated fluorescence line narrowing spectra of a bacteriochlorophyll dimer in the so-called B820 pigment-protein complex and found that the POP theory describes the experimental vibrational sideband, whereas within the COP theory significant deviations emerge at low frequencies.12 The inhomogeneous absorption spectra due to the average over disorder prevent seeing details of the vibrational sidebands, and therefore on the basis of the absorption spectra no conclusion was possible.12 The low-energy exciton state carries most of the oscillator strength due to the geometry of the transition dipoles of the pigments in the B820 complex. A comparison between the calculated and the inhomogeneously broadened experimental absorption spectra is rather difficult because of the low intensity of the high-energy exciton transition. In the experiment the latter is buried by other optical transitions (intramolecular vibrational transitions or transitions to higher excited states). The description of the high-energy part of a dimer spectrum is therefore always more ambiguous than the low-energy part because in the latter only the Qy transitions of the pigments contribute. Hence, a pigment-protein complex containing a dimer with different geometry, that leads to a strong high-energy exciton transition, together with a minimum spectral congestion by other pigments would be a highly desirable system. This condition is satisfied neither by the PPCs of the antenna systems nor by the “special pair” configurations in the reaction center of anoxygenic bacteria and photosystems I and II of oxygen-evolving organisms (for recent reviews, see refs 26-28). A suitable model system containing an excitonically coupled dimer with a strong high-energy transition was recently reported29 to be recombinant cauliflower WSCP (water-soluble chlorophyll binding protein) reconstituted with either Chl a or Chl d. The functional role of native WSCP is not fully understood. Since WSCPs are not located in thylakoids,30 they are very unlikely acting as light-harvesting proteins. WSCP formation is stimulated under stress conditions (heat, drought), and this protein is capable of extracting chlorophylls from solution and even from the thylakoid membrane. Furthermore, in spite of the lack of carotenoids, the formation of singlet oxygen by reaction of ground-state triplet oxygen with photoinduced triplet Chl’s in WSCP was found to be 4 times smaller than for Chl’s in solution.31 This effect (the underlying mechanism is not yet resolved) favors the idea of protective chlorophyll binding during either biosynthetic and/or degradative

Renger et al. pathways of chlorophyll in plants.30 Upon binding chlorophylls, WSCP oligomerizes. The phytyl chain of the chlorophylls was reported to be essential for this process.31 In cauliflower, WSCP forms tetramers that bind dimers of chlorophyll. Hughes et al.29 reconstituted recombinant cauliflower WSCP with Chl a and Chl d and characterized the optical properties by low-temperature spectroscopy. On the basis of a Gaussian band fitting of linear absorption and circular dichroism (CD) spectra at 1.7 K and a simple exciton model, Hughes et al.29 proposed an “open-sandwich” Chl a dimer with a tilt angle of 60° between the chlorin planes and a center-to-center distance of 7.6 Å. Different values for the absorption and CD band parameters were reported, reflecting approximations used in the simple model. We have measured absorption spectra at 77 and 298 K and the CD spectrum at 298 K on recombinant WSCP complexes that were reconstituted with Chl b instead of Chl a. In addition, heterodimers reconstituted with Chl a and Chl b were investigated. These results and the data of Hughes et al.29 provide a sound experimental basis for a thorough theoretical analysis on excitonic Chl dimers in WSCP. We adopted the idea of an “open-sandwich” dimer29 as the general structural feature of Chl binding to the WSCP complex but replaced the Gaussian band fitting procedure of Hughes et al.29 by a microscopic theory of optical spectra that results in vibrational sidebands and lifetime broadening of exciton transitions. In particular the latter requires a refinement of the structural model as outlined in Results and Discussion. Furthermore, the experiments on WSCP allow distinguishing between the predictive powers of the time-local POP and the time-nonlocal COP theory of optical spectra. We find compelling evidence that second-order POP contains a much more exact summation over the exciton-vibrational coupling than secondorder COP. The paper is organized in the following way. After the Introduction (part 1), part 2 summarizes the two theories (POP and COP) that are used for the description of the optical spectra. A generalized cumulant expansion is used to show how the partial summation of the exciton-vibrational coupling differs in the two approaches. Finally, the structural model and parameters used in the calculations are introduced. Part 3 contains a short summary of the preparation of recombinant cauliflower WSCP and the spectroscopical methods. In part 4 results are presented on (i) experimental and calculated optical spectra of WSCP reconstituted with Chl b homodimers, (ii) revisiting earlier experiments by Hughes et al.29 on WSCP with Chl a homodimers, (iii) experimental and calculated optical spectra of WSCP reconstituted with Chl a/Chl b heterodimers, (iv) prediction of exciton relaxation times in the different dimers using Redfield and modified-Redfield theory. The results are discussed in part 5, and finally, conclusions are given in part 6. 2. Theory 2.A. Linear Optical Spectra. The Hamiltonian of our pigment-protein complex, HPPC, consists of three parts

HPPC ) Hex+ Hvib + Hex-vib

(1)

an excitonic part Hex, a vibrational part Hvib, and an excitonvibrational part Hex-vib. The details of these three Hamiltonians are given later. Here it suffices to note that Hex contains just a few electronic degrees of freedom of the pigments, Hvib contains all the vibrational degrees of freedom of the protein and the

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J. Phys. Chem. B, Vol. 111, No. 35, 2007 10489

pigments, and Hex-vib contains the couplings between the electronic and vibrational degrees of freedom. The equation of motion for the statistical operator W ˆ of the system in the interaction representation reads

∂ tW ˆ ′ ) -iL ′ex-vib(t)W ˆ′

(2)

(t) is the Liouville super operator in the interacwhere L ex-vib ′ tion representation, L ex-vib ′ (t) ) ei(Lex+Lvib)tLex-vibe-i(Lex+Lvib)t. The action of the Liouville superoperator L on any operator O ˆ is LO ˆ ) p-1[H,O ˆ ] , as usual, where the brackets indicate a commutator and the Hamiltonian H is the one that corresponds to the Liouville operator, for example Hex-vib for Lex-vib. Prior to an interaction of the PPC with the external light field, the PPC is in its electronic ground state |0〉, and the vibrational degrees of freedom evolve according to their equilibrium statistical operators Wvib eq . Hence, the statistical operator at time zero, before the light field starts to interact, is given as W ˆ (0) ) . The interaction with an external field will bring the |0〉 〈0|Wvib eq PPC out of equilibrium, and the statistical operator of excitons and vibrations no longer factorizes. To describe the influence of the vibrations on the excitonic degrees of freedom, we use the reduced statistical operator32

ˆ (t)} Fˆ (t) ) Trvib{W

∑Mn(t))Fˆ (0)

(4)

n)1

where the nth order moment of the exciton-vibrational coupling is given as Mn(t) ) (i/p)n∫t0 dτ1‚‚‚∫τ0n-1 dτn〈L ′ex-vib(τ1) ‚‚‚ (τn)〉 and includes a thermal average 〈‚‚‚〉 ) Trvib{‚‚‚W ˆ vib L ex-vib ′ eq } with respect to the vibrational degrees of freedom. Kubo advised different ways to perform the sum over the moments in eq 4 by the help of a generalized cumulant expansion that assumes the following solution4

{∑ }

Kn(t) Fˆ (0)

(5)

n

where P denotes the prescription used in the construction of cumulants. Neglecting higher-order cumulants Kn for n larger than a certain value provides a systematic way to factorize the higher-order moments Mn(t). There exist different prescriptions to define the cumulants that result in different factorization schemes. In the chronological ordering prescription (COP) we have, assuming a linear coupling model in Hex-vib (given below), K1 ) 0, K2 ) M2, K3 ) 0, and K4 reads7,18

) M4 K(COP) 4

(p1) ∫ dτ ∫ 4

τ1

t

0

1

0

dτ2

∫0τ

2

dτ3

〈L

∫0τ

∫0t dτ ×

ex-vib(t)

e-iL

ex(t-τ)

L

ex-vib(τ)

eiL

exτ

〉 Fˆ (τ) (8)

whereas the equation, obtained from the second-order POP cumulant expansion is7,12

∞

Fˆ ′(t) ) expP

∂ 1 Fˆ (t)|COP ) - 2 ∂t p

(3)

which is obtained by integration from eq 2 as

Fˆ ′(t) ) (1 +

In the following we use a second-order cumulant expansion, i.e., assume Kn ≈ 0 for n > 2. The above fourth-order cumulants are given to illustrate the different factorization of moments Mn, for n > 2 , that a second-order cumulant expansion in POP and COP implies. By setting K4 ) 0 on the l.h.s. of eqs 6 and 7 different approximations for M4 result in POP and COP. In both theories M4 factorizes into a product of two-time correlation functions of the exciton-vibrational coupling. In COP there is a strict chronological ordering of this product, whereas in POP all permutations in time are included that still lead to a timeordered two-time correlation function. The higher-order moments n > 4 are factorized in the same way. Please note that a second-order cumulant expansion takes into account arbitrary high-order moments but in an approximate way. In principle, the sum in eq 4 can be performed now up to a certain finite order, assuming the above factorization schemes for the moments Mn for n > 2. However, an infinite sum can be computed by integrating instead an equation of motion for the reduced statistical operator. The equation of motion that corresponds to the second-order COP factorization reads in the Schro¨dinger picture7,12

3

dτ4 ×

∂ 1 Fˆ (t)|POP ) - 2 ∂t p 〈L

∫0t dτ ×

ex-vib(t)

e-iL

exτ

L

ex-vib(t

- τ) eiL

exτ

〉 Fˆ (t) (9)

Since the derivative of the statistical operator in eq 8 depends on the value of this operator at earlier times, it is sometimes called a time-nonlocal theory, whereas eq 9 is time-local in that respect. Note however, that also in the latter case there is some memory included due to the time-dependent coefficient. For further treatment, the details of the different parts of the Hamiltonian have to be outlined. These details are given in Appendix A. Expanding the reduced statistical operator Fˆ with respect to the delocalized exciton states |M〉, the line shape function DM(ω) for the optical transition between the ground state and the Mth exciton state is obtained as12,16

DM(ω) ) R

∫0∞ eiωt FM0(t)

(10)

where R denotes the real part of the integral and the density matrix FM0(t) ) 〈M|Fˆ (t)|0〉 has to be propagated with the initial condition FM0(0) ) 1. The COP-line shape function in secular approximation reads16

〈L ′ex-vib(τ1)L ′ex-vib(τ2)〉〈L ′ex-vib(τ3)L ′ex-vib(τ4)〉 (6)

D(COP) (ω) ) M

whereas in the alternative partial ordering prescription (POP) theory K1 ) 0, K2 ) M2, K3 ) 0 and K4 is obtained as7,11,18

ΓM(ω) (ω - ΣM(ω))2 + (ΓM(ω))2

(11)

with

K(POP) 4

()∫

∫

∫

∫

τ1 τ2 τ3 14 t ) M4 dτ1 0 dτ2 0 dτ3 0 dτ4 0 p × (〈L ′ex-vib(τ1)L ′ex-vib(τ2)〉〈L ′ex-vib(τ3)L ′ex-vib(τ4)〉 + 〈L ′ex-vib(τ1)L ′ex-vib(τ3)〉〈L ′ex-vib(τ2)L ′ex-vib(τ4)〉 + 〈L ex-vib ′ (τ1)L ex-vib ′ (τ4)〉〈L ex-vib ′ (τ2)L ex-vib ′ (τ3)〉) (7)

ΓM(ω) )

∑K γMKC˜ (Re) (ω - ωK0)

(12)

ΣM(ω) )

∑K γMKC˜ (Im) (ω - ωK0)

(13)

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Renger et al.

where ωK0 is the transition frequency between the Kth exciton state and the ground state. The γMK is obtained as

γMK )

∑ m,n

c(M) m

c(K) m

c(M) n

c(K) n

-Rmn/Rc

e

(14)

with the correlation radius of protein vibrations Rc that describes the correlation of modulation of transition energies at different sites12,33,34 that are separated by a distance Rmn. It is assumed that the correlation function of site energy fluctuations at different sites decays exponentially with intersite distance:12,33,34

Cmn(t) ) 〈δEm(t)δEn(t) ) C(t)e-Rmn/Rc

(15)

where the autocorrelation function C(t) ) Cmm(t) is assumed to be site independent. Usually in the literature, uncorrelated fluctuations of site energies (i.e., Rc ) 0) are assumed. Having in mind that, in particular, low-frequency protein vibrations modulate the transition energies of the pigments we introduced a correlation radius to take into account the nonlocal nature of low-frequency vibrational degrees of freedom. From a comparison of transient absorption data of D1D2-cytb559 complexes with calculations we inferred35 a value of Rc ) 5 Å that is used also in the present study. The function C ˜ (ω) in eqs 12 and 13 above and in eqs 22 and 25 below is the half-sided Fourier transform of C(t) in eq 15. Using a harmonic oscillator description for the vibrational degrees of freedom and a linear modulation of pigment transition energies as outlined in the Appendix A, the real part is obtained as12,16,34

C ˜ (Re)(ω) ) πω2{(1 + n(ω))J(ω) + n(-ω)J(-ω)}

(16)

where J(ω) is the spectral density

J(ω) )

∑ξ g2ξδ(ω - ωξ)

(17)

GM(t) ) γMM

∫0∞ dω{(1 + n(ω))J(ω) e-iωt + n(ω)J(ω) eiωt} (21)

ω ˜ M0 ) ωM0 - γMMλ/p +

γMKC ˜ (Im)(ωMK) ∑ K*M

where the reorganization energy λ is obtained from the spectral density J(ω) as

λ)

∫0∞ dωpωJ(ω) 1

τM-1 )

2

1 e

-1

(18)

˜ (Re)(ωMK) kMfK ) 2γMKC

C ˜

1 (ω j) ) F π

C ˜ (Re)(ω) dω -∞ ω j -ω

∫

∞

with

1 2π

∫-∞∞ dt ei(ω-ω˜

M0)t

(25)

(ω) D(Markoff) M

τ-1 M

)

(ω - ω ˜ M0)2 + τ-2 M

(26)

As will be shown in the Discussion section, a Markoff approximation neglects the vibrational side bands and therefore leads to less broadening of the spectra. The homogeneous absorption and CD spectra are obtained using the above line shape functions as

|µM|2DM(ω) ∑ M

(27)

rMDM(ω) ∑ M

(28)

and

where the transition dipole moment of the exciton transition is36

b µM )

c(M) µm ∑ m b m

(29)

and the rotational strength rM is given as36

(19)

where F denotes the principal part of the integral. The alternative POP line shape function, using a secular approximation and a Markoff approximation for the off-diagonal part of the excitonvibrational coupling, is obtained as12

(ω) ) D(POP) M

(24)

The function γMK (eq 14) contains the exciton coefficients and the correlation radius of protein vibrations, and the ωMK in eqs 22 and 25 is the transition frequency between exciton states |M〉 and |K〉, ωMK ) (M - K)/p. In Markoff approximation, both line shape functions, D(COP) (ω) and D(POP) (ω), become M M 12 equal to

CDhom(ω) ∝ ω

is the mean number of vibrational quanta that are excited at a given temperature. The same spectral density is used for all pigments, i.e., the local coupling constants g(m) ) gξ are ξ assumed to be the same for all sites m. We note that in eq 16 J(ω) ) 0 for ω < 0. The imaginary part C ˜ (Im)(ω j ) and the real part are related by a Kramers-Kronig relation12,16,34 (Im)

∑K kMfK

is determined by exciton relaxation described by the Redfield rate constant

Rhom(ω) ∝ ω

pω/kT

(23)

The dephasing time τM-1

and

n(ω) )

(22)

eGM(t)-GM(0) e-|t|/τM (20)

rM )

(M) Rmn‚µ bm × b µn ∑ c(M) m cn B

(30)

m>n

Here, b µm is the molecular transition dipole moment of the mth pigment, B Rmn is the distance vector from pigment m to pigment n, and ‚ and × are the usual scalar and vector products. Due to the linear combination in eq 29, the resulting magnitude of the exciton transition dipole moment b µM, and thereby the intensity of the respective line in the absorption spectrum, depends critically on the mutual orientation of the molecular transition dipole moments b µ m.

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J. Phys. Chem. B, Vol. 111, No. 35, 2007 10491

The exciton coefficients c(M) m and the exciton eigenenergies M are obtained by diagonalizing the exciton matrix that contains the site energies Em in the diagonal and the excitonic couplings Vmn in the off-diagonal. The site energy of a pigment is defined as the transition energy at the equilibrium position of nuclei in the electronic ground state in the absence of excitonic coupling with the other pigments. The excition coefficients c(M) and m energies M for the present dimer are given in Appendix B. While fast conformational motion of the protein is described by the spectral density J(ω), any conformational motion that is slow compared to the lifetime of excited states leads to an inhomogeneous broadening of the ensemble spectra. We take into account the static disorder in transition energies of the pigments by assuming a Gaussian distribution function P(Em -E h m) for the site energies Em. The absorption spectrum is then obtained as an average over the various homogeneous spectra Rhom(ω,E1,E2) that result for a particular realization of disorder, i.e., site energies E1 and E2

R(ω) )

∫ dE1 dE2P(E1 - Eh 1)P(E2 - Eh 2)Rhom(ω,E1,E2)

(31)

The same disorder average is used for the CD spectrum and also in the calculation of time constants for exciton relaxation. The average is performed numerically by using a Monte Carlo method. 2.B. Exciton Relaxation in Modified-Redfield Theory. In Redfield theory, the rate constant is obtained in perturbation theory in the exciton-vibrational coupling. In contrast, in modified-Redfield theory37-41 the diagonal part of the excitonvibrational coupling is described exactly, and a second-order perturbation theory is used only for the off-diagonal part. The general result for this rate constant is given in Appendix C. As shown there, in the case of a dimer the modified-Redfield rate constant takes a slightly simpler form than in the general oligomer case, reading

k2f1 )

∫-∞∞ dτ eiω

21τ

eφ21(τ)-φ21(0) [(G21(τ))2 + F21(τ)] (32)

where expressions for the time dependent functions φ21(τ), G21(τ), and F21(τ) are given in Appendix C. All these functions are determined by the spectral density J(ω), the exciton coefficients c(M) m , and the correlation radius of protein vibrations Rc. The rate constant in eq 32 will be used below to calculate exciton relaxation times which will be compared with values obtained from the Redfield rate constant in eq 25. We note that the Redfield rate constant in eq 25 can be obtained from eq 32 by setting φ21(τ) ) G21(τ) ) 0. These functions are related to the diagonal part of the exciton-vibrational coupling that describes a reorganization of nuclei upon exciton relaxation that is neglected in Redfield theory. In that respect modifiedRedfield theory might also be called generalized-Redfield theory. 2.C. Structural Model and Parameters. Following the proposal by Krausz and co-workers we assume a C2-symmetric pigment dimer in “open-sandwich” geometry.29 The dimer geometry is defined by the following three parameters, (i) the center-to-center distance r , where the center is defined as the center of the line connecting the NB and ND atoms (corresponding to N21 and N23, respectively, in IUPAC nomenclature42) of the chlorophylls, (ii) the tilt angle R between the planes formed by the macrocycles of the pigments, and (iii) the angle β between the projection of the C2-symmetry axis on the pigment planes and the NB-ND axis, as illustrated in Figure 1. The excitonic coupling is calculated with our TrEsp method43 from the Coulombic coupling between transition monopole charges,

Figure 1. Modified “open-sandwich” model for an excitonically coupled C2-symmetric chlorophyll dimer in WSCP. The parameter values of this model, given in Table 1 were obtained from the calculation of the homodimer spectra in Figures 3 and 5. The present calculations give an angle R ) 30°, whereas the analysis of Hughes et al.29 predicted R ) 60°. Graphics prepared with VMD (Humphrey, W.; Dalke, A.; Schulten, K. J. Mol. Graphics 1996, 14, 33.)

that were calculated by fitting the electrostatic potential of the transition density obtained with time-dependent density functional theory and the B3LYP exchange correlation functional. The transition charges44 are rescaled to yield an effective transition dipole moment of 3.6 D for Chl b and 4.0 D for Chl a.45 For the optimal geometries of the Chl a/Chl a and Chl b/Chl b homodimers and the Chl a/Chl b heterodimers the respective excitonic couplings V are 84, 72, and 76 cm-1. The spectral density J(ω) is described as

J(ω) ) SJ0(ω)

(33)

where the same functional form J0(ω) is assumed (with ∫ dω J0(ω) ) 1) as determined12 from fluorescence line-narrowing spectra of the B777 complex and the Huang-Rhys factor S ) 0.8 is obtained below from the temperature dependence of the line width in comparison with experimental data on WSCP. The functional form of the spectral density, determined earlier, reads12

J0(ω) )

1 s 1 + s2

∑ i)1,2

si

ω3 e-(ω/ωi)

1/2

4

(34)

7!2ωi

with s1 ) 0.8, s2 ) 0.5, pω1 ) 0.069 meV, and pω2 ) 0.24 meV. The maxima of the two contributions in eq 34 occur at frequencies 36 ωi, i.e., at 20 and 70 cm-1. The spectral density, eqs 33 and 34, is shown in the upper part of Figure 2. Although the spectral density was extracted originally for the B777 complex which contains just a single bacteriochlorophyll a pigment, it was successfully applied to larger complexes.35,40,41 In good approximation we find that the shape of the vibrational side bands in these larger complexes can be well described by the J0(ω) but that a fine-tuning of the integral coupling strength, i.e., the Huang-Rhys factor S ) ∫ dω J(ω) is necessary to

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Renger et al.

Figure 2. (Top) Spectral density J(ω) of the pigment-protein coupling. (Bottom) Function C ˜ (Re)(ω), eq 16, entering the Redfield rate constant of exciton relaxation in eq 25, together with a histogram of energy gaps between the high and low-energy exciton states of the Chl a/Chl a homo- and Chl a/Chl b heterodimers.

TABLE 1: Adjustable Parameters Used in the Calculations hc/EChla/nm

hc/EChlb/nm

S

675 (1.7 K) 658 (77 K) 0.8 677 (298 K) 660 (298 K)

∆inh/cm-1

R/deg

170

25 (a-a) 36 (b-b)

β/deg r /Å 6

7.4

describe the temperature dependence of the spectra.40,41 As described above, the function C ˜ (Re)(ω) that enters the Redfield rate constant k2f1 at the transition frequency ω ) ω21 between the upper and lower exciton state describes how well the protein is able to dissipate the excess energy of excitons by single vibrational quanta creation during exciton relaxation. In the lower part of Figure 2 the disorder-averaged probabilities are shown to find a certain energy difference between the exciton states in the homo (Chl b/Chl b) and the hetero (Chl a/Chl b) dimers. Obviously the protein can dissipate the smaller energy differences of the homodimers better than the larger ones of the heterodimers. As will be shown in detail in the Discussion section the modified-Redfield theory includes the possibility to dissipate excess energy by multiple quanta transitions, which become important for the heterodimers. Besides by the spectral density, the optical line broadening is determined by the width ∆inh (fwhm) of the Gaussian distribution function P(Ei - E h i) in eq 31 that is assumed to be the same for all pigments. The adjustable parameters of the present model and their values as determined from the calculation of homo Chl a/Chl a and Chl b/Chl b dimers are summarized in Table 1. The predictive power of the model is tested in calculations of optical spectra of the Chl a/Chl b heterodimers and in the calculation of exciton relaxation times. We note that, although the present theory might look complicated at first glance, the expression for the optical spectra contains fewer parameters than those used in the Gaussian band fitting29 as will be discussed in detail in the Discussion section. The key improvement is the use of a microscopic line shape theory that is based on a spectral density J(ω) extracted independently.12 3. Experimental Methods 3.A. Sample Material. Protein. Recombinant WSCP from cauliflower (Brassica oleracea var. Botrys) with an N-terminal

hexahistidyl (His) tag was expressed as previously described31 with some modifications. Following induction of WSCP expression with IPTG, the growth temperature was reduced to 28 °C, and protein expression was continued under these conditions overnight. After cell lysis and centrifugation, the supernatant containing soluble WSCP was divided into aliquots, stored at -20 °C and used for sample preparation. Pigment. Total pigment extract was isolated from pea plants as previously described.46 Purified Chl b was prepared as described in Hobe et al.47 Pigments were dried and stored at -20 °C in inert (nitrogen) atmosphere. Reconstitution of Chl-WSCP. Reconstitution and purification of soluble WSCP was performed simultaneously by binding the protein to a Chelating Sepharose Fast Flow (Amersham Biosciences) column charged with Ni2+ ions and equilibrated with 50 mM sodium phosphate (pH 7.8). The sample was filtered before loading on the column at 2 mg WSCP per mL of column material. After removal of contaminating bacterial proteins by washing with 10 column volumes of 50 mM sodium phosphate (pH 7.8) and following equilibration of the column with 5 column volumes of 50 mM sodium phosphate (pH 7.8)/12.5% sucrose/1% octyl-β-D-glucopyranosid (NaP-OG-buffer), the protein was incubated with a 5-fold molar excess of Chl (total pigment extract or purified Chl b), presolubilized in 100% ethanol at 10 mg/mL, and then diluted 20-fold with NaP-OGbuffer. The pigment solution was thoroughly dispersed with the column material and then incubated for 1-2 h at RT and protected from light. Unbound Chl was removed by stringently washing with 30 column volumes of the same buffer. After detergent removal by washing with 10 column volumes of 50 mM sodium phosphate (pH 7.8), the purified and reconstituted pigmented protein was eluted from the Ni2+ column with 20 mM sodium phosphate (pH 7.8)/300 mM imidazole/2 mM β-mercaptoethanol and subsequently concentrated in Centricon devices (30 kDa MW cutoff, Millipore), so that the sample had an OD at 673 nm of at least 1 at a path length of 2 mm. Molar Protein and Pigment Stoichiometries. To determine the molar Chl/protein ratio in Chl-WSCP, pigments were extracted with 2-butanol as described by Martinson and Plumley,48 except that prior to extraction at RT, Chl-WSCP complexes were denatured by adding 1% [w/v] SDS, and a final concentration of 330 mM NaCl was used. Pigments were quantified by HPLC analysis as described previously.31 Protein quantification was done photometrically using an extinction coefficient of 28910 M-1 cm-1 for recombinant WSCP, determined from the amino acid sequence by using the program Biopolymer Calculator available at http:// paris.chem.yale.edu/extinct.html and subtracting the chlorophyll contribution to the protein absorption at 280 nm. 3.B. Spectroscopic Analyses. Ground-state absorption spectra were recorded with a UV-vis-scanning spectrophotometer (UV2101PC, Shimadzu) in a range from 350 to 750 nm with medium scan speed and a bandwidth of 1 nm. Absorption spectra at 77 K were measured from 600 to 750 nm, and data from nine consecutive measurements of each sample were averaged. The deviation between a single measurement and the average was smaller than 0.2% at all wavelengths. The concentration of Chl a and Chl b molecules was chosen such that an optical density (OD) smaller than unity resulted. In the case of the Chl b homodimers the concentration of Chl b was 7.2 µM. For the Chl a/Chl b heterodimers, the sample with a Chl a/Chl b ratio of 2.6 contained 4.5 µM Chl a and 1.7 µM Chl b, and the sample with a Chl a/Chl b ratio of 4.2 contained 7.1 µM Chl a and 1.7 µM Chl b. The path length in the absorbance measurements

Refinement of a Model for a Pigment-Protein Complex

J. Phys. Chem. B, Vol. 111, No. 35, 2007 10493

Figure 3. Linear absorption at 77 and 298 K (top panel) and CD at 298 K (bottom panel) of recombinant WSCP reconstituted with Chl b. The symbols show the experimental data and the lines the calculations, using POP-line shape theory (eq 20) and the following site energies: E h1 ) E h 2 ) hc/658 nm at T ) 77 K and E h1 ) E h 2 ) hc/660 nm at T ) 298 K.

was 1 cm. In the case of a homogeneous sample with pigment concentration c and optical path length l the molar extinction coefficient is obtained as  ) OD/cl. In the CD experiments the samples were measured at a 5-fold higher concentration to obtain a good signal-to-noise ratio. Raw absorption and CD data of the WSCP spectra are presented without any smoothing but subtracting the absorption of the pure solvent (baseline correction). CD spectra were measured at RT with a Jasco J-810 spectropolarimeter using a quartz cuvette with a path length of 2 mm, scan speed 10 nm/min, bandwidth 1 nm, data pitch 0.5 nm, and response time 4 s. The CD spectra were recorded as ellipticity θ in mdeg. For a homogeneous sample the difference in molar extinction coefficients for left- and right-hand polarized light ∆ is related to the measured θ (in mdeg) by49

∆ )

θ1 f cl

(35)

with the constant f ) 1000 ln(10)180/(4π) ) 32982. In the case of the Chl b homodimers we present the absorption and CD spectra as molar extinction quantities  and ∆ with respect to the Chl b concentration, whereas in the Chl a/Chl b heterodimer case we provide the optical density OD and the ellipticity θ (because of the different concentrations of the Chl a and Chl b pigments). 4. Results 4.A. Linear Absorption and Circular Dichroism of Chl b/Chl b-WSCP. In Figure 3 the absorption spectra measured at 77 and 298 K and the experimental CD spectrum at 298 K are compared with calculated spectra using the POP line shape theory, site energies corresponding to wavelengths of 658 nm at 77 K and 660 nm at 298 K, and a Huang-Rhys factor S ) 0.8. These parameters are rather independent of the structural model. A shift of the site energy just leads to a displacement of the calculated spectrum along the wavelength axis, and a change of S ) 0.8 changes the thermal broadening of the line shape. The low-temperature absorption and the CD spectrum were used

Figure 4. Experimental absorption spectra taken from Figure 3 and calculations using different line shape theories. The experimental data are shown as symbols, the POP calculations (eq 20) are shown as solid lines, the Markoff approximation (eq 26), as dashed lines, and the COP calculations (eq 11), as dots-dashed lines.

to optimize the “open-sandwich” model of Hughes et al.29 From the CD spectrum the splitting between the two exciton states is obtained, and this splitting turns out to be most sensitive to the center-to-center distance between the pigments. The optimal distance is 7.4 Å, practically identical with the value of 7.6 Å reported by Hughes et al., who used a simple point dipole approximation for the coupling, but a smaller effective dipole strength.50 The relative height of the main absorption peak at 655 nm and of the shoulder at 665 nm is most sensitive to the tilt angle R between pigment planes. The best agreement with experimental data was obtained with a value of R ) 36°. The angle β between the projection of the C2-symmetry axis on the pigment planes and the NB-ND line determines the rotational strength, but has only a minor effect on the shape of the spectra. We estimated β ) 6°.51 The structural model is presented in Figure 1. The agreement between theory and experiment in Figure 3 could only be obtained by using the POP line shape theory. A comparison between the absorption spectra calculated with POP and the alternative COP theory or a simple Markoff approximation is given in Figure 4. At 77 K all three theories lead to a reasonable agreement; however, at 298 K only the POP theory permits a quantitatively satisfying description of the experimental data. Accordingly, in the following we will only discuss results obtained using this theory. There is a strong deviation for the angle R between the values gathered in the present and the earlier analysis by Krausz and co-workers.29 The present R ) 36° is only about half of the earlier value. To investigate whether this difference is due to the different pigment (Chl b instead of Chl a) or due to the different theoretical description, we revisited the experimental data of Hughes et al.29 4.B. Linear Absorption and Circular Dichroism of Chl a/Chl a-WSCP Revisited. In Figure 5 the optical spectra measured by Krausz and co-workers29 at 1.7 K are compared with the spectra obtained from our theory and structural model, determined above for the Chl b/Chl b-WSCP. When this model is used, slightly modified by decreasing the angle R from 36° to 25°, an excellent agreement between calculations and

10494 J. Phys. Chem. B, Vol. 111, No. 35, 2007

Figure 5. Linear absorption at 1.7 K (top panel) and CD at 1.7 K (bottom panel) of recombinant WSCP reconstituted with Chl a. The symbols represent the experimental data of Hughes et al.,29 and the lines represent the calculations, using POP-line shape theory (eq 20) and the following site energies: E h1 ) E h 2 ) hc/675 nm. The dashed and dotted lines show the contributions of the low- and high-energy exciton transition to the calculated absorption and CD signals.

experiments is obtained for both linear absorption and CD spectra. A site energy corresponding to 675 nm was used for both Chl a molecules. The (disorder averaged) functions |µ bM|2DM(ω) for both exciton states (M ) 1,2) are shown separately, in addition to their sum (eqs 27 and 31), in Figure 5. It is seen that there is a significant contribution of the high-energy exciton transition in the low-energy shoulder of the absorption spectrum. We note that the low-energy shoulder decreases to about half of its value in Figure 5 if the lifetime broadening of the highenergy exciton state is neglected in the calculations. 4.C. Linear Absorption and Circular Dichroism of Chl a/Chl b-WSCP. For an evaluation of the predictive power of our model, experiments and calculations were performed on a WSCP with a pigment heterodimer consisting of Chl a and Chl b. Due to the different affinities of the cauliflower WSCP to bind Chl a and Chl b,30 the sample contains a mixture of Chl a/Chl a homodimers and Chl a/Chl b heterodimers. A different ratio of homo- and heterodimers is obtained by varying the relative amounts of Chl a and Chl b in the reconstitution assay. Figure 6 shows the experimental and calculated spectra of samples reconstituted at Chl a/Chl b ratios of 2.6 and 4.2. It is assumed that the center-to-center distance between the pigments is the same as that in the homodimers and the tilt angle R is about 30°, in between the values 25° and 36° obtained for the Chl a/Chl a and Chl b/Chl b homodimers, respectively. Figure 6 shows the absorption (upper panels) and CD (lower panels) spectra that are obtained by using the site energies determined from the calculations of the homodimers (see above) and by adding the hetero- and homodimer spectra at a ratio that corresponds to the Chl a/Chl b stoichiometry.52 Here, we assume that the site energy of Chl a shifts virtually to the same extent as the one for Chl b if the temperature is increased to 298 K; i.e. the corresponding wavelength for the Chl a site energy is assumed to be 677 nm. Excellent agreement between experimental and calculated data is achieved for both experimental Chl a/Chl b ratios on the red side, but deviations emerge on the blue side of the spectra as before in the spectra of the homodimers in Figures 3 and 5.

Renger et al.

Figure 6. Linear absorption at 298 K (top panels) and CD at 298 K (bottom panels) of recombinant WSCP reconstituted with Chl a and Chl b. The symbols represent the experimental data, and the lines represent the calculations, using POP-line shape theory (eq 20). The spectra were measured on samples with a Chl a/Chl b ratio of 2.6 (left panels) and 4.2 (right panels). The following site energies were used in the calculations: E h Chla ) hc/677 nm and E h Chlb ) hc/660 nm.

TABLE 2: Disorder Averaged Exciton Relaxation Times -1 〈k2f1 〉dis/fs

Chla/Chla Chlb/Chlb Chla/Chlb

Redfield eq 25

mod. Redfield eq 32

experiment ref 53

77 80 5400

57 60 450