Delocalization-Enhanced Long-Range Energy ... - ACS Publications

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Delocalization-Enhanced Long-Range Energy Transfer between Cryptophyte Algae PE545 Antenna Proteins Hoda Hossein-Nejad,† Carles Curutchet,‡ Aleksander Kubica,§ and Gregory D. Scholes*,§ †

Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, M5S 1A7 Canada Institut de Química Computacional and Department de Química, Universitat de Girona, Campus Montilivi, 17071 Girona, Spain § Lash-Miller Chemical Laboratories, Institute for Optical Sciences and Centre for Quantum Information and Quantum Control, University of Toronto, 80 St. George Street, Toronto, Ontario, M5S 3H6 Canada ‡

bS Supporting Information ABSTRACT: We study the dynamics of interprotein energy transfer in a cluster, consisting of four units of phycoerythrin 545 (PE545) antenna proteins via a hybrid quantum-classical approach. Long-range exciton transport is viewed as a random walk in which the hopping probabilities are determined from a quantum theory. We apply two different formulations of the exciton transport problem to obtain the hopping probabilities, and find that a theory that regards energy transfer as relaxations among the excitonic eigenstates mediated by the vibrational bath, predicts the fastest dynamics. Our results indicate that persistent exciton delocalization is an important implication of the quantum nature of energy transfer on a multiprotein length scale, and that a hybrid quantum-classical approach is a viable starting point in studies of long-range energy transfer in condensed phase biological systems.

’ INTRODUCTION Photosynthesis is the process by which plants and algae convert carbon dioxide into organic compounds using the energy from the sun. A crucial step in photosynthesis is the transfer of the captured solar energy from the antenna proteins to the reaction center, where charge separation takes place.1 Owing to nature’s ability to find optimum solutions, it is hardly surprising that excitation transfer in photosynthesis occurs with near perfect quantum efficiency. This process, known as electronic energy transfer (EET) is often modeled as an incoherent hopping mechanism, where an excitation is initially localized on a donor molecule, and is dynamically driven toward a nearby acceptor. Despite offering a reasonable approximation, F€orster theory has proved inadequate in capturing the subtleties of EET in photosynthesis.2-4 The difficulty of the problem stems from the fact that in condensed-phase biological systems, bath-induced fluctuations in the site energies are of the same order as the electronic coupling between the pigments.5 It is therefore widely believed that a reliable theoretical description is one that treats the electronic interaction and the exciton-phonon coupling on the same footing. An early theoretical study of EET beyond the weak coupling (electronic coupling) approximation of the F€orster theory, is the non-Markovian Master Equation approach of Kenkre and Knox in which the transition probability is expressed as the integral of a memory kernel.6 Application of the Kenkre and r 2011 American Chemical Society

Knox theory to the spin-boson model captures both the wave-like and the diffusive aspects of exciton transport.7 A variation of the nonMarkovian Master equation was developed by Grover and Silbey to model the migration of Frenkel excitons in molecular crystals.8 An alternative diagonalization of the spin-boson Hamiltonian was adopted by Pererverzev and Bittner to study electronic relaxation in a mesoscopic electron-phonon system, with a finite number of bath modes.9 Recent experimental studies have revealed the existence of persistent coherence in light-harvesting antenna proteins,10-16 suggesting that quantum effects may contribute to the remarkable efficiency of these systems. This has revived theoretical interest in EET in light-harvesting antennas and, in particular, the influence of exciton delocalization and correlated energy fluctuations in boosting the efficiency of exciton transport.17-26 Until now two-dimensional photon echo (2DPE) experiments have been used to study exclusively single light-harvesting proteins. The conclusions regarding the existence of long-lasting Special Issue: Shaul Mukamel Festschrift Received: September 2, 2010 Revised: December 6, 2010 Published: January 20, 2011 5243

dx.doi.org/10.1021/jp108397a | J. Phys. Chem. B 2011, 115, 5243–5253

The Journal of Physical Chemistry B coherence are thus limited to the single protein scale, and the dynamics of energy transfer on a multiprotein level or the existence of interprotein coherence are, at present, matters of theoretical speculation. If quantum phenomena are indeed the driving force behind efficient excitation transfer, one might expect to find signatures of quantum behavior on a larger scale, as an excitation must journey through several units of closely packed protein molecules before sensitizing the reaction center. Moreover, to explore the consequences of coherent behavior on larger scales (ideally on the scale of the energy funnel in the lightharvesting complex), the different operating conditions of the experiments and the natural systems must be carefully considered. In 2DPE experiments femtosecond pulses are used to create coherent superpositions of the excitonic eigenstates, and the phase coherence of this state is observed to persist for hundreds of femtoseconds. Under natural conditions however, the excitation is created by broadband sunlight, and the corresponding initial state may be described as an incoherent mixture of the excitonic eigenstates. This poses the crucial question that if coherent initial states are unlikely to be created by sunlight, how can quantum phenomena, which are revealed in the experiments as phase coherence, be manifested under natural conditions? One implication of persistent phase coherence is that the excitonic eigenstates can maintain a finite delocalization in the steady state.27 In other words, as the eigenstate decoherence time is much shorter than the time scale over which sites lose their coherence and in light of the experimentally observed long (eigenstate) decoherence times, it is plausible to argue that in the thermodynamic limit the system relaxes toward states with finite delocalization, and that this delocalization can persist on the time scale of long-range energy transfer (∼100 ps). The second important question in long-range energy transfer is whether one needs to include all chromophores in the dynamics in order to obtain a reasonable estimate of the transfer time. If coherent phenomena are limited to the single protein scale, one might expect the excitation to flow toward the lowest energy sites on a single protein, and then hop from one protein to another without visiting the sites at higher energies. Within this F€orster-like approximation, each protein may be modeled by its lowest energy dimer, and interprotein energy transfer may be viewed as a random walk between the low energy dimers of neighboring proteins. If coherent phenomena are relevant on a multiprotein level (or if the transitions are strongly thermally activated), this approximation is unlikely to be valid, as an optimum path to the reaction center may include the higher energy sites. In this article we consider a molecular aggregate consisting of four units of phycoerythrin 545 (PE545) antenna proteins. PE545 is a water-soluble photosynthetic antenna protein isolated from certain species of cryptophyte algae. We use the atomic-resolution structural model of PE545 from Rhodomonas CS24,28 and study the interprotein transfer dynamics via a Monte Carlo simulation. The lowest order F€orster approximation is found to predict a slower transfer than a model in which all chromophores are included in the dynamics. We furthermore adopt different hopping models, based on various formulations of the exciton transfer problem,8,9 and find a transfer mechanism in which energy hops between delocalized eigenstates to yield the fastest dynamics. Our results indicate that close packing of proteins can delocalize an excitation between chromophores on adjacent units. This delocalization may be used advantageously in long-range exciton transport, suggesting that exciton delocalization is an important manifestation of quantum phenomena on a multiprotein scale.

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’ THEORETICAL METHODOLOGY The time-dependence of excitation transfer from an initial state i to a final state n under the influence of a non-Markovian bath, can be written as the integral of the memory kernel Gin(t), associated with the transfer process, that is Gin ðtÞ ¼ 2Re½eiΩin t ÆVin ðtÞVni ð0Þæ

ð1Þ

where Ωin is the frequency difference between the initial and final states (p = 1), Vin is the matrix element of the perturbation driving the transition, and the angle brackets indicate an averaging over the bath modes. The rate of energy transfer is subsequently obtained as the time integral of Gin(t) Z t kin ðtÞ ¼ Gin ðτÞ dτ ð2Þ 0

This is the Kenkre and Knox theory of non-Markovian energy transfer,6,7 which is reduced to the Fermi Golden rule in the limit of Markovian bath correlations. The above formulation gives rise to a subtlety that for a given Hamiltonian distinct memory kernels may be constructed, depending on the term that is identified as the perturbation driving the transition. This raises a question regarding the regimes of applicability of each formalism, as distinct constructions of the memory kernel predict different transition rates. We return to this point later on when we compare the two different constructions of the memory kernel for the spin-boson Hamiltonian. In this article we are interested in excitation transfer between chromophores of PE545 antenna proteins, as modeled by the linear exciton-phonon Hamiltonian. The linear excitonphonon Hamiltonian describing the interaction of N coupled two-level molecules and with a phonon bath is given by29 H ¼ -

X mn

Jmn a†m an þ

X n, k

a†n an ½εn þ φnk ωk ðb†k þ bk Þ þ

X

ωk b†k bk

k

ð3Þ where Jmn is the electronic coupling between the sites m and n, {a†n,an}

are the molecular raising and lowering operators for site n, εn is the electronic transition energy at site n, {b†k,bk} are the bosonic creation and annihilation operators for mode k of the bath, φnk = hnk/ωk is a dimensionless displacement quantifying the exciton-phonon coupling between site n and mode k, hnk is the exciton-phonon coupling in energy units, and ωk are the bath frequencies. Equation 1 may be applied to the linear exciton-phonon Hamiltonian in two inequivalent ways, without making explicit assumptions regarding the relative size of the electronic and the system-bath coupling. The first approach is the formulation of Grover and Silbey (henceforth referred to as the GS formulation) in which the shift in the nuclear coordinates in eq 3 is removed via a small-polaron transformation.29 The transformed Hamiltonian is found to be _ X X X εhn a†n an þ ωk b†k bk Jnm a†n Fn† Fm am ð4Þ H ¼ n

n6¼m

k

where Fn is the vibrational shift operator and εhn are the shifted energies " # X † φnk ðbk - bk Þ ð5Þ Fn ¼ exp k

εhn ¼ εn -

X hnk 2 k

ωk

ð6Þ

P The quantity k(hnk2/ωk) is the reorganization energy of the bath. The off-diagonal term in eq 4 may now be identified as the 5244

dx.doi.org/10.1021/jp108397a |J. Phys. Chem. B 2011, 115, 5243–5253

The Journal of Physical Chemistry B

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perturbation driving the transfer. The memory function associated with this perturbation can be written as 2 iΩ h int ÆF † ðtÞF ðtÞF † F æg Gsite n in ðtÞ ¼ 2jJin j Refe i n i

ð7Þ

where Ω h in is the frequency difference between the dressed sites, i and n. Notice that the perturbation is a simple product of contributions from the bath and the electronic part. The thermal expectation value is therefore taken with respect to the bath parameters only. After tracing over the bath modes, we arrive at the well-known result29 ( ) X 2 † † ÆFi ðtÞFn ðtÞFn Fi æth ¼ exp ðφik - φnk Þ ð1 þ 2nk Þ ( exp

k

X

2

ðφik - φnk Þ ½nk e

iωk t

Lastly, we eliminate the shift in the nuclear coordinates of the diagonal part of the Hamiltonian, and write the final expression as a diagonal zero-order term and an off-diagonal perturbation ~ ¼ H ~ 0 þ V~ H X X ~0 ¼ ER A†R AR þ ωk b†k bk H R

V~ ¼

þ ðnk þ 1Þe

A†β AR MRβk

where

) - iωk t

X

MβRk ¼ gRβk

ð8Þ

2gRRk b†k þ bk ωk

ð18Þ

k

R6¼β, k

ð17Þ

exp

8 > < = X ~ t eig iΩ Rβ ð22Þ GRβ ðtÞ ¼ 2Re e ÆMRβk ðtÞMβRq ð0Þæ > > ; : kq

Rβk

where

X gRRk 2

ð27Þ Henceforth this formulation is referred to as the PB (Pereverzev and Bittner) theory. Unlike eq 4 the perturbation 5245


The Journal of Physical Chemistry B of eq 20 is not a simple product of contributions from the system and the bath, and the system parameters also appear in the thermal averaging. The interpretation of this formulation is as follows: if the vibrational bath is sufficiently weak, the excitonic eigenstates retain their delocalized character. The bath influence will consequently be limited to inducing transitions between the dressed eigenstates. This formalism can thus be valid if (1) the reorganization energy of the bath as “seen” by the eigenstates is smaller than the typical value of the electronic coupling J, so that the eigenstates can maintain their delocalized character, and (2) the off-diagonal bath coupling which induces transitions between the eigenstates is smaller than the diagonal bath coupling, such that the perturbation of eq 16 remains small. To formalize the first condition we recall that the effective bath reorganization energy associated with the eigenstates |Ræ is given by 2 X gRRk R ER ¼ ð28Þ ωk k By substituting eq 12 into eq 28 and assuming that the bath modes at different sites are identical and independent, we arrive at the following equation for the renormalized bath energy X R ER ¼ ER TiR 4 ð29Þ i

P 2 where ER = P k(h4 nk /ωk) is the bare reorganization energy of the bath and iTiR is the inverse participation ratio (IPR) of the eigenstate |Ræ (for the electronic eigenstates all entries of the matrix T are real). IPR is an inverse measure of the delocalization length of the eigenstates.30 For maximally delocalized states IPR = 1/N, and for localized states IPR =1. If increasing the system size enhances exciton delocalization, the effective bath reorganization energy would be a size-dependent quantity, and would be reduced as the system size is increased. P P The second demands k|gRRk|2 > k|gRβk|2 and is P P condition 4 2 2 reduced to i|TiR| > i|TiR| |Tiβ| when the bath modes are identical and independent for all sites. There needs to be sufficient energetic disorder in the system for this condition to hold. If the energetic disorder is too large however, delocalization is destroyed, and the first condition would cease to hold. A balance of disorder and delocalization is thus necessary for the applicability of the PB diagonalization. In the PE545 tetramer, the bare reorganization energy of the bath is larger than the electronic coupling. The GS diagonalization is thus a viable procedure. Aggregation, however, delocalizes the excitation between sites on neighboring proteins, thereby lowering the effective bath reorganization energy. This renormalized bath energy is now of the same order as the electronic coupling. The first condition of the PB procedure is thus satisfied. There is also sufficient energetic disorder for the second condition to hold. It is therefore by no means obvious that which diagonalization precedure is best suited for the system. Motivated by these observations, we present a numerical comparison of the dynamics of energy transfer as obtained by each formulation. We find that the approaches do not converge in the limit of small electronic coupling, as even a small delocalization can generate correlations which are absent in a localized transfer model.

’ PE545 MOLECULAR CLUSTER In this article we study the interprotein energy transfer dynamics at ambient temperature among PE545 proteins that serve as the primary light-harvesting antenna complexes for the

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unicellular cryptophyte algae Rhodomonas CS24. PE545 contains eight light-absorbing bilin molecules covalently bound to the protein scaffold.31 Its structure and absorption spectrum is shown in Figure 1.28 The lowest energy chromphores are a pair of dihydrobiliverdin (DBV) bilins. There are additionally two pairs of phycoerythrobilin (PEB) chromphores with a single covalent bond to the protein and a dimer of doubly bound PEB chromphores, labeled PEB0 . In contrast to other antenna pigment-protein complexes, typically located in the thylakoid membrane, PE545 is a water-soluble protein, densely packed in the lumenal side of this membrane. These proteins are not known to have well-defined orientations relative to each other or the membrane,32 but there is evidence of associations.33,34 In order to study the energy transfer dynamics among PE545 units, we build a model system composed of four densely packed PE545 units. We first adopt the high-resolution crystal structure of PE545 (PDB ID 1XG0).28 Then, we use a rigid-body protein-protein docking algorithm in order to build a tetramer of PE545 units. In particular, we use the docking server ZDOCK to obtain a reasonable structure of the tetramer.35,36 ZDOCK algorithm starts by building a large number of possible orientations between two proteins (in this case two PE545 monomers). Then, the predicted structures are ranked according to a scoring function based on shape complementarity, desolvation energy, and electrostatics. In this work we always choose the first structure from this ranking. We start by predicting a suitable arrangement between two PE545 units. Then, we predict the orientation between this dimer and a new PE545 unit. This procedure is repeated until we find the structure of the PE545 tetramer, shown in Figure 2. We note that in order to predict the true binding mode of a protein-protein complex, it would be necessary to postprocess a considerable number of ZDOCK predicted structures using a flexible protein-protein method. Here, however, we aim at building a reasonable, representative arrangement between the four units among the many others that could occur in the lumen, so the rigid-body docking approach is sufficient for our purposes. PE545 Hamiltonian. Electronic couplings between all chromophores in the assembly were computed using quantum chemical calculations, whereas values for the diagonal site energies were taken from a fitting to steady-state spectra and transient absorption (model E from ref 37). The geometries of the chromophores were taken from the PE545 tetramer structure and hydrogens were added and optimized at the HF/6-31G(d) level of theory. We considered the fully protonated state of the two central pyrrole rings for all chromophores, consistent with previous observations,28 and substituted the cysteine residues bonded to the chromophores by the -S-CH3 groups. Electronic couplings were then obtained at the CIS/6-31G level of theory incorporating environment screening effects through the polarizable continuum model (PCM).38-40 PCM cavities were built from radii obtained by applying the united atom topological model to the atomic radii of the UFF force field.41 All quantum chemical calculations were performed using a local version of Gaussian 09 adapted to perform calculations of electronic couplings between excited states on molecules.42 The protein medium and surrounding water solvent was collectively modeled as a dielectric continuum with a relative static dielectric constant of 15 and optical dielectric constant of 2.43 The Hamiltonian of an isolated PE545 is listed in Table 2. The computed electronic couplings and diagonal energies of the tetramer are provided as Supporting Information. 5246



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Figure 1. (a) Chromophores from the structural model of PE545. (b) Absorption spectrum of PE545. The color of the bars correspond to the coloring of chromophores in panel a.

Figure 2. (a) PE545 molecular tetramer. The initial excitation was assumed to be on p2 DBV 19A. The time scale for the excitation to reach p4 DBV 19B was investigated. (b) Only the lowest energy chromophores (DBVs) are shown. (c) Arrangement of the DBV and the doubly bound PEB0 bilins. (d) Arrangement of the DBV and the singly bound PEB bilins. The chromophore labelings are listed in Table 1.

In addition, static disorder in the chromophore transition energies caused by slow fluctuations in the protein environment was modeled by assuming uncorrelated shifts in the site energies randomly taken from a Gaussian distribution with a width (fwhm) of Γσ = 400 cm-1,37 whereas off-diagonal disorder was neglected. PE545 Discrete Vibrational Modes. High-frequency intramolecular vibrational modes of the bilin chromophores were

described by a set of 14 discrete oscillators (DO).37 In addition, the coupling of excitations to low-frequency collective modes of the pigment-protein complex was described phenomenologically by an additional discrete mode (frequency 8 cm-1, HuangRhys factor 13.75) obtained by fitting to room temperature and 77 K fluorescence spectra. The fitting was performed using the emission spectra expression, eq 3 from ref 44 and adopting the 5247



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electronic couplings, site energies, and static disorder described in the previous section. Note however, that here the line shape is taken to be the sum of the above-mentioned 15 DO’s.

’ KINETIC MONTE CARLO ALGORITHM The disorder-generating algorithm stated in the previous section was used to generate 2000 realizations of the diagonal energies. For each realization two matrices quantifying the direct transfer rate between all pairs of chromophores in the site and the eigenstate basis were computed. To obtain an estimate of the time scale of long-range energy transfer, two chromophores were chosen as the initial and final sites to host the excitation. The initial site is the higher energy DBV chromphore on the second protein (p2 DBV 19 A) and the final site is the lower energy DBV chromophore on the fourth protein (p4 DBV 19 B). These two chromophores are the pair farthest-apart in the entire assembly, separated by a distance of 83.5 Å (Figure 2a). The computed transfer rates were subsequently used to generate a cumulative probability distribution (CPD) function. The CPD function associated with a transition i f n is defined to be n P kip p¼1 ð30Þ CPDi ðnÞ ¼ P kim m

where kim is the i f m transfer rate. The Monte Carlo hopping algorithm can be summarized as follows: 1. For each realization of the Hamiltonian compute the pairwise transfer rates and use that to calculate the CPD function for each site. 2. Start at site i. Choose a random number r and hop to site m, where m = CPDi-1(r). 3. Compute the waiting time tw, at site i as tw = -(ln(r0 )/Σmkim), where r0 is a newly generated random number. 4. Choose a new random number and continue the hop, while keeping track of the total accumulated waiting time. 5. Stop if the final site has been reached or if the total time exceeds a preset value tmax. In the latter case, record the hopping sequence as a failure. To gain an understanding of the role played by the PEB bilins, two transfer scenarios were considered. In the first scenario each protein was approximated by the low energy DBV molecules. The idea behind this approximation is that according to the F€orster theory the ratio of the downhill versus the uphill transfer as governed by the detailed balance is given by eβωin, where ωin is the transition energy. At room temperature and for a transition frequency of 1100 cm-1 (the smallest DBV to PEB transition frequency), this corresponds to a ratio of 266:1. We thus neglect all uphill transfer, and assume that if the excitation is initially on a low energy DBV, it can hop to other DBVs only. In the second scenario energy transfer in the full assembly was studied. Each scenario was investigated via both the site and the eigenstate model. The maximum time to complete the journey was set to tmax = 10 ns, which is an order of magnitude larger than the excited state lifetime in PE545.28 For each realization the time taken to complete the journey, the number of hops between the sites (or eigenstates), as well as the percentage of trajectories that fail to reach the destination chromophore, were recorded.

’ RESULTS AND DISCUSSION Figure 3 is a plot of the delocalization length of the excitonic eigenstates averaged over 2000 realizations of the Hamiltonian.

Figure 3. (a) Average delocalization length of the bilins in the PE545 tetramer computed over 2000 realizations of the Hamiltonian. The labelings of the chromphores (site index) are listed in Table 1. The five chromophores with the largest delocalization are labeled on the graph. (b) Delocalization length of chromophores in an isolated PE545 unit.

Delocalization length is defined to be ( )-1 N X 4 jcni j Ln ¼

ð31Þ

i

where cni are the normalized amplitudes of the eigenstate |ψnæ with respect to the site |iæ X jψn æ ¼ cni jiæ ð32Þ i N X

jcni j2 ¼ 1

ð33Þ

i

Chromophores with maximum delocalization are found to be those in the middle of the assembly and the ones with small delocalization are at the edge of the assembly. For comparison the exciton delocalization length of an isolated PE545 unit is also plotted (Figure 3b). The tetramer eigenstates are significantly more delocalized. This suggests that close packing of proteins inside the thylakoid lumen can result in substantial delocalization among chromphores on neighboring units. In order to understand the origin of any differences between the two formulations of exciton transfer, we first present a comparison of the memory kernel between arbitrarily chosen sites as obtained from each model. We then proceed to discuss the results of the Monte Carlo simulation. The memory kernel quantifies the temporal variations of the transfer rate. This variation can either arise from the bath nonMarkovianity, or from coherent electronic oscillations. In the former scenario (regime of interest in the present article) the 5248



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Table 1. Diagonal Elements: Transition Energies (cm-1), Model E from ref 37; Off-Diagonal Elements, Upper Half: Electronic Coupling (cm-1) between Chromphore Pairs, Lower Half: Center to Center Separation between Bilins (Å) DBV A

DBV B

PEB 158C

PEB 158D

PEB0 C

PEB0 D

PEB 82C

PEB 82D

DBV A

18008

-4.1

-31.9

2.8

2.1

-37.1

-10.5

45.9

DBV B PEB 158C

43.2 22.3

17973 48.3

-2.9 18711

30.9 -5.6

-35.4 -19.6

2.5 -16.1

-45.5 6.7

11.0 6.8

PEB 158D

47.6

21.8

45.6

18960

11.5

25.5

5.1

7.4

PEB0 C

31.6

25.3

25.7

23.5

18532

101.5

36.3

16.0 -38.6

PEB0 D

24.6

31.0

21.4

25.3

17.2

19574

17.6

PEB 82C

33.2

23.3

38.8

38.8

24.2

33.8

18040

2.6

PEB 82D

23.4

33.3

39.9

38.7

35.2

24

35.9

19050

Table 2. Chromophore Labelinga site index

a

bilin type

1 þ 8(n-1)

pn DBV 19 A

2 þ 8(n-1) 3 þ 8(n-1)

pn DBV 19 B pn PEB 158C 501

4 þ 8(n-1)

pn PEB 158D 504

5 þ 8(n-1)

pn PEB 5061C 500 (PEB0 )

6 þ 8(n-1)

pn PEB 5061D 503 (PEB0 )

7 þ 8(n-1)

pn PEB 82C 502

8 þ 8(n-1)

pn PEB 82D 505

The protein index n, can take the values 1-4.

time taken for the memory kernel to establish its steady state value is known as the bath memory. Non-Markovianity must be accounted for if bath memory is longer than, or comparable to, the time scale of the physical process that one wishes to study. In physical terms non-Markovianity implies the possibility of reversible information flow between the system and the bath, in contrast to Markovian dynamics where the flow of information is from the system to the bath only.45 For the system under investigation the memory time is found to be ∼20 fs. Since the bath memory is short in comparison to the time scale of energy transfer, the time dependence of the transfer rate may be neglected, and theR rate may be approximated by its steady state value, that is k = 0¥G(t) dt. This coarse-graining approximation of the non-Markovian transfer rate is in fact equivalent to the Markov approximation (Appendix A). Figure 4a is a plot of the memory kernel in the eigenstate and the site model, calculated between the sites i = 9 (p2 DBV 19A) and i = 26 (p4 DBV 19B) for the tetramer where all bilins are included in the simulation. These are the farthest pair in the assembly, separated by a distance of 83.5 Å, with a small electronic coupling of 0.2 cm-1. The steady state value of the transfer time is shorter by 2 orders of magnitude in the eigenstate model, 9.5 ns, compared to 5 μs in the site model. The negative values of the memory kernel in these plots have a vibrational origin, and indicate that the instantaneous rate of transfer may decrease as a function of time. Increasing the bath reorganization energy would eliminate any oscillations or negative values of the memory kernel, as one recovers the Markov limit for large reorganization energies. In these plots memory decays over the same time scale in both models. The enhancement in rate is therefore due to the larger value of the memory kernel at t = 0, which, in turn, is due to exciton delocalization, as well as a subtle interplay between the system and the bath. We elaborate on the role of these factors in the following paragraph.

Figure 4. Memory function versus time (a) between i = 9 and 26 in the eigenstate and the site model. (b) Corresponding plots between i = 1 and 27. (c) Memory function between i = 1 and 27 with the bath couplings reduced by a factor of 2.

To demonstrate how exciton delocalization can enhance energy transfer, consider a transfer scenario in which the donor (acceptor) state |φiæ (|φjæ), is delocalized over n (m) sites, labeled {i1, ..., in} ({j1, ..., jm}). We furthermore assume that each eigenstate is at least 60% localized on a particular chromophore, so that one can unambiguously assign each eigenstate to a unique site index (this is the case in the PE545 tetramer). We label the site with maximum amplitude in each set as i1 and j1 respectively. If the sites i1 and j1 are weakly coupled but there are strongly coupled pairs in the two sets, the contribution of these pairs to the transfer process can boost the overall transfer dynamics between the two excitonic states.46 Under these conditions, the effective coupling between two excitonic states is larger than the coupling between i1 and j1. Moreover, the difference between the two theories is maximal if many intermediate sites reside between the initial and final chromophores. In the site representation energy transfer is a two-body event, mediated by the direct coupling between i1 and j1. The existence of many intermediate sites often implies that the direct coupling and the corresponding 5249



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Figure 5. (a and c) Cumulative probability distribution (CPD) as defined by eq 30 in the eigenstate picture for site i with respect to sites n, for (a) only the DBV bilins included and (c) all chromophores included. (b and d) The corresponding plots in the site picture, (b) DBV bilins only and (d) all chromophores.

transfer rate are small. In the eigenstate representation however, energy transfer is a collective phenomenon, which accounts the influence of the intermediate sites. This co-operative nature of the transfer process is the key factor behind the fast dynamics in the eigenstate formulation. In other words, delocalization enhances energy transfer in much the same way that has been documented in the generalized F€orster theory as adopted to aggregated molecular assemblies.46-49 This argument suggests that the difference between the two models would assuage if the initial and final states were nearest neighbors. Figure 4b shows the corresponding plots for sites i = 1 and 27, with a coupling of 72.5 cm-1. The average transfer time between this pair is 1.5 ps in the eigenstate basis, compared to 27 ps in the site basis. The second factor influencing the disagreement between the two theories is with regards to the influence of the vibrational bath. In the site model the bath determines the rate at which the memory kernel decays, but has no effect on its initial amplitude. In the eigenstate model however, both the electronic and the bath part influence the initial value of the memory kernel. Figure 4c depicts the memory kernel between the sites i = 1 and 27, with the systembath coupling reduced by a factor of 2. In the site model this manifests itself only in the damping rate of the memory function, whereas in the eigenstate model the initial amplitude is also reduced by an order of magnitude. The average transfer time between the sites is now 281 fs in the eigenstate model, compared to 870 fs in the site model. The theories therefore appear to converge for nearestneighbor sites, for smaller bath couplings. In this limit the two distinguishing features of a delocalized model, namely long-range correlations and a reduced effective reorganization energy, have minimal influence and a local transfer model provides an adequate description. Beyond nearest-neighbor sites, collective effects persist, and the theories do not, in general, converge. We therefore conclude that in large systems with sufficient exciton delocalization and energetic disorder, the eigenstate formulation provides a more reliable picture of the long-range dynamics.

Figure 5, panels a and b, shows the CPD functions for the DBV chromophores, in the eigenstate and the site model respectively. Without the PEB bilins, the transfer probability from a given DBV is completely dominated by its nearest DBV neighbor. As the DBVs within a light-harvesting unit are positioned at the opposite ends of the protein, the nearest DBV neighbor is more likely to lie on an adjacent protein. This gives rise to strong energy traps between DBVs on adjacent proteins, and is manifested in the rapid rise of the CPD function. Once the PEB chromophores are introduced in the assembly, the CPD function is changed significantly. The PEB0 dimer is positioned in the middle of the DBV dimer, while the PEB bilins are positioned in an open arrangement on the outside of the protein. This arrangement of chromophores funnels the energy out of the DBV traps, and is manifested in Figure 5, panels c and d, by the gradual rise of the CPD function. This demonstrates the inadequacy of a F€orster mechanism in which energy is assumed to flow irreversibly toward the low energy sites. Unless energy transfer is strongly thermally activated, excitation is likely to be trapped at a local minima. Figure 6 shows the distribution of the transfer times and Table 3 summarizes the statistics of the transfer process in the four sets of data. With only the DBV chromphores present, the site model generates a large number of failed trajectories. In the eigenstates model, owing to faster pairwise transfer, the success rate is significantly larger. Strong trapping pairs in the eigenstate model give rise to large numbers of hops, as the energy is bounced back and forth many times in the trap before it can escape. With the PEB chromophores included in the dynamics, the total transfer time is reduced by an order of magnitude in the eigenstate model. In both models the failure rate is also substantially decreased. This demonstrates the importance of high energy bilins and static disorder in boosting the efficiency of long-range EET. Static disorder in the site energies can bring the higher energy sites in resonance with the low energy ones, mediating faster uphill transfer. The open arrangement of the PEB bilins creates an extended network of connections which 5250



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typically involves ∼200 molecules, and the energy is transferred across a total distance of ∼20-100 nm.16 Furthermore, the reaction center has a finite trapping rate which would further increase the transfer time. This suggests that on the scale of a natural energy funnel this simulation is likely to predict a slower transfer than the experimental findings. Further refinements in computation of the pairwise rates are therefore required if the experimental value of transfer time is to be approached. One way of refining the theory is by introducing spatial correlations in the vibrational modes. Spatial correlations reduce the reorganization energy of the bath, thus increasing the memory time and the transfer rate.17 The question therefore remains how correlated the bath fluctuations need to be in order to achieve sufficient speed up in the pairwise transfer rates.

Figure 6. Histograms representing the distribution of the transfer times over 2000 realizations of the Hamiltonian. (a) Eigenstate transfer model with only the DBV bilins present. (b) Site transfer model with only the DBV bilins present. (c) Eigenstate model with all chromophores included. (d) Site picture with all chromophores present.

Table 3. Statistics of the Transfer Process Averaged over 2000 Realizations of the Hamiltonian, Using the Eigenstate and the Site Transfer Models at T = 300 K 8 sites

eigenstate model

site model

average transfer time

1.58 ns

5.04 ns

average no. of hops

1460

44

fail

13.1%

75%

32 sites

eigenstate model

site model

average transfer time

247 ps

3.71 ns

average no. of hops fail

437 2.5%

93 21%

prevents the formation of local traps and enables the excitation to explore different regions of the aggregate. In summary, we find that three independent processes are important for efficient long-range energy transfer: (1) exciton delocalization, which induces long-range correlations in the aggregate, giving the transfer process a collective character, (2) high energy intermediate sites that bring chromophores from neighboring proteins within close proximity and prevent the formation of local energy traps, and (3) static disorder that brings the higher energy sites in resonance with the low energy ones. A transfer model that includes the PEB chromophores in the dynamics, takes into account the static disorder, and regards long-range transfer as a biased-random walk between delocalized eigenstates, utilizes the above factors maximally and predicts the fastest dynamics. Time-resolved fluorescence anisotropy experiments have revealed three time scales for the transfer of excitation from PE545 to the membrane associated chlorophylls at room temperature: 17, 58, and 113 ps,50 in comparison to the fastest transfer time of 247 ps obtained from this simulation. If our model were to be extended to the scale of an energy funnel, many more proteins would have to be included in the simulation as the funnel

’ CONCLUSION We have presented a computational study of long-range energy transfer in a cluster consisting of four units of PE545 proteins. Energy transfer is viewed as (1) transitions among the dressed excitonic eigenstates mediated by the vibrational bath and (2) a random walk between the dressed localized sites induced by mutual electronic interactions. We found that close packing of PE545 proteins inside the thylakoid lumen brings chromphores from adjacent proteins within close proximity of each other, thereby delocalizing the excitation between sites on neighboring units. This delocalization leads to a collective dynamics that can substantially boost long-range energy transfer. We have furthermore demonstrated that the arrangement of PEB chromphores in the PE545 tetramer, together with static disorder, prevents excitation trapping at the low energy dimers. These two factors help to create a large interconnected network, allowing the excitation to collectively explore different regions of the assembly and eventually sensitize the reaction center. We emphasize that in this simulation the initial and final sites are chosen arbitrarily, as we merely intend to acquire an insight regarding the dynamics of multiprotein energy transfer and do not aim to replicate a realistic transfer process, in which the excitation is funneled toward an external reaction center. Moreover, our results do not answer the question of quantum effects on an interprotein scale unequivocally, as the model adopted to simulate the transfer process is a semiclassical random walk. What the current work does establish, however, is the importance of exciton delocalization in long-range energy transfer: despite the small electronic couplings and the relatively small delocalization length of the excitonic eigenstates, a theory that includes delocalization predicts a much faster dynamics on an interprotein level, suggesting that persistent exciton delocalization is a necessary ingredient for fast long-range energy transport. ’ APPENDIX A. MARKOV LIMIT OF THE KENKREKNOX RESULT The non-Markovian Kenkre-Knox rate equation is given by Gin ðtÞ ¼ 2Re½eiΩin t ÆVin ðtÞVni ð0Þæ

ð34Þ

The Markov limit may be obtained by making the following substitution:6 Z ¥ Gin ðtÞ f δðtÞ Gin ðsÞ ds ð35Þ 0

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The Journal of Physical Chemistry B The rate in the Markov limit is thus given by Z ¥ Gin ðτÞ dτ kin ¼

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ð36Þ

0

Z ¼ 2Re

¥

Z

¥

δðτÞ dτ

0

eiΩin s ÆVin ðsÞVni ð0Þæ ds

ð37Þ

0

Z ¼ 2Re

¥

eiΩin s ÆVin ðsÞ Vni ð0Þæ ds

ð38Þ

0

which is the same as the steady-state non-Markovian solution.

’ ASSOCIATED CONTENT

bS

Supporting Information. Computed electronic couplings and diagonal energies of the tetramer. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT C.C. acknowledges support from the Comissionat per a Universitats i Recerca of the Department d’Innovacio, Universitats i Empresa of the Generalitat de Catalunya, Grant No. 2008BPB00108. We thank Benedetta Mennucci for the possibility to perform the quantum chemical calculations in collaboration with the Dipartimento di Chimica e Chimica Industriale of the Universita di Pisa using a local modified version of Gaussian 09. H.H. thanks Dr. A. Olaya-Castro and Dr. F. Fassioli for interesting discussions and Jonathan Dursi for computational support. A.K. acknowledges the Centre for Quantum Information and Quantum Control for the award of a summer studentship. The United States Air Force Office of Scientific Research is gratefully acknowledged for support of this research under contract number FA9550-10-1-0260. The Natural Sciences and Engineering Research Council of Canada is acknowledged for financial support. ’ REFERENCES (1) Blankenship, R. E., Molecular Mechanisms of Photosynthesis; Blackwell Publications: Chichester, U.K., 2002. (2) Beljonne, D.; Curutchet, C.; Scholes, G. D.; Silbey, R. J. J. Phys. Chem. B 2009, 113, 6583–99. (3) Scholes, G. D.; Fleming, G. R. Adv. Chem. Phys. 2005, 132, 57–129. (4) Clegg, R. M.; Sener, M.; Govindjee Opt. Biopsy VII 2010, 7561, 7561–12. (5) Jimenez, R.; Dikshit, S. N.; Bradforth, S. E.; Fleming, G. R. J. Phys. Chem. 1996, 100, 6825–6834. (6) Kenkre, V. M.; Knox, R. S. Phys. Rev. B 1974, 9, 5279–90. (7) Kenkre, V. M. Phys. Rev. B 1975, 12, 2150–60. (8) Grover, M.; Silbey, R. J. Chem. Phys. 1971, 54, 4843–51. (9) Pereverzev, A.; Bittner, E. R. J. Chem. Phys. 2006, 125, 104906. (10) Collini, E.; Wong, Y. C.; Wilk, K. E.; Curmi, P. M. G.; Brumer, P.; Scholes, G. D. Nature 2010, 463, 644–647. (11) Engel, G. S.; Calhoun, T. R.; Read, E. L.; Ahn, T.-K.; Mancal, T.; Cheng, Y.-C.; Blankenship, R. E.; Fleming, G. R. Nature (London) 2007, 446, 782–6. (12) Lee, H; Cheng, Y. C.; Fleming, G. R. Science 2007, 316, 1462–5.

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Delocalization-Enhanced Long-Range Energy ... - ACS Publications

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