Atomic-level structural and functional model ofa bacterial photosynthetic membrane vesicleMelih K. Sener*†‡, John D. Olsen§, C. Neil Hunter§, and Klaus Schulten*‡¶

*Beckman Institute and ¶Department of Physics, University of Illinois at Urbana–Champaign, Urbana, IL 61801; †Department of Physiology and Biophysics,Weill Medical College, Cornell University, New York, NY 10021; and §Department of Molecular Biology and Biotechnology, University of Sheffield,Sheffield S10 2TN, United Kingdom

Communicated by Karl Hess, University of Illinois at Urbana–Champaign, Urbana, IL, August 6, 2007 (received for review March 8, 2007)

The photosynthetic unit (PSU) of purple photosynthetic bacteriaconsists of a network of bacteriochlorophyll–protein complexesthat absorb solar energy for eventual conversion to ATP. Becauseof its remarkable simplicity, the PSU can serve as a prototype forstudies of cellular organelles. In the purple bacterium Rhodobactersphaeroides the PSU forms spherical invaginations of theinner membrane, �70 nm in diameter, composed mostly of light-harvesting complexes, LH1 and LH2, and reaction centers (RCs).Atomic force microscopy studies of the intracytoplasmic membranehave revealed the overall spatial organization of the PSU. In thepresent study these atomic force microscopy data were used toconstruct three-dimensional models of an entire membrane vesicleat the atomic level by using the known structure of the LH2complex and a structural model of the dimeric RC–LH1 complex.Two models depict vesicles consisting of 9 or 18 dimeric RC–LH1complexes and 144 or 101 LH2 complexes, representing a total of3,879 or 4,464 bacteriochlorophylls, respectively. The in silico re-constructions permit a detailed description of light absorption andelectronic excitation migration, including computation of a 50-psexcitation lifetime and a 95% quantum efficiency for one of themodel membranes, and demonstration of excitation sharing withinthe closely packed RC–LH1 dimer arrays.

excitation transfer � network kinetics � photosynthetic light harvesting �quantum efficiency � systems biology

Photosynthesis, the main source of energy for the biosphere(1, 2), is initiated when the thousands of pigments that

cooperate to form an interconnected photosynthetic unit (PSU)harvest and transfer solar energy before its conversion to acharge separation. Peripheral pigment–protein complexes de-liver energy to a reaction center (RC) (3–6), where it is used forthe transmembrane electron transfers (7, 8) that eventually driveATP synthesis. The structures of the key protein complexesinvolved in this process have been solved both for oxygenicphotosynthetic organisms, such as cyanobacteria, algae, andplants (9–14), and for anoxygenic photosynthetic bacteria (15–24). This recent progress makes it possible to study the biophys-ical processes involved in photosynthesis in atomic detail all theway down to the quantum mechanical level (25–31). However, itremains a challenge to understand how a biological membranecomprising hundreds of photosynthetic complexes functionswith great efficiency. To address this challenge we embarked onthe in silico construction of an entire photosynthetic membrane,namely the purple bacterial PSU (32), based on a combinationof cryo-EM (24, 33–35), NMR (22), x-ray crystallography (18, 19,23, 36), and atomic force microscopy (AFM) (37–41) data.

The purple bacterial PSU displays remarkable simplicitycompared with its eukaryotic, oxygenic analogues, being evolu-tionarily more primitive (42). It contains six different kinds ofproteins that work cooperatively: LH2 antenna complexes (18,19, 23) capture photons and transfer the resulting electronicexcitation to another antenna complex, LH1, and finally to a RC(22, 36, 40, 43, 44). Subsequently, the RC initiates transmem-

brane electron transfer to the small, membrane-diffusible elec-tron carrier quinone, reducing it to hydroquinone.

In Rhodobacter sphaeroides, LH1 and RC, together with thesmall polypeptide PufX, form dimeric supercomplexes as seen innegative-stain EM of intact membranes and in cryo-EM projec-tion maps of purified complexes (24, 33, 35, 39, 45). Thehydroquinones released from the RC diffuse through the cellmembrane to the ubiquinol–cytochrome c2 oxidoreductase (bc1complex), which oxidizes hydroquinone by transferring the elec-trons to the cytochrome c2 complex and at the same time pumpsprotons across the membrane. The resulting proton gradient isfinally used by ATP synthase (46–48) for the synthesis of ATP.Cytochrome c2 shuttles the electrons back to the RC, thusresetting the system. All aforementioned components are em-bedded in the cell membrane, with the exception of cyto-chrome c2.

EM studies revealed that the membrane-embedded PSUsform intracytoplasmic protrusions. The assembly of intracyto-plasmic membranes is a complex process, initiated by theclustering of the LH2 and RC–LH1–PufX pigment–proteincomplexes, which has been studied with radiolabeling, biochem-ical, and spectroscopic techniques (49–51). In R. sphaeroides andsome other bacteria a combination of aggregation of the proteincomplexes and lipid biosynthesis induces membrane curvature,thus causing the budding or invagination to form the intracyto-plasmic membrane (49, 52, 53).

Spectroscopic data on membrane vesicles suggested the exis-tence of quantified energy transfer domains comprising �3,000bacteriochlorophyll (BChl) molecules surrounding and intercon-necting �30 RCs (50). Modeling the collective behavior of thePSU requires knowledge of the relative stoichiometry and spatialdistribution of photosynthetic complexes. Recently, the su-pramolecular organization of the PSU of several purple bacteriahas been revealed through AFM (37, 38, 41) and linear dichroism(LD) (39) studies. The AFM data obtained for R. sphaeroidesmembranes (37) permits the construction of atomic models ofthe PSU needed for a detailed description of the successiveprocesses of light absorption, excitation migration, electrontransport, quinone diffusion, proton transport, and ATPsynthesis.

ResultsConstituents and Overall Architecture of the PSU. AFM images of R.sphaeroides membranes showed that the PSU consists of linear

Author contributions: M.K.S., J.D.O., C.N.H., and K.S. designed research; M.K.S., J.D.O.,C.N.H., and K.S. performed research; M.K.S. contributed new analytic tools; M.K.S. analyzeddata; and M.K.S., C.N.H., and K.S. wrote the paper.

The authors declare no conflict of interest.

Abbreviations: AFM, atomic force microscopy; bc1 complex, ubiquinol–cytochrome c2

oxidoreductase; BChl, bacteriochlorophyll; LD, linear dichroism; PSU, photosynthetic unit;RC, reaction center.

‡To whom correspondence may be addressed. E-mail: melih@ks.uiuc.edu or kschulte@ks.uiuc.edu.

© 2007 by The National Academy of Sciences of the USA

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arrays of RC–LH1–PufX dimers separated by domains of LH2complexes (37). However, the AFM data alone do not permit anunambiguous three-dimensional reconstruction of intact chro-matophore vesicles because the vesicles were flattened to formpatches onto a mica surface. The reconstruction process there-fore utilizes input from LD spectroscopy of native, intact mem-brane vesicles (39), revealing the orientation of the RC–LH1–PufX and LH2 complexes in the intact vesicle. This leads to amodel in which linear assemblies of dimeric RC–LH1–PufXcomplexes lie along the long axis of the vesicle connecting itsfarthest extremity with the cytoplasmic membrane (39).

The bc1 complex and the ATP synthase have eluded imagingby AFM and still have to be assigned. A recent model of the R.sphaeroides vesicle, which considers the location and stoichiom-etries of electron transport components (54), concludes that asfew as five bc1 complexes suffice to maintain adequate rates ofATP synthesis, but earlier studies suggest one bc1 complex pertwo RCs (55) and nine bc1 complexes per chromatophore (56).About one ATP synthase is present per vesicle with some vesicleslacking this complex (57).

In light of the evidence described, we propose the followingtentative picture for the architecture for an intracytoplasmicmembrane (chromatophore) vesicle. The vesicle, connected at its‘‘south pole’’ to the cell membrane, develops around lineararrays of RC–LH1–PufX dimers, oriented south–north consis-tent with ref. 39; LH2 complexes assemble around these arrays,consistent with the observed kinetics of LH2 incorporation (49,52, 53), and mainly fill space in the equatorial and ‘‘northern’’region. In mutants lacking LH2 continued formation of linearcore dimer arrays results in the formation of elongated tubes (33,35, 45). The proposed vesicle geometry is shown in Fig. 1.

Some bc1 complexes are proposed to populate the neck regionof the vesicle, because it is known that a noninvaginated cyto-plasmic membrane fraction can be purified from photosynthet-ically grown cells and that these membranes not only lackphotosynthetic complexes, but are also enriched in respiratorycomponents (58). However, other bc1 complexes may be locatedequatorially closer to the RC–LH1–PufX complexes. Becausepreparations of chromatophores do catalyze cyclic electrontransport and ATP synthesis, bc1 complexes and ATP synthasesmust be present in at least some of the vesicles. Their schematic,peripheral placement in Fig. 1 reflects the lack of their directdetection by AFM. The in silico reconstruction of intracytoplas-mic membrane vesicles presented herein is based on combiningmultiple small domains of AFM images consisting solely ofRC–LH1–PufX dimers and LH2 complexes with the help ofarea-preserving maps between planar and spherical patches (seeFig. 1 d–f and Methods).

Atomic-Level Structural Models of Chromatophore Vesicles. In addi-tion to the overall architecture elucidated by AFM and LD data,the construction of a model for an entire chromatophore vesiclerequires atomic-level structures for each of the constituentpigment–protein complexes shown in Fig. 1 b and c. The 6-Åprojection map of the R. sphaeroides LH2 complex shows that itis a nonameric structure (21), so, the 2.0-Å resolution structureof the homologous Rhodopseudomonas acidophila (23) LH2complex (PDB ID code 1NKZ) (21) was used in the absence ofa crystal structure for R. sphaeroides.

A structural model of the S-shaped dimeric RC–LH1–PufXcore complex of R. sphaeroides was constructed from the 8.5-Åcryo-EM projection maps reported in ref. 24, the highest reso-

Fig. 1. Architecture and constituents of a spherical chromatophore vesicle from R. sphaeroides constructed from AFM/LD data (37, 39). (a) Light-harvestingcomplexes, LH2 (green) and LH1 (red), absorb light and transfer the resulting excitation to the RC (blue), which subsequently initiates electron transfers reducingquinone to hydroquinone (not shown); the bc1 complex (yellow) oxidizes hydroquinone to create a proton gradient across the membrane, which in turn is usedby ATP synthase (orange) for ATP production. Electrons are shuttled back to the RC by cytochrome c2 (not shown). The current study focuses solely on thelight-harvesting process within the vesicle, and accordingly, bc1 complexes and ATP synthase are not considered, being depicted schematically peripheral to thechromatophore, although other bc1 complexes may be located within the vesicle closer to the RC–LH1–PufX complexes. The ratio of surface area covered byRC-LH1 versus LH2 complexes is 1:1.31 for the first vesicle (shown) and 1:3.23 for the second vesicle (Fig. 2 d and f ). (b) BChls (represented by their porphyrin rings)of the atomic model for the RC-LH1 complex constructed for this study based on cryo-EM data (24). The PufX polypeptide is not included. (c) BChls of the LH2complex based on R. acidophila (23). AFM images (d) (37) are used to identify the arrangement of pigment–protein complexes within planar patches (e). Anarea-preserving map from the plane on to the sphere, the inverse-Mollweide projection (89) (Eq. 1), is then used to position pigment–protein complexes on thevesicle surface (f). To minimize distortions, multiple planar patches were used, whose sizes are small compared with the inner diameter of the reconstructedvesicle (60 nm). [a–c were made with the program VMD (Visual Molecular Dynamics) (90).]

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lution data available, and from the monomeric RC–LH1 modelderived from cryo-EM, NMR, and AFM data (22, 34, 40), butwas not refined.

Several difficulties need to be faced before an atomic-levelmodel for the vesicle can be constructed. For example, thepreparation process distorts, stretches, and sometimes tearsvesicle patches. Therefore, the original f lattening process itselfis not likely to be area preserving. If unaccounted for, suchartificial stretching will reduce the pigment density and, as aresult, the calculated overall quantum efficiency of the system.The vesicle patches imaged by AFM (37) were aligned relativeto each other in a manner consistent with the LD data on intactmembrane vesicles (39).

Spherical vesicles were constructed by orienting planar stripscontaining the RC–LH1–PufX dimer arrays and their immediateLH2 neighbors along the north–south direction. The spacesbetween RC–LH1–PufX strips were filled with LH2-rich re-gions. An area-preserving map, the inverse-Mollweide transfor-

mation (see Methods, Eq. 1), was used to map planar regions ontospherical ones. Structural conflicts were subsequently removedand gaps that arose between the pigment–protein complexes asa result of the projection process were removed by manuallyshifting the center points of the proteins on the chromatophoresphere to achieve a tight packing. The ‘‘southern polar’’ regionwas left empty as a potential contact zone with the rest of themembrane. In this manner, two sample vesicles shown in Fig. 2were constructed that have an inner diameter of 60 nm, chosento be consistent with EM images. The first vesicle contains 18dimeric RC–LH1 complexes and 101 LH2 complexes with a totalof 3,879 BChls; the second vesicle contains nine dimeric RC–LH1 complexes and 144 LH2 complexes with a total of 4,464BChls. The total number of RCs is chosen to be consistent withearlier estimates (50) and to correspond with moderate- andlow-light growth conditions, respectively, for the two vesicles.The overall BChl:RC ratios for these vesicles of between 108 and248 are also consistent with those obtained for wild-type chro-matophores (59).

Fig. 2. Electronic interactions and excitation energy transfer across a chromatophore vesicle. BChls are represented by their porphyrin rings and colored asfollows: blue, LH2 B800; green, B850; red, LH1 B875; purple, RC/accessory; orange, RC/special pair. (a) Electronic couplings (see text) between BChls of thereconstructed chromatophore vesicle. For the sake of clarity, only couplings �3 cm�1 are shown on a logarithmic scale. (b) The rate of excitation transfer (Eq.2) between the BChl groups of the LH2 B850 ring (green) and the S-shaped LH1 assembly (red), represented as bonds connecting the respective center of massof each BChl group. For clarity, only strong connections are displayed on a logarithmic scale and the transfers involving other BChl groups, such as LH2 B800 BChlsor the RC BChls, are not shown. (c) Excitation lifetime as a function of the initially excited BChl for the first vesicle (compare with a and b). (d) Excitation lifetimeas a function of the initially excited BChl for the second, LH2-rich, vesicle. The cross-transfer probability between RCs, i.e., the probability that an excitation whichhas just been detrapped from a RC will be trapped at a given RC, is displayed in e and f for the first and second vesicles, respectively, for a detrapping event atthe RC pair shown at the center. The probability is color-coded according to the color bar shown. Notably, excitation sharing between RCs arises mainly betweenadjacent RCs. The distribution of excitation lifetimes (compare with c and d) are shown in g and h for the two vesicles as a function of distance to the nearestRC (filled, B850 BChls; open, B800 BChls). The continuity of the distributions in g and h indicates that all BChl clusters are functionally connected. The distributionof lifetimes is reminiscent of random walks on graphs. [a and b were made with the program VMD (Visual Molecular Dynamics) (90).]

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Excitation Migration over the Intracytoplasmic Membrane Vesicle.Reconstruction of the intracytoplasmic membrane vesicles al-lows a detailed description of the excitation transfer within thesemassive pigment networks, which is achieved in an effectiveHamiltonian formulation (28, 31, 60–68) (see Methods). Thecouplings between neighboring BChls in the B850 ring of LH2and the B875 ‘‘ring’’ of LH1 are strong enough for excitonicdelocalization and subsequent spectroscopic shifts to play aprominent role in the excitation migration process. Indeed, aMarkovian random walk of excitations localized on individualBChls fails to give a satisfactory account of excitation migrationin the pigment network and it is necessary to localize excitationson clusters of BChls by using Eq. 2. Such a description repro-duces observed excitation migration in small segments of thePSU (63–66) and in photosystem I in cyanobacteria and plants(29, 31, 68). An alternative formulation of excitation transferprocesses, the modified Redfield theory is expected to yield amore accurate description, especially of strong coupling, shorttime scale events (69–71).

The BChl clusters in LH2 consist of rings of 9 B800 BChls and18 B850 BChls; RC–LH1 dimers consist of an S-shaped array of 56B875 BChls, and the RCs each have two special pair and twoaccessory BChls. The effective Hamiltonian parameters describingthe site energies and the coupling strengths of these BChl clustersare based on earlier work conducted on smaller models of the PSU(27, 28, 63–66, 72). Specifically, the coupling between the specialpair BChls is assumed to be 500 cm�1 according to refs. 72 and 73.The alternating nearest-neighbor couplings of the LH2 B850 ringare taken to be 363 cm�1 and 320 cm�1 (27, 28), whereas thealternating nearest neighbor couplings of the LH1 B875 ‘‘S-band’’are taken to be 300 cm�1 and 233 cm�1 (72, 74). All other couplingsare computed in the dipolar approximation vij � C(di�dj/r ij

3 �3(rij�dj)(rij�dj)/r ij

5) with a coupling constant C of 348,000Å cm�1 chosen to reproduce LH2 exciton spectra (28, 75). Inter-and intraprotein pigment couplings are computed according toidentical formulae. Couplings between the BChls of the first vesicleare illustrated in Fig. 2a. Site energies of BChls are based on earlierspectroscopic data (28, 66, 72–74, 76).

The excitation transfer times between neighboring BChl clus-ters of B800, B850, and B875 BChls in LH2 and LH1 are similarto those obtained in earlier studies (63–66, 72). Excitation flowsquickly (�1 ps) from the B800 ring to the B850 ring in LH2 andis rapidly transferred to another LH2-B850 ring or to an LH1-B875 S-shaped assembly within �10 ps, depending on thedistance of the neighboring cluster. The cluster–cluster transferrates computed based on Eq. 2 are shown in Fig. 2b.

Notable differences exist between the dimeric arrangement ofthe RC–LH1–PufX core complexes and the circular cores con-sidered earlier (63–66, 72). First, for a circular LH1, mostoscillator strength is accrued in the doubly degenerate secondand third eigenstates as a result of symmetry (72), whereas forthe S-shaped dimer, the degeneracies are broken, even thoughthe third eigenstate still contributes the highest oscillatorstrength. The forward-transfer time (B8753RC) is 20 ps, whichis comparable to 15 ps computed in earlier studies (66, 72). Theserates are largely consistent with the trapping lifetime of 37 ps ofRC–LH1 complexes (77) that includes detrapping and retrap-ping events.

The back-transfer time (RC3 B875) of the S-shaped complexcomputed with the aforementioned Hamiltonian parametersadopted from the circular RC–LH1 complex is too short (1.4 ps)compared with the empirically supported value of 8 ps (66, 72).This back-transfer time is sensitive to the energies and couplingsof RC BChls, which are impossible to determine accuratelywithout a detailed atomic-level structure. Furthermore, theassumptions about Boltzmann equilibrium and delocalizationbecome increasingly more suspect as the size of the BChl clusterincreases and may not be the best suited tool for the description

of LH1 excitons, which are likely to undergo a higher degree ofthermal localization (65). Keeping these points in mind, weheuristically adopt a back-transfer time (RC3B875) of 8 ps, butconsider also the effects of a 1.4-ps back-transfer time. Thedetrapping probability that corresponds to a back-transfer timeof 8 ps is 27%, in agreement with the observed probability of�20% (78–80). Direct transfer events between clusters that donot have any BChls within 50 Å of each other are safelyneglected. The transfer network thus constructed permits com-putation of excitation lifetime and quantum efficiency for eachBChl initially excited in either of the model vesicles (see Fig. 2).

The average excitation lifetime associated with the first vesicleis 50 ps corresponding to a high quantum yield of 95%. Thesecond vesicle, which contains relatively more LH2 complexes,as seen in some AFM images, exhibits a longer lifetime, 162 ps,and a smaller quantum yield, 84%. The variant of the modelswith a 1.4-ps back-transfer rate results in lifetimes of 88 ps and222 ps and quantum yields of 91% and 78%, respectively, forthe two vesicle architectures. The increase in lifetime and thedecrease in quantum yield for this variant arise from theincreased prominence of detrapping events caused by fasterback-transfer from the RC. The continuous distribution oflifetimes and BChl–RC distances shown in Fig. 2 g and hindicates that none of the pigment clusters is functionallydisconnected from the rest, indicating that each vesicle consti-tutes an efficient energy transfer network.

Multiple detrapping and subsequent retrapping events resultin the excitation effectively being shared among neighboringRCs. Excitation sharing cannot stem from direct transfer be-tween RCs, which is observed to be rather slow (230 ps) even fornearest neighbor RCs; instead, it would arise from back-transfer(detrapping) events followed by subsequent migration of exci-tation to a nearby RC. Excitation sharing in a network of RCscan be analyzed by the sojourn expansion (29, 31) that expressesthe excitation transfer process in terms of an expansion ofrepeated detrapping events. In particular, the cross-transferprobability Qjk

X of an excitation that has just been detrapped fromRCk to be trapped at RCj provides a measure of connectivitybetween RCs. These probabilities are shown in Fig. 2 e and f forthe two constructed vesicles. The sum over all RCs, �jQjk

X, isapproximately equal to the quantum yield of the system. Theprobability of the excitation being detrapped from the RC shownat the center of the dimer array to be eventually trapped at thatsame RC is as low as 13% with the excitation being shared withthe neighboring RCs within the same dimer cluster. As a resultof the short transfer times between neighboring LH1 ‘‘rings’’ (aslow as 4 ps), the neighboring linear clusters of dimers share onlya very small portion of the detrapped excitation, as a comparisonbetween the two vesicles shows.

DiscussionThe in silico reconstruction of a chromatophore vesicle pre-sented here elucidates efficient energy transfer and trapping atthe scale of an entire PSU containing thousands of BChls.Experimental data going back �50 years had shown the inter-connected nature of energy transfer in bacterial photosyntheticmembranes, with 20 or more reaction centers interconnected bymany light-harvesting complexes, representing hundreds or eventhousands of BChl molecules (50, 81–84). The model presentedhere attempts to reconcile such data with the atomic-levelstructure of a whole membrane vesicle (chromatophore).

A remarkably short average excitation lifetime of 50 ps iscomputed across the entire vesicle corresponding to a quantumyield of 95%. This finding is in agreement with the observedexcitation lifetime and trapping efficiency in purple bacterialantenna systems, typically of 60 ps and 95%, respectively (re-viewed in ref. 25).

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This high efficiency is achieved by more than simply a highpigment density, namely, by spectral tuning of the variouspigment clusters in relation to one another. A significant degreeof excitation sharing is observed between the RCs of a RC–LH1–PufX dimer array as a result of the close packing of thedimer clusters. This arrangement might have implications formaintaining a high quantum efficiency when one or more RCsin a linear array are in the closed state, the likelihood of whichdepends on the coupling of RC photochemistry to downstreamprocesses such as quinol production and coupling to turnover atthe bc1 complex. It is possible that the organization of LH2 andcore complexes, which is now known to be highly variable (85),depends less on considerations of light harvesting than oncoupling to the bc1 complex and coping with RCs in a closedstate. This could dictate either the necessity for excitationsharing within rows of dimeric RC–LH1–PufX complexes, asseen in R. sphaeroides, or for the more diffuse arrangements ofmonomeric cores, each one surrounded by LH2, as visualized inAFM topographs of membranes from Rhodospirillum photomet-ricum (86).

Reconstruction of the supramolecular organization of cellulardomains is likely to become commonplace in connecting atomic-level structural information to cryo-EM cellular tomographydata (87, 88). The PSU model constructed here ably explainslight-harvesting kinetics and efficiency; it will also assist indescribing subsequent photosynthetic processes, such as quinonediffusion coupling to the bc1 complex, generation of a protongradient, and ATP synthesis. Eventually, it should be possible tointegrate the present level of information into an integral modelfor PSU assembly.

MethodsAn Area-Preserving Transformation Between Planar and SphericalRegions. We used the inverse-Mollweide projection (89) fromgeography to map planar regions to spherical patches. TheMollweide projection from a sphere onto a plane is an area-preserving projection that generates the familiar rendering ofthe globe where circles of latitude are mapped onto parallel lineson the plane. The (inverse) Mollweide projection maps infini-tesimal circles at the center of the transformed region ontocircles without distortion. It is defined by the following mappingfrom a point (x, y) on the plane onto a point (�, �) on the sphere,where � is the latitude and � is the longitude:

� � sin�1� 2� � sin�2��

�� , � �

�x/R

2 �2 cos���. [1]

Here, we define � � sin�1(y/�2R); R is the radius of the targetsphere.

The transformation (Eq. 1) was applied to determine thelocations of the center points and the relative orientations of theproteins involved and not directly to the individual atomiccoordinates. Thus, no artificial structural deformations of theconstituent pigment–protein complexes are introduced. Manysmall planar patches were used to construct a spherical model,which was then manually cleaned of steric clashes and gaps (seeFig. 1 e and f ).

Excitation Transfer Between Pigment Clusters. Excitation migrationwithin the pigment network, because of resonant energy transfer(60, 61), was described through an effective Hamiltonian as

applied earlier to smaller photosynthetic systems (28, 29, 31,63–66, 68). For the sake of completeness, we provide a briefsummary of this approach. The effective Hamiltonian is definedin terms of the site energies �i and the electronic couplings vij,which in the induced dipole-induced dipole approximation thatapplies typically for interpigment distances larger than 10 Å isgiven by vij � C((di�dj)/r ij

3 � 3(rij�di)(rij�dj)/r ij5). Coupling of closer

BChls is best determined via quantum chemistry methods (27,73, 74).

The rates TDA of excitation transfer between clusters ofchlorophylls were computed based on the assumption that thedonor-excited states are populated according to the Boltzmanndistribution leading to the expression (31, 66, 68)

TDA �2�

h �m�D

�n�A

e�EmD/kBT

� l�D e�ElD/kBT

�UmnDA�2 � dESm

D�E�SnA�E� .

[2]

D denotes the donor cluster and A the acceptor cluster; E mD and

E lD are eigenvalues of the Hamiltonians HD and HA correspond-

ing to D and A, respectively. The integral represents the spectraloverlap between donor (S m

D) and acceptor (S nA) emission and

absorption spectra, respectively. The coupling term U mnDA in Eq.

2 between the mth excited state of D and the nth excited stateof A is U mn

DA � ��D �A c m�D c n

A H�, where H� are matrixelements of the effective Hamiltonian that couple donor andacceptor pigments, c m�

D and c nA being the respective eigenvector

coefficients. The excitation lifetime and quantum yield q of thesystem for the initial state �i can be expressed in terms of thetransfer rates (Eq. 2) (29, 31, 66)

� �1M

�1M�K�1� i , q � �1M

kCS�RC�K�1� i , [3]

where M is the number of pigment clusters (M � 256 and M �315 for the first and second vesicle, respectively; B800 and B850BChls are considered separate clusters). The constant kCS � (3ps)�1 is the rate of electron transfer from the excited RC specialpair of BChls. K denotes the rate matrix with elements Kij � Tji� �ij(kCS�i,RC � kdiss � kTik), where kdiss � (1 ns)�1 is thedissipation rate in the system. The indices i, j run over pigmentclusters and not individual pigments.

The sojourn expansion (29, 31) expresses excitation transferprocesses in terms of repeated detrapping events from RCsfollowed by subsequent retrapping. In particular, the cross-transfer conditional probability between two RCs is given by

Q jkX � �kCS�RCj�K�1�Tk, [4]

where �Tk is the state populated immediately after detrappingfrom RCk. The sum jQ jk

X equals the quantum yield correspond-ing to an initial state given by �Tk.

We thank Tihamer Geyer for useful discussions and Olga Svinarski forthe hand-drawn portion of Fig. 1. This work was supported by NationalScience Foundation Grant MCB 0234938 and National Institutes ofHealth Grant PHS 2 P41 RR05969 (to K.S.) and by a HumboldtFoundation award (to K.S.). M.K.S. acknowledges the support of theDepartment of Physiology and Biophysics and Institute for Computa-tional Biomedicine of Weill Medical College, Cornell University. J.D.O.and C.N.H. acknowledge financial support from the Biotechnology andBiological Sciences Research Council (United Kingdom).

1. Blankenship RE (2002) Molecular Mechanisms of Photosynthesis (BlackwellScientific, Malden, MA).

2. Govindjee (2000) in Probing Photosynthesis: Mechanisms, Regulation, andAdaptation, eds Yunus M, Pathre U, Mohanty P (Taylor & Francis, New York),pp 9–39.

3. Emerson R, Arnold A (1932) J Gen Physiol 16:191–205.4. Forster T (1948) Ann Phys (Leipzig) 2:55–75.5. Oppenheimer JR (1941) Phys Rev 60:158.6. Arnold W, Oppenheimer JR (1950) J Gen Physiol 33:423–435.7. Marcus RA (1956) J Chem Phys 24:966–978.

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8. Marcus RA (1956) J Chem Phys 24:979–989.9. Krauss N, Schubert W-D, Klukas O, Fromme P, Witt HT, Saenger W (1996)

Nat Struct Biol 3:965–973.10. Zouni A, Witt H-T, Kern J, Fromme P, Krauss N, Saenger W, Orth P (2001)

Nature 409:739–743.11. Jordan P, Fromme P, Witt HT, Klukas O, Saenger W, Krauss N (2001) Nature

411:909–917.12. Ben-Shem A, Frolow F, Nelson N (2003) Nature 426:630–635.13. Ferreira KN, Iverson TM, Maghlaoui K, Barber J, Iwata S (2004) Science

303:1831–1838.14. Liu Z, Yan H, Wang K, Kuang T, Zhang J, Gui L, An X, Chang W (2004) Nature

428:287–292.15. Deisenhofer J, Epp O, Mikki K, Huber R, Michel H (1985) Nature 318:618–

624.16. Allen J, Yeates T, Komiya H, Rees D (1987) Proc Natl Acad Sci USA

84:6162–6166.17. Ermler U, Fritzsch G, Buchanan SK, Michel H (1994) Structure (London)

2:925–936.18. McDermott G, Prince SM, Freer AA, Hawthornthwaite-Lawless AM, Papiz

MZ, Cogdell RJ, Isaacs NW (1995) Nature 374:517–521.19. Koepke J, Hu X, Muenke C, Schulten K, Michel H (1996) Structure (London)

4:581–597.20. Karrasch S, Bullough P, Ghosh R (1995) EMBO J 14:631–638.21. Walz T, Jamieson SJ, Bowers CM, Bullough PA, Hunter CN (1998) J Mol Biol

282:833–845.22. Conroy MJ, Westerhuis W, Parkes-Loach PS, Loach PA, Hunter CN, Wil-

liamson MP (2000) J Mol Biol 298:83–94.23. Papiz MZ, Prince SM, Howard T, Cogdell RJ, Isaacs NW (2003) J Mol Biol

326:1523–1538.24. Qian P, Hunter CN, Bullough PA (2005) J Biol Chem 349:948–960.25. Sundstrom V, Pullerits T, van Grondelle R (1999) J Phys Chem B 103:2327–

2346.26. Scholes G, Gould I, Cogdell R, Fleming G (1999) J Phys Chem B 103:2543–

2553.27. Tretiak S, Middleton C, Chernyak V, Mukamel S (2000) J Phys Chem B

104:9540–9553.28. Sener M, Schulten K (2002) Phys Rev E 65:031916.29. Sener MK, Lu D, Ritz T, Park S, Fromme P, Schulten K (2002) J Phys Chem

B 106:7948–7960.30. Damjanovic A, Vaswani HM, Fromme P, Fleming GR (2002) J Phys Chem B

106:10251–10262.31. Sener MK, Park S, Lu D, Damjanovic A, Ritz T, Fromme P, Schulten K (2004)

J Chem Phys 120:11183–11195.32. Cogdell RJ, Gall A, Kohler J (2006) Q Rev Biophys 39:227–324.33. Jungas C, Ranck J, Rigaud J, Joliot P, Vermeglio A (1999) EMBO J

18:534–542.34. Jamieson SJ, Wang P, Qian P, Kirkland JY, Conroy MJ, Hunter CN, Bullough

PA (2002) J Mol Biol 21:3927–3935.35. Siebert CA, Qian P, Fotiadis D, Engel A, Hunter CN, Bullough P (2004) EMBO

J 23:690–700.36. Camara-Artigas A, Brune D, Allen JP (2002) Proc Natl Acad Sci USA

99:11055–11060.37. Bahatyrova S, Frese RN, Siebert CA, Olsen JD, van der Werf KO, van

Grondelle R, Niederman RA, Bullough PA, Otto C, Hunter CN (2004) Nature430:1058–1062.

38. Scheuring S, Sturgis JN, Prima V, Bernadac A, Levy D, Rigaud J-L (2004) ProcNatl Acad Sci USA 91:11293–11297.

39. Frese R, Siebert CA, Niederman RA, Hunter CN, Otto C, van Grondelle R(2004) Proc Natl Acad Sci USA 101:17994–17999.

40. Fotiadis D, Qian P, Philippsen A, Bullough PA, Engel A, Hunter CN (2004)J Biol Chem 279:2063–2068.

41. Scheuring S, Levy D, Rigaud J-L (2005) Biochim Biophys Acta 1712:109–127.42. Xiong J, Fischer WM, Inoue K, Nakahara M, Bauer CE (2000) Science

289:1724–1730.43. Visscher KJ, Bergstrom H, Sundstrom V, Hunter C, van Grondelle R (1989)

Photosynth Res 22:211–217.

44. Beekman LMP, van Mourik F, Jones MR, Visser HM, Hunter CN, vanGrondelle R (1994) Biochemistry 33:3143–3147.

45. Frese R, Olsen J, Branvall R, Westerhuis W, Hunter C, van Grondelle R (2000)Proc Natl Acad Sci USA 97:5197–5202.

46. Junge W, Lill H, Engelbrecht S (1997) Trends Biochem Sci 22:420–423.47. Fillingame RH, Jiang W, Dmitriev OY (2000) J Exp Biol 203:9–17.48. Fillingame RH (2000) Nat Struct Biol 7:1002–1004.49. Niederman RA, Mallon DE, Parks LC (1979) Biochim Biophys Acta 552:210–

220.50. Hunter CN, Kramer HJM, van Grondelle R (1985) Biochim Biophys Acta

807:44–51.51. Pugh R, McGlynn P, Jones M, Hunter C (1998) Biochim Biophys Acta

1366:301–316.52. Niederman RA, Mallon DE, Langan JJ (1976) Biochim Biophys Acta 440:429–

447.53. Hunter CN, Tucker JD, Niederman RA (2005) Photochem Photobiol Sci

4:1023–1027.54. Geyer T, Helms V (2006) Biophys J 91:927–937.55. Crofts AR, Meinhardt SW, Jones KR, Snozzi M (1983) Biochim Biophys Acta

723:202–218.56. Velasco F, Crofts AR (1991) Biochem Soc Trans 19:588–593.57. Feniouk BA, Cherepanov DA, Voskoboynikova NE, Mulkidjanian AY, Junge

W (2002) Biophys J 82:1115–1122.58. Parks LC, Niederman RA (1978) Biochim Biophys Acta 511:70–82.59. Aagaard J, Sistrom W (1972) Photochem Photobiol 15:209–225.60. Forster T (1948) Ann Phys (Leipzig) 2:55–75.61. Dexter D (1953) J Chem Phys 21:836–850.62. van Grondelle R, Dekker JP, Gillbro T, Sundstrom V (1994) Biochim Biophys

Acta 1187:1–65.63. Ritz T, Hu X, Damjanovic A, Schulten K (1998) J Luminesc 76–77:310–321.64. Hu X, Damjanovic A, Ritz T, Schulten K (1998) Proc Natl Acad Sci USA

95:5935–5941.65. Damjanovic A, Ritz T, Schulten K (1999) Phys Rev E 59:3293–3311.66. Ritz T, Park S, Schulten K (2001) J Phys Chem B 105:8259–8267.67. Sener M, Schulten K (2005) in Energy Harvesting Materials, ed Andrews DL

(World Scientific, Singapore), pp 1–26.68. Sener MK, Jolley C, Ben-Shem A, Fromme P, Nelson N, Croce R, Schulten K

(2005) Biophys J 89:1630–1642.69. Yang M, Fleming GR (2002) Chem Phys 282:163–180.70. Rutkauskas D, Novoderezhkin VI, Cogdell RJ, van Grondelle R (2005) Biophys

J 88:422–435.71. van Grondelle R, Novoderezhkin VI (2006) Phys Chem Chem Phys 8:793–807.72. Damjanovic A, Ritz T, Schulten K (2000) Int J Quantum Chem 77:139–151.73. Eccles J, Honig B, Schulten K (1988) Biophys J 53:137–144.74. Koolhaas MHC, Frese RN, Fowler GJS, Bibby TS, Georgakopoulou S, van der

Zwan G, Hunter CN, van Grondelle R (1998) Biochemistry 37:4693–4698.75. Knox RS, Spring BQ (2003) Photochem Photobiol 77:497–501.76. Groot M, Yu J, Agarwal R, Norris JR, Fleming GR (1998) J Phys Chem B

102:5923–5931.77. Bergstrom H, van Grondelle R, Sundstrom V (1989) FEBS Lett 250:503–508.78. Timpmann K, Freiberg A, Sundstrom V (1995) Chem Phys 194:275–283.79. Bernhardt K, Trissl H-W (2000) Biochim Biophys Acta 1457:1–17.80. Amesz J, Neerken S (2002) Photosynth Res 73:73–81.81. Duysens LNM (1952) PhD thesis (Univ of Utrecht, Utrecht, The Netherlands).82. Vredenberg W, Duysens LNM (1963) Nature 197:355–357.83. Monger T, Parson W (1977) Biochim Biophys Acta 460:393–407.84. Bakker JGC, van Grondelle R, Den Hollander WTF (1983) Biochim Biophys

Acta 725:508–518.85. Scheuring S (2006) Curr Opin Chem Biol 10:387–393.86. Scheuring S, Rigaud J-L, Sturgis JN (2004) EMBO J 23:4127–4133.87. Baumeister W (2005) Prot Sci 14:257–269.88. Baumeister W (2005) FEBS Lett 579:933–937.89. Snyder JP (1987) Map Projections: A Working Manual (US Government

Printing Office, Washington, DC), US Geological Survey Professional Paper1395.

90. Humphrey W, Dalke A, Schulten K (1996) J Mol Graphics 14:33–38.

15728 � www.pnas.org�cgi�doi�10.1073�pnas.0706861104 Sener et al.

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