Efficient energy transfer in light-harvesting systems, III: The influence of the eighth bacteriochlorophyll on the dynamics and efficiency in FMO
Jeremy Moix, Jianlan Wu, Pengfei Huo, David Coker, Jianshu Cao
aa r X i v : . [ phy s i c s . b i o - ph ] S e p Efficient energy transfer in light-harvesting systems, III: Theinfluence of the eighth bacteriochlorophyll on the dynamics andefficiency in FMO
Jeremy Moix
Department of Chemistry, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139Singapore MIT Alliance for Research and Technology,18 Medical Drive, Singapore 117456 andSchool of Materials Science and Engineering,Nanyang Technological University, Singapore 639798
Jianlan Wu
Department of Physics, Zhejiang University,38 ZheDa Road, Hangzhou, China, 310027 andSingapore MIT Alliance for Research and Technology,18 Medical Drive, Singapore 117456
Pengfei Huo and David Coker
Department of Chemistry, Boston University,590 Commonwealth Avenue, Boston, Massachusetts 02215, USA andDepartment of Physics, University College Dublin, Dublin 4, Ireland
Jianshu Cao ∗ Department of Chemistry, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139 andSingapore MIT Alliance for Research and Technology,18 Medical Drive, Singapore 117456 (Dated: November 13, 2018) bstract The most recent crystal structure of the Fenna-Matthews-Olson (FMO) protein complex indi-cates that each subunit contains an additional eighth chromophore. It has been proposed that thisextra site functions as a link between the chlorosome antenna complex and the remaining sevenchromophores in FMO [Schmidt am Busch et al, J. Phys. Chem. Lett., , 93 (2011)]. Here, weinvestigate the implications of this scenario through numerical calculations with the generalizedBloch-Redfield (GBR) equation and the non-interacting blip approximation (NIBA). Three keyinsights into the population dynamics and energy transfer efficiency in FMO are provided. First,it is shown that the oscillations that are often observed in the population relaxation of the dimercomposed of sites one and two may be completely suppressed in the eight site model. The pres-ence of the coherent oscillations is shown to depend upon the particular initial preparation of thedimer state. Secondly it is demonstrated that while the presence of the eighth chromophore doesnot cause a dramatic change in the energy transfer efficiency, it does however lead to a dominantenergy transfer pathway which can be characterized by an effective three site system arranged inan equally-spaced downhill configuration. Such a configuration leads to an optimal value of thesite energy of the eighth chromophore which is shown to be near to its suggested value. Finallywe confirm that the energy transfer process in the eight site FMO complex remains efficient androbust. The optimal values of the bath parameters are computed and shown to be closer to theexperimentally fitted values than those calculated previously for the seven site system. . INTRODUCTION The Fenna-Matthews-Olson (FMO) protein is one of the simplest and most well-studiedlight harvesting systems. The protein complex exists as a trimer of three identical subunitswhose function is to link the chlorosome antenna complex, where light-harvesting takes place,with the reaction center, where charge separation occurs. FMO is also one of the earliestlight harvesting systems for which a high resolution crystal structure has been obtained. This early crystal structure indicated that each of the three FMO subunits contains sevenBacteriochlorophyll (Bchl) chromophores which serve as the primary energy transfer path-way between the chlorosome and the reaction center.
Recently, however, a more carefulcrystallographic analysis of of FMO has been performed which demonstrates that the in-dividual subunits contain eight Bchls, not seven.
The eighth Bchl resides on the surfaceof the protein complex and it has been suggested that this additional chromophore is oftenlost during sample preparation.From an energy transfer perspective, the presence of an additional chromophore maychallenge current understanding of how exciton transfer occurs in FMO. For example, inmany previous studies on the seven Bchl complex, it is thought that two nearly independentenergy transfer pathways exist.
Sites one and six are approximately equidistant from theantenna complex and both are assumed to be possible locations for accepting the excitationfrom the chlorosome. From there, the energy is subsequently funneled either from site oneto two (pathway 1) or from site six to sites seven, five, and four (pathway 2). The terminalpoint through either route is site three where the exciton is then transferred to the reactioncenter (see Fig. 2(a)). The couplings within each of the pathways are much larger than thecouplings between the two which implies that the two routes are nearly independent.
Inthe second paper of this series, the two pathways and the respective probability of traversingeach have been quantified using a flux analysis. However, this two pathway picture is not entirely consistent with the recent experimentaldata. The new crystal structure indicates that the eighth chromophore resides roughly mid-way between the baseplate and the Bchl at site one.
Additionally, Renger and coworkersargued in Ref. 14 that the eighth Bchl provides the most efficient path for exciton injectioninto FMO as a result of its position and orientation with respect to the chlorosome. If thisis correct and site eight serves as the primary acceptor of excitation energy from the chloro-3ome, then a preferential energy transfer route emerges through pathway 1. Due to the weakinter-pathway couplings, the secondary channel involving the remaining four chromophoresin pathway 2 is largely bypassed in this scheme. This observation may have a significantimpact on the efficiency and robustness of the energy transfer process. The main objectiveof this work is to address this issue by exploring how the dynamics and the energy transferefficiency in FMO are affected by the presence of the eighth site and a realistic environment.The first major conclusion of the present study is related to the population dynamicsin FMO. In many of the the previous studies of the seven site model of FMO, the popula-tion relaxation dynamics are modeled with site one or site six initially populated.
Undereither of these initial conditions, pronounced oscillations in the short-time dynamics areobserved. However when site eight is initially excited, then the oscillations in the popula-tions are completely suppressed. This lack of oscillations has been independently observedin the dynamics recently reported in Refs. 14–16. Here, we provide a simple explanationfor this behavior. The eighth Bchl maintains a large energy gap with the other seven sitesin FMO in order to facilitate efficient directed energy transport. However, it is also ratherweakly coupled to the remaining Bchls. This leads to a slow incoherent decay of the initialpopulation at site eight, and hence a broad distribution of initial conditions at the dimer.The consequence of this result is that the population oscillations generally observed betweensites one and two are completely suppressed which illustrates the importance of the initialconditions on the dynamics of the dimer.The second key result of this work demonstrates that if the eighth Bchl is the primaryacceptor of excitation energy from the chlorosome as recently proposed then a primary energytransfer pathway in FMO does indeed emerge. Note that this situation is substantiallydifferent from the previous interpretations of the energy flow in the seven site models wheretwo independent pathways are generally assumed to exist. The extent of this effect andits impact on the energy transfer efficiency is quantified by introducing reduced models ofFMO that consist of only a subset of the sites in the full system. It is demonstrated thatsites eight, one, two and three which constitute pathway 1 provide the largest contributionto the dynamics of the full system. The remaining four sites of pathway 2 are seen to playa relatively small role. Despite the fact that only a single pathway dominates the energytransfer process, we also show that the presence of the extra Bchl does not significantlyimpact the efficiency or robustness of FMO. The eight site model leads to only a slight4ncrease in the transfer time as compared with the seven site system, and thus maintainsthe same high efficiency as observed in previous studies of FMO.Based upon energetic arguments, it has been suggested that the presence of the eighthBchl leads to optimal energy transfer in FMO. That is, its location near the chlorosomeallows for a large coupling to the antenna complex as well as substantial overlap of theabsorption spectrum of the eighth Bchl with the fluorescence spectrum of the chlorosome.These factors result in efficient transfer of the excitation energy into FMO while simultane-ously allowing the eighth chromophore to maintain a large energy gap with the remainingBchls and hence a favorable energy transport landscape. These features implicitly suggestthat there should be an optimal value of the site energy of the eighth Bchl. Here it is demon-strated that this observation is correct. However, in this case, the behavior is independentof the presence of the chlorosome, and can be understood by considering a further reductionpathway 1 to only three sites. The result of this procedure is a downhill configuration ofthree equally spaced sites (see Fig. 2(c)) which is known to allow for highly efficient energytransfer. Recently, several studies have shown that the environment does not have an entirelydestructive role in the energy transport properties of excitonic systems.
Instead, theenvironment can serve to enhance both the efficiency and robustness of the energy transferprocess. Optimal values have been shown to exist for the temperature as well as otherbath parameters which maximize the energy transfer efficiency in several light harvestingsystems. Moreover, the experimentally fitted model parameters for FMO are near optimalin many cases. An extensive search for the optimal environmental parameters has beenrecently presented in Refs. 11 and 21. In addition to the above findings, we also explorethe effect of the eighth Bchl on the environmentally assisted energy transport propertiesin FMO. It is found that the optimal values of the bath parameters are similar to thosefound in the seven site model, but are closer to the experimental values in general. As hasbeen observed before, the energy transfer efficiency is relatively stable over a broad rangeparameters illustrating the robustness of the network. In the next section, we present the average trapping time formalism which is used inthe remainder of the discussion as a measure of the energy transfer efficiency.
This isfollowed by a brief outline of the generalized Bloch-Redfield (GBR) approach and the modelHamiltonian for the eight site FMO complex used in the numerical calculations. The results5or the population dynamics and the development of the reduced models for FMO are thenpresented in Sec. III. This is followed by calculations of the trapping time as a function ofthe site energies and bath parameters. There it is demonstrated that optimal values exist formany of these parameters, and additionally that the experimentally fitted values for FMOare near-optimal.
II. METHODSA. Average Trapping time
The formalism for calculating the averaging trapping time in light harvesting systemshas been presented in detail previously in Refs. 17 and 11. Here we provide only the salientresults. The total system is characterized by a discrete N -site system Hamiltonian, H s and itsinteraction with the environment, H sb . Each site of the system is coupled to an independentbath of harmonic oscillators with the respective Hamiltonians, H b = P j (cid:0) p j + ω j x j (cid:1) . Thetotal Hamiltonian is then given by, H = N X n ǫ n | n i h n | + N X n = m V nm | n i h m | + N X n | n i h n | " H ( n ) b + X j c ( n ) j x ( n ) j , (1)where c ( n ) j denotes the coupling coefficient of site n to the j -th mode of its associatedbath. The values of the site energies, ǫ n , and couplings constants, V nm , are specified belowin Sec. II D.The time evolution of the reduced density matrix of this system can be convenientlydescribed in the Liouville representation as, ∂ρ ( t ) ∂t = − L tot ρ ( t )= − ( L s + L trap + L decay + L sb ) ρ ( t ) , (2)where L s ρ = i/ ¯ h [ H s , ρ ] describes the coherent evolution under the bare system Hamiltonian H s . In light harvesting systems, the energy flows irreversibly to the reaction center which ismodeled here through the trapping operator [ L trap ] nm,nm = ( k t m + k t n ) /
2, where k t n denotesthe trapping rate at site n . The energy transfer in FMO exhibits almost unit quantum yield.As a result, the decay rate of the excitation at any site to the ground state, k d , is expected6o be much smaller than the trapping rate, k t ≫ k d . This allows for the simplification L decay = 0. B. Generalized Bloch-Redfield Equation
It remains to account for the Liouville operator describing the system-bath coupling L sb in Eq. 2. For a harmonic bath linearly coupled to the system, the time correlation functionof the bath coupling operators is given by the standard relation C ( t ) = 1 π Z ∞ dω J ( ω ) (cid:18) coth (cid:18) ¯ hβω (cid:19) cos( ωt ) − i sin( ωt ) (cid:19) , (3)where β = 1 /k B T and J ( ω ) = π P j c j ω j δ ( ω − ω j ) is the spectral density of the bath. Forsimplicity, we assume the spectral density is the same for each of the independent baths andgiven by the Drude form J ( ω ) = 2 λω c ωω + ω c , (4)where λ is the bath reorganization energy and ω c is the Debye cutoff frequency. For this spe-cial choice, the correlation function may be expanded in terms of the Matsubara frequencies, ν j = πj ¯ hβ , as C ( t ) = λ ¯ hβ + 4 λω c ¯ hβ ∞ X j =1 ω c ω c − ν j − iλω c ! e − ω c t − λω c ¯ hβ ∞ X j =1 ν j ω c − ν j e − ν j t = ∞ X j =0 α j e − ν i t , (5)which defines the complex expansion coefficients α j with the condition ν = ω c .The dynamics in FMO have been computed using a variety of methods ranging in bothaccuracy and cost. Here, we choose the approximate generalized Bloch-Redfield(GBR) method which follows from a second order cumulant expansion in the system-bathinteraction. It provides an accurate, but computationally friendly approach for the propaga-tion of the density matrix over much of the physically relevant parameter space.
Due tothe decomposition of the bath autocorrelation function in Eq. 5, the system-bath interactionmay be accounted for through the introduction of auxiliary fields. The GBR equation ofmotion for the reduced density matrix is then given by ∂ρ ( t ) ∂t = − ( L sys + L trap ) ρ ( t ) − i N X n ∞ X j =0 [ A n , g n,j ( t )] . (6)7he coupling of each Bchl to an independent bath leads to the additional sum over the N sites where the system operator A n = | n i h n | and g n,j denotes the j th-auxiliary field coupledto site n . The auxiliary variables are subject to the initial conditions g n,j (0) = 0 and obeythe equations of motion, ∂g n,j ( t ) ∂t = − ( L sys + L trap + ν j ) g n,j ( t ) − i Re( α j ) [ A n , ρ ( t )] + Im( α j ) [ A n , ρ ( t )] + , (7)where the plus subscript denotes anti-commutation. C. Trapping Time
The mean residence time at site n is by definition h τ n i = Z ∞ dtρ nn ( t ) , (8)where ρ nn denotes the population at site n . The average trapping time is then given simplyas the sum of the residence times at each of the N sites, h t i = P Nn =1 h τ n i . Invoking thesteady state solution of Eq. 2, L tot h t i = ρ (0), then the average trapping time is given by thecompact expression, h t i = Tr (cid:0) L − ρ (0) (cid:1) , (9)where the trace is taken over the site populations of the reduced density matrix. D. Eight Site FMO model
The Hamiltonian for FMO is constructed from the crystal structure recently deposited inthe protein data bank (pdb code: 3eoj). The site energies are taken from those computed inRef. 14 and the coupling element between sites n and m is calculated from the dipole-dipoleapproximation, V nm = C (cid:18) d n · d m | r nm | − d n · r nm ) ( d m · r nm ) | r nm | (cid:19) . (10)Additional details and the explicit system Hamiltonian are given in the Appendix. Asidefrom the eighth site, the most significant difference between the present model Hamiltonianand the model previously derived by Cho et al. is in the energy difference between sites oneand two. In the current case, the energy transfer through pathway 1 is entirely energeticallyfavorable whereas a barrier is present between site one and two in the model of Ref. 26.8nless otherwise stated, the bath is characterized by the experimentally fitted valuesfor the reorganization energy of 35 cm − and Debye frequency ω − c = 50 fs (105 cm − ). Additionally the temperature is 300 K and the trap is located at site three with a trappingrate of k t = 1 ps − . III. NUMERICAL RESULTSA. Population Dynamics and suppression of the oscillations
The time evolution of the populations in the eight site model of FMO calculatedfrom Eq. 2 using the GBR is shown in Fig. 1(a) and (b) for the initial population locatedat site one and site eight, respectively. The bath parameters are taken at their fitted valuesspecified above and the trap at site three is not included. The most striking difference seenbetween Fig. 1(a) and (b) is the absence of the oscillations in the populations of the dimerwhen site eight is initially excited. Other recent studies of the eight site model of FMOhave also observed a similar lack of oscillations.
The origin of this effect may be tracedto the initial conditions at the dimer. The energy difference between site eight and theremaining sites is much larger than any of its respective couplings. This leads to the ratherslow incoherent exponential relaxation of the population of site eight seen in Fig. 1(b). Theresulting initial conditions at sites one and two are then given by a corresponding incoherentdistribution. It is this dephasing that suppresses the oscillations generally observed in thedynamics of the dimer.By applying the non-interacting blip approximation (NIBA) to the spin-boson model,Pachon and Brumer established that a necessary condition for the presence of the oscilla-tions in the dimer is an effective low temperature. The results of Fig. 1 demonstrate thatthe initial conditions impose an additional constraint on the observation of population os-cillations. Note that there are a variety of other initial preparations –such as starting froman eigenstate of the total system or exciting the system with incoherent light – which willalso suppress the oscillations in the dimer.In order to analyze the influence of the initial conditions in more detail, the dynamicsof the dimer calculated using the NIBA are presented in Fig. 1(c) and (d). The popula-tion dynamics described in Ref. 25 may be formulated as an equivalent generalized master9quation ∂P n ( t ) ∂t = Z t dt ′ N X m =1 K nm ( t − t ′ ) P m ( t ′ ) , (11)where P n ( t ) denotes the population of site n at time t . The elements of the time-dependenttransition matrix are constructed in the standard fashion K nm ( t ) = (1 − δ nm ) W mn ( t ) − δ nm X k W nk ( t ) (12)where δ nm is the Kronecker delta function and the individual rate kernels are given by theNIBA W nm ( t ) = 2 V nm e i ( ǫ n − ǫ m − λ ) t − C ( t ) . (13)As defined previously, the coupling between site n and m is denoted by V nm , ǫ n is theenergy of site n , λ is the reorganization energy, and C ( t ) is the bath correlation functiongiven in Eq. 5. The results shown in Fig. 1(c) are calculated from Eq. 11 with the initialpopulation located at site one and are seen to capture the key features of the full GBRdynamics shown in Fig. 1(a). The decay is accounted for by setting the transfer elements K n and K n to zero for all sites n > P ( t ) = exp( − γt ) where γ = 3 ps − . Assumingthat site eight decays only into site one, then the population of the latter is given by 1 − P ( t ),and the corresponding transition rate is W ( t ) = γ exp( − γt ). The influence of site eighton the oscillatory behavior of the dimer can then be captured by convoluting the dynamicsgiven in Fig. 1(c) (denoted by P n ( t )) with this initial condition, ¯ P n ( t ) = Z t dt ′ P n ( t − t ′ ) W ( t ′ ) . (14)The result of this procedure is shown in Fig. 1(d). As is evident, the oscillatory behavior hascompletely disappeared. For large γ , the transition rate becomes a delta function and the10ynamics of Fig. 1(c) are recovered. However, oscillations in the populations of both statescan be observed only if the decay rate out of site eight is increased fivefold. The presenceor absence of the trap simply effects the long time decay of the dimer populations and isirrelevant for the short time oscillatory behavior. These results demonstrate the importanceof the initial preparation of the populations on the oscillations in the dynamics.It should be noted that while the NIBA calculations lead to qualitatively similar popula-tion dynamics as those given by the GBR, neither of the two approaches are exact. In FMOand other light harvesting systems, many of the model parameters are of the same order ofmagnitude. For instance, the couplings, V nm , and energy differences, ǫ n − ǫ m , as well as thereorganization energy, λ , and thermal energy, β − , are all of comparable magnitude. As aresult, methods based upon second-order perturbation theory are, in general, insufficient toquantitatively describe the dynamics. A systematic procedure for computing higher-ordercontributions to the NIBA rates has been derived and recently implemented. This leadsto non-negligible corrections to the dynamics in the spin-boson model, FMO and LH2. Thus while the results in Fig. 1 and those of Ref. 25 capture the qualitative features thatare necessary to analyze the energy transfer behavior, there will be quantitative correctionsfrom higher order terms.
B. Pathway analysis and the ladder configuration
Regardless of the presence or absence of oscillations, it is readily seenfrom Fig. 1(a) and (b) that the population primarily flows through pathway 1. Amongthe sites in pathway 2, only site four ever accumulates more than ten percent of the popula-tion. Particularly for times less than 500 fs, the sites from pathway 2 are scarcely populated.Similar behavior of the population dynamics has been seen in many other simulations of theseven site model for FMO.
These observations lead to the first reduced model forFMO which consists of only the four sites in pathway one shown in Fig. 2(b). As demon-strated below, this model is able to accurately capture the key features of the energy transferprocess. One may proceed further by noting that sites one and two are coupled more stronglythan the energy difference between the two. Additionally, sites eight and three are widelyseparated from either site in the dimer. The couplings between the distant sites and dimerare also rather weak (see values in the model Hamiltonian in Eq. 15 and Fig. 2(b)). As11 result, there can be rapid coherent energy transfer between sites one and two, whereasthe transfer to sites eight or three will be comparatively slow. Therefore, when the initialpopulation is located at site eight we may simply assume that these two sites of the dimerbehave as one effective site with an energy that is the average of the two (270 cm − ). Asimilar “mean state” idea for developing this type of reduced model has been suggested inRef. 34 which explores the behavior of a dimer embedded in the PC645 photosynthetic net-work. Furthermore, the coupling between site eight and the terminal site, three, is negligiblysmall. This leads to a three site model for FMO where the couplings are determined by thenearest-neighbor values as shown in Fig. 2(c). C. Site Energy of Bchl 8
Fig. 3 displays the average trapping time calculated as a function of the site energy of Bchleight. It contains two additional key findings of this work. The first is that the trapping timebehavior, and hence the efficiency, seen in the eight site model of FMO is largely governedby the sites in pathway 1. The second feature is that an optimal value exists for the siteenergy of Bchl 8, and moreover, the optimum is near the fitted value determined in Ref. 14.The source of the latter is the highly efficient ladder configuration shown in Fig. 2(c).The main portion of Fig. 3 displays the average trapping time calculated with the fullHamiltonian given in Eq. 15 along with the corresponding results for the four site modelof FMO (see Fig. 2(a) and (b)). The bath parameters are taken at their experimentallyfitted values with the temperature of 300 K. The inset of Fig. 3 contains the results fromthe three sites model shown in Fig. 2(c). For this case, the exact trapping time calculatedfrom the hierarchical equation of motion method is also presented, as well as the results ofa F¨orster theory calculation. As can be seen, both the GBR and the F¨orster calculationspredict optimal values of the site energy that semi-quantitatively capture the behavior ofthe exact hierarchical results.These results demonstrate that the energy transport is dominated by a subset of the sitesin FMO and furthermore that the mechanism is correctly described by F¨orster theory. Thefour site model correctly describes the qualitative features seen in the full system and itaccounts for the majority of the trapping time. Of the remaining sites in Fig. 1, site fourwas seen to have the largest impact on the population relaxation. Calculations that consist12f pathway 1 plus site four are seen to capture almost all of the behavior seen in the fulleight site system.In Ref. 14, it was noted that the site energy of Bchl eight maximizes the overlap withthe chlorosome emission spectrum while simultaneously maintaining a large energy gap withthe remaining seven core Bchls. This indicates that there should be an optimal value of thesite energy of Bchl eight. In addition to the observation that the trapping time behaviormay be captured by a simplified model of FMO, there is another interesting feature seenin Fig. 3. The trapping time displays a minimum as a function of the energy of the eighthsite for all of the constructed models. Moreover, for the eight site model, the optimal valueof the site energy is rather close to the fitted value of 505 cm − . Increasing the energydifference between site eight and site one decreases the back-transfer rate, but also decreasesthe spectral overlap between the two. The position of the optimal value is no coincidence.The three-site model in Fig. 2(c) readily demonstrates that this particular choice for the siteenergy leads to a downhill configuration of three sites that are approximately equally-spacedand equally-coupled which is known to be very efficient. The qualitative behavior of thetrapping time in the full, complicated eight site system becomes obvious with the aid of thereduced models. Note also that the average trapping time varies little over a large range ofvalues of the energy of site eight indicating the robustness of the energy transfer process.Below it is demonstrated that many of the other fitted parameters are near optimal as well.
D. Optimal Bath Parameters
The average trapping time calculated as a function of the reorganization energy is shownin Fig. 4(a) and (b). Fig. 4(a) varies the reorganization of all eight sites simultaneously(in this model all chromophores are assumed to have identical environments) whereas (b)varies only that of site eight while keeping all of the others at the fitted value of 35 cm − .In order to demonstrate that the presence of the additional chromophore does not lead to adramatic increase in the trapping time, two different initial conditions are taken with eithersite one or site eight initially excited. As expected, a slightly larger trapping time is observedwith the initial population at site eight due to the larger distance to the trap. However,the difference between the two scenarios is not substantial. Additionally, the qualitativebehavior of the two initial conditions is quite similar and both lead to an optimal value of13he reorganization energy that is close to the experimentally fitted value of 35 cm − . It hasbeen proposed from recent numerical simulations that the reorganization energy in FMOshould be approximately twice as large as the experimentally fitted value used here. Theoptimal values of λ in Fig. 4(a) (55 cm − ) and (b) (40 cm − ) are consistent with a somewhatlarger value of the reorganization energy. The mean trapping time is more sensitive tovariations of the reorganization energy than was observed for the site energy in Fig. 3, butthere is still a large range of λ where the trapping time is near optimal.Finally, Fig. 4(c) and (d) display the results for the average trapping time calculated asa function of the Debye frequency and as a function of the temperature. As in Fig. 4(a) twoinitial conditions are shown with the initial population at either site one or site eight. Again,the average trapping time from site one is always faster than for site eight. Nevertheless,the two initial conditions display qualitatively similar behavior for both the temperatureand Debye frequency. Additionally an optimal value of the Debye frequency is observedthat is close to the experimentally fitted value of 105 cm − . These results for the averagetrapping time as a function of the bath parameters are similar to those observed previouslyfor the seven site model of FMO. There is one notable difference however. In all cases,the Hamiltonian for the eight site FMO model recently proposed by Renger et al leads tooptimal conditions that are closer to the experimentally fitted values than those calculatedpreviously using the Hamiltonian of Ref. 26. IV. CONCLUSIONS
The FMO protein serves as one of the model light harvesting systems, and the qualitativefeatures of the energy transfer process have been understood for some time. However, recentexperimental evidence has shown that many of the previously developed theoretical modelsare not entirely complete. An additional chromophore is present in each subunit of FMOthat resides between the chlorosome and site one. In this work we have shown that thepresence of the eighth site does not significantly alter the previous conclusions that havebeen reached with regards to environmentally-assisted exciton transport. Optimal valuesexist for many of the bath parameters and, moreover, the optimal conditions are generallycloser to their respective experimentally fitted values than in the seven site FMO models.Additionally, the dependence of the average trapping time with respect to variations of the14ath parameters is rather weak illustrating the overall robustness of the energy transferprocess.However, the presence of the eight Bchl may necessitate a reassessment of our under-standing of the energy transport process in FMO. Given that site eight is the primary entrypoint for the exciton into FMO, then Fig. 1 clearly exhibits a complete suppression of thepopulation oscillations that are generally observed in the seven site models of FMO. Thatis, the coherent population oscillations observed in previous studies depend upon the spe-cial choice of initial conditions. Here we have shown that the origin of the suppression isthe slow decay of the initial population at site eight which leads to an incoherent distribu-tion of initial conditions at the dimer. In the physical setting there will be an additionalsource of dephasing due to the extra step from the chlorosome to FMO. This will broadenthe distribution of initial conditions even more and further suppress the oscillations in thedimer.An additional feature of the eight site model that is markedly different from the previousseven site configuration is observed in the population dynamics shown in Fig. 1 and theaverage trapping times displayed in Fig. 3. These results demonstrate that the energy flowin the eight site model is dominated by a subset of the chromophores, whereas it has beenpreviously assumed that two independent pathways involving all of the Bchls are availablefor the energy transfer process. The qualitative features of the transport in the eight sitemodel are largely determined by the dynamics of pathway 1. Sites four, five, six and sevenprovide a rather small contribution to the overall efficiency in this case. The agreementbetween the results for the full eight site system and the reduced four- and three-site modelsshown in Fig. 2 provide further support to this claim. Nevertheless, the eight site model andthe seven site model display similar energy transport efficiencies. The origin of this behaviorin the former is evident from Fig. 2(c) which shows that the eighth Bchl forms an optimaldownhill ladder configuration with the dimer and site three. This result demonstrate theusefulness of the reduced models in providing an intuitive explanation of many of the keyfeatures present in the numerical results. 15 . ACKNOWLEDGMENT
This work was supported by grants from the National Science Foundation, DARPA, theCenter for Excitonics at MIT, the MIT energy initiative (MITEI), and the Singapore-MITalliance for research and technology (SMART)
VI. APPENDIX
Following the prescriptions used previously for constructing the dipole-dipoleinteractions, the unit vectors, d n , in Eq. 10 point along the axis connecting the N b and N d atoms of the n -th Bchl and r nm is the vector connecting the Mg atoms of Bchl n and m . Setting the constant C = 155000 cm − ˚A leads to an effective dipole strength of 30D . With these specifications, the system Hamiltonian (in cm − ) for the eight site model is H FMO = . − . . − . . − . − . . − . . . . . . . . . . . − . − . − . . . − . . − . . − . − . − . − . . . − . − . . . − . . − . . − . − . . . . − . − . . . − . − . . . − . . . . − . . − . − . . , (15)where the zero of energy is 12195 cm − . Note that there is an error in the sign of the couplingbetween sites one and two in the table provided in Ref. 14. Aside from this, these valuesreproduce all of the couplings listed therein to within 3 cm − . ∗ Electronic address: [email protected] R. E. Fenna and B. W. Matthews, Nature , 573 (1975). T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship, and G. R. Fleming, Nature , 625 (2005). G. S. Engel, T. R. Calhoun, E. L. Read, T. Ahn, T. Mancal, Y. Cheng, R. E. Blankenship, andG. R. Fleming, Nature , 782 (2007). G. Panitchayangkoon, D. Hayes, K. A. Fransted, J. R. Caram, E. Harel, J. Wen, R. E. Blanken-ship, and G. S. Engel, Proc. Natl. Acad. Sci. USA , 12766 (2010). E. Collini, C. Y. Wong, K. E. Wilk, P. M. G. Curmi, P. Brumer, and G. D. Scholes, Nature , 644 (2010). A. Ben-Shem, F. Frolow, and N. Nelson, FEBS Letters , 274 (2004). D. E. Tronrud, J. Wen, L. Gay, and R. E. Blankenship, Photosynth. Res. , 79 (2009). A. Ishizaki and G. R. Fleming, J. Chem. Phys. , 234111 (2009). P. Huo and D. F. Coker, J. Chem. Phys. , 184108 (2010). G. Tao and W. H. Miller, J. Phys. Chem. Lett. , 891 (2010). J. Wu, F. Liu, Y. Shen, J. Cao, and R. J. Silbey, New J. Phys. , 105012 (2010). J. Wu, F. Liu, J. Ma, X. Wang, R. Silbey, and J. Cao (2011), submitted. A. Ishizaki and G. R. Fleming, Proc. Natl. Acad. Sci. USA , 17255 (2009). M. Schmidt am Busch, F. M¨uh, M. E. Madjet, and T. Renger, J. Phys. Chem. Lett. , 93(2011). N. Renaud, M. A. Ratner, and V. Mujica, J. Chem. Phys. , 075102 (2011). G. Ritschel, J. Roden, W. T. Strunz, A. Aspuru-Guzik, and A. Eisfeld, arXiv:1108.3452 (2011). J. Cao and R. J. Silbey, J. Phys. Chem. A , 13825 (2009). M. Mohseni, P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik, J. Chem. Phys. , 174106 (2008). M. B. Plenio and S. F. Huelga, New J. Phys. , 113019 (2008). A. W. Chin, A. Datta, F. Caruso, S. F. Huelga, and M. B. Plenio, New J. Phys. , 065002(2010). M. Mohseni, A. Shabani, S. Lloyd, and H. Rabitz, arXiv:1104.4812 (2011). J. Cao, J. Chem. Phys. , 3204 (1997). U. Weiss,
Quantum Dissipative Systems (World Scientific, Singapore, 1999), 2nd ed. S. Mukamel,
Principles of Nonlinear Optical Spectroscopy (Oxford University Press, USA, 1999). L. A. Pachon and P. Brumer, arXiv:1107.0322 (2011). M. Cho, H. M. Vaswani, T. Brixner, J. Stenger, and G. R. Fleming, J. Phys. Chem. B ,10542 (2005). P. Brumer and M. Shapiro, arXiv:1109.0026 (2011). R. Silbey and R. A. Harris, J. Chem. Phys. , 2615 (1984). C. Aslangul, N. Pottier, and D. Saint-James, J. Phys. (Paris) , 5 (1986). J. Cao, J. Chem. Phys. , 6719 (2000). R. Zwanzig,
Nonequilibrium statistical mechanics (Oxford University Press, London, 2001). R. Kubo, M. Toda, and N. Hashitsume,
Statistical Physics II: Nonequilibrium Statistical Me-chanics (Springer-Verlag, Berlin, 1985), 2nd ed. J. Wu, R. J. Silbey, and J. Cao (2012), in preparation. P. Huo and D. F. Coker, J. Phys. Chem. Lett. , 825 (2011). C. Olbrich, J. Str¨umpfer, K. Schulten, and U. Kleinekath¨ofer, J. Phys. Chem. Lett. , 1771(2011). X. Hu, T. Ritz, A. Damjanovi´c, and K. Schulten, J. Phys. Chem. B , 3854 (1997). F. M¨uh, M. E. Madjet, J. Adolphs, A. Abdurahman, B. Rabenstein, H. Ishikita, E. Knapp, andT. Renger, Proc. Natl. Acad. Sci. USA , 16862 (2007). igures Time (ps) P o pu l a t i o n Site 1Site 2Site 3Site 4Site 8 (a)(b) (c)(d)
FIG. 1: Site populations of in the eight site model of FMO with Bchl one (a) or eight (b) initiallyexcited calculated with the GBR. The populations of the remaining sites five, six and seven arenever larger than 10% and not shown. The site populations of the dimer calculated using theNIBA of Eq. 11 calculated with site one initially excited (c) and from Eq. 14 (d). In all cases, thetemperature is 300 K with a reorganization energy of 35 cm − and cutoff frequency of ω − c = 50fs. E ( c m - ) (a) (c)(b) V = 30V = 38 ε - ε = 235 ε - ε = 270V = 98 ε - ε = 80V = 30V = 38 FIG. 2: Energy diagrams for the eight site model (a), the four site model (b), and three site model(c) used in the calculations of Fig. 3.
400 450 500 550 600 E (cm - ) 〈 t 〉 ( p s )
300 400 500 600E (cm -1 ) 〈 t 〉 ( p s ) IG. 3: The trapping time as a function of the site energy of site eight. The solid (black) lineand dashed (blue) line in the main figure correspond to the results calculated from the full eightsite Hamiltonian of Eq. 15 and with only the four sites of pathway 1, respectively. The red dotscorrespond to the optimal site energies and the vertical dashed line indicates the fitted value ofthe site energy of Bchl 8 of 505 cm − determined in Ref. 14. The inset contains results for thethree site model calculated with the GBR (solid black line), F¨orster theory (dotted red line), andhierarchical equation of motion (dashed blue line). The remaining parameters are the same asin Fig. 1. λ (cm - ¹)345 〈 t 〉 ( p s ) ω c (cm - ¹)345 〈 t 〉 ( p s )
100 200 300 400 500 T (K) 246 〈 t 〉 ( p s ) λ (cm - ¹) 3.944.14.2 〈 t 〉 ( p s ) (a) (b)(c) (d) IG. 4: The trapping time as a function of the reorganization energy of all Bchls (a), and as afunction of the reorganization of site eight only (b) while the remaining seven sites are fixed at theexperimentally fitted value of λ = 35 cm − . The trapping time as a function of the bath cutofffrequency and temperature are shown in figures (c) and (d) respectively. In all cases the trap rateat site three is 1 ps. The solid (black) and dashed (blue) lines correspond to initial excitation atsite eight or site one, respectively. The red dots indicate the optimal trapping times and the dottedvertical lines correspond to the respective experimentally fitted values. The remaining parametersare the same as in Fig. 1.. The trapping time as a function of the bath cutofffrequency and temperature are shown in figures (c) and (d) respectively. In all cases the trap rateat site three is 1 ps. The solid (black) and dashed (blue) lines correspond to initial excitation atsite eight or site one, respectively. The red dots indicate the optimal trapping times and the dottedvertical lines correspond to the respective experimentally fitted values. The remaining parametersare the same as in Fig. 1.