J. Phys. Chem. B 2007, 111, 6807-6814
6807
Multichromophoric Fo1 rster Resonance Energy Transfer from B800 to B850 in the Light Harvesting Complex 2: Evidence for Subtle Energetic Optimization by Purple Bacteria† Seogjoo Jang* Department of Chemistry and Biochemistry, Queens College of the City UniVersity of New York, 65-30 Kissena BouleVard, Flushing, New York 11367-1597
Marshall D. Newton Chemistry Department, BrookhaVen National Laboratory, Upton, New York 11973-5000
Robert J. Silbey Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 ReceiVed: January 6, 2007; In Final Form: March 1, 2007
This work provides a detailed account of the application of our multichromophoric Fo¨rster resonance energy transfer (MC-FRET) theory (Phys. ReV. Lett. 2004, 92, 218301) for the calculation of the energy transfer rate from the B800 unit to the B850 unit in the light harvesting complex 2 (LH2) of purple bacteria. The model Hamiltonian consists of the B800 unit represented by a single bacteriochlorophyll (BChl), the B850 unit represented by its entire set of BChls, the electronic coupling between the two units, and the bath terms representing all environmental degrees of freedom. The model parameters are determined, independent of the rate calculation, from the literature data and by a fitting to an ensemble line shape. Comparing our theoretical rate and a low-temperature experimental rate, we estimate the magnitude of the BChl-Qy transition dipole to be in the range of 6.5-7.5 D, assuming that the optical dielectric constant of the medium is in the range of 1.5-2. We examine how the bias of the aVerage excitation energy of the B800-BChl relative to that of the B850-BChl affects the energy transfer time by calculating the transfer rates based on both our MC-FRET theory and the original FRET theory, varying the value of the bias. Within our model, we find that the value of bias 260 cm-1, which we determine from the fitting to an ensemble line shape, is very close to the value at which the ratio between MC-FRET and FRET rates is a maximum. This provides evidence that the bacterial system utilizes the quantum mechanical coherence among the multiple chromophores within the B850 in a constructive way so as to achieve efficient energy transfer from B800 to B850.
I. Introduction The existence of life on earth relies on the conversion of light energy into chemical energy by bacteria and plants. These natural organisms execute the conversion with extremely high efficiency that has not been mimicked by any manmade system. The fact that the photosynthetic unit (PSU) of a bacterium or a plant consists of aggregates of chromophores had been known for a long time. However, it was only after pioneering X-ray crystallography experiments1-8 that the amazing degree of organization of those aggregates was truly recognized. Many spectroscopic9-26 and theoretical studies27-47 following the structural determination uncovered substantial information on the structure-function relationship and the mechanistic details of energy and charge transfer in the natural systems. However, even with efforts of many research groups, it is not yet clear how the specific structure and energetics adopted by the natural systems are tied to their high-energy conversion efficiency.49 In the present article, we report a theoretical result providing new insights into this issue for the case of the light harvesting complex 2 (LH2) of purple bacteria. LH2 consists of two structural units called B800 and B850,9 which were named after the positions of their absorption peak †
Part of the special issue “Norman Sutin Festschrift”. * E-mail:
[email protected].
maxima at room temperature. The B800 unit consists of 8 or 9 bacteriochlorophylls (BChls) and the B850 unit 16 or 18 BChls. Time-resolved spectroscopy and hole burning experiments14,22,50 showed that the photons absorbed by B800 are transferred to B850 in less than 2 ps. Soon, it was recognized that this time is much smaller (about a factor of 5) than the estimate based on the well-known Fo¨rster resonance energy transfer (FRET) theory.51-53 The possibility that carotenoids may be responsible for such fast energy transfer was examined, but its effect was found to be insignificant.9 On the basis of an approximate version of Sumi’s theory,54 Mukai, Abe, and Sumi (MAS)28 were the first to show that the multichromophoric (MC) effect can explain the discrepancy of the experimental rate from the prediction of FRET, but the phenomenological nature of their specification of the line shape of the acceptor made their conclusion lack a solid quantitative basis. Later, Scholes and Fleming (SF)30 presented a similar theoretical study using the same rate expression but employing line shape information determined from a photon echo experiment. They found that their best theoretical estimate was smaller than the experimental rate by about a factor of 2, which left the possibility that other effects need to be considered as well. Linnanto and KorppiTommola (LK)39 also provided calculations applying the same rate expression but employing a somewhat different set of parameters and assuming Lorentzian lineshapes for the exciton
10.1021/jp070111l CCC: $37.00 © 2007 American Chemical Society Published on Web 04/17/2007
6808 J. Phys. Chem. B, Vol. 111, No. 24, 2007
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states. They calculated the distribution of rates and provided theoretical results showing reasonable agreement with experimental data, but their conclusion suffers from an ambiguity similar to that of MAS,28 originating from the phenomenological nature of the line shape expressions used. Recently, we developed a theory generalizing the FRET for MC systems.34 Careful examination of Sumi’s paper54 showed that his general rate expression, which had not yet been used, is equivalent to the steady-state limit of our MC-FRET rate expression.34 We applied this steady-state rate expression to the energy transfer from B800 to B850, employing a full excitonbath model for B850 while representing B800 by a single BChlbath system. Our preliminary results showed that the MC enhancement is about a factor of 5, consistent with the experimental data. This article provides a more extensive account of our previous calculation and reports new calculations giving new insights into the physical implication of the spectral positions of the B800 and B850 bands. Our new results suggest that the relative spectral positions of these two bands are optimized so as to make the energy transfer from B800 to B850 as fast and irreversible as possible and that the MC effect of the B850 unit has a crucial role in achieving such optimization. In conjunction with a recent study by Cheng and Silbey44 showing that the MC effect of the B800 unit also has the positive contribution of making the distribution of rates less dispersive, our results demonstrate the active role played by the MC effects in the energy transfer within the LH2. The sections are organized as follows. Section II provides the model Hamiltonian, the detailed rate expression, and the model for the disorder. Section III provides the results of the calculations. Section IV concludes the paper by discussing the implications of the present results and open issues to be tackled in the future.
where |n〉 is the state where the n-th BChl of B850 is excited (Qy transition), whereas all other BChls are in the ground state, and En is the corresponding energy. We here use the convention that odd n represents an R-BChl and even n a β-BChl. There is disorder in En, and we assume that the disorder for each En has the same average and standard deviation.55 Further details of the disorder will be provided below. ∆(n - m) is the electronic coupling between states |n〉 and |m〉, for which we use the same set of values used in our previous study,33 assuming no disorder in these terms. H1A represents the coupling of the single exciton states with the bath and is assumed to be diagonal in the local excitation basis as follows:
II. Model and Theory
It is assumed that each BChl in B850 has the same spectral density. That is, the spectral density of the n-th BChl,
In this section, we present the model Hamiltonian, line shape expressions, and the MC-FRET rate expression used for the calculation. We assume that the LH2 has 9-fold symmetry, which is the case for Rhodopseudomonas (Rps.) acidophila and Rhodobacter (Rb.) sphaeroides. The modeling of the line shapes and the calculation of the rates are all performed at a lowtemperature limit of kBT ) 10 cm-1. A. Hamiltonian. The total Hamiltonian H consists of four terms as follows:
H ) Eg|g〉〈g| + HA + HD + Hc
(1)
where |g〉 is the ground electronic state where none of the BChls in B850 and B800 is excited and Eg is the corresponding energy. HA is the acceptor Hamiltonian and represents the entire B850 unit. HD is the donor Hamiltonian, which we model in terms of the basic symmetry element of the B800 unit containing a single BChl. Hc represents the electronic coupling between the donor and the acceptor. A detailed account of these terms is provided below. 1. Acceptor Hamiltonian, HA. The acceptor Hamiltonian consists of three terms as follows:
HA ) H0A + H1A + Hb,A
(2)
H0A represents the single exciton states in B850 and is given by 18
H0A )
∑En|n〉〈n| + n*m ∑ ∆(n - m)|n〉〈m| n)1
(3)
18
H1A )
∑ ∑ pωk gk,n(b†k + bk)|n〉〈n| n)1 k∈B
(4)
A
where ωk is the frequency of the k-th bath harmonic oscillator in the set of the acceptor bath BA, gk,n is the strength of its coupling to |n〉, and b†k and bk are corresponding raising and lowering operators. According to some experimental evidence,24-26 it may be necessary to include off-diagonal (in the local excitation basis) exciton-bath couplings as well. In the context of single molecule spectroscopy, two of us33 found that such off-diagonal couplings indeed cause nontrivial changes in the line shapes. However, whether they cause systematic changes in the rate distributions is not clear at this point, and more extensive study will be performed to address this issue. The remaining term in eq 2, Hb,A, is given by
Hb,A )
A
Jn(ω) ≡
(
)
1
∑ pωk b†kbk + 2 k∈B
∑ δ(ω - ωk)ωk2 gk,n2
(5)
(6)
k∈BA
is independent of n. Hereafter, we drop the subscript n in the spectral density. Renger and Marcus56 determined a quite reliable spectral density for the related B777 complex from a fluorescence line narrowing experiment. We employ the same spectral density but approximated with the following form:
ω2 ω3 J(ω) ) 0.22ωe-ω/ωc1 + 0.78 e-ω/ωc2 + 0.31 2e-ω/ωc3 ωc2 ωc3 (7) where ωc1 ) 170, ωc2 ) 34, and ωc3 ) 69 cm-1. The above form fits the spectral density of Renger and Marcus56 better than the simpler Ohmic form used in our previous study33 and still makes it possible to use a simple interpolation formula57 in calculating the bath correlation function. 2. Donor Hamiltonian, HD. The donor is defined as the basic symmetry element of the B800 unit, which contains a single BChl and its bath. This choice is based on the fact that the electronic couplings between BChls in B800 are much smaller than those in B850 and that the MC effect of the former makes a relatively minor contribution compared to that of the latter. This approximation is also supported by a recent study by Cheng and Silbey,44 which showed that the MC effect of the B800 does not affect the average rate significantly, although it makes the distribution of rates narrower.
FRET from B800 to B850 in LH2
J. Phys. Chem. B, Vol. 111, No. 24, 2007 6809
The donor Hamiltonian consists of three terms as follows:
HD ) H0D + H1D + Hb,D
(8)
for the |g〉 f |n〉 transition (the excitation of n-th BChl in B850). All the transition dipole vectors are assumed to have the same magnitude. We thus define the following magnitude of the transition dipole vector
The first term H0D represents the excited electronic state of the donor and is given by
H0D ) ED|D〉〈D|
(9)
where |D〉 is the state where the BChl representing the B800 is excited and ED is the corresponding energy. The second term in eq 8, H1D, represents the coupling of the B800-BChl to its bath and is given by
H1D )
∑ pωj gj(b†j + bj)|D〉〈D|
(10)
j∈BD
where ωj is the frequency of the j-th oscillator in the set of the donor bath BD, gj is the magnitude of its coupling to |D〉, and b†j and bj are the raising and lowering operators of the j-th oscillator. Finally, the last term Hb,D in eq 8 represents the donor harmonic oscillator bath and has the following form:
Hb,D )
(
D
)
1
pωj b†j bj + ∑ 2 j∈B
(11)
The environment around B800 is known to be different from that of B850. Therefore, the spectral density of the former is expected to be different from that of the latter. We assume that this can be accounted for by a different system-bath coupling strength and use the following approximation for the spectral density of the donor bath:
JD(ω) ≡
δ(ω - ωj)ω2j g2j ) ηJ(ω) ∑ j∈B
µ ) |µD| ) |µn|
and unit transition dipole vectors µˆ D ) µD/µ and µˆ n ) µn/µ. Other quantities in eq 14 are defined as follows. is the optical dielectric constant of the medium, Rn is the distance between the donor and the n-th acceptor, and R ˆ n is the corresponding unit distance vector. B. Line Shape Expressions. We here introduce the line shape expressions of the B850 unit (acceptor) and the B800 unit (donor). 1. Line Shape Expression for the Acceptor. The eigenstates and the line shape expression for the B850 unit have been described in detail in our previous study,33 which are summarized below with slightly different notation. Because the symmetry element of the B850 unit contains two BChls (one R and one β), the eigenstates of H0A consist of two bands,31 denoted upper and lower. Each band has nine electronic states. In the absence of disorder, the states in each band can be labeled according to their integer multiples, ranging from 0 to 8, of a wavenumber. The disorder existing in actual LH2 mixes these states, making the integer no longer a well-defined quantum number. However, the moderate magnitudes of the disorder still validate keeping this labeling convention.31 Thus, we denote the eigenstates of H0A as |ψl,p〉 and |ψu,p〉, where p ) 0, ..., 8 and l (u) represents the lower (upper) band. Then, H0A can be expressed as 8
H0A
)
(12)
D
where η is a dimensionless numerical factor and is smaller than unity because the bath coupling of the B800-BChl is weaker than that of the B850-BChl. On the basis of a series of tests to fit the experimental ensemble line shape, which will be described in more detail below, we find that η ) 0.7 (with an error range of ( 0.05) is an optimum value. Although this value is comparable to the values of the B800-BChl Debye-Waller factor recently found from single molecule spectroscopy,19 we expect our choice of η is a somewhat overestimated value due to our approximation to model the B800 in terms of a single BChl and its bath. 3. Coupling Hamiltonian, Hc. The electronic coupling Hamiltonian Hc in eq 1 represents the interaction between the excitation of the donor and the excitation of the acceptor and is given by
∑{El,p|ψl,p〉〈ψl,p| + Eu,p|ψu,p〉〈ψu,p|}
∑Jn(|D〉〈n| + |n〉〈D|)
(13)
n)1
where Jn represents the transition dipole interaction between |D〉 and |n〉 and has the following form:
Jn )
ˆ n)(µn‚R ˆ n) µD‚µn - 3(µD‚R Rn3
(14)
where, for p < p′, El,p < El,p′ and Eu,p > Eu,p′. Let us introduce a transformation matrix C relating the local excitation states to the above eigenstates as follows: 8
|n〉 )
n n {C l,p |ψl,p〉 + C u,p |ψu,p〉} ∑ p)0
(17)
n where C l,p ) 〈ψl,p|n〉 and C nu,p ) 〈ψu,p|n〉. Given the param0 eters of HA for a specific B850 unit, the eigenvalues and eigenvectors in eq 16 and the transformation matrix in eq 17 can be determined simultaneously through numerical matrix diagonalization of H0A. Within the approximation of the second-order Quantum Master Equation (QME),32 the ideal absorption line shape (IAL) of a single B850 unitsthe line shape of a single B850 unit in the ideal situation without quasistatic disorder32,33sis given by
IA(ω) ≈ -
1 3π
Im
∑ TrA eˆ )xˆ ,yˆ ,zˆ
{
eˆ ‚|µ〉〈µ|‚eˆ
}
ω + (g - H0A)/p + iK ˆ (ω) (18)
where eˆ is the unit polarization vector of the radiation, |µ〉 ) 18 ∑n)1 µˆ n|n〉, “Im” implies the imaginary part of a complex number, TrA implies the trace over the basis of H0A, and 18
In this expression, µD is the transition dipole for the |g〉 f |D〉 transition (the excitation of the BChl in B800) and µn is that
(16)
p)0
18
Hc )
(15)
K ˆ (ω) )
8
u
n 2 | |n〉〈n| ∑ ∑ ∑ κˆ (ω - Eb,p/p)|Cb,p n)1 p)0 b)l
(19)
6810 J. Phys. Chem. B, Vol. 111, No. 24, 2007
Jang et al.
where
18
ks )
κˆ (ω) ≡
∫0∞ dt eiωt ∫0∞ dω J(ω)
{ ( ) coth
}
pω cos(ωt) - i sin(ωt) 2kBT (20)
As has been addressed in the works of Jang and Silbey,32,33 the actual single molecule absorption line shape (SMAL) of the B850 unit is different from the above IAL, involving additional averaging over slowly varying quasistatic disorder that occurs during the scanning time of each single molecule. On the other hand, the ensemble absorption line shape (EAL) is the average of the IAL over all possible realizations of the disorder in the ensemble of the B850 units. 2. Line Shape Expression for the Donor. Because the donor in our model corresponds to a single B800-BChl and its bath, the standard Fermi golden rule (FGR) can be used for its line shapes. The IAL of the donor is given by
∫
∞
-∞
∞dω(J
dt eiωt-i(D-g)t/p e-η∫0
(ω)/ω2){coth(pω/2kBT)(1-cos(ωt))+isin(ωt)}
(21)
For the purpose of calculating the FRET and MC-FRET rates, we also need to calculate the ideal emission line shape (IEL) of the donor. Application of the FGR leads to the following expression: LD(ω) )
∫
∞
-∞
n,n′ )1
∞dω(J
dt eiωt-i(D-g)t/p e-η∫0
(ω)/ω2){coth(pω/2kBT)(1-cos(ωt))-isin(ωt)}
(22)
C. Multichromophoric FRET Rate. According to our MCFRET theory,34 the time-dependent rate of energy transfer from the donor to the acceptor, for an impulsive excitation of the donor at t ) 0, is given by 18
k(t) )
∑
n,n′)1
JnJn′ 2πp
2
∫-∞∞ dω IA,nn′(ω) LD(t,ω)
(23)
where
IA,nn′(ω) ≡
∫-∞∞ dτ eiωτ TrbA{〈n|eiH
LD(t;ω) ) 2Re[
b,Aτ/p
∫0t dτ e-iωτ TrbD{〈D|e-iH
e-iHAτ/pFbA|n′〉} (24)
b,Dτ/p
e-iHD(t-τ)/p ×
|D〉〈D|FbDeiHDt/p|D〉}] (25) In eq 24, TrbA is the trace over the basis of the acceptor bath Hamiltonian Hb,A and FbA is the corresponding canonical density operator e-Hb,A/(kBT)/TrbA{e-Hb,A/(kBT)}, which represents the acceptor bath in canonical equilibrium with the acceptor in the ground electronic state. In eq 25, TrbD is the trace over the basis of the donor bath Hamiltonian Hb,D and FbD is the corresponding canonical density operator e-Hb,D/(kBT)/TrbD{e-Hb,D/(kBT)}, which represents the donor bath in canonical equilibrium with the ground electronic state donor. Let us assume that the donor bath becomes equilibrated with respect to the excited electronic state of the donor during ts. Then, for t > ts, eq 23 reduces to the following time independent steady-state limit expression:34,58
JnJn′ 2πp
2
∫-∞∞ dω IA,nn′(ω) LD(ω)
(26)
where LD(ω) is the steady-state limit of LD(t;ω) and is expressed as eq 22. Let us introduce the following linear combination of the acceptor states weighted by Jn: 18
|J〉 )
∑Jn|n〉
(27)
n)1
Then, invoking the approximation equivalent to the secondorder QME as was used in deriving the IAL of eq 18,33 one can show that the double summation over n and n′ in eq 23 can be approximated as 18
∑
JnJn′IA,nn′ ≈
n,n′)1
-
1 π
ID(ω) )
∑
Im TrA
{
|J〉〈J|
ω + (Eg - H0A)/p + iK(ω)
}
≡ JA(ω) (28)
Inserting the above expression into eq 26, we obtain the following compact expression for the steady-state MC-FRET rate from a B800-BChl to the B850:
ks ≈
1 2πp2
∫-∞∞ JA(ω) LD(ω)
(29)
This is the final form of the rate expression used in our previous work34 and also in the present paper. D. Disorder and Ensemble Line Shape. Disorder exists in LH2 at all temperatures. Although moderate (comparable to the magnitudes of electronic couplings between nearest neighbor BChls), it is quite significant and needs to be accounted for properly in order to explain ensemble line shape and the distribution of the energy transfer rates. Jang and Silbey31,33,59 have worked extensively on this issue for the case of the B850 unit and found that many different types of disorder can be consistent with experimental results. For example, both the disorder in the excitation energies and the disorder in the electronic couplings (due to structural disorder) can exist. As yet, there is no decisive information on which of the plausible types of disorder really correspond to the B850 unit. We here assume the simplest type of disordersGaussian disorder in three energies, En, Eg, and ED. Choosing other types of disorder consistent with the ensemble line shape may change some quantitative details, but it is likely that the major conclusion of the present paper remains intact. We calculated theoretical ensemble line shapes for the B850 and B800 bands separately. Thus, our theoretical fit is valid near the peak regions of the B850 and B800 bands but does not have significant meaning in the intermediate region (near 0 cm-1 in Figure 1) and in the far blue side of the B800 peak. For the B850 band, we first calculated eq 18 at each frequency by numerical matrix inversion of the denominator for each realization of the parameters, which were then averaged over the ensemble (40 000 realizations) of the disorder in En and Eg. For the B800 band, eq 21 calculated using numerical fast Fourier transforms was averaged over the ensemble (40 000) of the disorder in ED - Eg. The experimental ensemble absorption line shape by the Vo¨lker group for a mutant of Rhodobacter sphaeriodes, which has similar character to the wild type, was used for the fitting. For the B850 unit, we found that standard deviations of 250 and 40 cm-1 in En and Eg, respectively, give reasonable fitting. For the B800 unit, we found that 260 cm-1
FRET from B800 to B850 in LH2
J. Phys. Chem. B, Vol. 111, No. 24, 2007 6811 are, respectively, 21.2, 17.6, and 18.3 Å. On the basis of simple trigonometry for these data, we found the following values of the three coordinate parameters in eq 30: RD ) 31 Å, νD ) 23.55°, and zD ) 16.6 Å. The transition dipole vector of the B800-BChl was assumed to be in the plane of the B800 unit, which is parallel to the xy-plane. Thus, it can be specified by the following twodimensional vector parallel to the xy-plane:
µD ) µ(cos(φD), sin(φD))
Figure 1. Experimental ensemble line shape for a mutant of Rhodobacter sphaeroides by Vo¨lker group and its theoretical fit. The x-axis is the wavenumber (cm-1) relative to that of the average excitation energy of the B850-BChl. The y-axis is in arbitrary units.
for 〈ED - En〉, 54 cm-1 for the standard deviation in ED - Eg, and the choice of η ) 0.7, which was defined in eq 12, give a reasonable fitting of the peak. Among these three parameters for B800 band, the first two were determined first to reproduce the red side of the B800 peak. Then, the best value of η was determined on the basis of the fitting of the blue side of the peak was chosen. The error range of η is (0.05. Figure 1 compares the theoretical fitting with the experimental line shape. The x axis is the energy (wavenumbers) in units of inverse centimeters relative to that for 〈En〉 - 〈Eg〉. The IAL of the B850 unit was calculated in the range60 between -1100 and -100 cm-1. The falloff of the theoretical B850 ensemble line shape in the blue tail is due to this finite range of fitting. On the other hand, the deviation in the blue tail of the theoretical B800 line shape seems to originate from the neglect of MC quantum coherence within the B800 unit. At least dimeric modeling of B800 seems necessary to reproduce this part correctly as shown in the study by Cheng and Silbey.44 E. Model Parameters for the Rate Calculation. For the calculation of the rate, it is necessary to specify the positions and dipole vectors of BChls in both the B850 and B800 units. The data for Rps. Acidophila, for which the X-ray crystallography structure is known,1 were used for this purpose. For the acceptor (B850 unit), we used the same set of parameters employed in our previous study.33 For the donor, we only need to specify the position of a B800-BChl and the direction of its transition dipole moment. III. Results In specifying the parameters of the B850 unit,33 we assumed a coordinate convention that the B850 lies in the xy-plane and the positive x-axis bisects an R(n ) 1)-β(n ) 2) pair of BChls. In the same convention, the position vector of the B800-BChl, which we denote as rD, can be represented in the cylindrical coordinate as follows:
rD ) (RD, νD, zD)
(30)
where RD is the radius of the ring of the B800 unit, νD is the azimuthal angle between the vector from the origin to the center of the B800-BChl and the x-axis, and zD is the distance between the B800 and the B850 planes. We employed the structural parameters provided by Krueger et al.,29 where the distances from the above B800-BChl to its nearest neighbor B800-BChl, to the nearest R B850-BChl, and to the nearest β B850-BChl
(31)
where µ is the magnitude of the Qy transition as stated before. We found that the choice of φD ) 244.57° leads to values for the orientational factors consistent with those provided by Krueger et al.29 That is, the resulting orientational factors between the above dipole and those of the nearest neighbor B800-BChl, next nearest neighbor B800-BChl, and the nearest neighbor R B850-BChl are, respectively, -1.33, -1.08, and 0.974. We have calculated the distribution of the B800 f B850 rates employing the model parameters described above. That is, for each realization of the Gaussian disorder, we have calculated the rate employing eq 29 by performing numerical integration over the range of wavenumbers (relative to that of the average excitation energy of the B850-BChl) between -1100 and 1100 cm-1. In this rate expression, LD(ω) given by eq 22 was calculated through fast Fourier transforms61 and JA(ω) given by eq 28 was calculated by matrix inversion at each frequency as done in the calculation of the line shape.33 We have also calculated the distribution of FRET rates using eqs 18 and 22 and the nearest B800-BChl and B850-BChl distance as the distance between the donor and the acceptor. The resulting distributions of MC-FRET and FRET rates have been reported in our previous publication.34 In that work, by introducing an unknown scaling factor common to both FRET and MC-FRET and depending on the value of µ,62 we showed that the MC-FRET rate at the maximum of its distribution is about a factor of 5 larger than that for FRET, which was consistent with the experimental evidence. We also showed that the ensemble decay of the excited donor is nonexponential due to inhomogeneous broadening even though we assume each transfer at single LH2 level follows the rate description. Even with the nonexponentiality in the decay of the excited B800 ensemble, it is still meaningful to calculate the average rate and use it as the measure of the transfer time. In order to get the rough estimate of the magnitude of the transition dipole moment, let us assume that the average rate from our calculation is equal to the experimental rate determined by Pullerits et al. at 4 K.22 That is, assuming that the average rate calculated from our distribution34 is equal to the experimental rate,22(1.5 ps)-1, we find that
µ ) 5.3 Debye (D) x
(32)
where is the optical dielectric constant of the medium that appears in eq 14 and no local field correction has been considered. Figure 2 shows the resulting distributions of rates in units of inverse picoseconds. As discussed in detail by SF,30 more theoretical studies are needed on the issues such as the nature of the dielectric medium near the BChls, possible local field effects, and the effects of carotenoids. Assuming that these uncertainties do not make significant quantitative contributions, the estimate ) 1.5-2 in eq 32 results in µ ) 6.5-7.5 D. This range is somewhat
6812 J. Phys. Chem. B, Vol. 111, No. 24, 2007
Figure 2. Distribution of rates (ps-1). Solid line is the distribution of rates based on MC-FRET, and red dashed line is the distribution of rates based on FRET. The height of the FRET distribution is 1/7 of the actual height.
larger than 6.4 D, the magnitude of the transition dipole moment of a BChl in acetone22,63 and is less than the value of 8.4 D used by Monshouwer et al.21 Thus, eq 32 produces a reasonable value of the transition dipole moment, which suggests that the B800 f B850 energy transfer can be understood on the basis of our MC-FRET theory34 alone. There have been discussions about where the difference in the energy of the B800 band and the B850 band comes from. Within our model, about two-thirds of this come from the excitonic delocalization of the B850 band, and the rest (about 260 cm-1 according to our estimate34) comes from the bias of the average excitation energy of B800-BChl from that of B850BChl. This is an example of energy funneling, which sets the directionality of the overall energy flow. However, whether the value of 260 cm-1 is indeed the optimum value providing efficient energy funneling has not been clear. Additional calculations employing our model can provide new insights into this issue. We have calculated the distributions of both MC-FRET rates and the original FRET rates for six other values of the bias of the aVerage excitation energy of the B800-BChl from that of B850-BChl. The resulting distributions are shown in Figure 3. The distributions for different values of bias have quite different shapes. Still, at the lowest level of approximation, one can use the inverse of the average rate as the measure of the transfer time. For each value of bias, we calculated τ ) 1/kaV, where kaV is the average of the distribution of kMC-FRET’s or kFRET’s as shown in Figure 3. The upper panel of Figure 4 shows the calculated transfer times. The values at 260 cm-1 are also shown. The lower panel of Figure 4 shows the ratio of the two. It is important to note that the transfer time based on MCFRET is quite insensitive to the bias up to about 400 cm-1, whereas that on the basis of FRET varies over an order of magnitude. Thus the multichromophoric effects tend to make the transfer time insensitive to changes in energy bias. This indicates that the purple bacteria are utilizing the MC effect to a great extent to guarantee the irreversibility of the energy flow from B800 to B850 while not affecting the transfer time significantly. IV. Discussion We conclude this paper by discussing the implications of our results and some issues that need to be clarified or resolved. A. Effects of Multichromophoric Cooperation, Disorder, and Electron-Phonon Coupling. On the basis of a model
Jang et al.
Figure 3. Distribution of rates for different values of the bias, which are written in the upper right-hand corner of each panel. Solid lines are results based on the MC-FRET, and the red dashed lines are those based on FRET. Except for the case with the bias of -200 cm-1, the height of each FRET distribution was made comparable to that of the MC-FRET by multiplying the numerical factor shown near the peak maximum of the former.
Figure 4. (a) Theoretical estimates of the energy transfer time from B800 to B850, based on FRET and MC-FRET, with the variation of the bias of the average excitation energy of the B800-BChl from that of the B850-BChl. (b) The ratios of the transfer times.
Hamiltonian that provides reasonable description of LH2 with independently determined parameters, we showed that the MC effect alone can explain the fast energy transfer rate from B800 to B850. In addition, we showed that the excitation energies of BChls in B800 and B850 are optimized such that the energy transfer is both fast and irreversible. Our results provide a solid quantitative basis for the original suggestion made by MASSF,28,30 and establish deeper theoretical understanding of the band positions of the B800 and B850.
FRET from B800 to B850 in LH2 The theoretical and computational details of the present work differ from those of the works of MAS-SF28,30 in the following two respects: (i) The rate expression eq 29 includes all the intraacceptor quantum coherence terms whereas the expression employed by MAS-SF neglects the off-diagonal terms in the exciton basis; (ii) The present work employs a more satisfactory determination of the acceptor line shape functions based on the solution of quantum Master equation for the exciton-bath model (with all the parameters determined independent of the rate calculation), which accounts for different line shape functions for different exciton states and different realizations of disorder. In our previous paper,34 we compared the rate distributions calculated by eq 29 and the approximate expression employed by MAS-SF, both based on our exciton-bath model. The difference between the two was quite small, although noticeable. Similar distributions have been calculated (not shown) for different values of bias, and we confirmed the two rate expressions result in similar distributions regardless of the bias. Thus, we conclude that the neglect of off-diagonal coherence terms (in the exciton basis) as was assumed by MAS-SF does not alter the rate distribution significantly, at least, in the lowtemperature limit considered in this work. Rather, the major factor that brings quantitative difference to our result is the second one (ii) described above, which indicates that the assumption of uniform homogeneous line shape as assumed by MAS-SF is too simplistic for the acceptor, B850. Whether similar conclusions can be made at room temperature is an interesting theoretical issue and will be the subject of future work. Figure 4 summarizes the major result of this paper that the relative spectral positions of the B800 and B850 band positions have important implications for the efficient energy flow from the former to the latter. In LH2 complexes, the coupling of the electronic excitation of each BChl to environmental phonon modes is weak or moderate. This prevents significant loss of excitation energy through phonon relaxation. However, at the same time, it does not provide significant spectral overlap between B800 and B850, if the transfer rate is governed by the original FRET. However, the MC-FRET mechanism makes it possible for relatively fast energy transfer to occur by widening the spectral overlap from the donor to the acceptor, selectively. In addition, we find that the MC effect makes the rate quite insensitive to the disorder or to the energetic bias of the B800BChls relative to B850-BChls, thus serving the role of a spectral buffer. B. Quantum Coherence between B800 and B850. In the present work, the coherence between B800 and B850 or back transfer from the latter to the former is not taken into consideration. As the bias of the B800-BChl excitation energy from that of the B850-BChl decreases (approaching the value of zero), the effect of such coherence or back transfer is expected to alter the overall transfer time significantly. For example, a theoretical study48 suggests that the MC-FRET theory employed by MAS and SF does not reproduce the trend of the experimental data23 obtained from a series of mutants with different energy biases. These authors48 explained64 the experimental trend23 by employing a generalized master equation approach that includes the back transfer from B850 to B800, but their use of solution-phase spectral densities and neglect of inhomogeneity need further theoretical verification. Yang and co-workers41 also showed the importance of back transfer, by employing a quantum master equation approach. It is possible to use the MC-FRET theory and our model Hamiltonian for the calculation of the kernels in the quantum
J. Phys. Chem. B, Vol. 111, No. 24, 2007 6813 master equation, and the calculation results will provide a clearer understanding of the role of B800-B850 coherence. Alternatively, one may apply a recent theory developed by Knoester and co-workers,65 which can be applied to general exciton systems including the nonperturbative regime of donor-acceptor coupling given that the effects of phonon bath can be treated perturbatively. C. Quantum Coherence within B800. Recently, Cheng and Silbey44 made a detailed study of the MC effect on the B800 unit. On the basis of the modeling of the ensemble line shape and the intra-B800 energy transfer dynamics, they found that a dimeric representation provides a reasonable description of the quantum coherence within the B800 unit. Employing the dimeric model of the B800 unit, they also calculated the B800 f B850 energy transfer rate and found that the quantum coherence (dimeric nature) within the B800 narrows the distribution of rates, making the transfer rate less sensitive to disorder. This is consistent with our finding that the MC effect within the B850 also makes the energy transfer less sensitive to the disorder. Quantitatively, the study of Cheng and Silbey44 suggests that the MC effect of B800 can also change the average B800 f B850 energy transfer rate somewhat, although it is believed to be small, so as not to affect the major conclusions of this paper. For a fully quantitative understanding of the MC effect of the B800, a thorough study based on the full model of the B800 unit should be done. Acknowledgment. The authors acknowledge the fundamental contribution Dr. Norman Sutin has made in our understanding of charge-transfer processes in condensed phases. S.J. acknowledges a start-up fund from Queens College of the City University of New York. Partial summer support of S.J. and the support of M.D.N. come from the contract DEAC0298CH10886 with the U.S. Department of Energy and from its Division of Chemical Sciences, Office of Basic Energy Sciences. R.J.S. acknowledges the support of NSF. References and Notes (1) McDermott, G.; Prince, S. M.; Freer, A. A.; HawthornthwaiteLawless, A. M.; Paplz, M. Z.; Cogdell, R. J.; Issacs, N. W. Nature 1995, 374, 517. (2) Koepke, J.; Hu, X.; Muenke, C.; Schulten, K.; Michel, H. Structure 1996, 4, 581. (3) Papiz, M. Z.; Prince, S. M.; Howard, T.; Cogdell, R. J.; Isaacs, N. W. J. Mol. Biol. 1995, 46, 701. (4) Roszak, A. W.; Howard, T. D.; Southall, J.; Gardiner, A. T.; Law, C. J.; Isaacs, N. W.; Cogdell, R. J. Science 2003, 302, 1969. (5) Zouni, A.; Witt, H. T.; Kern, J.; Fromme, P.; Krauss, N.; Saenger, W.; Orth, P. Nature 2001, 409, 739. (6) Jordan, P.; Fromme, P.; Witt, H. T.; Klukas, O.; Saenger, W.; Krauss, N. Nature 2001, 411, 909. (7) Kamiya, N.; Shen, J.-R. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 98. (8) Ben-Shem, A; Frolow, F.; Nelson, N. Nature 2003, 426, 630. (9) Sundstrom, V.; Pullerits, T.; van Grondelle, R, J. Phys. Chem. B 1999, 103, 2327. (10) Koolhaas, M. H. C.; van der Zwan, G.; van Grondelle, R. J. Phys. Chem. B 2000, 104, 4489. (11) Wu, H.-M.; Reddy, N. R. S.; Small, G. J. J. Phys. Chem. B 1997, 101, 651. (12) Wu, H.-M.; Ratsep, M.; Jankowiak, R.; Cogdell. R. J.; Small, G. J. J. Phys. Chem. B 1997, 101, 7641. (13) Wu, H.-M.; Small, G. J. J. Phys. Chem. B 1998, 102, 888. (14) Matsuzaki, S.; Zazubovich, V.; Frase, N. J.; Cogdell, R. J.; Small, G. J. J. Phys. Chem. B 2001, 105, 7049. (15) Freiberg, A.; Jackson, J. A.; Lin, S.; Woodbury, N. W. J. Phys. Chem. A 1998, 102, 4372. (16) Freiberg, A.; Timpmann, K.; Ruus, R.; Woodbury, N. W. J. Phys. Chem. B 1999, 103, 10032. (17) Timpmann, K.; Woodbury, N. W.; Freiberg, A. J. Phys. Chem. B 2000, 104, 9769.
6814 J. Phys. Chem. B, Vol. 111, No. 24, 2007 (18) Tietz, C.; Chekhlov, O.; Dra¨benstedt, A.; Schuster, J.; Wrachtrup, J. J. Phys. Chem. B 1999, 103, 6328. (19) Hofmann, C.; Michel, H.; van Heel, M.; Ko¨hler, J. Phys. ReV. Lett. 2005, 94, 195501. (20) Groot, M. L.; Breton, J.; van Wilderen, J. G. W.; Dekker, J. P.; van Grondelle, R. J. Phys. Chem. B 2004, 108, 8001. (21) Monshouwer, R.; Abrahamsson, M.; van Mourik, F.; van. Grondelle, R. J. Phys. Chem. B 1997, 101, 7241. (22) Pullerits, T.; Hess, S.; Herek, J. L.; Sundstro¨m, V. J. Phys. Chem. B 1997, 101, 10560. (23) Herek, J. L.; Fraser, N. J.; Pullerits, T.; Martinsson, P.; Polivka, T.; Scheer, H.; Cogdell, R. J.; Sundstro¨m, V. Biophys. J. 2000, 78, 2590. (24) Creemers, T. M. H.; De Caro, C. A.; Visschers, R. W.; van Grondelle, R.; Vo¨lker, S. J. Phys. Chem. B 1999, 103, 9770. (25) Timpmann, K.; Ra¨tsep, M.; Hunter, C. N.; Freiberg, A. J. Phys. Chem. B 2004, 108, 10581. (26) Urbonien, V.; Vrublevskaja, O.; Gall, A.; Trinkunas, G.; Robert, B.; Valkunas, L. Photosynt. Res. 2005, 86, 49. (27) Hu, X.; Ritz, T.; Damjanovic, A.; Autenrieth, F.; Schulten, K. Quart. ReV. Biophys. 2002, 35, 1. (28) Mukai, K.; Abe, S.; Sumi, H. J. Phys. Chem. B 1999, 103, 6096. (29) Krueger, B. P.; Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 1998, 102, 5378. (30) Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 2000, 104, 1854. (31) Jang, S.; Dempster, S. E.; Silbey, R. J. J. Phys. Chem. B 2001, 105, 6655. (32) Jang, S.; Silbey, R. J. J. Chem. Phys. 2003, 118, 9312. (33) Jang, S.; Silbey, R. J. J. Chem. Phys. 2003, 118, 9324. (34) Jang, S.; Newton, M. D.; Silbey, R. J. Phys. ReV. Lett. 2004, 92, 218301. (35) Ray, J.; Makri, N. J. Phys. Chem. A 1999, 103, 9417. (36) Herman, P.; Barvik, I. J. Phys. Chem. B 1999, 103, 10892. (37) Ritz, T.; Park, S.; Schulten, K. J. Phys. Chem. B 2001, 105, 8259. (38) Linnanto, J.; Korppi-Tommola, J. E. I.; Helenius, V. M. J. Phys. Chem. B 1999, 103, 8739. (39) Linnanto. J.; Korppi-Tommola, J. E. I. Phys. Chem. Chem. Phys. 2002, 4, 3453. (40) Linnanto, J.; Korppi-Tommola, J. J. Phys. Chem. . 2004, 108, 5872. (41) Yang, M.; Agarwal, R.; Fleming, G. R. J. Photochem. Photobiol., A 2001, 142, 107. (42) Agarwal, R.; Yang, M.; Xu, Q.-H.; Fleming, G. R. J. Phys. Chem. B 2001, 105, 1887. (43) Agarwal, R.; Rizvi, A. H.; Prall, B. S.; Olsen, J. D.; Hunter, C. N.; Fleming, G. R. J. Phys. Chem. A 2002, 106, 7573. (44) Cheng, Y. C.; Silbey, R. J. Phys. ReV. Lett. 2006, 96, 028103. (45) Janosi, W.; Kosztin, I.; Damjanovic, A. J. Chem. Phys. 2006, 125, 014903.
Jang et al. (46) Sener, M. K.; Park, S.; Lu, D.; Damjanovic´, A.; Ritz, T.; Fromme, P.; Schulten, K. J. Chem. Phys. 2004, 120, 11183. (47) Bru¨ggemann, B.; Sznee, K.; Novoderezhkin, V.; van Grondelle, R.; May, V. J. Phys. Chem. B 2004, 108, 13536. (48) Kimura, A.; Kakitani, T. J. Phys. Chem. B 2003, 107, 7932. (49) Fleming, G. R.; Scholes, G. D. Nature 2004, 431, 256. (50) Jimenez, R.; Dikshit, S.; Bradforth, S. E.; Fleming, G. R. J. Phys. Chem. 1996, 100, 6825. (51) Fo¨rster, Th. Discuss. Faraday Soc. 1953, 27, 7. (52) Fo¨rster, Th. In Modern Quantum Chemistry, Part III; Sinanoglu, O., Ed., Academic Press, New York, 1965. (53) Agranovich, V. M.; Galanin, M. D. Electronic excitation energy transfer in condensed matter; North-Holland: Amsterdam, 1982. Andrews, D. L., Demidov, A. A., Eds. Resonance Energy Transfer; John Wiley & Sons: Chichester, 1999. Scholes, G. D. Annu. ReV. Phys. Chem. 2003, 54, 57. (54) Sumi, H. J. Phys. Chem. B 1999, 103, 252. (55) There is some evidence that there is a difference in the average excitation energy of R (odd n) and β (even n) BChls. However, we find that the effects of such a difference on the ensemble line shape can be offset by slight changes in the magnitudes of disorder and the value of En - Eg. (56) Renger, T.; Marcus, R. A. J. Chem. Phys. 2002, 116, 9997. (57) Jang, S.; Cao, J.; Silbey, R. J. J. Phys. Chem. B 2002, 106, 8313. (58) Jang, S.; Jung, Y. J.; Silbey, R. J. Chem. Phys. 2002, 275, 319. (59) Jang, S.; Silbey, R. J. In preparation. (60) This range is limited to the fitting of the ensemble line shape only, and a larger range of -1100 to 1100 cm-1 is used for the integration of the FRET and MC-FRET rate expressions. (61) The calculation needs to be made only for one single reference value of D - g. Once the emission lineshape is stored, the line shapes for other values of the energy can be obtained by shifting the reference value only. (62) There is a typo in the unit definition related to scaling factor in the Figure 2 caption of this paper. p ) 1 should read as h ) 1. (63) Scherz, A.; Parson, W. W. Biochim. Biophys. Acta 1984, 766, 666. (64) Kimura and Kakitani in ref 48 also used different spectral densities with longer tails and bumps in the high frequency phonon modes, which seems necessary for the understanding of the experimental rates for mutuants with large energy bias (data points e and f in their Figure 1). This shows that the use of more accurate spectral densities including high frequency phonon modes may be necessary for satisfactory theoretical description of all the experimental data in ref 23 even if we include the effect of back transfer reactions. (65) Didraga, C.; Malyshev, V. A.; Knoester, J. J. Phys. Chem. B 2006, 110, 18818.
