17950
J. Phys. Chem. 1996, 100, 17950-17956
Singlet-Singlet Annihilation and Local Heating in FMO Complexes Vidmantas Gulbinas,† Leonas Valkunas,*,† Darius Kuciauskas,† Evaldas Katilius,† Vladas Liuolia,† Wenli Zhou,‡ and Robert E. Blankenship‡ Institute of Physics, A. Gostauto 12, 2600 Vilnius, Lithuania, and Department of Chemistry and Biochemistry, Center for the Study of Early EVents of Photosynthesis, Arizona State UniVersity, Tempe, Arizona 85287-1604 ReceiVed: May 3, 1996; In Final Form: August 30, 1996X
Energy transfer in a trimeric Bchl a containing FMO pigment-protein complex from the green sulfur bacterium Chlorobium tepidum has been studied by means of picosecond transient absorption spectroscopy under highexcitation conditions. At room temperature the excited state absorption spectrum of the FMO complex was found to be similar to that of noninteracting Bchl molecules in solution, which suggests that the influence of exciton coupling on the spectroscopic properties of the FMO complex at room temperature is not substantial. Analysis of the excited state relaxation kinetics in singlet-singlet annihilation conditions shows that the energy transfer from the excited monomer to another excited monomer is independent of the oxidationreduction state of the complex and is slower than the intermonomer excitation migration rate. The difference spectrum at 77 K resembles the absorption spectrum, showing three exciton subbands. In addition to the singlet-singlet annihilation, the 7 ps rate of which is similar to that at room temperature, and to the intrinsic exciton decay, which is also temperature independent, energy redistribution between exciton states with a mean time of 26 ps is evident. This redistribution is explained as being due to local heating/cooling kinetics stimulated by the excitation pulses.
Introduction Establishing a relationship between the structural build-up and spectral properties of pigment-protein complexes is one of the goals of spectroscopic research of photosynthetic systems. Despite numerous investigations, it has turned out to be a nontrivial task to correlate the structure and spectra.1 The best understood photosynthetic pigment-protein complex confirming this statement is a bacteriochlorophyll a (Bchl a) containing pigment-protein complex from green sulfur bacteria, known as the FMO protein after Fenna, Matthews, and Olson.2 The structure of this complex has been determined to 1.9 Å resolution,3,4 and the amino acid sequence of the protein is known as well.5 The FMO complex is a trimer (C3 symmetry) composed of identical protein subunits of 47 kDa, each of which hosts seven Bchl a molecules arranged in a highly nonregular fashion. A number of spectroscopic studies6-8 have been performed on FMO complexes, and some theoretical efforts9,10 have been applied. Despite the absence of the exact coincidence of the theoretical calculations and the spectroscopic results, some qualitative conclusions were formulated. By analyzing lowtemperature absorption and CD spectra on the basis of the exciton theory, it was concluded9 that the longest wavelength absorption spectrum subband was determined by the absorption of Bchl a No. 7 (according to the numbering scheme of Matthews and Fenna3) on which the excitation resides for most of the excited state lifetime even at room temperature. The hole-burning experiments6 as well as the singlet-triplet difference absorption measurements7,8 supported the assumption of the exciton interaction between the pigment molecules in the monomer and between Bchl a No. 7 of different monomers in the trimer. However, the purely excitonic picture is not so straightforward when applied at room temperature. Disappearance of the fine structure of the absorption spectrum or temperature dependence of the excitation lifetime †
Institute of Physics. Arizona State University. X Abstract published in AdVance ACS Abstracts, October 15, 1996. ‡
S0022-3654(96)01272-5 CCC: $12.00
could indicate the absence of excitonic interactions at higher temperature.11 The excitation lifetime in FMO complexes is strongly dependent on the redox state of the protein. Recently it has been concluded that the excitation lifetime under strongly reducing conditions is about 2 ns, while in the samples, which are in neutral or oxidizing conditions, it is approximately 60 ps.12 Such a short excitation lifetime seems to be due to the quenching of excitations and has been proposed as a means of protection of the green sulfur bacteria from the oxygen-mediated damage under the oxygenic conditions.13 However, the origin of the quenchers is far from obvious. It is noteworthy that this process is essentially suppressed by lowering the temperature.12 This could be due to the slowing down of the rate of the excitation migration, to the changes in the occupations of the excitonic states stimulating the localization of the excitons on Bchl a pigments No.7, or to a temperature dependence of the chemical reaction thought to be responsible for quenching the excited state. The excitation migration through the domain of pigments is a driving force of the nonlinear annihilation of excitations. Singlet-singlet14,15 and singlet-triplet14,16,17 annihilation studies have been used to determine the energy migration parameters in various light-harvesting antennae, demonstrating the sensitivity of these methods to the degree of coupling of the pigments in the domain and to the spectral composition of the pigment molecules. Singlet-singlet annihilation measurements in the FMO complexes have shown that at room temperature efficient energy transfer takes place within the trimer.18 The singlettriplet annihilation measurements8 demonstrated the temperature sensitivity of the coupling between pigment molecules. This paper is focused on singlet-singlet annihilation studies in the FMO complexes. We also address the question of excitonic coupling in the FMO complexes at room temperature and at 77 K, since many of the experimental studies and theoretical analyses were performed at low temperatures. © 1996 American Chemical Society
Singlet-Singlet Annihilation in FMO Complexes
J. Phys. Chem., Vol. 100, No. 45, 1996 17951
Figure 1. FMO absorption (solid line), emission (dashed line), and transient absorption spectra (open circles) at room temperature (295 K). The dotted line shows the difference absorption spectrum calculated according to eq 1. Triangles with connecting line are the ESA spectrum approximation.
Experimental Section Fenna-Matthews-Olson protein from the green sulfur bacterium Chlorobium tepidum has been purified as described by Savikhin et al.19 Investigations were carried out at room temperature and at 77 K. For the measurements at 77 K the samples were diluted with glycerol (approximately two volumes of glycerol per one volume of the protein solution) and cooled in a liquid nitrogen cryostat. Investigations at room temperature have been performed on the reduced and oxidized forms of the protein. The reduction was performed by adding about 20 mmol sodium dithionite solution. Steady state absorption spectra were measured on a Beckman UV 5270 spectrophotometer. A pump-probe absorption spectrometer with 2 ps time resolution, based on the passively modelocked feedback controlled 1 Hz repetition rate Nd-glass laser, was used for transient absorption measurements. The samples were excited at 760 nm by the second Stokes component obtained by Raman scattering in ethanol of the second harmonics of the laser radiation. A white light continuum generated in water was used for probing. The optical density of the samples at the Qy band maximum was A ) 1.1 for the room temperature measurements and A ) 0.3 for measurements at 77 K. The optical path length of the cell was 3 mm. Experimental Results Room Temperature. Figure 1 shows the FMO protein difference absorption (∆A) spectrum at zero delay between excitation and probing. No changes of the spectrum on the picosecond time scale except a proportional decrease of the intensity were observed when the delay time was increased. The spectrum contains contributions from the photobleaching (PB) of the ground state (ground state absorption cross section, σG), stimulated emission (SE) from the excited state (SE cross section, σSE), and excited state absorption (ESA cross section, σESA). The change of the optical density can be expressed as
∆A(λ) ) (σESA(λ) - σG(λ) - σSE(λ))dn
(1)
where d is the optical path length of the cell and n is the concentration of the excited molecules. In the Qy band region (λ > 770 nm) the difference spectrum is dominated by a combination of the PB and SE. At shorter wavelengths the difference spectrum is mainly determined by the ESA. The stimulated emission cross section, σSE, could be calculated from the normalized steady state fluorescence spectrum, F(λ),13
Figure 2. FMO bleaching at 810 nm decay kinetics for different excitation intensities at room temperature. The right-hand axis indicates the calculated excitation concentration normalized to the total Bchl a molecule concentration.
according to σSE ∝ λ4F(λ).20 To calculate the ESA spectrum as it follows from eq 1, the absolute values of the corresponding cross sections and the concentration of the excited molecules should be determined. Unfortunately, it is difficult to define these values with a sufficient accuracy. Therefore, the ESA could be determined only at the spectral points where σG, σSE, or ∆A are equal to zero or are negligibly small, i.e. at 870, 650, and 770 nm. Within the Qy band the sum of the ground state and the stimulated emission spectra coincide with the ∆A spectrum, which is a strong indication that the ESA spectrum in this region is weak and has no well-expressed structure. Therefore, we approximate the ESA spectrum simply as a line connecting the defined values at 870, 770, and 650 nm. By means of such an approximation the spectrum calculated according to eq 1 explains the experimentally measured ∆A spectrum as it is shown in Figure 1. It is noteworthy that the ESA spectrum of the chlorophyll in solution is broad and structureless and thus is similar to that obtained here, but is of higher intensity.20 On the other hand, calculations of the ESA spectrum of the FMO monomeric complex based on exciton theory21 showed that in the strong coupling case the ESA spectrum in the Qy band region should be quite strong and exhibit a well-defined structure. For example, it should result in the induced absorption in the 789-800 nm region. Thus, the ESA spectrum of the FMO complex at room temperature closely resembles the ESA spectrum of the Bchl a molecules in organic solvents, but not the spectrum of the excitonically coupled Bchl a aggregate. The decay kinetics of the absorption changes (∆A) at 810 nm at various excitation intensities are shown in Figure 2. The kinetics are more informative when presented as the ratio of the excited molecules density n to the total density of the pigment molecules N. This ratio is depicted on the right-side ordinate axis of Figure 2. Such renormalization was based on the following relationship:
∆A n σESA - σSE - σG ) A N σG
(2)
Relative values of cross sections σSE(λ), σG(λ), and σESA(λ) at the probe wavelength were estimated from the data presented in Figure 1. Thus, at 820 nm σSE/σG ) 1.43 and σESA/σG ) 0.12. Due to the spectral heterogeneity of the complex, eq 2 is a simplification that may be justified by taking into account that the homogeneous absorption bands are rather broad (fwhm more than 200 cm-1 ) 13 nm as found by Lu and Pearlstein10
17952 J. Phys. Chem., Vol. 100, No. 45, 1996
Gulbinas et al.
Figure 4. FMO transient absorption kinetics at different probing wavelengths at 77 K.
Figure 3. 77 K FMO absorption spectrum (a) and transient difference absorption spectra at various delay times after excitation (b).
already at 77 K compared to about 30 nm fwhm of the whole Qy absorption band). The global analysis of the excitation decay kinetics gives three relaxation phases with lifetimes equal to τ1 ) 7 ps, τ2 ) 60 ps, and τ3 ) 2000 ps. At low excitation intensity, when less than one excitation per trimer is generated (n/N < 1/21) and mainly linear relaxation processes are responsible for the excitation decay, two phases corresponding to the longer relaxation times remain. The amplitude of the fast relaxation phase (7 ps) increases with increase of the excitation intensity, while the amplitudes of the middle and slow phases remain unchanged. This is a direct indication that this fast relaxation is caused by the singlet-singlet annihilation. Two slower relaxation times (60 ps and 2 ns) are the same as obtained by Zhou et al.,12 while the relative amplitudes of these exponents are different: in our case the 60 ps component is more pronounced. It was already shown12 that 60 ps decay dominates in the oxidized and neutral complexes, while the 2 ns component is dominant in the reduced complexes. Thus, we also assume that at the particular redox potential there is a dynamic equilibrium between the oxidized (and neutral) and the reduced complexes. 77 K Temperature. The transient difference absorption spectrum of the FMO complex in the Qy transition absorption band at 77 K (see Figure 3) is similar to that of the steady state spectrum that has been attributed to the excitonic intermolecular interactions within the monomer or even within the whole trimer.9,10,21 Three subbands in the ∆A spectrum correspond to the three lowest energy subbands of the Qy molecular transition absorption spectrum. Within experimental accuracy no bleaching was observed at the shorter wavelengths. The bleaching spectrum can be approximated quite well by the sum of three Gaussian functions assuming constant widths and maxima positions and time dependent amplitudes. The decay kinetics depend both on the excitation intensity and on the probing wavelength, as is seen in Figure 4 and Figure 5. In addition to the relaxation in time of the entire ∆A spectrum, redistribution of the bleaching intensities between subbands takes place; that is, the bleaching subband of the longest
Figure 5. Averaged transient absorption kinetics at different excitation intensities at 77 K. The kinetics were obtained by averaging bleaching decays at 807, 814, and 825 nm. Right-hand axis indicates the calculated excitation concentration normalized to the total Bchl a molecule concentration. Excitation wavelength λex ) 760 nm.
wavelength (maximum at 825 nm) gains at the expense of the two others. Thus, at least three processes taking place simultaneously on the picosecond time scale can be distinguished: (i) intrinsic excitation decay, (ii) singlet-singlet annihilation, and (iii) excitation energy redistribution between the subbands. The redistribution effect between the subbands can be excluded by averaging the excitation decay kinetics measured at the three subbands maxima, with only the decay kinetics of the integral signal of the entire Qy band remaining (see Figure 5). In the case when the subbands originate from the excitonic interactions,9,10,21 this integral signal has to be proportional to the exciton density generated in the system, and the absolute value of this density can be determined from the comparison of the integral bleaching signal with the integral ground state absorption. For such estimations the decrease of the transition dipole moment corresponding to the Qy absorption band by 1/7 of its steady state value when one excitation per monomer is created has to be taken into account, as it has been done calculating the ESA spectrum.7,8,21 The same value of the integral absorption decrease has to be assumed as being due to the stimulated emission. Thus, for the coherent exciton model of the FMO monomer the ESA spectrum within the whole Qy band has to be smaller by 2/7 of its steady state value. As indicated above, the calculated relative exciton density decay kinetics at different excitation intensities are shown in Figure 5 (right scale). Three exciton lifetimes are obtained by means of the global analysis: 7.4, 104, and 1020 ps. Due to the dependency of the amplitude of the 7.4 ps component on the excitation intensity, this component can be attributed to the exciton-exciton
Singlet-Singlet Annihilation in FMO Complexes
Figure 6. Time evolution of the bleaching of the different subbands normalized to the total bleaching spectrum area. λex ) 760 nm, T ) 77 K. The bleaching subband intensities were obtained from the approximation of the bleaching spectra at different delay times by the sum of three Gaussian functions with constant widths and maxima positions and time dependent amplitudes. Lines show the singleexponential approximation with 26 ps mean time.
annihilation. At high excitation intensity the annihilation reduces the excitation density to approximately one excitation per FMO trimer, while at lower intensity annihilation input agrees well with the two-excitation creation probability in the FMO trimer described by the Poisson statistical distribution. To distinguish the exciton redistribution dynamics from the decay kinetics, amplitudes of the Gaussian functions used to approximate the difference spectra were analyzed. Time dependencies of each Gaussian component normalized to the integral area of the bleaching band (which is proportional to the exciton density) are presented in Figure 6. The 825 nm component grows at the expense of the 807 nm component, while the intermediate component at 814 nm does not change significantly. In the single-exponential approximation this redistribution kinetics at 77 K can be defined by the characteristic time of 26 ( 3 ps. Discussion Let us explore now the annihilation process considering both the structural data of the FMO complexes and the spectral peculiarities of the pigments. The biochemical assays and the spectroscopic analysis8 show that the complexes found in solution are arranged in trimers; thus, a natural assumption of the absence of the energy transfer between different unaggregated trimers can be made. As it was already determined, the mean times of the interpigment energy migration within the monomer is on the order of 100-900 fs19,22 at room temperature. This conclusion is also in line with the assumption of the presence of excitonic interactions in the monomers at lower temperatures.6-10 Thus, taking into account the initial distribution of the excitations between trimers as being Poissonian, which means that the annihilation occurs only in the trimers in which more than one excitation is created, the 7 ps relaxation component evidently corresponds to the energy transfer between excited monomers within the trimer. Thus, because the energy migration within a monomer is much faster than the energy transfer between monomers at least at room temperature, the whole energy migration process can be separated into two stages: the excitation equilibration within a separate monomer and the energy transfer between monomers. The excitation equilibration within the monomer leads to a localization of the excitation, presumably on Bchl a No. 7.9,10 Thus, the approximation that the energy transfer between monomers proceeds mainly between Bchl a No. 7 molecules
J. Phys. Chem., Vol. 100, No. 45, 1996 17953 seems to be reasonable. According to the Boltzmann distribution, at room temperature the excitation resides on Bchl a No. 7 molecule about 40% of its lifetime; therefore, the annihilation between excitations located on different monomers could proceed by two different ways: (i) the annihilation is a onestep process that is due to the interaction of two excited Bchl a molecules located on different monomers; (ii) the excitation energy is transferred to the same monomer due to the interaction of the excited and unexcited molecules from the different monomers, and then annihilation between two excitations within the same monomer takes place. For the incoherent (Fo¨rstertype) energy transfer between molecules the ratio of rates of the two ways can be estimated by analyzing the spectral overlap of the donor molecule fluorescence and the acceptor molecule absorption.15 Rough deconvolution of the difference spectrum at room temperature presented in Figure 1 leads us to conclude that the emission spectrum overlap with the ESA spectrum is about 7 times smaller than that with the ground state absorption spectrum, and correspondingly, the rate of the first way is much slower. Thus the annihilation proceeds mainly by the second way and, due to the probability factors, should be slower than the excitation energy transfer observed in depolarization experiments. This finding is in agreement with the conclusions of the intermonomer energy transfer being the order of several picoseconds obtained from the femtosecond excitation kinetic measurements.19 At intermediate excitation intensities, when not more than two excitations are created within the trimer, the annihilation is a single-exponential process, while at higher intensities when three excitations per trimer are created the annihilation rate becomes a complex function of excitation intensity and time (see the Appendix). Thus, the kinetic processes taking place on the picosecond time scale at room temperature are completely explained within the context of the inhomogeneous energy migration. At 77 K the absorption spectrum is more complex; it contains three subbands within the Qy molecular transition band, which is most probably due to the coherent interaction between the Bchl a molecules in the monomer.9,10 However, according to the excitonic calculations,8-10 the excitation of the longest wavelength subband (825 nm) corresponds also to the exciton localization presumably on Bchl a No. 7. This could be the reason that the annihilation rate at 77 K is the same as it is at room temperature. Due to the exciton interaction within the monomer, the relaxation process and, respectively, the excitation localization on Bchl a No. 7 have to be fast. Femtosecond spectral equilibration studies showed that downhill energy transfer at room temperature has several stages with lifetimes that range from ∼100 to ∼900 fs.19,22 However, evolution of the transient difference spectra at 77 K suggest a 26 ps equilibration time. The equilibration rate can become slower by lowering the temperature due to the changes of the density of vibrational modes involved in the relaxation process. However, it is very unlikely that these changes could be so pronounced. Moreover, the annihilation time is found to be 7 ps, which implies that the excitation equilibration should be even faster, because excitations, as it was discussed above, should be localized on Bchl a No. 7 prior to annihilation. Thus, within our time resolution the spectral changes have to be related to the thermalized exciton redistribution. According to the Boltzmann distribution, at 77 K about 95% of the excitons should be situated on the lowest excitonic level when the system is thermalized. However, as has already been shown,7-9 the lowest excitonic state corresponds mainly to the localization of the excitation on Bchl a No. 7. Therefore, the difference spectrum, when the energy
17954 J. Phys. Chem., Vol. 100, No. 45, 1996 redistribution process within the whole trimer is over, has to correspond to the difference spectrum mainly determined by the bleaching of the longest wavelength band with a slight (∼5%) mixing of the higher exciton states. However, this is not the case in the difference spectrum even at 90 ps delay (see Figure 3). The bleaching bands of the higher excitonic states cannot be related to the shift of the exciton bands caused by the occupation of the lowest exciton state, because in this case increase of the absorption in some spectral regions has to appear, and the bleaching spectrum could hardly resemble the steady state absorption spectrum so closely. The estimated local temperatures that according to the Boltzmann distribution could correspond to the experimentally measured intensities of the bleaching bands at 814 and 807 nm in comparison with the lowest one at 825 nm equal approximately 260 K at 0 ps, 220 K at 5 and 10 ps, 170 K at 30 ps, and 150 K at 90 ps. It is noteworthy that estimations obtained from the ratio of the bleaching intensities at 825 and 814 nm bands and that at 814 and 807 nm bands give very similar temperature values and confirm that some sort of thermalization is really taking place. The maximum possible thermal energy excess supplied to the Bchl a molecules in the FMO complex during the relaxation can be estimated by taking into account the heat capacity of the Bchl a molecule. However, we do not know the exact values of the heat capacity of these molecules at 77 K; therefore, for estimations we used known heat capacity values of similar organic molecules, and by taking into account the number of atoms in the molecule, we assumed that the Bchl a molecule heat capacity approximately equals 100-150 J/(mol K).23 Thus, for the excitation at 760 nm, the increase of the local temperature by ∼10 K for all seven Bchl a molecules can be estimated assuming that the thermal energy excess created during the excitation relaxation to the lowest exciton state is concentrated only on Bchl a molecules and is equally distributed over all seven Bchl a molecules. However, much higher values of the local temperature can be stimulated by the singlet-singlet annihilation. Thermal energy created during a single excitonexciton annihilation event is enough to raise the temperature of all seven Bchl a molecules by more than 100 K. If we assume that only two excited Bchl a No. 7 molecules are involved in the nonlinear annihilation, then the increase of the temperature for the excited molecule can be as high as 300-500 K, even if the increase of the heat capacity with temperature is also taken into account. This result is not very surprising. Similar values of molecular heating were achieved in numerous organic molecules in solutions24 and even in solid state.25 Equilibration of the vibrational energy within the molecular modes takes place on the femtosecond time scale, while the further heat relaxation of the Bchl a molecules should be defined by the vibrational energy transfer from the “heated” molecular modes to the vibrational modes of the surrounding protein. Typical times for energy redistribution between soluted molecules and the nearest solvent surrounding range between 15 and 40 ps, while further energy redistribution over the entire solvent is even much slower and can reach a nanosecond time scale.24 Thus, nonequilibrium temperature calculated from the ratio of the bleaching subbands assuming the Boltzmann distribution between the exciton energy states of the monomer reasonably corresponds to the value of the temperature obtained from the thermal energy excess estimates. The 26 ps spectral evolution time is also in good agreement with typical cooling rates of the molecules in solutions. As discussed above, the model which allows us to understand the redistribution kinetics at 77 K based on the heating/cooling
Gulbinas et al. processes of the Bchl a molecules is as follows. Two stages of the heating process can be distinguished. The first one determines the heating of the Bchl a molecules during the excitation equilibration through the excitonic states. During this stage, which last ∼1 ps, the temperature of all seven pigment molecules in the monomer increase by up to 10 K. The second stage determines the heating stimulated by the singlet-singlet annihilation. During this stage, which lasts about 10 ps, the Bchl a No. 7 molecules that are involved in the nonlinear annihilation are heated up to 300-500 K. Heating rates are comparable to excitation pulse duration; therefore, the largest heat values are achieved already during the excitation pulse action. Further redistribution of the thermal energy to the nearest protein surrounding is defined by the mean time of 26 ps, when the local temperature of the Bchl a No. 7 molecule and that of its nearest surrounding reach the same values, which could be estimated as being equal to about 150 K. This local temperature relaxes within the nanosecond time scale. A more detailed analysis of the local heating by singlet-singlet annihilation and corresponding relaxation processes was previously established.25,26 It is also noteworthy that the kinetics on the order of a few hundreds picoseconds was observed at room temperature at high excitation intensities using 25 ps pulses of a Nd:YAG laser (results not shown). This result could be considered as the evidence of the local heating at room temperature. The local heating/cooling process was considered by analyzing room temperature femtosecond energy equilibration in FMO complexes as well;22 nevertheless, local heating influence at room temperature is not so evident. The heating can influence the difference absorption spectrum as follows: (i) it can change the ground state absorption, and (ii) it can change the distribution of exciton states population. So far we considered only the population changes; nevertheless, the ground state absorption changes also have to be taken into consideration by analyzing the experimental data.25 Qualitatively it is evident that the heating should lead to the broadening of the absorption bands and, consequently, to the bleaching at the absorption maxima and the induced absorption at the minima. Estimation of the spectral changes based on comparison of 300 and 77 K spectra shows that, nevertheless, this influence should be smaller than that due to the changes of the exciton states population. Conclusions By analyzing the difference spectrum of the FMO complex at room temperature, the ESA spectrum in the Qy band region was defined as being broad and structureless, while the corresponding cross section was much smaller than that of the ground state. The analysis of the singlet-singlet annihilation and the linear kinetics is based on a few assumptions: (i) all complexes are trimers; (ii) the solution of the FMO complexes is heterogeneous in the sense of linear energy relaxation rates, i.e. part of the complexes are in the neutral-oxidized state and part of them are in the reduced state; (iii) the annihilation rate is independent of the redox state of the protein. Using the model of static annihilation in a trimeric complex, the excitation energy transfer onto the excited monomer was estimated to be slower than the energy transfer rate between monomers. This result is in line with the ratio of the integrals of spectral overlap between the fluorescence spectrum with the ESA and ground state absorption spectra, respectively. At 77 K the absorption and transient spectra are evidently structured and the excited state dynamics is more complicated. The excitation decay kinetics determined by the singlet-singlet annihilation is the same as at room temperature. In addition to
Singlet-Singlet Annihilation in FMO Complexes
J. Phys. Chem., Vol. 100, No. 45, 1996 17955
this, a slower phase of the excitation redistribution between the absorption subbands is present. The kinetic phase is explained by assuming that local heating/cooling processes take place after the excitation pulse. Acknowledgment. The work was supported by the U.S. Department of Energy Contract W-31-109-ENG-38 and by the U.S. Department of Energy Division of Energy Biosciences Grant DE-FG-85ER133388 (to R.E.B.). This is publication 307 from the Arizona State University Center for the Study of Early Events in Photosynthesis. We are also grateful to Dr. F. van Mourik for valuable discussions. Appendix To describe the statistical ensemble of N monomers, we determine the N-particle distribution function fN(r1, r2, ..., rN, t) normalized as follows:
1
∫ fN (br 1,br 2,...,br N, t)∏ dbr i ) 1 N!
(1a)
i
For noninteracting trimers it follows
g3(a,t) )
1 f3(1,2,3) n3(t)
where a is the distance between monomers within the trimer. Thus, for the symmetric trimer and defining λ(a) ) λij, we get the following equations:
∂ n(t) ) 3I(t) - kn(t) - 2γ(t) n2(t) ∂t γ(t) ) λ(a) g(a,t)
(5a)
and
I(t) ∂ g(a,t) ) 2 (1 - g(a,t)) - λ(a) g(a,t) + ∂t n(t) 4γ(t) n(t) g(a,t) - 3λ(a) n(t) g3(t) (6a) where n(t) is the averge number of excitation in a monomer (the concentration of excitation normalized to the monomer) and g(a,t) is the pair-correlation function of two excitations situated in the same trimer. g3(t) is the three-particle correlation function, which corresponds to the following evolution equation:
N/3
r 1, b r 2,...,b r N,t) ) ∏ f3(r1m,r2m,r3m) fN ( b
(2a)
m
where m enumerates the trimer. Assuming detailed balance, a naturally cut chain of kinetic equations with immobile threeparticle distribution functins f3 (1,2,3) can be obtained:
∂ ∂t
f1(i) ) I(t) -
1 τ0
f1(i) + ∑ (wjif1(j) - wijf1(i)) j*i
(λijf2(i,j) + λjif2(i,j)) ∑ j*i
∂ 2 f (i,j) ) I(t)(f1(i) + f1(j)) - f2(i,j) + {wkif2(i,k) + ∂t 2 τ0 wkjf2(k,j) - f2(i,j)(wik + wjk)} - (λij + λji)f2(i,j) (λik + λjk + λki + λkj)f3(1,2,3) (3a) ∂
f3(1,2,3) ) I(t){f2(1,2) + f2(1,3) + f2(2,3)} -
∂t
3 τ0
f3(1,2,3) - ∑ wijf(1,2,3)
n(t) )
g2(a,t) )
3
∑i f1(i)
1 2
6n (t)
∑ f2(i,j) i,j i*j
Equations 5a-7a can be solved numerically. However, some functional time dependence is evident directly from the analysis of eq 6a. After the excitation pulse action, i.e. when I(t) ) 0, and by neglecting the last two terms on the right-hand side of eq 6a, the following analytical function for g(a,t) can be obtained:
g(a,t) ) g0(a)e-λ(a)t
(8a)
The last two terms on the right-hand side of eq 6a describe the higher correlation aspects and are negligible indeed for our intermediate intensities. The initial value g0(a) can be determined from the solution of eq 6a during the excitation pulse action. As it has already shown,15 during the excitation pulse action the higher correlations are weakened, and thus, by assuming the pulse shape being rectangle, this value can be evaluated as follows:
g0(a) )
i,j l*j
where k * i, j; i, j ) 1-3; f1(i) is a single-excitation function; f2(i,j) is the two-excitation function (i and j enumerate the excited monomers within the trimer); τ0 is the linear lifetime; I(t) is the excitation generation function; wij is the excitation energy transfer rate between the ith and jth monomers within the trimer; λij is the excitation energy transfer rate from the excited monomer onto another excited monomer in the trimer; and number m and time dependence of the functions are omitted for clarity. Introducing normalized one-, two-, and three-particle distribution functions for the entire trimer,
1
I(t) I(t) ∂ g3(t) ) g(a,t) - 9 g (t) - 3λ(a) g3(t) + ∂t n(t) n(t) 3 2γ(t) n(t) g3(t) (7a)
I/n I/n + λ(a)
(9a)
Thus, the annihilation rate γ(t) according to eq 5a equals
γ(t) ) γ0(a)e-λ(a)t
(10a)
giving the exponential time dependence. Taking into account that the annihilation process is much faster than the linear decay rates, after the excitation pulse action eq 5a can be rewritten thus:
∂ n(t) ) -2γ0(a)e-λ(a)tn2(t) ∂t
(11a)
and the corresponding solution of eq 11a equals
(4a)
n0
n(t) ) 1 + n0
γ0(a) λ(a)
(12a) -λ(a)t
(1 - e
)
17956 J. Phys. Chem., Vol. 100, No. 45, 1996
Gulbinas et al.
At low initial excitation concentrations eq 12a can be rewritten thus:
(
n(t) ) n0 1 - n0
)
γ0(a) λ(a)
+ n02
λ0(a) -λ(a)t e λ(a)
(13a)
giving a similar result as that obtained from a simplified Poissonian model presented in the Discussion section. References and Notes (1) Van Grondelle, R.; Dekker, J. P.; Gillbro, T.; Sundstro¨m, V. Biochim. Biophys. Acta 1994, 1187, 1 and references therein. (2) Olson, J. M. Biochim. Biophys. Acta 1980, 594, 33. (3) Matthews, B. W.; Fenna, R. E. Acc. Chem. Res. 1980, 13, 390. (4) Tronrud, D. E.; Schmid, M. F.; Olson, J. M. J. Mol. Biol. 1986, 188, 443. (5) Dracheva, S.; Williams, J. C.; Blankenship, R. E. In Research in Photosynthesis; Murata, N., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992; Vol. I, p 53. (6) Johnson, S. G.; Small, G. J. J. Phys. Chem. 1991, 95, 471. (7) Van Mourik, F.; Verwijst, R. R.; Mulder, J. M.; Van Grondelle, R. J. Phys. Chem. 1994, 98, 10307. (8) Van Mourik, F. Ph.D. Thesis, Vrije Universiteit, Amsterdam, 1993. (9) Pearlstein, R. M. Photosynth. Res. 1992, 31, 213. (10) Lu, X. Y.; Pearlstein, R. M. Photochem. Photobiol. 1993, 57, 86.
(11) Tars, M.; Kukk, P.; Freiberg, A.; Miller, M. Private communication. (12) Zhou, W.; Lo Bruto, R.; Lin, S.; Blankenship, R. E. Photosynth. Res. 1994, 41, 89. (13) Blankenship, R. E.; Cheng, P.; Causgrove, T. P.; Brune, D. C.; Wang, H.-H.; Chon, J.-U.; Wang, J. Photochem. Photobiol. 1993, 57, 103. (14) Van Grondelle, R. Biochim. Biophys. Acta 1985, 811, 147. (15) Valkunas, L.; Trinkunas, G.; Liuolia, V.; Van Grondelle, R. Biophys. J. 1995, 69, 1117. (16) Monger, T. G.; Parson, W. W. Biochim. Biophys. Acta 1977, 460, 393. (17) Valkunas, L.; Liuolia, V.; Freiberg, A. Photosynth. Res. 1991, 27, 83. (18) Van Grondelle, R.; Hunter, C. N.; Bakker, J. G. C.; Kramer, H. J. M. Biochim. Biophys. Acta 1983, 723, 30. (19) Savikhin, S.; Zhou, W.; Blankenship, R. E.; Struve, W. S. Biophys. J. 1994, 66, 110. (20) Becker, M.; Nagarajan, V.; Parson, W. W. J. Am. Chem. Soc. 1991, 113, 6840. (21) Van Amerongen, H.; Struve, W. S. J. Phys. Chem. 1991, 95, 9020. (22) Savikhin, S.; Struve, W. S. Biochemistry 1994, 33, 11200. (23) CRC Handbook of Chemistry and Physics, 72 ed.; Linde, D. R., Ed.; CRC Press: Boca Raton, FL, 1992. (24) Elsaesser, T.; Kaiser, W. Annu. ReV. Phys. Chem. 1991, 42, 83. (25) Gulbinas, V.; Valkunas, L.; Gadonas, R. Lithuanian J. Phys. 1994, 34, 348. (26) Gulbinas, V.; Zebrauskiene, A. Lithuanian J. Phys. 1992, 32, 93.
JP961272K
