Photodynamics of Light Harvesting Systems
Ruijter, Ward Piet Frans de
Citation
Ruijter, W. P. F. de. (2005, November 7). Photodynamics of Light Harvesting
Systems. Retrieved from https://hdl.handle.net/1887/4332
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The photodynamics of light-harvesting (LH) systems lie at the basis of photosynthesis. They de-termine the light-absorption properties and the rapid and efficient energy transfer that are essential for the photosynthetic process. The efficiency of the processes in the LH systems is especially im-portant when a photosynthetic organism is grown at low light intensities. A detailed understanding of the LH processes will elucidate the foundations of nature’s machinery.
In this thesis, the results are presented of single-particle spectroscopy at low temperature on four different systems. With this technique, it is possible to observe the details and heterogeneity of the spectra of the individual LH systems. These parameters are otherwise hidden by the averaging ef-fect of an ensemble measurement.
Fluorescence-excitation spectroscopy on LH4 is presented in Chapter 2 along with structure-based simulations. The experimental spectra are reproduced by structure-based simulations which con-firm the structure model of the complex. Predictions are made on the localization of the excited state of LH4. In Chapter 3, the fluorescence-emission characteristics of LH2 are compared to those of LH4. It is shown that spectral diffusion of the emitting state plays a key role in explaining the width of the spectral features of the individual complexes. It is shown that the large structural dif-ferences between LH2 and LH4 can explain the variation in fluorescence bandwidth that is ob-served. For the fluorescence spectra of LH3, investigated in Chapter 4, increased spectral diffusion can be due to the lack of an H-bond as compared to LH2. In the fifth and final chapter, the largest known LH antennae – the chlorosomes – are studied. Heterogeneity between chlorosomes is shown by size differences as measured by AFM and optical spectra of individual chlorosomes. A relation between the heterogeneities is proposed. A comparison with theoretical calculations is made that il-lustrates the heterogeneity of the internal organization of the chlorosome pigments.
This introduction provides an overview of the components of the bacterial photosynthetic unit in section 1.1. The functional as well as structural aspects of LH2, LH3, LH4 and of chlorosomes are outlined. In section 1.2, the experimental techniques that have been used to study the LH systems are described. The final section of the introduction deals with the theoretical basis that has been used for the analysis of the experimental data.
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The largest part of the terrestrial photosynthetic activity takes place in green plants. However, our planet’s surface is covered for nearly 75 % by oceans that contain a large diversity of marine phytoplankton. Although the organisms in the phytoplankton constitute only 0.2 % of the global biomass, they account for close to 50 % of the net primary production of biomass (Field et al., 1998). Photosynthetic bacteria contribute to this production and therefore a study of their photosyn-thetic properties can provide insights into their efficient and effective production.
The photosynthetic apparatus typically consists of light-harvesting antennae and a reaction center (RC). The size and composition of the photosynthetic apparatus differs between plants and bacteria. It is advantageous to study the bacterial photosynthetic processes over those of plants due to their relative simplicity1. For the purple bacteria, the antennae consist of circular LH complexes, while
1 An exception are the cyanobacteria that have a photosynthetic mechanism which is essentially the same as that
for green sulfur and green filamentous bacteria they are formed by the vesicular chlorosomes. These antennae are described in the following two paragraphs.
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The photosynthetic unit (PSU) of bacteria consists of the RC and the neighboring LH complexes. Purple bacteria have only one type of reaction center which is enclosed by the main LH complex, LH1 (Roszak et al., 2003). Additional peripheral LH complexes surround the RC-LH1 complex in order to enhance energy collection for the photosynthetic reactions, especially when light intensi-ties are relatively low. They transfer energy toward the reaction center very rapidly at timescales of the order of femtoseconds for intra-complex energy transfer and of picoseconds in the case of inter-complex transfer (van Grondelle et al., 1994). The type of LH inter-complexes that are synthesized de-pends on the species involved as well as on its growth conditions (Gardiner et al., 1993). Three types of peripheral LH complexes have been discovered and crystallized in the last few decades. They are referred to as LH2, LH3 and LH4 (Hartigan et al., 2002; McDermott et al., 1995; McLuskey et al., 2001). By expressing these different antenna complexes in significant quantities, the bacterium increases the effective absorption cross-section of the RC. In this way, the size of the PSU can range from approximately 30 bacteriochlorophylls (Bchls) to more than 300 Bchls per RC (Drews, 1985; Sistrom, 1978). In addition, the peripheral LH complexes each have their own spe-cific absorption wavelengths, thus optimizing absorption of the available wavelengths in the light spectrum. The bacteria thereby enlarge their chances for survival in relatively dark environments.
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The crystal structure of the LH3 complex has been published recently (McLuskey et al., 2001). Its structure is very similar to the LH2 complex of the previous paragraph. However, its absorption and emission spectrum differ from LH2, and therefore LH3 may be called a ‘spectroscopic variant’ of the LH2 complex. The main NIR absorption bands of LH3 lie at 800 nm and at 820 nm while fluorescence is emitted around 875 nm (Angerhofer et al., 1986). Fluorescence is thus emitted close to the absorption band of the LH1 Bchls, just like in the case of the LH2 emission. However, the energy difference between the red-most absorption of the LH3 complex and the absorption of the LH1 complex is larger than for LH2. Calculations on the kinetics of excitation migration and trap-ping in the photosynthetic unit have shown that this causes an increase in the energy transfer effi-ciency from LH3 to the RC-LH1 complex by 3-6 % (Ritz et al., 2001). This finding is supported by the increased expression of LH3 at very low light conditions (Gardiner et al., 1993). This complex therefore allows a bacterium to sustain itself at even lower light intensities or less spectral availabil-ity than when it only synthesizes LH2 as peripheral LH complex.
Just as in the case of LH2, most investigations have focused on the absorption properties of the complex. Single-complex experiments by Ketelaars et al. (2005) have indicated that spectral diffu-sion of the exciton states is light-induced. Furthermore, this spectral diffudiffu-sion occurs more often in LH3 than in LH2. Spectral diffusion of the lowest exciton state is likely to influence the fluores-cence emission properties of LH3. Extensive fluoresfluores-cence measurements and simulations on indi-vidual LH3 complexes that show this influence are presented in Chapter 4. The chapter provides a comparison between the results on LH2 and LH3 and elaborates on the influence of small structural differences on spectral and energetic behavior.
Organization of the Bchl pigments in the LH2 complex. The phytol chains of the pigments are
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The LH4 complex is very different from LH2 and LH3. It was discovered in the photosynthetic bacterium 5KRGRSVHXGRPRQDVSDOXVWULV due to its distinct spectral characteristics. In contrast to LH2 and LH3, it has only a single absorption band in the NIR, located around 800 nm. Genetic evidence for its existence has been provided (Larimer et al., 2004) and its structure model is cur-rently based on the 7.5 Å electron density map, which is displayed in Fig. 2 (Hartigan et al., 2002). From biochemical analysis, it was concluded that the LH4 complex contains 32 Bchl Dpigments (Hartigan et al., 2002). Their positions have been modeled in an arrangement as indicated in Fig. 3. From this figure it is clear that there are two large differences with the two previously described LH complexes. First of all, there is an additional pigment ring, the B800-2 ring, which is located at an intermediate position between the other two rings, that are called the B800-1 and B-D/B-E ring in LH4. Secondly, the pigments in the densest ring, the B-D/B-Ering, are oriented in a radial fashion rather than the tangential orientation of LH2 and LH3. Although the structure model is defined in a very precise manner, the resolution of the electron density map of the complex is insufficient for an exact determination of the positions and orientations of the pigments. The model was refined by simulating the absorption and CD spectra and comparing the calculated spectra with the data ob-tained from the purified complex (Hartigan et al., 2002). However, there is still room for discussion on the validity of the structure model. The experiments and structubased simulations that are re-ported in Chapter 2 have been performed in order to test the structural model and to investigate the excitonic behavior of the complex. Further evidence is provided to support the structural model, and it appears that the subunits play a dominant role in the structure of the exciton manifold. This is corroborated by the fluorescence experiments and simulations of Chapter 3.
Organization of the Bchl pigments in the LH4 complex. The phytol chains of the pigments
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Chlorosomes are the light-harvesting antennae of green sulfur and green filamentous bacteria. They are vesicular bodies that contain many more Bchl pigments than the LH complexes of the purple bacteria. When chlorosomes were first observed in 1964, they were named ‘chlorobium vesicles’ (Cohen-Bazire et al., 1964). At a Gordon conference, it was decided that this name was inappropri-ate and the term ‘chlorosome’ was coined (Staehelin et al., 1978). Chlorosome (FOZURVZPD) means ‘green body’ , which refers to the vesicular nature and color of the chlorosomes. They are es-timated to contain 200,000 Bchl F pigments and 20,000 carotenoid molecules. This large number of pigments allows them to sustain a bacterial metabolism even when each Bchl absorbs a photon only once every 8 hours (Overmann et al., 1992). The majority of studies on the organization of the Bchl F pigments assumes that they are organized in rod-like structures where they are held together by their non-covalent interactions (Holzwarth and Schaffner, 1994), see Fig. 4. No proteins are in-volved. Recently, however, an alternative model has been proposed in which the pigments are ar-ranged in lamellae that grow from the baseplate (Pšencík et al., 2004). The prevailing evidence that has been obtained by NMR (de Boer et al., 2004), vibrational spectroscopy (Hildebrandt et al., 1994) and aggregation experiments (Huber et al., 2005) supports the rod model, which will be used for evaluation of the results in Chapter 5.
The overall arrangement of the rods and the chlorosome with respect to the cytoplasmic membrane, in which the RC is situated, is shown in Fig. 5. The chlorosome is supposedly attached to the mem-brane by the Fenna-Mathews-Olson (FMO) complex. The arrows in Fig. 5 indicate the direction of energy transfer. Energy from light is absorbed by the Bchl F and the carotenoid molecules in the chlorosome. From there it is transferred to the reaction center via the Bchl D containing baseplate of the chlorosome and the FMO complex and ultimately to the RC. In the RC, charge separation takes place, which drives the chemical pathway of photosynthesis.
Electron densities as derived from the LH4 x-ray data. (A) Top view of the complex. (B) View of the complex in the membrane plane. The center of the complex is on the right and the -terminus is at the
Chlorosomes are fou nd in the green sulfur and the green filamentous bacteria (Olson, 1998). In this thesis, chlorosomes from both types of bacteria have been studied. The green sulfur bacteria are represented by &KORURELXPWHSLGXP and 3URVWKHFRFKORULVDHVWXDULL, and the green filamentous bac-teria by &KORURIOH[XVDXUDQWLDFXV. The habitats of &WHSLGXPand&IODXUDQWLDFXV are the extreme environments of hot spring effluents at temperatures from 50-70ºC (Madigan, 2003; Pierson and Castenholz, 1974; Wahlund et al., 1991). A different environment, being hypersaline marine, is in-habited by 3WFDHVWXDULL (Gorlenko, 1970; Pierson and Castenholz, 1974).
In Chapter 5 an investigation of individual chlorosomes is presented. Optical experiments have been performed in order to be able to compare the spectra of single chlorosomes with that of an en-semble (van Amerongen et al., 1988). Furthermore, it is possible to test the predictions that have been made by theoretical calculations about the optical spectra (Didraga et al., 2002; Didraga and Knoester, 2004; Prokhorenko et al., 2003). Finally, the optical experiments are combined with atomic force microscopy (AFM) measurements to determine the heterogeneity of the sizes of chlo-rosomes and how this influences the optical spectra.
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Spectroscopy on a large ensemble of molecules, as is performed in bulk spectroscopic techniques, is not able to unravel the contributions of the individual molecules. It provides only average values for their properties. However, these properties are not identical for every molecule but are distrib-uted over a range of possible values. Single-molecule spectroscopy is the technique that can assess the distributions (Moerner and Kador, 1989; Moerner and Orrit, 1999; Orrit and Bernard, 1990). The heterogeneity of the distributions unveils details of local parameters such as the uniformity of the overall structure of the individual particles (Chapter 5) and the involved pigment-protein inter-actions (Chapters 2-5).
The heterogeneity in light harvesting particles has two origins, static and dynamic disorder. Dy-namic disorder is caused by interactions of the individual particles with the environment. At room temperature, the spectra of the individual particles are broadened by fluctuations in the environment
Graphical representation of the double-walled rod-like model for Bchl organization in the chlo-rosomes of ! . The top part of the figure displays the arrangement of the Bchls for a single rod. In
due to solvent dynamics (Tamarat et al., 2000). At low temperature these dynamics are frozen, and a snapshot of the conformations at room temperature can be studied. The experiments in this thesis have all been performed at low temperature and thus provide observations of the static het-erogeneity between the particles. The static disorder is observed as variations in the band shape and position of the LH2, LH3 and LH4 fluorescence spectra (Chapters 3 & 4) as well as changes in the reduced linear dichroism of chlorosomes (Chapter 5).
In order to measure these properties, a number of techniques need to be combined in a single ex-perimental setup. We perform confocal laser microscopy, detect the weak signals of single mole-cules, and utilize two types of spectroscopy. The first requires scanning of the wavelength of laser
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Model of the PSU of green sulfur and green filamentous bacteria. The elliptical structure at the top part of the figure is the chlorosome with its boundary of a single layer of lipids. The chlorosome is viewed at a section perpendicular to its long axis. The interior of the chlorosome is filled with rod elements, which are the circular objects in the figure. The Bchl- $containing baseplate of the chlorosome is located at
excitation in order to obtain an excitation spectrum, while the second, fluorescence emission spec-troscopy, needs spectral dispersion of the emitted light. In addition, cryogenic techniques are used to cool the sample. Finally, automation of the selection of molecules and the acquisition of their spectra enables rapid accumulation of data, but it also increases the reproducibility and objectivity of the experiment.
The home-built setup that combines all these techniques is depicted schematically in Fig. 6. The solid and dotted black trajectories in the figure correspond to excitation light, while the grey trajec-tory depicts the path of the fluorescence that is emitted by the sample. The solid black trajectrajec-tory corresponds to the confocal excitation beam path and the dotted line depicts the wide field excita-tion path. Switching between the two configuraexcita-tions is facilitated by a moveable mirror in the exci-tation path. In the wide-field beam path, residual fluorescence from the laser is removed by a short-pass filter after which the light is focused to an approximately 100 Pm large spot on the sample by a planoconvex lens with a large focal length (f = 140 mm). In this way a large number of molecules are excited simultaneously. The fluorescence that is emitted by these molecules is collected by a microscope objective with an N.A. of 0.9 (Microthek GmbH, Germany) and directed through a set of band-pass filters (Dr. Hugo Anders, Germany) in order to eliminate the excitation light from the fluorescence beam. Subsequently, it is focused onto a charge-coupled-device (CCD) camera (Roper Scientific, USA) where the fluorescence of the individual particles is recorded as a pattern of bright spots in the image, see Fig. 7. From such an image, the surface coverage of the sample can be
as- &%
sessed and individual molecules can be selected for further study. The latter is performed in the confocal mode of the experimental setup, where excitation follows the solid black trajectory in Fig. 6. In this mode, the beam is directed towards the sample by the combination of a beam splitter and a scanning mirror. The scanning mirror can accurately move the position of the laser spot on the sample (steps smaller than the diffraction limit). Telecentric lenses are placed between the scanner mirror and the objective to minimize the displacement of the beam at the opening of the microscope objective. The objective focuses the beam to a diffraction-limited spot on the sample. Fluorescence from the sample is again collected by the microscope objective, passes through the beam splitter and the filters and is directed by a moveable mirror to either the avalanche photodiode (APD) (Perkin Elmer SPCM-AQR-16) for fluorescence-excitation measurements, or to the CCD camera for fluorescence-emission measurements.
The confocal mode of the setup allows for a large reduction of the background by spatial selection of photons. The confocal arrangement limits detection to photons that originate only from the molecules in the focal point. Fig. 8 provides a schematic overview of the principle of the confocal microscope. This figure illustrates that only light that originates from the focus of the objective is projected onto the APD. Light that originates from outside the focus is either projected wide of the APD, or is out of focus so that only a negligible fraction will go through.
The focal volume of the microscope objective is not infinitely small, but is limited by diffraction. It therefore has dimensions that are larger than the single particles under study. All particles in this study were incorporated into thin films on a substrate surface. The thickness of these films was less than 1 Pm, which is slightly smaller than the diffraction limit of the focus in the longitudinal direc-tion of the incident beam. The confocal volume thus extends throughout the sample depth. As a consequence the dimensions of the confocal volume in the sample plane determine the resolution. For an ideal microscope objective, the intensity profile of the focal plane has the shape of an Airy
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disk (see e.g. Lauterborn et al. (1995)). The full width at half maximum (FWHM) of this disk is de-termined by the numerical aperture 1$of the microscope objective and the wavelength O of the in-cident light. The FWHM of the Airy disk defines the resolution U of the microscope, which is equal to:
0.61
U
1$
O
(1.1)Values of 800 nm and 0.9 for O and the NA, respectively, are parameters that are typical for the ex-periments. Inserting these values into Eq. 1.1 leads to a resolution of 542 nm. Since all of the stud-ied particles are much smaller than the resolution of the microscope, it is important to have a sam-ple which is diluted sufficiently to ensure optical separation of the particles in the film. In order to achieve this, the concentration of our samples has typically been around 10-11 M prior to deposition
on the substrate.
After selection of an optically separated individual particle, a spectrum is measured by polarization-dependent fluorescence-excitation spectroscopy or by fluorescence-emission spectroscopy. A fluo-rescence-excitation spectrum is obtained by scanning the wavelength of excitation and detecting the red-shifted fluorescence. In order to obtain information on the orientation of the transition-dipole
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moments of the excited states of the molecules, the polarization of the excitation light is rotated (see Chapters 2 & 5). Experimentally, this is achieved by the insertion of a ½O plate in the excita-tion beam. The ½O plate is rotated by steps of 3.6º after each scan, so the details of the bands in the spectrum can be studied over a large range of polarization angles. Typically 200 spectra were taken consecutively, one for each rotation step.
Complementary to the information that is obtained by fluorescence-excitation spectroscopy, the fluorescence-emission spectrum reports on the characteristics of the lowest excited state. In Chap-ters 3 & 4 we will see that heterogeneity of the fluorescence spectra of LH complexes reports on the sensitivity of the complexes to environmental conditions. Selection and excitation of LH sys-tems for fluorescence-emission spectra is conducted with the same procedure as in the fluores-cence-excitation experiments. There are, however, two differences. First of all, the excitation wave-length is held constant for the fluorescence-emission spectra. Secondly, the detection beam path is altered. This beam path is drawn in Fig. 6 as a grey line. The fluorescence which is emitted by the systems under study is directed onto a grating (235 lines/mm, Richardson Gratings, USA) which re-flects and disperses the incident light. Interference maxima occur at the angles - according to the formula which is known as the grating equation:
1
sin
Q
sin
L
G
O
-
+§
·
¨
¸
©
¹
(1.2)where Ois the wavelength, L is the angle of the incident beam with respect to the normal of the grat-ing surface, the order of the interference maximum is indicated by Q, and Gis the distance between the lines on the grating. In the setup, Q, Gand L are constant, thus the angle - is only dependent on the incident wavelength. The resolution of the grating is solely determined by the number of illu-minated grating lines 1 according to the formula:
U
1
O
(1.3)In the case of the fluorescence-emission experiments in Chapters 3 and 4 the spectral resolution was 2 nm. The dispersed photons are projected onto the CCD camera by a lens (f = 50 mm, Ed-mund Optics, USA) of which the focal length is chosen such that the fluorescence spectra are fo-cused into a narrow line on the CCD camera, see Fig. 9. The wavelength is calibrated by tuning the laser wavelength into the transmitting part of the detection filters and determining the position of the laser focus on the CCD chip.
fluorescence emission spectrum is thus always a trade-off between high resolution and short inte-gration times.
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The spectra that are obtained by single molecule spectroscopy (SMS) contain significant informa-tion on the excited states of the studied particles. However, their interpretainforma-tion is usually not straightforward and hence simulations of the observed spectra become necessary to test the predic-tions that have been made, based on structural models. Both the fluorescence-excitation as well as the fluorescence-emission spectra have been simulated.
The spectrum of an LH system depends largely on the type of pigment molecules that are incorpo-rated and on their positioning and orientation in the protein scaffold. The most abundant pigment molecule in the LH complexes is bacteriochlorophyll D (Bchl D) in the case of purple bacteria and Bchl F in the chlorosomes. In solution, Bchl D has a strong absorption in the near infrared around 770 nm. The absorption at 800 nm of the LH complexes is shifted 30 nm to the red side with re-spect to that of BChl D in solution. The absorption maximum of Bchl F in solution at 660 nm is also at a very different spectral location than the 750 nm absorption band of the chlorosomes of & WHSLGXP (Hoff and Amesz, 1991). These large differences have two origins. The first is the value of the so-called site energy of the individual pigments, which is the energy at which the pigment would absorb if intermolecular interactions were absent. The site energy is affected by specific pigment-protein interactions in the binding pocket. The second cause for the observed red-shifts is the occurrence of strong pigment-pigment interactions that lead to excitonic coupling. Due to this dipole-dipole coupling, the excited state becomes delocalized over multiple pigments. Therefore, the individual excited states are no longer associated with specific pigments but form a so-called
800 850 900 950 0 5
Fl
uo
re
sc
en
ce
(A
.U
.)
Wavelength (nm)
#,exciton manifold. The distribution of energy levels in this manifold depends on the strength of the interactions; a stronger interaction is accompanied by a wider spread. In addition, the oscillator strength is redistributed and hence, depending on symmetry, the oscillator strength concentrates in only a few exciton states. Direct evidence of this mechanism is observed for the LH2 complex. In this complex, the absorption at 800 nm is caused by absorptions of Bchls with weak interactions while the absorption at 850 nm is caused by Bchls with strong interactions.
The Bchls in the LH complexes and chlorosomes are in most cases arranged in sufficiently close proximity to each other that their interaction energy plays an important role in the electronic struc-ture of the systems. In the NIR, the electronic strucstruc-ture is dominated by the so-called Qy transition
dipole moment of Bchl D and F. The interaction energy 9(
5
*
) of two dipoles with dipole moments
-P* and .
/0 at a relative distance
5
*
(unit vector
5ˆ
) in a medium with dielectric constant H is given by: 3ˆ
ˆ
3(
)(
)
1
( )
4
132 15
25
9 5
5
P P
P
P
SH
*
*
*
*
*
(1.4)This equation determines the off-diagonal elements of the Hamiltonian in the point-dipole ap-proximation. This is a relatively crude approximation since it treats the dipole moment of each Bchl as a point. When the pigments are located very close to each other, the point-dipole approximation is not valid and the so-called point-monopole approximation may be applied (Philipson et al., 1971; Weiss, 1972). In this approximation the dipole is considered to consist of two (or more) monopoles, representing the Qy transition dipole in the case of Bchl pigments. The Hamiltonians in the present
4 5
Schematic drawing of the effect of strong electrostatic interactions and exciton delocalization on the excited states of a dimer. In the absence of interactions the states 6 and 7 are localized on the
in-dividual molecules (left side of the figure). In the case of strong interactions 8!9:, the excitation is
study were obtained by the point-dipole approximation for LH3 (Ketelaars et al., 2005) and by the point monopole approximation for LH2 and LH4 (Hartigan et al., 2002).
Both the interaction energies and the site energies need to be taken into account in simulations of the experimental spectra. For the simulations, the excited-state properties of the LH systems can be described by the following Hamiltonian (+):
< => < =?>
+
( Q Q
9 L M
@¦
¦
(1.5)In this formula, (A are the site energies of the individual pigments and 9BC are the interactions
be-tween the pigments. Notice that the site energies constitute the diagonal entries of the Hamiltonian while the interactions are located in the off-diagonal elements. Diagonalization of the Hamiltonian provides the energy levels of the electronic transitions of the involved pigments. When the interac-tions are sufficiently strong, the excited states become delocalized over the pigments and are called exciton states (van Amerongen et al., 2000). Consequently, the wave function D of the excited
state is a combination of the wave functions of the individual pigments. This is schematically indi-cated in Fig. 10 for the case of two interacting pigments with wave functions E and F . The
fig-ure shows that the lowest state of the exciton manifold has lower energy than the site energies of the individual pigments, as has been mentioned above.
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In Chapter 2 the results are presented of fluorescence excitation spectroscopy on individual LH4 complexes. Those spectra have been simulated using the procedures as described in this section.
In the first step a Hamiltonian matrix was calculated for the LH4 complex, by applying a point-monopole approximation to the structure model that has been published by Hartigan et al. (2002). Next, this Hamiltonian was diagonalized in order to obtain the eigenvalues, which are the energy levels of the exciton manifold, as well as the eigenvectors. From these data, the strength of the tran-sition dipole moment (1)
G
% of the N-th exciton state can be calculated according to the formula:
2 2 (1) (0) (1) 2 1 1 2 2
ˆ
ˆ
Hˆ
Hˆ
Hˆ
H H IKJML NON%
F P \
P
F
P
F
P
F
P
(1.6) in whichF
(0) and (1) P\ are the ground and single-exciton state, respectively, Pˆˆ is the dipole mo-ment of the full complex, PQRS is the monopole moment of an individual Bchl pigment,
F
TU is theeigenvector component of the O-th pigment from the eigenvector corresponding to the N-th exciton state and, finally,
P
ˆ
V is the unit-dipole vector of the O-th pigment. Now that we have the energyho-mogeneous line shape. We have approximated this line shape by a Lorentzian. This approximation induces small differences with the actual line shape of a single molecule (Jang and Silbey, 2003), but the approximation is sufficient for the purposes in this thesis. The area of the Lorentzians is proportional to the dipole strengths and their width is determined by the lifetime of the associated exciton state. The lifetime is the reciprocal of the energy transfer rate,
*
W , which can be calculatedaccording to a formula which has been derived by Leegwater et al. (1997). We only consider downhill energy transfer since all experiments have been performed at low temperature (1.2 K), which simplifies the original equation of Leegwater et al. to:
2 2 ,
2
(
)
X Y Z Z X [ [ [ XF
F
J Z Z
*
¦
(1.7)In this formula,
J Z Z
(
\ ])
accounts for energy relaxation through vibronic coupling of theexci-ton state O with the surrounding medium. We have utilized the empirical function for
J Z
( )
that has been described by Vulto et al. (1999).The spectrum that we can calculate at this point is the spectrum of the complex in one specific rep-resentation of local disorder representing a particular conformation. However, every complex is slightly different in reality, including alterations of the binding pockets of the pigments. In the simulations this is represented by variations of the site energies of the individual pigments. These variations are approximated by the addition of a Gaussian random distribution to the site energy values. Just as the experimental bulk spectrum is the result of the contributions of a large number of complexes in different conformations, the simulated bulk spectrum is constructed by summing a large number of simulated single-complex spectra with added disorder.
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The fluorescence-emission spectra in Chapters 3 & 4 have been simulated with the assumption that at low temperatures fluorescence is always emitted from the lowest exciton state, the k = 0 state. The energy of the lowest state can be determined by diagonalization of the Hamiltonian as has been described in the previous section. The emitted fluorescence consists of two contributions: photons that are emitted by transitions between vibrationless states constitute the so-called zero phonon line (ZPL) and photons that involve vibronic states of the pigment-protein matrix give rise to the asso-ciated phonon side-band (PSB) (Toutounji and Small, 1999). Dressing of the individual k = 0 sticks now consists of two Lorentzian contributions that represent the ZPL and the PSB. The width of the ZPL is determined by the lifetime of the N = 0 state and the dephasing time of the excitation. Hole-burning, pump-probe as well as single-molecule experiments on LH2 have shown that the ZPL has a width of approximately 5 cm-1
while
for the PSB a width of around 35 cm-1 was determined(Small, 1995; Wendling et al., 2003). More details on these contributions are provided in Chapter 4. Gaussian random disorder was added to the diagonal elements of the Hamiltonian (the site ener-gies) to simulate the conformational variations. In order to reproduce the experimental summed single-molecule and bulk spectra, a large number of simulated spectra (typically 105) were
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