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* A. N. Belozersky Institute of Physico-Chemical Biology, Moscow State University, 119992 Moscow, Russia; and Department of Biophysics and Physics of Complex Systems, Division of Physics and Astronomy, Faculty of Sciences, Vrije Universiteit, 1081 HV Amsterdam, The Netherlands
Correspondence: Address reprint requests to Danielis Rutkauskas, Fax: 31-20-5987999; E-mail: danielis{at}nat.vu.nl.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORYRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORYRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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). These initial and ultrafast events have been extensively investigated in photosynthetic purple bacteria. The bacterial light-harvesting complexes comprise a number of packed pigments circularly arranged in a protein scaffold. The peripheral light-harvesting complex 2 (LH2) from Rhodopseudomonas (Rps.) acidophila, whose crystal structure has been determined with high resolution (2,3), is a highly symmetric ring of nine pigment-protein subunits, each containing two transmembrane polypeptide helixes and three bacteriochlorophyll (BChl) molecules, with the 
-polypeptide on the inner and the ß-polypeptide on the outer side of the ring. The hydrophobic terminal of the protein binds a ring of 18 tightly coupled BChl molecules with a center-to-center distance of <1 nm between neighboring pigments. This ring is responsible for the intense absorption of LH2 peaking at 
850 nm (B850 ring). A second ring of nine weakly interacting BChls is located in the polar region of the protein and is responsible for the absorption at 800 nm (B800 ring).
From nonlinear spectroscopic studies a consistent physical picture of the excitation dynamics in the B850 ring has emerged (4–6). All basic spectroscopic features can be understood on the basis of a model that includes both the excitonic coupling between pigments and the disorder induced by nuclear motion. Strong excitonic interactions between the chromophores within the B850 ring tend to delocalize the excitation over a number of pigments, whereas the nuclear motions (slow conformational changes of the pigment-protein matrix, fast intrapigment vibrations, and phonon modes) break the symmetry of the complex producing more localized states (7–9). Conventional bulk experiments reveal the details of excitonic relaxation and excitation energy transfer dynamics on a femtosecond/picosecond timescale, but these observations and conclusions are associated with an averaging over a large number of complexes with different realizations of the energetic disorder, which is manifested as inhomogeneous spectral line broadening. On the other hand, although restricted to visualization of dynamics on millisecond/second timescale, single-molecule experiments circumvent this ensemble averaging.
Based on room- and low-temperature single-molecule experiments it has been proposed that the LH2 ring can deviate from the ideally circular structure (10–13). Room-temperature polarized fluorescence (FL) experiments were interpreted in terms of an elliptical absorber and emitter with ellipticity and directions of the principal axis varying as a result of the B800 and/or B850 distortion destroying the rotational symmetry and traveling around the ring on a timescale of seconds (10). The anomalously large splitting of the two major orthogonal excitonic transitions observed in low-temperature polarized FL excitation spectra was attributed to a modulation of the coupling strength in the B850 ring that was asserted to be associated with an elliptical deformation (11–13). Spectral fluctuations of different magnitude observed on different timescales were associated with a hierarchical structure of the protein conformational landscape (14). Smaller structural ellipticities were found for loosely packed LH2s in membranes, and were probably partly due to an interplay of the disrupting tip and stabilizing lipid environment (15).
In general, the observed variation of the spectral and functional properties of LH2 suggests that the complex can undergo a variety of deformations or evolve through a number of conformational substates. It is reasonable to assume that such conformational substates or in other words realizations of nuclear coordinates underlie the pattern of the static disorder of pigment site energies and interaction energies between pigments that play a key role in our current understanding of the spectroscopic and energy transfer properties of LH2. In a recent experiment (16,17) we have observed a spectral evolution of LH2 between different conformations at room temperature occurring on a second timescale (corresponding to slow nuclear motions) and manifested by abrupt movements of the fluorescence peak wavelength of individual LH2 complexes between long-lived quasistable levels differing by up to 30 nm. We accounted for these spectral fluctuations on the basis of the modified Redfield theory by modeling the FL profiles for different realizations of the static disorder of the pigment transition energies.
In this article we study in greater detail the mechanism of how different realizations of static disorder are associated with the observed spectral changes of LH2 (16,17). 1), We have found that the experimentally observed spectral fluctuations reflect realizations of the disorder in the B850 ring with different degrees of exciton delocalization and different effective coupling of excitons to phonon modes. 2), The positions and spectral shapes of the main exciton components are determined by the disorder-induced shift of the exciton eigenvalues in combination with a phonon-induced shift and broadening. Being dependent on the realization of the disorder, these factors produce different shapes of the emission profile. 3), In addition, different delocalization and effective coupling to phonons determine a different type of excitation dynamics for each of these realizations. We conclude that experimentally observed quasistable conformational states are characterized by different excitation energy transfer regimes varying from a wavelike motion of a delocalized exciton to a noncoherent hopping of a localized wavepacket and an immobile self-trapped excitation.
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THEORY |
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TOPABSTRACTINTRODUCTIONTHEORYRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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) to calculate spectral responses in the presence of strong exciton-phonon coupling. This method allows for an exact solution of spectral line shapes and excitation energy dynamics in a molecule with a single electronic transition (18). In a generalization for molecular aggregates (19) the line shapes are calculated including explicitly a diagonal coupling of nuclear modes to one-exciton states (inducing fluctuations of the energies of excitonic transitions), whereas off-diagonal coupling (inducing relaxation between the exciton eigenstates) is taken into account perturbatively. Linear spectra, i.e., absorption (OD) and nonselective steady-state fluorescence (FL) can be expressed as:
 | (1) |
k denote the steady-state population, transition dipole moment, and frequency (first moment of the absorption spectrum) of the k-th one-exciton state, gkkkk is the line-broadening function, 
kkkk is the reorganization energy value for the k-th state. The wavefunction amplitude 
(participation of the n-th site in the k-th exciton state) associates the transition dipole of the exciton state with the molecular transition dipole dn. It is important to note that our expressions for the linear spectra take into account a relaxation-induced broadening of the exciton states given by their inverse lifetimes, i.e., Rkkkk. This term was not present in the original theory of Zhang et al. (19), but applications of the theory demonstrated its importance to fit quantitatively the spectra of molecular assemblies such as J-aggregates (20), the FMO complex (21), the PSII reaction center (22–24), bacterial LH2 antenna (16,17), and the LHCII complex from higher plants (25,26). The inverse lifetime of the k-th state is given by a sum of the relaxation rates within the one-exciton manifold (Eq. 1). The rate of the k' 
k population transfer is given by (19):
 | (2) |
kk' = k – k'. The relaxation tensor in the form of Eq. 2 is denoted as the modified Redfield tensor (27) to distinguish it from the standard Redfield tensor obtained in the limit of weak exciton-phonon coupling. The g-functions and -values in Eqs. 1 and 2 are:
 | (3) |
) is the matrix of the spectral densities in the eigenstate (exciton) representation, which reflects coupling of one-exciton states to a manifold of nuclear modes (see next section). In principle, coupling to nuclear motions with arbitrary timescales can be included explicitly in the g-function. But it is convenient to consider separately the action of fast and slow nuclear motions.
Fast modes
Electronic transitions in a pigment molecule (or collective excitonic transitions in a cluster of strongly coupled pigment molecules) are affected by fast intra- and intermolecular vibrations, phonon modes of the protein environment, etc., that is all these factors cause the so-called dynamic disorder. In the frequency range of 10–2000 cm–1 (as revealed by fluorescence line-narrowing (28), hole-burning (29), and molecular dynamic simulations (30)) these modes determine the homogeneous line broadening (represented by gkkkk) of the exciton transitions, a red shift of the zero-phonon lines (ZPL) (which is equal to the reorganization energy value kkkk), and the Stokes shift of the emission maximum of the k-th exciton state (equal to 2kkkk). Equation 3 relates these quantities to the spectral density in the eigenstate basis Ckkkk(). The latter (as well as the Ckk'k''k'''() quantities needed to calculate the relaxation tensor) can be obtained from the matrix of the spectral densities in the site representation Cnmn'm'().
To simplify the matter we assume only a diagonal electron-phonon coupling in the site representation, i.e., nuclear modes induce fluctuations of the pigment site energies without acting on the intermolecular couplings. Generally, there may be some correlations between fluctuations acting on different sites, but we restrict to the simplest case of uncorrelated dynamic disorder. This model implies that each molecule has its own independent thermal bath. We further suppose that the spectral density function of each such bath is site independent, i.e., spectral density in the site representation is Cnmn'm'() = 
nmnn'mm'C(). Transformation to the eigenstate representation yields:
 | (4) |
Equation 4 shows that diagonal phonon coupling in the site representation results in both diagonal and off-diagonal couplings in the exciton basis. To construct the spectral density profile we use the sum of an overdamped Brownian oscillator and resonance contributions due to high-frequency modes:
 | (5) |
is the reorganization energy in the site representation, Sj is the Huang-Rhys factor of the j-th vibrational mode in the site representation. Parameters of the temperature-independent spectral density (frequencies j, couplings j, and damping constants 
j for high-frequency vibrations together with the coupling 0, and damping 0 for Brownian oscillator) are obtained from the fluorescence line-narrowing spectra (FLN), and further adjusted from the simultaneous fit of OD/FL spectra at different temperatures.
Note that according to Eqs. 3–5 the line-broadening functions and reorganization energies of the k-th exciton state in the eigenstate representation (gkkkk and kkkk) are smaller than in the site representation by a factor of 1/
()4 (this factor connects Ckkkk() and C() in Eq. 4). The latter quantity is known as the inverse participation ratio (PR) and is equal to the delocalization length of individual exciton state (31–33). In the literature there exist many other definitions of delocalization length reflecting different aspects of the exciton dynamics, and therefore yielding different "exciton sizes" for the same system. The relation between them has been studied in great detail (7–9).
Slow modes
Slow nuclear motions associated with conformational changes of the pigment protein in a native membrane or when immobilized in a gel, on a mica surface, etc., and occurring on the microsecond to second timescale, result in the so-called static disorder of the pigment electronic transition energies (i.e., the nonequivalence of pigment site energies—transition energy of each pigment in the complex deviates from some average value by an amount specific to that pigment) manifested as inhomogeneous broadening in conventional bulk spectroscopic experiments or the time dependence of fluorescence spectral parameter traces in single-molecule measurements (16,17).
We model different realizations of the static disorder (combinations of pigment site energy deviations from the nondisordered value), corresponding to different complexes measured simultaneously in the bulk, or one complex at different moments in time in single-molecule experiments, by random uncorrelated shifts of the site energies (diagonal disorder). Numerical diagonalization of the one-exciton Hamiltonian yields eigenstate energies k and eigenfunctions , required to calculate OD/FL spectra for one realization of the static disorder. Such calculations with a number of realizations of static disorder allows for a statistical characterization and comparison of the resulting spectra with the results of single-molecule experiment or with the bulk spectra after averaging of spectra calculated for different realizations of the disorder.
Combined action of slow and fast modes
Notice that in the absence of exciton-phonon coupling the k value corresponds to ZPL position of the k-th exciton level. In the weak coupling limit ZPLs are slightly broadened due to bath-induced exciton relaxation but the k-value still corresponds to the ZPL position and coincides with the maximum of the absorption and emission of the k-th state.
In the case of strong coupling with phonons (Eq. 1) the ZPL position is k – kkkk, i.e., red-shifted with respect to the eigenvalues of a free-exciton Hamiltonian k. The first moment of the OD is at k, whereas the first moment of FL is k – 2kkkk. Static disorder induces random shifts of the eigenvalues k. In their turn, the line-broadening function gkkkk (i.e., phonon-induced broadening) and reorganization energy kkkk (together with the Stokes shift of the k-th level, i.e., 2kkkk) are also affected by the static disorder being proportional (see Eqs. 3 and 4) to a disorder-dependent participation ratio PRk = n()4 (PRk value is dependent on the specific realization of the static disorder through the wavefunction amplitudes ). It has been well established (31–33) that an increase in static disorder (given by the full width at half-maximum (FWHM) of the Gaussian distribution from which the pigment site energy shifts are drawn) on the average increases the PRk values for different realizations of the static disorder, thus increasing the effective dynamic disorder value, i.e., the phonon induced broadening of the exciton states and reorganization shift. Furthermore, an increase in static disorder induces a larger spread of the PRk values corresponding to different realizations, thus increasing the spread of the line broadenings and reorganization shifts.
Thus, both the positions of the exciton levels (given by k – kkkk) and the linewidths (determined by gkkkk) depend on a combined action of static and dynamic disorder. In particular, the experimentally measured distribution of the ZPL positions (as obtained from a hole-burning experiment for k = 0, or studied by a single-molecule technique) includes the disorder of the purely exciton eigenvalues k, in combination with the variation of reorganization-induced shifts kkkk. The amount of this additional reorganization-induced disorder increases in proportion to the exciton-phonon coupling strength j and the amount of static disorder that makes the reorganization effects more pronounced in a more disordered system with larger site inhomogeneity.
In the case of the LH1 antenna or the B850 ring from LH2 the static disorder has its most prominent effect on the lowest exciton states, especially k = 0 ((8,33,34), and examples given below). Realizations of strong disorder produce more localized and more red-shifted states. The reorganization energy for these realizations is also larger, inducing a further red shift. Moreover, these red-shifted states feature a more pronounced phonon/vibrational wing (because the line-broadening function gkkkk increases for localized states with bigger PRk value). Thus, the character of the slow conformational changes of the complex significantly influences the effective coupling of the exciton states with the fast nuclear motions (given by Eq. 4), producing different phonon-induced broadening and reorganization energy effects. Thus one can expect a red shift and broadening of the FL spectrum for conformations of the complex with broken symmetry that are associated with a large disorder of the site energies.
Complexes in a more symmetric configuration are expected to conform with the classical excitonic picture (no static disorder). The corresponding spectral profiles are not shifted and as will be shown later are broadened predominantly due to exciton relaxation.
These described spectral trends were observed in our study of fluorescence fluctuations of single LH2 (16,17). Single complex fluorescence spectra exhibited large red and blue shifts occurring on a second timescale that were accompanied by spectral broadening. These observations are interpreted on the basis of the proposed model.
Model of LH2 antenna
The LH2 spectral lineshapes and their dynamics are determined by the coupling of electronic excitations to a manifold of nuclear modes (both fast and slow). Fast nuclear modes define the optical line shapes both in conventional (18) and in single-molecule spectroscopy (35). Slow nuclear motions associated with conformational changes of the pigment protein on a microsecond to second timescale are responsible for the change of realizations of static disorder. It is reasonable to suppose that the experimentally observed quasistable states of single LH2 fluorescence spectral traces, characterized by different line shape and peak position, can be treated as different equilibrium positions of the nuclear coordinates, i.e., different realizations of the static disorder (36,37). We hypothesize that transitions between these states occur due to thermal nuclear motion. We do not consider here the dynamics of such transitions but rather restrict ourselves to a modeling of fluorescence line shapes for different realizations of the static disorder. Our model takes into account excitonic interactions within the B850 ring, the presence of static disorder of pigment site energies, and strong coupling of electronic excitations to phonons.
To construct a one-exciton Hamiltonian, the unperturbed transition energies of BChl 850() and BChl 850(ß) were taken as 12,275 and 12,125 cm–1, respectively. The energies of the 11ß, 1ß2, 12, 1ß2ß, and 12ß pigment-pigment interactions were taken to be 291, 273, –50, –36, and 12 cm–1, respectively, according to Sauer et al. (38). The static disorder was described by uncorrelated perturbations of the transition energies randomly chosen from a Gaussian distribution with the FWHM of 
. Numerical diagonalization of the Hamiltonian (for each realization of the disorder) gives the energies of the exciton states k and the wavefunction amplitudes (participation of the n-th site in the k-th exciton state).
The absorption and fluorescence line shapes are calculated using Eqs. 1–5. To reproduce the peak positions and the low-frequency wings of the OD/FL spectra, it is not necessary to include the high-frequency part of the spectral density function (second term in Eq. 5). Thus, we assume that the spectral density function C() has the form of a single overdamped Brownian oscillator with coupling parameter = 0 and relaxation time 
= 1/0 (keeping just the first term in Eq. 5). In our model and (as well as ) are site-independent parameters adjusted from the fit of the bulk OD and FL spectra. All experimental data presented below were acquired at room temperature and the simulation of the data was performed with the corresponding parameters.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHEORYRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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,17). In this experiment complexes were immobilized by the electrostatic interaction with a glass coverslip treated with the poly-l-lysine, and submerged in an oxygen-free buffer containing detergent. Complexes were excited with linearly polarized laser light at 800 nm spectrum of LH2. The polarization sensitivity of the detection was found to be insignificant and thus no correction was required.
Fig. 1 presents an average of single-molecule FL spectra in time and over particles as well as the distribution of the FL peak positions together with corresponding results of the modeling. The averaged FL profile has a maximum near 870 nm. Realizations with the FL peak near this wavelength occur with the highest probability. The data are reproduced with parameters = 370 cm–1, = 390 cm–1, and = 50 fs. Notice that the OD spectrum of the B850 ring of LH2 complex in vivo can be reproduced with the same and , and = 220 cm–1 (data not shown). Most probably, this difference in reflects the different strength of exciton-phonon coupling for immobilized and in vivo complexes.
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PR
, defined as an average of PRk values of the individual exciton states weighted with the steady-state populations of those states. PR values corresponding to the blue, intermediate, and red-shifted FL positions shown in Fig. 2 are 0.098, 0.118, and 0.245, respectively, reflecting the increasingly localized character of the exciton states contributing to the redder FL spectrum.
Participation ratio for different realizations of static disorder
Fig. 3 shows the PRk = n()4 values for the five lowest exciton levels (i.e., k = 0, –1, 1, –2, and 2) of LH2 calculated for 2000 realizations of the disorder. Each point corresponds to the PRk value of one exciton state for one realization of the static disorder as a function of the wavelength of the ZPL of this state (the position of ZPL is k – kkkk on the energy scale). Averaging of the PR values corresponding to exciton states with ZPLs within narrow wavelength intervals results in a smooth PR curve well known for LH2/LH1 complexes, i.e., almost constant in the middle of the long-wavelength absorption band with an increase on the red side (8,33,34,39). Fig. 4 shows the distributions of PRk values for each of the exciton states taken from the 2000 realizations shown in Fig. 3 corresponding to spectral profiles with peak positions in the ranges 858–862 nm, 869–871 nm, and 887–894 nm. Note that this is a statistical representation of data similar to Fig. 2 although the intervals of blue- and red-shifted spectral profile maxima are wider to collect more occurrences.
The PR values of the exciton states corresponding to blue-shifted FL spectra are 0.085–0.13 for the higher states (k = ±1, ±2) and 0.065–0.12 for the lowest one (k = 0) (Fig. 4). These values are close to the homogeneous limit. That is, for a circular aggregate of N molecules PR = 1/N for the lowest and 3/2N for the higher states in the absence of the static disorder. For LH2 with N = 18 this results in 0.056, and 0.083, respectively. So, we conclude that the exciton states that result from realizations of static disorder corresponding to blue-shifted spectra are not significantly destroyed by the disorder. As a result in this case the exciton structure is also not too different from that of the homogeneous ring, i.e., most of the dipole strength is concentrated in two degenerate levels (k = ±1), whereas the lowest state (k = 0) is almost forbidden due to the symmetry of the ring. The room-temperature emission originates mostly from the degenerate k = ±1 pair giving rise to a blue-shifted FL (Fig. 2, top frame).
FL profiles with an intermediate peak position result from realizations of stronger static disorder (Fig. 4, middle frame). This situation corresponds to an increase in the splitting between the exciton levels and a more localized character of the exciton states yielding PRk values that have increased up to 0.15–0.20 for the k = ±1 and up to 0.30 for the lowest k = 0 state. This red-shifted and more localized k = 0 state also becomes more radiant borrowing some of the dipole strength from higher levels so its contribution to the FL profile becomes more significant (Fig. 2, middle frame).
The bottom frame of Fig. 4 illustrates PRk values of exciton states corresponding to red-shifted FL profiles. In this case the exciton structure is very strongly perturbed by the disorder, which induces a large splitting between the energy levels, in particular, the splitting between the k = ±1 and the lowest k = 0 state is increased significantly (as compared with the blue-shifted and intermediate spectral profiles). Due to this large splitting between excitonic energy levels FL mostly originates from the lowest state, which is now strongly allowed and red-shifted. Its localized character (PR = 0.2–0.4) gives rise to an increased strength of the effective exciton-phonon coupling (proportional to the PRk value). Such an increased coupling results in the broadening of the FL profile together with an additional red shift (Fig. 2, bottom frame).
Relaxation-induced broadening
We have seen that an increase in the red shift (because of the low exciton eigenvalue and reorganization shift) is accompanied by enhanced exciton-phonon coupling in the exciton representation (Eq. 4), which produces an increasingly larger broadening of the FL spectra. Another line-broadening factor that determines the width of the blue-shifted FL spectra is exciton relaxation. Typically due to predominant downhill relaxation the inverse lifetime increases for the higher levels. In our model Rk = 16, 29, 44, and 57 ps–1 for the k = 0, –1, 1, and –2 levels, respectively. The more pronounced relaxation broadening of the k = ±1 levels results in a larger width of the blue-shifted FL spectra (determined mostly by the k = ±1 emission). Notice that in the absence of relaxation a blue shift would be always accompanied by a narrowing of the FL line because the k = ±1 levels are always narrower than the lowest one (for a disordered complex) due to their smaller PR values and, as a result, a smaller value of the line-broadening function. Disregarding exciton relaxation it would be impossible to explain the observed broadening of the blue-shifted FL spectra (Fig. 5).
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Exciton wavefunctions in the site representation
Analysis of the PRk values suggests a different degree of delocalization for each of the three spectral profiles shown in Fig. 2. This can also be illustrated by a straightforward visualization of the corresponding exciton wavefunctions. As an example we selected one typical realization of the static disorder with a blue-shifted FL peak position at 858 nm, an intermediate one at 871 nm, and two red-shifted ones at 894 nm. The corresponding thermally averaged PR values are 0.087, 0.14, 0.25, and 0.35, corresponding to a delocalization over 11, 7, 4, and 3 pigment molecules, respectively. Fig. 6 (left frames) shows the shifts of the transition energies of the sites from n = 1 to n = 18 for these realizations. For the same realizations we show the squared wavefunction amplitudes ()2 for k = 0, i.e., the participation of the n-th site in the lowest k = 0 exciton state (Fig. 6, middle frames). To visualize the shapes of the exciton wavepacket (given by a superposition of the exciton wavefunctions) we calculate the density matrix in the site representation, i.e., 
. In the steady-state limit (after exciton relaxation) 
kk' = kk'Pk, where Pk is the Boltzmann distribution of the exciton populations. In Fig. 6 (right frames) we show the steady-state density matrices at room temperature for the chosen realizations. The distribution in the diagonal direction, i.e., nn shows the area of a noncoherent delocalization of the excitation. The width of the density matrix in the antidiagonal direction gives the coherence length of the exciton (which is always less than the width of the distribution in the diagonal direction).
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5–6 molecules due to thermal mixing of the states. The distribution along the diagonal direction shows no preferred localization site on the ring. So, for the dynamics we can expect the motion of a wavepacket delocalized over 5–6 pigments around the whole ring (as we will show below).
In case of the second realization of static disorder (intermediate FL peak) the site energy shifts are not significantly larger, but they are less correlated. For example, there is a number of neighboring pigments (like n = 4–5, 6–7, 9–10, and 11–12 in this example) with opposite signs of the shift. The difference in energies starts to exceed the coupling M between these sites. For these intermediate realizations the lowest exciton state becomes more localized. The corresponding wavefunction has a pronounced maximum at the red-most pigment, n = 10. In the steady-state limit the wavepacket (with a coherence length of 3–4 molecules) is localized near the sites n = 10 or n = 4.
The third realization (red FL peak) is similar to the second one, but with larger and uncorrelated site shifts that exceed significantly the M value. The k = 0 wavefunction is localized with a main peak at n = 2 and a smaller one at n = 4. The wavepacket is localized near n = 2 with some small coherence between the n = 2 and n = 4 sites.
The forth realization demonstrates yet another scenario of localization. In this case the major part of the pigments exhibits a correlated shift (as in the first realization), but one pigment (n = 4) has a very large red shift. This determines the almost complete localization of the k = 0 state on the n = 4 site and localization of the wavepacket on that same site without sizable coherence with other sites.
Coherent dynamics of the density matrix
The diagonal elements of the steady-state density matrix nn (Fig. 6) display the probability of population of the n-th site of the B850 ring. The population distribution is different for different realizations of the disorder. Thus, a small amount of disorder is characterized by a more or less uniform distribution, i.e., the excitation (coherently delocalized over a few molecules) can be found on any part of the ring. On the other hand, a larger amount of disorder leads to predominant localization of the excitation on a group of pigments, or even on a single site.
We wish to study the dynamics of the excitation for these limiting cases. To this end we calculate the time evolution of the density matrix for the same realizations as in Fig. 6. We restrict to coherent dynamics, i.e., we calculate the motion of the initially prepared wavepacket without taking into account the relaxation of the one-exciton populations and the decay of coherences between one-exciton states. The coherent dynamics in the eigenstate basis is calculated using the Liouville equation for kk'(t) (without including a relaxation tensor), with the initial conditions corresponding to the coherent wavepacket kk'(t = 0) = (Pk Pk')1/2 exp(i
k – ik'), with the steady-state populations kk(t = 0) = Pk and arbitrarily fixed phases k of the exciton states (in principle such a wavepacket could be created using a specially shaped laser pulse). The time evolution of the density matrix in the site representation is 
. Notice that the initial phases determine the location of the wavepacket within the ring at t = 0, but do not influence the character of its time evolution. The time evolution of the site populations nn(t) during the 0–1.1-ps and 0–200-fs time intervals is shown in Figs. 7 and 8, respectively.
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kk'(t) = kk'(0)exp(ikt – ik't), induces increasingly large dephasing at large delays. Due to this additional maxima can appear at larger delays instead of a single main maximum at t = 0. Secondly, even without such a dephasing, the distribution of the site populations nn(t) may have a complicated form (with a lot of maxima) due to the disorder that destroys the shape of the exciton wavefunctions. As a result we obtain a complex time-dependent redistribution of the excitation density between many maxima instead of a perfect aberration-free motion of a single wave. But some wavelike features are still clearly distinguishable even at larger delays. Some maxima exhibit a well-pronounced motion around the ring (in both directions) with the same time constant (100 fs to pass over half of the ring).
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350 fs. Thus, between 0 and 150 fs the excitation is on the n = 2–4 group, then from 150 to 500 fs the sites n = 10–13 are populated, between 500 and 850 fs the excitation is again on the n = 2–4 group. Within the n = 2–4 group there are oscillations between the sites n = 2 and n = 4 with a time constant of 50 fs (corresponding to a jump from n = 2 to n = 4 or vise versa). Similar oscillations occur between the site n = 10 and sites n = 11–13. The wavelike motion observed for the first realization with small disorder is almost absent in this case, but it is possible to recognize some wavelike flow of excitation density from one group to another through the intermediate site n = 7. Due to the much stronger disorder the wavelike motion is destroyed by scattering on impurities, producing a lot of secondary waves moving in both directions. This results in a complicated nonuniform standing-wave pattern with oscillating populations of some sites (whereas other sites are almost not populated). Generally for a disordered system we always get a superposition of propagating waves and standing waves. For example, in the first realization besides the pronounced wavelike motion one can see some oscillations between the n = 10–16 and n = 17–18, 1–3 group with the time constant (half of period) of 350 fs.
In an ensemble experiment the combination of the wavelike motion and the hopping-type dynamics with time constants of 100 and 350 fs should give rise to similar components in the anisotropy decay kinetics. This prediction is in surprisingly good agreement with the observed biexponential polarization decay with 100 and 400 fs components in fluorescence upconversion experiments for LH1 (40) and the B850 ring of LH2 (41).
In the third realization of static disorder the excitation stays on the n = 2 site with some hopping to the n = 4 site. But the average population of the n = 4 site remains relatively small.
For the fourth realization the excitation is completely localized at one site, i.e., n = 4 without any migration to the other sites. But there is still some oscillatory modulation of the n = 4 population due to small coherences between the n = 4 and neighboring sites.
All of these excitation dynamics scenarios are associated with specific realizations of the static disorder. Thus, in a single-molecule experiment we observe realizations corresponding to physically different limits of the excitation dynamics, i.e., coherent wavelike motion of a delocalized exciton (with a 100-fs pass over half of the ring), hopping-type motion of the wavepacket (with 350-fs jumps between separated groups of 3–4 molecules), and self-trapping of an excitation that does not move from its localization site (Figs. 7–9
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Circular and elliptical antenna models
In our model we assume random Gaussian static energetic disorder. Alternatively, a partially correlated disorder due to elliptical structural deformation has been proposed to interpret the low-temperature polarized fluorescence measurements (12,37,42). In the following we compare the exciton structure of the 850-nm band for these two possibilities.
In a homogeneous ring only the doubly degenerate k = ±1 exciton level is allowed, whereas the lowest k = 0 and other levels are forbidden by the symmetry of the ring. Introducing the disorder induces a splitting between the k = ±1 levels in proportion to /M, where is the FWHM of the Gaussian distribution of uncorrelated disorder of pigment site energies, and M is the pigment-pigment interaction energy (33). The disorder value required to explain a broad absorption of the B850 band in the LH2 and LH1 is 400–500 cm–1 (4–9,33,34). This is associated with a splitting between the k = ±1 levels of 70–80 cm–1. Additionally, these levels borrow significant part of their dipole strength to the nearest exciton states. In particular, the lowest k = 0 level becomes strongly allowed (even superradiant). It is remarkable that the experimental superradiance value (2.8 for LH2 (43)) is very close to that predicted by the disordered model with = 420–450 cm–1 (8,34).
Alternatively, the low-temperature single-molecule polarized fluorescence studies gave the splitting between the k = ±1 levels of 110 cm–1 (13), which is difficult to explain with the disordered model. Naturally, such a splitting could be interpreted by further increasing the disorder, but in the case of uncorrelated disorder this will broaden the spectrum due to the increase of the dipole strength and the splitting between the other levels. In principle, a splitting between the k = ±1 levels without significant influence on the other levels is possible with any type of spectral or structural disorder correlated with the shape of the k = –1 and k = 1 wavefunctions. The latter have a form of cos(2
n/N) and sin(2n/N) in the homogeneous limit, reaching their maximal amplitudes at the opposite sides of the ring in the x and y directions, respectively. Thus, any perturbation proportional to sin(4n/N) has different signs near the k = –1 and k = 1 maximums, shifting these levels in opposite directions. Such a perturbation can be created by a correlated shift of the site energies in an unperturbed ring or a modulation of the coupling strength (and/or site energies) associated with an elliptical deformation (12,37,42). Due to a smaller distortion of the exciton states (compared to the random disordered model) this model predicts larger exciton delocalization and only weakly allowed k = 0 exciton level (12,13,42). The latter, however, is in contradiction with the experimentally observable superradiance of the k = 0 (43).
Disordered ring versus elliptical deformations: modeling of single-molecule spectral profiles
In the following we investigate how the room-temperature fluorescence spectral profile shapes with different peak wavelengths calculated with the two models of energetic disorder compare with our experimental results.
As we have seen (Fig. 2) the disordered model requires relatively large uncorrelated disorder values to obtain the splitting between the main exciton components necessary to simulate the absorption spectrum. In our model with the uncorrelated disorder value of = 370 cm–1 the splitting between the k = ±1 energy levels is 40, 65, and 100 cm–1 for realizations with the FL peak at 860, 870, and 890 nm, respectively (Fig. 2). The important feature is the increasingly bigger splitting between the k = ±1 levels and the intense k = 0 transition for more disordered realizations producing the specific shape of the red-shifted FL spectra.
To study the effect of ellipticity we restrict ourselves to the case of the deformation-induced modulation of the site energies 
En in combination of the reduced amount of random (uncorrelated) disorder of the site energies . We suppose that En = Esin(4(n – n0)/N), where E is taken from a Gaussian distribution with the FWHM of corr and with E = 0. Generally the first moment E can be nonzero, but this does not change significantly the results discussed below. Because we do not consider the polarization of the emission, the n0 value (that determines the principle axes of ellipses) is taken arbitrary. Generally, both the n0 and E values are fluctuating according to experimental observations (10). With such a model the bulk absorption and averaged FL spectra can be explained by taking corr = 275 cm–1 and = 125 cm–1 (instead of corr = 0 and = 370 cm–1 in the disordered model). The averaged k = ±1 splitting is now increased up to 120 cm–1. The emission profiles calculated for occurrences with different FL peak positions are shown in Fig. 10.
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In the elliptical model the blue-shifted (860 nm) emission originates from the B850 ring weakly perturbed both by random and correlated shifts of the site energies. In this case the k = ±1 levels are almost degenerated and contribute equally to the emission (Fig. 10, top).
Realizations with a bigger elliptical deformation produce an increased splitting between the k = ±1 levels, so that the emission is determined mostly by the lower, k = –1 state giving FL profiles peaking at 870 nm (Fig. 10, middle). Due to the anomalously big k = ±1 splitting the k = 1 level is only weakly populated, whereas the lowest k = 0 state is almost forbidden. Thus, the contribution of the k = 1 and k = 0 states to the emission is much lower (as compared with the disordered model), thus producing narrower FL profiles at 870 nm.
The k = 0 level becomes more intense for the red-shifted realizations, but its relative contribution is still not so much pronounced as in the disordered model (as well as its shift from the k = –1 level is not as large). Thus, the FL spectra peaking at 890 nm are determined mostly by the k = –1 level with some contribution of the almost nonshifted k = 0 level. Such a configuration of the exciton levels makes it impossible to reproduce the broad experimental FL profile (Fig. 10, bottom).
We conclude that the exciton structure predicted by the elliptical model considered here (which is in fact typical for all the models with elliptical deformations) is not consistent with the observed changes in FL spectra.
Comparison of different experimental approaches
From the previous sections it appears that the interpretation of the different experimental data requires assuming different disorder models. In the following we seek the possible explanation of this apparent discrepancy in the variation of the sample preparation and measurement procedure.
In the room-temperature experiment by Bopp et al. FL spectra peak position exhibits a distribution with a FWHM of 100 cm–1 (see Fig. 8 in Bopp et al. (10)), which is similar to our measurements (see histogram in Fig. 1). The width of the FL spectrum fluctuates by >200 cm–1 (10) again in agreement with our data. However, the authors do not investigate if the elliptical deformation is required to explain the observed FL spectral line shapes. At the same time the ellipticity hypothesis in the low-temperature studies (11–13) is based solely on the polarization of the fluorescence-excitation spectra, whereas the shapes of the FL spectra were not studied. On the other hand, our measurements were indiscriminate for the spectral polarization. Thus, the direct comparison of the two types of measurements cannot be made at the moment.
It is possible that the difference in the model assumptions required to interpret the different experimental data originates from the sample preparation conditions. In our ambient temperature experiment complexes are immobilized on a "carpet" of amino groups of poly-l-lysine and submerged in a physiological buffer and thus are in a radically different environment than in the case where they are spin-coated in a PVA matrix at cryo-temperature as in van Oijen et al. (13).
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TOPABSTRACTINTRODUCTIONTHEORYRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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The alternative models that assume elliptical deformations of the ring fail to explain the spectral shapes of the experimentally observed emission profiles corresponding to different FL peak positions.
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TOPABSTRACTINTRODUCTIONTHEORYRESULTS AND DISCUSSIONCONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Submitted on August 15, 2005; accepted for publication January 3, 2006.
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