Photosynthesis Research 71: 99–123, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
99
Regular paper
The quantitative relationship between structure and polarized
spectroscopy in the FMO complex of Prosthecochloris aestuarii: refining
experiments and simulations
Markus Wendling1,∗ , Milosz A. Przyjalgowski1,4, Demet Gülen2 , Simone I. E. Vulto3,5, Thijs
J. Aartsma3 , Rienk van Grondelle1 & Herbert van Amerongen1
1 Vrije
Universiteit, Faculty of Sciences, Division of Physics and Astronomy, Department of Biophysics and Physics of Complex Systems, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands; 2 Department of Physics,
Middle East Technical University, TR-06531 Ankara, Turkey; 3 Rijksuniversiteit Leiden, Department of Biophysics,
Huygens Laboratory, P.O. Box 9504, 2300 RA Leiden, The Netherlands; 4 Present address: National University of
Ireland, Department of Physics, Galway, Ireland; 5 Present address: Philips Research Laboratories, Building WB6,
Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands; ∗ Author for correspondence (e-mail: markus@nat.vu.nl;
fax: +31-20-4447999)
Received 10 May 2001; accepted in revised form 14 June 2001
Key words: circular dichroism, electron-phonon coupling, excitons, homogeneous line shape, inhomogeneous
broadening, lifetime broadening, light harvesting, linear dichroism, modeling, photosynthesis, pigment–protein
complex, polarized fluorescence, polarized spectroscopy
Abstract
New absorption, linear dichroism (LD) and circular dichroism (CD) measurements at low temperatures on the
Fenna–Matthews–Olson complex from Prosthecochloris aestuarii are presented. Furthermore, the anisotropy of
fluorescence excitation spectra is measured and used to determine absolute LD spectra, i.e. corrected for the degree
of orientation of the sample. In contrast to previous studies, this allows comparison of not only the shape but also
the amplitude of the measured spectra with that calculated by means of an exciton model. In the exciton model,
the point-dipole approximation is used and the calculations are based on the trimeric structure of the complex. An
improved description of the absorption and LD spectra by means of the exciton model is obtained by simply using
the same site energies and coupling strengths that were given by Louwe et al. (1997, J Phys Chem B 101: 11280–
11287) and including three broadening mechanisms, which proved to be essential: Inhomogeneous broadening in a
Monte Carlo approach, homogeneous broadening by using the homogeneous line shape determined by fluorescence
line-narrowing measurements [Wendling et al. (2000) J Phys Chem B 104: 5825–5831] and lifetime broadening.
An even better description is obtained when the parameters are optimized by a global fit of the absorption, LD and
CD spectra. New site energies and coupling strengths are estimated. The amplitude of the LD spectrum is described
quite well. The shape of the CD spectrum is modelled in a satisfactory way but its size can only be simulated by
using a rather large value for the index of refraction of the medium surrounding the chromophores. It is shown
that the estimated coupling strengths are compatible with the value of the dipole strength of bacteriochlorophyll a,
when using the empty-cavity model for the local-field correction factor.
Abbreviations: BChl – bacteriochlorophyll; Cb. tepidum – Chlorobium tepidum; CD – circular dichroism; FLN
– fluorescence line-narrowing; FMO – Fenna–Matthews–Olson; fwhm – full width at half maximum; IDF –
inhomogeneous distribution function; LD – linear dichroism; Pc. aestuarii – Prosthecochloris aestuarii; PW –
phonon wing; T-S – triplet-minus-singlet; ZPL – zero-phonon line
100
Introduction
Photosynthesis is the synthesis of organic compounds
by the use of photons: the energy of sunlight is converted into chemical energy. The primary steps in
this process are ‘light harvesting’ and charge separation. For light harvesting, all photosynthetic organisms
have so-called antenna complexes. The function of
these pigment–protein complexes is to absorb light
and to transport the excited-state energy finally to a
special pigment–protein complex – the photosynthetic
reaction centre, where charge separation occurs (van
Grondelle et al. 1994; Sundström et al. 1999).
In green sulphur bacteria such as Prosthecochloris
(Pc.) aestuarii, Chlorobium (Cb.) tepidum and Cb.
limicola, energy transfer between the main chlorosome antenna and the reaction centres is believed
to be mediated by a bacteriochlorophyll (BChl) acontaining protein known as the Fenna–Matthews–
Olson (FMO) protein (Olson 1980, 1998). Being
water-soluble, the FMO protein of Pc. aestuarii was
the first photosynthetic pigment-protein complex determined structurally at high resolution (2.8 Å) (Fenna
and Matthews 1975). Later the structure was refined
to 1.9 Å resolution (Tronrud et al. 1986). More recently the structure of the FMO protein from Cb.
tepidum was determined at 2.2 Å resolution (Li et
al. 1997). Each FMO complex in the crystal structure is a C3 -symmetric trimer and each subunit in the
trimer encloses seven BChl a molecules with nearest
neighbour Mg-Mg distances between 11 and 15 Å.
In this article, we will focus on the spectroscopic
properties of the FMO complex from Pc. aestuarii in
the Qy absorption region. Owing in part to the early
structure determination this FMO complex has been
one of the best studied photosynthetic antennae (Pearlstein 1991; van Grondelle et al. 1994; Blankenship et
al. 1995; Vulto et al. 1997; Savikhin et al. 1998, 1999).
Pearlstein and co-workers (Pearlstein and Hemenger
1978; Pearlstein 1991, 1992; Lu and Pearlstein 1993)
recognized early on that the FMO protein is an excellent system for modeling the electronic excited-state
structure-spectroscopy relationship in photosynthetic
antennae. It resisted, however, a consistent explanation over many years.
Recently, significant progress in the description of
this system has been achieved (Louwe et al. 1997a,
b; Iseri and Gülen 1999; Owen and Hoff 2001). Currently there exist good qualitative explanations for a
number of spectra, such as absorption, linear dichroism (LD), triplet-minus-singlet absorption difference
(T–S) and to a lesser extent for circular dichroism (CD)
of Pc. aestuarii. Although the currently available models vary in details, the parameter space has narrowed
down considerably and there is partial consensus about
the overall character of the electronic excited states of
the complex. Spectral features in the 780–830 nm region are mainly due to a combination of site shifts and
dipolar interactions among the lowest singlet transitions (Qy ) of the seven BChl a molecules contained
in each subunit. The lowest-energy band (around 825
nm) is mainly localized on a single BChl molecule
and this molecule is BChl a 3 [numbered according
to Fenna and Matthews (1975)]. The remaining Qy
states are mini-excitons with excitations delocalized
over two or three molecules at most.
It is believed that one of the reasons for the failure
of the early models has been the choice to put strong
emphasis on simulating (only) CD spectra which are
very sensitive to small changes in BChl organization
(Gülen 1996; Koolhaas et al. 1997a, 1998, 2000). The
large differences in published CD spectra, obtained
in different laboratories (Philipson and Sauer 1972;
Olson et al. 1976; Vasmel et al. 1983; Vulto et al.
1998; present study) have certainly contributed to the
complications in the (earlier) modeling.
Secondly, it appears that the coupling strengths
between the pigments are significantly lower when
compared to previous estimates (Pearlstein and Hemenger 1978; Pearlstein 1992; Lu and Pearlstein
1993; Gülen 1996), while the lowest-energy pigment
is now assumed to be BChl a 3 (Louwe et al. 1997a,
b; Iseri and Gülen 1999; Vulto et al. 1999; Owen and
Hoff 2001). It is generally agreed upon that the dipole
strength is reduced by the dielectric screening of the
protein but the question of how effective the screening
is still remains unanswered.
Considerable improvement has been obtained by
Louwe et al. (1997a, b) by reducing the coupling
strengths by ∼50% compared to earlier studies and
by taking the lowest-energy pigment to be BChl a 3.
However, the two most recent studies by Iseri and
Gülen (1999) and by Owen and Hoff (2001) suggest
an increase of around 20–25% with respect to the
coupling strengths of Louwe et al. (1997b).
The exciton calculations are often limited to the
monomeric subunit of the FMO complex (Louwe et al.
1997b; Iseri and Gülen 1999; Vulto et al. 1999; Owen
and Hoff 2001). The reason is that the largest interaction between BChl a molecules belonging to different
subunits is about one order of magnitude smaller than
the largest intrasubunit interaction. Therefore, most of
101
the excitations are localized within a subunit (see e.g.
Savikhin et al. 1998). However, despite of being relatively small, these long-range interactions between
different subunits in the trimer can affect the CD
spectrum much more than the absorption spectrum
(Pearlstein 1992; Savikhin et al. 1999).
Most of the current models use the point-dipole approximation to account for the excitonic interactions.
The site energies are treated as empirical fit parameters and excitonic stick spectra are calculated and
dressed afterwards with Gaussians to account for the
spectral broadening (Pearlstein 1991); homogeneous
and inhomogeneous broadening were not explicitly
accounted for.
In all the current models, only the shape of the
spectra was simulated, whereas the amplitude (relative to the absorption spectrum) was not considered.
For the (polarized) T–S spectra, this was not possible
because the average number of triplets per complex
has not been determined. The size of the LD could
not be simulated, because the degree of orientation of
the complexes in the LD measurements was unknown.
The description of the CD spectra faces an additional
problem because published spectra show significant
differences.
In the present study, we present new absorption,
LD, CD and polarized fluorescence excitation spectra over a range of cryogenic temperatures. The LD
and polarized fluorescence results are combined to
deduce the absolute LD spectrum, i.e. the spectrum
that would be measured if the complexes were perfectly oriented. The combination of these techniques
allows a much more detailed investigation of the relationship between structure and spectroscopy (van
Amerongen et al. 1991, 1994). It was for instance
concluded by van Amerongen et al. (1991) that the
Qy transition dipole moments of the light-harvesting
complex LH 2 from Rhodobacter sphaeroides are
nearly parallel to the membrane plane. When the structure of the supposedly very similar LH 2 complexes
from Rhodopseudomonas acidophila (McDermott et
al. 1995) and Rhodospirillum molischianum (Koepke
et al. 1996) were determined later, they indeed showed
such an organization.
We simulate all spectra for the trimeric FMO complex and at a higher level of sophistication than the
recent models mentioned above by including inhomogeneous, homogeneous and lifetime broadening. To
account for the homogeneous broadening, we make
use of recent fluorescence-line-narrowing (FLN) data
and the analysis of the temperature dependence of the
lowest-energy band (Wendling et al. 2000).
Materials and methods
Sample preparation and spectroscopy
The FMO complex of Pc. aestuarii was prepared
as described elsewhere (Francke and Amesz 1997).
Samples were diluted in a buffer containing 50 mM
Tris-HCl (pH = 8.3), 200 mM NaCl and 66.6% (v/v)
glycerol. For the measurement of LD and fluorescence
excitation spectra the samples were embedded in a
polyacrylamide gel. The same buffer as above was
used, containing in addition 15% (w/v) acrylamide
and 0.5% (w/v) polyacrylamide. The gel was polymerized by adding 0.3% (v/v) TEMED and 0.03%
(w/v) ammonium persulphate. For the fluorescence
measurements, the polymerized gel was not squeezed.
For the LD measurements, the sample was oriented. This was done by 2-D squeezing the polymerized gel with a home-built press in the x- and
y-direction into a 1.0 × 1.0 cm acrylic cuvette and
allowing it to expand in the z-direction. The difference
() in absorption (A) A = Az −Ax is measured. The
reduced LD (Ared ) is defined as:
A(λ)
=
(1)
3Aiso(λ)
Az (λ) − Ax (λ)
Az (λ) − Ax (λ)
=
Az (λ) + Ax (λ) + Ay (λ)
Az (λ) + 2Ax (λ)
Ared (λ) ≡
where Aiso(λ) is the isotropic absorption spectrum.
Absorption, LD, CD and fluorescence excitation
measurements were performed on home-built setups.
The excitation light for the measurement of fluorescence excitation spectra was provided by a tungsten halogen lamp via a 1/2-meter monochromator
(Chromex 500SM, spectral resolution 3 nm). A 600
nm highpass filter behind the monochromator was
used to block second order light. The excitation light
was modulated with a mechanical chopper. Fluorescence was detected in a 90◦ geometry via a 905 nm
bandpass filter (20 nm full width at half maximum
(fwhm)) by a photodiode and fed into a lock-in amplifier. The polarizations of the excitation and of the
emission were selected by polarizers. The fluorescence excitation anisotropy spectrum r(λ), which is
detected at wavelength λdet and excited at different
102
wavelengths λ, is given by:
r(λ) =
Fvv (λ, λdet ) − Fvh (λ, λdet )
Fvv (λ, λdet ) + 2Fvh (λ, λdet )
(2)
where F is the measured fluorescence intensity, the
subscripts v (vertical) and h (horizontal) refer to the
polarization directions of the excitation (first index)
and detection (second index). The fluorescence intensity F was corrected for the polarization dependent
differences in detection sensitivity.
The experiments were performed in a liquidhelium flow cryostat (Utreks) which allowed measurements at different temperatures.
Theory for obtaining absolute linear-dichroism
spectra
The absolute LD spectrum with respect to the molecular frame of the complex, Amol(λ), is defined
by
Amol (λ) = Ac (λ) − Aa (λ)
(3)
Ac indicates the absorption of light polarized along
the c-axis (C3 -symmetry axis) of the complex and
Aa indicates the absorption of light polarized perpendicular to it, i.e. parallel to the plane perpendicular
to the C3 -axis. Note that rotation of the molecules
around the c-axis will not alter Aa (λ) because of the
C3 -symmetry. Such a Amol (λ) spectrum would be
measured, if one would align all molecules with their
c-axis parallel to each other. One should note that this
is the spectrum that one can calculate using exciton
theory and theoretical and experimental spectra can
quantitatively be compared (see below).
The reduced-LD spectrum with respect to the molecular frame of the complex is given by
Amol (λ)
≡
≡
3Aiso(λ)
Ac (λ) − Aa (λ)
= P2 (cos βµ )
λ
Ac (λ) + 2Aa (λ)
Amol
red (λ)
(4)
βµ is the angle between the transition dipole moment
and the C3 -symmetry axis of the disc-like trimeric
complex. P2 (x) = (3x2−1)/2 is the second order Legendre polynomial and the brackets reflect a weighted
average over the different pigments (or excitonic
states) contributing to the absorption at wavelength λ
(see Equation (A3)). The term P2 (cos βµ )
λ depends
entirely on the transition dipole moment orientation
within the trimer and is an intrinsic property of the
complex.
In Appendix A it is shown how P2 (cos βµ )
λ can
be obtained from the reduced-LD spectrum Ared (λ)
(see Equation (1)) and the fluorescence excitation anisotropy spectrum r(λ) (see Equation (2)). It turns
out that the shape of P2 (cos βµ )
λ can be determined
experimentally by measuring Ared (λ). Furthermore,
Ared (λ) is proportional to r(λ), provided that the
equilibration within the trimer is much faster than
the excited-state lifetime in the complex. Once the
value of r(λ) is known at (over) a suitable wavelength
(range), the entire experimental reduced-LD spectrum
can be scaled. From this scaled spectrum, P2 (cos βµ )
can be determined for the lowest-energy state, because
in the case of the FMO complex at low temperature
this state is well separated from the next lowest state.
Since the isotropic absorption spectrum is also known,
one can directly obtain Amol (λ). Now it is straightforward to calculate Ac (λ) and Aa (λ) using Equation
(3) and
3Aiso(λ) = Ac (λ) + 2Aa (λ)
(5)
which leads to:
Ac (λ) =
1
(3Aiso(λ) + 2Amol(λ))
3
(6)
Aa (λ) =
1
(3Aiso(λ) − Amol (λ))
3
(7)
Note that the experimental spectra Amol (λ),
Ac (λ) and Aa (λ) are given in ‘absorption units’, i.e.
relative to the experimental absorption spectrum.
Numerical simulations
The excitonic interaction Vkm (in cm−1 ) between the
transition dipoles of the (chemically identical BChl a)
pigments k and m was calculated assuming dipole–
dipole coupling in the point-dipole approximation (see
e.g. van Amerongen et al. 2000):
Vkm = C
5.04κkmd
3
rkm
(8)
where rkm is the distance (in nm) between the dipoles,
d is the dipole strength (in D2 ) and C is a factor
that accounts for the dielectric environment (protein
screening) and which is often neglected (i.e. set to 1).
Here we use it initially as a scaling parameter, and later
103
the physical relevance of the obtained parameter will
be discussed. The term screened dipole strength will
be used for the product C ∗ d. κkm is an orientational
factor which is given by
κkm = µˆk · µˆm − 3(µˆk · rkm
ˆ )(µˆm · rkm
ˆ )
(9)
where µˆk , µˆm and rkm
ˆ are the unit vectors in the
direction of the transition dipoles of the pigments k
and m and the line joining the centres of the pigments,
respectively.
For the calculation of the exciton coupling positions and directions of the transition dipoles of the
pigments are required. The positions of the nitrogen atoms of the BChl a molecules were taken from
the published structure (Tronrud and Matthews 1993),
which is deposited at the Protein Data Bank (Berman
et al. 2000, PDB identifier 4BCL). For the position of
each pigment the positions of its four nitrogen atoms
were averaged. The direction of the Qy transition dipole moment was taken from ND to NB according to
the nomenclature of the Protein Data Bank.
The calculation of absorption, LD and CD spectra
on the basis of an exciton model in a Monte Carlo
simulation is described in the literature (Fidder et al.
1991; Monshouwer et al. 1997; van Amerongen et
al. 2000; Kleima et al. 2000b). A summary is given
in Appendix C. We note that our method is essentially the same as Pearlstein’s (Pearlstein 1991) except
that we used a Monte Carlo approach (typically 10000
iterations) to account for the inhomogeneous broadening (see below). Pearlstein used Gaussians to broaden
the calculated stick spectra for absorption and CD.
However, it was shown by Somsen et al. (1996), that
dressing CD stick spectra with the same Gaussians as
used for the absorption stick spectra, will result in CD
bands which are too broad (see also Buck et al. 1997).
the complex. Thus, there are seven different IDF’s
corresponding to the seven different sites per subunit.
Because of its statistical origin, the IDF is generally taken to be a Gaussian. In every Monte Carlo
step, each of the site energies for the 21 BChl a molecules is randomly taken from its own Gaussian IDF,
independently from the site energies of the other pigments. We stress that although the IDF’s are equal for
symmetry-related pigments in different subunits, the
random variations for the site energies are assumed
to be uncorrelated. The width of the Gaussians was
assumed to be equal for all pigments. Each exciton
band of (strongly) coupled pigments will in general be
narrower than the original distributions of site energies
due to exchange narrowing (see e.g. Knapp 1984) and
the relevance for the FMO complex was demonstrated
by van Amerongen et al. (2000). Note, that this procedure does not lead to an increase in the number of
parameters compared to the simulation approach by
Louwe et al. (1997b).
Homogeneous broadening
Up to now, only (accumulated) stick spectra were discussed. Homogeneous broadening is accounted for in
the following way. Instead of sticks, for every exciton
state a homogeneous line shape is used. A homogeneous line generally consists of a zero-phonon line
(ZPL) and a phonon wing (PW). Recently, we used
temperature-dependent absorption and FLN measurements on the FMO complex of Pc. aestuarii to extract the one-phonon-vibration profile (spectral density) and to determine the overall Huang–Rhys factor
S to be 0.45 (Wendling et al. 2000). This information
is used to construct a homogeneous spectrum by the
formalism described (Pullerits et al. 1995; Wendling
et al. 2000). In short, the homogeneous absorption
spectrum I (ω) at 0 K can be written as
Inhomogeneous broadening
In each subunit of the FMO complex, the site energies
of the seven BChl a molecules are essentially different
due to heterogeneity in the direct protein environment.
Moreover, there is variability in each site energy from
subunit to subunit and from complex to complex due
to slight variations in the protein surrounding at that
site. This is called inhomogeneous broadening and the
distribution of site energies per pigment is referred to
as the inhomogeneous distribution function (IDF). Because of the trimeric structure of the FMO complex, it
is assumed that the IDF is equal for symmetry-related
BChl a molecules in the three different subunits of
I (ω) =
∞
S i · e−S
· li (ω)
i!
(10)
i=0
where i runs over the Franck–Condon progression of
phonons and vibrations, l0 is the ZPL and li is the convolution of li−1 with the one-phonon-vibration profile
φ(ν). The area of φ(ν) is normalized to unity. The ZPL
l0 is assumed to be a Lorentzian with a width determined by the lifetime of the exciton state (see below).
We assume the spectral density and the Huang–Rhys
factor to be equal for all exciton states.
From a mathematical point of view, the outlined
procedure is equivalent to starting with a sharp line (δ-
104
peak) as ZPL, calculating the homogeneous spectrum
I (ω) with Equation (10) and convoluting I (ω) afterwards with a Lorentzian of the desired width. This will
give a broadened ZPL and a broadened PW. Note again
that this does not increase the number of parameters in
the model.
Lifetime broadening
Finally, we account for lifetime broadening: depending on the (calculated) lifetime of each exciton state,
the homogeneous line will be broadened. In a first approach Vulto et al. (1999) have described the excitedstate dynamics in the FMO complex at very low
temperatures by population relaxation induced by vibronic coupling. Therefore, only the diagonal terms of
the density matrix (Leegwater et al. 1997) are relevant:
d red
red
ρJ J (t) = −
'J J,KK ρKK
(t)
dt
(11)
K
ρ J J red (t) is the probability of finding a molecule at
time t in the exciton state J and ' J J,KK is the population transfer rate from exciton state J to exciton state
K. ' J J,KK is given by (Leegwater et al. 1997):
'J J,KK = 2δJ,K
γ (ωK − ωL )|cLm |2 |cKm |2
m,L
−2
(12)
γ (ωK − ωJ )|cJ m |2 |cKm |2
m
where δ is the Kronecker delta and cJ m is the mth component of the eigenvector for exciton state
J . Therefore, |cJ m |2 reflects the probability density of excitations on pigment m in exciton state J .
γ (ω) is the spectral density describing the frequencydependent coupling of the electronic states of the
complex to their environment. The spectral density
is extracted from the experimental PW as described
in Wendling et al. (2000). It is known up to a scaling factor γ 0 as pointed out by Vulto et al. (1999).
Therefore,
γ (ν) = γ0 · φ(ν)
(13)
This scaling factor γ 0 is an additional parameter
in the simulations. Note that compared to Vulto et
al. (1999), we do not use a generic function but the
experimental spectral density including vibronic lines.
In every Monte Carlo iteration, the decay rate
' J J,J J of each exciton state J is calculated using
Equation (12) and the excited-state lifetime is equal
to 1/'J J,J J . The Lorentzian ZPL of that exciton
state will have a fwhm given by 'J J,J J /(2πc), with
c being the speed of light. As a result of using a
(lifetime-)broadened ZPL in Equation (10), also the
PW will become broader.
As there is no depopulation of the lowest exciton
state in this model, this state has an infinite lifetime (therefore, the ZPL is a sharp δ-peak). This
is appropriate because the lifetime of the lowest exciton state is in the order of nanoseconds compared to
(sub)picoseconds for the other states.
Instead of using experimental estimates, we have
chosen to calculate the decay rates within our exciton
model. In this way, a different rate can be assigned
to every individual exciton state in every random realization of diagonal disorder, correctly exploring all
implication of the Monte Carlo method. Dispersive
kinetics are therefore naturally included in the algorithm. This is not possible using the (averaged)
experimental values.
Scaling and fitting the calculated to the measured
spectra
The experimental isotropic absorption, LD, Ac , Aa
and CD spectra are given in ‘absorption units’, i.e.
relative to the experimental absorption spectrum. All
simulated spectra are scaled by using a common scaling factor. This factor is determined by scaling the
simulated isotropic absorption spectrum (or a combination of absorption, LD and CD spectra, see ‘Results
and discussion’) to the experiment: It translates the
simulations into ‘absorption units’. In this way, we are
able to compare not only the shapes of simulated and
experimental spectra, but also their amplitudes.
The simulation of exciton spectra in our model
includes inhomogeneous, homogeneous and lifetime
broadening. The calculation depends on the fwhm of
the IDF, the seven site energies, the screened dipole
strength C ∗ d, the scaling factor γ 0 and the homogeneous line shape. This line shape and the fwhm of the
IDF are fixed. For the other parameters suitable starting values are chosen (see ‘Results and discussion’)
and a fitting routine is used for parameter optimization. We used the non-linear least-squares routine
from S-PLUS (www.insightful.com), which calculates
finite-difference gradients of the model function with
respect to the parameters and uses a Gauss–Newton
algorithm.
105
Figure 1.1. Absorption spectra of the FMO complex from Pc. aestuarii at 4 (solid), 25 (dashed), 50 (dotted) and 77 K (long-dashed).
The spectra are normalized to a maximal absorption of 1 in the 4 K
spectrum.
Figure 1.2. LD spectra of the FMO complex from Pc. aestuarii at
4 (solid), 25 (dashed), 50 (dotted) and 77 K (long-dashed).
Results and discussion
Experiments
The absorption spectra of the FMO complex from Pc.
aestuarii at 4, 25, 50 and 77 K are given in Figure 1.1.
They show a gradual increase of the intensities and a
concomitant sharpening of the absorption bands upon
cooling to 4 K. Especially at 4 K, five partially overlapping absorption bands can be distinguished with
peaks near 790, 800, 804, 813 and 824 nm. The spectra
at 4 and 77 K are very similar to those reported earlier
(Olson et al. 1976; Whitten et al. 1980; van Mourik et
al. 1994; Wendling et al. 2000).
The LD spectra at 4, 25, 50 and 77 K are presented in Figure 1.2. Like for the absorption spectra, an
increase in intensity is accompanied by a narrowing
of the bands upon cooling. The shape of the spectrum
at 4 K is similar to the one presented by van Mourik
et al. (1994) and clearly, five contributing bands can
be discerned at similar (although not identical) positions as the ones observed in the absorption spectrum
(see also Figure 3). The shapes of the absorption and
LD spectra of the lowest-energy band above 820 nm
Figure 1.3. Reduced-LD spectra of the FMO complex from Pc. aestuarii at 4 (solid), 25 (dashed), 50 (dotted) and 77 K (long-dashed).
106
Figure 2. CD spectra of the FMO complex from Pc. aestuarii at 4
(solid) and 77 K (dashed). The 4 K spectrum corresponds to that
of a sample with an absorption maximum of 1 at 4 K. The other
spectrum is scaled with the same factor.
are very similar, in line with the idea, that this red
band is due to one transition per monomer (Louwe et
al. 1997b). Just to the blue of the maximum of the
lowest-energy band these spectra start to deviate due
to the contributions of different electronic transitions,
indicating that very low temperatures are needed to get
selective fluorescence from the lowest-energy state.
The reduced-LD spectra at 4, 25, 50 and 77 K
are presented in Figure 1.3. Noteworthy is the small
but reproducible drop of the reduced LD at the long
wavelength side, i.e. upon going to 830 nm. The 4 K
spectrum is again very similar to the one reported by
van Mourik et al. (1994) where this ‘drop’ was also
observed.
In Figure 2 the CD spectra at 4 and 77 K are
presented. Significantly different CD spectra have
been reported in the literature. Our 4 K spectrum is
very similar in shape to the one published recently for
6 K (Vulto et al. 1998), although in the present study
the positive feature near 805 nm is more pronounced.
In all other studies this feature was not observed and
neither did it appear in the modeling study by Louwe
et al. (1997b). The 77 K spectrum is reminiscent of
the one measured by Vasmel et al. (1983), although in
that spectrum the negative 815 nm band is clearly less
pronounced. In the past, rather different spectra at 77
K were published (Philipson and Sauer 1972; Olson et
al. 1976). The difficulty with measuring CD spectra of
the FMO complex was already noted by Philipson and
Sauer (1972), who observed some variation in peak
positions and amplitudes from sample to sample.
We note that it was first the experimental CD
spectrum published by Philipson and Sauer (1972),
that Pearlstein and coworkers tried to simulate using the crystallographic structure of the FMO complex from Pc. aestuarii (Pearlstein and Hemenger
1978; Pearlstein 1991, 1992). Later Lu and Pearlstein (1993) used both experimental data sets to fit
simultaneously absorption and CD spectra. The best
agreement was obtained for the spectra measured by
Olson et al. (1976). Those were measured in a cryosolvent containing glycerol but not glycerophosphate
as in the case of the spectra of Philipson and Sauer
(1972). It was argued by Lu and Pearlstein that the
glycerophosphate-containing cryosolvent could perturb the protein structure altering the BChl–BChl
interactions.
In Figure 3 the absorption, LD and CD spectra at
4 K are shown together. All spectra have bands near
790, 800 and 824 nm in common, indicated with numbers 1, 2 and 5. Bands near 804 and 813 nm in the
absorption and LD spectrum, which are numbered 3
and 4, appear to have a clearly composite character,
indicated by two CD (sub)bands around both bands 3
and 4.
In Figure 4 the fluorescence excitation anisotropy
spectra are given for 4, 25 and 50 K. Whereas the
shape of the spectrum is more or less independent of
temperature, the size decreases upon increasing the
temperature. The reason for this is that at higher temperatures the fluorescence does not only stem from
the lowest state but also from a state which is higher
in energy and which tends to change the sign of the
anisotropy (see Appendix A).
In Figure 5 the fluorescence excitation anisotropy
spectra at 4, 25 and 50 K are compared to the reducedLD spectra. The reduced-LD spectra have been scaled
to the anisotropy spectra in the wavelength region
between 800 and 815 nm. Up to 818 nm, the reducedLD and the anisotropy spectra do coincide, which is
expected if efficient subunit equilibration takes place.
Also the temperature dependence is as expected, as
discussed above, and it appears that even at 25 K
still some fluorescence arises from states whose transition dipole moments have orientations that differ from
107
Table 1. Parameter sets used for the exciton simulations of steady-state
spectra of the FMO complex from Pc. aestuarii shown in Figures 7, 9.1,
9.2, 9.3, 10 and 11
(A),(B),(C)
site energy of BChl a
1
2
3
4
5
6
7
C ∗ d (in D2 )
γ0 (in ps−1 )
fwhmIDF (in cm−1 )
(D)
(E)
12350 (809.7)
12465 (802.2)
12160 (822.4)
12350 (809.7)
12600 (793.7)
12480 (801.3)
12460 (802.6)
28.7
–
3050
80
12430 (804.5)
12405 (806.1)
12175 (821.4)
12315 (812.0)
12625 (792.1)
12500 (800.0)
12450 (803.2)
30.6
3050
(A), (B), (C), (D) and (E) refer to the different simulation approaches
(see Figure 7) and correspond to the subpanels of Figures 7, 9.1, 9.2, 9.3
and 10. The site energies are given in cm−1 . The values in brackets are
the corresponding wavelengths in nm. The site energies for the simulations (A), (B), (C) and (D) are those of Louwe et al. (1997b) but shifted
to the blue by 30 cm−1 .
Table 2. Used coupling strengths in cm−1 for the exciton simulation approach (E) of steady-state spectra of the FMO complex from
Pc. aestuarii shown in Figures 7, 9.1, 9.2, 9.3 and 10. These were
calculated from the structure and scaled with the screened dipole
strength C ∗ d of 30.6 D2
Subunits
BChl a
(1,1)
(1,2)
(1,3)
(1,1)
(1,2)
(1,3)
(1,1)
(1,2)
(1,3)
(1,1)
(1,2)
(1,3)
(1,1)
(1,2)
(1,3)
(1,1)
(1,2)
(1,3)
(1,1)
(1,2)
(1,3)
1
1
2
3
4
−102
6 −6
7
1 −1
0
4
2
2
0
1
32
8
1
2
0 −4 −3
12
1
0
1
1
1
−56 −2
2
1 −4
8
7
−1 −4 −4
1
2
−69
0
1
1
3 −1
0 −3
8
3
0
2
2
2
3
4
5
5
6
7
−15
2
0
14
7
1
−10
3
1
−19
−1
2
89
3
0
3 −2
1
4
1
12
2
7
0
−1
0
2
1
7
1
3
2
−1
−2
0
1
2
1
4
0
8
8
3
−2
−1
6
−14
2
1
9
4
1
2
8
0
−60
3
8
−4
−1
−2
37
−2
3
−2
0
7
0
3
10
10
The column ‘Subunits’ labels the subunits (i, j ) of the interacting BChl a molecules. The interaction matrix can be constructed
by using pigment-pair symmetry and the cyclic symmetry of the
trimeric FMO complex, i.e. (1,1)=(2,2)=(3,3); (1,2)=(2,3)=(3,1);
(1,3)=(2,1)=(3,2).
108
Figure 3. The absorption (solid), LD (dashed) and CD (dotted)
spectra of the FMO complex from Pc. aestuarii at 4 K. The spectra
are all scaled in such a way that their shapes can easily be compared.
The main features in the spectra are numbered (see text for details).
those corresponding to the lowest-energy band. Therefore, in order to obtain the anisotropy that corresponds
to excitation and emission of the lowest state we can
only use the 4 K data.
It was pointed out in ‘Materials and methods’
(see also Appendix A), that the fluorescence excitation anisotropy and reduced-LD spectra have the
same shape, provided that fluorescence arises from
a subunit-equilibrated system. But the fact that the
anisotropy and reduced-LD spectra differ in the longwavelength region indicates that such a fast subunit
equilibration does not take place. We ascribe the
difference between both types of spectra to inhomogeneous broadening of the lowest-energy band. The
coupling between the monomeric subunits in the FMO
complex is weak and, therefore, each monomeric subunit has an absorption band around 825 nm. But due
to inhomogeneous broadening, the exact positions of
these bands are not identical, some are to the blue of
the average peak of the lowest-energy band and some
are to the red (see e.g. Rätsep et al. 1999). Excitation on the blue side (∼820 nm) leads to preferential
excitation of the monomers that have a higher lowestenergy state followed by transfer to a subunit with
Figure 4. Fluorescence excitation anisotropy spectra of the FMO
complex from Pc. aestuarii at 4 (solid), 25 (dashed) and 50 K
(dotted) detected at 905 nm.
a band at somewhat lower energy. As a result, the
fluorescence will mainly arise from this lowest-energy
subunit. Therefore, the anisotropy will be lower than
in the case of perfect subunit equilibration. On the
other hand, excitation on the red side of the band leads
to preferential emission by the pigments (or states)
that have been excited directly and the anisotropy is
higher than expected in the case of subunit equilibration. These characteristics can be recognized upon
comparing the anisotropy and reduced-LD spectra.
However, the reduced-LD spectrum has the shape
that the fluorescence anisotropy spectrum would have
had in the entire range of the spectrum (i.e. also
above 818 nm), if subunit equilibration would have
taken place. Scaling the reduced-LD spectrum to the
anisotropy spectrum below 818 nm provides the anisotropy value after subunit equilibration for the lowest
state. Thus the equilibrated anisotropy value for excitation/emission in/from the lowest-energy band, that
we were looking for, at 4 K is determined to be 0.039
± 0.002 at the maximum of the band (see Figure 5).
Note that the anisotropy is 0.032 and 0.023 at 25 and
50 K, respectively. In Appendix B it is shown how the
values at higher temperatures can approximately be
corrected for the contribution from the emission from
109
Figure 5. Fluorescence excitation anisotropy spectra (solid) at 4,
25 and 50 K (see Figure 4) are compared to the reduced-LD spectra
(dashed) at the same temperatures (see Figure 1.3). The reduced-LD
spectra have been scaled to the anisotropy spectra between 800 and
815 nm.
higher states. Applying this method leads to corrected
values for r of 0.034, 0.027 and 0.038 for 25, 50 and
77 K (not shown), respectively. From these values one
can now calculate P2 (cos βµ1 ) for the lowest-energy
band with Equation (A7) and one arrives at −0.31,
−0.29, −0.26 and −0.31 for 4, 25, 50 and 77 K,
respectively. Note, that the positive LD of this band
corresponds to a negative value for P2 (cosβ 1µ ), since
the degree of orientation is negative. The results at
different temperatures are fairly consistent.
One can now construct Amol (λ), Amol
red (λ),
Ac (λ), and Aa (λ), using the method outlined in ‘Materials and methods’. The results are given in Figure 6. The difference in P2 (cosβ 1µ ) between −0.31 and
−0.26 only leads to small differences in the constructed spectra. In the following, we will use the average
value of −0.29 for P2 (cosβ 1µ ).
Simulations
In this study, we want to compare in a quantitative way
the experimental absorption, LD and CD spectra of the
FMO complex from Pc. aestuarii with the spectra that
can be simulated using exciton theory, taking also into
Figure 6. The constructed polarized absorption spectra of the FMO
complex from Pc. aestuarii for light polarized parallel to the C3 -axis
(Ac (λ), solid and dotted) and perpendicular to it (Aa (λ), dashed
and long-dashed). The spectra were constructed from a linear combination of the absorption and LD spectra as explained in the
text, assuming that P2 (cos βµ )
is either −0.31 (solid, dashed) or
−0.26 (dot, long-dashed). The spectra correspond to an isotropic
absorption spectrum Aiso with a maximum of 1.
account the different broadening mechanisms. In most
of the current models, excitonic stick spectra are calculated and the sticks then are dressed with Gaussians
for qualitative comparison with experimental spectra
(e.g. Louwe et al. (1997b) used Gaussians with a
fwhm of 80 cm−1 ). For a detailed comparison, however, the inhomogeneous distribution of the site energies has to be considered, since it is well known that
this may especially affect the polarized spectra (van
Mourik et al. 1992; Koolhaas et al. 1994; Buck et al.
1997). Moreover, the coupling of the electronic transitions to intramolecular vibration and phonon modes
was not explicitly considered. Finally, also lifetime
broadening can affect the absorption (difference) spectra. Recently, we determined with the use of FLN
spectroscopy the contributions from electron-phonon
coupling and in addition the intensities of the vibronic
bands for the lowest electronic transition in the FMO
complex of Pc. aestuarii (Wendling et al. 2000). It
can now be examined to what extent these contributions affect the Qy absorption (difference) spectra. To
110
(Gaussian, fwhm of 80 cm−1 ) stick spectra (solid, fat
line) that are calculated using the approach and parameters of Louwe et al. (1997b). It should be noted that
all our experimental spectra for some unknown reason
appear to be blue-shifted with respect to the simulated
spectra of Louwe et al. (1997b) and, therefore, all site
energies were increased by 30 cm−1 in order to compare experiments and simulations. We note that this
shift is entirely irrelevant for the remaining part of the
discussion.
For illustration the simulation based on the whole
trimeric complex (solid, fat line) is shown together
with the one calculated using only a monomeric subunit of the FMO complex (solid, thin line) as in the
original work of Louwe et al. (1997b). Noting that the
largest intersubunit coupling is about 10 times smaller
than the largest intrasubunit coupling (see Table 2),
it is surprising how much the absorption spectrum is
influenced by including intersubunit interactions. The
red most band around 825 nm and the band around
800–805 nm gain some oscillator strength, when instead of the monomer the whole trimer is used. The
bands around 815 nm and 790 nm decrease. Remarkable is the blue shift of the 815 nm and 800-805 nm
bands. It seems as if the monomer spectrum fits the
absorption spectrum a little better. This is not surprising as Louwe el al. (1997b) adjusted the parameters
in a simulation for the monomer. The global features
of the experimental spectrum are reproduced by both
simulations. The most notable differences between
Figure 7.
illustrate the effects of the above-mentioned broadening mechanisms, we will use the parameters of Louwe
et al. (1997b) (see Table 1). We then offer a quantitative comparison of the experiment and theory by taking
several published parameter sets as starting values in a
global fit routine (see below).
In Figure 7A the experimental absorption spectrum
at 4 K is given (dotted line) together with the dressed
Figure 7. Comparison of the experimental 4 K absorption spectrum
(dotted lines) and simulated absorption spectra Aiso using the trimeric structure of the FMO complex from Pc. aestuarii (solid, fat
lines). The site energies, screened dipole strength C ∗ d and scaling factor γ0 for the spectral density (when applicable) are given
in Table 1. In (A), (B) and (C) the simulation was scaled to the
lowest-energy band of the experimental absorption spectrum. In (D)
and (E) the scaling factor was determined by the global fit (see text).
(A) The excitonic stick spectrum was convoluted with a Gaussian
with a fwhm of 80 cm−1 . This is the method used by Louwe et al.
(1997b). For comparison we also show the simulation using a monomeric subunit only (solid, thin line). (B) Monte Carlo simulation: the
site energies are randomly taken from Gaussian distributions, each
with a fwhm of 80 cm−1 , centred at the site energies as used in
(A). (C) Monte Carlo simulation and homogeneous line shape: The
spectrum from (B) was convoluted with the homogeneous absorption spectrum derived by Wendling et al. (2000) (see also Figure 8).
(D) Monte Carlo simulation and lifetime broadened homogeneous
line shape: Each excitonic stick transition in (B) is dressed with a
line shape which is broadened due to the calculated lifetime of that
exciton state. (E) Monte Carlo simulation and lifetime broadened
homogeneous line shape: Same as in (D) but the parameters were
adjusted as shown in Table 1 to improve the agreement with the
experiment.
111
Figure 8. Shown are the mirror image of the FLN spectrum at 4 K
(A), which resembles the homogeneous absorption spectrum. Note
that the ZPL is masked by scattered laser light. From the smoothed
PW (B) the one-phonon-vibration profile (C) was calculated for S =
0.45. The one-phonon-vibration profile is proportional to the spectral density. For clarity of representation, the spectra were given
equidistant offsets. The curves were taken from Wendling et al.
(2000).
experiment and simulations are the deviation below
800 nm, where the contribution of electron-vibrational
coupling is largest (vide infra), and the fact that the experimental spectrum shows two distinct features near
800 nm.
In order to study the effect of inhomogeneous
broadening, a Monte Carlo approach was used. The
site energies were randomly varied (using a Gaussian
IDF) around the values that were used for calculating the dressed stick spectra and the same coupling
strengths were used as in Figure 7A. The results are
shown in Figure 7B.
An IDF with a fwhm of 80 cm−1 was used in
(B). Note that the bands sharpen compared to (A).
This well-known effect is called exchange narrowing
(see e.g. Knapp 1984; van Amerongen et al. 2000).
In order to reproduce the widths of the experimental
absorption bands around 815 and 825 nm with the
Monte Carlo simulation (B) an IDF with a larger
width, approximately 110 cm−1 , would be required
(not shown).
Figure 9.1. Comparison of the experimental 4 K absorption spectrum Ac for light polarized parallel to the C3 -symmetry axis of
the FMO complex (dotted lines) and simulated spectra using the
trimeric structure (solid, fat lines). The spectra correspond to the
isotropic absorption spectra Aiso as shown in Figure 7 and we note
again that the amplitudes of the spectra can be compared. The different approaches (A) to (E) are the same as described for Figure 7.
For approach (A) we also show for comparison the simulation using
a monomeric subunit only (solid, thin line).
The inclusion of inhomogeneous broadening in (B)
does not alter the overall shape of the spectrum sig-
112
Figure 9.2. Same as Figure 9.1 but for the absorption spectrum Aa
for light polarized perpendicular to the C3 -symmetry axis of the
FMO complex.
nificantly. However, the band around 800 nm now
partly splits into two contributions, reminiscent of the
two features that are observed in the experimental
spectrum although their relative intensities are not
identical. The simulated 800 nm band is too high in
intensity.
Figure 9.3. Same as Figure 9.1 but for the LD spectrum with respect
to the molecular frame of the complex.
Recently, we determined the FLN spectrum of the
FMO complex from Pc. aestuarii upon excitation in
the red wing of the 825 nm band (Wendling et al.
2000). A representative spectrum (mirror image) is
given in Figure 8A, which is the presumed single-site
absorption spectrum. It consists of a ZPL and a PW
and vibronic bands. It was shown that the temperature-
113
Figure 10. Comparison of the experimental 4 K CD spectrum of
the FMO complex (dotted lines) and simulated CD spectra using
the trimeric structure (solid, fat lines). The spectra correspond to the
isotropic absorption spectra Aiso as shown in Figure 7 and we note
again that the amplitudes of the spectra can be compared. However,
no scaling was performed in order to account for the refractive index
n in (A), (B), (C) and (D); the spectrum in (E) was scaled by a
factor 2.6 (see text). The different approaches (A) to (E) are the
same as described for Figure 7. For approach (A), we also show for
comparison the simulation using a monomeric subunit only (solid,
thin line).
Figure 11. Shown are the simulated CD spectra using the trimeric
structure of the FMO complex (solid lines), where the direction
of the Qy transition dipole moment of a selected BChl a (and the
symmetry-related molecules in the trimer) was rotated within the
plane of the porphyrin ring. The number of the BChl a and the angle
are indicated in the figure. The simulations were done with the parameter set (E) shown in Table 1. The unaltered CD spectrum is also
shown in every panel (dashed lines) together with the experimental
CD spectrum (dotted lines). All simulated CD spectra have been
scaled with a factor n = 2.6.
114
dependence of the 825 nm absorption band can be well
fitted by assuming that the lowest-energy band has an
inhomogeneous width of 80 cm−1 and the Huang–
Rhys factor S is 0.45 ± 0.05, presuming that electron–
phonon coupling and vibronic coupling are the same in
the ground and excited states. The Huang–Rhys factor
reflects the relative areas of the ZPL (areaZPL) and
the PW including vibronic bands (areaPW ) according
to areaZPL /(areaZPL + areaPW ) = e−S . In Figures 8B
and C the phonon-vibronic wing and the one-phononvibration profile (spectral density) are shown. The
latter can be calculated from the former by using the
estimated Huang–Rhys factor of 0.45.
The effect of vibrational coupling on the calculated
absorption spectrum is illustrated in Figure 7C where
all calculated electronic transitions have been convoluted with the same single-site absorption spectrum,
i.e. the same spectral density, the same value of S
= 0.45 and the same ZPL (δ-peak) were used for all
electronic transitions. Although the spectral densities
and the S-values are not necessarily identical for all
electronic transitions, there is not an obvious indication that they are very dissimilar and the simulation
demonstrates that in this way the integrated absorption
strength below 800 nm can largely be explained.
Finally, we consider the effect of homogeneous
broadening due to dephasing effects. This effect
broadens sharp δ-transitions to Lorentzian bands with
line width (fwhm, in cm−1 ) of 1/(πT2 c). T2 (in s) is
the total dephasing time and c is the speed of light
(in cm s−1 ). T2 is given by 1/T2 = 1/(2T1) + 1/T2 where T2 is the pure dephasing time and T1 is the lifetime of an excited state. Excited-state relaxation in the
FMO complex takes at most several picoseconds and
at 4.2 K the pure dephasing time is much longer, and
therefore it does not contribute significantly to the line
broadening. Only for the lowest-energy state the pure
dephasing time may dominate since relaxation to the
ground state takes several nanoseconds. But the broadening effect is so small that it may safely be neglected.
Therefore, if the depopulation rate of exciton state J
is denoted as 'J J,J J = 1/T1 (see Equation (11)), the
line width is given by 'J J,J J /(2πc). As described in
‘Materials and methods’ this value was used as fwhm
of a Lorentzian ZPL to calculate the homogeneous line
shape for every exciton state in each Monte Carlo iteration. We note again that a lifetime broadened ZPL in
Equation (10) will also broaden the PW.
The resulting absorption spectrum is shown in
Figure 7D (γ 0 was determined in a global fit as described below). Including lifetime broadening leads to
a significant improvement of the simulated spectrum.
The amplitude of the 815 nm band is underestimated.
However, all features present in the experimental absorption spectrum can be reproduced. Especially the
two features around 800–805 nm are now reasonably
well described and their relative intensities fit better to
the experimental observation.
In general, exciton equilibration depends on many
factors like the exciton interaction energies, the homogeneous line, the inhomogeneous broadening and
the temperature, which makes its description extremely complicated. However, if it is assumed that
the electron-phonon coupling is weak compared to the
exciton coupling, the equilibration can be described at
arbitrary temperatures in a relatively straightforward
way as pointed out by Leegwater et al. (1997). Thus,
these authors were able to describe energy transfer
kinetics in Photosystem II reaction centres of green
plants, using phonon-assisted relaxation between exciton levels. The same approach was used by Vulto
et al. (1999) to describe exciton relaxation in the
FMO complexes from Cb. tepidum and Pc. aestuarii, for which it is known that the electron-phonon
coupling is weaker than the exciton coupling (see
e.g. Rätsep et al. 1999; Wendling et al. 2000). This
approach requires one scaling parameter (γ 0 ) to describe the (sub)picoseconds kinetics for all exciton
states. A good description was found for the FMO
complex from Cb. tepidum. For the FMO complex
from Pc. aestuarii the situation was less perfect but
still quite reasonable. We use the same approach
here. The scaling parameter γ0 is now varied to
optimize the lifetime broadening such that the best
agreement is found between observed and measured
spectra (frequency domain). If the description of both
the (sub)picoseconds kinetics and (polarized) spectra would be perfect, the scaling parameter should
be identical for the time- and frequency domain.
Given the approximate nature of the formalism, perfect agreement is not expected. But we find that both
values are at least in the same order of magnitude.
Up to now, our simulations were done with the
parameter set of Louwe et al. (1997b), i.e. an IDF
of 80 cm−1 fwhm, the given site energies, a screened
dipole strength C ∗ d of 28.7 D2 , and for Figure 7D
a γ 0 of 3050 ps−1 (see Table 1). γ 0 was used as a
fitting parameter as was done by Vulto et al. (1999)
to simulate the excited state dynamics in the FMO
complex. For comparison, they used a value of ∼7000
ps−1 taking a generic function as the spectral density.
115
As the simulation strongly depends on the above
mentioned parameters, a non-linear least-squares fitting routine was used to get an optimal resemblance between simulated and experimental spectra.
Again inhomogeneous broadening (fwhmIDF , fixed to
80 cm−1 ), a homogeneous line shape and lifetime
broadening were included. We noticed that the absorption spectrum alone could be fitted quite well using
different parameter sets. Therefore, a global fit of the
absorption, LD and CD spectra was performed, making use of three independent measurements. In this
case the common scaling factor is determined in the
fit by scaling all three calculated spectra together to
the experimental spectra. However, scaling the CD
spectrum requires one additional factor which is given
by the refractive index n (see Equation (C6) and below). The spectra were weighted in the fit, so that the
contribution of each spectrum to the final residual sum
of squares was approximately equal. The fit was performed from 787.4 nm to 834.1 nm to exclude baseline
effects which usually are more pronounced on the blue
side of each spectrum.
We have tried several parameter sets as starting values for the fit. These were the parameter sets of Louwe
et al. (1997b), Owen and Hoff (2001), Vulto et al.
(1999) and two sets of Iseri and Gülen (1999) (in that
reference see Table 1: BChl 3 model for absorption
strength 29.5 and 36 D2 ). Appropriate shifting of the
site energies as described above was performed.
Except for the set of Vulto et al. (1999), the final
residual sums of squares were approximately equal
for the different starting value sets and the final fitted values were similar. When the set of Vulto et
al. (1999) was used for the starting values, a larger
residual sum of squares was found and the fitted parameters clearly deviated compared to the other sets.
Therefore, the average of each parameter was calculated disregarding those given by the starting value
set of Vulto et al. The global fit of absorption, LD
and CD puts a considerable constraint on the parameter space. The best simultaneous description of all
spectra is therefore clearly a compromise. The final
rounded averaged parameters are given in Table 1 and
the coupling strengths are listed in Table 2. The resulting absorption spectrum is shown in Figure 7E. In
comparison to Figure 7D, a much improved fit has
been obtained through the adjustment of site energies,
coupling strengths and lifetime broadening. The main
band around 815 nm has increased in amplitude but
still does not perfectly fit the experiment. The sub-
band structure around 800–805 nm is now very nicely
reproduced by the calculation.
In conclusion, the experimental absorption spectrum is better described, if the various broadening
mechanisms are taken into account, even without adjusting the site energies and coupling strengths as
given by Louwe et al. (1997b). Adjustments of these
parameters leads to an additional improvement. Below
we will show the LD and CD spectra, which were calculated in a similar way, i.e. after performing Monte
Carlo calculations to account for the inhomogeneous
distribution of site energies, the spectra are convoluted with a ‘single-site absorption spectrum’ and then
additionally lifetime broadening is considered.
LD
In this study, a lot of attention was paid to obtain the absolute LD spectrum (corrected for the degree of orientation) by using polarized fluorescence
excitation spectroscopy. For the analysis, it is required that all fluorescence arises from the lowest
electronic state, i.e. the temperature should be sufficiently low. On the other hand, the subunit equilibration should occur on a time scale much faster than
the excited-state lifetime. It is known from literature that subunit equilibration occurs on a time scale
of several tens of picoseconds over a wide temperature range (Louwe and Aartsma 1997; Matsuzaki et
al. 2000; for an overview see also van Amerongen
et al. 2000) whereas the excited-state lifetime is on
the order of nanoseconds (Rätsep et al. 1999). Therefore, it is expected that the shape of the reduced-LD
spectrum and the fluorescence excitation anisotropy
spectrum are identical, which is indeed observed at
wavelengths shorter than 818 nm at various temperatures. It was discussed that the temperature dependence
of the height of the anisotropy spectrum should be
ascribed to temperature-dependent Boltzmann equilibration over different electronic levels. At wavelengths
above 818 nm the LD and anisotropy spectra differ.
This can be attributed (at least at a qualitative level) to
inhomogeneous broadening, i.e. the three monomers
all have a lowest-energy state, and these states are
similar in energy but not identical. Excitations at low
temperatures end up predominantly in the monomer
with the lowest energy state.
In Figures 9.1 and 9.2, the experimental and simulated polarized absorption spectra are presented (parallel and perpendicular to the C3 -symmetry axis, respectively) and in Figure 9.3 the LD spectra are shown.
116
The experimental spectra were calculated with P2 =
−0.29 as described above.
For the polarized absorption (difference) spectra,
the difference in the simulations between monomer
and trimer is similar as for the isotropic absorption.
A redistribution of oscillator strengths and a little blue
shift of the bands around 800–805 and 815 nm for the
Ac and LD spectra can be observed. This leads finally
to an apparent blue shift of the main LD feature when
the whole trimer is used.
We note again that because of the absolute character of the experimental Ac , Aa and LD spectra, we
are able to compare not only the shapes of simulated
and experimental spectra, but also their amplitudes. As
is already obvious from the isotropic absorption, the
simulations with dressed stick spectra (A) or inhomogeneous broadening (B) have a rather poor agreement,
especially in the blue region of the spectrum below
800 nm. This is expected as no homogeneous line
shapes have been included up to that point. Taking
these line shapes into account (C, D, E) improves the
agreement in this spectral region.
Interesting is the 2-band-feature around 800–805
nm, which was already discussed for the isotropic absorption. Also in the polarized spectra this feature is
not visible in the dressed stick spectra (A). The composite character of the spectrum is only reproduced
by including inhomogeneous broadening (B, C, D, E).
The intensity of the simulated 805 nm band in the Ac
spectrum in (B) is too weak. When in addition the
homogeneous line shape is taken into account (C, D,
E) the intensity of the 805 nm band is reasonably well
described, but this is not the case for the 800 nm band.
This band is too pronounced in (C) and (E) in the Ac
and LD spectra.
In general, apart from small deviations around
800–805 nm and below 790 nm, the overall resemblance is rather good when inhomogeneous broadening,
homogeneous line shape and lifetime broadening are
considered. The shape and the amplitude of the polarized (difference) spectra are best described with the
refined parameter set in (E) (see Tables 1 and 2).
CD
In Figure 10, the experimental 4 K CD spectrum is
given together with the simulations. In (A) the simulations for monomer and trimer are shown. This comparison illustrates the sensitivity of the CD spectrum
to intersubunit interactions. These couplings clearly
result in shifts of features, see e.g. the 815 nm and
800–805 nm bands. The change of the CD spectrum
is more pronounced than that of the absorption spec−→
trum. The reason is the rkm factor in Equation (C6) for
the calculation of the rotational strength (compare to
Equation (C2)).
The difficulties in simulating the exact CD spectrum for the FMO complex are well-known. Comparing the simulation to the experiment it can nevertheless
be stated that the resemblance in the long wavelength
region is reasonable, reproducing the features near
825, 815 and 813 nm with the correct signs. The feature around 805 nm was not observed experimentally
at 77 K and was only clearly present at 4 K (see Figure 2). In most spectra published so far it was therefore
not observed, the only exception being the paper by
Vulto et al. (1998) where it was less pronounced. The
simulated spectrum in Louwe et al. (1997b) using the
monomeric subunit did not produce such a feature nor
does the simulation based on the whole trimer (Figure 10A). However, by taking the same site energies as
Louwe et al. (1997b) in a Monte Carlo simulation and
using also an IDF of 80 cm−1 for all site energies, such
a feature also appears in the simulations with the correct sign and position (B). Including a homogeneous
line shape in (C) does not change the CD spectrum significantly. Lifetime broadening ‘smears out’ the bands,
especially on the blue side of the spectrum (D, E).
The most conspicuous discrepancy in the shape of the
experimental and simulated spectra is around 800 nm
where a positive experimental band is observed which
is apparently split into two bands in the simulation (A,
B, C, D). Moreover, the simulated band near 790 nm
is somewhat overestimated although the sign is correct. The overall resemblance between the shapes of
the calculated and measured spectra is reasonable but
definitely not perfect.
When the global fit is performed the agreement
between experiment and simulated CD spectra is considerably improved (Figure 10E). The bands around
825, 813, 805 and 800 nm appear with the correct signs and their relative intensities are reasonable.
Moreover, the bands appear at the correct wavelengths
in contrast to the simulations in (A, B, C, D). The main
disagreement of the experimental CD spectrum and
simulation (E) is the negative band around 815 nm,
which is merely seen as a little dip in the simulation.
It should be noted, however, that in the 77 K spectrum
measured by Vasmel et al. (1983), this 815 nm band
is clearly less pronounced than in our spectrum. The
band around 790 nm is overestimated in size in sim-
117
ulation (E). However, this region is also not perfectly
described in the (polarized) absorption and LD spectra
(Figures 7, 9.1, 9.2 and 9.3).
We have shown that it is possible to get a reasonable description of the shape of the CD spectrum by
using the trimeric structure of the FMO complex. We
note that this was not possible by using the monomeric
subunit only (not shown).
Amplitude of the CD spectrum
In spite of the reasonable shape of the simulated CD
spectrum, there is a strong deviation between its intensity and that of the experimental CD spectrum,
which was determined relative to the experimental
absorption spectrum. This discrepancy was not observed/discussed by Louwe et al. (1997b). One reason
why the simulated spectrum is too low is the fact that
so far the index of refraction n in Equation (C6) was
taken to be 1 (see Figure 10), similar to all previous CD calculations on the FMO complex. Increasing
the value of n leads to a concomitant increase of the
intensity of the calculated CD spectrum without affecting its shape in any way. Therefore, in principle
the height of the spectrum can easily be adjusted by
varying the value of n.
The fitting procedure suggests n = 2.6 in Figure 10E. However, a realistic value for n should lie
somewhere between the value for the solvent (1.33
for water) and the value for the immediate protein
environment. For the likewise water-soluble lightharvesting peridinin–chlorophyll-a-protein a value of
n = 1.6 ± 0.1 was found (Kleima et al. 2000a). The
experimental uncertainty in the amplitude of the CD
spectrum is approximately 20% due to the error margins in the calibration of our setup. Therefore, our
estimated n seems to be too large. One might speculate that intrinsic CD of some BChl a molecules
might also contribute to the experimental CD spectrum, whereas the calculated CD spectrum is purely
excitonic. Possibly, the interactions of the Qy with
higher-energy transitions (Qx , Soret) also might play
a role (Koolhaas et al. 1997b).
Koolhaas et al. (1998) have investigated the influence of small changes in the directions of the transition
dipole moments on the excitonic CD spectrum of
the light-harvesting complex LH 2 of the purple bacterium Rhodopseudomonas acidophila. Starting from
the crystallographic structure, the transition dipole
moment vectors of all BChl a molecules were rotated
by a small angle. It was observed that such small
rotations have a very small effect on the absorption
spectrum but dramatically change the excitonic CD
spectrum.
In Figure 11, this is demonstrated for the FMO
complex from Pc. aestuarii. The dipole moment vector of a selected BChl a molecule was rotated within
the plane of the porphyrin ring by +5 or −5◦. This was
done for all three symmetry-related BChl a molecules
in the trimer. The directions of the other dipoles remained unaltered. This gives an impression of the
changes in the excitonic CD spectrum, which are induced by just one change. Especially the changes
induced by rotating BChl a 1 by +5◦ and BChl a 4 by
−5◦ are remarkable. In these cases, the negative CD
band around 815 nm becomes stronger and shows a
better agreement with the experimental CD spectrum.
It seems tempting to try to simulate the CD spectrum by changing some or all of the directions of the
transition dipole moments. However, being aware of
the simplifications in our model, this is not very useful. We present the simulations in Figure 11 simply to
bring this effect to the attention of the reader.
Screened dipole strength C∗d
The coupling strength Vkm between two point dipoles
is given by Equation (8). The dipole strength d of BChl
a (corrected for the solvent) has been reported to be
39.9 ± 0.3 D2 (Alden et al. 1997) for the main band
without the vibronic tail. Including this tail would increase the value to ∼50 D2 . For the simulations by
Louwe et al. (1997b) a value of 28.7 D2 was used for
C ∗ d, where C was identified as 1/εr and εr is the
relative dielectric constant (being 1 in vacuum), which
is equal to n2 at optical frequencies. For d = 39.9
D2 this would imply that C = 0.72. However, as was
also pointed out by Renger and May (1998), C is not
identical to 1/εr but in addition a local field factor fl
should be taken into account, i.e. C = fl2 /εr . Renger
and May used the Lorentz factor:
fl =
εr + 2
3
(14)
However, it was shown by Alden et al. (1997) that the
use of the empty cavity model with
fl =
3εr
2εr + 1
(15)
gives a much better description of the refractive index dependence of the absorption of BChl a. Using
this expression, n = 1.6 leads to a value of C = 0.62,
118
whereas n = 1.33 leads to C = 0.77. Therefore, from
a physical point of view, values of C ∗ d can easily
range between 25 and 40 D2 , given the uncertainty in
the dipole strength and the refractive index. According
to recent simulations by Iseri and Gülen (1999) using
values of C ∗ d between 30 and 40 D2 , good fits can be
obtained for the various spectra, provided that BChl a
3 is the lowest-energy pigment. This is in reasonable
agreement with our estimate of 30.6 D2 for C ∗ d (see
Table 1).
Conclusions
An improved description of the absorption and LD
spectrum is obtained by simply using the same site
energies and coupling strengths that were presented
by Louwe et al. (1997b), for which the inclusion of
the three broadening mechanisms mentioned proved
to be essential. An even better agreement of experiment and theory is obtained, when the site energies
and the coupling strengths are adjusted in a global fit
to the experimental absorption, LD and CD spectra.
Also the amplitude of the LD spectrum was described
rather well, and the same is true for the shape of the
CD spectrum.
The calculated CD spectrum was too small, using conventional calculation methods like presented
for instance by Pearlstein (1991) in combination with
the parameters of Louwe et al. (1997b). Part of the
discrepancy is explained by the fact that conventional
calculations do not account for the index of refraction
of the medium surrounding the chromophores, which
leads to an effective reduction of the wavelength of the
light.
It was shown that the estimated screened dipole
strength is compatible with the value of the dipole
strength of BChl a, when using the empty-cavity
model for the local-field correction factor.
Appendix A: theory for obtaining absolute
linear-dichroism spectra
The following theory is an excerpt from van Amerongen and Struve (1995) and it is tailored to applications
in this paper. In an LD experiment, one measures the
difference () in absorption (A) A = Az − Ax , where
the z-axis is the axis along which the gel expands
after squeezing in both the x- and y-direction. The
reduced LD (Ared ) is defined as A/(3Aiso) where
Aiso is the isotropic absorption. For a disc-like trimeric
complex, which contains only one type of pigment
with one specific absorption band and one specific
angle β µ between the transition dipole moment and
the C3 -symmetry axis, the following relation holds:
Ared ≡
A
Az − Ax
=
=
3Aiso
Az + Ax + Ay
(A1)
Az − Ax
= P2 (cos β)
P2 (cos βµ )
Az + 2Ax
where P2 (x) = (3x2 − 1)/2 is the second order Legendre polynomial. The second term P2 (cosβµ ) on
the right-hand side depends entirely on the transition
dipole moment orientation within the trimer and is an
intrinsic property of the complex. The first term on the
right-hand side describes the ‘average’ orientation of
the complexes in the squeezed gel, β being the angle
between the C3 -symmetry axis of the complex and the
z-axis of the gel. The brackets indicate averaging over
all possible orientations of the complexes in the gel.
P2 (cos β)
is also called the degree of orientation or
orientation factor. It equals 0 in the absence of compression, whereas it is −0.5 in the case of maximal
ordering upon 2-D squeezing, i.e. the C3 -axis of all
complexes is oriented perpendicularly to the z-axis and
randomly within the x−y plane of the gel. In the conventional LD measurements, the value of P2 (cos β)
is unknown. Therefore, the molecular parameter of
interest, namely P2 (cos βµ ), cannot be determined
without additional information (vide infra).
In case several pigments are present per complex
with different absorption bands and different orientations of the transition dipole moments with respect
to the symmetry axis, Ared becomes wavelength (λ)
dependent in contrast to the case treated above, and it
is given by
Ared (λ) = P2 (cos β)
P2 (cos βµ )
λ
(A2)
The second term now reflects a weighted average over
the different pigments (or excitonic states) contributing at wavelength λ. Their contributions are weighted
by their (absorption) dipole strength at λ and the
formal expression for P2 (cos βµ )
λ is
N
P2 (cos βµ )
λ =
P2 (cos βµi )Ai (λ)
i=1
N
i=1
(A3)
Ai (λ)
119
where i enumerates the N various pigments (or exciton states) and Ai (λ) is the absorption of pigment or
exciton state i at λ.
Note that upon squeezing P2 (cos β)
changes, but
P2 (cos βµ )
λ remains the same. Therefore, the shape
of the reduced-LD spectrum Ared (λ) remains unaltered. We note that in most of the recent exciton
models only the shape of the LD spectrum could be
compared, since its amplitude (i.e. corrected for the
degree of orientation of the sample) had not been
determined experimentally. One of the aims of the
present study is to compare the spectra in a quantitative way. For this purpose, additional use is made of
polarized fluorescence excitation spectra at cryogenic
temperatures.
The fluorescence excitation anisotropy spectrum
r(λ) (see Equation (2)) is related to the spectroscopic
properties of the complex as follows:
r(λ) =
2
P2 (cos βν )
λdet P2 (cos βµ )
λ
5
P2 (cos βν )
λdet =
N
(A5)
Fi (λdet )
i=1
where Fi (λdet ) is the fluorescence intensity of pigment or state i at wavelength λdet for an excited-state
equilibrated system. βν is the angle between the emitting transition dipole moment and the C3 -axis of the
complex.
Equation (A5) is a relatively complicated expression which depends for instance on the temperature. At sufficiently low temperatures, all fluorescence stems from the lowest-energy band and
P2 (cos βν )
λdet becomes equal to P2 (cos βν1 ), where
the lowest-energy band has been numbered 1.
For Equation (A4) to hold the equilibration within
the trimer should be much faster than the excited-state
lifetime in the complex, i.e. the excitation that finally
leads to fluorescence from one of the monomeric subunits was absorbed with equal probability by either
one of the three subunits. It thus can be seen, that
for a particular choice of the detection wavelength,
P2 (cos βν )
λdet is just a constant, independent of the
(A6)
This means that the spectral shape of r(λ) is
identical to that of Ared (λ) (see Equation (A2)),
provided that subunit equilibration, i.e. energy exchange between subunits, is indeed fast enough.
In the case of the FMO complex at low temperatures, one ‘isolated’ absorption band is present around
825 nm, which is much lower in energy than the next
lowest one around 815 nm. Therefore, at sufficiently
low temperature (e.g. 4 K) all fluorescence arises from
this band, which we call state 1 or band 1. If one selectively excites this lowest-energy band, then absorption
and emission occur by/from the same state or from one
of the two other equivalent states in the trimer, and
if subunit equilibration takes place and is fast enough
one obtains:
r(λ) =
P2 (cos βνi )Fi (λdet )
i=1
r(λ) ∼ P2 (cos βµ )
λ
(A4)
where λ is the wavelength of excitation and
P2 (cos βν )
λdet reflects a weighted average over the
emitting pigments or states at detection wavelength
λdet . It is formally given by
N
excitation wavelength λ, leading to
2
P2 (cos βν1 )P2 (cos βµ1 )
5
(A7)
for this special case. Since absorption and fluorescence
correspond to the same electronic transition, βν1 = βµ1
and the anisotropy provides P2 (cosβµ1 ) for the lowestenergy state. This allows absolute calibration of the
LD and reduced-LD spectra. However, it is possible
that at these ultra low temperatures subunit equilibration is not obtained. In order to avoid too slow
equilibration one can raise the temperature. However,
in that case fluorescence will also start to arise from
states that are higher in energy. Since these states have
a different value for P2 (cosβν ) which is opposite in
sign to that of the lowest-energy state, the absolute
value of the anisotropy will drop (see also ‘Results
and discussion’). Therefore, the anisotropy excitation
spectrum r(λ) has to be measured at different temperatures in order to assess the degree of equilibration
but at the same time to estimate the amount of emission arising from states lying higher in energy. It is
shown in ‘Results and discussion’ that the shape of
Ared (λ) and r(λ) are identical in the largest part of
the spectrum but that differences exist in the lowestenergy band, which are ascribed to inhomogeneous
broadening. Moreover, the absolute size of the anisotropy excitation spectrum decreases upon increasing
the temperature whereas its shape remains essentially
unchanged. This reflects the expected temperature dependence of r(λ) based on the Boltzmann equilibration
over the excited-state levels (see Appendix B).
120
Appendix B: temperature dependence of the
anisotropy
The anisotropy is temperature dependent if at different temperatures the equilibration over pigments or
states with contrasting transition dipole orientations
varies. In the case of two pigments/states per monomer
with excited electronic state energies hν1 and hν2 ,
which have equal transition dipole strength, the relative population of these states at temperature T will be
e−hν1 /(kB T ) /e−hν2 /(kB T ) where h is Planck’s constant
and kB is the Boltzmann constant. The anisotropy in
that case is given by
2
r=
ri e−hνi /(kB T )
i=1
2
(B1)
e−hνi /(kB T )
i=1
where use has been made of the additive property of the anisotropy (see e.g. van Amerongen and
Struve 1995) and ri is the anisotropy if state i would
emit. Note that ri varies as a function of excitation
wavelength but the expression is valid for any excitation wavelength as long as it is kept fixed. If
different molecules/states have different emitting dipole strengths, then the contributions of both states
to the emission have to be weighted by these dipole strengths. Assuming that the energy separation
between both states is identical for absorption and
emission, one obtains
2
r=
ri Aiso,i e−hνi /(kB T )
i=1
2
(B2)
Aiso,i
e−hνi /(kB T )
i=1
where use has been made of the fact that the dipole
strength for absorption scales linearly with that for
emission for electronic transitions.
In case of real pigment–protein complexes, many
bands are present, all having a finite width and in that
case Equation (B2) generalizes to
rλdet =
r(ν)Aiso(ν)e−hν/(kB T ) dν
Aiso (ν)e−hν/(kB T ) dν
(B3)
where ν runs over the states that contribute to the
absorption (and emission).
Since in the case of subunit equilibration r(ν) =
C A(ν)/(Aiso(ν)) (see Appendix A) one can write
Aiso(ν)e−hν/(kB T ) dν
C AA(ν)
iso (ν)
rλdet =
= (B4)
Aiso(ν)e−hν/(kB T ) dν
C A(ν)e−hν/(kB T ) dν
Aiso(ν)e−hν/(kB T ) dν
where C is a scaling constant, whose value is not important for the T dependence. For A(ν) and Aiso(ν)
the spectra at the respective temperature have to be
taken. Therefore, from the T dependence of the absorption and LD spectra, the T dependence of the
anisotropy can be directly calculated. In this way,
anisotropy results at higher temperatures, where efficient equilibration occurs, can be extrapolated to
low temperatures where emission stems from the
lowest-energy state with equal probability from all
monomers.
Appendix C: exciton simulations
Absorption, LD and CD spectra are calculated on the
basis of an exciton model in a Monte Carlo simulation (Fidder et al. 1991; Monshouwer et al. 1997; van
Amerongen et al. 2000; Kleima et al. 2000b). In each
Monte Carlo iteration, a Hamiltonian is generated with
the diagonal elements representing the site energies
of the pigments and the other matrix elements the interactions between the pigments. This Hamiltonian is
numerically diagonalized (Press et al. 1992) and the
eigenvalues and eigenvectors are used to calculate the
energies and transition dipoles of the exciton states.
Typically 10 000 iterations are performed. The site
energy for each pigment is allowed to vary around
a mean value according to a Gaussian distribution
function.
If cJ m is the m-th element of the eigenvector for the
J-th exciton state, then the transition dipole moment
−→
µJ of that state is given by
−→
µJ
=
N
−→
(C1)
c J m µm
m=1
−→
where µm is the transition dipole moment of pigment
m. The (absorption) dipole strength DJ of that exciton
state is
−→2
DJ = µJ =
N
k,m=1
−→ −→
c J k c J m ( µk · µm )
(C2)
121
−→
From the calculated transition dipole moments µJ
of the exciton states the absolute (difference) absorption spectra with respect to the molecular frame of
the complex Ac , Aa and LD can be calculated in a
straightforward way (see also Pearlstein 1991). When
ĉ is the unit vector in the direction of the C3 -symmetry
axis, the dipole strengths DJc and DJa of the components of the J -th exciton state which are parallel and
perpendicular to this axis, respectively, are given by
−→
DJc = ( µJ ·ĉ)2
DJa =
(C3)
1
(DJ − DJc )
2
(C4)
The difference provides the absolute LD:
LDJ = DJc − DJa
(C5)
The CD spectrum is the difference in absorption of
left and right circularly polarized light, CD = AL −AR .
The difference in dipole strength of left and right
circularly polarized light for the exciton state J is
determined by the rotational strength RJ :
CDJ = DJL − DJR = 4RJ =
−
−→
2πn
λ
N
−→
(C6)
−→
−→
cJ k cJ m rkm ·( µk × µm )
k,m=1
where rkm is the vector from the centre of pigment
k to that of pigment m. λ is the wavelength of the
light (in vacuum). To correct for the effect of the medium on the wavelength, the refractive index n has
been included. But as it is a priori not evident, which
value has to be used, we initially have used n = 1 (see
‘Results and discussion’).
Note, that the dipole strength DJ in Equation
(C2) corresponds to 3∗Aiso , CDJ in Equation (C6)
to 3∗(AL − AR ) and DJc , DJa and LDJ in Equation
(C3) to Equation (C5) to Ac , Aa and LD, respectively.
The factor 3 in absorption and CD arises from orientational averaging in an isotropic sample; Ac , Aa and
LD are, however, calculated in the molecular frame.
As can be seen from Equation (C2) to Equation (C6)
the absorption, Ac , Aa , LD and CD spectra can all
be expressed in ‘absorption units’, i.e. for scaling the
theoretical absorption (difference) spectra to the experimental spectra a common scaling factor has to be
used for all of them.
Acknowledgements
The authors thank Ivo van Stokkum for stimulating
comments on the data fitting procedure. This research
was supported by the Netherlands Foundation for Scientific Research (NWO) via the Foundation for Earth
and Life Sciences (ALW). M. W. received a MarieCurie Fellowship (EC grant ERB FMBICT 960842).
M. A. P. was a guest student from Wroclaw University of Technology, supported by a Tempus grant.
T. J. A. acknowledges support by the EC (contract
FMRX-CT960081).
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