Minimal Model of Quantum Kinetic Clusters for the Energy-Transfer
Network of a Light-Harvesting Protein Complex
Jianlan Wu,*
,†
Zhoufei Tang,
†
Zhihao Gong,
†
Jianshu Cao,
‡
and Shaul Mukamel
¶
†
Physics Department, Zhejiang University, 38 ZheDa Road, Hangzhou, Zhejiang 310027, China
‡
Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139,
United States
¶
Department of Chemistry, University of California, Irvine, California 92697, United States
*
S
Supporting Information
ABSTRACT: The energy absorbed in a light-harvesting protein complex is
often transferred collectively through aggregated chromophore clusters. For
population evolution of chromophores, the time-integrated effective rate
matrix allows us to construct quantum kinetic clusters quantitatively and
determine the reduced cluster−cluster transfer rates systematically, thus
defining a minimal model of energy-transfer kinetics. For Fenna−
Matthews−Olson (FMO) and light-havrvesting complex II (LCHII)
monomers, quantum Markovian kinetics of clusters can accurately
reproduce the overall energy-transfer process in the long-time scale. The
dominant energy-transfer pathways are identified in the picture of
aggregated clusters. The chromophores distributed extensively in various clusters can assist a fast and long-range energy transfer.
W
ith almost unity transfer efficiency, the early energy-
transfer step in light-harvesting protein complexes has
attracted great interest from both experimental and theoretical
perspectives.
1
The two-dimensional electronic spectroscopy
technique can explore major energy-transfer pathways from the
detailed time-res olved spectros copic studies
2,3
and has
demonstrated long-lived quantum coherence in various light-
harvesting systems, suggesting its role in the energy-transfer
process.
4−6
Theoretical studies of energy transfer are carried
out under the quantum dynamic framework, beyond the
conventional incoherent hopping picture. The bath-induced
quantum dissipation is found to result in an optimal transfer
efficiency in the intermediate dissipation regime,
7−12
which is
close to the physiological condition of light-harvesting protein
complexes.
The light-harvesting protein complexes are rich in their
spatial structures and energy-transfer dynamics. The bacterio-
chlorophyll (BChl) molecules in light-harvesting I (LHI) and II
(LHII) complexes of purple bacteria are organized into
symmetric rings. On the short-time scale, the intraring energy
transfer can behave coherently due to symmetric and non-
negligible interactions between BChl molecules.
13,14
On the
long-time scale, the inter-ring energy transfer becomes
incoherent, and multichromophoric Fo
rster resonance energy
transfer (MC-FRET) is proposed for calculating the dynamics
of aggregates.
15−17
For many other light-harvesting systems, the
spatial arrangement of chromophores is not symmetric, but
aggregated clusters can be still formed because interactions
between chromophores can be partitioned into groups. The
overall energy-transfer process is separated into fast intracluster
and slow intercluster dynamics. For example, the major two-
steppathwayinaneight-BChlFenna−Matthews−Olson
(FMO) monomer can be approximated in a three-cluster
system, where BChls 1 and 2 are simplified into a pre-
equilibrated dimer (see Figure 1a).
8,18,19
In the light-harvesting
complex II (LHCII) system, previous studies have suggested
three spatial clusters on the stromal layer, chlorophylls (Chls)
a602-a603, a610-a611-a612, and b601′-b608-b609, and two
clusters on the lumenal layer, Chls b606-b607-b605-a604 and
a613-a614.
20−22
The characterization of clusters in previous studies is mainly
based on the spatial separation, which is a static picture and
structure-based. However, for an energy-transfer network, one
site can extensively interact with multiple sites as a bridge
connecting various spatially separated clusters.
6
The fast, long-
range energy transfer assisted by such a hub site modulates
energy-transfer pathways in light-harvesting protein complexes.
To illustrat e this important contribu tion of d elocalized
quantum coherence, we will treat each cluster as a linear
combination of sites with fractional coefficients in this Letter.
As demonstrated in two example systems, FMO and LHCII, an
effective site−site rate matrix helps us construct clusters
systematically and quantitatively, and quantum Markovian
kinetics of clusters can reproduce the overall population
evolution and the equilibrium distribution accurately. This
method can also identify relevant hub sites with large partition
coefficients in connecting clusters. These hub sites are crucial
for maintaining the stability of the energy-transfer network.
Received: February 2, 2015
Accepted: March 19, 2015
Published: March 19, 2015
Letter
pubs.acs.org/JPCL
© 2015 American Chemical Society 1240 DOI: 10.1021/acs.jpclett.5b00227
J. Phys. Chem. Lett. 2015, 6, 1240−1245
For an energy-transfer network in light-harvesting systems,
the total Hamiltonian is given by H
tot
= H
S
+ H
B
+ H
SB
, where
H
S
and H
B
represent the bare Hamiltonians of the system and
bath, respectively. Within the single-excitation manifold, the
system Hamiltonian is written as H
S
= ∑
n
ϵ
n
|n⟩⟨n| + ∑
n≠m
J
nm
|
n⟩⟨m|, where ϵ
n
defines the electronic excitation energy at site n
and J
nm
defines the electronic coupling between sites n and m.
The system−bath interaction H
SB
is approximated in a bilinear
form, H
SB
= ∑
n
|n⟩⟨n|B
n
, where B
n
is a bath operator. Under a
system−bath facto rized initial conditi on, we assume an
incoherent site population preparation for the system. The
time evolution of the system population P(t) reduced from full
quantum dynamics rigorously follows a time-convolution
equation
∫
τττ
=− −
P
ttP() ( ) ()d
t
0
2
(1)
where
t()
2
is the non-Markovian rate kernel matrix. Despite
relevant non-Markovian features in the short-time scale, the
quantum Markovian kinetics approximately describe t he
population evolution, P
(t)=−KP(t), where t he time
integration
∫
=
∞
Ktt()d
0
2
(2)
leads to effective rates between different chromophores
quantitatively.
23
The effective rate matrix K can be obtained
by the block matrix inversion method in the hierarchy equation
of motion (HEOM).
24,25
Our first example is the FMO protein complex in green
sulfur bacteria. Recent experiments have revealed the eighth
BChl molecule locating on the surface of the FMO protein
complex. The 8-site Hamiltonian from refs 8 and 19 is applied
to the FMO monomer. The system−bath coupling is modeled
by the Debye spectral density, J(ω)=(2λ/π)ωω
D
/(ω
2
+ ω
D
2
),
with the reorganization energy λ =35cm
−1
and the cutoff
frequency ω
D
−1
= 50 fs.
7−9
At room temperature (T = 300 K),
the high-T approximation leads to a single-exponential decay
for the bath−time correlation function C(t). Because energy is
transported from the chlorosome, we set the initial population
at BChl 8. The full quantum dynamics is calculated using the
HEOM, which is numerically converged at the hierarchic
truncation order of H = 7. The resulting population dynamics is
plotted in Figure 2, consistent with previous studies.
8,19
Next,
we extract the major energy-transfer pathways. The population
initially localized at BChl 8 is first transferred to BChl 1 due to
their relatively strong electronic coupling strength (37.5 cm
−1
).
A much stronger coupling strength (−97.9 cm
−1
) quickly
synchronizes the population of BChls 1 and 2, and these two
chromophores behave like a dimer.
8,18,19,26,27
After the
population of the dimer reaches its maximum value within 1
ps, the population starts to accumulate at BChl 3 with the
lowest excitation energy. The other chromophores also
gradually increase their populations. The FMO monomer
reaches full equilibrium in less than 5 ps.
Following the block matrix inversion method, we calculate
the time-integrated rate matrix K defined in eq 2 (see the
Supporting Information). This effective rate matrix is
independent of the initial system population. As shown in
Figure 2, Markovian kinetics, P(t) = exp(−Kt)P(0), agree with
non-Markovian dynamics for t > 2 ps, predicting exactly the
same equilibrium population. Only a small non-Markovian
effect can be observed for t < 2 ps. The energy-transfer rates are
not evenly distributed between different pairs of chromophores,
and the population evolution exhibits a time scale separation;
sites under fast local equilibrium aggregate into quantum
kinetic clusters, and the excitation energy is transferred
collectively through the clusters, especially in the long-time
scale. From Markovian kinetics, the population of each jth BChl
molecule is written as
Figure 1. Schematic pictures of (a) four quantum kinetic clusters in the 8-site FMO and (b) five quantum kinetic clusters in the 14-site LHCII. Each
cluster is labeled and identified by a gray ellipse. The major reduced cluster−cluster transfer rates are provided.
Figure 2. Time evolution curves of the 8-site FMO with the initial
population at BChl 8: (a) the site populations; (b) the cluster
populations. The dotted and solid lines are calculated by the non-
Markovian HEOM and Markovian kinetics, respectively. The results of
each BChl site in (a) and each kinetic cluster in (b) are labeled and
shown in a specific color.
The Journal of Physical Chemistry Letters Letter
DOI: 10.1021/acs.jpclett.5b00227
J. Phys. Chem. Lett. 2015, 6, 1240−1245
1241
∑
γ=−
=
P
tc t() exp( )
j
k
jk
k
0
7
(3)
where γ
k
is the kth eigenvalue of the rate matrix K, satisfying
γ
0
(=0) < γ
1
< ··· < γ
7
. The coefficient c
j0
associated with the
zero eigen rate is the equilibrium population p
j
eq
of the jth BChl
molecule. For the final site, BChl 3, the coefficients c
3k(≤3)
associated with the four smallest eigen rates γ
k(≤3)
are at least 30
times larger than the other four coefficients (see eq 5 of the
Supporting Information). Thus, a four-cluster minimal model
can be constructed to represent the energy-transfer network if
we are mainly interested in long-time dynamics with population
accumulation at BChl 3.
To separate the fast and slow components of population, we
introduce a linear transformation to combine the population of
each ith cluster as P
i
C
(t)=∑
j
S
ij
P
j
(t)=∑
j,k
S
ij
c
jk
exp(−γ
k
t),
where the elements S
ij
of the transformation matrix need to be
determined. For the four quantum kinetic clusters, the partition
coefficients associated with fast eigen rates must vanish, that is,
∑
j
S
ij
c
jk
= 0 for k > 3. The system Hamiltonian provides a crude
estimation of kinetic clusters. For instance, the weak couplings
between BChl 8 and the other sites suggest a single-site cluster.
Our cluster partition procedure (see more details in the
Supporting Information) is described as follows: (i) For each
cluster, we keep five sites and neglect the three least relevant
sites based on the electronic couplings. Each of the initial (BChl
8) and final (BChl 3) sites is assigned to an individual cluster,
while a dominant site is selected as the reference site in each
intermediate cluster. (ii) For each cluster, we set a unity
partition coefficient for the reference site, and the remaining
four partition coefficients are determined relatively by solving
the four-equation array, ∑
j
S
ij
c
jk
=0(k =4−7), from fast
kinetics. (iii) The normalization condition ∑
i=1
4
S
ij
= 1 resolves
the transformation matrix S uni quely. Consequently, we
construct the following four quantum kinetic clusters
=+
=+++
=+++
=+ + +
Pt Pt Pt
Pt Pt Pt Pt Pt
Pt Pt Pt Pt Pt
P t Pt Pt Pt Pt
() () 0.06 ()
( ) 0.92 ( ) 0.96 ( ) 0.06 ( ) 0.07 ( )
( ) 0.93 ( ) 0.93 ( ) 0.62 ( ) 0.47 ( )
( ) ( ) 0.53 ( ) 0.37 ( ) 0.07 ( )
I
C
81
II
C
1256
III
C
5674
IV
C
3472
where only components with |S
ij
| ≥ 0.05 are shown for clarity.
Although the cluster construction exhibits flexibility, our
approach assisted by the system Hamiltoni an separation
captures physically appealing clusters in the minimal model.
The partition coefficients for the fast population kinetics can be
similarly calculated. The transformed rate matrix K′ = SKS
−1
becomes block diagonal, with two 4 × 4 submatrices separating
fast and slow eigen rates. The reduced rate matrix K
C
for the
quantum kinetic clusters is given by
=
−−
−−−
−− −
−−
−
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
K
2.401 0.410 0.058 0.053
2.092 1.635 0.140 0.468
0.354 0.230 1.725 0.762
0.044 0.996 1.528 1.176
ps
C1
(4)
The eigen rates of K
C
are exactly the same as those for the four
relevant ones of the site−site rate matrix K (see eq 4 in the
Supporting Information). The quantum kinetic clusters in
FMO and their dominant effective rates are plotted in Figure
1a. We observe a major energy-transfer pathway, clusters I
(BChl 8) → II (BChls 1 and 2) → IV (BChls 3, 4, and 7),
whereas cluster III (BChls 5, 6, 4, and 7) modulates the energy-
transfer process through a minor pathway, clusters I → III →
IV. Next, we plot the cluster population evolution from the
Markovian kinetics, using P
C
(t) = exp(−K
C
t)P
C
(0), compared
with the result of the non-Markovian HEOM in Figure 2b. For
kinetic clusters, the results of Markovian kinetics and non-
Markovian dynamics agree well with each other. The major
pathway is revealed by the subsequent population accumulation
along clusters I → II → IV. The dominant intermediate, cluster
II, is identified with its maximum population within 1 ps. The
two-step energy-transfer scenario has been proposed pre-
viously,
8,19
and our current work provides a more systematic
and quantitative description using the concept of quantum
kinetic clusters.
Our constructed clusters are not limited to a special initial
state at BChl 8. As an example, we prepare the initial
population at BChl 6 and calculate the new population
evolution. Figure 3 demonstrates that Markovian kinetics
from the same cluster rate matrix in eq 4 agree well with the
exact non-Markovian dynamics, although the site rate matrix
fails to reproduce a short-time quantum oscillation of BChls 6
and 5. The time evolution of aggregated clusters can be reliably
described by quantum Markovian kinetics. With the same
cluster rate matrix, the energy-transfer pathway changes with
the initial condition. With the initial population at BChl 6, the
pathway, clusters III → IV, becomes dominant, consistent with
the previous site-based pathway studies in the 7-site FMO
system.
9,18,26,27
The “hub” site, BChl 4, extensively distributed
in clusters III and IV controls the population flow along the
major energy-transfer pathway and realizes a fast (∼1 ps) long-
range energy transfer from BChls 6 to 3. BChl 4 is crucial for
the stability of the FMO network, and a permanent damage on
this hub site can significantly delay the energy-transfer process
(∼5 ps) when the initial population is at BChl 6.
Our second example is the LHCII, which is an important
antenna protein complex, initiating the excitation energy-
transfer process in the photosystem II (PSII). Each LHCII
monomer is composed of 14 chlorophyll (Chl) molecules, 8
Chl-a and 6 Chl-b. The effective 14-site Hamiltonian is taken
from ref 20. The model bath spectral density includes a low-
Figure 3. Time evolution curves of the 8-site FMO with the initial
population at BChl 6: (a) the site populations; (b) the cluster
populations. The dotted and solid lines are calculated by the non-
Markovian HEOM and Markovian kinetics, respectively. The results of
each BChl site in (a) and each kinetic cluster in (b) are labeled and
shown in a specific color.
The Journal of Physical Chemistry Letters Letter
DOI: 10.1021/acs.jpclett.5b00227
J. Phys. Chem. Lett. 2015, 6, 1240−1245
1242
frequency Debye mode and 48 high-frequency Brownian
oscillator modes. For simplicity, we only focus on the low-
frequency Debye spectral density (λ =85cm
−1
and ω
D
−1
= 53.08
fs) and ignore all of the high-frequency components. This
approximation neglects population dynamics from resonant
electron−phonon coupling,
20−22,28,29
which is however accept-
able considering our main purpose of demonstrating aggregated
kinetic clusters. LHCII is a membrane protein complex, with
both stormal and lumenal sides. Here, we set the initial
population at Chl b606 on the lumenal side and inspect the
population accumulation at Chls a610-a611-a612 on the
stormal side. With the high-temperature approximation for T
= 300 K, the HEOM is truncated at H = 6, where the average
numerical error of energy-transfer rates is less than 3%.
30
The population evolutions from the non-Markovian HEOM
and the Markovian kinetics of the time-integrated rate matrix
(see the Supporting Information) are plotted in Figure 4. The
result of population dynamics is qualitatively consistent with
previous studies.
20−22,28,29
These two approaches lead to almost
the same population dynamics, except for observable differ-
ences at short times. The dynamical behaviors in the two panels
of Figure 4 reveal a strong time scale separation.
20−22,28,29
In
the short-time regime (<0.5 ps), population is quickly
transferred from Chl b606 to Chls a604 (interband) and
b607 (intraband) due to their strong interactions. After
populations of Chls a604 and b07 reach maximum values,
these three sites on the lumenal side are in local equilibrium.
Subsequently, the population slowly diffuses to the entire
LHCII monomer and accumulates on the stromal side. In the
long-time regime, population dynamics become much more
complicated, involving multiple energy-transfer pathways.
We now apply our approach used in FMO to construct
quantum kinetic clusters in the LHCII monomer. By expanding
the multiexponential functions of Markovian kinetics, we find
that the population evolution of the final sites, Chls a610-a611-
a612, are dominated by the five smallest eigen decay rates (see
eqs 10−12 of the Supporting Information). The 14-site LHCII
monomer is kinetically reduced into a five-cluster minimal
model. Following the same procedure in FMO, we set the
initial cluster of Chl b606 and the final cluster of Chls a610-
a611-a612. As for the intermediate clusters, Chl b605 must be
isolated to avoid an unphysically large partition coefficient in
other clusters. Unlike previous studies,
20−22
Chls b601-b608-
b609 do not form an isolated kinetic cluster due to their
extensive interactions with multiple clusters. After solving the
relative coefficients and applying the normalization condition,
the five quantum kinetic clusters (see Figure 1b) are
determined as follows
=+++
+
=
=+
=+++
+−
=+ + +
+
Pt Pt Pt Pt Pt
Pt
Pt Pt
Pt Pt Pt
Pt Pt Pt Pt Pt
Pt Pt
Pt Pt Pt Pt Pt
Pt
( ) 1.09 ( ) ( ) 0.99 ( ) 0.13 ( )
0.10 ( )
() ()
( ) 0.99 ( ) 0.99 ( )
( ) 0.97 ( ) 0.96 ( ) 0.87 ( ) 0.76 ( )
0.57 ( ) 0.05 ( )
( ) ( ) 1.02 ( ) 1.02 ( ) 0.29 ( )
0.21 ( )
I
C
46 7 8
9
II
C
5
III
C
13 14
IV
C
2391
84
V
C
10 11 12 8
1
where only components with |S
ij
| ≥ 0.05 are shown for clarity.
The effective rate matrix for this five-cluster system is given by
=
−−−−
−
−− −−
−−− −
−−−−
−
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
K
0.164 0.203 0.006 0.024 0.002
0.045 0.218 0.000 0.001 0.000
0.015 0.002 0.081 0.017 0.016
0.081 0.007 0.024 0.236 0.080
0.024 0.006 0.051 0.195 0.097
ps
C1
(5)
The quantum kinetic clusters in LHCII and their dominant
effective rates are plotted in Figure 1b. As shown in Figure 4,
the short-time population evolution (<0.5 ps) mainly occurs in
cluster I (Chls a604-b606-b607) for the local equilibrium. Next,
we plot the cluster population evolution from non-Markovian
dynamics and Markovian kinetics in Figure 5. The results of
these two approaches are almost identical, except for very small
differences in the short-time regime (<1 ps). The overall
dynamics of clusters in the LHCII monomer can be accurately
described by quantum Markovian kinetics. Following the
Figure 4. Site population evolution curves of the 14-site LHCII with
the initial population at Chl b606. Panels (a) and (b) represent short-
time and long-time results, respectively. The dotted and solid lines are
calculated by the non-Markovian HEOM and Markovian kinetics,
respectively. The result of each Chl site is labeled and shown in a
specific color.
Figure 5. Cluster population evolution curves in the 14-site LHCII
with the initial population at Chl b606. The short-time (<1 ps) results
are magnified in the inset. The dotted and solid lines are calculated by
the non-Markovian HEOM and Markovian kinetics, respectively. The
result of each cluster is labeled and shown in a specific color.
The Journal of Physical Chemistry Letters Letter
DOI: 10.1021/acs.jpclett.5b00227
J. Phys. Chem. Lett. 2015, 6, 1240−1245
1243
population flux analysis technique (see the Supporting
Information), the major pathway is a sequential energy transfer
of clusters I → IV (∼8 ps) → V(∼15 ps after initialized around
1 ps). The large ratio of k
V←I
/k
I←V
leads to a minor but
unidirectional pathway, clusters I → V(∼10 ps). The
intermediate interaction between Chls b606 and b605 allows
a fast energy transfer, clusters I → II, but the population
accumulated at the isolated cluster II is mainly transferred back
to cluster I after 6.5 ps to facilitate the two pathways above.
Cluster III (Chls a613-a614) can transfer its population to
cluster V with an intermediate rate (0.048 ps
−1
), but its weak
interaction with clusters I and IV determines that cluster III
plays a minor role in the overall energy-transfer process.
The above construction of quantum kinetic clusters and the
population evolution are qualitatively consistent with previous
studies, especially on the major energy-transfer pathways.
20−22
In our approach, the two major hub sites, Chls b608 and b609,
are extensively distributed into three clusters, I, IV, and V,
controlling the population flow. For example, the population
flowed from Chl b606 is almost 100% transferred to Chls a602-
a603 through the intermediate site Chl b609 (see the
Supporting Information). The same behavior occurs for the
population flow of Chls b606 → b608 → a610. The major
cluster pathway, I → IV → V, in the minimal model
corresponds to the pathway Chls b606-(b607-a604) → b609-
(b608-b601) → a603-a602 → a610-a611-a612 in previous
studies.
20−22
Similarly, the minor cluster pathway, I → V,
corresponds to the pathway Chls b606-(b607-a604) → b608-
(b609-b601) → a610-(a611-a612). For the subnetwork of
clusters I, IV, and V, the removal of Chls b608 and b609 can
significantly delay the overall transfer process from 10 to 70 ps.
The minimal model can thus identify the importance of these
two hub sites in the whole energy-transfer network. With high-
frequency bath modes, the cluster structure and the energy-
transfer pathways are qualitatively consistent but require a
quantitative modification.
Artificial and natural photosynthetic systems take advantage
of the Coulombic interaction to enhance the stability and
efficiency of energy transfer. The excitation electronic coupling
results in the delocalization of the wave function and hence the
formation of coherent clusters that function as the basic units of
thermal hopping. In this Letter, we propose a strategy to
construct these clusters from dynamic trajectories instead of
static structure and thus rigorously reproduce the energy flow
in between clusters. With the time-integrated effective site−site
rate matrix from full quantum dynamics, we apply the block
diagonalization approach to construct quantum kinetic clusters
in a minimal model. The proposed approach is demonstrated in
two example systems, 8-site FMO and 14-site LHCII
monomers, which are reduced to four- and fi ve-cluster
networks, respectively. The quantum Markovian kinetics of
aggregated clusters can accurately reproduce the same long-
time evolution as that from the full non-Markovian quantum
dynamics. The major energy-transfer pathways are determined
in the cluster description. More importantly, our method
reveals that energy is transferred collectively through kinetic
clusters, assisted by extensive interactions of hub chromo-
phores. This minimal model analysis can be applied to other
artificial and natural photosynthetic systems.
31,32
For complex
multisite light-harvesting systems, our investigation suggests
that full quantum dynamics may be replaced by coarse-grained
methods based on kinetic clusters determined self-consistently.
■
ASSOCIATED CONTENT
*
S
Supporting Information
Time-integrated effective rate matrix of the 8-s ite FMO,
quantum kinetic cluster construction in the 8-site FMO,
time-integrated effective rate matrix of the 14-site LHCII,
quantum kinetic cluster construction in the 14-site LHCII, and
Markovian population flux in the 14-site LHCII. This material
is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail: jianlanwu@zju.edu.cn.
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
The work reported here is supported by the Ministry of Science
and Technology of China (MOST-2014CB921203) and the
National Nature Science Foundation of China (NSFC-
21173185).
■
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