In the chapter 3, we can build CGM at some options. First, if the MBT need to rear-range? Second if the weight of minimum-cut need to normalize? And after modifying the MBT, choosing a method to build the coarse-grained model. Each method would give us a difference CGM. Fig. 4.3
By the simplest strategy, we can choose the lowest cost method. In additionally, we also check what the order of the cluster that the given method cut out. But FMO networks is too small, so all these methods do not show obviously difference. Many CGM are the same although we use different method to build. And the SR method can only build three coarse-grained model whether we tune the cut-off ratio. So we would focus on the SC and AC methods.
We can choose any number of clustering. For instance we can test the population
Figure 4.4: FMO 4 clusters population dynamics: The solid lines are the CGM dynamics, and the dash lines are the exact MRT dynamics.
dynamics with 4 clusters built by method SC, norm-SC, or norm-AC, see Fig. 4.4 The solid lines are the original population dynamics and the dashed lines are dynamics by the quasi-thermal equilibrium approximation. We can see that they are almost the same.
For a small network like FMO, all the clustering do not show obviously different. The coarse-grained model of FMO may be good in 4(or more) clusters. In Fig. 4.5, we show the simpler network of FMO. And in Fig. 4.6, we map the coarse-grained model into real space. We can see the energy flow can go through two pathways from highest energy exciton state to lowest energy exciton state.
Figure 4.5: FMO 4 clusters coarse-grained model with abstract diagram, we can see the EET flow simply.
Figure 4.6: FMO 4 clusters coarse-grained model in real space: Each ball’s centor denote the centor of each exciton state. And each color denote the cluster of CGM. The size of ball shows the relative population in the cluster base on the eigenenergies.
Chapter 5
Light Harvesting Complex II
LHCII complex is a membrane protein, it participates in the 1st steps of photosynthesis by harvesting sunlight and transfer excitation energy to the core complex. The structure of LHCII is a trimer. To simplified the system, we only consider the monomer of LHCII.
5.1 Effective Hamiltonian
To describe EET dynamics in a monomeric subunit of the LHCII complex, we follow the simulations carried out by Wu et al. [16] and adopt the effective Hamiltonian (Ta-ble 4.1) proposed by Novoderezhkin et al. [9]. Excitonic couplings in this Hamiltonian were calculated based on a 2.72 Å crystal structure [7] using a transition-dipole-transition-dipole interaction model for chlorophylls, and the site energies were determined by fitting to various experimental linear spectra of the LHCII systems. This model Hamiltonian has been tested and shown to yield time-resolved transient-absorption and fluorescence spec-tra that are in excellent agreement with the experiments, hence the model should reproduce accurate population dynamics in the LHCII monomer system. Note that although a more recent modification to the model Hamiltonian has bee proposed to improve the simulations of two-dimensional electronic spectra [1], the differences in the updated Hamiltonian are minor and should not affect the conclusion of our study.
5.2 Rate constant matrix
For simplicity, we only consider the low-frequency Debye spectral density and ignore all the high-frequency components. This approximation neglects population dynamics from resonant electron-phonon couplings[9, 11, 15, 10, 6], which is acceptable considering our main purpose of calculating excitation energy transfer rate. The parameter of MRT is from previous work[16]. T = 300.0 K−1, λ0 = 85.0 cm−1, Γ = 628.4 cm−1(Tabel 5.2)
5.3 MRT population dynamics
We also compare the MRT population dynamics with previous Wu et al. work[16].
In Fig. 5.1, we can see the transfer rate of MRT is smaller, but this difference would not impact on the coarse-grained.
5.4 Network analysis
The MBT of LHCII monomer is Fig. 5.2 And we also compare all the methods and check their costs. Fig. 5.4
As the CGM has more clusters, the reproduced population dynamics should close to the exact population dynamics, and has lower cost. But we can see that some methods has small cost when number of clusters is small, but get larger when the number of clusters is larger. Fig. 5.4 So we think that may not obtain suitable CGM. The method AC, norm-SC, norm-SR and norm-AC (Fig. 5.5 ) are apparently decrease as the numbers of cluster increase.
In additionally, comparing these methods, three methods are modified with normalized MBT. That maybe mean that the transfer rate constant has extensive property. Because SR and SC methods are not able to build the coarse-grained model with an arbitray number of clusters, so we focus on method 002 and method 012. The only different between them is if we normalize the MBT.
We can check the CGMs. In first cut, they both cut out exciton state 1, 2 and 5. And in
Figure 5.1: Check the rate constant matrix by comparing the population dynamics of MRT result(a,b) to Wu et al. work(c,d).
Figure 5.2: LHCII monomer MBT (unit: ps−1 ): The ellipses denote the subgraphs and each number α in the ellipse denotes the α-th exciton state.
Figure 5.3: The modified MBT of LHCII: The capacity of minimum-cut is normalized.
Figure 5.4: LHCII cost comparing: Each color denotes different methods of building grained model. And some methods like SC and norm-SC can not build coarse-grained model with an arbitrary number of clusters.
Figure 5.5: LHCII cost comparing: The methods have decreasing cost as increasing the number of clusters.
next cut, method 012 would cluster exciton state 4 and 7, but the method 002 would just cut out the node one-by-one. Exciton state 4 and 7 are localized at a613 and a614, and the EET transfer rate from state 7 to state 4 is 1.0 ps−1. So it’s reasonble to merge them together.
Finally, we choose method 012. And this result also consists with Novoderezhkin et al.
and Wu et al. works[8, 16].