Optimization of energy transfer through a high-energy barrier pigment in LHCs.

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Optimization of energy transfer through a high-energy

barrier pigment in LHCs

Weiyi Ding

August 31, 2017

Studentnummer 10907777

Daily supervisor Pavel Mal´y

Supervisor prof. dr. Rienk van Grondelle Second examinator dr. Raoul Frese

ABSTRACT

In photosynthesis, energy coming from the sunlight is converted into chemical energy. It starts with the absorption of a photon by a pigment, then the pigment gets into an electronic excited state, which is called an exciton. The formed exciton transfers further through a network of other pigments to the reaction centre. The efficiency of this energy transfer is very high, it is therefore interesting to study for improving solar energy sources, where the transport is an important factor determining the overall efficiency. Efficient transport implies high speed and directionality. High speed can be achieved by avoiding large energy gaps and ensuring strong coupling between the pigments. At short distances, such as within one light-harvesting complex, the directionality can be ensured by a downhill energy transfer. However, this would lead to energy loss at longer distances, there are necessarily high-energy pigments to overcome as barriers. Historically, energy transfer is described as ”hopping” of localized excitons between states on single pigments by the F¨orster theory. According to this theory, overcoming a barrier is a big bottleneck and therefore a problem for localized excitons. Recent works have shown that excitons are more delocalized in space because of the strong electronic coupling between pigments. Then the dynamics cannot be calculated with the F¨orster theory. Delocalized excitons are superpositions of excited states, the transfer of these excitons will be calculated with population dynamics according to the Redfield theory. To study the excitonic effects we use a minimal 5-pigment donor-barrier-acceptor model system which occurs in light-harvesting complexes. The influence of exciton delocalization on the transfer through the barrier and the mechanisms behind it are investigated. Results show how couplings and energies of the pigments quantitatively contribute to exciton delocalization, and to the optimization of exciton transfer. Ranges and conditions are found for the optimal exciton transfer. In general, too much delocalization of excitons on the barrier pigment decreases the transfer rate. On the donor side, more delocalized excitons and a higher pigment energy attached

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to the barrier lead to faster transfer. In transfer through the barrier, the effect of coupling to the barrier dominates over the effect of other couplings. Weak coupling on the acceptor side leads to more overall exciton transfer.

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1 Introduction

Within the developments of renewable energy sources, increasing the efficiency of solar cells is an enormous challenge. One of the main aspects of this challenge is transporting energy and charges over large distances without significant recombination losses. This has already been achieved by plants, algae and bacteria in the biological process photosynthesis. It leads energy coming from sunlight across the photo−systems and then provides chemical energy which other organisms rely on, as a food source. The light harvesting system in photosynthesis involves the absorption of photons by pigments, subsequently electronic excitations, which are called excitons. Then the energy is transferred to the reaction centre through other pigments. The quantum efficiency is close to 100%, so this mechanism is interesting to study for improving the technology around solar energy.

To describe this energy transfer, models of population dynamics are applied. The way in which excitons migrate depend on the electronic coupling between pigments and the coupling between the pigments and the environment, which is called the bath. It is calculated using quantum mechanics. Due to the complexity of the problem, perturbative theories are tipically used. One of the used theory is the F¨orster theory, which is for the regime where coupling between the pigments and the environment dominates. The other one is the Redfield theory, which assumes stronger electronic couplings between the pigments themselves and allows for coherent oscillations of the populations at individual pigment sites(). From the results of recent publications(Novoderezhkin et al, 2016; Qin et al,2015), we will look at the regime of Redfield theory and our interest is in excitonic effects. Optimizing the energy transfer in the Redfield regime implicates maximizing the exciton transfer rate avoiding large energy gaps and ensuring a strong coupling between the pigments. At short distances the energy transfer is easier because of the downhill energy transfer through a funnel energy landscape(). However, this is not feasible at longer distances like between light−harvesting complexes, because of the energy loss through high energy pigments. Therefore, constructing a long−distance transfer network with as little energy loss as possible through high energy barriers is a big challenge.

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Figure 1: Schematic system of 5 pigments

The parameters which influence the localization of excitons and the transfer rate in this case are: the vibrations(phonons) of the bath, which are described by spectral density; the energy levels of the pigments themselves; and the electronic coupling between them. The three parameters will be investigated in what way they influence the energy transfer from pigments on the donor side through a high energy barrier pigment, to the pigments on the acceptor-side. By theoretical numerical simulations the relevant parameter ranges will be assessed for optimization. Results will be applied to a concrete example of energy transfer between light−harvesting complexes, where such barriers exist.

2 Theory

2.1 Excitons in light-harvesting complexes: LHC

Photosynthesis starts with the capture of photons from sunlight by specialized pigmentprotein light harvesting complexes. On the protein structures light-absorbing pigments are attached (Fassioli, 2014). When a photon gets absorbed by the pigment, the energy which correspond to the transition energy of one molecular orbital to another one gets absorbed. This causes the excitation of electron from the ground state to an electronic excited state, which is called the exciton. The formed exciton will be transferred within the photosystems which are art of the photosynthetic machinery tool that in the end uses its energy to produce ATP and NADPH. The energy of this electronic excitation must be harvested very fast before its relaxation to the ground state. That is, transferring the exciton through space among the pigments until it reaches a reaction centre where it initiates charge separation.

The total Hamiltonian of the whole system is given by formula 1.

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Where Helecis the electronic Hamiltonian, Hphis the Hamiltonian of the bath vibrations(phonons),

and Helec-ph is their interaction. In the regime of Redfield theory, only the electronic

Hamilto-nian will be considered. HamiltoHamilto-nian of electrons and phonons from the bath is the perturbation, Hamiltonian of the phonons will not be considered.

In the basis of the pigments, the site-basis, Helec is given below:

HS ₌            e1 J12 . . . J1n J21 e2 . .. . . . J2n J31 J32 . .. . .. ... .. . . .. . .. ... Jn1 . . . en            (2)

The pigment energies are on the diagonal, the couplings between them on the rest. The pigment is considered as a two-level system; ground state and excited state. When pigments are photo-excited, they interact with each other by a coulombic interaction between their transition densities. This coupling exchanges electronic excitation between them, the interaction can be described by a dipoledipole coupling between transition dipole moments, given by formula 3.

Jnm= f 4πε0 µn· µm− 3 µn· ˆR µm· ˆR R3 (3)

Where µ is the transition dipole moment, R is the distance between the pigments, f is the screening−factor.

In site basis, Helec-phis diagonal. The vibrations(phonons) from the bath and their coupling

to the electronic transitions of the pigments is characterized by the spectral density;

C(ω) = h1 + coth ω 2kBT

i

· C00_(ω), _where _C00_{(ω) =} 2λΛω

ω2_+Λ2 (4)

The bath around the pigments in LCHs is described by the re-organization energy λ, and the inverse bath correlation time Λ. λ determines the coupling to the bath, associated with equilibration of the environment after excitation. Λ determines the width of the spectrum of the phonons. In the strong electronphonon coupling regime(λ >> J ), energy transfer is described by the theory of F¨orster. It corresponds to a second−order perturbation theory with respect to the electronic coupling and describes a ”classical” hopping of localized excitons between states on single pigments. The transfer happens through the mechanism that is known as resonant energy transfer. The exciton has an eigen−energy which matches the spectral absorption of another pigment, there is spectral overlap(resonance) between pigments.

When the coupling is large compared to the re-organization energy(J > λ), the interaction modifies the stationary states of the system relative to the isolated pigments. It is not possible to say where the excitation is, it becomes delocalized. In some LCHs these couplings can become significant because pigments are closer to each other for improving the energy absorption. In this regime, excitons are delocalized and F¨orster theory is replaced by Redfield theory. The coupling between the electronic states with the bath induces disorder in the system, which can

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influence the extent of exciton delocalization and dynamics, this coupling to the bath is treated as perturbative(2000, Van Amerongen).

Many calculations have shown how excitons can substantially change the electronic structure of an antenna as well as the energy transfer dynamics. Due to the strong coupling between molecules, this shifts the absorption resonance frequency to the new eigen-energy of the exciton. Strong coupling results in delocalized excitons, the absorption resonance energy of the pigments shifts to a new energy level. In this case, the electronic excited state is delocalized over two or more pigments. This means that the exciton is in a quantum superposition of the electronically excited states of different pigments. Excitons can extend over multiple pigments and can have a profound impact on the electronic structure and hence optical properties of the system, and also on energy transfer dynamics(2014,Fassioli).

In the Redfield regime the Hamiltonian can be written in terms of localized pigment states(site basis) or exciton basis. The reason is that the perturbation theory is not applicable in site-basis according Redfield theory, only in exciton-basis. In the general form:

HE_{= C}†_HS_C ₍₅₎

, where C is the rotation-matrix and the Hamiltonian in exciton-basis is diagonal:

HE _{= diag {1, ...,}

n} (6)

Eigen-energy states are given by superposition of pigment energies: |ii =

X

n

C_ni|eni (7)

In the case of 2 pigments dimer:

HS₌ " e1 J12 J21 e2 # (8) C = " cos(θ) − sin(θ) sin(θ) cos(θ) # (9) The values of excitons is calculated by:

1,2 =e1+e₂ 2 ± q (e1−e2)2 4 + 4J2, tan(2θ) = 2J e1−e2 (10)

2.2 Dynamics according to Redfield Theory

Using Redfield theory for calculating exciton dynamics, we get the following master equation for the exciton population Pi:

∂Pi ∂t = P j kijPj− P j kji ! Pi (11)

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kji=P n Cmi 2 · Cnj 2 · C (ωij) , where ωij = i−j ~ (12)

A smaller energy gap between the pigments increase delocalization, energy transfer goes more ”downhill”, this increases transfer rate. Redfield theory also assumes that the environment equi-librates very fast after an electronic transition from the ground to the excited state. Therefore, transfer of excitation happens from equilibrium phonon states.

The detailed balance is:

k21= k12· e−

2−1

kB T ₍₁₃₎

These calculations are in exciton basis, converting into site-basis one can use formula 6 obtain-ing the population:

Pn(t) = X n C_ni 2 · Pi(t) (14)

For the state of 5 pigments in the D - B - A toy-model:

HS₌          ed1 Jdd 0 0 0 Jdd ed2 Jdb 0 0 0 Jdb eb Jba 0 0 0 Jba ea1 Jaa 0 0 0 Jaa ea2          (15)

The superposition of each exciton:

The localization of each exciton on each pigment is given by the square of C-matrix. The probability of exciton localization on pigments:

P | he1|1i |2= |C11|2 P | he1|2i |2= |C11| 2 .. . (18)

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2.3 Initial condition

We have to choose initial condition for the population dynamics. Absorption of sunlight and also energy transfer from neighbouring group of pigments populates exciton states. We therefore formulate the initial condition in the exciton basis. In this basis, the initial condition is that all population start from the lowest exciton; P1(t = 0) = 1. All the other excitons have no population

at t=0;P2,3,4,5(t = 0) = 0. In site basis, the pigment with the lowest energy has most the of the

excitons(with the lowest energy) at t=0, according to formula 2.2.

For a two pigment dimer example, the exciton transfer from pigment 1 to pigment 2 is given below(population of exciton 1 at t=0, both exponentially:

P1(0) = 1, P1(t) = e−kt= e− t τ, P˙₁(t) = −kP₁(t) P2(0) = 0, P2(t) = 1 − e−kt= 1 − e− t τ, P˙₂(t) = kP₂(t) (19) Where the lifetime is given by:

k = 1

τ (20)

In the D-B-A model, most of the excitation starts from the pigment with the lowest energy on the donor side(D1). When the transfer begins, excitons start moving to the next pigment D2, Pd2

increases, but some excitons are also moving away to the barrier, and so on, till the equilibrium occurs. With different localizations and energy gaps, the overall rate of population transfer is not easy to determine. We therefore use as a measure the amount of population transferred to the acceptor side at a fixed time:

PA= Pa1(0.5ps) + Pa2(0.5ps) − (Pa1(0) + Pa2(0)) (21)

The results are not qualitatively sensitive to the exact time, as long as it is in the intermediate range between initial dynamics and equilibrium.

2.4 Parameters of the pigments

As previously mentioned, designing a network of pigments for energy transfer includes high-energy barriers, this occurs also in nature. Excitons move faster to a pigment with lower site energy than one with higher site energy.

The system of pigments zoomed out a bit, is regulated by the protein. It adjust the pigment conditions, their site energies, and couplings between them. For the energy transfer inside a cluster of pigments with a high-energy barrier in the middle, there is relation between the transfer-rate and values of site energies and couplings. The mathematical approach of population dynamics will show that in some ranges of these parameters the transfer becomes maximized.

The system is built by a donor-side, high energy barrier and acceptor-side, in site-basis. The pigment parameter ranges are chosen according to recent publications of Qin et al(2015), and Novoderezhkin et al(2016).

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Figure 2: 5 pigments and the parameters

The donor-side contains two pigments with two energy levels ed1 and ed2. Because the

cal-culations are about energy differences between the pigments, the first pigment with the lowest energy-level ed1 is set to 0, ed2is to variate between 0 and eb. For the high−energy barrier, eb has

the range 350 − 400cm−1.

The acceptor-side contains also two pigments, ea1 and ea1. The quantitative conditions for

these two are: ea2 < ea1 < eb; ed1 < ea2 < ea1. Because of the initial condition, starting from

the lowest exciton, we must take care not to start already at the acceptor side, see results section below.

The pigments are linked in a serie to each other through couplings(Jdd, Jdb, Jba, Jaa. If two

pigments are strongly coupled, the energy-levels will split in a upper and down level. Excitons on these pigments will get more delocalized, which contributes to the faster transfer of the excitons. The other way around, excitons will get more localized on the pigments if the coupling is getting weaker.

In exciton-basis, the initial condition is that the population transfer starts at lowest exciton.

3 Results & Discussion

Using Mathematica, the parameters have been iterated relative to each other for the simulations of exciton transfer. Within their ranges as mentioned in Chapter 2, the next optimizations are found for transfer rates.

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3.1 Phonons from the bath

In the Redfield regime, vibrations from the bath determine the transfer rate(see formula 12). The spectral density according to formula 4) is shown in figure 3.

Figure 3: The spectral density of the bath.

Λ indicates the width of the peak, as long as this peak is broad enough, the rate only weakly depends on the shape of the spectral density. Because the pigments are weakly coupled to the bath, λ only scales the rate. If λ goes to 2λ, k will be 2k (τ will be τ₂).

For these reasons simulations Λ and λ are set on 50 cm−1() and not further varied.

3.2 Energies and couplings

As previously mentioned, the eight parameters to vary and iterate are ed2, eb, ea1, ea2, Jdd, Jdb,

Jba, Jaa. Mechanisms behind exciton (de)localization and transfer lead to the next results for the

optimization:

Excitons moving from donor side through the barrier to the acceptor side as fast as possible means for the donor side: losing populations; for the acceptor side: gaining populations. And this in a period as short as possible. The contourplots show in which region of the parameters the population on the acceptor side is high enough, for analysing the transfer rate. A bigger surface is more robust because of the less fluctuations of excitons, less energetic disorder from protein motion.

3.2.1 Energy of the barrier

If ebincreases with respect to the rest of the system, transfer rate decreases, because it takes more

energy for the excitons to get through the higher level. The range for eb was 350 − 400cm−1, for

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3.2.2 Couplings

It turns out that when a pigment is coupled to two other pigments, the two couplings are inversely proportional for the same transfer rate. If both couplings are too small, the excitons are not delocalized enough to move. If both are too strong, there is less overall population transfer.

3.2.3 Donor side + Barrier Optimization occurs when:

ed2 is close to eb; (eb− ed2) ∈ (10, 100)

Jdd< Jdb, Jdd ∈ (40, 80); Jdb∈ (40, 80)

Or: Jdb< Jdd, Jdd∈ (100, 150); Jdb∈ (30, 30) (Only if Jba is strong enough.)

If ed2 is lower, JDD can be increased for the same overall transfer, but the rate decreases

because the excitons are more delocalized, it takes more time to reach the barrier. If both ed2 are

JDD decreased, more excistons stay on the donor side. Increasing JDB will not have significant

influence.

(a) The exciton delocalization on D1 and D2 _{(b) The exciton delocalization on D2 and B}

Figure 4: Excitons between donor side and the barrier.

(a) Jdb on y, Jdd on x. (b) Ed2 on y, Jdd on x.

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3.2.4 Barrier + Acceptor

Optimization occurs when: ea2 ∈ (10, 30) is very low but not too close to ed1 ( = 0 ), otherwise

the lowest exciton will also locate on A2.

ea1 ∈ (50, 90) is close to ea2, Jaa ∈ (30, 60) < Jba ∈ (90, 160)), Jaa has to be weak for more

exciton localizaton on the acceptor side. Varying Jba in the range depends on Jdb, the couplings

can be changed inversely relative to each other, taking into account Jdd and Jaa.

(a) The exciton delocalization between B and A1 (b) The exciton delocalization between A1 and A2

3.2.5 Overall picture optimization: Donor + Barrier + Acceptor Donor side

On the donor side, there is more delocalization for the excitons to move up in energy level to reach the barrier. The ideal case is when ed2 is almost as high as eb. For a lower ed2, Jdd can

be increased for more delocalization, which will move more exciton around D2 to get lifted to the barrier. Both couplings on the donor side must be changed inversely for the same transfer rate.

Competition for the barrier

Jdband Jbaare inversely proportional to each other. If Jdbis increased, Jbamust be decreased,

and conversely, shown in figure 7b. If both couplings are large, excitons are too much delocalized for transfer. If they are both small, the splitted energy levels are not high enough, excitons will stay on the donor side. It seems to be a big ”competition” for the barrier, the effects of changing Jdb and Jbadominate over the effects from other two couplings.

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(a) The exciton delocalization on the barrier

(b) Inverse relation between donor barrier and barrier acceptor.

Acceptor side

On the acceptor side the inverse effect of the couplings also applies. But the overall delocaliza-tion on this side must be lower than donor side, Jaahas to stay low, otherwise excitons on A1 get

more delocalized, there is more population on the barrier. When the ea1is relatively low, excitons

move easier downwards in energy levels. If the energy is too high, excitons get localized and it will take much more time for the equilibrium occurs. But if the energy is too low, less excitation will start from donor side. The splitted energy-levels by the coupling on the acceptor side must be higher than the donor side, This is to prevent the low energy excitons also also being localized on the acceptor side from the beginning, meaning that the excitons skip the barrier and the problem gets inversed. In the contourplots this situation is recognized by a negative sign of the transferred pupolations.

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(a) Jba on y, Jdd on x, Jdb is small. (b) Jba on y, Jdd on x, Jdb is large.

Figure 8: Population transferred to the acceptor side at a fixed transfer time, PA.

(a) Ed2 on y, Jba on x.

(b) Couling on acceptor side must be small.

Figure 9: Population transferred to the acceptor side at a fixed transfer time, PA.

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Figure 11: Optimal transfer dynamics, larger coupling to the donors.

3.3 Results applied to LHCa1/LHCa4

(final picture and calculations follows)

Applying these results, using data from Qin et al. and Novoderezhkin for calculation examples of some high energy barriers in LHC, the couplings and pigment energies match with the result. But these are only pigments which are strongly coupled. The most pigments as shown in figure ?? are weakly coupled. Then the calculations according to the Redfield theory will not work.

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Figure 12: Results applied to paper of Qin et al. 2015

4 Conclusion

The design a network of pigments with high-energy barriers:

In general, too much delocalization on the same pigment decreases the transfer rate. On the donor side, more delocalized excitons and a higher pigment energy attached to the barrier lead to faster transfer. Excitons get lifted up to the barrier. Transfer effects from couplings to the barrier dominates over the effects from other couplings. Weak couplings on the acceptor side lead to more overall exciton transfer.

5 References

Novoderezhkin, V., Croce, R., Wahadoszamen, M., Polukhina, I., Romero, E. and van Grondelle, R. (2016). Mixing of exciton and charge-transfer states in light−harvesting complex Lhca4. Phys. Chem. Chem. Phys., 18(28), pp.19368− 19377.(Novoderezhkin et al., 2016)

Amerongen, H. (2000). Photosynthetic excitons. Singapore: World Scientific. (Amerongen, 2000)Molecular mechanisms of photosynthesis

Blankenship, R. (2014). Molecular mechanisms of photosynthesis. Chichester: Wiley Black-well.(Blankenship, 2014)(Amerongen, 2000)

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Croce, R. and van Amerongen, H. (2013). Light−harvesting in photosystem I. Photosynthesis Research, 116(2−3), pp.153−166.(Croce and van Amerongen, 2013)

Novoderezhkin, V. and van Grondelle, R. (2010). Physical origins and models of energy transfer in photosynthetic light-harvesting. Physical Chemistry Chemical Physics, 12(27), p.7352.(Novoderezhkin and van Grondelle, 2010)

Optimization of energy transfer through a high-energy barrier pigment in LHCs.