ianto cannon long vacation project

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Using The Hierarchy Equations of Motion to Simulate

Exciton Transfer in Photosynthesis

January 17, 2016

Contents

1 Introduction 2

2 Theoretical background 3

2.1 The antenna complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Hierarchy equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Two dimensional spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Method and Results 7

3.1 System bath model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Writing the program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Simulating exciton transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.4 Creating electromagnetic spectra . . . . . . . . . . . . . . . . . . . . . . . 11

3.5 Two dimensional spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Discussion 15

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4.1 Exciton transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Electromagnetic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Two dimensional spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4 Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Conclusions 18

6 Acknowledgements 19

Abstract

A computer model was created to simulate pigment molecules in the photosyn-thesis system. This model is based on the hierarchy equations of motion, a set ofquantum mechanical equations. The model was used to show that the probability ofexciton transfer is larger if the excitation is being passed from a high energy moleculeto a lower energy molecule. It also revealed that the effectiveness of exciton transferis reduced if the molecules are tightly packed within the chloroplast. Investigationsof the electromagnetic spectra of the simulated pigment molecules showed that therange of absorbance of the photosynthesis system can be increased by using pig-ments with different excitation energies. We saw that it can also be increased bytighter molecular packing.

1 Introduction

In the past 10 years, advances in femtosecond pulse lasers and x-ray crystallographyhave allowed us to view the shapes and positions of the molecules in the photosynthesissystem [8]. These discoveries, combined with improvements in computing power, enableus to simulate the system at work.

The aim of my project is to simulate how energy is transferred during the early stages ofphotosynthesis. I hope to use this simulation to investigate which factors affect energy

absorption and transfer. The Hierarchy Equations of Motion (HEOM) will form thebasis of my simulation. Simulations of the photosynthesis system have previously beencarried out by Dr. Alex Chin and others at the University of Cambridge [10], but outsideof the Tanimura group no simulations have been made using the HEOM.

The HEOM are a set of quantum mechanical equations which describe the time evolutionof areduced density matrixin the canonical ensemble. Unlike the other reduced equations

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of motion approaches, the HEOM are numerically exact and can be used for a finitetemperature bath [7]. See section 2.2 for an introduction to the HEOM.

Photosynthesis provides all of the food and most of the energy consumed by the humanrace [8]. Discoveries in this field may lead to the production of genetically modifiedplants, which have a greater capacity to produce food. These plants would enable usto feed the growing world population. Ultra cheap solar panels could also be based onknowledge gleaned from photosynthetic reactions.

The theoretical background of the experiment is summarised in the next section. Sec-tion 3 describes the method used and the results of each part of the experiment. InSection 4, we discuss the results of the experiment, and potential sources of error. Sec-tion 5 states the conclusions we draw from the experiment.

2 Theoretical background

2.1 The antenna complex

The first stage of photosynthesis occurs in the antenna complex, here sunlight is absorbedand energy is transported to the reaction centre (see figure 1). This energy drives a seriesof redox reactions known as the Z scheme, which move protons across a membrane. Theproton gradient is then used to generate ATP, which can be used elsewhere in the plantto build sugars.

The antenna complex consists of rings ofpigment molecules, which surround the reactioncentre. The chlorophyll pigment molecules absorb light in the blue and red regions of thevisible spectrum, giving rise to the green colour of plants. When a photon is absorbedby a pigment, it causes an electron to move from the orbital to the orbital withinthe pigment. This excitonis passed along a chain of neighbouring pigments, eventuallyreaching the reaction centre. Each pass occurs by a process known as exciton transfer.

2.2 Hierarchy equations of motion

The HEOM were developed by Y. Tanimura and R. Kubo in 1989 [9]. These equationsdescribe the time evolution of thereduced density matrixof a system in thermal contactwith a heat bath. The heat bath is characterised by the brownian oscillator model.I.e., the bath radiates electromagnetic energy with a frequency distribution given byJ() = 2/(2 +2), where is the frequency of radiation, is the reorganisationenergy and 1/ represents the timescale of energy dissipation within the bath. TheHEOM are given by the following set ofN equations;

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Figure 1: A drawing of the antenna complex. The green disks repre-sent chlorophyll molecules and the orange disks represent carotenoidmolecules.

tn=

i

HS + n

n+n+1 nn1, (1)

where 0 n N and N >1/. Note that only 0 is equivalent to the reduced densityoperator. The other levels of the hierarchy 1, 2, ... Nare auxiliary operators whichtake into account the fluctuation of energy levels within the system and the dissipationof energy due to the heat bath. In all of my simulations, these matrices 1, 2, ... Nare set to zero for the initial condition.

The relaxation operators are given by

= iV,

= i

2

2V i

V

, (2)

where we have introduced the hyper-operator notations, AB = AB B A, and AB =AB + BA. In equation (2), = 1

kT, where k is Boltzmanns constant and T is the

temperature of the bath in Kelvin. is Plancks constant.

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Figure 2: Structure of the LH2 complex in purple photosynthetic bac-teria. The chlorophyll pigments are shown in red, and the carotenoidsin orange. The yellow and grey molecules are structural proteins, whichhold the pigments in place. Figure taken from [8], p79.

The operator HS is the system Hamiltonian, it describes the energies ofstationary stateswithin the system. V represents the system bath coupling, and is determined by theshapes of dipoles within the system.

The HEOM overcomes many limitations which the other reduced equations of motionapproaches suffer from. For example, the equations based on perturbation theory canexhibit thepositivity problem, whereby diagonal elements of the density matrix become

negative. This is unphysical, as the diagonal elements must express a probability betweenzero and one. Another alternative is the Redfield equation, but this can only be usedfor high temperatures.

The HEOM method can model the effect of a finite temperature bath in a non per-turbative, numerically exact manner [1]. It has previously been applied in the areas ofquantum information [3] and multidimensional spectroscopy [2].

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2.3 Two dimensional spectroscopy

Measurement of 2D spectra

Two dimensional spectroscopy allows us to observe the extent of coupling between dif-ferent vibrational energy levels in a system, it can also be used to determine how excitedmolecules behave over time. As mentioned in section 1, this measurement technique hasonly become practical in the past 10 years, with the advancement of femtosecond lasertechnology.

In order to make a 2D spectrum, the sample is injected with 3 pulses from a femtosecondlaser (see figure 3). Next, the light emitted by the relaxation of the molecules is observed.The process is repeated several times, with different values of the excitation time t1.From these measurements, we can plot the response time t3 against t1, giving a 2Dresponse function(figure 4). The 2D spectrum is found by taking a Fourier transformof the 2D response function.

Figure 3: Measurement of 2D spectra. The three solid peaks representlaser pulses incident on the sample, and the dashed line shows theemitted light. The first two pulses are known as pump pulses, andare used to excite the sample. The third pulse called the probe pulse,it causes a transition back to the ground state. Figure courtesy ofHironobu Ito, Kyoto University, with permission.

Interpretation of 2D spectra

It is instructional to think of1 as the frequencies at which the system absorbs electro-magnetic energy, and 3 as the frequencies at which this energy is emitted. Hence, apeak on the line1 = 3corresponds to energy which is absorbed by a vibrational state,and re-emitted by that same state. These are the bottom left and top right peaks in fig-ure 5. On the other hand, the off diagonal peaks are produced when energy is absorbedby a particular vibrational state, and re-emitted from a different vibrational state. I.e.,the energy is transferred between states after absorption, and before emission. These

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Figure 4: A simulation of the 2D response function. Blue and redregions correspond to positive and negative fluctuations in the electricfield. The linet1 = 0 is equivalent to the 1D impulse response functionmentioned in section 3.4

states are said to be coupled. 2D spectra are a central part of my investigation, as thesize of the off diagonal peaks is directly related to the speed of excitation transfer. Seechapter 12 of [4] for a more complete introduction to 2D spectroscopy.

3 Method and Results

3.1 System bath model

I employed the system bath model in my simulations, whereby the state of the systemis calculated at every increment in time, and the bath is treated simply as a source andsink of energy for the system. Let us assume that the time taken for the FORTRAN

MATMUL algorithm to carry out an n n matrix multiplication is O(n3

) [11]. Addingone additional molecule to the system would introduce at least one extra stationary stateto my system, therefore the size of the reduced density matrixwould increase fromn nto at least (n + 1) (n + 1). Hence, the computation time for my simulation is no betterO(n3), where n is the number of molecules in the system. In other words, using moremolecules than necessary would greatly increase the computation time, forcing me toincrease the length of the time step, which would result in larger computational errors.

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Figure 5: A 2D spectrum of two adjacent pigment molecules. E1 andE2 are the excitation energies of pigment 1 and pigment 2. 1 and 3are stated in arbitrary units.The purple regions correspond to peaksin the spectrum. The bottom left peak is caused by the response ofpigment 1, and the top right peak is caused by the response frompigment 2. The two other peaks are the result of coupling betweenthese states.

I chose to use two pigment molecules to make up my system. Two molecules is the min-imum needed in order to simulate an exciton transfer, so it is the most computationallyefficient choice. Also, it is simple to quantify the extent of exciton transfer when thereare only two molecules involved.

As shown in figure 7, the system has three energy states;

|g The HOMOs of pigments A and B are both fully occupied with two electrons. Thisstate is defined to have energy zero.

|e1 Pigment A possesses one electron in its HOMO and one in its LUMO. Pigment Bhas a fully occupied HOMO and an empty LUMO. This state has energy e1 .

|e2 Pigment B possesses one electron in its HOMO and one in its LUMO. Pigment Ahas a fully occupied HOMO and an empty LUMO. This state has energy e2 .

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Figure 6: The system is made up of two pigment molecules, all of thesurroundings are modelled using a bath.

Figure 7: The three basis states of my system,|e1 , |e2, and|g. Thestates|e1 and |e2can interchange by a mechanism known as exciton

transfer (XT). Each of the diagrams shows the HOMO and LUMO ofpigment A and pigment B. The green dots represent electrons. Figurecourtesy of Hironobu Ito, Kyoto University, with permission.

The system Hamiltonian and reduced density matrix are as follows,

H=

e1 J 0J e2 0

0 0 0

, =

e1e1 e1e2 e1ge2e1 e2e2 e2g

ge1 ge2 gg

,

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where e1 and e2 are the excitation energies from ground state to states|e1and |e2respectively, and Jis the coupling energy between |e1and |e2.

The system bath coupling was set to,

V =

1 0 00 1 0

0 0 0

.

3.2 Writing the program

I wrote my code in the FORTRAN 90 programming language. I chose to use FORTRANas it is faster than other languages when working with arrays of data, and its long historyof use in science means there is a wealth of code that can be downloaded easily.

The Hierarchy Equations of Motion, (HEOM) were solved using a fourth order Runge-Kutta algorithm (RK4). This method is easy to use and understand. For a givenaccuracy, it requires less computer time than other single step methods, such as theEuler method. Also, unlike the multi step methodsmethods, it requires only one initialcondition.

I ensured that my calculation errors were small by reducing the time step by half andre-running the simulation. If the final state of both the simulations was the same, thenI could be sure the size of the time step was not affecting my results.

3.3 Simulating exciton transfer

Once the program had been written, I put it to work in order to investigate which factorsaffect the exciton transfer process. The density matrix was set to

=

1 0 00 0 0

0 0 0

for the initial condition. Which means that, at time t = 0, there is a 100% chance thatpigment A is found in the excited state and pigment B is found in the ground state.

This represents the quantum mechanical state of an antenna complex at the instant oneof its pigments is excited.

The time step,dt, was set to 1030 seconds and the constants in equation (1) were giventhe following values,

J N

1.00 0.50 0.25 -0.20 4

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The results of two simulations are shown in figure 8. In simulation 1, pigment A has anexcitation energy of 1 = 5 and pigment B has an excitation energy of 2 = 2. Insimulation 2, both pigments have an excitation energy of1= 2 = 2.

Figure 8: These results show how the excitation energies of the twopigments affects the probability of exciton transfer. They axis showsthe probability of finding the system in a particular state. Time isshown on the x axis, in arbitrary units. The blue line corresponds toelement e1e1 of the density matrix, and the green line corresponds toe2e2 . Created using pyplot. See section 4.1 for a discussion of theseresults.

3.4 Creating electromagnetic spectra

EM spectra are the primary method of observing organic molecules. Therefore it isuseful to calculate the spectra of my simulated pigments, allowing me to compare myresults with real experimental data.

The EM spectrum of a system is equivalent to the Fourier transform of its impulseresponse function(IRF). The IRF can be measured by injecting the system with a shortpulse of laser light, and observing the subsequent changes in the electromagnetic field(see figure 9).

The IRF of my system was calculated using the following formula,

R(t) = i

Tr

G(t)eq

, (3)

where eq is the equilibrium state of the density matrix, For non extreme tempera-tures, we can assume that the system is in equilibrium when in the ground state1, i.e.,

1The temperatureTrequired for a 50% probability of excitation can be found by solving the equation

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Figure 9: A method of obtaining the impulse response function (IRF).They axis shows electric field strength, and time is on the x axis. Theinitial spike is caused by a femtosecond laser pulse, after which we seethe IRF of the system.

eq =|g g|. G(t) is the Greens function of equation (1), it represents the time propa-gation of any matrix to the right of it. For example,

G(t)(0) = (t)

In the case of equation (3), The expression G(t)eq is calculated by setting eq asthe initial condition, and running the simulation for time t.

The transition dipole moment, , represents the action of an electric field on the system,it causes the transition between the ground state and an excited state;

= |e1 g| + |g e1| + |e2 g| + |g e2|=

0 0 10 0 11 1 0

.

I modified my code so that the value of R(t) is calculated at every time step, usingequation (3).

I used a Fast Fourier Transform (FFT) algorithm in order to create an EM spectrumfrom the impulse response function. The source code fft.f90 was taken from the bookNumerical Recipes in Fortran 90 [6], which also contains an explanation of how fastFourier transform works. I decided a FORTRAN program was the easiest method formaking Fourier transforms of my data. As unlike Matlab FFT and other applications, itcan be run from the command line. Therefore I could create a batch file, which requiresonly one command in order to compile and execute both my simulation and the FFT

code. Also, the code fft.f90 can be easily modified to carry out multi dimensional Fouriertransforms, which are used in section 4.3.

The step by step process for creating an EM spectrum is as follows;

1. Choose desired values for parameters, , , J , N, 1, 2, and dt.

exp() = 12

. For a 650nm transition, this gives T= 30 000 K.

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2. Compile and run the simulation, starting with initial condition eq. This outputsvalues R(t) to a text file named R.txt.

3. Run the FFT algorithm, this reads R.txt and writes the Fourier transform to a filecalled spec.txt.

4. Plot spec.txt using Pyplot.

Figure 10 shows a set of EM spectra, for various pigment excitation energies. Figure 11shows how the dissipation, affects the EM spectrum of my system.

Figure 10: The calculated absorbance spectrum of a system of twopigments. E1 is the energy required to excite pigment A, and E2 isthe energy required to excite pigment B. The wavelength is given inarbitrary units. Notice how the width of the peak increases as thedifference between E1 and E2 increases.

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Figure 11: The calculated absorbance spectrum of a system of twopigments for various values of, the dissipation parameter. The wave-length is given in arbitrary units. Notice how the the peaks becomebroader as increases.

3.5 Two dimensional spectroscopy

The following relationship was used to compute the nonlinear response function,

R(t3, t2, t1) = i

3

Tr G(t3)G(t2)G(t1)eq .In order to reduce the computation time of my simulation, t2 was set to zero, this givesa response function which depends on two variables, t3 and t1,

R(t3, 0, t1) =

i

3Tr

G(t3)G(t1)

eq

.

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The computation of this function requires a nested do loop, here is the structure of myprogram;

DO

propagate eq to time t1, giving matrix M(t1)

DO

propagate M(t1) to time t3, giving matrix N(t3, t1)

calculate R(t3, 0, t1) = (i

)3Tr[N(t3, t1)]

END DO

END DO.

Propagations are carried out in the same way as in sections 3.3 and 3.4, using the HEOM.

The FFT program was modified to carry out a 2D Fourier transform of the nonlinearresponse function, giving the 2D Spectrum of the system. Figure (12) shows my results.

4 Discussion

4.1 Exciton transfer

Figure (8) shows my results for two exciton transfer simulations. A photon is absorbedby pigment A (shown in blue) at time t = 0. In the first simulation, pigment A has agreater excitation energy than pigment B. After 80 time units, there is an 80% chanceof exciton transfer. Whereas in the second simulation, both pigments have the sameexcitation energy and the chance of exciton transfer is only 50% after 80 time units.These results show that exciton transfer is more effective if the exciton is being passedfrom a higher energy molecule to a lower energy molecule.

The excitation energy of a pigment is affected by its molecular structure. It can also beincreased by pigment protein interactions [8], p80.

It has been found that this principle is employed in nature, the pigments furthest fromthe reaction centre have the highest energy excited states (see figure 13). This means thesystem acts as a funnel; excitons are preferentially passed toward the reaction centre.However, there is a trade off. When an exciton is passed to a lower energy molecule, theextra energy is dissipated as heat. This results in a loss of efficiency. It is likely that3000 million years of photosynthesis evolution has led to an optimum energy difference

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Figure 12: The effect of on the 2D spectrum of the system. 1 and3 are given in arbitrary units. Notice how the purple coloured peaksbecome less well defined as increases.

between adjacent pigments.

4.2 Electromagnetic spectra

Figure 10 shows my calculated absorption spectrum of two pigment molecules. It can

be seen that a narrow peak is produced when the molecules have the same excitationenergy. The range over which absorption occurs is increased if the two molecules havedifferent excitation energies.

A broader absorption spectrum can allow the organism to absorb more light and pho-tosynthesise more effectively. This is a second evolutionary reason for pigments to havedifferent excitation energies (see previous section). A broad absorption spectrum isparticularly useful in organisms which do not have access to direct sunlight, such as

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Figure 13: The positioning of pigments around the reaction centre(RC) in purple bacteria. The pigments are labelled according to thefrequencies of light that they absorb. For example, the B850 pig-ment absorbs 850nm light. Notice how the wavelength increases withproximity to the reaction centre. Reproduced from Fleming and vanGrondelle (1997).

plants living on the forest floor. Another example is Prochlorococcus, a genus of deepsea plankton which contains a unique form of chlorophyll not found in any other organ-ism ([8], p20).

A similar spectral broadening effect is seen in figure 11, in this case it is due to anincrease in the dissipation parameter . The dissipation parameter describes the effectof bath on the system. If was set to zero, the peaks would be unrealistically narrow.This shows that it is essential to incorporate a bath in the simulations.

In a real organism, can be increased by moving the surrounding molecules closer to thepigments, which could be achieved by using genetic modification to change the shapeof supporting proteins in the antenna complex (see figure 2). Hence the results shownin figure 11 would be useful for a food scientist attempting to change the absorption

characteristics of a crop.

4.3 Two dimensional spectra

Figure 12 shows a series of 2D spectra for two pigments with various values for the dissi-pation parameter . It is a two dimensional version of the results shown in figure 11.

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The size of the top left and bottom right purple regions corresponds to the amount ofenergy transferred from one state to the other. (See section 2.3 for an explanation of thisprocess). These regions become smaller as the energy difference between the molecules

increases, suggesting the effectiveness of exciton transfer is reduced by a larger dissipa-tion parameter. The fourth plot shows an extreme case of large dissipation, the peaksare indistinguishable from each other and very little energy is absorbed.

It can be seen that increasing also causes the peaks to lose their diamond appearanceand become more circular. This is due to the fact that the dissipation term causesthe energy levels of the system to fluctuate. Therefore the energy of the absorbed andemitted photons become more spread out, forming a less defined peak on the plot.

4.4 Improvements

In its current form, the model cannot make predictions of the values of excitation transfertime, wavelength absorption, and broadness of the absorption peak. The result of thesequantities are given in arbitrary units.

In order to make quantitative predictions, the model needs to include real values for theall of the parameters used. The values ofE1, E2, and are well known. However, dueto the juvenility of the HEOM, the values of, , andJhave not been measured for anantenna system.

, , and Jcould be found by reverse engineering the model. I.e, it could be used to

make qualitative predictions about a simple system, and the parameters adjusted untilthese predictions are correct.

5 Conclusions

A computer model of pigment molecules within the photosynthesis system was created.This system behaves similarly to the real system, and was used to create realistic 1Dand 2D spectra.

It was found that exciton transfer is more probable when passing from a high energymolecule to a lower energy molecule. Exciton transfer is also more probable if thereis a small dissipation parameter, which can be achieved with a loosely packed antennacomplex.

The spectra show that the frequency range of light absorption can be increased byincorporating molecules with different excitation energies into the antenna complex, orby increasing the dissipation parameter.

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Further work needs to be done in this area in order to determine the values , , andJ. This would enable the model to make qualitative predictions, such as the time takenfor an excitation transfer, or the broadness of the absorption peak of the pigments.

6 Acknowledgements

I would like to thank Prof. Y. Tanimura, H. Ito, and J. Y. Jo for teaching me therelevant areas of quantum mechanics and spectroscopy required for this investigation.

The project was made possible with funding from the Amgen scholars program.

Glossary

ATP adenosine triphosphate, a molecule used for energy storage in living organisms. 3

exciton a bound state of an electron and an electron hole, excitons are produced whenground state electrons move to a higher state. 1, 3, 7, 8, 10, 11, 15, 17

multi step method a method of numerical integration which uses more than one pre-viously calculated solution, xn, xn1... in order to calculate the next solution xn.10

pigment molecule a molecule which absorbs in a region of the visible spectrum. Chloro-phyll and carotenoids are the main pigment molecules involved in photosynthesis..3, 6, 7, 16

reduced density matrix a density matrix which describes the state of the system,but not the bath. 2, 3, 7, 8

single step method a method of numerical integration which uses only the most re-cently calculated solution xn in order to calculate the next step xn+1. 10

stationary state a state of a quantum system which has only one possible energy,particles in a stationary state have a probability density function which does notvary in time. 4, 7

time step the time between successive calculations in a computer simulation. Using alarge time step requires fewer calculations, but produces larger errors in the results.7, 10, 12

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Z scheme a series of redox reactions occurring in photosynthesis. These reactions splitH2O into O2and H

+, as well as converting ADP into ATP. The Z scheme is a formofelectron transport chain. 3

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[8] Blakenship, R. E., 2002. Molecular Mechanisms of Photosynthesis Blackwell Sci-ence.

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[10] Caruso, F., Chin, A. W., Datta, A., Huelga, S. F., Plenio, M. B., 2009. Highly

efficient energy excitation transfer in light-harvesting complexes: The fundamentalrole of noise-assisted transportJ. Chem. Phys. 131, p105106

[11] Robinson, S., 2005. Toward an Optimal Algorithm for Matrix MultiplicationSIAM News38 (9)

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