There’sonlyonepersonIcandedicatethisworkto... You.iii Dedication

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Dedication

There’s only one person I can dedicate this work to... You.

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Acknowledgements

Opportunities are seized by those who dare to take them, but I couldn’t have dared to take this opportunity if it wasn’t for the unconditional patience and love of all those who have stood by me through sunny days and stormy nights.

This thesis has only been possible thanks to the guidance, resilience, generosity and patience of my advisor, Francisco Delgado, who I have come to admire as a mentor, as a person, and as a friend. I truly hope one day I can repay him.

I want to thank all my family and friends for their unconditional support. I would like to express my special gratitude to:

My parents, Patricia Soria and John Gonz´alez, who have carried me and encouraged me to keep on going

My sister, Marisa Gonz´alez, who endured me and my desperation while pushing me forward

My grandmother, Pilar Chor´en, who never stopped believing in me, even when I didn’t believe in myself

My aunt, Pilar Soria, who gave me the strength and motivation to work with a smile

My cousins, Jos´e Manuel Mart´ınez, Diego Mart´ınez and Aranza Mart´ınez, who inspired me to be better and aim higher

And my friends, Aranza S´anchez, Nicol´as Ram´ırez, Cecilia Loyola, Mauricio Vargas, Laura Sanjurjo, Luis Arias, and Kimberly Estrada, who accompanied me during these years providing me with invaluable advice and words of encouragement.

I thank the Tecnol´ogico de Monterrey for its support on tuition and the CONACyT for its support on living expenses.

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Quantum effects in the efficiency of Fenna-Matthews-Olson light-harvesting complexes

by

Bruno Gonz´alez Soria Abstract

Abstract.- Renewable energy continues to be the fastest growing industry among the power sector, the same is true for the development of quantum computing for data analysis. There is a stretch relationship between these two branches of research being brought together by yet another branch of science: biology. Biological light-harvesting complexes (LHCs) involved in the photosynthetic process present energy transfer efficiencies of almost 100%, providing a source of inspiration for the development new technologies that could mimic these characteristics. One of the most extensively studied LHCs is the Fenna-Matthews-Olson (FMO) complex. This work is focused on the development of a comprehensible model of excitation energy transfer dynamics in the FMO light-harvesting complex. Considering the research branches involved in this study and the different perspectives from which this complex has been analysed, this work will be taking into account some biological considerations at the molecular, genetic and organism levels to avoid unsubstantiated assumptions.

The presence of quantum coherence between electronic states of the bacteriochlorophylls concealed inside the FMO complex during the photosynthetic process of green sulfur bacteria has inspired researchers to attempt computer simulations to understand its complexity. Although several methods have been explored to model this quantum phenomenon in the domains of open quantum systems, the traditional methods used do not take into account the memory effects of the surroundings, which is commonly approximated as a phonon bath on thermal equilibrium. A popular solution to overcome this limitation is the application of the hierarchical equations of motion (HEOM) method, a non-Markovian approach also used to analyze the dynamics of such a complex, for the modeling of the system evolution. The proposed variation of the parameters that govern the HEOM method in this study provides a new form of characterization for the FMO system. A parametric analysis of some physical features involved during the excitation energy transfer process is performed to better understand its non-trivial dependence on operation parameters in the quantum realm. The analysis is con- ducted in terms of the parameters of temperature, relocation energy, and dephasing rate in the system to track the complex global behavior of coherence, entanglement, decoherence times, transference times, and efficiency of the main process of energy transfer. Complementarily, a comparison between two different species is made as a suggestive possible road map to track genetic differences in the photosynthetic performance of the complex through its biological nature.

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List of Figures

1.1 Possible paths for the reader to explore this work according his or her interests. Coloured arrows indicate recommended path for the reader interested in biological (green) or physical (red) concepts and backgrounds. Blue arrows are to be followed by any reader. . . 3 1.2 Taxonomy of Green Sulfur Bacteria (GSB) based on 16S. Generated using

Type (Strain) Genome Server TYGS with the GBDP method. Branch lengths (red) indicate genetic change. Confidence values (blue). . . 8 1.3 Structural comparison between bacteriochlorophyll a (left) and chlorophyll a

(right). . . 10 1.4 Simplified energy level diagram of and absorption (black) and fluorescence

spectrum (green) of bacteriochlorophyll a in diethyl ether. Modified figure fromBlankenship, 2014 . . . 11 1.5 Schematic map of the multidisciplinary study surrounding the FMO research

and some of the works that have provided the basis of this article. . . 13 2.1 Schematic representation of the spatial arrangement of the Chlorosome, the

FMO protein and the reaction center in the cytoplasmic membrane. Light is absorbed by photopigments in the chlorosome and its excitation energy is transfered down to the baseplate which then transfers it to the FMO complex and sequentially to the reaction center. . . 16 2.2 FMO from Prosthecochloris aestuarii. (A) Complete structure of the protein

trimer. (B) Close-up to a single monomer. (C) The eight numbered chromophores in the monomer exhibiting their localized dipolar momenta. (D) A single BChl molecule, one of the eight chromophores in each monomer.

Figure produced from Protein Data Bank file 3EOJ [1]. . . 17 2.3 Diagram of the excitation energy transfer pathways in the FMO complex de-

picting the energies of each site and considering the participation of the additional eighth bacteriochlorophyll (blue). . . 19 3.1 Schematic representation of excitation energy transfer conditions and interac-

tions in GSB. . . 22

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5.1 Population evolution in [0, 4] ps for the 7 BChls in the FMO monomer with γk = 50 cm⁻¹. λk = 35, 65 cm⁻¹reported in columns, and T = 77, 185, 293^◦K in rows. Note the differences in the coherence times, transference times and oscillation amplitudes. (A) λ_k = 35cm⁻¹, T = 77^◦K; (B) λ_k = 65cm⁻¹, T = 77^◦K; (C) λk = 35cm⁻¹, T = 185^◦K; (D) λk = 65cm⁻¹, T = 185^◦K; (E) λ_k = 35cm⁻¹, T = 293^◦K; (F) λ_k = 65cm⁻¹, T = 293^◦K. . . 44 5.2 Contour plots for the equilibrium populations ρ_ii_∞ behaviour in color for each

BChl as function of T, λk, γk. (A–C) those with the largest variability, and (D–

G) those with the lowest. Insets are alternative equivalent plots in transparent colors in the analysis region. Curves on the contours are cuts with constant T (green), λ_k(red) and γ_k(yellow). . . 46 5.3 (A) Maximum coherence Cl1(ρ) in the entire dynamics, and (B) half-life co-

herence time T_d_50%as function of T, λ_k and γ_k. . . 48 5.4 Maximum coherence C_l₁(ρ) versus ∆T_ch as function of T, λ_k and γ_k. . . 49 5.5 Entanglement color chart for γk = 50 cm⁻¹ with (A) λk = 35 cm⁻¹, and (B)

λ_k = 65 cm⁻¹. In the center, concurrence C{k}evolution for each BChl, and in the outer tracks C{kl} between different pairs in the crossing of each circular track and each angular sector. Time grows counterclockwise in [0, 2] ps on each angular sector. Temperature grows radially outwards on each track in [77, 347]^◦K. . . 52 5.6 Evolution of coherence for the first 2.5 ps for γ = 50 cm⁻¹as function of T in

color, for (A) λ = 35 cm⁻¹, and (B) λ = 65 cm⁻¹showing the transition from concurrence between pair (1, 2)—solid line—and (3, 4)—dashed line—as the main provider of coherence. . . 53 5.7 (A) Eficiency η in the entire dynamic, and (B) transference time T_t_95%as func-

tion of T, λ_kand γ_k. . . 54 6.1 Population evolution in the first 6ps (10ps in the upper inset) for (A-D) N = 7

and (E-F) N = 8 BChls in the FMO monomer of C. tepidum with T = 293^◦K, γk = 50cm⁻¹. First and second columns for λk = 35, 65cm⁻¹ respectively, and initial conditions: |1ih1|, |6ih6| and ρ⁸_{F RET}, in rows. Note the differences in the coherence and transference times. . . 59 6.2 Population evolution in the first 6ps (10ps in the upper inset) for the 7 and 8

BChls in the FMO monomer of P. aestuarii with T = 293^◦K, γ_k = 50cm⁻¹. First and second columns for λ_k = 35, 65cm⁻¹respectively, and initial conditions: |1ih1|, |6ih6| and ρ⁸_{F RET}, in rows. Note the differences in the coherence and transference times. . . 60 6.3 Evolution of concurrence C_l₁(ρ) versus ρ₃₃ + ρ₄₄ converging to η (in log-

scale) for C. tepidum (blue) and P. aestuarii (red) for λ = 35cm⁻¹(light) and λ = 65cm⁻¹ (dark), for the cases (A) ρ(0) = |1ih1|, (B) ρ(0) = |6ih6|, and (C) ρ(0) = ρ⁸_FRET. . . 62

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6.4 Overlapped protein structure and amino acid sequence for the fmoA gene in FMO complexes of C. tepidum and P. aestuarii. (A) Location of peptide differences (blue) and similarities (red) between both structures. (B) BChl orga- nization overlapping. (C) Linear representation of protein structures showing residues belonging to β-sheets (green arrows) and α-helices (pink curves).

Punctual differences in the amino acid sequence (blue) in (C) match the sites illustrated in A). Protein Data Bank files 3EOJ (P. aestuarii) and 3BSD (C.

tepidum). Generated using iCn3D. . . 64

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List of Tables

5.1 Summary of some operative values for each BChl through the parametric analysis for γ = 50 cm⁻¹with λ = 35 cm⁻¹, and λ = 65 cm⁻¹. . . 45 6.1 Summary of featuring values for C. tepidum and P. aestuarii under several

initial conditions for λ = 35cm⁻¹ and λ = 65cm⁻¹ (T = 293^◦K, γ = 50cm⁻¹, times are given in ps). . . 63

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Chapter 1 Introduction

As the world’s energy demand continues to grow, so does the awareness of the global climate impacts of meeting this demand with traditional methods. Despite accounting for only 8% of the total energy generation of the world in 2017, the growth rate of renewable energy continues to be the highest among the power sector, representing half of the total growth in power generation on that same year [2]. During the year 2018 the growth rate of global energy consumption surpassed the 2.9%, representing the fastest growth in the last decade [3]. Nevertheless, even with the increasing generalized concern for climate impact, carbon emissions rose at their highest rate in the past seven years [3]. Power generation from renewable sources has shown steady and continuous growth for the past years [2, 3]. Year 2018 reported an increase of 71 mtoe, with solar generation representing the 40% of such increase [3]. Solar energy is the most rapidly growing form of energy generation but, though promising, it still faces some mayor challenges: cost, scalability and intermittency [4]. Additionally, even though current photovoltaic cells have continuously improved over the years, one of the main concerns around this technology is the availability of its materials and the environmental impact their recycling and re-usability imply [5].

Effective solutions to these challenges are actively being applied every single day in our planet. In nature, photosynthetic organisms possess light-harvesting complexes where the photochemical quantum yield of products formed per photon absorbed are very close to 100%

[6, 7, 8, 9]. The fact that nature has evolved to efficiently exploit solar energy has fueled inspiration towards deeper understanding of the process and its possible applications for new technological developments. Already many of the most recent and significant improvements to photovoltaics have been inspired by nature [5]. From nanostructures found in the human eye and on butterfly wings, to algorithms inspired by wasp behaviour as well as self-cleaning and antireflective surfaces present on lotus leaves [5]. The correct implementation of biocom- patible materials to the production of solar cells may represent a significant advance towards more sustainable energy generation methods. Furthermore, the study of energy dynamics within these complexes has demonstrated possible applications in a diversity of fields such as quantum computing.

Interest in quantum computing comes from the possibility of performing extremely com- plicated tasks that might take up to thousands of years to be accomplished by means of our current most powerful supercomputers in a matter of seconds. This would be possible by taking advantage of the laws of quantum physics and superposition for improving computing

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CHAPTER 1. INTRODUCTION 2

efficiency [10]. One of the main challenges for quantum computers is the development of suit- able high-fidelity processors able to run quantum algorithms [11, 10]. Research being done in this field has recently resulted in the experimental validation of quantum supremacy [11], finally confirming the true potential of quantum computing. Once more, nature provides possible solutions that may serve as inspiration for the improvement of our technologies. Just as with solar power, photosynthesis and its efficient energy transfer could help us understand the mechanics behind quantum processing and develop practical tools based on natures design.

As a result of this search for understanding the phenomena behind the high energy transfer efficiency observed in photosynthesis, an emerging branch of biology known as quantum biology has been gaining momentum in recent years. Although speculations made about quantum biology date as early as a century ago, it has only been during recent decades that there has been an increasing interest into its applications [12]. Due to the quantum behaviour of light, it is only natural to venture into quantum mechanics to explain the light harvesting phenomena in photosynthetic organisms. This work will propose a model based on quantum mechanics to describe the behaviour of the light-harvesting complex Fenna–Mathews–Olson (FMO) in green sulfur bacteria and its reaction to varying environmental conditions.

Introductory Chapter 1 aims to familiarize the reader with the basic concepts of photosynthesis, light-harvesting pigment-protein complexes, green sulfur bacteria and quantum mechanics that will be further discussed throughout this thesis but are necessary for a complete understanding of the problem and the approach to the problem being proposed. Chapter 2 provides a complete description and characterization of the FMO complex. Chapter 3 de- fines quantum open systems (QOS) and deals with the different modeling approaches used to describe them and which will be the basis of the resulting model of this thesis. In Chapter 4, the proposed model using the hierarchical equations of motion method for excitation energy transfer dynamics in the FMO complex is explained, including data analysis. Chapter 5 de- scribes quantum entanglement quantification and efficiency in the FMO complex. In Chapter 6, the relation between the behaviour of the complex and the different species and strains that contain it will be addressed considering their genetic diversity. Conclusions and further investigation are presented in Chapter 7. An outline of this work is represented in Figure 1.1, illustrating the recommended paths to be taken by each reader depending on his or her prefer- ence to approach the problem. Next is a list of articles related with this investigation and that might help the reader continue his or her exploration into the topic:

• B. Gonz´alez-Soria and F. Delgado, “Quantum entanglement in Fenna-Matthews- Olson photosynthetic light-harvesting complexes: A short review of analysis methods,” J.Phys.: Conf. Ser., vol. 1540, p. 012026, 2020.

• B. Gonz´alez-Soria, F. Delgado, and A. Anaya-Morales, “Predicting entanglement and coherent times in FMO complex using the HEOM method,” arXiv e-prints, p. arXiv:2008.07580, 2020. (Accepted to be Published in J. of Phys. Conf. Series)

• B. Gonz´alez-Soria, F. Delgado, and A. Anaya-Morales, “Parametric Mapping of Quan- tum Regime in Fenna–Matthews–Olson Light-Harvesting Complexes: A Synthetic Review of Models, Methods and Approaches,” Applied Sciences 10, 2020.

These articles have been presented in the following events:

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• International Conference on Quantum Phenomena, Quantum Control and Quantum Op- tics, October 28-November 1, 2019, Physics Department, Cinvestav, Mexico.

• 50 Congreso de Investigaci´on y Desarollo, February 25-28, 2020, ITESM, Mexico.

• 9th International Conference on Mathematical Modeling in Physical Sciences, Septem- ber 7-10, 2020, Tinos island, Greece.

• Quantum 2020, October 19-22, 2020, IOP Publishing.

Figure 1.1: Possible paths for the reader to explore this work according his or her interests.

Coloured arrows indicate recommended path for the reader interested in biological (green) or physical (red) concepts and backgrounds. Blue arrows are to be followed by any reader.

1.1 Problem and Motivation

A variety of different materials and designs have been implemented trying to obtain more energy- and cost-efficient photovoltaic (PV) cells. The main factors affecting the conversion efficiency (the ratio of the electricity produced by a photovoltaic PV cell by the amount of sunlight that reaches such cell), are wavelength, recombination, temperature and reflection [13]. While crystalline silicon continues to be the most common material used in solar cells, there have been other developments like thin-layer, organic, tandem and concentrated PV cells trying an infinite variety of material combinations [4, 14, 15]. Nevertheless, there is no PV technology that excels in high power conversion efficiency while keeping a low material us- age, manufacturing complexity and cost [4]. Regardless of these obstacles, the world’s global necessity to reduce CO₂emissions and the broad distribution of the solar resource around the planet are two very important motivators for the continuous development and improvement

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of this technology [4, 16, 2, 3]. There are two trends for improvement in PV technology:

solar cell efficiency and energy storage. Both of which are receiving tremendous attention by markets, governments and researchers [4]. While highly complex and expensive III-V multi- junction solar cells used for space applications have a record power conversion efficiency of 46% [4], photosynthetic organisms present a remarkable near unity quantum efficiency even under low radiance conditions [17]. The understanding of how quantum dynamics present in light-harvesting complexes like the FMO can help improve the energy conversion efficiency of solar cells through efficient excitation energy transfer (EET).

The same applies to the development of quantum processors which so far have only been able to operate at close to absolute zero temperatures and are made of rare metals, which makes their manufacturing and operation little accessible [11]. The amazing fact that LHCs operate at room temperature is alone an incomparable advantage against our current technology. Being able to bring some of natures features into our quantum computing developments might make this kind of computing much more accessible.

1.1.1 Problem Statement

This project will present a quantum model of electronic energy transfer dynamics in the Fenna-Matthews-Olson light-harvesting complex, present in green sulfur bacteria, and a comparison of the effects of varying conditions between species C. tepidum and P. aestuarii to determine its optimal operation conditions.

1.1.2 Research Questions

Several questions arise as one adventures into the depths of the world of photosynthetic organisms and the incredible tools they have developed to survive. Due to growing interest in efficient energy transfer within these structures, the physics, biochemistry and genetics behind this phenomena are under intense investigation. From the sea of intricacies and mysteries that surround these complexes, a few questions have been determined as a guide to determine and limit the aim of this work. The questions to be addressed within the scope of this particular investigation are the following:

1. What role do parameters like temperature, protein structure and inner interactions play in the characterization of efficient coherent energy transfer within the complex?

2. How can such parameters be enhanced or modified to maintain long-lived coherence and entanglement?

3. How do different initial conditions depending on excitation paths, site couplings and number of interacting pigment molecules affect the energy transfer in the two different species C. tepidum and P. aestuarii?

These questions establish the boundaries of this research and trace a logical path for the journey to be undertaken. Starting by the necessary definition of the parameters to be evaluated, then finding a way to perform such evaluation, and finally comparing and analysing the results. Following this path will lead to the conclusion of this journey but, to accomplish such task the objectives for this research must be determined based on these questions.

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1.1.3 Research Objectives

The aim of this work is to provide an integral and up-to-date perspective of our understanding of the quantum phenomena that take place within the FMO light-harvesting complex and push this understanding forward by modeling the behaviour of the complex and evaluating key parameters that have an effect on such behaviour. Definitely, the first objective of this work is to perform a parametric model of energy transfer dynamics within the FMO complex. This model is to be repeatedly applied under various conditions in order to obtain a complete data set that will provide a complete image of the behaviour of the energy transfer under a range of conditions which will be determined according to those reported in literature and those the organism faces in vivo for obtaining realistic and comparable results. The presence of coherence must also be identified and quantified to determine its role in the resulting efficiency of the process. This work will also have as an objective to compare the diverse initial condition scenarios for the different structures that have been considered in literature for this complex.

Finally, the affiliation between the obtained results and the genetic encoding of the complex will be identified.

1.2 Photosynthesis

Photosynthesis is the process by which phototrophic organisms like plants, algae and certain types of bacteria are able to capture and store light energy to be used in energy-requiring processes [18, 19, 9]. Photosynthesis is a fundamental process for life on Earth. It results in the production of food, fuels and fiber which are necessary for the survival of almost all living organisms on the planet and plays an important role in regulating global atmospheric conditions [18, 20]. Photosynthesis can be oxygenic, meaning water is used as an electron donor to produce oxygen, or anoxygenic, when sulfide or other organic compounds are used instead [18]. Photosynthesis is composed by two stages of chemical reactions: the ”light”

stage, during which photochemical reactions occur to produce energy storage compounds, and the ”dark” stage, during which enzymatic reactions take place to, among other things, perform carbon fixation [18]. Although the process requires highly complex biological structures that interact with one another in a series of chemical reactions, it can be generally summarized into this single reaction [18, 19]:

CO₂+ 2H₂A−−→ CH^light ₂O + A₂+ H₂O,

where A is an electron acceptor (oxygen in green plants and sulfur in bacteria), and the energy of light is used to separate the hydrogen ions (H⁺) from the electron donor molecules. The reaction also shows that carbon dioxide (CO₂) from the atmosphere is reduced to a carbohy- drate; this is known as carbon fixation [18, 19].

The electrons removed during the oxidation of the electron donor molecules are incorpo- rated into an electron transfer chain which results in the formation of (NADPH), the reduced form of nicotinamide adenine dinucleotide phosphate (NAD(P)⁺) [18]. Meanwhile, the resulting concentration gradient of hydrogen ions is exploited for the formation of adenosine triphosphate (ATP) from adenosine diphosphate (ADP) and inorganic phosphate (P_i) [18].

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These are the compounds that store the chemical energy produced during the light stage and that can now be used by the organism to fuel its metabolism.

The success of the photosynthetic process is the result of millions of years of evolution during which highly specialized structures able to capture, transfer and store light energy by a series of physical and chemical transformations have been developed. An important group of these structures are the light-harvesting complexes (LHC), responsible for capturing photons and transferring their energy to the reaction centers (RC), where charge separation takes place. Since their early discovery, LHCs have captured the attention of researchers due to their remarkable nearly 100% photochemical quantum yield [6, 7, 8, 9]. The system of interest to be studied in this thesis, the Fenna-Matthews-Olson (FMO) pigment-protein, is a LHC that contains only bacteriochlophyll a photopigments which act as antennas capturing photons and transferring their energy through the complex to the bacteria reaction center. This complex is found in green sulfur bacteria (GSB) and is responsible for the very first step of the light stage of photosynthesis in these organisms. This work will be focused in deeply explaining and understanding the process of light harvesting by the FMO complex, further steps and reactions corresponding to the process of photosynthesis are out of the scope of this investigation.

1.3 Green Sulfur Bacteria

The presence of sulfur in protein-building amino acids and other biological molecules make it an essential element to all life forms for anabolic processes [21]. However, sulfur bacteria have evolved to also use sulfur in higher amounts as an electron donor or electron acceptor for energy production, taking advantage of its properties as redox reaction mediator [21]. Sulfur bacteria have three main ways of using sulfur in their metabolism: (1) reduced sulfur compounds as the main electron donor for photoautotrophic growth, (2) sulfur-oxidation as energy source for growth, and (3) oxidized sulfur compounds as electron acceptors for anaerobic res- piration [21]. In the case of photosynthetic sulfur bacteria, which are divided into two main groups due to the photosynthetic pigments they present - green and purple sulfur bacteria -, their distribution and growth are determined by the light availability and and sulfide concentrations in their media [21]. Their anoxygenic photosynthesis reaction can be summarized in any of the two following reactions depending on the oxidized sulfur compounds formed [21]:

2H2S + CO2 −→ CH2O + H2O + 2S⁰ H₂S + 2CO₂+ 2H₂O −→ 2CH₂O + H₂O + H₂SO₄

Green sulfur bacteria (GSB) of the Chlorobiaceae family are anoxygenic photosynthetic sulfur bacteria that use light as energy source for carbon fixation, mainly using hydrogen sulfide (H₂S) as electron donor [21]. This kind of bacteria are obligate anaerobic photolithotrophs that only grow under strict anoxic conditions and use (depending on species) sulfide, elemen- tal sulfur, thiosulfate, molecular hydrogen or even reduced iron and other organic substances as electron donors for anoxygenic photosynthesis [21]. The electrons from the reduced form of sulfur are used for CO₂fixation via the reverse tricarboxylic acid cicle, while the oxidation of sulfide results in the formation of sulfure globules deposited outside the cell [21, 9]. GSB

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have been found in a variety of environments ranging from sulfur springs and deep-sea hy- drothermal vents to the anoxic hypolimnia of lakes, where they grow under either high or low but stable sulfide concentrations [22, 21]. The depths at which these bacteria are found, where there is little if any access to light, and the vast range of temperatures at which they grow and adapt, contributes to the possibility that the efficiency of their photosynthetic mechanisms take advantage of quantum effects at relatively high temperatures. GSB, as any other living organisms, possess regulatory mechanisms to better adapt to the changing circumstances of their environment. For instance, the bacteriochlorophyll synthesis is strongly regulated by light intensity, meaning that pigments and chlorosomes can be multiplied under light-limiting conditions or their production can be stopped under rich light conditions [21].

Some GSB strains have a high content of photopigments like bacteriochlorophyll c or d, and the carotenoid chlorobactene, giving them a green colour, while other GSB strains mainly contain bacteriochlorophyll e and caroteinoids isorenieratene or β-isorenieratene, which makes them brown-coloured [23, 21, 9]. In low light environments, GSB dominate over purple sulfur bacteria due to their higher quantum yield and lower ATP requirements for CO2fixation, the larger size of antenna light-harvesting complexes, and the range of wavelengths absorbed by its variety of photopigments [21]. While bacteriochlorophyll a (main photopigment in purple sulfur bacteria) presents an absorption maxima at 830nm, carotenoids isorenieratene and chlorobactene present absorption maxima at 435nm and 459nm, repectively, which better correspond to the light availability in deeper waters [21]. The combination of various pigments results in the absorption of a wider range of the light spectrum and an increase in their optical cross section [21]. Also, GSB present a higher sulfur affinity and sulfide concentration for growth inhibition than purple sulfur bacteria, which explains their dominance in low and high sulfide concentration habitats [21, 9].

The two main species of GSB that will become the center of attention further on are Prosthecochloris aestuariiand Chlorobaculum tepidum. Although the taxonomic classifica- tion of Chlorobiaceae was typically based on morphological and phenotypic characteristics, they are now better identified by phylogenetic relationships using their 16S rRNA and fmoA gene (gene that codes for the Fenna-Matthews-Olson protein) sequences [23, 21, 24]. Genetic distancing between these two species and others belonging to the group of GSB can be found in Figure 1.2. The type strain of P. aestuarii, the nomenclatural type of the species described by Gorlenko in 1970 [25], is the DSM 271 (also SK 413) with a G+C DNA content is in the range 52.0 − 56.1mol% and which 16S rDNA sequence is identified by the GenBank accession number: Y07837. This strain is a non-motile, spherical to ovoid GSB of 0.5 − 0.8µm that lacks gas vesicles [23]. It has been reported that the bacteriochlorophylls c, d or e, and chlorobactene carotenoids are its major pigments [23]. This organism requires salt and vita- min B₁₂ as growth factor [23]. In the other hand, C. tepidum is a moderately thermophilic freshwater species without a salt requirement and with a DNA G+C content of 56.5mol%

[23]. The type strain is ATCC 49652^T (also DSM 12025^T) with a GenBank/EMBL accession number for the 16S rDNA sequence of M58468.

1.3.1 Light-Harvesting Complexes in GSB

Capturing light photons in order to convert and store their energy is possible thanks to the light-harvesting complexes present in the photosynthetic organism. The absorbed energy

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Figure 1.2: Taxonomy of Green Sulfur Bacteria (GSB) based on 16S. Generated using Type (Strain) Genome Server TYGS with the GBDP method. Branch lengths (red) indicate genetic change. Confidence values (blue).

needs to be transferred to and used by other structures that require it for the following steps of the process of photosynthesis. Light-harvesting antenna complexes absorb light and transfer its energy to the reaction center that acts as an energy sink or trap. Almost all antenna complexes have strong pigment-protein interactions with one one exception, the chlorosome antenna complex, found in the green photosynthetic bacteria, in which mostly pigment–pigment interactions occur with very little protein involvement [9]. Figure 2.1 shows the positioning of the chlorosome on a schematic representation of the cell membrane. For pigments to suc- cessfully transfer light’s energy, their spectral and spatial cross sections must be optimized for light absorption, and rapidly funnel the absorbed sunlight to avoid heat energy losses [9].

Since pigments require a protein structure to support them, these complexes are known as pigment-protein complexes. The light-harvesting ability of these complexes relies on the or- ganization of the pigments in their protein scaffold [9]. For instance, the spatial cross section for light absorption can be increased by increasing the number of pigment molecules in an antenna complex or the number of antenna complexes in an organism [9]. Due to their specific structural and chemical nature, proteins may form local environments within themselves.

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This local protein environment surrounding the chlorophyll pigment molecules can significantly alter their spectral and redox properties through specific pigment–protein interactions [7]. However, also short-range dispersion and repulsion interactions strongly influence the properties of the chromophores [7]. As an example of the effect of this pigment-protein interactions, the maximum absorption wavelength of a pigment can be shifted by hydrogen bonds between the pigment and the protein amino acids, affecting the mechanism and efficiency of the light-harvesting process [7].

The FMO protein was the first chlorophyll-containing protein to have its structure determined thanks to its water-soluble characteristics, and since then it has been thoroughly studied by spectroscopic and theoretical techniques. The FMO protein was also the first antenna complex found to exhibit quantum coherence effects [9, 26]. It belongs to the group of extrinsic antenna complexes that are associated with other components in the membrane, but are not themselves inserted in the membrane [9]. For photosynthetic antennas to function, the light absorbed by one pigment may subsequently be transferred to other pigments by means of excitation, and it is this excitation energy that is eventually transferred to the reaction center.

Tracing this energy transfer process is possible by irradiating a sample with light that is selec- tively absorbed by one set of pigments and monitoring the fluorescence that originates from a different set of pigments [9, 7]. If light is absorbed by one set of pigments and emitted by another set, energy transfer must have taken place between the two groups of pigments [9, 7].

Bacteriochlorophyll a

Bacteriochlorophyll (BChl) a is the main type of photopigment present in the majority of anoxygenic photobacteria [9]. Its chemical structure compared to that of chlorophyll a, as shown in Figure 1.3, is less symmetrical and has a reduced degree of conjugation in the macrocycle, differences that have an important effect on the spectral properties of the molecule [9].

Notice the acetyl group at the C − 3 position and the single bond in ring B between C − 7 and C − 8 (IUPAC nomenclature) in bacteriochlorophyll a, instead of the double bond found in chlorophylls. The x and y axes are shown in the bacteriochlorophyll a molecule in Figure 1.3 for future reference. The phytyl tail label in the bottom left of (a) receives that name after its isoprenoid alcohol precursor (phytol) attached during biosynthesis. All chlorophyll-type molecules share common structural elements: a stable ring-shaped molecule with a magne- sium atom in the center and a long carbon–hydrogen side chain (phytyl chain).

These molecules possess a delocalized π electron system extended over most of the molecule, which allows electrons to freely migrate around the alternating single and double bonds in the macrocycle of the molecule. Chlorophylls have normally three chiral carbon atoms, C − 13⁰⁰, C − 17, and C − 18. BChl a has two additional chiral carbon atoms, C − 7 and C − 8. Both, chlorophylls and bacteriochlorophylls contain a metal atom, usually Mg, coordinated with the four nitrogen atoms of the molecule [9].

Spectroscopic properties of bacteriochlorophyll a

All chlorophyll pigment molecules contain two major absorption bands, one in the near ultra violet (UV) region and one in the near infra red (IR) region as illustrated in Figure 1.4. [9].

The highest excited state (blue) loses energy as heat in matter of ps to produce the lowest

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(a) Bacteriochlorophyll a (b) Chlorophyll a

Figure 1.3: Structural comparison between bacteriochlorophyll a (left) and chlorophyll a (right).

excited state (red). This lowest excited state, generally long-lived (ns), is used for electron transfer and energy storage in photosynthesis [9]. The green colour observed in the green strains of GSB results from the lack of absorption in the green region of the light spectrum.

The absorption and fluorescence spectra of bacteriochlorophyll a are shown together in Figure 1.4. The two lowest-energy transitions are called the Q bands, and the two highest are known as the B bands or Soret bands.

Electronic transitions have transition dipole momenta with different strengths and orien- tations [9]. The longest-wavelength transition, known as the Qy transition, is polarized along the y-axis while the Qxtransition is polarized along the x-axis. The electric vector of plane- polarized exciting light must be parallel to the y-axis of the pigment for strongest absorption.

In such case of a Qytransition the exciting light couples to the π electrons of the molecule and rearranges them during the transition which causes a shift in electron density directed along the y-axis of the molecule.

The fluorescence spectrum of all chlorophylls peaks at slightly longer wavelengths than the absorption maximum (illustrated in Figure 1.4) that is emitted from the Qy transition.

Molecular vibrations that are activated during electronic absorption are also likely to be activated upon fluorescence emission since ground and excited states have similar shapes [9].

However, in fluorescence, the initial state is the ground vibrational state of the excited electronic state, and the final state is the excited vibrational state of the ground electronic state, thus shifting the emission to the longer-wavelength side of the main transition known as the Stokes shift [9]. When embedded in the protein scaffold, the spectral properties of these pigments are significantly altered, causing a shift of up to 100nm [9]. In the pigment-protein structure, the shape of the Qy absorption peak of the BChl a molecule is sensitive to exciton coupling, while the Qxabsorption peak is more sensitive to the number and chemical nature of the coordinating ligand to the central Mg atom of the pigment [27].

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Figure 1.4: Simplified energy level diagram of and absorption (black) and fluorescence spectrum (green) of bacteriochlorophyll a in diethyl ether. Modified figure from Blankenship, 2014

1.4 Quantum Mechanics in Photosynthesis

Quantum mechanics is based on the uncertainty principle, wave-particle duality, quantization of energies and the modification of classical probability laws [17]. It has been discussed for more than a century, but recently, these quantum features have been taking a more important role in biology, as they might explain the nature of some biological processes. Quantum mechanics is an inherently probabilistic theory, meaning that all predictions derived from it are of a probabilistic character [28]. As such, the statistical interpretation of quantum mechanics implies the predictions about the behaviour of ensembles [28]. It is important to state that as far as it is understood, there is no deterministic theory from which the quantum probabilities could be deduced [28]. Just as in classical physics, neglecting the influence of the environment on a system is considering an ideal non-realistic scenario. Thus, it is necessary to consider the effects of the surroundings on the system to properly model and understand its behaviour. This is specially necessary for biological systems which are completely exposed to environmental conditions and interference.

The light-harvesting complex II is the most abundant photosynthetic antenna complex on the planet, containing over 50% of the world’s chlorophyll molecules [29]. Evidence of quantum entanglement and superposition in this complex explains how excitation energy may be transferred from the photons to the electrons as a quantum wave through photosynthetic proteins to keep their phase coherent. Some studies suggest the presence of entanglement is strongly correlated with transport efficiency. The more entangled the system is, the higher the transport efficiency of the system will be [30]. This same effect is also present in smaller complexes like the FMO complex, a pigment-protein light-harvesting component [30]. The

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presence of entanglement in biological systems represents a whole different opportunity for the application of this phenomenon, since it was believed to be a fragile property that required highly controlled environments to manufacture, but it has been demonstrated that such properties can exist in “noisy” natural environments like cells and protein complexes at normal temperatures and pressures [30, 31, 29]. Light harvesting in photosynthetic organisms have a remarkable quantum efficiency of nearly 100% for initiation of charge separation per absorbed photon [17]. This is true even under conditions of low irradiance due to fast excited state dynamics in energy transfer and charge separation, where quantum superposition and coherence play important roles [17]. The FMO, as a biological complex is inherently affected by environmental interactions [31]. This causes the coherence properties of the FMO protein to be a result between coherent dynamics of the complex and the decoherent effects of the environment. Thus, since the environment must be considered for the prediction of quantum properties such as entanglement [31], these biological systems belong to the study of quantum open systems (QOS) which will be addressed in Chapter 3.

1.5 Multidisciplinary Study of FMO Complexes

The study of photosynthesis continued as theories on electron transfer in biological systems matured [32, 33], and the FMO complex took the spotlight as it became the first mapped LHC structure [34, 35]. Multiple disciplines are actively collaborating in the race for unraveling the mechanism behind this energy transfer efficiency and its large range of possible applications that range from photovoltaics to quantum computing [36, 37]. New and improved information about the structure and its behaviour continues emerging [6, 38, 1]. While theoretical physics has provided the basis for advanced femtosecond laser techniques necessary for tracing EET dynamics [39, 40], biochemical studies on the species [21, 22] and genetic methods are being used for purifying, characterizing and even modifying the protein structure [41, 24, 42, 43].

Additionally, improving X-ray crystallographic procedures have been key for obtaining high resolution electron density maps of protein complexes which allow insight into existing interactions within the macro molecule [44, 34, 35].

This merge of disciplines has resulted in the introduction of physical concepts into a biological context for the wide study of photosynthesis. For instance, in solid-state physics the exciton refers to a collective electronic excitation with well-defined quantum properties in a molecular crystal. Though lacking the same ordered and static structure, photosynthetic proteins also share excitations among their pigment molecules in a more ordered manner than pure pigments in solution [44]. Thus, these systems have been settled to be somewhere between crystals and fluids [44]. The complexity of the system has required the use of well- known developments of other disciplines as tentative approaches to the problem. A diagram of the main scientific branches involved in the study of the FMO complex is depicted in Figure 1.5 remarking just some emblematic works contributing to the understanding of the FMO complex. This map shows the complexity and richness of the problem which should be addressed in the understanding and collaborative research around it.

As a matter of fact, the first widely studied light-harvesting complex was the FMO pigment-protein. Its study has become an example of a multidisciplinary effort to understand an entire behaviour departing from quantum properties of biological systems.

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Figure 1.5: Schematic map of the multidisciplinary study surrounding the FMO research and some of the works that have provided the basis of this article.

There are some promising applications to these systems. Light-harvesting complexes have inspired the development of new solar cell designs that have taken advantage of their extremely high quantum yields. The work published by Ravi et al. [36] discusses an in- novative method for protein-based photovoltaic cell manufacturing using a phase-changing electrolyte to build solid-state photobiolectrochemical devices without jeopardizing the protein structure. Although poor photocurrent densities and stability are strong limitations, there is increasing interest in developing this kind of technology due to its multiple environmental, abundance and efficiency advantages. Quantum computing has also started to inquire into the possibilities posed by these complexes in efficient information transfer. It has been theorized that computers based on artificial light-harvesting complexes could operate with a processing capacity of two to three levels of magnitude higher than current computers at room temperature [37]. Understanding how light-harvesting complexes work represents a multidisciplinary challenge. A variety of researchers from all backgrounds are working to find answers to this

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thrilling problem. This thesis aims to find a balance between the different approaches that surround the FMO complex and come out with an objective and well founded base for further research.

The next chapter will provide a detailed description of the FMO complex, its structure and the interactions that take place within the protein and between the molecule and its environment. For the reader’s ease, the abbreviation BChl will be used from now on as reference to the bacteriochlorophyll a chromophores in the FMO complex.

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Chapter 2 The Fenna-Mathews-Olson Complex

This chapter provides a comprehensive description of the FMO complex to help the reader get a better idea of the molecule, its structure and interactions. This description will lay the basis for the parameters to be considered and evaluated in further chapters and is fundamental for understanding the excitation energy transfer process subject of this investigation.

2.1 Structure

Initially known as the ”bacteriochlorophyll a protein”, the Fenna-Matthews-Olson (FMO) protein gets its name after Roger Fenna and Brian Matthews, who first determined its structure [34, 35], and John Olson, who discovered the protein [45]. This LHC is located between the chlorosome (another LHC) and the reaction center as shown in the recreation included in Figure 2.1. It participates in the excitation energy transfer process during photosynthesis by funneling these excitations from the baseplate of the chlorosome to the reaction center [9].

This is possible due to its pigment-protein couplings which tune the optical properties of the complex [6].

2.1.1 Protein Characteristics

The FMO is a water-soluble protein with a molecular weight of 150, 000Da [34, 35] with a maximum diameter around of 8.3nm. This rare property has facilitated its crystallization for high resolution spectroscopic studies [35, 1]. Trimer structure of FMO includes monomer sequences working together still with certain independence to gather the light excitation. Each monomer is a ∼ 360 amino acid sequence of the complex which folds into a ”bag” containing 8 BChl molecules with an average distance between nearest-neighbor pigments of 12 ˚A[41]

(Figure 2.1). The FMO structure, shown in Figure 2.2, consists of an homotrimeric protein complex, which subunits are related by a 3-fold symmetry axis. The side chains of the amino acids conforming the protein structure interact with one another through Van der Waals forces, dipole-dipole interactions, ions and hydrogen bonding creating structures called α-helices, β-sheets and random coils resulting in the monomer shown in Figure 2.2 B. The β-sheet ribbons compose the large surface wall of the complex, and its amphipathic nature (possessing both hydrophilic and hydrophobic properties) provides shielding for the non-polar chlorophyll

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CHAPTER 2. THE FENNA-MATHEWS-OLSON COMPLEX 16

Figure 2.1: Schematic representation of the spatial arrangement of the Chlorosome, the FMO protein and the reaction center in the cytoplasmic membrane. Light is absorbed by photopigments in the chlorosome and its excitation energy is transfered down to the baseplate which then transfers it to the FMO complex and sequentially to the reaction center.

core from the surroundings, while α-helices and random coils make up the contact region between subunits, Figure 2.2 (A). These nanostructures have an important effect in BChl interactions[40].

The reader must remember that the FMO complex is present in all species of GSB, being C. tepidum and P. aestuarii the two most widely studied cases. Genetic diversity between species and strains (genetic variants of the species) results in slight structural differences between their FMO complex. The amino acids that conform the protein differ from one another in chemical and physical properties that are responsible for the characteristic folding of the complex and its resulting functionality, stability and structure altogether. The amino acids in the FMO peptide sequence share a general structure with an α carbon covalently bonded to an amine (-NH₂) and a carboxyl (-COOH) functional groups, together with a specific side chain which determines the characteristic properties of each amino acid. Amino acids may be polar, nonpolar or amphipathic, as well as amphoteric which makes them interact in different ways depending on the properties of their surroundings. To build the polypeptide chain, amino acids are covalently bonded to one another by peptide bonds which polar nature provides a certain degree of torsion to the chain. Nevertheless, this type of bond presents a partial double bond behaviour between the carbon, oxygen and nitrogen atoms involved in the bonding site, making them coplanar and limiting the free rotation of the chain to the α carbons. The side chains may interact with one another through van der Waals forces, dipole-dipole interactions, ions and hydrogen bonding creating α-helices, β-sheets and random coils. Most importantly, the folding of the protein will result in the formation of local environments determined by its own 3-dimensional structure in which the amino acids may be subject to different conditions and thus present a different behaviour than in solvent [46, 47, 48].

Some amino acid side chains can be protonated or not, depending on the pH they are exposed to. These amino acids are referred to as titratable amino acids which can belong to the

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Figure 2.2: FMO from Prosthecochloris aestuarii. (A) Complete structure of the protein trimer. (B) Close-up to a single monomer. (C) The eight numbered chromophores in the monomer exhibiting their localized dipolar momenta. (D) A single BChl molecule, one of the eight chromophores in each monomer. Figure produced from Protein Data Bank file 3EOJ [1].

acidic group (glutamic acid, aspartic acid and cysteine) or basic group (arginine, lysine and histidine). The side chains of these amino acids exist in an acid-base equilibrium in solution.

The set of titratable groups of a protein is completed by the amino group of the first and the carboxyl group of the last amino acid of a chain. They are not involved in a peptide bond and therefore are also capable of protonation and deprotonation. Considering these interactions is important for the modeling and understanding of the protein function.

2.1.2 Chromophore Sites

It was initially reported that each of its subunits contains seven bacteriochlorophylls [34], but an eighth BChl molecule was later discovered between subunits, bringing the total number of BChls in the protein to 24 [38, 49, 1, 6]. The established nomenclature for numbering the pigments is still the originally proposed by Fenna and Mathews [34], to which the eighth BChl was added. Following this nomenclature, BChl 3 and BChl 4 are the nearest to the cytoplasmic membrane while BChl 1, BChl 6 and BChl 8 are closer to the chlorosome baseplate (as seen in Figure 2.2) [9, 49, 6, 50]. The eighth additional pigment discovered in each subunit of the protein is positioned towards the chlorosome and is probably the entry point for excitations from the chlorosome baseplate [9, 49, 6]. Within each subunit center-center separations of nearest-neighboring BChls are ∼ 12 ˚A. The position of the chromophores is highly related to their excitation energy and their role in the energy transfer path. For instance, the highest exciton state of the FMO which has the largest contribution from the excited BChl 8 and its strongly excitonic coupled BChl 1 are the two main entrance routes for excitation energy coming from the baseplate [6]. The process of relaxation of the excitation energy is funnelled to BChl 3 along the branches formed by BChl 2 and by BChl 4 to BChl 7 [6].

Whereas relaxation/equilibration via the latter is completed within 500fs, BChl 2 forms a bottleneck that limits the overall relaxation/equilibration time to about 1.5ps. The short relaxation/equilibration times and the large energy gap between the entrance BChl 8 and the sink

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BChl 3 prevents recombination and leaking [6]. The distances between the bacteriochlorophylls were found to range from 4 to 11 ˚A within each subunit and the distances between bacteriochlorophylls from different subunits are over 20 ˚A [51].

As previously mentioned, the position and excitation energy of each of these pigments are strongly dependent on their interactions with the protein scaffold. The nature of the protein structure holding the bacteriochlorophylls, in the FMO and in any other LHC, provides localized environments and interactions that influence the optical transition energies of the pigments. These interactions are known as pigment-protein interactions. Electrostatic interactions with titratable groups, Mg²⁺-ligation, H-bonding, and structural conformation of the pigment, are some examples of these interactions. It is challenging to consider all the interactions within such complex structures to directly determine the site energies of each BChl molecule in the protein. This is why site energies are often treated as parameters that are determined from the fit of optical spectra [39]. Further information about this topic will be addressed in Chapter 3.

2.2 FMO and the Environment

The folding of the protein will result in the formation of local environments in which the amino acids may be subject to different conditions thus modifying their behaviour. These localized environments influence the optical transition energies of the pigments [6]. These interactions are known as pigment-protein interactions, and their resulting influence on the whole quantum site energies in the FMO complex are schematized in Figure 2.3 without a detailed quantum description which will be addressed in the development below. The position and orientation of the chromophores is highly related to their excitation energy and their role in the energy transfer path, shifting the absorption energy of the pigments facing the outer antenna towards blue compared to those linked to the reaction center shifted towards the red [6].

Due to their structural complexity and behaviour, proteins are systems with thousands of degrees of freedom [52]. On the other hand, the functional subsystem (the active site where the BChl cofactors are bound) involves only a few quantum states [52]. The change in the quantum state as the transition from the ground state to the excited electronic state is associated with a change in the electric dipole momentum of the subsystem [52]. The polar residues contained in the protein and its highly polar solvent (water) surroundings, result in a strong interaction between the functional subsystem and its environment. The protein itself may undergo structural and electrostatic changes depending on its environmental conditions, meaning that the environment must be included in the model for a correct approximation to the system [40, 6, 52].

The quantum mechanics approach to the modeling of FMO complexes is first given through a sufficiently accurate Hamiltonian HSreproducing the main interactions in the complex. It means, those dipole-dipole electrical components among BChls considered as the system, S. This main Hamiltonian is built by the individual excitation energies along the diagonal and the dipole-dipole coupling terms in the off-diagonal positions. Diagonaliza- tion of this Hamiltonian yields to a set of eigenenergies corresponding to the eigenstates in the exciton basis [44]. The energies of these excitons correspond to the observable transitions in a linear absorption spectrum, nevertheless, due to homogeneous and inhomogeneous

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Figure 2.3: Diagram of the excitation energy transfer pathways in the FMO complex de- picting the energies of each site and considering the participation of the additional eighth bacteriochlorophyll (blue).

broadening the energies of the entire seven or eight excitons have never been experimentally measured, although an enormous effort in theoretical work has been done to model the system [50], departing from the main interactions obtained by spectroscopy in order to be predicted.

Thus, this interacting set of BChls states a quantum system with well defined energy levels.

2.2.1 Main dipole interactions among BChls

Changes in the absorption spectrum of chromophores are usually different for excitonically coupled chromophores compared to the isolated ones [44, 53]. Both, in the ground and excited state, the molecule has a permanent dipole momentum, µ_g and µ_e respectively, which can differ significantly for asymmetric pigments like bacteriochlorophylls [44]. In the presence of a linear electric field FE, the energy levels Egand Eefor the ground and excited states change (together with the difference ∆E between them). Those changes are:

E_g,e = E_g,e⁰ − F_E· µ_g,e → ∆E = ∆E⁰− F_E∆µ cos θ (2.1) where ∆E = E_e−E_g, ∆E⁰ = E_e⁰−E_g⁰, ∆µ = µ_e−µ_g, and θ is the angle between F_Eand ∆µ.

The superscript 0 refers to the case FE = 0. The spatial form of fields and forces FE for the dipole-dipole interactions is avoided since it is common in electromagnetism literature [54].

From here, one must gather that knowledge about the FMO structure is essential to understand the energy transfer dynamics and performance between chromophores. Nevertheless, the protein itself may undergo structural and electrostatic changes depending on its environment conditions. For this reason, to understand the EET dynamics within the molecule, one must first understand the coherent distribution of excitation among the different BChl sites.

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This is usually achieved by taking two assumptions into consideration: 1) BChl sites are modelled as two-level systems, and 2) only one site can be excited at a time for EET to occur, making negligible the probabilities of bi-excitonic and other higher states compared to the single excitation [55]. Those assumptions can be attributed to the dipole blockade, an effect in which excitation of one site shifts other sites out of resonance due to the interaction energy being added to or subtracted from, the excitation energy for attractive and repulsive interactions, respectively [56, 55]. The strongest forces are exerted by the electric dipole interaction, arising from the closely spaced arrangement of the BChl sites [56, 55]. Since only exciton temporal dynamics have been experimentally observed through spectroscopy techniques, site excitation dynamics still must be indirectly modelled [31, 55, 57, 58, 50] in order to get an entire description of the Hamiltonian.

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Chapter 3 FMO Modeling in the Quantum Regime

Understanding how the protein environment affects the behaviour and energy dynamics of the chromophores inside it is important to understand their light-harvesting mechanism and efficiency. As previously mentioned in Chapters 1 and 2, the protein structure has an important effect on the light-harvesting capacity of the chromophores due to the local environments within itself. This chapter will provide a detailed description of the relation between the FMO complex and the quantum realm. It will begin by addressing all the interactions that govern the behaviour of the complex. Then, a brief explanation on some of the theories and approaches surrounding this complex is given for a global perspective of the different angles of the problem that have been explored.

3.1 FMO pigment-protein interactions

The protein scaffold that holds the BChl molecules in the FMO complex constrains the con- figurational space of the BChls, and modulates the energy levels responsible for their spectral and redox properties depending on the characteristics of the binding pocket holding the chromophore, [7, 59]. In other words, the protein can modulate the energy and charge migration pathways in order to deliver the energy in the right place by manipulating the chlorophylls’

energy levels [7]. In the case of the FMO complex, pigment–protein interactions shift the energies of the eight BChl chromophores in slightly different ways leading to an overall range of ∼ 500cm⁻¹covered by the energies of the different sites [6]. Electrostatic pigment–protein interactions are considered to be the leading mechanism for shifting the site energies of the chromophores [7].

Site energy disorder is of fundamental importance to appropriately direct the captured energy toward the reaction centers. The proximity of energy levels among chlorophylls pro- motes the appearance of so-called excitonic states, which can be delocalized over several coupled pigments. In this case, the whole mechanism of energy transfer among chlorophylls is modified, because the energy-transferring states are now coherently shared among several molecules. This effect leads to significant shifts in the transition energies of such states, as well as in a redistribution of dipole strength among transitions, thereby allowing for potential manipulation of the probability for light absorption associated to a particular wavelength. In turn, also the mechanisms of energy migration can be affected by exciton delocalization.

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CHAPTER 3. FMO MODELING IN THE QUANTUM REGIME 22

Excitation energy transfer in GSB within the FMO and including other photosynthetic complexes can be pictured as a multi-layer cascade process for transfer of energy first coming from the environmental photons (Figure 3.1). There, the FMO could be conceived as a multipartite funnel with a spring scaffolding where energy is collected at a top large surface to excite some of its components (the pigments) and then being transferred and concentrated towards a sink, while the process is damped by its scaffolding structure to reboot the FMO.

The energy transfer happening in this funnel involves the migration of excited states being sequentially shared by the pigments. In order to bring this analogy closer to reality one must think of this light-harvesting funnel as an arrangement of pigment particles (the BChls) with dipole like charge distributions and springs holding them to the surrounding structure and connecting them to one another. The photons (energy packets) from which the energy is being absorbed and transferred can be visualized as a rain of pebbles of different sizes (their energy or frequency). To know how many photons are absorbed by a pigment particle one must only know the photon flux and the effective cross-sectional area of the pigment [9]. This is not a physical size but rather an effective size considering factors such as the wavelength of excitation. As energy transfer occurs in this system, one can imagine the springs movement causing slight changes to the original arrangement at the time they partially damp the process.

It can be understood as the changes in the protein structure as the transfer occurs, which is labeled as the ”reorganization energy”, λ. The driving force for energy transfer is more or less equal but in opposite direction to the reorganization energy [44]. Key quantities are the electronic coupling between the BChls, the electron-phonon coupling (reorganization energy λ_k and time scale γ_k⁻¹), the temperature T , and the disorder [44]. Finally, such energy transfer boosts the reduction/oxidation chemical reactions taking place in the RC, involving other biological components and continuing the photosynthetic process. This loony picture (Figure 3.1) is useful to capture the general process formally depicted below in a much more technical description.

Figure 3.1: Schematic representation of excitation energy transfer conditions and interactions in GSB.