# Stochastic processes applied to classical physics and quantum mechanics

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(2) Précis The dynamics in physical systems when subject to environmental fluctuations is usually described by ensemble averages, providing information on the probability of the described system to reside in a possible state. Classical description differs from quantum mechanical, since the latter includes state superpositions. However, the progression among both remains a subject of current research, and the investigation beyond averages, concerning the distributions upon which they are calculated, can unveil the features involved in quantum-to-classical descriptions. For instance, stochastic realizations in the classical realm are used to provide insight on the chromatic adaptation of purple bacteria. The membranes include harvesting complexes and Reaction Centers, where a metabolically useful charge separation is accomplished and that, even though photons are scarcer under low light intensity conditions, are synthesized in a smaller amount compared to growing under high light intensity. It is found that the trade-off between extended (exciton kinetics) and local (Reaction Center charge separation) dynamics is the most important feature to improve efficiency requirements of low light intensity membranes, and to use dissipation to safeguard the bacteria of burning in high light intensity conditions. Interestingly, metabolism of bacteria (a macroscopic feature) induces adaptations of complexes involving many body transfer of quantum mechanical excitations. Without loosing sight on the optics-matter interaction involved in photosynthetic membranes, the quantum realm is studied with use of stochastic realizations for the description of photon time traces from fluorescent quantum multilevel systems, consistent with continuous monitoring of the quantized surrounding electromagnetic field. We study non-renewal statistics, happening when a fluorescent system maintains information of its state after photon detection occurs. When a process is renewal, the distribution of consecutive inter-photon times factors out, and allows quantification of memory after photon detection, relying in the moments of inter-photon distributions. We study the feasibility to experimental verify this measure. Finally, physical realizations show that quantum systems are vulnerable to ensemble averaging. In particular, it is shown that entanglement is modeled on equal footing in very different systems at the level of a density operator due to the averaging effect, and that at such resolution, a prediction of quantum correlations is in general, misleading. Continuous environmental observation is shown to lift this ambiguity, while opens the possibility to witness entanglement on situations where the ensemble averaged state predicts none. It is shown that entanglement can be experimentally verified there when none is expected from mixed state available measure, or enhanced by “decoherent” mechanisms and witnessed through observables when pure dephasing is involved. The quantum-classical middle realm investigated in photosynthesis will provide insight on this progression and on the consequences of quantum properties at the macroscopic scale. 1.

(3) Contents 1 Purple Bacteria Photosynthesis 1.1 Structure of Harvesting Membranes . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 LH2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 LH1 and RC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Processes and rates involved in bacterial Photosynthesis . . . . . . . . . . . . 1.2.1 Photon Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Excitation transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Special pair P ionization . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Quinone-quinol cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Excitation dynamics in photosynthetic membranes . . . . . . . . . . . . . . . 1.3.1 Excitation dynamics through a Master Equation . . . . . . . . . . . . 1.3.2 Master equation: Single excitation Green’s function approach . . . . . 1.3.3 Stochastic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Comparison between Master Equation and stochastic populations results 1.4 Concluding Remarks of this Chapter. . . . . . . . . . . . . . . . . . . . . . . .. 8 8 11 11 12 12 13 18 19 19 20 20 22 27 30 34. 2 Chromatic Adaptation of Purple Bacteria: Exciton dynamics 2.1 Atomic Force Microscopy Results . . . . . . . . . . . . . . . . . . 2.2 Excitation dynamics in complete chromatophore vesicles . . . . . 2.2.1 Characterization of HLI and LLI membranes . . . . . . . 2.2.2 Single excitation dynamics in complete chromatophores . 2.2.3 Multiple excitation dynamics . . . . . . . . . . . . . . . . 2.3 Conclusion of excitation dynamics. . . . . . . . . . . . . . . . . .. . . . . . .. 35 35 37 41 42 59 65. 3 Excitation and RC cycling coupled dynamics 3.1 Effective Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Dynamical behavior of membranes due to RC cycling: τ variation . . 3.1.2 Metabolic requirements of bacteria: Light intensity variation. . . . . .. 67 69 70 72. 2. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . ..

(4) 3.2 3.3. 3.1.3 Discussion: Arrangement of complexes . . 3.1.4 Robustness of photosynthetic membranes Global excitation Analytical Model . . . . . . . . 3.2.1 Coupled excitation-RC dynamics model . Metabolic requirements: Stoichiometry prediction. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 4 Memory in the Photon Statistics of Quantum Multilevel Systems 4.1 Quantum Optics: Density operator approach . . . . . . . . . . . . . . . . . . 4.2 Quantum optics: Quantum jump approach . . . . . . . . . . . . . . . . . . . 4.2.1 Perfect photon detection . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Observables of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Non-exclusive detection rate I(t) and second order correlation function g(2) (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Photon time traces statistics . . . . . . . . . . . . . . . . . . . . . . . 4.4 Non-Renewal Statistics (NRS) in Multi-Level Systems . . . . . . . . . . . . . 4.4.1 Three-level Λ-system (I) . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Three level cascade system (II) . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Four level cascade system (III) . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Four level non-cascade system (IV) . . . . . . . . . . . . . . . . . . . . 4.5 Experimental feasibility of M measurement . . . . . . . . . . . . . . . . . . . 4.6 Experimental features: inclusion of background noise and detector’s dead time 4.6.1 Background noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Detector’s dead time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Super-operator description of photon time traces statistics from single quantum systems: Q Mandel parameter . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Photo-electron counting ditributions . . . . . . . . . . . . . . . . . . . 4.7.2 Explicit calculation of Madel’s Q parameter . . . . . . . . . . . . . . . 4.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74 75 78 78 80 85 88 89 91 93 93 95 98 98 99 107 108 109 114 114 117 118 119 122 125. 5 Entanglement beyond decoherence 126 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.1.1 Measures of entanglement: Concurrence on pure and mixed states . . 128 5.1.2 The Schrödinger-HJW theorem . . . . . . . . . . . . . . . . . . . . . . 132 5.1.3 Interaction Hamiltonian and Master Equation . . . . . . . . . . . . . 133 5.2 Pure dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.3 Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.1 Entanglement on one excitation initial states . . . . . . . . . . . . . . 136 5.3.2 More general initial superpositions: Entanglement Sudden Death (ESD) 138. 3.

(5) 5.3.3. Witnessing entanglement beyond ESD: Total angular momentum S 2 . 146. 6 Use of decoherent mechanisms to enhance entanglement 150 6.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.2 Entanglement of Two particles subject to Brownian motion . . . . . . . . . . 155 6.2.1 Static configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.2.2 Quantum dynamics and Brownian motion through an adiabatic approximation: populations and concurrence insights . . . . . . . . . . . 155 6.2.3 Concurrence distribution . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2.4 Concurrence with variation of Brownian motion constants . . . . . . . 158 6.3 Use of dephasing to entangle particles . . . . . . . . . . . . . . . . . . . . . . 162 6.4 Witnessing entanglement from a dephasing mechanism: Magnetic susceptibility 168 7 Prespectives. 171. A M rate matrix 173 A.1 System A: LH2, and system B: LH1 . . . . . . . . . . . . . . . . . . . . . . . 173 A.2 System A: RC, and system B: LH1 . . . . . . . . . . . . . . . . . . . . . . . . 174 A.3 System A: LH1, system B: LH1 . . . . . . . . . . . . . . . . . . . . . . . . . . 174 B Determination of moments from PCD and waiting time distributions 175 B.1 Calculation of moments of inter-photon waiting times distributions with finite detector’s dead time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 B.2 Calculation of PCD distributions. . . . . . . . . . . . . . . . . . . . . . . . . . 177. 4.

(6) Introduction The study of systems that comprise many degrees of freedom, or where observables are modeled only by statistical averages, both classically and quantum mechanically, is opportune, as rare events can generate outstanding consequences, being an illustrative example, Life on earth. Furthermore, stochastic modeling can be used to include processes long time ago left aside, thought to be too complex to be modeled, reason why the effort to include them was not balanced by the importance believed they had. Nowadays, it is a growing concern to check whether these processes have been correctly neglected, or if indeed, as now heuristically believed, may play an interesting role in the evolution of the systems in which they are involved. Stochastic processes are used to describe situations where full details are too difficult to model, or ignorance on their behavior leave us with the only hope to describe them in a statistical fashion. In this Thesis, we use stochastic modeling with emphasis in optics-matter interaction, to study phenomena in both classical and quantum domains. First, the problem of classical excitation kinetics in photosynthetic bacteria is studied in order to understand how these organisms adapt to different illumination regimes, and more exactly, how these adaptations change the function of harvesting photosynthetic membranes. We find that constraints on the charge carriers responsible of bacterial metabolism, become of paramount importance when light intensity illumination is changed, to determine the relative amount of harvesting complexes in photosynthetic membranes. Stochastic modeling allows to include the complex trade-off between charge carriers and exciton dynamics. It was recently experimentally observed that the photosynthetic membranes in Rsp. Photometricum adapt to the light intensity conditions under which they grow. Illuminated under High Light Intensity ≈ 100Watt/m2 , membranes grow with a ratio of antenna-core complexes (i.e. stoichiometry) LH2/LH1≈ 3.5-4. For Low Light Intensity ≈ 10Watt/m2 , this ratio increases to 7-9. Here we develop a quantitative theory to explain this adaptation in terms of a dynamical interplay between excitation kinetics and reaction-center dynamics. Although the transport is treated as a noisy classical process, the underlying element being transfered are quantum mechanical many-body excitations. The membrane architecture effectively acts as a background network which loosely coordinates the entire process. The main results of this subject were presented 5.

(7) in [1] and published in [2,3]. Secondly, we focus our attention in spontaneous emission, a phenomenon stochastic in nature. We study the photon statistics of electromagnetic fields created from emission of quantum multi-level systems, where a measure of the memory left from states before photon detection, to the state after it happens, is proposed. Here, it is highlighted the importance to study distributions beyond the usual mean value analysis, to explore further the properties on different quantum systems with discrete level schemes. The use of stochastic realizations does not only provide computational benefits. If the set of realizations (decompositions) are chosen wisely, they can describe the state of the system under continuous environment observation. Accordingly, under direct photo-detection of radiation from fluorescent systems, decompositions rely on the splitting in conditioned to no-photon and photon detection evolution. We study non-renewal statistics, happening when information of the system state remains after photon detection occurs. When the process is renewal, the average of the product of consecutive inter-photon times factors out. We make use of the just outlined property, to propose and study measures that supply information on the memory after photon emission of quantum multi-level systems. The main results of this subject were published in [4], and presented in [5]. We also show how the super-operator description used for study of non-renewal after photon detection, allows treatment of other quantities, in particular, the analytical calculation of Mandel Q parameter. Finally, also within the quantum mechanics realm, we study a measure of non-locality in quantum mechanics: the entanglement. We find that individual outcomes on experimental realizations, are able to witness entanglement predicted by concurrence (a measure of entanglement) calculated at such level of resolution. It will be shown that predictions of individual realizations significantly differ from the outcome of an averaged state. The stochastic realization reproduce the dynamical behavior of non-locality in a closer and faithful manner as compared to ensemble averaged states [6]. [1] 15th International Workshop on Single Molecule Spectroscopy & Ultrasensitive analysis. Berlin-Adlershof, Germany, Sept. 15-18, 2010. APS March meeting 2008, New OrleansLouisiana, USA, March 2008. APS March meeting 2007, Denver-Colorado, USA, March 2007. [2] Felipe Caycedo-Soler, Ferney J. Rodrı́guez, Luis Quiroga, and Neil F. Johnson. LightHarvesting Mechanism of Bacteria Exploits a Critical Interplay between the Dynamics of Transport and Trapping. Phys. Rev. Lett. 104, 158302 (2010) [3] Felipe Caycedo-Soler, Ferney J. Rodrı́guez, Luis Quiroga, and Neil F. Johnson. Interplay. 6.

(8) of excitation transfer and Reaction Center dynamics in purple bacteria adaptation. Special issue in Quantum effects and noise in biomolecules Focus in Quantum aspects and noise in Photosynthesis. New. J. Phys. To be published. [4] Felipe Caycedo-Soler, Ferney J. Rodrı́guez, and Gert Zumofen. Memory in the photon statistics of Quantum multi-level systems. Phys. Rev. A 78, 053813 (2008). [5] Single molecule study in Life related Sciences. Aussois, France, 2006. APS March meeting 2006, Baltimore-Virginia, USA. March 2006. [6] Felipe Caycedo-Soler, Ferney J. Rodrı́guez and Luis Quiroga. Entanglement beyond decoherence. To be submitted (2010).. 7.

(9) Chapter 1. Purple Bacteria Photosynthesis The study of light conversion can give us insight on how nature, through millions of years of evolution, has selected organisms capable of using solar energy as their fuel provider. Photosynthesis, contrary to fossil fuel burning, preserves oxigen and carbon cycles equilibrium [1, 2], in complete accordance with conservation policies we certainly need. Interest in natural photosynthesis might help us improve current alternative energy sources in a rapid energy consuming world. In the following Chapters, we consider the excitation dynamics of photosynthetic membranes in Rps. Photometricum purple bacteria. We assume Förster excitation transfer to construct stochastic realizations consistent with master equation populations, where observables concerning membrane’s efficiency are calculated. We use data from Ref.[3] to compare the behavior of Low Light Intensity (LLI) and High Light Intensity (HLI) adapted membranes under respective regimes of illumination.. 1.1. Structure of Harvesting Membranes. Purple bacteria sustain their metabolism using photosynthesis in anaerobic conditions and under the dim light excitation proper of several metters deeps at ponds, lagoons and streams [4]. As depicted in Fig.1.1, aerobic organisms are present near the surface of water reservoirs, collect blue and red spectral components of sun’s light, leaving only the green and far red (> 750nm) components, from where purple bacteria must fulfill their energy requirements. The light energy absorption is accomplished through intracytoplasmic membranes where different pigment-protein complexes accommodate. Light Harvesting complexes (LHs) have the function of absorbing light and transfer it to Reaction Centers (RCs), where a charge separation process is initiated [5]. The unpaired charge reduces a quinone, which using a periplasmic hidrogen, converts to quinol (QB H2 ). RCs neutrality is restablished thanks to cytochrome c2 charge carrier, which after undocking from the RC, must find the bc1 complex 8.

(10) Figure 1.1: Representation of a lake containing both aerobic and anaerobic phototrophic organisms. Note that purple bacterial photosynthesis is restricted to the lower anaerobic layer and so they only receive solar energy that has been filtered, mainly by chlorophylls belonging to algae, cyanobacteria and plants.. to receive an electron and start its cycle all over again. The electron in bc1 is given due to cytoplasmic Quinol delivery in bc1 . The proton gradient induced by the charge carriers cycling becomes the precursor of adenosyn triphosphate (ATP) synthesis: ADP+P→ ATP+Energy, where ADP and P refer respectively, to adenosine diphosphate and phosphorous. The cycle is depicted in Fig.1.2. Light absorption occurs through organic molecules, known as chromophores, inserted in protein complexes. Bacteriochlorophyls (BChl) and carotenoids (Car) chromophores are the main absorbers in purple bacteria photosynthesis, absorbing maximally in the far red and green, respectively. The light absorption process occurs through chromophore’s Qy electronic transition excitation. Several chromophores are embedded in protein helices, named α and β apoproteins, inside complexes, classified by their absorption spectral maximum. Light Harvesting complex 1 (LH1) absorbs maximally at 883 nm, and contain 32 BChls, arranged in 16 bi-chromophore units. Light Harvesting complex 2 (LH2) reveals two concentric subunits that according to their absorption maxima are called B800 or B850. The B800 is composed of nine chromophores having dipole moments parallel to the membrane plane, while B850 include eighteen BChla chromophores. Each αβ subunit supports three BChls, one B850 highly interacting dimer and a single B800 chromophore. The αβ units have in average 9-. 9.

(11) Figure 1.2: Schematic representation of the photosynthetic apparatus in the intracytoplasmic membrane of purple bacteria. The reaction center (RC, red) is surrounded by the lightharvesting complex 1 (LH1, green) to form the LH1+RC complex, which is surrounded by multiple light-harvesting complexes (LH2s) (blue), forming altogether the photosynthetic unit (PSU). Photons are absorbed by LHs and excitation is transferred to the RC initiating a charge (electron-hole) separation. Electrons are shuttled back by the cytochrome c2 charge carrier (blue) from the ubiquinone-cytochrome bc1 complex (yellow) to the RC. The electron transfer across the membrane produces a proton gradient that drives the synthesis of ATP from ADP by the ATPase (orange). Electron e− is represented in blue, and quinones QB , likely confined to the intramembrane space, in black.. 10.

(12) Figure 1.3: LH2 complex structure. The position of chromophores B800 (red dots) and B850 (blue dots) along their Qy dipole moment orientation is schematically shown, relative to α (black) and β (green) helices. Details in the text. fold symmetry (occasional octameric or decameric symmetries). Extended LH2 absorption between 839 and 846 nm come along with symmetry heterogeneity.. 1.1.1. LH2. The LH2 complex is a circular structure composed of nine pairs of apoproteins [6]. Each of these apoproteins has a single transmembrane α-helix and a single β-helix. These helices run through a membrane where complexes are arranged, and cross such membrane from the periplasm to the cytoplasm. The inner wall of the complex is formed by the 9 α-apoprotein helices, while the outer wall is formed by the nine β-apoprotein helices. Pigments are located in the space between the α and β helices. The central Mg+ 2 atom of a B800 BChla chromophore, is fixed along the outer ring of α helices. These set of chromophores have their dipole moment perpendicular to the α helix, therefore paralell to the intracytoplasmic membrane [6]. Further inside the proteins scaffold, eighteen BChla chromophores with center to center distance of 9.5 Å within an αβ-apoprotein pair and by 8.8 Å to the closest BChl molecule in the next αβ-apoprotein pair. This close distance induces a collective effect responsible of the B850 absorption band of the LH2. Fig.1.3 summarizes schematically positions of αβ helices and orientation of chromophore dipole moments. Raman spectra using different excitation wavelength [7] and stoichiometry analysis [8], indicated that one carotenoid (Car) is present per each αβ heterodimer.. 1.1.2. LH1 and RC. In a 8.5 Å resolution projection map, the LH1 complex was shown to be a circle of 16 αβ heterodimers, with a hole in the middle that was large enough to accommodate the RC [9]. 11.

(13) The LH1 is subject of ongoing interest, as individual bacterium species display different core complex structures. Recently, Atomic Force Microscopy (AFM) was used to nanodissect membranes of Rsp. viridis where LH1 monomers (only one LH1 and its RC within) were observed, while Rb. sphaeroides and Rb. blasticus [10] display dimeric LH1 complexes. Chromophores are dimeric within an αβ unit, to sum up 32 within the LH1 ring. The best known structure of RC containing BChla chromophores, has been accomplished in Rb. sphaeroides [11, 12, 13]. Multiple pigment molecules are bound to protein subunits, named L (light) and M (medium). Along the subunits, two symmetric branches (A and B) arrange a set of molecules that sequentially produce charge transfers. Excitation of a highly interacting dimer, the special pair (P), in close proximity to BA accessory BChla pigments, initiates charge separation along the A branch, continued by reduction of bacteriopheophytin HA , followed by reduction of quinone QA . If available, quinone QB is the final receiver of the sequential tranfer, to use the accepted electron to form firstly, QB H, and secondly (after another charge separation chain), quinol QB H2 .. 1.2 1.2.1. Processes and rates involved in bacterial Photosynthesis Photon Absorption. Absorption occurs in harvesting complexes, through Car and BChl molecules, proportional to the complexes’ cross section. This quantity has been calculated for LH1 and LH2 complexes, where all absorbing molecules and extinction coefficients [14] have been taken into account. A photon of wavelentht λ, is part of the power spectrum of the source E(λ)n(λ). Normalized to 18 W/m2 intensity, the rate of photon absorption for circular LH1 complexes in Rb. sphaeroides [15]: Z γ1A =. n(λ)σLH1 (λ)dλ = 18s−1 .. (1.1). The same procedure applied to LH2 complexes, yields a photon capture rate of γ2A = 10s−1 . Since these rates are normalized to 18 W/m2 , the extension to other intensity regimes is straightforward. The rate of photon absorption normalized to 1 W/m2 intensity, will be γ1(2) = 1(0.55)s−1 . From now on, subindexes 1 and 2 relate to quantities of LH1 and LH2 complexes, respectively. Vesicles containing several hundreds of complexes will have an absorption rate: γA = I(γ1 N1 + γ2 N2 ) (1.2) where N1(2) is the number of LH1 (LH2) complexes in the vesicle.. 12.

(14) 1.2.2. Excitation transfer. Theoretical overview Excitation transfer happens through Coulomb interaction of electrons, excited to the Qy electronic transition in chromophores. The interaction energy can be formally written [16]: Vij =. 1 X X + hφm φn |V |φp φq ic+ mσ cpσ′ cqσ cnσ , 2 m,n,p,q ′. (1.3). σ,σ. where c+ mσ , cnσ′ are fermion creation and annihilation operators of electrons with spin σ and ′ σ , in the mutually orthogonal atomic orbitals φm and φn . The overlap hφm φn |V |φp φq i is the Coulomb integral: Z Z e2 φ∗ (r~2 − r~j )φq (r~2 − r~j ) (1.4) hφm φp |V |φn φq i = d~r1 dr~2 φ∗m (r~1 − r~i )φp (r~1 − r~i ) |r~1 − r~2 | n Intra-molecular contributions arise when m, n, p, q are all only donnor or only acceptor states. The inter-molecular contributions, where de-excitation of the donnor at r~i = ~rD and excitation of the acceptor at r~j = ~rA , has been ussually stated to happen through two different mechanisms. If φm and φp describe a set of states from the donnor, and φn , φq of the acceptor, the mechanism is named direct Coulomb interaction. It can be related with an electron that makes a transition between φm and φp , having both a finite value near the position of the donnor r~1 ≈ ~rD , while another electron is excited between φn and φq at the acceptor coordinate r~2 ≈ ~rA . On the other hand. if φm and φn describe donnor states, while φp and φq are acceptor states, a permutation of coordinates between acceptor and donnor states in the integral eq.(1.4) gives rise to the exchange mechanism. Here, the interaction will only be important if the states of donnor and acceptor have a substantial overlap, due to their close proximity. Accordingly, the exchange mechanism can happen only if donnor and acceptor are at a distance comparable to the extent of the molecules. Equation (1.4) describes the bare Coulomb. The fact that states involved in excitation transfer in molecules include vibrational degrees of freedom, makes enormous the Hilbert space over which the sums of eq.(1.4) is performed. When the transfer occurs with chromophores that belong to different complexes, thermalization happens rapidly as compared to excitation transfer, (no phase relationship between donnor and acceptor electronic states). The Coulomb excitation transfer mechanism is responsible of the RC ionization from a photon initially absorbed in harvesting complexes. The hermitian nature of excitonic transfer is replaced due to decoherence, by a rate of excitation transfer to describe the process within the vibrational manifold. According to Fermi’s golden. 13.

(15) rule, adapted to the vibrational continuum, the rate of transfer is given by [17]: Z Z Z 2π ∗ kDA = dwD dwA dE ~ ED ∗ EA 0. ×. 0. ∗ (w∗ ) exp(−w∗ /kT ) gD gA (wA ) exp(−wA /kT ) D D |ŨDA |2 . ∗ ZD ZA. (1.5) ∗ depend on Boltzmann factor (k is Botlzmann constant if Partition functions ZA and ZD otherwise not stated) and multiplicity of vibrational levels g(w) at electronic ground acceptor ∗ . The inferior limits on the levels of energy wA , and at excited donnor levels of energy wD partition function and in the two inner integrals in eq.(1.5) are the zero phonon line energies in the electronic acceptor ground (EA0 ) and donnor excited (ED0∗ ) levels. Therefore, Boltzmann factor and multiplicity are weighting factors to the different contributions of the interaction energy ŨDA = hΨD∗ ΨA |VDA |ΨD ΨA∗ i, where |Ψi are assumed as products of electronic |φi and vibrational |χi molecular states. With the Born-Oppenheimer approximation [16, 17]: ∗ ∗ )i )|χ(wD )ihχ(wA )|χ(wA ÛDA ≈ hφD∗ φA |VDA |φD φA∗ i × hχ(wD ∗ ∗ ≈ UDA hχ(wD )|χ(wD )ihχ(wA )|χ(wA )i. (1.6). where UDA = hφD∗ ψA |VDA |ψD ψA∗ i.. (1.7). where VDA is the matrix whose values are electrostatic interations between the donnor and acceptor chromophores, calculated from eq.(1.3). Using the approximation of eq.(1.6), the expresion (1.5) now including the overlap between vibrational levels can be cast in a more illustrative form: 2π kDA = |UDA |2 JDA (1.8) ~ where Z JDA = dE GD (E)GA (E). (1.9) Here, GD (E) and GA (E) are often called the Franck-Condon weighted and thermally averaged combined density of states. Explicitly: Z ∗ ∗ ∗ ∗ 2 ∗ gD (wD ) exp(−wD /kB T )|hχ(wD )|χ(wD )i| dwD GD (E) = ∗ ZD ED ∗ 0 Z ∗ )i|2 gA (wA ) exp(−wA /kB T )|hχ(wA )|χ(wA dwA GA (E) = ZA EA 0. schematically presented in Fig.1.4, where ∗ wD = ED0∗ + wD −E. ∗ wA = −EA0 + wA + E.. 14.

(16) Figure 1.4: Energy level scheme of donnor and acceptor molecules. Although the zero phonon line might be different between both, energy conservation on the transfer applies due to vibrational levels.. Förster showed [18] that these distributions are related to extinction coefficient ǫ(E) and fluorescence spectrum fD (E): ǫ(E) = fD (E) =. 2πNO |DA |2 EGA (E) 3 ln 10~2 nc 3~4 c3 τ0 |DD |2 E 3 GD (E) 4n. (1.10). where N0 = 6.022 × 1020 is the number of molecules per mol per cm3 , n is the refractive index of the molecule sample, c the speed of light, and τ0 the mean fluorescence time of the donnor excited state. If spectra are normalized: ǫˆA (E) = fˆD (E) = the expression eq.(1.9) can be cast: JDA =. Z. ǫA (E) dE ǫA (E)/E fD (E) R dE fD (E)/E 3. (1.11). ǫ̂A (E)fˆD (E) . E4. (1.12). R. dE. Apart from the factor E −4 , the integral represents the spectral overlap between fluorescence donnor and absorption acceptor spectra. Therefore, whenever fluorescence and absorption spectra are available, an estimate for the excitation transfer rate can be calculated. Inter-complex tranfer rates Changes on delocalization degree will manifest on the nature of the states from which the total interaction energy UDA is to be calculated. If decoherence sources are not important 15.

(17) within the time-scale of excitation dynamics on donnor or acceptor aggregates, they can be known from the eigenvector problem H|ñi = Eñ |ñi.. (1.13). The Hamiltonian H in the chromphore site basis |ii, if no inhomogenuous broadening is included, has diagonal elements hi|H|ii = ǫ . For non-neighboring chromophores the dipoledipole approximation is ussualy used, ! ~i ·D ~j ~ i )(~rij · D ~ j) D 3(~rij · D hi|H|ji = C − 3 5 rij rij with i 6= j, i 6= j ± 1. (1.14). ~ i is the dipole moment and r is the distance between the interacting dipoles. Neighwhere D boring chromophores are too close to neglect their charge distribution. Its interaction is determined such that the effective Hamiltonian spectrum matches the spectrum of an extensive quantum chemistry calculation [19]. In the LH2 complex, the B850 ring, with nearest neighbors whithin an αβ unit hi|H|i + 1i = ν1 = 806 cm−1 , and in different units ν2 = 377 cm −1 ; other constants: ǫ = 13059 cm−1 and C = 519 044 Å3 cm−1 . The LH1 complex has a different spectrum maximum at 875 nm, and although LH1 is bigger than LH2, it is also conformed by αβ units, housing two chromophores, with intercomplex distances and nearest neighbor interactions equal to B850 LH2 ring. In order to calculate VDA , states have been supposed to be delocalized over the complete rings. The B800 LH2 ring chromophores are further apart, implying smaller coupling that leads to longer transfer time-scale than the required for decoherence through vibrational dephasing. Although complete localization was thought to extend on just a chromophore, detailed study has opened the possibility of delocalization over 2-3 pigments, to improve robustness of B800→B850 energy transfer [20]. Hence, delocalization requires a quantum description for tranfer among the ring eigenstates, that follow. The eigenstate |ñi will be a superposition of the individual Qy electronic transitions |ii: X |ñi = añ,i |ii. (1.15) i. The inter-complex interaction energy between LH1s and LH2s complexes is mainly ascribed to the B850 and B875 rings. Within these structures a Hamiltonian H=ǫ. X i. |iihi| +. X i,j. Vij |iihj|. (1.16). yields completely delocalized exciton eigenstates |ñi. On the time-scale of inter-complex (a few picoseconds) energy transfer, exciton states thermalize, to become a statistical ensemble. 16.

(18) h from\ to. LH1. LH2. RC. LH1. 20.0 ps. 15.5 ps. 15.8 ps. LH2. 7.7 ps. 10.0 ps. N.A. RC. 8.1 ps. N.A. N.A. Table 1.1: Theoretical estimation of inter-complex transfer times in picoseconds. N.A are not available calculations. of eigenstates |ñi weighted by their Boltzmann factors: ρ=. 1 X exp(−Eñ /kT ) m̃ihm̃| Tr {·} |. (1.17). ñ. where Tr {·} is trace of the numerator operator, used to normalize the state. From this state, the net inter-complex interaction can be found UDA = Tr{ρVDA } XX 1 exp(−Em̃ /kT )hm̃|VDA |ñi = P ñ exp(−Eñ /kT ) ñ. m̃. (1.18). now, hm̃|VDA |ñi are the elements of interaction among exciton states in molecules on different complexes. They can be found from individual chromophore inter-complex interactions, making use of eq.(1.15) X hm̃|VDA |ñi = an,i a∗m,j Vij (1.19) i,j. Therefore, an estimate of rate transfer can be calculated from eq.(1.19), using equations (1.18) and (1.9). Table 1.1 shows the results of a calculation of mean transfer times presented in Ref.[21]. Experimental evidence The difference in the complexes main absorption band allows pump probe spectroscopy experimental techniques, in order to find the transfer rates. Since LH1↔ LH1 and LH2↔ LH2 transfer steps involve equal energy transitions, no experimental evidence is available regarding the rate at which these transitions occur. The experimentally determined B800→ B850 rate was 1/700fs [22]. The inter-complex transfer rate between LH2→ LH1 have been determined experimentally to be 1/3.3ps [23]. Experimentally, LH1↔ RC forward transfer rate ranges between 1/50ps and 1/35ps, while back-transfer rate ranges between 1/12ps and 17.

(19) h from\ to. LH1. LH2. RC. LH1. N.A. N.A. 30-50 ps. LH2. 3.3 ps. N.A. N.A. RC. 8 ps. N.A. N.A. Table 1.2: Experimental evidence of inter-complex transfer times in picoseconds. N.A are not available data. 1/8ps [24, 25, 26]. It is interesting to note that exists a two fold difference in the experimental and theoretical determined LH2→LH1, ascribed to BChla Qy dipole moment underestimation. It is assumed for theoretical calculation a value of 6.3 Debye, while a greater BChla Qy dipole moment in Prostecochloris aestuarii (not a purple bacterium) of 7.7 Debye has been determined [21]. On the other hand, LH1→RC theoretical calculation gives a greater value for tranfer rate, thought to arise due to an overestimate of LH1 exciton delocalization [21]. This rate decreases when delocalization is assumed over fewer BChl’s, therefore, further research is needed to understand the effect of decoherence sources (static inhomogeneities and dynamical disorder due to thermal fluctuations) on the delocalization length. In summary, experimental evidence, to our knowledge, has determined the transfer times presented in Table 1.2.. 1.2.3. Dissipation. Excitation in chromophores might be dissipated by two main mechanisms. The first is fluorescence, where the electronic excited state has a finite lifetime on the nanosecond time-scale, due to its interaction with the electromagnetic vacuum [27]. The second is internal conversion, where the electronic energy is transferred to vibrational degrees of freedom. Within the Born-Oppenheimer approximation, the molecular state Ψ, can be decomposed into purely electronic φ and (nuclear) vibrational χ states. The transition probability between initial state Ψi and final state Ψf , is proportional to hΨi |H|Ψf i ∝ hχi |χf i. Note that χi (χf ) are vibrational levels in the ground (excited) electronic state manifold (see Fig.1.5). If the energy difference is small, and the overlap between vibrational levels of different electronic states is appreciable, the excitation can be transferred from the excited electronic state, to an excited vibrational level in the ground electronic state. This overlap increases with decreasing energy difference between electronic states. As higher electronic levels have smaller energy difference among their zero phonon lines, internal conversion process is more probable the higher energy electronic states have. Fluorescence and internal conversion between first excited singlet and ground electronic states, induce dissipation in a range of hundreds of 18.

(20) Figure 1.5: Dissipation mechanisms. In (a), the electron de-excites due to its interaction with the quantized electromagnetic vacuum field through a fluorescent photon. In (b) internal conversion mechanism, where the vibrational levels overlap induces a transition between electronic excited and ground states. Dissipation overcomes when thermal equilibrium is reached in the vibrational manifold of electronic ground state. picoseconds and a few nanoseconds. Numerical simulations are performed with a dissipation time including both fluorescence and internal conversion of 1/λD = 1 ns, also used in [21].. 1.2.4. Special pair P ionization. Excitation reach may from LH1 complexes, the special pair (P), a dimer located at the Reaction Center protein complex. The excitation can be transferred back to its surrounding LH1, or initiate a chain of ionizations along only the A branch, probably, due to a tyrosine residue strategically positioned instead of a phenylalanine present in the B branch [28]. Once the special pair is excited, it has been determined experimentally [29] that takes 3-4 ps − , in a reaction for the special pair to ionize and produce a reduced bacteriopheophytin, HA − ∗ + P → P HA . This reaction initiates an electron hop, to a quinone QA in about 200 ps, and to a second quinone, QB if available. When reduced twice, QB takes two introcytoplasmic protons to form quinol QB H2 .. 1.2.5. Quinone-quinol cycling. Once quinol is created, the affinity of this new product to the RC is lowered, and allows the required unbinding of quinol to start the migration to the bc1 complex. The time before the special pair becomes neutral again, quinol unbinds, and a new QB is available, is within milliseconds [30, 31]. This processes leave the RC in a state where no electronic excitation can be used to enhance the required proton gradient of the bacteria. This stage will be better explored in Chapter 2.. 19.

(21) 1.3. Excitation dynamics in photosynthetic membranes. Photosynthesis is initiated by photon absorption at harvesting complexes, followed by transfer to RCs, where charge separation on the special pair, sustain an electrochemical potential gradient. Each step on all these processes can be described stochastically, in order to smear out a minuocious study, but still include the probabilistic nature of excitation absorption and transfer, and variations that due to local inhomogeneities, the photosynthetic membranes may have. The stochastic nature of these processes is tied to the constraints pointed out in Section 1.2, by the mean values related with all the processes involved in photosynthesis of purple bacteria. An excitation can be used to induce charge separation at a special pair, or be dissipated by internal conversion or fluorescence. The special pair must be neutral and a quinone must be available for the excitation to be used to initiate charge separation at RCs, to form quinol. The final fate of excitations depend on how easy an available RC is reached. The processes involved are shown schematically in Fig.1.6.. 1.3.1. Excitation dynamics through a Master Equation. A master equation describes the evolution of the probability to find a given system in a certain state. Therefore, probabilities only have physical meaning as a result of performing multiple experiments, the master equation is able to describe mean values of observables as the outcome of ensemble averaging. The excitation transport is thought to occur through incoherent mechanisms as pointed out in Section 1.2.2. If pi (t) is the probability of an excitation residing in the ith site, its master equation is dpi (t) X = (kij pj (t) − kji pi (t)) − λDi pi (t) + γi . dt. (1.20). j. These probabilities canbe added to find populations of numberof excitations in a given complex, when more than one excitation in such node is allowed. Therefore, its addition is not normalized. Here, kij are rate constants for transfer among states, found from inter-pigment energy tranfer rates, and γDi is dissipation rate (mainly due to fluorescence and vibrational relaxation) for all decay processes other than energy transfer. The rate γi is the excitation rate from incoming photons. If absorption is not accounted, i.e. γi = 0, one can set the system of eqs.(1.20) as ˙ = M̃ p̃(t) p̃(t) (1.21) where M̃ is the matrix of pairwise rates Mij = (1 − δij )kij − δij. 20. X l. klj − δij γdi .. (1.22).

(22) Figure 1.6: Inter-complex mean transfer times in picoseconds [21], and charge carrier cycling [31, 30] time-scale in Rsp. Photometricum. The system described by eq.(1.21) is solved subject to an initial probability vector p̃(0) ˜ t)p̃(0) p̃(t) = exp(M. (1.23). if M̃ does not depend on time. Otherwise, in a more general fashion, p̃(t) = G̃(t, t0 )p̃(t0 ), where the Green function of the system G̃(t, t0 ), has elements Gij (t, t0 ) to be interpreted as the conditional probability that an excitation starting at site j at time t = t0 reaches site i at time t. Any complex can be excited from the ground state to an excitonic state. p̃(t) describes a state of N complexes, {c1 , ., ci , ..cN } where ci = 0 (1) represents the ground (one exciton state) in complex i. Although two exciton states might happen in harvesting complexes, due to the light regime at which purple bacteria live, in physiological conditions, those events are rare. The photosynthetic unit dynamics can be understood by means of the excited populations on the complexes involved. When an excitation reaches a RC, and quinol is formed, the RC is no longer able to produce charge separation, for a time τ as shown in figure 1.6 on the millisecond range, setting it ”closed”. The open and closed states of the RC must be included in the excitation dynamics since they change the excitation fate. Therefore, in the case of RCs, it is not only sufficient to state whether the RC is excited or not, since it can be open or closed. If ci represents the state of a RC, since it can be excited or unexcited while being open (closed), we make distinction between open and closed states, labeling 1 (1+ ) if it is excited, or 0 (0+ ) if it is in the ground state, respectively. The rate from open to close states will be half the rate of special pair oxidation, since two excitations are required to close the RC, while the rate from closed to open is the inverse of the mean time it takes quinones to cycle.. 21.

(23) h. from\ to. LH1. LH2. RC. LH1. 20 ps. 15.5 ps. 25 ps. LH2. 3.3 ps. 10.0 ps. -. RC. 8 ps. -. -. Table 1.3: Transfer times (in picoseconds) used for numerical simulation. The hyphen are second to nearest neighbors transfer times.. Transfer Rates involved in simulations Given the reviewed agreement between pump-probe spectroscopy and theoretical effective Hamiltonian - Fermi’s Golden Rule transfer rates, inter-complex rates that are used in the numerical simulations, follow the theoretically calculated ones, whenever no experimental determination has been obtained. Since experimentally, the LH1→ RC transfer rate has such 1 a wide range, a conservative value, considering the faster theoretical transfer, is 25 ps. To summarize, the mean transfer times that will be used in numerical simulations are presented in Table 1.3. The unavailable rates represent transfer to next-neighbor complexes, and are of no importance if simulations are performed to nearest neighbors only.. 1.3.2. Master equation: Single excitation Green’s function approach. In order to understand the dynamical behavior of photosynthetic membranes, the green function is discussed for different architectures. Our aim is to know where are excitations expected to be found within the membrane. The special case to be treated is restricted to one excitation in the membrane, and no interest is focused in the RC cycling dynamics. LH2-LH1-RC Photosynthetic Unit First, let us consider the minimal photosynthetic unit, conformed by an LH2, an LH1 with its RC. Gi,j (t) will be the conditional probability of an excitation starting at site i, to be at time t in site j. The master equations follow dpRC dt dp1 dt dp2 dt. = −kRC,1 pRC + k1,RC p1 − (t+ )−1 pRC. (1.24). = kRC,1 pRC − k1,RC p1 + k2,1 p2 − k1,2 p1 − λD p1 + γ1. (1.25). = −k2,1 p2 + k1,2 p1 − λD p2 + γ2 ,. (1.26). with labels 1, 2 and RC for LH1, LH2 and RC complexes, respectively. The sum of the above equations dp/dt = d(pRC + p1 + p2 )/dt is the probability to have an excitation within the 22.

(24) membrane, and reads: dp = −(t+ )−1 pRC − λD p1 − λD p2 + γ1 + γ2 (1.27) dt In this preliminary study, we are interested to know where will excitations probably be found at stationary state. This situation is conditioned to have the excitation within the membrane. As presented in eq.(1.27), dissipation and RC photosynthesis are mechanisms that reduce p(t), while external excitation, represented by γ1(2) increase it, such that p(t) will therefore quantify the probability for the excitations to remain in the membrane. As pointed out, incoming photons produce an excitation rate γ1(2) on the millisecond time-scale, that for the purpose of this discussion will be set to zero, and a solution of the form eq.(1.21) is expected. On the other hand, dissipation happens in nanoseconds time-scale and rarely will two excitations be present at a given time inside the membrane. The rate matrix M adopts the form: −λD − k2,1 k1,2 0 (1.28) M = k21 −λD − k1,2 − k1,RC kRC,1 0. k1,RC. −(t+ )−1 − kRC,1. The probability Gij (t) to find at a given complex j, an excitation starting at a complex i, t will be eW̃ ij . Since our interest is conditioned on having at least one excitation along the membrane, Gij (t) is normalized such that Exp[M̃ t]ij Ĝij (t) = P , j Exp[M̃ t]ij. (1.29). to account all possible paths the excitation might explore. From Figs.1.7(a)-(c), it is apparent that excitations will mostly be found in the LH1 complex, followed by occurrences at the LH2 and lastly at the RC. This implies that excitation funneling concept, widely spread on this community, is not favored, as excitations do not move towards the RC more preferably than to harvesting complexes. It could be stated that excitations will not be found in the RC, due to their possibility to initiate charge transfer along the RCs active branch within 3 ps. Nevertheless, the LH1 continues to be the most populated, even though, the photosynthesis transfer rate tRC is set to zero as shown in Figs.1.7(b) and (e). Complexes arrangement The basic structure in photosynthetic membranes has been regarded to be one LH2, one LH1 and its RC, namely, the photosynthetic unit (PSU). Once it has been determined how excitation dynamics can be described, as the chromatophore has many complexes whose size is way beyond the PSU’s, the question arises on whether the arrangement of complexes might change the tendency of excitations to populate certain complexes. If the arrangement of complexes might induce a given tendency to populate specific complexes, purple bacteria could 23.

(25) ` Gi,j HtL 1 0.8 0.6 0.4 0.2 ` Gi,j HtL 1 0.8 0.6 0.4 0.2. ` Gi,j HtL 1 0.8 0.6 0.4 0.2. HaL. 5 10 15 20 25 30. t @psD. ` Gi,j HtL 1 0.8 0.6 0.4 0.2. HdL. 5 10 15 20 25 30. t @psD. ` Gi,j HtL 1 0.8 0.6 0.4 0.2. HbL. 5. 10 15 20 25 30. t @psD. HeL. 5 10 15 20 25 30. t @psD. ` Gi,j HtL 1 0.8 0.6 0.4 0.2. HcL. 5 10 15 20 25 30. t @psD. HfL. 5 10 15 20 25 30. t @psD. Figure 1.7: Top panel t+ = 3ps, bottom panel t+ = 0. Conditional probabilities Gij , to find the excitation at the LH2 ((a) and (d)), at the LH1 ((b) and (e)) or at the RC((c) and (f)), when excitations start at the RC (green), LH1 (blue) or LH2 (red) use a given architecture to enhance a behavior according to its environmental illumination conditions. As stated, our interest is to check whether excitations tendency to visit a given complex type is changed with their relative arrangement. For instance, absorption has not been taken into account. In order to study the effect of continual light excitation, the probability Pk to populate a complex type k, will take into account the complex i absorption cross section σi : N X 1 σi Gij (t) (1.30) Pk (t) = P (t) i=1,∀jǫk. P where P (t) is a normalization constant P (t) = N i,j σi Gij (t), required to account only for the states conditioned to have an excitation, with an initial value P (0) = N ; k is restricted to complexes of the same type. Fig.1.8 shows three model architectures for the same set of harvesting complexes (16 LH2s, 2 LH1-RCs). In Fig.1.9 the x-axis has not been plotted, to highlight that the RCs population is very close to zero, due to: 1. small absorption cross section, 2. slow LH1→RC rate, 3. the special pair ionization process.. It should be highlighted that harvesting function increases LHs and reduces the RC populations. It can also be seen that greater connectedness between an LH1 and LH2s in architectures of Fig.1.8, induces the LH1 complexes to become more populated. Therefore 24.

(26) HbL HaL. HcL. Figure 1.8: (a)-(c) Different model architectures with equal number and type of complexes. LH1s (red big circles) and LH2s (green small circles). pk 1. pk 1. HaL. pk 1. HbL. 0.8. 0.8. 0.8. 0.6. 0.6. 0.6. 0.4. 0.4. 0.4. 0.2. 0.2. 0.2. 5. 10. 15. 20. t @psD. 5. 10. 15. 20. t @psD. HcL. 5. 10. 15. 20. t @psD. Figure 1.9: In (a), (b) and (c) probabilities Pk (t) of excitations to be found in LH1s (continuous), LH2 (dashed) and RCs (dotted), for architectures of Fig.1.8, respectively. 25.

(27) connectedness can become a tool for bacteria to produce higher populations in LH1s, to set excitations closer to become a RC ionization and improve the membrane’s efficiency. Relative number of complexes: stoichiometry The relative number of complexes might also change the population of available states. Fig.1.10 shows small networks of LH-RC nodes, where the relative amount of LH2 and LH1 complexes, quantified by stoichiometry s = N2 /N1 , with N1(2) as the number of LH1(2) complexes, is changed in order to study the exciton dynamics. In Fig.1.11(a) the population ratio at stationary state of LHs demonstrate that as stoichiometry s becomes greater, the population of LH1s, becomes smaller, since their amount is reduced. It is apparent that RC population is quite small, and although their abundance increases the exciton trend to be found in any RC (Fig.1.11(b)), generally, excitations will be found in harvesting complexes. The population of LHs should be dependent on the ratio of complexes type. As verified in Fig.1.11(b), RCs have almost no population, and for the discussion below, they will not be taken into account. Populations can be written as: N1 f1 (s) PLH1 (t → ∞) = f1 (s) = N1 + N2 1+s sf2 (s) N2 = PLH2 (t → ∞) = f2 (s) N1 + N2 1+s. (1.31) (1.32). k where the dependence on the amount of complexes is made explicit with the ratio N1N+N , 2 and where f1 (s) and f2 (s) are enhancement factors. This factor provides information on how population on individual complexes changes, apart from the abundance of complexes. With use of eqs.(1.31-1.32), f1 (s) and f2 (s) can be numerically calculated provided that Pk (t → ∞) can be known from the Green’s function, while s is a parameter given for each network. The results for enhancement factors are presented in Fig.1.11(c). The enhancement factor f2 (s) for LH2 seems to saturate at values below one, as a consequence of the trend of excitations to remain in LH1s. This means that increasing further the number of LH2s will not enhance further the individual LH2 populations. On the other hand f1 (s) has a broader range, and increases with s. This result reflects the fact that more LH2s will increase the cross section related to a given PSU, and predicts that population of individual LH1s will become greater as more LH2 complexes surround a given LH1. An unconventional architecture (last in second row in Fig.1.10) has an outermost line of LH1 complexes, whose connectedness to LH2s is compromised. In all the results in figure 1.11 (sixth point), this architecture does not follow the trends just pointed out, as LH1 and RC population, and enhancement factors, are clearly reduced. The population of LH1 complexes depends on their neighborhood and connectedness. Whenever connectedness of LH1 complexes is lowered, their population will also be reduced. Hence, deviations from populations trend with variation of stoichiometry,. 26.

(28) are a consequence of different degrees of connectedness of LH1s. Until this moment Green’s function approach has helped to understand generally the effect of stoichiometry and architecture in small networks. Two conclusions can be made 1. Connectedness of LH2 complexes to LH1s, facilitate transfer to RCs 2. The relative amount of LH2/LH1 complexes, namely, stoichiometry s = N2 /N1 , when augmented, induces smaller population of LH1-RC complexes, while its reduction, increases the total cross section of individual PSUs. It is important to mention that results for Green’s function calculation require several minutes of calculation in a standard computer to yield the results shown in Fig.1.11, and that these networks have an amount of nodes an order of magnitude smaller than the actual chromatophore vesicles. Dynamics concerning the RC cycling have not been described yet, fact that would increase further the dimension of possible membrane’s states. To circumvent this problem, further analysis will proceed from stochastic simulations, and observables will be obtained from ensemble averages.. 1.3.3. Stochastic Model. Advantages of an stochastic model The vesicles where harvesting, excitation transfer and charge separation occurs, include several hundreds of complexes. The master equation treatment is straightforward, but has a shortcoming, since the vector describing populations involve a great amount of states. A vesicle has N2 LH2 harvesting complexes, N1 LH1 complexes and an equal amount N1 of RCs. In total, there are N2 + 2N1 complexes to be included in the excitation kinetics model. Each LH2 and each LH1 is supposed to have just two possible states (excited and unexcited), while RCs have four possible states (excited and unexcited, while simultaneously can be open or closed, as will be stated), fact that determines a total number of 2N2 2N1 4N1 = 2N2 +3N1 states. A typical membrane having ≈300-400 harvesting complexes, will have ≈ 10100 probable states. Taking into account that W matrix will have a size of ≈ 10100 × 10100 , an exact master equation model is not practical within a complete chromatophore vesicle. Simulation scheme In order to circumvent this shortcoming, stochastic realizations were performed. In a general case the starting point is to obtain coordinates of LH1 (Xi , Yi ), with diameter φ1 , LH2 (xi , yi ) and diameter φ2 . The neighbors i, j are defined when complex i is within a distance (φi + φj )/2 + δ from complex j, where δ is set from experimental evidence at 30 Å [32]. An adjacency matrix A, Ai,j = Θ((φi + φj )/2 + δ) − rij ), (Θ(x) is the step function) is found. A 27.

(29) Figure 1.10: Networks with different stoichiometries, from left to right, top to bottom, s={1.04, 2.06, 3.08, 4.44, 5.125, 6, 7.16, 8.8, 11.25, 15.33, 23.5, 48}, and equal number of harvesting complexes.. 28.

(30) pk. pRC 0.006 0.005 0.004 0.003 0.002 0.001. HaL. 1 0.8 0.6 0.4 0.2 10. 20. 30. s. 40. HbL. 10. 20. 30. 40. s. fHsL 5. HcL. 4 3 2 1 10. 20. 30. 40. s. Figure 1.11: In (a) stationary state populations for LH2s (circles), LH1s (diamonds) and RCs (crosses), as a function of the stoichiometry of membranes presented in Fig.1.10. In (b) a zoom of RC populations is made, and in (c) the enhancement factors f1 (s) (diamonds) and f2 (s) (circles) are presented. rate matrix Γ with Γi,j = ki,j Ai,j , where ki,j are the experimentally measured or theoretically calculated rates for complex i →complex j excitation transfer, reviewed in Section 1.2. After A is determined, excitation kinetics can be modeled. Photons excite harvesting complexes according to absorption rate γA from eq.(1.2). Excitation of the membrane by individual photons happens in a time interval dt with probability γA dt. A random complex of LH1(2) type is chosen with probability P LH1(2) σ (1.33) p1(2) = k k σT OT where σT OT is the total cross section of the membrane. Once excitation is absorbed, the ith excited LH complex may transfer the excitation to its {j} available neighbors (LH1s, LH2s or RCs in ground state) with probability X Pi→{j} = dt γij (1.34) j. if Pi→{j} is greater that a pseudo-aleatory number in the [0,1) domain, transfer occurs. To know to which of these neighbors the excitation migrates, a second pseudo-aleatory number [0,1) a is generated. The acceptor j is chosen if this number belong to the interval γi(j+1) dt γij dt ≤a< . Pi→{j} Pi→{j} 29. (1.35).

(31) The sum in eq.(1.34) is performed including the dissipation channel with a rate γD in any complex. The rate of charge separation at the special pair in a RC is also included to happen at the rate (t+ )−1 . Therefore, these channels are included as extra neighbors and treated as such in eq.(4.2), with the consequence that if chosen, the excitation leaves the membrane. To summarize, the steps of the numerical simulation are: 1. Obtain coordinates and type of LHs, and numerate complexes. Choose neighbors according to a maximum center to center distance, including in the position j an array of all numbers corresponding to neighbor complexes of complex j. Rates are assigned according to the kind of couple. 2. Choose dt such that approximately only one event per excitation occurs, i.e. Pi→{j} ≪ 1. 3. Send photons to the network, to be absorbed with probability γA dt according to eq.(1.2), and add one to absorbed excitation nA . Excite randomly LH2s or LH1s, according to the total cross section of type of complexes from eq.(1.33). Assign the position of the excitation according to the excited complex. 4. In case that RC cycling is taken into account, check if a time τ has elapsed for any of the closed RCs, and if so, open that RC. 5. If the ith complex is excited, excitation can be transferred to the available neighbors with probability Pi→{j} from equation (1.34), or to other channels (dissipation, charge transfer) according to equation (4.5). Change internal states of acceptor and donor. 6. Check state of the excitation. (If RC cycling is accounted, when excitations are processed in a RC, the excitation disappears and internal state of the RC is set closed for a stochastically generated time of mean τ , if this excitation is the second after RC was initially opened). 7. Add one excitation to processed excitations nRC if it is used as such in a RC, or add one to dissipated excitations ndiss , if within dt a dissipation event happened. 8. Once the kinetics of all existing excitation are calculated, internal states are saved permanently, start from step 3.. 1.3.4. Comparison between Master Equation and stochastic populations results. In order to test stochastic simulations and master equation schemes, interacting transfer among a few complexes will be considered. In the cases illustrated one excitation can be in 30.

(32) pHtL 1 0.8 0.6 0.4 0.2. pHtL 1 0.8 0.6 0.4 0.2. HaL. 10. 20. 30. 40. 50. t @psD. HbL. 2. 4. t @msD. Figure 1.12: Time evolution of population on the LH2-LH2 system with light pumping. Continuous lines are analytical solutions, and crosses, numerical simulations. In (a) it is shown the picosecond range dynamics, due to excitations transfer, while in (b) is shown the long time behavior due to photon absorption. Numerical simulation are the result of 10000 runs. p2 (0) = 1. the LH2s and RCs, and two at the LH1s at most. Dirac notation will be used to label states, although no quantum mechanical calculation is done. For instance, dissipation rates will be set to zero, to capture better absorption dynamics. LH2-LH2 The possible states are (00), (01), (10), (11), correspond to subscripts 1 to 4 in the probability vector of the following discussion. Light can excite both LH2s with rate γ2 , and interaction between LH2s gives rise to hopping from A to B at a rate kLH2,LH2 , and back transfer at kLH2,LH2 , both equal to t−1 22 . W matrix is: −2γ2 0 0 0 γ2 −(k + γ ) k 0 LH2,LH2 2 LH2,LH2 M = (1.36) γ kLH2,LH2 −(kLH2,LH2 + γ2 ) 0 2 0 γ2 γ2 0. The population of one LH2, named A, is pA = p1 + p3 and, the one of the other LH2, named B, is pB = p2 + p3 . These populations will follow the differential equations: ṗA = γ2 (1 − pA ) − kLH2,LH2 pA + kLH2,LH2 pB. (1.37). ṗB = γ2 (1 − pB ) − kLH2,LH2 pB + kLH2,LH2 pA .. (1.38). Solving the above two equations for pA and pB , is equivalent to solving the full four equations (1.36) for p1 , p2 , p3 and p4 .. 31.

(33) pHtL 1 0.8 0.6 0.4 0.2. pA pB 0.125. HaL. HbL. 0.12 0.115 0.11 0.105. 1. 2. 3. 4. 5. t @psD. 0.095. 5. 10. 15. 20. 25. 30. t @psD. Figure 1.13: Time evolution in (a), of population of excited states of LH2 (red, dotted), namely system A, and LH1 (black, continuous), known as system B, for two initial conditions p5 (0) = 1; p3 (0) = 1, along stochastic simulation results (crosses, 100000 runs). In (b) the ratio of excited state populations of A and B, pA /pB (continuous) is a shown along kBA nA /(kAB nB ) (dashed). LH2-LH1 We label the LH2 complex as system A, and the LH1 complex as system B. In order to study the dynamics as a function of the available states, it will be supposed that a maximum of one excitation is present at the LH2 (nA = 1) and a maximum of two excitations might be present at the LH1 (nB = 2). In the Dirac notation for states, the first index corresponds to complex A (LH2), the second and third to complex B (LH1), and the 23 possible states are numbered from 1 to 8 in this order (000), (001), (010), (011), (100), (101), (110), (111), transitions respond to light absorption from LH1 (LH2) γ1(2) , forward transfer from LH2 to LH1 at rate kAB and back transfer kBA , intra LH1 transfer kBB . Dynamics are described with the transition M shown in the Appendix. Agreement between both calculations is achieved, as shown in Fig.1.13(a). The stationary state populations depend on the ratio between inter-complex rates and available states to perform a transition. If the excitation is in the LH1, it has only a possible state to occupy at the neighboring LH2 (nA = 1). On the other hand, if the excitation is at the LH2, it will have two reachable states at the LH1 (nB = 2). Analytically, using transfer matrix M of eq.(1.21), the stationary state value for the ratio between populations is: pA kBA nA = pB kAB nB. (1.39). Therefore, stationary state populations depend only on the rates involved and on the available states per complex, as shown with the dashed line in Fig.1.13(b).. 32.

(34) pHtL 1 0.8 0.6 0.4 0.2. pHtL 2 HaL. HbL. 1.5 1 0.5. 10. 20. 30. 40. t @psD 50. 2. 4. 6. 8. 10. t @psD. pHtL 1 0.8 0.6 0.4 0.2. HcL. 20. 40. 60. 80. t @psD 100. Figure 1.14: In (a) time evolution of population on the LH1 (blue)-RC(red) interaction (p2 (0) = 1, one excitation initially in the LH1). In (b) LH1-LH1 excited state population comparison between solution involving the 16 probability equations (black) for the initial state p4 (0) = 1 (two excitations initially in the LH1). Here, are also shown the comparisons between the reduced system described by eqs.(1.40) (red) and stochastic simulations (crosses in all cases) for p3 (0) = 1. In (c) the complete photosynthetic LH2(green)-LH1(blue)-RC(red) unit excited state populations for p2 (0) = 0. RC-LH1 Now that LH1 model has been investigated, we follow with RC dynamics. Again, let us suppose the RC has a maximum allowed number of one excitation, and therefore the available states are the same as LH2-LH1 interaction. The Reaction Center is not directly excited by light, since its absorption cross section is much smaller than the one of harvesting complexes, due to lack of carotenoids and fewer chromophores. Also differently, the RC once excited might start the sequence of charge transfers along A branch, with a rate (t+ )−1 , with value of 1 3 ps, as already mentioned. The M matrix is presented in the Appendix A.1. The confirmation among stochastic and master equation is presented in Fig.1.14(a).. 33.

(35) LH1-LH1 The possible states (0000), (0001), (0010), (0011), (0100), (0101), (0110), (0111), (1000), (1001), (1010), (1011), (1100), (1101), (1110), (1111), numbered from 1 to 16. A and B correspond to each LH1 considered. As LH1s are all equal, symmetry on rates kAB = kBA = k is justified. Under this condition the set of 16 equations reduce to 4: ṗA1(2) = γ1 (1 − pA1(2 )) + k(pB1 + pB2 − pA1(2) ) ṗB1(2) = γ1 (1 − pB1(2) ) + k(pA1 + pA2 − pB1(2) ). (1.40). confirmed numerically and shown in figure 1.14(b). RC-LH1-LH2 Labeling systems A as the RC, system B as the LH1, and system C as the LH2, the same model as the LH1-LH1 interaction can be used, with the replacements kAA → kAB , kBB → kBC and kAB → kBB , as well as back transfer rates, when sub-indexes 1 ↔ 2 are interchanged. kAC = kBB = 0. It is apparent on Fig.1.14(c) as already pointed out in Section 1.7, that excitation funneling is not the general rule, as an excitation starting at an LH2, takes some hundreds of picoseconds to be processed at the RC, to be reminded as a consequence of slow LH1→RC rate: 1/25 ps.. 1.4. Concluding Remarks of this Chapter.. 1. Stochastic numerical simulations agree with master equation results 2. The state space grows exponentially, and even though the studied architectures have only a few complexes, the required computational time is already important 3. Excitation kinetics have been investigated in small networks. It has been shown generally, that RCs are not highly visited by excitations, since their probability of occupancy is quite small. It is mandatory to analyze the RC population in complete vesicles since the available state space should change dynamics of excitations compared to small networks. The funneling to LH1-RC complexes can be slowed down due to available states from the set of antenna complexes. 4. Adaptation of purple bacteria presents bigger cluster sizes, and it is important to understand the RC cycling, that will increase further the state space size in a master equation approach. 34.

(36) Chapter 2. Chromatic Adaptation of Purple Bacteria: Exciton dynamics The first studies on purple bacteria photosynthetic membranes were interested on each complex conformation and detailed structure [6, 33, 34, 35]. The interested reader might find helpful the last chapter where a summary on structures and processes that allow photosynthesis in purple bacteria, is presented with a discussion on the dynamics of excitations in a few model architectures, in order to understand the consequences on dynamics, due to connectedness and the amount of harvesting complexes, recall, stoichiometry. Within the last few years, it has been possible to observe the relative position of complexes, by nanodissection of photosynthetic membranes with Atomic Force Microscopy (AFM), a technique that allows the topography of biological samples to be aquired in buffer solution at room temperature and normal pressure. These studies are done, in order to illustrate the changes induced on different bacteria species, or when environmental conditions change [36, 37, 32]. Explanations of the consequences or the reasons underlying these adaptations have been given only qualitatively. This chapter intends to explain in a quantitative fashion, the changes and trends involved in purple bacteria adaptation to light intensity.. 2.1. Atomic Force Microscopy Results. Photosynthetic purple bacteria contain RC-LH1 core and LH2 antenna complexes, with both light harvesting structures comprising roughly circular arranged αβ apo-protein units, with bound Carotenoids and Bacterio-chlorophyll pigments [6, 33, 34, 35]. To investigate the function of the structures involved in light harvesting, AFM has been used to understand the macromolecular organization of the photosynthetic membranes. Through AFM, it has been found that photosynthetic membranes in Rsp. Photometricum change when external con-. 35.

(37) HaL. HbL. Figure 2.1: In (a) LLI and (b) HLI grown membranes, display LH2 complexes (small, blue circles), LH1 (big, green circles) and RCs (dots, orange) in configurations resembling the chromatophores of Rsp. Photometricum [3].. ditions vary. Illuminated under high light intensity (HLI) (≈ 100W/m2 ), membranes grow with a ratio or stoichiometry of antenna-core complexes LH2/LH1≈ 3.5-4. When growth is achieved with LLI (≈ 10W/m2 ), this ratio increases to 7-9. The same type of adaptation was found in Rhodopseudomonas Palustris [36]. Furthermore, a general trend of clustering of LH1s and LH2s, was apparent from density correlation measures, to evidence formation of parachrystalline domains [3, 37, 38]. Specifically, in Rsp. Photometricum, there is a combination of PSUs as traditionally believed, having several LH2s around an LH1, and no core complexes having direct contact; and regions where aggregation of several LH1 complexes happens, all surrounded as a set, by antenna complexes [37]. It is also found that, parachrystalline domains of LH2 complexes form, when bacteria are grown with LLI [3, 37]. Such trend indeed persist with various purple bacteria species. Dimeric LH1 complexes appear in Rhodobacter Sphaeroides [32] and Rhodobacter Blasticus [36], while in Rhodopseudomonas Palustris[36] it is evident that under HLI, also LH1 para-crystalline domains form. All features observed experimentally in Rsp. Photometricum are reproduced empirically (Fig.2.1) with relaxation of networks having stoichiomtries consistent with found evidence, done through a Metropolis simulation scheme [39], with attractive long range potential wells: LH1-LH1=5 kB T, LH2-LH1=2 kB T and LH2-LH2=3 kB T [3]. Problems that remain unsolved The changes in architecture of membranes respond to adaptations of Purple Bacteria with the given excitation intensity. Qualitative reasons have been proposed in order to explain the 36.

(38) changes found: 1. More antenna complexes LH2 increase the cross section of the membrane as a need of higher absorption rates under LLI conditions. 2. When core clustering occurs, one excitation at a closed RC has to travel a smaller distance to reach an available RC. With respect to the first remark, the change in stoichiometry of complexes is consistent with doubling the amount of LH2 complexes. Along with the fact that LH1 cross section is almost double the one of LH2 (16 vs 8-10 in-membrane plane αβ-BChl units, respectively), is not able to increase membrane’s cross section in a way consistent with the ten fold light intensity difference. On the other hand, as has been stated in the previous chapter, it is less probable for an excitation to reach a RC when fewer are available. The second qualitative remark might be intuitively correct but lacks on a quantitative support on the behavior of multiple excitation dynamics in photosynthetic membranes.. 2.2. Excitation dynamics in complete chromatophore vesicles. To explain the changes found on complete chromatophore vesicles, as a first step, single excitation dynamics can be considered, and due to the global nature of adaptations, it is possible that a description of only a PSU is no longer complete, therefore, an analysis of greater membrane sections is in order. As will be found, the stoichiometry change seems not intuitive if vesicles adapt only due to single excitation kinetics, since fewer RCs makes less probable the ionization of any special pair, (remind they are named P). Multiple excitation effects can be of importance to optimize the membrane function. The membrane might behave differently when multiple excitation dynamics are accounted, as annihilation or blockade mechanisms become important, and can alter the RC supply of excitations. Global observables of interest These observables describe globally the excitation dynamics in the photosynthetic membranes. Although the quantities presented below are calculated numerically, a review is presented along formal expressions for their calculation: Connectivity The connectivity C is the mean number of connections that each node has within the network. A link between two nodes i and j is created if their center to center distance is smaller than (φi + φj )/2 + δ, where φi is the complex diameter at site i and δ is an additional distance used to give a criterion for a neighboring site. Diameter of LH2s and LH1s are 60 nm and 115 nm respectively. 37.

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