Toward Reliable Prediction of the Energy Ladder in Multichromophoric Systems: A Benchmark Study on the FMO Light-Harvesting Complex

Electric Field Susceptibility of Chlorophyll c Leads to Unexpected Excitation Dynamics in the Major Light-Harvesting Complex of Diatoms

Diatoms are one of the most abundant photosynthetic organisms on earth and contribute largely to atmospheric oxygen production. They contain fucoxanthin and chlorophyll-a/c binding proteins (FCPs) as light-harvesting complexes with a remarkable adaptation to the fluctuating light on ocean surfaces. To understand the basis of the photosynthetic process in diatoms, the excitation energy funneling within FCPs must be probed. A state-of-the-art multiscale analysis within a quantum mechanics/molecular mechanics framework has been employed. To this end, the chlorophyll (Chl) excitation energies within the FCP complex from the diatom Phaeodactylum tricornutum have been determined. The Chl-c excitation energies were found to be 5-fold more susceptible to electric fields than those of Chl-a pigments and thus are significantly lower in FCP than in organic solvents. This finding challenges the general belief that the excitation energy of Chl-c is always higher than that of Chl-a in FCP proteins and reveals that Chl-c molecules are much more sensitive to electric fields within protein scaffolds than in Chl-a pigments. The analysis of the linear absorption spectrum and the two-dimensional electronic spectra of the FCP complex strongly supports these findings and allows us to study the excitation transfer within the FCP complex.

Predicting Mutation-Induced Changes in the Electronic Properties of Photosynthetic Proteins from First Principles: The Fenna-Matthews-Olson Complex Example

Multiscale molecular modeling is utilized to predict optical absorption and circular dichroism spectra of two single-point mutants of the Fenna-Matthews-Olson photosynthetic pigment-protein complex. The modeling approach combines classical molecular dynamics simulations with structural refinement of photosynthetic pigments and calculations of their excited states in a polarizable protein environment. The only experimental input to the modeling protocol is the X-ray structure of the wild-type protein. The first-principles modeling reproduces changes in the experimental optical spectra of the considered mutants, Y16F and Q198V. Interestingly, the Q198V mutation has a negligible effect on the electronic properties of the targeted bacteriochlorophyll a pigment. Instead, the electronic properties of several other pigments respond to this mutation. The molecular modeling demonstrates that a single-point mutation can induce long-range effects on the protein structure, while extensive structural changes near a pigment do not necessarily lead to significant changes in the electronic properties of that pigment.

Hybrid Quantum Mechanical/Molecular Mechanical Methods For Studying Energy Transduction in Biomolecular Machines

Hybrid quantum mechanical/molecular mechanical (QM/MM) methods have become indispensable tools for the study of biomolecules. In this article, we briefly review the basic methodological details of QM/MM approaches and discuss their applications to various energy transduction problems in biomolecular machines, such as long-range proton transports, fast electron transfers, and mechanochemical coupling. We highlight the particular importance for these applications of balancing computational efficiency and accuracy. Using several recent examples, we illustrate the value and limitations of QM/MM methodologies for both ground and excited states, as well as strategies for calibrating them in specific applications. We conclude with brief comments on several areas that can benefit from further efforts to make QM/MM analyses more quantitative and applicable to increasingly complex biological problems.

A Multiconfigurational Wave Function Implementation of the Frenkel Exciton Model for Molecular Aggregates

We present an implementation of the Frenkel exciton model into the OpenMolcas program package enabling calculations of collective electronic excited states of molecular aggregates based on a multiconfigurational wave function description of the individual monomers. The computational protocol avoids using diabatization schemes and, thus, supermolecule calculations. Additionally, the use of the Cholesky decomposition of the two-electron integrals entering pair interactions enhances the efficiency of the computational scheme. The application of the method is exemplified for two test systems, that is, a formaldehyde oxime and a bacteriochlorophyll-like dimer. For the sake of comparison with the dipole approximation, we restrict our considerations to situations where intermonomer exchange can be neglected. The protocol is expected to be beneficial for aggregates composed of molecules with extended π systems, unpaired electrons such as radicals or transition metal centers, where it should outperform widely used methods based on time-dependent density functional theory.

Recent progress in atomistic modeling of light-harvesting complexes: a mini review

In this mini review, we focus on recent advances in the atomistic modeling of biological light-harvesting (LH) complexes. Because of their size and sophisticated electronic structures, multiscale methods are required to investigate the dynamical and spectroscopic properties of such complexes. The excitation energies, in this context also known as site energies, excitonic couplings, and spectral densities are key quantities which usually need to be extracted to be able to determine the exciton dynamics and spectroscopic properties. The recently developed multiscale approach based on the numerically efficient density functional tight-binding framework followed by excited state calculations has been shown to be superior to the scheme based on pure classical molecular dynamics simulations. The enhanced approach, which improves the description of the internal vibrational dynamics of the pigment molecules, yields spectral densities in good agreement with the experimental counterparts for various bacterial and plant LH systems. Here, we provide a brief overview of those results and described the theoretical foundation of the multiscale protocol.

Thermal site energy fluctuations in photosystem I: new insights from MD/QM/MM calculations

Cyanobacterial photosystem I (PSI) is one of the most efficient photosynthetic machineries found in nature. Due to the large scale and complexity of the system, the energy transfer mechanism from the antenna complex to the reaction center is still not fully understood. A central element is the accurate evaluation of the individual chlorophyll excitation energies (site energies). Such an evaluation must include a detailed treatment of site specific environmental influences on structural and electrostatic properties, but also their evolution in the temporal domain, because of the dynamic nature of the energy transfer process. In this work, we calculate the site energies of all 96 chlorophylls in a membrane-embedded model of PSI. The employed hybrid QM/MM approach using the multireference DFT/MRCI method in the QM region allows to obtain accurate site energies under explicit consideration of the natural environment. We identify energy traps and barriers in the antenna complex and discuss their implications for energy transfer to the reaction center. Going beyond previous studies, our model also accounts for the molecular dynamics of the full trimeric PSI complex. Via statistical analysis we show that the thermal fluctuations of single chlorophylls prevent the formation of a single prominent energy funnel within the antenna complex. These findings are also supported by a dipole exciton model. We conclude that energy transfer pathways may form only transiently at physiological temperatures, as thermal fluctuations overcome energy barriers. The set of site energies provided in this work sets the stage for theoretical and experimental studies on the highly efficient energy transfer mechanisms in PSI.

An efficient protocol for excited states of large biochromophores

Efficient energy transport in photosynthetic antenna is a long-standing source of inspiration for artificial light harvesting materials. However, characterizing the excited states of the constituent chromophores poses a considerable challenge to mainstream quantum chemical and semiempirical excited state methods due to their size and complexity and the accuracy required to describe small but functionally important changes in their properties. In this paper, we explore an alternative approach to calculating the excited states of large biochromophores, exemplified by a specific method for calculating the Q_y transition of bacteriochlorophyll a, which we name Chl-xTB. Using a diagonally dominant approximation to the Casida equation and a bespoke parameterization scheme, Chl-xTB can match time-dependent density functional theory's accuracy and semiempirical speed for calculating the potential energy surfaces and absorption spectra of chlorophylls. We demonstrate that Chl-xTB (and other prospective realizations of our protocol) can be integrated into multiscale models, including concurrent excitonic and point-charge embedding frameworks, enabling the analysis of biochromophore networks in a native environment. We exploit this capability to probe the low-frequency spectral densities of excitonic energies and interchromophore interactions in the light harvesting antenna protein LH2 (light harvesting complex 2). The impact of low-frequency protein motion on interchromophore coupling and exciton transport has routinely been ignored due to the prohibitive costs of including it in simulations. Our results provide a more rigorous basis for continued use of this approximation by demonstrating that exciton transition energies are unaffected by low-frequency vibrational coupling to exciton interaction energies.

Electronic Structure Effects Related to the Origin of the Remarkable Near-Infrared Absorption of Blastochloris viridis' Light Harvesting 1-Reaction Center Complex

Various photosynthetic organisms have evolved to absorb light in different regions of the visible light spectrum, thus adapting to the various lighting conditions available on Earth. While most of these autotrophic organisms absorb wavelengths around the 700-800 nm region, some are capable of red-shifted absorptions above this range, but none as remarkably as Blastochloris viridis whose main absorption is observed at 1015 nm, approximately 220 nm (0.34 eV) lower in energy than their main constituent pigments, BChl-b, whose main absorption is observed at 795 nm. The structure of its light harvesting 1-reaction center was recently elucidated by cryo-EM; however, the electronic structure details behind this red-shifted absorption remain unattended. We used hybrid quantum mechanics/molecular mechanics (QM/MM) calculations to optimize one of the active centers and performed classical molecular dynamics (MD) simulations to sample conformations beyond the optimized structure. We did excited state calculations with the time-dependent density functional theory method at the CAM-B3LYP/cc-pVDZ level of theory. We reproduced the near IR absorption by sequentially modifying the number of components involved in our systems using representative structures from the calculated MD ensemble. Natural transition orbital analysis reveals the participation of the BChl-b fragments to the main transition in the native structure and the structures obtained from the QM/MM and MD simulations. H-bonding pigment-protein interactions play a role on the conformation stabilization and orientation; however, the bacteriochlorin ring conformations and the exciton delocalization are the most relevant factors to explain the red-shifting phenomenon.

Computational elucidations on the role of vibrations in energy transfer processes of photosynthetic complexes

Coupling between pigment excitations and nuclear movements in photosynthetic complexes is known to modulate the excitation energy transfer (EET) efficiencies. Toward providing microscopic information, researchers often apply simulation techniques and investigate how vibrations are involved in EET processes. Here, reports on such roles of nuclear movements are discussed from a theory perspective. While vibrations naturally present random thermal fluctuations that can affect energy transferring characteristics, they can also be intertwined with exciton structures and create more specific non-adiabatic energy transfer pathways. For reliable simulations, a bath model that accurately mimics a given molecular system is required. Methods for obtaining such a model in combination with quantum chemical electronic structure calculations and molecular dynamics trajectory simulations are discussed. Various quantum dynamics simulation tools that can handle pigment-to-pigment energy transfers together with their vibrational characters are also touched on. Behaviors of molecular vibrations often deviate from ideality, especially when all-atom details are included, which practically forces us to treat them classically. We conclude this perspective by considering some recent reports that suggest that classical descriptions of bath effects with all-atom details may still produce valuable information for analyzing sophisticated contributions by vibrations to EET processes.

Multiscale Modeling of Electronic Spectra Including Nuclear Quantum Effects

Theoretical prediction of electronic absorption spectra without input from experiments is no easy feat, as it requires addressing all of the factors that affect line shapes. In practice, however, the methodologies are limited to treat these ingredients only to a certain extent. Here, we present a multiscale protocol that addresses the temperature, solvent, and nuclear quantum effects as well as anharmonicity and the reconstruction of the final spectra from individual transitions. First, quantum mechanics/molecular mechanics (QM/MM) molecular dynamics is conducted to obtain trajectories of solute-solvent configurations, from which the corresponding quantum-corrected ensembles are generated through the generalized smoothed trajectory analysis (GSTA). The optical spectra of the ensembles are then produced by calculating vertical transitions using time-dependent density-functional theory (TDDFT) with implicit solvation. To obtain the final spectral shapes, the stick spectra from TDDFT are convoluted with Gaussian kernels where the half-widths are determined by a statistically motivated strategy. We have tested our method by calculating the UV-vis spectra of a recently discovered acridine photocatalyst in two redox states. Vibronic progressions and broadenings due to the finite lifetime of the excited states are not included in the methodology yet. Nuclear quantization affects the relative peak intensities and widths, which is necessary to reproduce the experimental spectrum. We have also found that using only the optimized geometry of each molecule works surprisingly well if a proper empirical broadening factor is applied. This is explained by the rigidity of the conjugated chromophore moieties of the selected molecules, which are mainly responsible for the excitations in the spectra. In contrast, we have also shown that other parts of the molecules are flexible enough to feature anharmonicities that impair the use of other techniques such as Wigner sampling.

Assessment of the Ab Initio Bethe-Salpeter Equation Approach for the Low-Lying Excitation Energies of Bacteriochlorophylls and Chlorophylls

Bacteriochlorophyll and chlorophyll molecules are crucial building blocks of the photosynthetic apparatus in bacteria, algae, and plants. Embedded in transmembrane protein complexes, they are responsible for the primary processes of photosynthesis: excitation energy and charge transfer. Here, we use ab initio many-body perturbation theory within the GW approximation and Bethe-Salpeter equation (BSE) approach to calculate the electronic structure and optical excitations of bacteriochlorophylls a, b, c, d, and e and chlorophylls a and b. We systematically study the effects of the structure, basis set size, partial self-consistency in GW, and the underlying exchange-correlation approximation and compare our calculations with results from time-dependent density functional theory, multireference RASPT2, and experimental literature results. We find that optical excitations calculated with GW+BSE are in excellent agreement with experimental data, with an average deviation of less than 100 meV for the first three bright excitations of the entire family of (bacterio)chlorophylls. Contrary to state-of-the-art time-dependent density functional theory (TDDFT) with an optimally tuned range-separated hybrid functional, this accuracy is achieved in a parameter-free approach. Moreover, GW+BSE predicts the energy differences between the low-energy excitations correctly and eliminates spurious charge transfer states that TDDFT with (semi)local approximations is known to produce. Our study provides accurate reference results and highlights the potential of the GW+BSE approach for the simulation of larger pigment complexes.

Spectral tuning of chlorophylls in proteins - electrostatics vs. ring deformation

In photosynthetic complexes, tuning of chlorophyll light-absorption spectra by the protein environment is crucial to their efficiency and robustness. Recombinant type II water soluble chlorophyll-binding proteins from Brassicaceae (WSCPs) are useful for studying spectral tuning mechanisms due to their symmetric homotetramer structure, and the ability to rigorously modify the chlorophyll's protein surroundings. Our previous comparison of the crystal structures of two WSCP homologues suggested that protein-induced chlorophyll ring deformation is the predominant spectral tuning mechanism. Here, we implement a more rigorous analysis based on hybrid quantum mechanics and molecular mechanics calculations to quantify the relative contributions of geometrical and electrostatic factors to the absorption spectra of WSCP-chlorophyll complexes. We show that when considering conformational dynamics, geometry distortions such as chlorophyll ring deformation accounts for about one-third of the spectral shift, whereas the direct polarization of the electron density accounts for the remaining two-thirds. From a practical perspective, protein electrostatics is easier to manipulate than chlorophyll conformations, thus, it may be more readily implemented in designing artificial protein-chlorophyll complexes.

How Can We Predict Accurate Electrochromic Shifts for Biochromophores? A Case Study on the Photosynthetic Reaction Center

Protein-embedded chromophores are responsible for light harvesting, excitation energy transfer, and charge separation in photosynthesis. A critical part of the photosynthetic apparatus are reaction centers (RCs), which comprise groups of (bacterio)chlorophyll and (bacterio)pheophytin molecules that transform the excitation energy derived from light absorption into charge separation. The lowest excitation energies of individual pigments (site energies) are key for understanding photosynthetic systems, and form a prime target for quantum chemistry. A major theoretical challenge is to accurately describe the electrochromic (Stark) shifts in site energies produced by the inhomogeneous electric field of the protein matrix. Here, we present large-scale quantum mechanics/molecular mechanics calculations of electrochromic shifts for the RC chromophores of photosystem II (PSII) using various quantum chemical methods evaluated against the domain-based local pair natural orbital (DLPNO) implementation of the similarity-transformed equation of motion coupled cluster theory with single and double excitations (STEOM-CCSD). We show that certain range-separated density functionals (ωΒ97, ωΒ97X-V, ωΒ2PLYP, and LC-BLYP) correctly reproduce RC site energy shifts with time-dependent density functional theory (TD-DFT). The popular CAM-B3LYP functional underestimates the shifts and is not recommended. Global hybrid functionals are too insensitive to the environment and should be avoided, while nonhybrid functionals are strictly nonapplicable. Among the applicable approximate coupled cluster methods, the canonical versions of CC2 and ADC(2) were found to deviate significantly from the reference results both for the description of the lowest excited state and for the electrochromic shifts. By contrast, their spin-component-scaled (SCS) and particularly the scale-opposite-spin (SOS) variants compare well with the reference DLPNO-STEOM-CCSD and the best range-separated DFT methods. The emergence of RC excitation asymmetry is discussed in terms of intrinsic and protein electrostatic potentials. In addition, we evaluate a minimal structural scaffold of PSII, the D1-D2-CytB559 RC complex often employed in experimental studies, and show that it would have the same site energy distribution of RC chromophores as the full PSII supercomplex, but only under the unlikely conditions that the core protein organization and cofactor arrangement remain identical to those of the intact enzyme.

Artificial Photosynthesis: Is Computation Ready for the Challenge Ahead?

A tremendous effort is currently devoted to the generation of novel hybrid materials with enhanced electronic properties for the creation of artificial photosynthetic systems. This compelling and challenging problem is well-defined from an experimental point of view, as the design of such materials relies on combining organic materials or metals with biological systems like light harvesting and redox-active proteins. Such hybrid systems can be used, e.g., as bio-sensors, bio-fuel cells, biohybrid photoelectrochemical cells, and nanostructured photoelectronic devices. Despite these efforts, the main bottleneck is the formation of efficient interfaces between the biological and the organic/metal counterparts for efficient electron transfer (ET). It is within this aspect that computation can make the difference and improve the current understanding of the mechanisms underneath the interface formation and the charge transfer efficiency. Yet, the systems considered (i.e., light harvesting protein, self-assembly monolayer and surface assembly) are more and more complex, reaching (and often passing) the limit of current computation power. In this review, recent developments in computational methods for studying complex interfaces for artificial photosynthesis will be provided and selected cases discussed, to assess the inherent ability of computation to leave a mark in this field of research.

DFTB/MM Molecular Dynamics Simulations of the FMO Light-Harvesting Complex

Because of the size of light-harvesting complexes and the involvement of electronic degrees of freedom, computationally these systems need to be treated with a combined quantum-classical description. To this end, Born-Oppenheimer molecular dynamics simulations have been employed in a quantum mechanics/molecular mechanics (QM/MM) fashion for the ground state followed by excitation energy calculations again in a QM/MM scheme for the Fenna-Matthews-Olson (FMO) complex. The self-consistent-charge density functional tight-binding (DFTB) method electrostatically coupled to a classical description of the environment was applied to perform the ground-state dynamics. Subsequently, long-range-corrected time-dependent DFTB calculations were performed to determine the excitation energy fluctuations of the individual bacteriochlorophyll a molecules. The spectral densities obtained using this approach show an excellent agreement with experimental findings. In addition, the fluctuating site energies and couplings were used to estimate the exciton transfer dynamics.

Predictive First-Principles Modeling of a Photosynthetic Antenna Protein: The Fenna-Matthews-Olson Complex

High efficiency of light harvesting in photosynthetic pigment-protein complexes is governed by evolutionary-perfected protein-assisted tuning of individual pigment properties and interpigment interactions. Due to the large number of spectrally overlapping pigments in a typical photosynthetic complex, experimental methods often fail to unambiguously identify individual chromophore properties. Here, we report a first-principles-based modeling protocol capable of predicting properties of pigments in protein environment to a high precision. The technique was applied to successfully uncover electronic properties of the Fenna-Matthews-Olson (FMO) pigment-protein complex. Each of the three subunits of the FMO complex contains eight strongly coupled bacteriochlorophyll a (BChl a) pigments. The excitonic structure of FMO can be described by an electronic Hamiltonian containing excitation (site) energies of BChl a pigments and electronic couplings between them. Several such Hamiltonians have been developed in the past based on the information from various spectroscopic measurements of FMO; however, fine details of the excitonic structure and energy transfer in FMO, especially assignments of short-lived high-energy sites, remain elusive. Utilizing polarizable embedding quantum mechanics/molecular mechanics with the effective fragment potentials, we computed the electronic Hamiltonian of FMO that is in general agreement with previously reported empirical Hamiltonians and quantitatively reproduces experimental absorption and circular dichroism spectra of the FMO protein. The developed computational protocol is sufficiently simple and can be utilized for predictive modeling of other wild-type and mutated photosynthetic pigment-protein complexes.

FMOxFMO: Elucidating Excitonic Interactions in the Fenna-Matthews-Olson Complex with the Fragment Molecular Orbital Method

In order to study Förster resonance energy transfer (FRET), the fragment molecular orbital (FMO) method is extended to compute electronic couplings between local excitations via the excited state transition density model, enabling efficient calculations of nonlocal excitations in a large molecular system and overcoming the previous limitation of being able to compute only local excitations. The results of these simple but accurate models are validated against full quantum calculations without fragmentation. The developed method is applied to a very important photosynthetic pigment-protein complex, the Fenna-Matthews-Olson complex (FMOc), that is responsible for the energy transfer from a chlorosome to the reaction center in the green sulfur bacteria. Absorption and circular dichroism spectra of FMOc are simulated, and the role of the molecular environment on the excitations is revealed.

Benchmark and performance of long-range corrected time-dependent density functional tight binding (LC-TD-DFTB) on rhodopsins and light-harvesting complexes

The chromophores of rhodopsins (Rh) and light-harvesting (LH) complexes still represent a major challenge for a quantum chemical description due to their size and complex electronic structure. Since gradient corrected and hybrid density functional approaches have been shown to fail for these systems, only range-separated functionals seem to be a promising alternative to the more time consuming post-Hartree-Fock approaches. For extended sampling of optical properties, however, even more approximate approaches are required. Recently, a long-range corrected (LC) functional has been implemented into the efficient density functional tight binding (DFTB) method, allowing to sample the excited states properties of chromophores embedded into proteins using quantum mechanical/molecular mechanical (QM/MM) with the time-dependent (TD) DFTB approach. In the present study, we assess the accuracy of LC-TD-DFT and LC-TD-DFTB for rhodopsins (bacteriorhodopsin (bR) and pharaonis phoborhodopsin (ppR)) and LH complexes (light-harvesting complex II (LH2) and Fenna-Matthews-Olson (FMO) complex). This benchmark study shows the improved description of the color tuning parameters compared to standard DFT functionals. In general, LC-TD-DFTB can exhibit a similar performance as the corresponding LC functionals, allowing a reliable description of excited states properties at significantly reduced cost. The two chromophores investigated here pose complementary challenges: while huge sensitivity to external field perturbation (color tuning) and charge transfer excitations are characteristic for the retinal chromophore, the multi-chromophoric character of the LH complexes emphasizes a correct description of inter-chromophore couplings, giving less importance to color tuning. None of the investigated functionals masters both systems simultaneously with satisfactory accuracy. LC-TD-DFTB, at the current stage, although showing a systematic improvement compared to TD-DFTB cannot be recommended for studying color tuning in retinal proteins, similar to some of the LC-DFT functionals, because the response to external fields is still too weak. For sampling of LH-spectra, however, LC-TD-DFTB is a viable tool, allowing to efficiently sample absorption energies, as shown for three different LH complexes. As the calculations indicate, geometry optimization may overestimate the importance of local minima, which may be averaged over when using trajectories. Fast quantum chemical approaches therefore may allow for a direct sampling of spectra in the near future.

Successes & challenges in the atomistic modeling of light-harvesting and its photoregulation

Light-harvesting is a crucial step of photosynthesis. Its mechanisms and related energetics have been revealed by a combination of experimental investigations and theoretical modeling. The success of theoretical modeling is largely due to the application of atomistic descriptions combining quantum chemistry, classical models and molecular dynamics techniques. Besides the important achievements obtained so far, a complete and quantitative understanding of how the many different light-harvesting complexes exploit their structural specificity is still missing. Moreover, many questions remain unanswered regarding the mechanisms through which light-harvesting is regulated in response to variable light conditions. Here we show that, in both fields, a major role will be played once more by atomistic descriptions, possibly generalized to tackle the numerous time and space scales on which the regulation takes place: going from the ultrafast electronic excitation of the multichromophoric aggregate, through the subsequent conformational changes in the embedding protein, up to the interaction between proteins.

Macrocycle ring deformation as the secondary design principle for light-harvesting complexes

Natural light-harvesting is performed by pigment-protein complexes, which collect and funnel the solar energy at the start of photosynthesis. The identity and arrangement of pigments largely define the absorption spectrum of the antenna complex, which is further regulated by a palette of structural factors. Small alterations are induced by pigment-protein interactions. In light-harvesting systems 2 and 3 from Rhodoblastus acidophilus, the pigments are arranged identically, yet the former has an absorption peak at 850 nm that is blue-shifted to 820 nm in the latter. While the shift has previously been attributed to the removal of hydrogen bonds, which brings changes in the acetyl moiety of the bacteriochlorophyll, recent work has shown that other mechanisms are also present. Using computational and modeling tools on the corresponding crystal structures, we reach a different conclusion: The most critical factor for the shift is the curvature of the macrocycle ring. The bending of the planar part of the pigment is identified as the second-most important design principle for the function of pigment-protein complexes-a finding that can inspire the design of novel artificial systems.

Calculation of the excited states properties of LH1 complex of Thermochromatium tepidum

Calculation of the excited states properties of pigment complexes is one of the key problems in the photosynthesis research. The excited states of LH1 complex of Thermochromatium tepidum were studied by means of the high-precision quantum chemistry methods. The influence of different parameters of the calculation procedure was examined. The optimal scheme of calculation was chosen by comparison of calculated results with the experimental data on absorption, electronic and magnetic circular dichroism spectra. The high importance of the account of the second excited states of bacteriochlorophylls and of site heterogeneity was shown. © 2018 Wiley Periodicals, Inc.

Excited state energy fluctuations in the Fenna-Matthews-Olson complex from molecular dynamics simulations with interpolated chromophore potentials

We analyze the environment-induced fluctuation of pigment excitation energies in the Fenna-Matthews-Olson (FMO) complex from various perspectives, by employing an interpolation-based all-atom potential energy model for describing realistic pigment vibrations. We conduct molecular dynamics simulations on a 100 ns timescale, which is an extent that can enclose the effect of static disorder, and demonstrate its timescale separation from fast dynamic disorder. We extract the spectral densities of the complex by considering both the site and the exciton bases. We show that exciton delocalization reduces the effective environmental fluctuation and rationalize this aspect based on a model of fluctuating molecular aggregates. We also obtained the spectral density of the lowest exciton state under low temperature conditions and show that it reasonably well reproduces the experimental result. Finally, by additionally performing non-equilibrium excited state trajectory simulations, we show that the system lies well within the linear response regime after photo-absorption and that the pigments do not visit anharmonic regions of the potential surface to a significant extent. This indicates that methodologies based on harmonic bath models are indeed reasonable approaches for describing the excited state dynamics of the FMO complex.

Simple Quantum Dynamics with Thermalization

In this paper, we introduce two simple quantum dynamics methods. One is based on the popular surface-hopping method, and the other is based on rescaling of the propagation on the bath ground-state potential surface. The first method is special, as it avoids specific feedback from the simulated quantum system to the bath and can be applied for precalculated classical trajectories. It is based on the equipartition theorem to determine if hops between different potential energy surfaces are allowed. By comparing with the formally exact Hierarchical Equations Of Motion approach for four model systems we find that the method generally approximates the quantum dynamics toward thermal equilibrium very well. The second method is based on rescaling of the nonadiabatic coupling and also neglect the effect of the state of the quantum system on the bath. By the nature of the approximations, they cannot reproduce the effect of bath relaxation following excitation. However, the methods are both computationally more tractable than the conventional fewest switches surface hopping, and we foresee that the methods will be powerful for simulations of quantum dynamics in systems with complex bath dynamics, where the system-bath coupling is not too strong compared to the thermal energy.

Photosynthesis, pigment-protein complexes and electronic energy transport: simple models for complicated processes

In this review, we discuss our recent work on modelling biological pigment-protein complexes, such as the Fenna-Matthews-Olson complex and light-harvesting complex-II, to explain their electronic energy transport properties. In particular, we highlight how a network-based analysis approach, where the light-absorbing pigments are treated as a network of interconnected nodes, can provide a qualitative picture of quantum dynamic energy transport. With this in mind, we demonstrate how other properties such as robustness to environmental changes can be assessed in a simple and computationally tractable manner. Such analyses could prove useful for the design of artificial energy transport networks such as those which might find application in solar cells.

Intermolecular Modes between LH2 Bacteriochlorophylls and Protein Residues: The Effect on the Excitation Energies

Light-harvesting system 2 (LH2) executes the primary processes of photosynthesis in purple bacteria; photon absorption, and energy transportation to the reaction center. A detailed mechanistic insight into these operations is obscured by the complexity of the light-harvesting systems, particularly by the chromophore-environment interaction. In this work, we focus on the effects of the protein residues that are ligated to the bacteriochlorophylls (BChls) and construct potential energy surfaces of the ground and first optically excited state for the various BChl-residue systems where we in each case consider two degrees of freedom in the intermolecular region. We find that the excitation energies are only slightly affected by the considered modes. In addition, we see that axial ligands and hydrogen-bonded residues have opposite effects on both excitation energies and oscillator strengths by comparing to the isolated BChls. Our results indicate that only a small part of the chromophore-environment interaction can be associated with the intermolecular region between a BChl and an adjacent residue, but that it may be possible to selectively raise or lower the excitation energy at the axial and planar residue positions, respectively.

Theoretical description of protein field effects on electronic excitations of biological chromophores

Photoinitiated phenomena play a crucial role in many living organisms. Plants, algae, and bacteria absorb sunlight to perform photosynthesis, and convert water and carbon dioxide into molecular oxygen and carbohydrates, thus forming the basis for life on Earth. The vision of vertebrates is accomplished in the eye by a protein called rhodopsin, which upon photon absorption performs an ultrafast isomerisation of the retinal chromophore, triggering the signal cascade. Many other biological functions start with the photoexcitation of a protein-embedded pigment, followed by complex processes comprising, for example, electron or excitation energy transfer in photosynthetic complexes. The optical properties of chromophores in living systems are strongly dependent on the interaction with the surrounding environment (nearby protein residues, membrane, water), and the complexity of such interplay is, in most cases, at the origin of the functional diversity of the photoactive proteins. The specific interactions with the environment often lead to a significant shift of the chromophore excitation energies, compared with their absorption in solution or gas phase. The investigation of the optical response of chromophores is generally not straightforward, from both experimental and theoretical standpoints; this is due to the difficulty in understanding diverse behaviours and effects, occurring at different scales, with a single technique. In particular, the role played by ab initio calculations in assisting and guiding experiments, as well as in understanding the physics of photoactive proteins, is fundamental. At the same time, owing to the large size of the systems, more approximate strategies which take into account the environmental effects on the absorption spectra are also of paramount importance. Here we review the recent advances in the first-principle description of electronic and optical properties of biological chromophores embedded in a protein environment. We show their applications on paradigmatic systems, such as the light-harvesting complexes, rhodopsin and green fluorescent protein, emphasising the theoretical frameworks which are of common use in solid state physics, and emerging as promising tools for biomolecular systems.

Atomistic non-adiabatic dynamics of the LH2 complex with a GPU-accelerated ab initio exciton model

We present GPU-accelerated ab initio molecular dynamics simulations of nonadiabatic dynamics in the LH2 complex in full atomistic detail.

Nonlinear network model analysis of vibrational energy transfer and localisation in the Fenna-Matthews-Olson complex

Collective protein modes are expected to be important for facilitating energy transfer in the Fenna-Matthews-Olson (FMO) complex of photosynthetic green sulphur bacteria, however to date little work has focussed on the microscopic details of these vibrations. The nonlinear network model (NNM) provides a computationally inexpensive approach to studying vibrational modes at the microscopic level in large protein structures, whilst incorporating anharmonicity in the inter-residue interactions which can influence protein dynamics. We apply the NNM to the entire trimeric FMO complex and find evidence for the existence of nonlinear discrete breather modes. These modes tend to transfer energy to the highly connected core pigments, potentially opening up alternative excitation energy transfer routes through their influence on pigment properties. Incorporating localised modes based on these discrete breathers in the optical spectra calculations for FMO using ab initio site energies and excitonic couplings can substantially improve their agreement with experimental results.

An Ab Initio Description of the Excitonic Properties of LH2 and Their Temperature Dependence

The spectroscopic properties of light-harvesting (LH) antennae in photosyntehtic organisms represent a fingerprint that is unique for each specific pigment-protein complex. Because of that, spectroscopic observations are generally combined with structural data from X-ray crystallography to obtain an indirect representation of the excitonic properties of the system. Here, an alternative strategy is presented which goes beyond this empirical approach and introduces an ab initio computational description of both structural and electronic properties and their dependence on the temperature. The strategy is applied to the peripheral light-harvesting antenna complex (LH2) present in purple bacteria. By comparing this model with the one based on the crystal structure, a detailed, molecular level explanation of the absorption and circular dichroism (CD) spectra and their temperature dependence is achieved. The agreement obtained with the experiments at both low and room temperature lays the groundwork for an atomistic understanding of the excitation dynamics in the LH2 system.

Machine learning exciton dynamics

Obtaining the exciton dynamics of large photosynthetic complexes by using mixed quantum mechanics/molecular mechanics (QM/MM) is computationally demanding. We propose a machine learning technique, multi-layer perceptrons, as a tool to reduce the time required to compute excited state energies. With this approach we predict time-dependent density functional theory (TDDFT) excited state energies of bacteriochlorophylls in the Fenna-Matthews-Olson (FMO) complex. Additionally we compute spectral densities and exciton populations from the predictions. Different methods to determine multi-layer perceptron training sets are introduced, leading to several initial data selections. In addition, we compute spectral densities and exciton populations. Once multi-layer perceptrons are trained, predicting excited state energies was found to be significantly faster than the corresponding QM/MM calculations. We showed that multi-layer perceptrons can successfully reproduce the energies of QM/MM calculations to a high degree of accuracy with prediction errors contained within 0.01 eV (0.5%). Spectral densities and exciton dynamics are also in agreement with the TDDFT results. The acceleration and accurate prediction of dynamics strongly encourage the combination of machine learning techniques with ab initio methods.

Quantum Chemical Studies of Light Harvesting

The design of optimal light-harvesting (supra)molecular systems and materials is one of the most challenging frontiers of science. Theoretical methods and computational models play a fundamental role in this difficult task, as they allow the establishment of structural blueprints inspired by natural photosynthetic organisms that can be applied to the design of novel artificial light-harvesting devices. Among theoretical strategies, the application of quantum chemical tools represents an important reality that has already reached an evident degree of maturity, although it still has to show its real potentials. This Review presents an overview of the state of the art of this strategy, showing the actual fields of applicability but also indicating its current limitations, which need to be solved in future developments.

Contribution of bacteriochlorophyll conformation to the distribution of site-energies in the FMO protein

The structural data for the Fenna-Matthews-Olson (FMO) protein indicate that the bacteriochlorophylls (BChls) display a significant degree of conformational heterogeneity of their peripheral substituents and the protein-induced nonplanar skeletal deformations of the tetrapyrrole macrocycle. As electronic properties of chromophores are altered by such differences, a conformational effect may influence the site-energies of specific pigments and thus play a role in mediating the excitation energy transfer dynamics, but this has not yet been established. The difficulty of assessing this question is shown to be partly the result of the inability of the sequential truncation approach usually employed to account for interactions between the conformations of the macrocycle and its substituents and an alternative approach is suggested. By assigning the BChl atoms to meaningful atom groups and performing all possible permutations of partial optimizations in a full-factorial design, where each group is either frozen in the crystal geometry or optimized in vacuo, followed by excited state calculations on each resulting structure (PM6//ZIndo/S), the specific effects of the conformations of each BChl component as well as mutual interactions between the molecular fragments on the site-energy can be delineated. This factorial relaxation procedure gives different estimates of the macrocycle conformational perturbation than the approach of sequentially truncating the BChl periphery. The results were evaluated in the context of published site-energies for the FMO pigments from three species to identify how conformational effects contribute to their distribution and instances of cross-species conservation and functional divergence of the BChl nonplanarity conformational contribution are described.

Hybrid QM/MM study of FMO complex with polarized protein-specific charge

The Fenna-Matthews-Olson (FMO) light-harvesting complex is now one of the primary model systems for the study of excitation energy transfer (EET). However, the mechanism of the EET in this system is still controversial. In this work, molecular dynamics simulations and the electrostatic-embedding quantum-mechanics/molecular-mechanics single-point calculations have been employed to predict the energy transfer pathways utilizing the polarized protein-specific charge (PPC), which provides a more realistic description of Coulomb interaction potential in the protein than conventional mean-field charge scheme. The recently discovered eighth pigment has also been included in this study. Comparing with the conventional mean-field charges, more stable structures of FMO complex were found under PPC scheme during molecular dynamic simulation. Based on the electronic structure calculations, an exciton model was constructed to consider the couplings during excitation. The results show that pigments 3 and 4 dominate the lowest exciton levels whereas the highest exciton level are mainly constituted of pigments 1 and 6. This observation agrees well with the assumption based on the spatial distribution of the pigments. Moreover, the obtained spectral density in this study gives a reliable description of the diverse local environment embedding each pigment.

Limits and potentials of quantum chemical methods in modelling photosynthetic antennae

A critical overview of quantum chemical approaches to simulate the light-harvesting process in photosynthetic antennae is presented together with a perspective on the developments that need to be introduced to reach a quantitative predictive power.

Theoretical study on excited states of bacteriochlorophyll a in solutions with density functional assessment

The excited-state properties of bacteriochlorophyll (BChl) a in triethylamine, 1-propanol, and methanol are investigated with the time-dependent density functional theory by using the quantum mechanical and molecular mechanical reweighting free energy self-consistant field method. It is found that no prevalent density functionals can reproduce the experimental excited-state properties, i.e., the absorption and reorganization energies, of BChl a in the solutions. The parameter μ in the range-separated hybrid functional is therefore optimized to reproduce the differences of the absorption energies in the solutions. We examine the origin of the differences of the absorption energies in the solutions and find that sensitive balance between contributions of structural changes and solute-solvent interactions determines the differences. The accurate description of the excitation with the density functional with the adjusted parameter is therefore essential to the understanding of the excited-state properties of BChl a in proteins and also the mechanism of the photosynthetic systems.

The Fenna-Matthews-Olson protein revisited: a fully polarizable (TD)DFT/MM description

We report a combined molecular dynamics and quantum mechanics (QM)/molecular mechanics (MM) analysis of the excitonic properties of the Fenna-Matthews-Olson (FMO) protein by using a polarizable MM model combined with a time-dependent density functional theory description. Overall, our results indicate that structural fluctuations, electrostatic interactions, and short-range quantum effects can significantly modulate the model Hamiltonian parameters (site energies and couplings). We find that the specific interactions with the axial ligand and the hydrogen-bonded residues are responsible for the energy ladder, with their effects being mainly due to electrostatic interactions, but with short-range quantum contributions that are not negligible. In addition, a striking modulation of the screening effects experienced by the BChl pairs, due to the heterogeneous polarizability of the FMO and solvent environment, is observed. Finally, we find that the exciton model gives a reliable description of the delocalized excited states in the complex.