Quantitative Evaluation of Site Energies and Their Fluctuations of Pigments in the Fenna–Matthews–Olson Complex with an Efficient Method for Generating a Potential Energy Surface

Cavity-Mediated Enhancement of the Energy Transfer in the Reduced Fenna-Matthews-Olson Complex

Strong light-matter interaction leads to the formation of hybrid polariton states and can alter the light-harvesting properties of natural photosynthetic systems without modifying their chemical structure. In the present study, we computationally investigate the effect of the resonant cavity on the efficiency and the rate of the population transfer in a quantum system coupled to the cavity and the dissipative environment. The parameters of the model system were chosen to represent the Fenna-Matthews-Olson natural light-harvesting complex reduced to the three essential sites. The dynamics of the total system was propagated using the hierarchical equations of motion. Our results show that the strong light-matter interaction can accelerate the population transfer process compared to the cavity-free case but at the cost of lowering the transfer efficiency. The transition to the strong coupling regime was found to coincide with the degeneracy of polariton eigenvalues. Our findings indicate the potential and the limit of tuning the energy transfer in already efficient natural light-harvesting systems.

PigmentHunter: A point-and-click application for automated chlorophyll-protein simulations

Chlorophyll proteins (CPs) are the workhorses of biological photosynthesis, working together to absorb solar energy, transfer it to chemically active reaction centers, and control the charge-separation process that drives its storage as chemical energy. Yet predicting CP optical and electronic properties remains a serious challenge, driven by the computational difficulty of treating large, electronically coupled molecular pigments embedded in a dynamically structured protein environment. To address this challenge, we introduce here an analysis tool called PigmentHunter, which automates the process of preparing CP structures for molecular dynamics (MD), running short MD simulations on the nanoHUB.org science gateway, and then using electrostatic and steric analysis routines to predict optical absorption, fluorescence, and circular dichroism spectra within a Frenkel exciton model. Inter-pigment couplings are evaluated using point-dipole or transition-charge coupling models, while site energies can be estimated using both electrostatic and ring-deformation approaches. The package is built in a Jupyter Notebook environment, with a point-and-click interface that can be used either to manually prepare individual structures or to batch-process many structures at once. We illustrate PigmentHunter's capabilities with example simulations on spectral line shapes in the light harvesting 2 complex, site energies in the Fenna-Matthews-Olson protein, and ring deformation in photosystems I and II.

Potential energy interpolation with target-customized weighting coordinates: application to excited-state dynamics of photoactive yellow protein chromophore in water

Interpolation of potential energy surfaces (PESs) can provide a practical route to performing molecular dynamics simulations with a reliability matching a high-level quantum chemical calculation. An obstacle to its widespread use is perhaps the lack of general and optimal interpolation settings that can be applied in a black-box manner for any given molecular system. How to set up the weights for interpolation is one such task, and we still need to diversify the approaches in order to treat various systems. Here, we develop a new interpolation weighting scheme, which allows us to choose the weighting coordinates in a system-specific manner, by amplifying the contribution from specific internal coordinates. The new weighting scheme with an appropriate selection of coordinates is proved to be effective in reducing the interpolation error along the reaction pathway. As a demonstration, we consider the photoactive yellow protein chromophore system, as it constitutes itself as an interesting target that bears long-standing questions related to excited-state dynamics inside protein environments. We build its two-state diabatic interpolated PES with the new weighting scheme. We indeed see the utility of our scheme by conducting nonadiabatic molecular dynamics simulations with the required semi-global PES based on a limited number of data points.

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.

Bioexcitons by Design: How Do We Get There?

Biological pigment-protein complexes (PPCs) exhibit a remarkable ability to tune the optical properties of biological excitons (bioexcitons) through specific pigment-protein interactions. While such fine-tuning allows natural systems (e.g., photosynthetic proteins) to carry out their native functions with near-optimal performance, native function itself is often suboptimal for applications such as biofuel production or quantum technology development. This perspective offers a look at near-term prospects for the rational reoptimization of PPC bioexcitons for new functions using site-directed mutagenesis. The primary focus is on the "structure-spectrum" challenge of understanding the relationships between structural features and spectroscopic properties. While recent examples demonstrate that site-directed mutagenesis can be used to tune nearly all key bioexciton parameters (e.g., site energies, interpigment couplings, and electronic-vibrational interactions), critical challenges remain before we achieve truly rational design of bioexciton properties.

Theoretical Study on Thermal Structural Fluctuation Effects of Intermolecular Configurations on Singlet Fission in Pentacene Crystal Models

Singlet fission (SF) occurs as a result of complex excited state relaxation dynamics in molecular aggregates, where a singlet exciton (FE) state is converted into a double-triplet exciton (TT) state through the interactions with several other degrees of freedom, such as nuclear motions. In this study, we combined quantum dynamics simulation based on the quantum master equation approach with all-atom-based classical molecular mechanics/molecular dynamics to examine the thermal structural fluctuation (i.e., static disorder) effects of intermolecular configuration on SF in pentacene crystal models. In particular, we considered two types of static-disordered models, in which excited states are assumed to interact with nuclear motions of intermolecular modes in the classical mechanical/statistical manner. We found that the introduction of static disorder effects leads to a faster decay of coherence between the FE and charge transfer (CT) states in the early stage of SF, contributing to the accelerations of several FE → TT relaxation pathways. Such acceleration in these models is shown to be attributed to fluctuations in the energies and electronic coupling of the CT states based on relative relaxation factor analysis. The present study is expected to contribute to further development of bottom-up materials design for efficient SF in condensed phases where the exitonic system interacts with nuclear motions in various coupling strengths.

Coherence and Efficient Energy Transfer in Molecular Wires: Insights from Surface Hopping Simulations

Understanding the dynamics of electronic energy transfer through a molecular wire is essential to understand the working of natural processes like photosynthesis. We investigate simpler 2 and 3-site model Hamiltonians in this work to understand the importance of coherence to efficient energy transfer. We compare the results of surface hopping simulation with that of numerically exact results and rate theories. Different parameters are analyzed, motivated by a photosynthetic molecular wire - the FMO complex. A comparison of results from different theories shows that coherence can play an important role towards efficient energy transfer for certain parameters. When these coherences are important, even small couplings (of the order of 5 cm^-1 ) in the Hamiltonian can significantly affect rates. Surface hopping simulations capture all the results correctly qualitatively. Rate theories, on the other hand, can differ significantly from numerically exact results when coherences become important. The results of this work should provide design guidelines for efficient energy transfer in molecular wires.

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.

Accurate prediction of mutation-induced frequency shifts in chlorophyll proteins with a simple electrostatic model

Photosynthetic pigment-protein complexes control local chlorophyll (Chl) transition frequencies through a variety of electrostatic and steric forces. Site-directed mutations can modify this local spectroscopic tuning, providing critical insight into native photosynthetic functions and offering the tantalizing prospect of creating rationally designed Chl proteins with customized optical properties. Unfortunately, at present, no proven methods exist for reliably predicting mutation-induced frequency shifts in advance, limiting the method's utility for quantitative applications. Here, we address this challenge by constructing a series of point mutants in the water-soluble chlorophyll protein of Lepidium virginicum and using them to test the reliability of a simple computational protocol for mutation-induced site energy shifts. The protocol uses molecular dynamics to prepare mutant protein structures and the charge density coupling model of Adolphs et al. [Photosynth. Res. 95, 197-209 (2008)] for site energy prediction; a graphical interface that implements the protocol automatically is published online at http://nanohub.org/tools/pigmenthunter. With the exception of a single outlier (presumably due to unexpected structural changes), we find that the calculated frequency shifts match the experiment remarkably well, with an average error of 1.6 nm over a 9 nm spread in wavelengths. We anticipate that the accuracy of the method can be improved in the future with more advanced sampling of mutant protein structures.

Excitation Energy Transfer Efficiency in Fluctuating Environments: Role of Quantum Coherence in the Presence of Memory Effects

Several recent studies have interrogated the role of quantum coherence in affecting the transfer efficiency of an optical excitation to the designated "trap" state where the energy can be used for subsequent reactions, as in photosynthesis. However, these studies invoke a Markovian approximation for the time correlation function describing the environment-induced stochastic fluctuations. Here, we employ Kubo's quantum stochastic Liouville equation (QSLE) to include memory effects. We extend the existing QSLE scheme to introduce decay of a newly created excitation due to radiative and nonradiative channels and also by desired trapping toward the targeted chromophore. We show that the theoretical formalism based on the QSLE correctly reproduces the rate equation description in the Markovian limit, with the rate constants determined by an appropriate quantum limiting procedure. We find that under certain conditions, the efficiency of excitation transfer to the trap gains from the combined presence of quantum coherence and temporally correlated stochastic fluctuations. We work out different limiting situations in order to discover and quantify the optimum conditions for the energy transfer to the trapped state. We find that maximum energy transfer efficiency is achieved in the intermediate limit between coherent and incoherent transport.

Exciton Lifetime Distributions and Population Dynamics in the FMO Protein Complex from Prosthecochloris aestuarii

Significant protein rearrangement upon excitation and energy transfer in Fenna-Matthews-Olson protein of Prosthecochloris aestuarii results in a modified energy landscape, which induces more changes in pigment site energies than predicted by the "standard" hole-burning theory. The energy changes are elucidated by simulations while investigating the effects of site-dependent disorder, both static (site-energy distribution widths) and dynamic (spectral density shapes). The resulting optimized site energies and their fluctuations are consistent with relative differences observed in inhomogeneous widths calculated by recent molecular dynamic simulations. Two sets of different spectral densities reveal how their shapes affect the population dynamics and distribution of exciton lifetimes. Calculations revealed the wavelength-dependent distributions of exciton lifetimes (T ₁) in the femtosecond to picosecond time frame. We suggest that the calculated multimodal and asymmetric wavelength-dependent T ₁ distributions offer more insight into the interpretation of resonant hole-burned (HB) spectra, kinetic traces in two-dimensional (2D) electronic spectroscopy experiments, and widely used global analyses in fitting data from transient absorption experiments.

Static Disorder in Excitation Energies of the Fenna-Matthews-Olson Protein: Structure-Based Theory Meets Experiment

Inhomogeneous broadening of optical lines of the Fenna-Matthews-Olson (FMO) light-harvesting protein is investigated by combining a Monte Carlo sampling of low-energy conformational substates of the protein with a quantum chemical/electrostatic calculation of local transition energies (site energies) of the pigments. The good agreement between the optical spectra calculated for the inhomogeneous ensemble and the experimental data demonstrates that electrostatics is the dominant contributor to static disorder in site energies. Rotamers of polar amino acid side chains are found to cause bimodal distribution functions of site energy shifts, which can be probed by hole burning and single-molecule spectroscopy. When summing over the large number of contributions, the resulting distribution functions of the site energies become Gaussians, and the correlations in site energy fluctuations at different sites practically average to zero. These results demonstrate that static disorder in the FMO protein is in the realm of the central limit theorem of statistics.

Normal mode analysis of spectral density of FMO trimers: Intra- and intermonomer energy transfer

The intermolecular contribution to the spectral density of the exciton-vibrational coupling of the homotrimeric Fenna-Matthews-Olson (FMO) light-harvesting protein of green sulfur bacteria P. aestuarii is analyzed by combining a normal mode analysis of the protein with the charge density coupling method for the calculation of local transition energies of the pigments. Correlations in site energy fluctuations across the whole FMO trimer are found at low vibrational frequencies. Including, additionally, the high-frequency intrapigment part of the spectral density, extracted from line-narrowing spectra, we study intra- and intermonomer exciton transfer. Whereas the intrapigment part of the spectral density is important for fast intramonomer exciton relaxation, the intermolecular contributions (due to pigment-environment coupling) determine the intermonomer exciton transfer. Neither the variations of the local Huang-Rhys factors nor the correlations in site energy fluctuations have a critical influence on energy transfer. At room temperature, the intermonomer transfer in the FMO protein occurs on a 10 ps time scale, whereas intramonomer exciton equilibration is roughly two orders of magnitude faster. At cryogenic temperatures, intermonomer transfer limits the lifetimes of the lowest exciton band. The lifetimes are found to increase between 20 ps in the center of this band up to 100 ps toward lower energies, which is in very good agreement with the estimates from hole burning data. Interestingly, exciton delocalization in the FMO monomers is found to slow down intermonomer energy transfer, at both physiological and cryogenic temperatures.

Analysis of Photosynthetic Systems and Their Applications with Mathematical and Computational Models

In biological and life science applications, photosynthesis is an important process that involves the absorption and transformation of sunlight into chemical energy. During the photosynthesis process, the light photons are captured by the green chlorophyll pigments in their photosynthetic antennae and further funneled to the reaction center. One of the most important light harvesting complexes that are highly important in the study of photosynthesis is the membrane-attached Fenna–Matthews–Olson (FMO) complex found in the green sulfur bacteria. In this review, we discuss the mathematical formulations and computational modeling of some of the light harvesting complexes including FMO. The most recent research developments in the photosynthetic light harvesting complexes are thoroughly discussed. The theoretical background related to the spectral density, quantum coherence and density functional theory has been elaborated. Furthermore, details about the transfer and excitation of energy in different sites of the FMO complex along with other vital photosynthetic light harvesting complexes have also been provided. Finally, we conclude this review by providing the current and potential applications in environmental science, energy, health and medicine, where such mathematical and computational studies of the photosynthesis and the light harvesting complexes can be readily integrated.

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.

Delocalization and Quantum Entanglement in Physical Systems

Quantum coherence and entanglement in an extended interacting system where energy levels are nondegenerate and coupled to a dissipative environment is a common occurrence in nature, like in photosynthetic reaction systems and conjugated polymers. The temperature dependence of quantum coherence in a trimer complex (first three subunits of the Fenna-Matthews-Olson complex) is studied using a temperature-dependent quantum stochastic Liouville equation. In the non-Markovian limit, the lowering of temperature induces long-lasting quantum coherence that, in turn, leads to delocalization, whose length grows. The entanglement and coherence length determine the nature of the dynamic localization.

Intramolecular Vibrations Complement the Robustness of Primary Charge Separation in a Dimer Model of the Photosystem II Reaction Center

The energy conversion of oxygenic photosynthesis is triggered by primary charge separation in proteins at the photosystem II reaction center. Here, we investigate the impacts of the protein environment and intramolecular vibrations on primary charge separation at the photosystem II reaction center. This is accomplished by combining the quantum dynamic theories of condensed phase electron transfer with quantum chemical calculations to evaluate the vibrational Huang-Rhys factors of chlorophyll and pheophytin molecules. We report that individual vibrational modes play a minor role in promoting charge separation, contrary to the discussion in recent publications. Nevertheless, these small contributions accumulate to considerably influence the charge separation rate, resulting in subpicosecond charge separation almost independent of the driving force and temperature. We suggest that the intramolecular vibrations complement the robustness of the charge separation in the photosystem II reaction center against the inherently large static disorder of the involved electronic energies.

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.

Theoretical approaches for dynamical ordering of biomolecular systems

BACKGROUND

Living systems are characterized by the dynamic assembly and disassembly of biomolecules. The dynamical ordering mechanism of these biomolecules has been investigated both experimentally and theoretically. The main theoretical approaches include quantum mechanical (QM) calculation, all-atom (AA) modeling, and coarse-grained (CG) modeling. The selected approach depends on the size of the target system (which differs among electrons, atoms, molecules, and molecular assemblies). These hierarchal approaches can be combined with molecular dynamics (MD) simulation and/or integral equation theories for liquids, which cover all size hierarchies.

SCOPE OF REVIEW

We review the framework of quantum mechanical/molecular mechanical (QM/MM) calculations, AA MD simulations, CG modeling, and integral equation theories. Applications of these methods to the dynamical ordering of biomolecular systems are also exemplified.

MAJOR CONCLUSIONS

The QM/MM calculation enables the study of chemical reactions. The AA MD simulation, which omits the QM calculation, can follow longer time-scale phenomena. By reducing the number of degrees of freedom and the computational cost, CG modeling can follow much longer time-scale phenomena than AA modeling. Integral equation theories for liquids elucidate the liquid structure, for example, whether the liquid follows a radial distribution function.

GENERAL SIGNIFICANCE

These theoretical approaches can analyze the dynamic behaviors of biomolecular systems. They also provide useful tools for exploring the dynamic ordering systems of biomolecules, such as self-assembly. This article is part of a Special Issue entitled "Biophysical Exploration of Dynamical Ordering of Biomolecular Systems" edited by Dr. Koichi Kato.

Probing the excitonic landscape of the Chlorobaculum tepidum Fenna-Matthews-Olson (FMO) complex: a mutagenesis approach

In this paper we report the steady-state optical properties of a series of site-directed mutants in the Fenna-Matthews-Olson (FMO) complex of Chlorobaculum tepidum, a photosynthetic green sulfur bacterium. The FMO antenna complex has historically been used as a model system for energy transfer due to the water-soluble nature of the protein, its stability at room temperature, as well as the availability of high-resolution structural data. Eight FMO mutants were constructed with changes in the environment of each of the bacteriochlorophyll a pigments found within each monomer of the homotrimeric FMO complex. Our results reveal multiple changes in low temperature absorption, as well as room temperature CD in each mutant compared to the wild-type FMO complex. These datasets were subsequently used to model the site energies of each pigment in the FMO complex by employing three different Hamiltonians from the literature. This enabled a basic approximation of the site energy shifts imparted on each pigment by the changed amino acid residue. These simulations suggest that, while the three Hamiltonians used in this work provide good fits to the wild-type FMO absorption spectrum, further efforts are required to obtain good fits to the mutant minus wild-type absorption difference spectra. This demonstrates that the use of FMO mutants can be a valuable tool to refine and iterate the current models of energy transfer in this system.

Modeling Electronic-Nuclear Interactions for Excitation Energy Transfer Processes in Light-Harvesting Complexes

An accurate approach for computing intermolecular and intrachromophore contributions to spectral densities to describe the electronic-nuclear interactions relevant for modeling excitation energy transfer processes in light harvesting systems is presented. The approach is based on molecular dynamics (MD) calculations of classical correlation functions of long-range contributions to excitation energy fluctuations and a separate harmonic analysis and single-point gradient quantum calculations for electron-intrachromophore vibrational couplings. A simple model is also presented that enables detailed analysis of the shortcomings of standard MD-based excitation energy fluctuation correlation function approaches. The method introduced here avoids these problems, and its reliability is demonstrated in accurate predictions for bacteriochlorophyll molecules in the Fenna-Matthews-Olson pigment-protein complex, where excellent agreement with experimental spectral densities is found. This efficient approach can provide instantaneous spectral densities for treating the influence of fluctuations in environmental dissipation on fast electronic relaxation.