Fragment Quantum Mechanical Method for Excited States of Proteins: Development and Application to the Green Fluorescent Protein

Excitation configuration analysis for divide-and-conquer excited-state calculation method using dynamical polarizability

The authors previously developed a divide-and-conquer (DC)-based non-local excited-state calculation method for large systems using dynamical polarizability [Nakai and Yoshikawa, J. Chem. Phys. 146, 124123 (2017)]. This method evaluates the excitation energies and oscillator strengths using information on the dynamical polarizability poles. This article proposes a novel analysis of the previously developed method to obtain further configuration information on excited states, including excitation and de-excitation coefficients of each excitation configuration. Numerical applications to simple molecules, such as ethylene, hydrogen molecule, ammonia, and pyridazine, confirmed that the proposed analysis could accurately reproduce the excitation and de-excitation coefficients. The combination with the DC scheme enables both the local and non-local excited states of large systems with an excited nature to be treated.

Analysis of Site Energies and Excitonic Couplings: The Role of Symmetry and Polarization

An excitonic coupling model is developed based on an equation-of-motion coupled cluster combined with the fragment molecular orbital method. The effects of polarization and excitonic coupling on the splitting of quasi-degenerate levels in systems containing multiple chromophores are elucidated on dimers of formaldehyde, water, formic acid, hydrogen fluoride, and carbon monoxide. It is shown that the level structure is mainly determined by the mutual polarization of chromophores and to a lesser extent by the excitonic coupling. The role of symmetry in excitonic coupling in dimers is discussed. The excitonic coupling between all residues in the photoactive yellow protein (PDB: 2PHY) is analyzed.

Excited-state dynamics and fluorescence lifetime of cryogenically cooled green fluorescent protein chromophore anions

Time-resolved action spectroscopy together with a fs-pump probe scheme is used in an electrostatic ion-storage ring to address lifetimes of specific vibrational levels in electronically excited states. Here we specifically consider the excited-state lifetime of cryogenically cooled green fluorescent protein (GFP) chromophore anions which is systematically measured across the S0-S1 spectral region (450-482 nm). A long lifetime of 5.2 ± 0.3 ns is measured at the S0-S1 band origin. When exciting higher vibrational levels in S1, the lifetime changes dramatically. It decreases by more than two orders of magnitude in a narrow energy region ∼250 cm-1 (31 meV) above the 0-0 transition. This is attributed to the opening of internal conversion over an excited-state energy barrier. The applied experimental technique provides a new way to uncover even small energy barriers, which are crucial for excited-state dynamics.

Deciphering Interactions between Potential Inhibitors and the Plasmodium falciparum DHODH Enzyme: A Computational Perspective

Malaria is a parasitic disease that, in its most severe form, can even lead to death. Insect-resistant vectors, insufficiently effective vaccines, and drugs that cannot stop parasitic infestations are making the fight against the disease increasingly difficult. It is known that the enzyme dihydroorotate dehydrogenase (DHODH) is of paramount importance for the synthesis of pyrimidine from the Plasmodium precursor, that is, for its growth and reproduction. Therefore, its blockade can lead to disruption of the parasite's life cycle in the vertebrate host. In this scenario, PfDHODH inhibitors have been considered candidates for a new therapy to stop the parasitic energy source. Given what is known, in this work, we applied molecular fractionation with conjugated caps (MFCC) in the framework of the quantum formalism of density functional theory (DFT) to evaluate the energies of the interactions between the enzyme and the different triazolopyrimidines (DSM483, DMS557, and DSM1), including a complex carrying the mutation C276F. From these results, it was possible to identify the main features of each system, focusing on the wild-type and mutant PfDHODH and examining the major amino acid residues that are part of the four complexes. Our analysis provides new information that can be used to develop new drugs that could prove to be more effective alternatives to present antimalarial drugs.

Site-Specific Ionization Potentials and Electron Affinities in Large Molecular Systems at Coupled Cluster Level

A many-body expansion of ionization potentials and electron affinities is developed based on a combination of the fragment molecular orbital method and equation-of-motion coupled-cluster (EOM-CC). In addition to site-specific values, obtained as one-body properties, pair and triple corrections are added to account for nonlocal EOM-CC contributions of the molecular environment of a chromophore. The developed method is applied to carboxylic acids, alkyl cations, a protein ubiquitin (Protein Data Bank ID 1UBQ), and a nano ribbon of white graphene elucidating the effect of environment on ionization potential and electron affinity.

Fast and Accurate Calculation of the UV-Vis Spectrum with the Modified Local Excitation Approximation

The local excitation approximation (LEA), a method for the calculation of electronic excitations localized in a specific region of a molecule, has been modified with new approaches to enhance the accuracy of the original method. The primary concept behind LEA involves isolating the region of interest as a submolecule from the full molecule using a localization method, followed by calculating electronic excitations solely within this submolecule. In this study, we examined approaches that improve the accuracy in describing the region of interest, particularly its molecular orbital energies. Additionally, the localization method was extended with a new projection technique to accelerate calculations. These approaches were studied in time-dependent density functional theory (TDDFT) calculations applied to four testing systems with a chromophore as the region of interest: two basic linear molecules, acrolein surrounded by 24 water molecules, and a model of a green fluorescent protein. For all studied systems, the results of TDDFT calculations combined with LEA exhibited near-zero error when groups of atoms adjacent to the chromophore were explicitly included in the submolecule. This was achieved with at least a quadratic speedup of the calculation time as a function of the submolecule size.

Machine-learning assisted scheduling optimization and its application in quantum chemical calculations

Easy and effective usage of computational resources is crucial for scientific calculations, both from the perspectives of timeliness and economic efficiency. This work proposes a bi-level optimization framework to optimize the computational sequences. Machine-learning (ML) assisted static load-balancing, and different dynamic load-balancing algorithms can be integrated. Consequently, the computational and scheduling engine of the ParaEngine is developed to invoke optimized quantum chemical (QC) calculations. Illustrated benchmark calculations include high-throughput drug suit, solvent model, P38 protein, and SARS-CoV-2 systems. The results show that the usage rate of given computational resources for high throughput and large-scale fragmentation QC calculations can primarily profit, and faster accomplishing computational tasks can be expected when employing high-performance computing (HPC) clusters.

Fragment-Based Calculations of Enzymatic Thermochemistry Require Dielectric Boundary Conditions

Electronic structure calculations on enzymes require hundreds of atoms to obtain converged results, but fragment-based approximations offer a cost-effective solution. We present calculations on enzyme models containing 500-600 atoms using the many-body expansion, comparing to benchmarks in which the entire enzyme-substrate complex is described at the same level of density functional theory. When the amino acid fragments contain ionic side chains, the many-body expansion oscillates under vacuum boundary conditions but rapid convergence is restored using low-dielectric boundary conditions. This implies that full-system calculations in the gas phase are inappropriate benchmarks for assessing errors in fragment-based approximations. A three-body protocol retains sub-kilocalorie per mole fidelity with respect to a supersystem calculation, as does a two-body calculation combined with a full-system correction at a low-cost level of theory. These protocols pave the way for application of high-level quantum chemistry to large systems via rigorous, ab initio treatment of many-body polarization.

Analytic Gradient for Time-Dependent Density Functional Theory Combined with the Fragment Molecular Orbital Method

The analytic energy gradient of energy with respect to nuclear coordinates is derived for the fragment molecular orbital (FMO) method combined with time-dependent density functional theory (TDDFT). The response terms arising from the use of a polarizable embedding are derived. The obtained analytic FMO-TDDFT gradient is shown to be accurate in comparison to both numerical FMO-TDDFT and unfragmented TDDFT gradients, at the level of two- and three-body expansions. The gradients are used for geometry optimizations, molecular dynamics, vibrational calculations, and simulations of IR and Raman spectra of excited states. The developed method is used to optimize the geometry of the ground and excited electronic states of the photoactive yellow protein (PDB: 2PHY).

Extending multi-layer energy-based fragment method for excited-state calculations of large covalently bonded fragment systems

Recently, we developed a low-scaling Multi-Layer Energy-Based Fragment (MLEBF) method for accurate excited-state calculations and nonadiabatic dynamics simulations of nonbonded fragment systems. In this work, we extend the MLEBF method to treat covalently bonded fragment ones. The main idea is cutting a target system into many fragments according to chemical properties. Fragments with dangling bonds are first saturated by chemical groups; then, saturated fragments, together with the original fragments without dangling bonds, are grouped into different layers. The accurate total energy expression is formulated with the many-body energy expansion theory, in combination with the inclusion-exclusion principle that is used to delete the contribution of chemical groups introduced to saturate dangling bonds. Specifically, in a two-layer MLEBF model, the photochemically active and inert layers are calculated with high-level and efficient electronic structure methods, respectively. Intralayer and interlayer energies can be truncated at the two- or three-body interaction level. Subsequently, through several systems, including neutral and charged covalently bonded fragment systems, we demonstrate that MLEBF can provide accurate ground- and excited-state energies and gradients. Finally, we realize the structure, conical intersection, and path optimizations by combining our MLEBF program with commercial and free packages, e.g., ASE and SciPy. These developments make MLEBF a practical and reliable tool for studying complex photochemical and photophysical processes of large nonbonded and bonded fragment systems.

Generalized Energy-Based Fragmentation Approach for the Electronic Emission Spectra of Large Systems

The excited-state (ES) geometry optimization and electronic emission (fluorescence and phosphorescence) spectra and the ES vibrational spectra of large systems are great challenges in quantum chemistry. In this work, we develop a generalized energy-based fragmentation (GEBF) approach to compute the localized ES structures and vibrational frequencies of large systems. In this approach, the ES energy derivatives (gradients or Hessians) for a localized ES of a large system can be obtained by combining the ES energy derivatives of the corresponding active subsystems (including local excitation center) and the ground-state energy derivatives of inactive subsystems. Two strategies are adopted to overcome two difficulties from state-classification and state-tracking for treating specific ESs. First, for state-classification, we develop an improved density-based spatial clustering applied with noise algorithm with a modified transition orbital projection (TOP) algorithm, which allow a certain ES energy and energy derivatives of the whole system to be calculated with different ES energies and energy derivatives of active subsystems. Furthermore, we also employ the TOP algorithm for tracking the ESs in their geometry optimizations at the time-dependent density functional theory (TDDFT) level. Then, the GEBF approach is applied to investigate the optimized ES geometries or ES vibrational frequencies for two typical systems. Our results show that the cost-effective GEBF approach can accurately reproduce the TDDFT fluorescence spectra of the cytosine derivative and the experimental phosphorescence spectra of the β-cyclodextrin derivative. The GEBF approach is expected to be routinely applied to investigate the electronic emission spectra of very large systems with local chromophores.

Conical Intersections in Solution with Polarizable Embedding: Integral-Exact Direct Reaction Field

A common strategy to exploring the properties and reactivity of complex systems is to use quantum mechanics/molecular mechanics (QM/MM) embedding, wherein a QM region is defined and treated with electronic structure theory, and the remainder of the system is treated with a force field. Important to the description of electronic excited states, especially those of charge-transfer character, is the treatment of the coupling between the QM and MM subsystems. The state of the art is to use a polarizable force field for the MM region and mutually couple the QM wavefunction and MM induced dipoles, in addition to the usual electrostatic embedding, yielding a polarizable embedding (QM/MM-Pol) approach. However, we showed previously that current popular QM/MM-Pol approaches exhibit issues of root flipping and/or incorrect descriptions of electronic crossings in multistate calculations [J. Chem. Theory Comput. 14, 2137 (2018)]. Here, we demonstrate a solution to these problems with an integral-exact reformulation of the direct reaction field approach of Thole and Van Duijnen (QM/MM-IEDRF). The resulting embedding potential includes one- and two-electron operators and many-body dipole-induced dipole interactions and thus includes a natural description of the screening of electron-electron interactions by the MM induced dipoles. Pauli repulsion from the environment is mimicked by effective core potentials on the MM atoms. Inherent to the DRF approach is the assumption that MM dipoles respond instantaneously to the positions of the QM electrons; therefore, dispersion interactions are captured approximately. All electronic states are eigenfunctions of the same Hamiltonian, while the polarization induced in the environment and the associated energetic stabilization are unique to each state. This allows for a consistent definition of transition properties and state crossings. We demonstrate QM/MM-IEDRF by exploring the influence of a (polarizable) inert xenon matrix environment on the conical intersection underlying the photoisomerization of ethylene.

Transition orbital projection approach for excited state tracking

Quantitively comparing the features between different electronic excited states (ESs) is a crucial task in both potential energy surface (PES) studies and excited-state fragmentation approaches. However, it is still a challenging problem in regard to the comparison of complex and highly degenerate systems. Herein, we present a transition orbital projection (TOP) method to calculate the similarity of different ESs based on the configuration vectors of two types of transition densities. It fully considers four significant problems, including phase, hole-particle bijectivity, orbital permutation, and sign of configuration coefficients. TOP state-tracking-based excited-state optimization shows high robustness in several high-symmetric systems, which are difficult to describe with traditional state-tracking approaches. The TOP state-tracking method is expected to be widely applied to the PES of photochemical reactions, ES molecular dynamics to track the diabatic states, and fragmentation approaches for local excitation of large systems.

Low-Scaling Excited State Calculation Using the Block Interaction Product State

We develop an automatic and efficient scheme for the accurate construction of the bases for excitonic models, which can enable "black-box" excited state structure calculations for large molecular systems. These new and optimized bases, which are named the block interaction product state (BIPS), can be expressed as the direct products of the local states for each chromophore. Each chromophore's local states are selected by diagonalization of its reduced density matrix, which is obtained by quantum chemical calculation of the small subsystem composed of the chromophore and its nearest neighbors. We implemented the BIPS framework with fragment-based calculations considering two- and three-body interactions. Test calculations for eight different molecular aggregates demonstrate that this framework provides an accurate description of not only the excitation energies but also the first-order wave function properties (dipole moment and transition dipole moment) of the low-lying excited states at a low-scaling computational cost.

Fragment-Based Quantum Mechanical Calculation of Excited-State Properties of Fluorescent RNAs

Fluorescent RNA aptamers have been successfully applied to track and tag RNA in a biological system. However, it is still challenging to predict the excited-state properties of the RNA aptamer–fluorophore complex with the traditional electronic structure methods due to expensive computational costs. In this study, an accurate and efficient fragmentation quantum mechanical (QM) approach of the electrostatically embedded generalized molecular fractionation with conjugate caps (EE-GMFCC) scheme was applied for calculations of excited-state properties of the RNA aptamer–fluorophore complex. In this method, the excited-state properties were first calculated with one-body fragment quantum mechanics/molecular mechanics (QM/MM) calculation (the excited-state properties of the fluorophore) and then corrected with a series of two-body fragment QM calculations for accounting for the QM effects from the RNA on the excited-state properties of the fluorophore. The performance of the EE-GMFCC on prediction of the absolute excitation energies, the corresponding transition electric dipole moment (TEDM), and atomic forces at both the TD-HF and TD-DFT levels was tested using the Mango-II RNA aptamer system as a model system. The results demonstrate that the calculated excited-state properties by EE-GMFCC are in excellent agreement with the traditional full-system time-dependent ab initio calculations. Moreover, the EE-GMFCC method is capable of providing an accurate prediction of the relative conformational excited-state energies for different configurations of the Mango-II RNA aptamer system extracted from the molecular dynamics (MD) simulations. The fragmentation method further provides a straightforward approach to decompose the excitation energy contribution per ribonucleotide around the fluorophore and then reveals the influence of the local chemical environment on the fluorophore. The applications of EE-GMFCC in calculations of excitation energies for other RNA aptamer–fluorophore complexes demonstrate that the EE-GMFCC method is a general approach for accurate and efficient calculations of excited-state properties of fluorescent RNAs.

Computational and data driven molecular material design assisted by low scaling quantum mechanics calculations and machine learning

Electronic structure methods based on quantum mechanics (QM) are widely employed in the computational predictions of the molecular properties and optoelectronic properties of molecular materials. The computational costs of these QM methods, ranging from density functional theory (DFT) or time-dependent DFT (TDDFT) to wave-function theory (WFT), usually increase sharply with the system size, causing the curse of dimensionality and hindering the QM calculations for large sized systems such as long polymer oligomers and complex molecular aggregates. In such cases, in recent years low scaling QM methods and machine learning (ML) techniques have been adopted to reduce the computational costs and thus assist computational and data driven molecular material design. In this review, we illustrated low scaling ground-state and excited-state QM approaches and their applications to long oligomers, self-assembled supramolecular complexes, stimuli-responsive materials, mechanically interlocked molecules, and excited state processes in molecular aggregates. Variable electrostatic parameters were also introduced in the modified force fields with the polarization model. On the basis of QM computational or experimental datasets, several ML algorithms, including explainable models, deep learning, and on-line learning methods, have been employed to predict the molecular energies, forces, electronic structure properties, and optical or electrical properties of materials. It can be conceived that low scaling algorithms with periodic boundary conditions are expected to be further applicable to functional materials, perhaps in combination with machine learning to fast predict the lattice energy, crystal structures, and spectroscopic properties of periodic functional materials.

Protein Response Effects on Cofactor Excitation Energies from First Principles: Augmenting Subsystem Time-Dependent Density-Functional Theory with Many-Body Expansion Techniques

We investigate the possibility of describing protein response effects on a chromophore excitation by means of subsystem time-dependent density-functional theory (sTDDFT) in combination with a many-body expansion (MBE) approach. While sTDDFT is in principle intrinsically able to include such contributions, addressing cofactor excitations in protein models or entire proteins with full environment-response treatments is currently out of reach. Taking different model structures of the green fluorescent protein (GFP) and bovine rhodopsin as examples, we demonstrate that an embedded-MBE approach based on sTDDFT in its simplest version leads to a good agreement of the predicted protein response effect already at second order. To reproduce reference response effects from nonsubsystem TDDFT calculations quantitatively (error ≤ 5%), however, a third- or even fourth-order MBE may be required. For the latter case, we explore a selective inclusion of fourth-order terms that drastically reduces the computational burden. In addition, we demonstrate how this sTDDFT-MBE treatment can be utilized as an analysis tool to identify residues with dominant response contributions. This, in turn, can be employed to arrive at smaller structural models for light-absorbing proteins, which still feature all of the main characteristics in terms of photoresponse properties.

Ab initio multi-level layered elongation method and its application to local interaction analysis between DNA bulge and ligand molecules

A multi-level layered elongation method was developed for efficiently analyzing the electronic states of local structures in large bio/nano-systems at the full ab initio level of theory. The original elongation method developed during the last three decades in our group has focused on the system in one direction from one terminal to the other terminal to sequentially construct the electronic states of a polymer, called a theoretical synthesis of polymers. In this study, an important region termed the central (C) part is targeted in a large polymer and the remainder are terminal (T) parts. The electronic structures along with polymer elongation are calculated repeatedly from both end T parts to the C central part at the same time. The important C part is treated with large basis sets (high level) and the other regions are treated with small basis sets (low level) in the ab initio theoretical framework. The electronic structures besides the C part can be reused for other systems with different structures at the C part, which renders the method computationally efficient. This multi-level layered elongation method was applied to the investigation on DNA single bulge recognition of small molecules (ligands). The reliability and validity of our approach were examined in comparison with the results obtained by direct calculations using a conventional quantum chemical method for the entire system. Furthermore, stabilization energies by the formation of the complex of bulge DNA and a ligand were estimated with basis set superposition error corrections incorporated into the elongation method.

Active orbital preservation for multiconfigurational self-consistent field

We introduce Active Orbital Preservation for Multiconfigurational Self-Consistent Field (AOP-MCSCF), an automated approach to improving the consistency of active space orbitals over multiple molecular configurations. Our approach is based on maximum overlap with a reference set of active space orbitals taken from a single geometry of a chromophore in the gas phase and can be used to automatically preserve the appropriate orbitals of the chromophore across multiple thermally sampled configurations, even when the chromophore is solvated by quantum-mechanically treated water molecules. In particular, using the singular value decomposition of a Molecular Orbital (MO) overlap matrix between the system and reference, we rotate the MOs of the system to align with the reference active space orbitals and use the resulting rotated orbitals as an initial guess to a MCSCF calculation. We demonstrate the approach on aqueous p-hydroxybenzylidene-imidazolinone (HBI) and find that AOP-MCSCF converges to the "correct" orbitals for over 90% of 3000 thermally sampled configurations. In addition, we compute the linear absorption spectrum and find excellent agreement with new experimental measurements up to 5.4 eV (230 nm). We show that electrostatic contributions to the solvation energy of HBI largely explain the observed state-dependent solvatochromism.

Advances, challenges and perspectives of quantum chemical approaches in molecular spectroscopy of the condensed phase

The purpose of this review is to demonstrate advances, challenges and perspectives of quantum chemical approaches in molecular spectroscopy of the condensed phase. Molecular spectroscopy, particularly vibrational spectroscopy and electronic spectroscopy, has been used extensively for a wide range of areas of chemical sciences and materials science as well as nano- and biosciences because it provides valuable information about structure, functions, and reactions of molecules. In the meantime, quantum chemical approaches play crucial roles in the spectral analysis. They also yield important knowledge about molecular and electronic structures as well as electronic transitions. The combination of spectroscopic approaches and quantum chemical calculations is a powerful tool for science, in general. Thus, our article, which treats various spectroscopy and quantum chemical approaches, should have strong implications in the wider scientific community. This review covers a wide area of molecular spectroscopy from far-ultraviolet (FUV, 120-200 nm) to far-infrared (FIR, 400-10 cm-1)/terahertz and Raman spectroscopy. As quantum chemical approaches, we introduce several anharmonic approaches such as vibrational self-consistent field (VSCF) and the combination of periodic harmonic calculations with anharmonic corrections based on finite models, grid-based techniques like the Numerov approach, the Cartesian coordinate tensor transfer (CCT) method, Symmetry-Adapted Cluster Configuration-Interaction (SAC-CI), and the ZINDO (Semi-empirical calculations at Zerner's Intermediate Neglect of Differential Overlap). One can use anharmonic approaches and grid-based approaches for both infrared (IR) and near-infrared (NIR) spectroscopy, while CCT methods are employed for Raman, Raman optical activity (ROA), FIR/terahertz and low-frequency Raman spectroscopy. Therefore, this review overviews cross relations between molecular spectroscopy and quantum chemical approaches, and provides various kinds of close-reality advanced spectral simulation for condensed phases.

Cluster many-body expansion: A many-body expansion of the electron correlation energy about a cluster mean field reference

The many-body expansion (MBE) is an efficient tool that has a long history of use for calculating interaction energies, binding energies, lattice energies, and so on. In the past, applications of MBE to correlation energy have been unfeasible for large systems, but recent improvements to computing resources have sparked renewed interest in capturing the correlation energy using the generalized nth order Bethe-Goldstone equation. In this work, we extend this approach, originally proposed for a Slater determinant, to a tensor product state (TPS) based wavefunction. By partitioning the active space into smaller orbital clusters, our approach starts from a cluster mean field reference TPS configuration and includes the correlation contribution of the excited TPSs using the MBE. This method, named cluster MBE (cMBE), improves the convergence of MBE at lower orders compared to directly doing a block-based MBE from a RHF reference. We present numerical results for strongly correlated systems, such as the one- and two-dimensional Hubbard models and the chromium dimer. The performance of the cMBE method is also tested by partitioning the extended π space of several large π-conjugated systems, including a graphene nano-sheet with a very large active space of 114 electrons in 114 orbitals, which would require 10⁶⁶ determinants for the exact FCI solution.

Accurate Prediction of Absorption Spectral Shifts of Proteorhodopsin Using a Fragment-Based Quantum Mechanical Method

Many experiments have been carried out to display different colors of Proteorhodopsin (PR) and its mutants, but the mechanism of color tuning of PR was not fully elucidated. In this study, we applied the Electrostatically Embedded Generalized Molecular Fractionation with Conjugate Caps (EE-GMFCC) method to the prediction of excitation energies of PRs. Excitation energies of 10 variants of Blue Proteorhodopsin (BPR-PR105Q) in residue 105GLN were calculated with the EE-GMFCC method at the TD-B3LYP/6-31G* level. The calculated results show good correlation with the experimental values of absorption wavelengths, although the experimental wavelength range among these systems is less than 50 nm. The ensemble-averaged electric fields along the polyene chain of retinal correlated well with EE-GMFCC calculated excitation energies for these 10 PRs, suggesting that electrostatic interactions from nearby residues are responsible for the color tuning. We also utilized the GMFCC method to decompose the excitation energy contribution per residue surrounding the chromophore. Our results show that residues ASP97 and ASP227 have the largest contribution to the absorption spectral shift of PR among the nearby residues of retinal. This work demonstrates that the EE-GMFCC method can be applied to accurately predict the absorption spectral shifts for biomacromolecules.