Towards the Realization of Ab Initio Dynamics at the Speed of Molecular Mechanics: Simulations with Interpolated Diabatic Hamiltonian

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.

LVC/MM: A Hybrid Linear Vibronic Coupling/Molecular Mechanics Model with Distributed Multipole-Based Electrostatic Embedding for Highly Efficient Surface Hopping Dynamics in Solution

We present a theoretical framework for a hybrid linear vibronic coupling model electrostatically embedded into a molecular mechanics environment, termed the linear vibronic coupling/molecular mechanics (LVC/MM) method, for the surface hopping including arbitrary coupling (SHARC) molecular dynamics package. Electrostatic embedding is realized through the computation of interactions between environment point charges and distributed multipole expansions (DMEs, up to quadrupoles) that represent each electronic state and transition densities in the diabatic basis. The DME parameters are obtained through a restrained electrostatic potential (RESP) fit, which we extended to yield higher-order multipoles. We also implemented in SHARC a scheme for achieving roto-translational invariance of LVC models as well as a general quantum mechanics/molecular mechanics (QM/MM) interface, an OpenMM interface, and restraining potentials for simulating liquid droplets. Using thioformaldehyde in water as a test case, we demonstrate that LVC/MM can accurately reproduce the solvation structure and energetics of rigid solutes, with errors on the order of 1-2 kcal/mol compared to a BP86/MM reference. The implementation in SHARC is shown to be very efficient, enabling the simulation of trajectories on the nanosecond time scale in a matter of days.

Prediction of Fluorophore Brightness in Designed Mini Fluorescence Activating Proteins

The de novo computational design of proteins with predefined three-dimensional structure is becoming much more routine due to advancements both in force fields and algorithms. However, creating designs with functions beyond folding is more challenging. In that regard, the recent design of small beta barrel proteins that activate the fluorescence of an exogenous small molecule chromophore (DFHBI) is noteworthy. These proteins, termed mini fluorescence activating proteins (mFAPs), have been shown to increase the brightness of the chromophore more than 100-fold upon binding to the designed ligand pocket. The design process created a large library of variants with different brightness levels but gave no rational explanation for why one variant was brighter than another. Here, we use quantum mechanics and molecular dynamics simulations to investigate how molecular flexibility in the ground and excited states influences brightness. We show that the ability of the protein to resist dihedral angle rotation of the chromophore is critical for predicting brightness. Our simulations suggest that the mFAP/DFHBI complex has a rough energy landscape, requiring extensive ground-state sampling to achieve converged predictions of excited-state kinetics. While computationally demanding, this roughness suggests that mFAP protein function can be enhanced by reshaping the energy landscape toward conformations that better resist DFHBI bond rotation.

Surface Hopping Dynamics on Vibronic Coupling Models

The simulation of photoinduced non-adiabatic dynamics is of great relevance in many scientific disciplines, ranging from physics and materials science to chemistry and biology. Upon light irradiation, different relaxation processes take place in which electronic and nuclear motion are intimately coupled. These are best described by the time-dependent molecular Schrödinger equation, but its solution poses fundamental practical challenges to contemporary theoretical chemistry. Two widely used and complementary approaches to this problem are multiconfigurational time-dependent Hartree (MCTDH) and trajectory surface hopping (SH). MCTDH is an accurate fully quantum-mechanical technique but often is feasible only in reduced dimensionality, in combination with approximate vibronic coupling (VC) Hamiltonians, or both (i.e., reduced-dimensional VC potentials). In contrast, SH is a quantum–classical technique that neglects most nuclear quantum effects but allows nuclear dynamics in full dimensionality by calculating potential energy surfaces on the fly. If nuclear quantum effects do not play a central role and a linear VC (LVC) Hamiltonian is appropriate—e.g., for stiff molecules that generally keep their conformation in the excited state—then it seems advantageous to combine the efficient LVC and SH techniques. In this Account, we describe how surface hopping based on an LVC Hamiltonian (SH/LVC)—as recently implemented in the SHARC surface hopping package—can provide an economical and automated approach to simulate non-adiabatic dynamics. First, we illustrate the potential of SH/LVC in a number of showcases, including intersystem crossing in SO₂, intra-Rydberg dynamics in acetone, and several photophysical studies on large transition-metal complexes, which would be much more demanding or impossible to perform with other methods. While all of the applications provide very useful insights into light-induced phenomena, they also hint at difficulties faced by the SH/LVC methodology that need to be addressed in the future. Second, we contend that the SH/LVC approach can be useful to benchmark SH itself. By the use of the same (LVC) potentials as MCTDH calculations have employed for decades and by relying on the efficiency of SH/LVC, it is possible to directly compare multiple SH test calculations with a MCTDH reference and ponder the accuracy of various correction algorithms behind the SH methodology, such as decoherence corrections or momentum rescaling schemes. Third, we demonstrate how the efficiency of SH/LVC can also be exploited to identify essential nuclear and electronic degrees of freedom to be employed in more accurate MCTDH calculations. Lastly, we show that SH/LVC is able to advance the development of SH protocols that can describe nuclear dynamics including explicit laser fields—a very challenging endeavor for trajectory-based schemes. To end, this Account compiles the typical costs of contemporary SH simulations, evidencing the great advantages of using parametrized potentials. The LVC model is a sleeping beauty that, kissed by SH, is fueling the field of excited-state molecular dynamics. We hope that this Account will stimulate future research in this direction, leveraging the advantages of the SH/VC schemes to larger extents and extending their applicability to uncharted territories.

Efficiently Transplanting Potential Energy Interpolation Database between Two Systems: Bacteriochlorophyll Case with FMO and LH2 Complexes

Generating a reliable potential energy surface (PES) is an important issue for studying the dynamics of any system with computational simulations. Interpolation mechanics/molecular mechanics (IM/MM) based on a PES interpolation scheme is a useful tool in that regard as it provides an accuracy of a quantum chemistry (QC) level while maintaining its computational cost comparable to conventional MM force fields. Despite this benefit, constructing the database for interpolation itself is still challenging and time-consuming. Here, we present a method with which we can construct the IM database of one system based on a preexisting data set for another related system. We adopt the case of constructing bacteriochlorophyll PESs for the light-harvesting 2 (LH2) complex by utilizing already available IM database for the BChls from the Fenna-Matthews-Olson (FMO) complex. In this method, the IM database from FMO is first transplanted to LH2 by considering BChl displacement vectors that take into account the geometry differences induced by the protein scaffolds. From this transplanted primitive database entries, a relatively small number of effective ones are selected by a survival process based on a genetic algorithm such that the IM energies evaluated at geometries in a conveniently collected prediction set can closely match with the reference QC energies. The selection process is expedited by using two different levels of basis sets for the QC calculations. To demonstrate the utility of the PES thus constructed, we carry out 1 ns of IM/MM dynamics simulations with the finally optimized BChl database for LH2. Indeed, the energy profiles of the snapshots are found to be closely matching with the reference QC calculation data, with only ∼0.07 eV of errors in the ground- and excited-state energies and ∼0.008 eV of errors in the transition energies. We also show that properly selecting data points is actually quite important for generating an IM PES toward performing molecular dynamics simulations.

Modeling Charge Flux by Interpolating Atomic Partial Charges

Although the charge flux effect or the geometric dependence of the atomic partial charges have been known for a long time, how it can be effectively handled is not yet established. Here, we present a charge interpolation scheme as a new general tool for representing the charge flux in an analytically well-defined manner. By applying it to the anionic GFP chromophore with the diabatically represented atomic charges, we show that the charge interpolation provides a substantial improvement on the accuracy of the geometry-dependent changes in the molecular dipole moments in the gas phase. We also test the scheme toward describing the electrostatic term in the solvation energy in the aqueous environment and observe that it is also improved but that the extent of the improvement is somewhat limited. We show that the remaining errors can be largely corrected by introducing atomic polarizabilities. Overall, our results show that charge interpolation is an amenable approach for describing the charge flux effect and that its description in the condensed phase should be accompanied by proper treatments of polarization effects.

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.

Constructing an Interpolated Potential Energy Surface of a Large Molecule: A Case Study with Bacteriochlorophyll a Model in the Fenna-Matthews-Olson Complex

Constructing a reliable potential energy surface (PES) is a key step toward computationally studying the chemical dynamics of any molecular system. The interpolation scheme is a useful tool that can closely follow the accuracy of quantum chemical means at a dramatically reduced computational cost. However, applying interpolation to building a PES of a large molecule is not a straightforward black-box approach, as it frequently encounters practical difficulties associated with its large dimensionality. Here, we present detailed courses of applying interpolation toward building a PES of a large chromophore molecule. We take the example of S₀ and S₁ electronic states of bacteriochlorophyll a (BChla) molecules in the Fenna-Matthews-Olson light harvesting complex. With a reduced model molecule that bears BChla's main π-conjugated ring, various practical approaches are designed for improving the PES quality in a stable manner and for fine-tuning the final surface such that the surface can be adopted for long time molecular dynamics simulations. Combined with parallel implementation, we show that interpolated mechanics/molecular mechanics (IM/MM) simulations of the entire complex in the nanosecond time scale can be conducted readily without any practical issues. With 1500 interpolation data points for each chromophore unit, the PES error relative to the reference quantum chemical calculation is found to be ∼0.15 eV in the thermally accessible region of the conformational space, together with ∼0.01 eV error in S₀ - S₁ transition energies. The performance issue related to the use of a large interpolation database within the framework of our parallel routines is also discussed.

Electric Field Keeps Chromophore Planar and Produces High Yield Fluorescence in Green Fluorescent Protein

The green fluorescent protein and its designed variants fluoresce efficiently. Because the isolated chromophore is not fluorescent in a practical sense, it is apparent that the protein environment plays a crucial role in its efficiency. Because of various obstacles in studying excited state dynamics of complex systems, however, the detailed mechanism of this efficiency enhancement is not yet clearly elucidated. Here, by adopting excited state nonadiabatic molecular dynamics simulations together with an interpolated quantum chemical potential model of the chromophore, we find that the strong electric field from the protein matrix contributes dominantly to the motional restriction of the chromophore. The delay in twisting motion subsequently obstructs the nonradiative decay that competes with fluorescence, leading naturally to an enhancement in light-emitting efficiency. Surprisingly, steric constraints make only a minor contribution to these aspects. Through residue specific analyses, we identify a group of key residues that control the excited state behavior. Testing a series of mutant GFPs with different brightnesses also supports the view regarding the importance of protein electrostatics. Our findings may provide a useful guide toward designing new fluorescent chemical systems in the future.

Emission shaping in fluorescent proteins: role of electrostatics and π-stacking

We obtained the fluorescence spectrum of the GFP with trajectory simulations, and revealed the role of the protein sidechains in emission shifts.

Diabatic Population Matrix Formalism for Performing Molecular Mechanics Style Simulations with Multiple Electronic States

An accurate description of nonbonded interactions is important in investigating dynamics of molecular systems. In many situations, fixed point charge models are successfully applied to explaining various chemical phenomena. However, these models with conventional formulations will not be appropriate in elucidating the detailed dynamics during nonadiabatic events. This is mainly because the chemical properties of any molecule, especially its electronic populations, significantly change with respect to molecular distortions in the vicinity of the surface crossing. To overcome this issue in molecular simulations yet within the framework of the fixed point charge model, we define a diabatic electronic population matrix and substitute it for the conventional adiabatic partial charges. We show that this matrix can be readily utilized toward attaining more reliable descriptions of Coulombic interactions, in combination with the interpolation formalism for obtaining the intramolecular interaction potential. We demonstrate how the mixed formalism with the diabatic charges and the interpolation can be applied to molecular simulations by conducting adiabatic and nonadiabatic molecular dynamics trajectory calculations of the green fluorescent protein chromophore anion in aqueous environment.

Effect of Chromophore Potential Model on the Description of Exciton-Phonon Interactions

System-bath interactions in nonadiabatic simulations are often depicted by first performing molecular dynamics calculations and then by evaluating excitation energies at the trajectory snapshots. Usually, molecular mechanics models and quantum chemical calculations are used in a mixed manner toward a trade-off between efficiency and accuracy. Here we investigate how this mixing scheme affects that depiction by using various potential energy surfaces (PESs) of coumarin-153 chromophore, with the help of interpolated PESs that can closely match the accuracies of quantum chemical calculations. We find that although spectral densities are computed only with second stage data the PES characteristics during the first sampling stage can still prevail in the densities, with limited influences on related reorganization energies. Our results suggest that using the mixed scheme can be acceptable when dynamics is mainly governed by the integrated effect of all phonon modes, but care must be taken for understanding detailed effects from individual modes.