Quantum versus classical electron transfer energy as reaction coordinate for the aqueous Ru2+/Ru3+ redox reaction

Redox Properties of Flavin in BLUF and LOV Photoreceptor Proteins from Hybrid QM/MM Molecular Dynamics Simulation

Flavins play an important role in many oxidation and reduction processes in biological systems. For example, flavin adenine dinucleotide (FAD) and flavin mononucleotide (FMN) are common cofactors found in enzymatic proteins that use the special redox properties of these flavin molecules for their catalytic or photoactive functions. The redox potential of the flavin is strongly affected by its (protein) environment; however, the underlying molecular interactions of this effect are still unknown. Using hybrid quantum mechanics/molecular mechanics (QM/MM) simulation techniques, we have studied the redox properties of flavin in the gas phase, aqueous solution, and two different protein environments, in particular, a BLUF and a LOV photoreceptor domain. By mapping the changes in electrostatic potential and solvent structure, we gain insight into how specific polarization of the flavin by its environment tunes the reduction potential. We find also that accurate calculation of the reduction potentials of these systems by using the hybrid QM/MM approach is hampered by a too limited sampling of the counterion configurations and by artifacts at the QM/MM boundary. We make suggestions for how these issues can be overcome.

Electron transfer reaction of TEMPO-based organic radical batteries in different solvent environments: comparing quantum and classical approaches

In this study, we delve into the complex electron transfer reactions associated with the redox-active (2,2,6,6-tetramethylpiperidin-1-yl)oxyl (TEMPO), a common component in organic radical batteries (ORBs). Our approach estimates quantum electron-transfer (ET) energies using Density Functional Theory (DFT) calculations by sampling from structures simulated classically. This work presents a comparative study of reorganization energies in ET reactions across different solvents. Furthermore, we investigate how changes in the electrolyte environment can modify the reorganization energy and, consequently, impact ET dynamics. We also explore the relationship between classical and quantum vertical energies using linear regression models. Importantly, this comparison between quantum and classical vertical energies underscores the role of quantum effects, like charge delocalization, in offering added stabilization post-redox reactions. These effects are not adequately represented by the classical vertical energy distribution. Our study shows that, although we find a significant correlation between the vertical energies computed by DFT and the classical force field, the regression parameters depend on the solvent, highlighting that classical methods should be benchmarked by DFT before applying them to novel electrolyte materials.

Hybrid Functional and Plane Waves based Ab Initio Molecular Dynamics Study of the Aqueous Fe2+ /Fe3+ Redox Reaction

Kohn-Sham density functional theory and plane wave basis set based ab initio molecular dynamics (AIMD) simulation is a powerful tool for studying complex reactions in solutions, such as electron transfer (ET) reactions involving Fe²⁺ /Fe³⁺ ions in water. In most cases, such simulations are performed using density functionals at the level of Generalized Gradient Approximation (GGA). The challenge in modelling ET reactions is the poor quality of GGA functionals in predicting properties of such open-shell systems due to the inevitable self-interaction error (SIE). While hybrid functionals can minimize SIE, standard plane-wave based AIMD at that level of theory is typically 150 times slower than GGA for systems containing ∼100 atoms. Among several approaches reported to speed-up AIMD simulations with hybrid functionals, the noise-stabilized MD (NSMD) procedure, together with the use of localized orbitals to compute the required exchange integrals, is an attractive option. In this work, we demonstrate the application of the NSMD approach for studying the Fe²⁺ /Fe³⁺ redox reaction in water. It is shown here that long AIMD trajectories at the level of hybrid density functionals can be obtained using this approach. Redox properties of the aqueous Fe²⁺ /Fe³⁺ system computed from these simulations are compared with the available experimental data for validation.

Mixed Sulfur/Selenium Anions Weaken Electron-Vibrational Interaction in Cu2ZnSn(S,Se)4 Photoabsorber

Kesterite Cu₂ZnSn(S,Se)₄ (CZTSSe) solar absorbers have attracted intensive investigations for next-generation photovoltaic applications. Here, by using ab initio static and molecular dynamics simulations, we investigated the anion compositional dependence of electron-vibration interaction in CZTSSe materials. We found that the conduction band fluctuates more than the valence band, and as a result, the band gap variation is more sensitive to the change of the former, which can be understood in terms of p-d hybridization in the valence bands. Electron-phonon coupling is smaller in CZTSSe alloy compared to pure S- or Se-containing structures, as evidenced by the smaller fluctuation of excitation energy, and can be attributed to the weaker structural dynamics of the metal-anion bond. Small electron-phonon coupling strength may lead to better charge transport in these materials. We also elucidated the interplay between disordered structures and S/Se stoichiometry through analysis of optical line width. The results highlight the importance of anion composition engineering and provide new insights into the rational design of high-performing kesterite absorbers for solar cells.

Implementation and Validation of Constrained Density Functional Theory Forces in the CP2K Package

Constrained density functional theory (CDFT) is a powerful tool for the prediction of electron transfer parameters in condensed phase simulations at a reasonable computational cost. In this work we present an extension to CDFT in the popular mixed Gaussian/plane wave electronic structure package CP2K, implementing the additional force terms arising from a constraint based on Hirshfeld charge partitioning. This improves upon the existing Becke partitioning scheme, which is prone to give unphysical atomic charges. We verify this implementation for a variety of systems: electron transfer in (H₂O)₂⁺ in a vacuum, electron tunnelling between oxygen vacancy centers in solid MgO, and electron self-exchange in aqueous Ru²⁺–Ru³⁺. We find good agreement with previous plane-wave CDFT results for the same systems, but at a significantly lower computational cost, and we discuss the general reliability of condensed phase CDFT calculations.

Ewald sum corrections in simulations of ion and dipole solvation and electron transfer

Periodic boundary conditions and Ewald sums used in standard simulation protocols require finite-size corrections when the total charge of the simulated system is nonzero. Corrections for ion solvation were introduced by Hummer, Pratt, and García, [J. Chem. Phys. 107, 9275 (1997)]. The latter approach is extended here to derive finite-size correction for the Stokes-shift and reorganization energy applied to electron-transfer reactions. The same correction term, scaling inversely with the box size, adds to the reorganization energy from the energy-gap variance but is subtracted from the reorganization energy calculated from the Stokes shift. Finite-size corrections thus widen the gap between these two quantities, which were recently found to diverge for protein electron transfer. Corrections to the free energy of dipole solvation and the variance of the electric field scale as m²/L³ with the solute dipole m and the box size L.

The Influence of Spectator Cations on Solvent Reorganization Energy Is a Short-Range Effect

In this manuscript, we use classical molecular dynamics simulation to explore the origin of specific cation effects on the rates of bulk-phase aqueous electron transfer (ET) reactions. We consider 0.6 M solutions of Cl^- and a series of different cations: Li⁺, Na⁺, K⁺, Rb⁺, and Cs⁺. We evaluate the collective electrostatic fluctuations that drive Marcus-like ET and find that they are essentially unaffected by changes in the cationic species. This finding implies that the structure making/breaking properties of various cations do not exert a significant influence on bulk-phase ET reactions. We evaluate the role of ion pairing in these specific cation effects and find, unsurprisingly, that model redox anions that are more highly charged tend to pair more effectively with spectator cations than their monovalent counterparts. We demonstrate that this ion pairing significantly affects local electrostatic fluctuations for the anionic redox species and thus conclude that ion pairing is one of the likely sources of rate-dependent cation effects in aqueous ET reactions.

Ab Initio QM/MM Simulation of Ferrocene Homogeneous Electron-Transfer Reaction

Here, we demonstrate the feasibility of hybrid computational methods to predict the homogeneous electron exchange between the ferrocene and its oxidized (ferrocenium) state. The free energy for ferrocene oxidation was determined from thermodynamic cycles and implicit solvation strategies within density functional theory (DFT) methods leading to no more than 15% of deviation (in the range of 0.1-0.2 eV) when compared to absolute redox free energies obtained experimentally. Reorganization energy, as defined according to the Marcus theory of electron-transfer rate, was obtained by sampling the vertical ionization/electron affinity energies using hybrid quantum/classical (QM/MM) Born-Oppenheimer molecular dynamics trajectories. Calculated reorganization energies show a subtle but noteworthy dependence with the nature and the localization of the compensating countercharge. We concluded that the adopted hybrid computational strategy, to simulate homogeneous redox reactions, was successfully demonstrated and it further permits applications in more complex systems (required in daily life applications), where the electron transfer occurs heterogeneously.

Frontiers in molecular simulation of solvated ions, molecules and interfaces

This themed collection is a collection of articles on frontiers in molecular simulation of solvated ions, molecules and interfaces.

Revisiting nuclear tunnelling in the aqueous ferrous–ferric electron transfer

We find that golden-rule quantum transition-state theory predicts nearly an order of magnitude less tunnelling than some of the previous estimates. This may indicate that the spin-boson model of electron transfer is not valid in the quantum regime.

Electron transfer in nonpolar media

Electron transfer in nonpolar media violates the temperature scaling predicted by the fluctuation–dissipation theorem.

Microscopic Picture of the Solvent Reorganization During Electron Transfer to Flavin in Water

The redox potential of molecular species is largely modulated by its molecular environment so that a change of the environment will lead to a different redox potential. However, a detailed molecular picture of reorganization of the environment upon reduction is still unclear. To unravel the details of the solvent reorganization during electron transfer, we have performed density functional theory-based molecular dynamics (DFT-MD) and hybrid quantum mechanics/molecular mechanics (QM/MM) simulations of the reduction of lumiflavin. Previously, we have calculated the reduction free energy curves of the redox half reactions of lumiflavin in water as a function of the instantaneous gap energy (ΔE) ( J. Chem. Theory Comput. 2013 , 9 , 3889 - 3899 ). In this work, we focus on finding the changes in the solvent environment that correlate with this ΔE reaction coordinate. Comparing the QM/MM simulations, in which the solvent is modeled with an empirical force field, with the (full) DFT-MD simulations, we find that the response through electronic polarization plays a significant role in the latter case. Also a small charge transfer between flavin and solvent is observed in the full DFT treatment. As a result, we find only in the case of the QM/MM model a strong correlation between ΔE and the (pairwise computed) electrostatic potential (ESP) at the flavin due to the solvent. By analyzing the contribution of the ESP at the flavin per solvent molecule, we cannot only distinguish between the different modes of hydration by solvent molecules that coordinate at the hydrophilic and hydrophobic sides of the flavin molecule but also quantify their contribution to the reorganization free energy by measuring the ESP fluctuations per solvent molecule.

Electron transfer in extended systems: characterization by periodic density functional theory including the electronic coupling

We describe a new computer implementation of electron transfer (ET) theory in extended systems treated by periodic density functional theory (DFT), including the calculation of the electronic coupling transition element V_AB. In particular, the development opens up the full characterization of electron transfer in the solid state. The approach is valid for any single-determinant wavefunction with localized character representing the electronic structure of the system, from Hartree-Fock (HF) theory, to density functional theory (DFT), hybrid DFT theory, DFT+U theory, and constrained DFT (cDFT) theory. The implementation in CP2K reuses the high-performance functions of the code. The computational cost is equivalent to only one iteration of an HF calculation. We present test calculations for electron transfer in a number of systems, including a 1D-model of ferric oxide, hematite Fe₂O₃, rutile TiO₂, and finally bismuth vanadate BiVO₄.

2'-Deoxy-2'-fluoro-arabinonucleic acid: a valid alternative to DNA for biotechnological applications using charge transport

The non-biological 2'-deoxy-2'-fluoro-arabinonucleic acid (2'F-ANA) may be used as a valid alternative to DNA in biomedical and electronic applications because of its higher resistance to hydrolysis and nuclease degradation. However, the advantage of using 2'F-ANA in such applications also depends on its charge-transfer properties compared to DNA. In this study, we compare the charge conduction properties of model 2'F-ANA and DNA double-strands, using structural snapshots from MD simulations to calculate the electronic couplings and reorganization energies associated with the hole transfer steps between adjacent nucleobase pairs. Inserting these charge-transfer parameters into a kinetic model for charge conduction, we find similar conductive properties for DNA and 2'F-ANA. Moreover, we find that 2'F-ANA's enhanced chemical stability does not correspond to a reduction in the nucleobase π-stack structural flexibility relevant to both electronic couplings and reorganization free energies. Our results promote the use of 2'F-ANA in applications that can be based on charge transport, such as biosensing and chip technology, where its chemical stability and conductivity can advantageously combine.

A molecular density functional theory approach to electron transfer reactions

Beyond the dielectric continuum description initiated by Marcus theory, the standard theoretical approach to study electron transfer (ET) reactions in solution or at interfaces is to use classical force field or ab initio molecular dynamics simulations. We present here an alternative method based on liquid-state theory, namely molecular density functional theory, which is numerically much more efficient than simulations while still retaining the molecular nature of the solvent. We begin by reformulating molecular ET theory in a density functional language and show how to compute the various observables characterizing ET reactions from an ensemble of density functional minimizations. In particular, we define within that formulation the relevant order parameter of the reaction, the so-called vertical energy gap, and determine the Marcus free energy curves of both reactant and product states along that coordinate. Important thermodynamic quantities such as the reaction free energy and the reorganization free energies follow. We assess the validity of the method by studying the model Cl⁰ → Cl⁺ and Cl⁰ → Cl^- ET reactions in bulk water for which molecular dynamics results are available. The anionic case is found to violate the standard Marcus theory. Finally, we take advantage of the computational efficiency of the method to study the influence of a solid-solvent interface on the ET, by investigating the evolution of the reorganization free energy of the Cl⁰ → Cl⁺ reaction when the atom approaches an atomistically resolved wall.

Accurate Quantum Mechanical/Molecular Mechanical Calculations of Reduction Potentials in Azurin Variants

Understanding the regulation mechanism and molecular determinants of the reduction potential of metalloprotein is a major challenge. An ab initio quantum mechanical/molecular mechanical (QM/MM) method combining the minimum free energy path (MFEP) and fractional number of electron (FNE) approaches has been developed in our group to simulate the redox processes of large systems. The FNE scheme provides an efficient unique description for the redox process, while the MFEP method provides improved conformational sampling on complex environments such as protein in the QM/MM calculations. The reduction potentials of wild-type and seven mutants of azurin, a type 1 copper metalloprotein, were simulated with the QM/MM-MFEP+FNE approach in this paper. A range of 350 mV for the variations of the reduction potentials of these azurin proteins was reproduced faithfully with relative errors around 20 mV. The correlation between structural interactions and reduction potentials observed in simulations provides in-depth insight into the regulation of reduction potentials, which potentially can also be very useful to the engineering of metalloprotein-based electrocatalysts in artificial photosynthesis. The excellent accuracy and efficiency of the QM/MM-MFEP+FNE approach demonstrate the potential for simulations of many electron transfer processes in condensed phases and biochemical systems.

Effects of tunnelling and asymmetry for system-bath models of electron transfer

We apply the newly derived nonadiabatic golden-rule instanton theory to asymmetric models describing electron-transfer in solution. The models go beyond the usual spin-boson description and have anharmonic free-energy surfaces with different values for the reactant and product reorganization energies. The instanton method gives an excellent description of the behaviour of the rate constant with respect to asymmetry for the whole range studied. We derive a general formula for an asymmetric version of the Marcus theory based on the classical limit of the instanton and find that this gives significant corrections to the standard Marcus theory. A scheme is given to compute this rate based only on equilibrium simulations. We also compare the rate constants obtained by the instanton method with its classical limit to study the effect of tunnelling and other quantum nuclear effects. These quantum effects can increase the rate constant by orders of magnitude.

Ergodicity breaking of iron displacement in heme proteins

We present a model of the dynamical transition of atomic displacements in proteins. Increased mean-square displacement at higher temperatures is caused by the softening of the force constant for atomic/molecular displacements by electrostatic and van der Waals forces from the protein-water thermal bath. Displacement softening passes through a nonergodic dynamical transition when the relaxation time of the force-force correlation function enters, with increasing temperature, the instrumental observation window. Two crossover temperatures are identified. The lower crossover, presently connected to the glass transition, is related to the dynamical unfreezing of rotations of water molecules within nanodomains polarized by charged surface residues of the protein. The higher crossover temperature, usually assigned to the dynamical transition, marks the onset of water translations. All crossovers are ergodicity breaking transitions depending on the corresponding observation windows. Allowing stretched exponential relaxation of the protein-water thermal bath significantly improves the theory-experiment agreement when applied to solid protein samples studied by Mössbauer spectroscopy.

Electron-transfer chain in respiratory complex I

Complex I is a part of the respiration energy chain converting the redox energy into the cross-membrane proton gradient. The electron-transfer chain of iron-sulfur cofactors within the water-soluble peripheral part of the complex is responsible for the delivery of electrons to the proton pumping subunit. The protein is porous to water penetration and the hydration level of the cofactors changes when the electron is transferred along the chain. High reaction barriers and trapping of the electrons at the iron-sulfur cofactors are prevented by the combination of intense electrostatic noise produced by the protein-water interface with the high density of quantum states in the iron-sulfur clusters caused by spin interactions between paramagnetic iron atoms. The combination of these factors substantially lowers the activation barrier for electron transfer compared to the prediction of the Marcus theory, bringing the rate to the experimentally established range. The unique role of iron-sulfur clusters as electron-transfer cofactors is in merging protein-water fluctuations with quantum-state multiplicity to allow low activation barriers and robust operation. Water plays a vital role in electron transport energetics by electrowetting the cofactors in the chain upon arrival of the electron. A general property of a protein is to violate the fluctuation-dissipation relation through nonergodic sampling of its landscape. High functional efficiency of redox enzymes is a direct consequence of nonergodicity.

Theory and Electrochemistry of Cytochrome c

Extensive simulations of cytochrome c in solution are performed to address the apparent contradiction between large reorganization energies of protein electron transfer typically reported by atomistic simulations and much smaller values produced by protein electrochemistry. The two sets of data are reconciled by deriving the activation barrier for electrochemical reaction in terms of an effective reorganization energy composed of half the Stokes shift (characterizing the medium polarization in response to electron transfer) and the variance reorganization energy (characterizing the breadth of electrostatic fluctuations). This effective reorganization energy is much smaller than each of the two components contributing to it and is fully consistent with electrochemical measurements. Calculations in the range of temperatures between 280 and 360 K combine long, classical molecular dynamics simulations with quantum calculations of the protein active site. The results agree with the Arrhenius plots for the reaction rates and with cyclic voltammetry of cytochrome c immobilized on self-assembled monolayers. Small effective reorganization energy, and the resulting small activation barrier, is a general phenomenology of protein electron transfer allowing fast electron transport within biological energy chains.

Dehydrogenation Free Energy of Co2+(aq) from Density Functional Theory-Based Molecular Dynamics

Electron and proton transfers are important steps occurring in chemical reactions. The often used approach of calculating the energy differences of those steps using methods based on geometry optimizations neglects the influence of dynamic effects. To further investigate this issue and inspired by research in water oxidation, we calculate in the present study the dehydrogenation free energy of aqueous Co²⁺, which is the free energy change associated with the first step of the water oxidation reaction mechanism of recently investigated model Co(II)-aqua catalysts. We employ a method based on a thermodynamic integration scheme with strong ties to Marcus theory to obtain free energy differences, solvent reorganization free energies, and dynamic structural information on the systems from density functional theory-based molecular dynamics. While this method is computationally orders of magnitude more expensive than a static approach, it potentially allows for predicting the validity of the approximation of neglecting dynamic effects.

Diabat Interpolation for Polymorph Free-Energy Differences

Existing methods to compute free-energy differences between polymorphs use harmonic approximations, advanced non-Boltzmann bias sampling techniques, and/or multistage free-energy perturbations. This work demonstrates how Bennett's diabat interpolation method ( J. Comput. Phys. 1976, 22, 245 ) can be combined with energy gaps from lattice-switch Monte Carlo techniques ( Phys. Rev. E 2000, 61, 906 ) to swiftly estimate polymorph free-energy differences. The new method requires only two unbiased molecular dynamics simulations, one for each polymorph. To illustrate the new method, we compute the free-energy difference between face-centered cubic and body-centered cubic polymorphs for a Gaussian core solid. We discuss the justification for parabolic models of the free-energy diabats and similarities to methods that have been used in studies of electron transfer.

Molecular-Based Theory for Electron-Transfer Reorganization Energy in Solvent Mixtures

Using statistical-field techniques, we develop a molecular-based dipolar self-consistent-field theory (DSCFT) for charge solvation in liquid mixtures under equilibrium and nonequilibrium conditions, and apply it to compute the solvent reorganization energy of electron-transfer reactions. In addition to the nonequilibrium orientational polarization, the reorganization energy in liquid mixtures is also determined by the out-of-equilibrium solvent composition around the reacting species due to preferential solvation. Using molecular parameters that are readily available, the DSCFT naturally accounts for the dielectric saturation effect and the spatially varying solvent composition in the vicinity of the reacting species. We identify three general categories of binary solvent mixtures, classified by the relative optical and static dielectric permittivities of the solvent components. Each category of mixture is shown to produce a characteristic local solvent composition profile in the vicinity of the reacting species, which gives rise to the distinctive composition dependence of the reorganization energy that cannot be predicted using the dielectric permittivities of the homogeneous solvent mixtures.

Quantum Mechanics/Molecular Mechanics Method Combined with Hybrid All-Atom and Coarse-Grained Model: Theory and Application on Redox Potential Calculations

We developed a new multiresolution method that spans three levels of resolution with quantum mechanical, atomistic molecular mechanical, and coarse-grained models. The resolution-adapted all-atom and coarse-grained water model, in which an all-atom structural description of the entire system is maintained during the simulations, is combined with the ab initio quantum mechanics and molecular mechanics method. We apply this model to calculate the redox potentials of the aqueous ruthenium and iron complexes by using the fractional number of electrons approach and thermodynamic integration simulations. The redox potentials are recovered in excellent accordance with the experimental data. The speed-up of the hybrid all-atom and coarse-grained water model renders it computationally more attractive. The accuracy depends on the hybrid all-atom and coarse-grained water model used in the combined quantum mechanical and molecular mechanical method. We have used another multiresolution model, in which an atomic-level layer of water molecules around redox center is solvated in supramolecular coarse-grained waters for the redox potential calculations. Compared with the experimental data, this alternative multilayer model leads to less accurate results when used with the coarse-grained polarizable MARTINI water or big multipole water model for the coarse-grained layer.

Explicit solvent simulations of the aqueous oxidation potential and reorganization energy for neutral molecules: gas phase, linear solvent response, and non-linear response contributions

First principles simulations were used to predict aqueous one-electron oxidation potentials (Eox) and associated half-cell reorganization energies (λaq) for aniline, phenol, methoxybenzene, imidazole, and dimethylsulfide. We employed quantum mechanical/molecular mechanical (QM/MM) molecular dynamics (MD) simulations of the oxidized and reduced species in an explicit aqueous solvent, followed by EOM-IP-CCSD computations with effective fragment potentials for diabatic energy gaps of solvated clusters, and finally thermodynamic integration of the non-linear solvent response contribution using classical MD. A priori predicted Eox and λaq values exhibit mean absolute errors of 0.17 V and 0.06 eV, respectively, compared to experiment. We also disaggregate Eox into several well-defined free energy properties, including the gas phase adiabatic free energy of ionization (7.73 to 8.82 eV), the solvent-induced shift in the free energy of ionization due to linear solvent response (-2.01 to -2.73 eV), and the contribution from non-linear solvent response (-0.07 to -0.14 eV). The linear solvent response component is further apportioned into contributions from the solvent-induced shift in vertical ionization energy of the reduced species (ΔVIEaq) and the solvent-induced shift in negative vertical electron affinity of the ionized species (ΔNVEAaq). The simulated ΔVIEaq and ΔNVEAaq are found to contribute the principal sources of uncertainty in computational estimates of Eox and λaq. Trends in the magnitudes of disaggregated solvation properties are found to correlate with trends in structural and electronic features of the solute. Finally, conflicting approaches for evaluating the aqueous reorganization energy are contrasted and discussed, and concluding recommendations are given.

Reactive trajectories of the Ru2+/3+ self-exchange reaction and the connection to Marcus' theory

Outer sphere electron transfer between two ions in aqueous solution is a rare event on the time scale of first principles molecular dynamics simulations. We have used transition path sampling to generate an ensemble of reactive trajectories of the self-exchange reaction between a pair of Ru²⁺ and Ru³⁺ ions in water. To distinguish between the reactant and product states, we use as an order parameter the position of the maximally localised Wannier center associated with the transferring electron. This allows us to align the trajectories with respect to the moment of barrier crossing and compute statistical averages over the path ensemble. We compare our order parameter with two typical reaction coordinates used in applications of Marcus theory of electron transfer: the vertical gap energy and the solvent electrostatic potential at the ions.

Redox potentials of aryl derivatives from hybrid functional based first principles molecular dynamics

We report the redox potentials of a set of organic aryl molecules, including quinones, juglone, tyrosine and tryptophan, calculated using a first principles molecular dynamics (FPMD) based method.

'Dynamic Distance' Reaction Coordinate for Competing Bonds: Applications in Classical and Ab Initio Simulations

A versatile reaction coordinate, the "dynamic distance", is introduced for the study of reactions involving the rupture and formation of a series of chemical bonds or contacts. The dynamic distance is a mass-weighted mean of selected distances. When implemented as a generalized constraint, the dynamic distance is particularly suited for driving activated processes by controlled increase during a simulation. As a single constraint acting upon multiple degrees of freedom, the sequence of events along the resulting reaction pathway is determined unambiguously by the underlying energy landscape. Free energy profiles can be readily obtained from the mean constraint force. In this paper both theoretical aspects and numerical implementation are discussed, and the unique and diverse properties of this reaction coordinate are demonstrated using three examples:  In the framework of Car-Parrinello molecular dynamics, we present results for the prototypical double proton-transfer reaction in formic acid dimer and the photocycle of the guanine-cytosine DNA base pair. As a classical mechanical example, the opening of the binding pocket of the enzyme rubisco is analyzed.

Protein electron transfer: is biology (thermo)dynamic?

Simple physical mechanisms are behind the flow of energy in all forms of life. Energy comes to living systems through electrons occupying high-energy states, either from food (respiratory chains) or from light (photosynthesis). This energy is transformed into the cross-membrane proton-motive force that eventually drives all biochemistry of the cell. Life's ability to transfer electrons over large distances with nearly zero loss of free energy is puzzling and has not been accomplished in synthetic systems. The focus of this review is on how this energetic efficiency is realized. General physical mechanisms and interactions that allow proteins to fold into compact water-soluble structures are also responsible for a rugged landscape of energy states and a broad distribution of relaxation times. Specific to a protein as a fluctuating thermal bath is the protein-water interface, which is heterogeneous both dynamically and structurally. The spectrum of interfacial fluctuations is a consequence of protein's elastic flexibility combined with a high density of surface charges polarizing water dipoles into surface nanodomains. Electrostatics is critical to the protein function and the relevant questions are: (i) What is the spectrum of interfacial electrostatic fluctuations? (ii) Does the interfacial biological water produce electrostatic signatures specific to proteins? (iii) How is protein-mediated chemistry affected by electrostatics? These questions connect the fluctuation spectrum to the dynamical control of chemical reactivity, i.e. the dependence of the activation free energy of the reaction on the dynamics of the bath. Ergodicity is often broken in protein-driven reactions and thermodynamic free energies become irrelevant. Continuous ergodicity breaking in a dense spectrum of relaxation times requires using dynamically restricted ensembles to calculate statistical averages. When applied to the calculation of the rates, this formalism leads to the nonergodic activated kinetics, which extends the transition-state theory to dynamically dispersive media. Releasing the grip of thermodynamics in kinetic calculations through nonergodicity provides the mechanism for an efficient optimization between reaction rates and the spectrum of relaxation times of the protein-water thermal bath. Bath dynamics, it appears, play as important role as the free energy in optimizing biology's performance.

Practically Efficient QM/MM Alchemical Free Energy Simulations: The Orthogonal Space Random Walk Strategy

The difference between free energy changes occurring at two chemical states can be rigorously estimated via alchemical free energy (AFE) simulations. Traditionally, most AFE simulations are carried out under the classical energy potential treatment; then, accuracy and applicability of AFE simulations are limited. In the present work, we integrate a recent second-order generalized ensemble strategy, the orthogonal space random walk (OSRW) method, into the combined quantum mechanical/molecular mechanical (QM/MM) potential based AFE simulation scheme. Thereby, within a commonly affordable simulation length, accurate QM/MM alchemical free energy simulations can be achieved. As revealed by the model study on the equilibrium of a tautomerization process of hydrated 3-hydroxypyrazole and by the model calculations of the redox potentials of two flavin derivatives, lumichrome (LC) and riboflavin (RF) in aqueous solution, the present OSRW-based scheme could be a viable path toward the realization of practically efficient QM/MM AFE simulations.

Computational electrochemistry: prediction of liquid-phase reduction potentials

This article reviews recent developments and applications in the area of computational electrochemistry. Our focus is on predicting the reduction potentials of electron transfer and other electrochemical reactions and half-reactions in both aqueous and nonaqueous solutions. Topics covered include various computational protocols that combine quantum mechanical electronic structure methods (such as density functional theory) with implicit-solvent models, explicit-solvent protocols that employ Monte Carlo or molecular dynamics simulations (for example, Car-Parrinello molecular dynamics using the grand canonical ensemble formalism), and the Marcus theory of electronic charge transfer. We also review computational approaches based on empirical relationships between molecular and electronic structure and electron transfer reactivity. The scope of the implicit-solvent protocols is emphasized, and the present status of the theory and future directions are outlined.

Multiscale Free Energy Simulations: An Efficient Method for Connecting Classical MD Simulations to QM or QM/MM Free Energies Using Non-Boltzmann Bennett Reweighting Schemes

The reliability of free energy simulations (FES) is limited by two factors: (a) the need for correct sampling and (b) the accuracy of the computational method employed. Classical methods (e.g., force fields) are typically used for FES and present a myriad of challenges, with parametrization being a principle one. On the other hand, parameter-free quantum mechanical (QM) methods tend to be too computationally expensive for adequate sampling. One widely used approach is a combination of methods, where the free energy difference between the two end states is computed by, e.g., molecular mechanics (MM), and the end states are corrected by more accurate methods, such as QM or hybrid QM/MM techniques. Here we report two new approaches that significantly improve the aforementioned scheme; with a focus on how to compute corrections between, e.g., the MM and the more accurate QM calculations. First, a molecular dynamics trajectory that properly samples relevant conformational degrees of freedom is generated. Next, potential energies of each trajectory frame are generated with a QM or QM/MM Hamiltonian. Free energy differences are then calculated based on the QM or QM/MM energies using either a non-Boltzmann Bennett approach (QM-NBB) or non-Boltzmann free energy perturbation (NB-FEP). Both approaches are applied to calculate relative and absolute solvation free energies in explicit and implicit solvent environments. Solvation free energy differences (relative and absolute) between ethane and methanol in explicit solvent are used as the initial test case for QM-NBB. Next, implicit solvent methods are employed in conjunction with both QM-NBB and NB-FEP to compute absolute solvation free energies for 21 compounds. These compounds range from small molecules such as ethane and methanol to fairly large, flexible solutes, such as triacetyl glycerol. Several technical aspects were investigated. Ultimately some best practices are suggested for improving methods that seek to connect MM to QM (or QM/MM) levels of theory in FES.

On the Origins of the Linear Free Energy Relationships: Exploring the Nature of the Off-Diagonal Coupling Elements in S(N)2 Reactions

Understanding the relationship between the adiabatic free energy profiles of chemical reactions and the underlining diabatic states is central to the description of chemical reactivity. The diabatic states form the theoretical basis of Linear Free Energy Relationships (LFERs) and thus play a major role in physical organic chemistry and related fields. However, the theoretical justification for some of the implicit LFER assumptions has not been fully established by quantum mechanical studies. This study follows our earlier works(1,2) and uses the ab initio frozen density functional theory (FDFT) method(3) to evaluate both the diabatic and adiabatic free energy surfaces and to determine the corresponding off-diagonal coupling matrix elements for a series of S(N)2 reactions. It is found that the off-diagonal coupling matrix elements are almost the same regardless of the nucleophile and the leaving group but change upon changing the central group. Furthermore, it is also found that the off diagonal elements are basically the same in gas phase and in solution, even when the solvent is explicitly included in the ab initio calculations. Furthermore, our study establishes that the FDFT diabatic profiles are parabolic to a good approximation thus providing a first principle support to the origin of LFER. These findings further support the basic approximation of the EVB treatment.

Charge transfer in model peptides: obtaining Marcus parameters from molecular simulation

Charge transfer within and between biomolecules remains a highly active field of biophysics. Due to the complexities of real systems, model compounds are a useful alternative to study the mechanistic fundamentals of charge transfer. In recent years, such model experiments have been underpinned by molecular simulation methods as well. In this work, we study electron hole transfer in helical model peptides by means of molecular dynamics simulations. A theoretical framework to extract Marcus parameters of charge transfer from simulations is presented. We find that the peptides form stable helical structures with sequence dependent small deviations from ideal PPII helices. We identify direct exposure of charged side chains to solvent as a cause of high reorganization energies, significantly larger than typical for electron transfer in proteins. This, together with small direct couplings, makes long-range superexchange electron transport in this system very slow. In good agreement with experiment, direct transfer between the terminal amino acid side chains can be dicounted in favor of a two-step hopping process if appropriate bridging groups exist.

Kinetics of the terminal electron transfer step in cytochrome c oxidase

Cytochrome c oxidase (cco) catalyzes the oxygen reduction reaction in most aerobically respiring organisms. Decades of research have uncovered many aspects relating to structure and function of this enzyme. However, the origin of the unusually fast terminal electron transfer step from heme a to heme a(3) in cco has been the subject of intense discussions over recent years. Yet, no satisfactory consensus has been achieved. Carrying out large-scale molecular dynamics simulation of the protein embedded in a solvated membrane, we obtain a reorganization free energy λ = 0.57 eV. Evaluation of the quantized single-mode rate equation using the experimental rate and the computed reorganization free energy gives a value of 1.5 meV for the average electronic coupling (H(ab)) between heme a and heme a(3). Thus, according to our calculations, the nanosecond electron transfer (ET) is due to a small but significant activation barrier (ΔG(‡) = 0.12 eV) in combination with effective electronic coupling between the two cofactors. The activation free energy is caused predominantly by collective reorganization of protein residues. We show that our results are consistent with the weak temperature dependence observed in experiment if one allows for very minor variations in the donor-acceptor distance as the temperature changes.

Reorganization energy for internal electron transfer in multicopper oxidases

We have calculated the reorganization energy for the intramolecular electron transfer between the reduced type 1 copper site and the peroxy intermediate of the trinuclear cluster in the multicopper oxidase CueO. The calculations are performed at the combined quantum mechanics and molecular mechanics (QM/MM) level, based on molecular dynamics simulations with tailored potentials for the two copper sites. We obtain a reorganization energy of 91-133 kJ/mol, depending on the theoretical treatment. The two Cu sites contribute by 12 and 22 kJ/mol to this energy, whereas the solvent contribution is 34 kJ/mol. The rest comes from the protein, involving small contributions from many residues. We have also estimated the energy difference between the two electron-transfer states and show that the reduction of the peroxy intermediate is exergonic by 43-87 kJ/mol, depending on the theoretical method. Both the solvent and the protein contribute to this energy difference, especially charged residues close to the two Cu sites. We compare these estimates with energies obtained from QM/MM optimizations and QM calculations in a vacuum and discuss differences between the results obtained at various levels of theory.