Two-dimensional Energy Surface Research Articles

Solvent Control of Chemical Identity Can Change Photodissociation into Photoisomerization.

In solution-phase chemistry, the solvent is often considered to be merely a medium that allows reacting solutes to encounter each other. In this work, however, we show that moderate locally specific solute-solvent interactions can affect not only the nature of the solute but also the types of reactive chemistry. We use quantum simulation methods to explore how solvent participation in solute chemical identity alters reactions involving the breaking of chemical bonds. In particular, we explore the photoexcitation dynamics of Na2+ dissolved in liquid tetrahydrofuran. In the gas phase, excitation of Na2+ directly leads to dissociation, but in solution, photoexcitation leads to an isomerization reaction involving rearrangement of the first-shell solvent molecules; this isomerization must go to completion before the solute can dissociate. Despite the complexity, the solution-phase reaction dynamics can be captured by a two-dimensional energy surface where one dimension involves only the isomerization of the first-shell solvent molecules.

Non-Markovian rate theory on a multidimensional reaction surface: Complex interplay between enhanced configuration space and memory

A theory of barrier crossing rate on a multidimensional reaction energy surface is presented. The theory is a generalization of the earlier theoretical schemes to higher dimensions, with the inclusion of non-Markovian friction along both the reactive and the nonreactive coordinates. The theory additionally includes the bilinear coupling between the reactive and the nonreactive modes at the Hamiltonian level. Under suitable conditions, we recover the rate expressions of Langer and Hynes and establish a connection with the rate treatment of Pollak. Within the phenomenology of generalized Langevin equation description, our formulation provides an improvement over the existing ones because we explicitly include both the non-Markovian effects along the reaction coordinate and the bilinear coupling at the Hamiltonian level. At intermediate-to-large friction, an increase in dimensionality by itself tends to reduce the rate, while the inclusion of the memory effects increases the rate. The theory predicts an increase in rate when off-diagonal friction terms are included. We present a model calculation to study isomerization of a stilbene-like molecule using the prescription of Hochstrasser and co-workers on a two-dimensional reaction energy surface, employing Zwanzig-Bixon hydrodynamic theory of frequency-dependent friction. The calculated rate shows a departure from the predictions of Langer's theory and also from the two-dimensional transition state theory.

Benchmarking Cation−π Interactions: Assessment of Density Functional Theory and Möller–Plesset Second-Order Perturbation Theory Calculations with Optimized Basis Sets (mp2mod) for Complexes of Benzene, Phenol, and Catechol with Na+, K+, Rb+, and Cs+

The reliability of several DFT functionals and of the Möller-Plesset Second Order Perturbation Theory Calculations with Modified Basis Sets (MP2mod) approach in describing cation-π interactions is systematically investigated by benchmarking their performances with respect to high quality reference CCSD(T) calculations of the binding energies between first group alkali ions (Na+, K+, Rb+ and Cs+ ) and three aromatic species (benzene, phenol and catechol). For this class of non-covalent interaction, MP2mod delivers, in average, results in very good agreement with the reference CCSD(T) data, yet at a very small computational cost, exploiting the reduced dimensions of the modified basis set. Conversely, the tested DFT functionals show a more erratic behaviour, with different performances depending on both the investigated system and the combination of the employed functional and basis set. The MP2mod computational convenience is further exploited to extensively sample two-dimensional interaction energy surfaces of all investigated cation-π systems, which allow for a deeper insight on the effect of the increasing number of hydroxyl substituents, revealing the insurgence, upon substitution, of alternative minima, evident in particular for the smaller cations. The present results are very encouraging for further applications of the MP2mod method to study a larger variety of aromatic/metal cation species, relevant both in biological processes and in technological applications.

Ab Initio Molecular Dynamics Simulation and Energetics of the Ribulose-1,5-biphosphate Carboxylation Reaction Catalyzed by Rubisco: Toward Elucidating the Stereospecific Protonation Mechanism.

In the carboxylation reaction catalyzed by ribulose 1,5-bisphosphate (RuBP) carboxylase-oxygenase (Rubisco), which is fundamental to photosynthesis, scission of a C-C bond in the six-carbon gemdiolate intermediate forms a carbanion that must be protonated stereospecifically to form product. It is thought that a conserved lysine side chain (LYS175 in spinach Rubisco), in the immediate vicinity of the carbanion, provides the necessary proton. Here, we endeavor to determine from the electronic-structure calculations whether protonation via this route is energetically possible. The two-dimensional energy surface was mapped to determine the minimum energy path (MEP) using density functional theory (B3LYP) and incorporating basis set superposition and classical (London) dispersion corrections. The potential of mean force (free energy) was then calculated from ab initio molecular dynamics simulations with umbrella sampling in the vicinity of the MEP on the scission-protonation reaction coordinate. MEP calculations were also carried out to evaluate the possibility of an active-site water near the phosphate (P1) of RuBP, with an excess proton positioned at P1, as an alternative facilitator of stereospecific protonation via a classical Grotthuss mechanism. In both cases, the C-C bond scission in the six-carbon intermediate and proton transfer from the donor was found to be concerted and highly asynchronous, without a stable carbanion intermediate. However, the free energy change was unfavorable for direct protonation by the LYS175 side chain. In contrast, the Grotthuss mechanism yielded stable products and an activation energy in good agreement with experiment. It also provides a plausible mechanism for alternative product formed in enzyme mutations at the LYS175 position and is consistent with the observed deuterium isotope effects.

Mechanobiology: protein refolding under force.

The application of direct force to a protein enables to probe wide regions of its energy surface through conformational transitions as unfolding, extending, recoiling, collapsing, and refolding. While unfolding under force typically displayed a two-state behavior, refolding under force, from highly extended unfolded states, displayed a more complex behavior. The first recording of protein refolding at a force quench step displayed an initial rapid elastic recoil, followed by a plateau phase at some extension, concluding with a collapse to a final state, at which refolding occurred. These findings stirred a lively discussion, which led to further experimental and theoretical investigation of this behavior. It was demonstrated that the polymeric chain of the unfolded protein is required to fully collapse to a globular conformation for the maturation of native structure. This behavior was modeled using one-dimensional free energy landscape over the end-to-end length reaction coordinate, the collective measured variable. However, at low forces, conformational space is not well captured by such models, and using two-dimensional energy surfaces provides further insight into the dynamics of this process. This work reviews the main concepts of protein refolding under constant force, which is essential for understanding how mechanotransducing proteins operate in vivo.

All-Atom Internal Coordinate Mechanics (ICM) Force Field for Hexopyranoses and Glycoproteins.

We present an extension of the all-atom internal-coordinate force field, ICMFF, that allows for simulation of heterogeneous systems including hexopyranose saccharides and glycan chains in addition to proteins. A library of standard glycan geometries containing α- and β-anomers of the most common hexapyranoses, i.e., d-galactose, d-glucose, d-mannose, d-xylose, l-fucose, N-acetylglucosamine, N-acetylgalactosamine, sialic, and glucuronic acids, is created based on the analysis of the saccharide structures reported in the Cambridge Structural Database. The new force field parameters include molecular electrostatic potential-derived partial atomic charges and the torsional parameters derived from quantum mechanical data for a collection of minimal molecular fragments and related molecules. The ϕ/ψ torsional parameters for different types of glycosidic linkages are developed using model compounds containing the key atoms in the full carbohydrates, i.e., glycosidic-linked tetrahydropyran–cyclohexane dimers. Target data for parameter optimization include two-dimensional energy surfaces corresponding to the ϕ/ψ glycosidic dihedral angles in the disaccharide analogues, as determined by quantum mechanical MP2/6-31G** single-point energies on HF/6-31G** optimized structures. To achieve better agreement with the observed geometries of glycosidic linkages, the bond angles at the O-linkage atoms are added to the internal variable set and the corresponding bond bending energy term is parametrized using quantum mechanical data. The resulting force field is validated on glycan chains of 1–12 residues from a set of high-resolution X-ray glycoprotein structures based on heavy atom root-mean-square deviations of the lowest-energy glycan conformations generated by the biased probability Monte Carlo (BPMC) molecular mechanics simulations from the native structures. The appropriate BPMC distributions for monosaccharide–monosaccharide and protein–glycan linkages are derived from the extensive analysis of conformational properties of glycoprotein structures reported in the Protein Data Bank. Use of the BPMC search leads to significant improvements in sampling efficiency for glycan simulations. Moreover, good agreement with the X-ray glycoprotein structures is achieved for all glycan chain lengths. Thus, average/median RMSDs are 0.81/0.68 Å for one-residue glycans and 1.32/1.47 Å for three-residue glycans. RMSD from the native structure for the lowest-energy conformation of the 12-residue glycan chain (PDB ID 3og2) is 1.53 Å. Additionally, results obtained for free short oligosaccharides using the new force field are in line with the available experimental data, i.e., the most populated conformations in solution are predicted to be the lowest energy ones. The newly developed parameters allow for the accurate modeling of linear and branched hexopyranose glycosides in heterogeneous systems.

간접천이대를 갖는 2차원 소자에서 성장방향에 따른 Hall 인수의 이방성 연구

The Hall factor in a two-dimensional device with indirect conduction valleys is calculated for several growth on various strain conditions. In the [001] or [111] growth direction, the two-dimensional constant energy surfaces of occupied valleys are shown to be isotropically distributed. However, in the [110] growth direction, the distribution of occupied valleys on the plane is not isotropic. This fact is the reason for the anisotropic Hall factor on the sample plane.

Polymer escape from a confining potential

The rate of escape of polymers from a two-dimensionally confining potential well has been evaluated using self-avoiding as well as ideal chain representations of varying length, up to 80 beads. Long timescale Langevin trajectories were calculated using the path integral hyperdynamics method to evaluate the escape rate. A minimum is found in the rate for self-avoiding polymers of intermediate length while the escape rate decreases monotonically with polymer length for ideal polymers. The increase in the rate for long, self-avoiding polymers is ascribed to crowding in the potential well which reduces the free energy escape barrier. An effective potential curve obtained using the centroid as an independent variable was evaluated by thermodynamic averaging and Kramers rate theory then applied to estimate the escape rate. While the qualitative features are well reproduced by this approach, it significantly overestimates the rate, especially for the longer polymers. The reason for this is illustrated by constructing a two-dimensional effective energy surface using the radius of gyration as well as the centroid as controlled variables. This shows that the description of a transition state dividing surface using only the centroid fails to confine the system to the region corresponding to the free energy barrier and this problem becomes more pronounced the longer the polymer is. A proper definition of a transition state for polymer escape needs to take into account the shape as well as the location of the polymer.

간접천이대를 갖는 양자우물 구조에서의 Hall 상수

The Hall factor in a quantum well structure with X or L-type indirect conduction valleys is calculated for various strain conditions. The two-dimensional constant energy surfaces of occupied valleys are proven to be identical. As a result, the Hall factor depends on the relative direction of occupied valleys to the growth direction, regardless of the number of occupied valleys. This work is widely applicable to the two-dimensional structure with indirect conduction minima for any growth direction and under different strain conditions.

Microscopically computing free-energy profiles and transition path time of rare macromolecular transitions

We introduce a rigorous method to microscopically compute the observables which characterize the thermodynamics and kinetics of rare macromolecular transitions for which it is possible to identify a priori a slow reaction coordinate. In order to sample the ensemble of statistically significant reaction pathways, we define a biased molecular dynamics (MD) in which barrier-crossing transitions are accelerated without introducing any unphysical external force. In contrast to other biased MD methods, in the present approach the systematic errors which are generated in order to accelerate the transition can be analytically calculated and therefore can be corrected for. This allows for a computationally efficient reconstruction of the free-energy profile as a function of the reaction coordinate and for the calculation of the corresponding diffusion coefficient. The transition path time can then be readily evaluated within the dominant reaction pathways approach. We illustrate and test this method by characterizing a thermally activated transition on a two-dimensional energy surface and the folding of a small protein fragment within a coarse-grained model.

CHARMM Additive All-Atom Force Field for Phosphate and Sulfate Linked to Carbohydrates

Presented is an extension of the CHARMM additive all-atom carbohydrate force field to enable the modeling of phosphate and sulfate linked to carbohydrates. The parameters are developed in a hierarchical fashion using model compounds containing the key atoms in the full carbohydrates. Target data for parameter optimization included full two-dimensional energy surfaces defined by the glycosidic dihedral angle pairs in the phosphate/sulfate model compound analogs of hexopyranose monosaccharide phosphates and sulfates, as determined by quantum mechanical (QM) MP2/cc-pVTZ single point energies on MP2/6-31+G(d) optimized structures. In order to achieve balanced, transferable dihedral parameters for the dihedral angles, surfaces for all possible anomeric and conformational states were included during the parametrization process. In addition, to model physiologically relevant systems both the mono- and di-anionic charged states were studied for the phosphates. This resulted in over 7000 MP2/cc-pVTZ//MP2/6-31G+(d) model compound conformational energies which, supplemented with QM geometries, were the main target data for the parametrization. Parameters were validated against crystals of relevant monosaccharide derivatives obtained from the Cambridge Structural Database (CSD) and larger systems, namely inositol-(tri/tetra/penta) phosphates non-covalently bound to the pleckstrin homology (PH) domain and oligomeric chondroitin sulfate in solution and in complex with cathepsin K protein.

Modeling the Transition State of Enzyme-catalyzed Phosphoryl Transfer Reaction Using QM/MM Method

Reversible phosphorylation of proteins is a post-translational modification that regulates diverse biological processes. The molecular mechanism underlying phosphoryl transfer catalyzed by enzymes, in particular the nature of transition state (TS), remains a subject of active debate. Structural evidence supports an associative TS, whereas physical organic studies point to a dissociative character. In this article, we briefly introduce our recent effort using the hybrid quantum mechanics/molecular mechanics (QM/MM) simulations to resolve the controversy. We perform QM/MM simulations for the reversible phosphorylation of phosphoserine phosphatase (PSP), which belongs to one of the largest phosphotransferase families characterized to data. Both phosphorylation and dephosphorylation reactions are investigated based on the two-dimensional energy surfaces along phosphoryl and proton transfer coordinates. The resultant structures of the active site at TS in both reactions have compact geometries but a less electron density of the phosphoryl group. This suggests that the TS of PSP has a geometrically associative yet electronically dissociative character and strongly depends on proton transfer being coupled with phosphoryl transfer. Structure and literature database searches on phosphotransferases suggest that such a hybrid TS is consistent with many structures and physical organic studies and likely holds for most enzymes catalyzing phosphoryl transfer.

Peierls-Nabarro energy surfaces and directional mobility of discrete solitons in two-dimensional saturable nonlinear Schrödinger lattices

We address the problem of directional mobility of discrete solitons in two-dimensional rectangular lattices, in the framework of a discrete nonlinear Schrödinger model with saturable on-site nonlinearity. A numerical constrained Newton-Raphson method is used to calculate two-dimensional Peierls-Nabarro energy surfaces, which describe a pseudopotential landscape for the slow mobility of coherent localized excitations, corresponding to continuous phase-space trajectories passing close to stationary modes. Investigating the two-parameter space of the model through independent variations of the nonlinearity constant and the power, we show how parameter regimes and directions of good mobility are connected to the existence of smooth surfaces connecting the stationary states. In particular, directions where solutions can move with minimum radiation can be predicted from flatter parts of the surfaces. For such mobile solutions, slight perturbations in the transverse direction yield additional transverse oscillations with frequencies determined by the curvature of the energy surfaces, and with amplitudes that for certain velocities may grow rapidly. We also describe how the mobility properties and surface topologies are affected by inclusion of weak lattice anisotropy.

Theoretical Insights into the Magnetostructural Correlations in Mn3-Based Single-Molecule Magnets

Density functional theory (DFT) and the valence bond configuration interaction (VBCI) model have been applied to the oximato-based Mn(III)(3)O single-molecule magnets (SMMs), allowing one to correlate the Mn(III)-Mn(III) exchange coupling energy (J) with the bridging geometry in terms of two structural angles: the Mn-O-N-Mn torsion angle (γ) and the Mn(3) out-of-plane shift of O (angle δθ). Using DFT, a two-dimensional (γ, δθ) energy surface of J is derived and shown to yield essentially good agreement with the reported J values deduced from magnetic susceptibility data on trigonal oximato-bridged Mn(3) SMMs. VBCI is used to understand and analyze the DFT results. It is shown that the exchange coupling in these systems is governed by a spin-polarization mechanism inducing a pronounced and dominating ferromagnetic exchange via the oximato bridge as opposed to kinetic exchange, which favors a weaker and antiferromagnetic exchange via the bridging oxide. In the light of these results, a discussion of the exchange coupling in the Mn(6) family of the SMM with a record demagnetization barrier is given.

Geometrically associative yet electronically dissociative character in the transition state of enzymatic reversible phosphorylation

Reversible phosphorylation of proteins is a post-translational modification that regulates diverse biological processes. The molecular mechanism underlying phosphoryl transfer catalyzed by enzymes remains a subject of active debate. In particular, the nature of transition state (TS), whether it has an associative or dissociative character, has been one of the most controversial issues. Structural evidence supports associative TS, whereas physical organic studies point to a dissociative character. Here we perform hybrid quantum mechanics/molecular mechanics simulations for the reversible phosphorylation of phosphoserine phosphatase (PSP) to study the nature of the TS. Both phosphorylation and dephosphorylation reactions are investigated based on the two-dimensional energy surfaces along phosphoryl and proton transfer coordinates. The structures of the active site at TS in both reactions reveal compact geometries, consistent with crystal structures of PSP with phosphate analogues. However, the electron density of the phosphoryl group in both TS structures slightly decreases compared with that in the reactant states. These findings suggest that the TS of PSP has a geometrically associative yet electronically dissociative character and strongly depends on proton transfer being coupled with phosphoryl transfer. Structure and literature database, which searches on phosphotransferases, suggest that such a hybrid TS is consistent with many structures and physical organic studies and likely holds for most enzymes catalyzing phosphoryl transfer.

Controlling the secondary-structure-forming tendencies of proteins by a backbone torsion-energy term

We examined a new backbone torsion-energy term proposed by us in the force field for protein systems. This torsion-energy term is represented by a double Fourier series in two variables, namely the backbone dihedral angles φ and ψ. It gives a natural representation of the torsion energy in the Ramachandran space in the sense that any two-dimensional energy surface periodic in both φ and ψ can be expanded by the double Fourier series. We can then easily control secondary-structure-forming tendencies by modifying the torsion-energy surface. For instance, we can increase or decrease the α-helix-forming-tendencies by lowering or raising the torsion-energy surface in the α-helix region and likewise increase or decrease the β-sheet-forming tendencies by lowering or raising the surface in the β-sheet region in the Ramachandran space. We applied this torsion-energy modification method to six force fields, AMBER parm94, AMBER parm96, AMBER parm99, CHARMM27, OPLS-AA and OPLS-AA/L, and demonstrated that our modifications of the torsion-energy terms resulted in the expected changes of secondary-structure-forming tendencies by performing folding simulations of α-helical and β-hairpin peptides.

A two-dimensional energy surface of the phosphoryl transfer reaction catalyzed by phosphoserine phosphatase

The phosphoryl transfer reaction from phospho- L-serine (pSer), catalyzed by phosphoserine phosphatase, is investigated using the hybrid quantum mechanics/molecular mechanics calculations. The two-dimensional energy surface along the phosphoryl and proton transfer distances reveals early protonation of the leaving group oxygen of pSer, prior to the transition state (TS), which triggers subsequent phosphoryl transfer reaction. Calculated electronic properties of the phosphoryl group at the active site suggest significant metaphosphate-like character of TS, which is consistent with kinetic experiments on related phosphatases. The features are not obtained with a one-dimensional search along the phosphoryl transfer coordinate, due to inadequate description of proton movement.

CHARMM Additive All-Atom Force Field for Glycosidic Linkages between Hexopyranoses.

We present an extension of the CHARMM hexopyranose monosaccharide additive all-atom force field to enable modeling of glycosidic-linked hexopyranose polysaccharides. The new force field parameters encompass 1→1, 1→2, 1→3, 1→4, and 1→6 hexopyranose glycosidic linkages, as well as O-methylation at the C(1) anomeric carbon, and are developed to be consistent with the CHARMM all-atom biomolecular force fields for proteins, nucleic acids, and lipids. The parameters are developed in a hierarchical fashion using model compounds containing the key atoms in the full carbohydrates, in particular O-methyl-tetrahydropyran and glycosidic-linked dimers consisting of two molecules of tetrahyropyran or one of tetrahydropyran and one of cyclohexane. Target data for parameter optimization include full two-dimensional energy surfaces defined by the Φ/Ψ glycosidic dihedral angles in the disaccharide analogs as determined by quantum mechanical MP2/cc-pVTZ single point energies on MP2/6-31G(d) optimized structures (MP2/cc-pVTZ//MP2/6-31G(d)). In order to achieve balanced, transferable dihedral parameters for the Φ/Ψ glycosidic dihedral angles, surfaces for all possible chiralities at the ring carbon atoms involved in the glycosidic linkages are considered, resulting in over 5000 MP2/cc-pVTZ//MP2/6-31G(d) conformational energies. Also included as target data are vibrational frequencies, pair interaction energies and distances with water molecules, and intramolecular geometries including distortion of the glycosidic valence angle as a function of the glycosidic dihedral angles. The model-compound optimized force field parameters are validated on full disaccharides through comparison of molecular dynamics results to available experimental data. Good agreement is achieved with experiment for a variety of properties including crystal cell parameters and intramolecular geometries, aqueous densities, and aqueous NMR coupling constants associated with the glycosidic linkage. The newly-developed parameters allow for the modeling of linear, branched, and cyclic hexopyranose glycosides both alone and in heterogenous systems including proteins, nucleic acids and/or lipids when combined with existing CHARMM biomolecular force fields.

Improved transition path sampling methods for simulation of rare events

The free energy surfaces of a wide variety of systems encountered in physics, chemistry, and biology are characterized by the existence of deep minima separated by numerous barriers. One of the central aims of recent research in computational chemistry and physics has been to determine how transitions occur between deep local minima on rugged free energy landscapes, and transition path sampling (TPS) Monte-Carlo methods have emerged as an effective means for numerical investigation of such transitions. Many of the shortcomings of TPS-like approaches generally stem from their high computational demands. Two new algorithms are presented in this work that improve the efficiency of TPS simulations. The first algorithm uses biased shooting moves to render the sampling of reactive trajectories more efficient. The second algorithm is shown to substantially improve the accuracy of the transition state ensemble by introducing a subset of local transition path simulations in the transition state. The system considered in this work consists of a two-dimensional rough energy surface that is representative of numerous systems encountered in applications. When taken together, these algorithms provide gains in efficiency of over two orders of magnitude when compared to traditional TPS simulations.

Dependence of the l-Alanyl-l-Alanine Conformation on Molecular Charge Determined from Ab Initio Computations and NMR Spectra

The l-alanyl-l-alanine (AA) molecule behaves differently in acidic, neutral, and basic environments. Because of its molecular flexibility and strong interaction with the aqueous environment, its behavior has to be deduced from the NMR spectra indirectly, using statistical methods and comparison with ab initio predictions of geometric and spectral parameters. In this study, chemical shifts and indirect spin-spin coupling constants of the AA cation, anion, and zwitterion were measured and compared to values obtained by density functional computations for various conformers of the dipeptide. The accuracy and sensitivity of the quantum methods to the molecular charge was also tested on the (mono)-alanine molecule. Probable AA conformers could be identified at two-dimensional potential energy surfaces and verified by the comparison of the computed parameters with measured NMR data. The results indicate that, whereas the main-chain peptide conformations of the cationic (AA+) and zwitterionic (AAZW) forms are similar, the anion (AA-) adopts also another, approximately equally populated conformer in the aqueous solution. Additionally, the NH2 group can rotate in the two main chain conformations of the anionic form AA-. According to a vibrational quantum analysis of the two-dimensional energy surfaces, higher-energy conformers might exist for all three charged AA forms but cannot be detected directly by NMR spectroscopy because of their small populations and short lifetimes. In accord with previous studies, the NMR parameters, particularly the indirect nuclear spin-spin coupling constants, often provided an excellent probe of a local conformation. Generalization to peptides and proteins, however, has to take into account the environment, molecular charge, and flexibility of the peptide chain.