The roto-conformational diffusion tensor as a tool to interpret molecular flexibility.

Stochastic modeling approaches can be used to rationalize complex molecular dynamical behaviours in solution, helping to interpret the coupling mechanisms among internal and external degrees of freedom, providing insight into reaction mechanisms and extracting structural and dynamical data from spectroscopic observables. However, the definition of comprehensive models is usually limited by (i) the difficulty in defining - without resorting to phenomenological assumptions - a representative reduced ensemble of molecular coordinates able to capture essential dynamical properties and (ii) the complexity of numerical or approximate treatments of the resulting equations. In this paper, we address the first of these two issues. Building on a previously defined systematic approach to construct rigorous stochastic models of flexible molecules in solutions from basic principles, we define a manageable diffusive framework leading to a Smoluchowski equation determined by one main tensorial parameter, namely the scaled roto-conformational diffusion tensor, which accounts for the influence of both conservative and dissipative forces and describes the molecular mobility via a precise definition of internal-external and internal-internal couplings. We then show the usefulness of the roto-conformational scaled diffusion tensor as an efficient gauge of molecular flexibility through the analysis of a set of molecular systems of increasing complexity ranging from dimethylformamide to a protein domain.

A multiscale approach to coupled nuclear and electronic dynamics. I. Quantum-stochastic Liouville equation in natural internal coordinates.

Multiscale methods are powerful tools to describe large and complex systems. They are based on a hierarchical partitioning of the degrees of freedom (d.o.f.) of the system, allowing one to treat each set of d.o.f. in the most computationally efficient way. In the context of coupled nuclear and electronic dynamics, a multiscale approach would offer the opportunity to overcome the computational limits that, at present, do not allow one to treat a complex system (such as a biological macromolecule in explicit solvent) fully at the quantum mechanical level. Based on the pioneering work of Kapral and Ciccotti [R. Kapral and G. Ciccotti, J. Chem. Phys.110, 8919 (1999)], this work is intended to present a nonadiabatic theory that describes the evolution of electronic populations coupled with the dynamics of the nuclei of a molecule in a dissipative environment (condensed phases). The two elements of novelty that are here introduced are (i) the casting of the theory in the natural, internal coordinates, that are bond lengths, bond angles, and dihedral angles; (ii) the projection of those nuclear d.o.f. that can be considered at the level of a thermal bath, therefore leading to a quantum-stochastic Liouville equation. Using natural coordinates allows the description of structure and dynamics in the way chemists are used to describe molecular geometry and its changes. The projection of bath coordinates provides an important reduction of complexity and allows us to formulate the approach that can be used directly in the statistical thermodynamics description of chemical systems.

Parameter free evaluation of SN2 reaction rates for halide substitution in halomethane.

We estimate the kinetic constants of a series of archetypal SN2 reactions, i.e., the nucleophilic substitutions of halides in halomethane. A parameter free, multiscale approach recently developed [Campeggio et al., Phys. Chem. Chem. Phys., 2020, 22, 3455] is employed. The protocol relies on quantum mechanical calculations for the description of the energy profile along the intrinsic reaction coordinate, which is then mapped onto a reaction coordinate conveniently built for the reactive process. A Kramers-Klein equation is used to describe the stochastic time evolution of the reaction coordinate and its velocity; friction is parameterized using a hydrodynamic model and Kramers theory is used to derive the rate constant of the reaction. The method is here applied to six SN2 reactions in water at 295.15 K, which differ in the nucleophile and the leaving group. The computed reaction rates are in good agreement with the experimental data and correlate well with the trends observed for the activation energies.

Ethanol electro-oxidation reaction on the Pd(111) surface in alkaline media: insights from quantum and molecular mechanics.

The ethanol electro-oxidation catalyzed by Pd in an alkaline environment involves several intermediate reaction steps promoted by the hydroxyl radical, OH. In this work, we report on the dynamical paths of the first step of this oxidation reaction, namely the hydrogen atom abstraction CH3CH2OH + OH → CH3CHOH + H2O, occurring at the Pd(111) surface and address the thermodynamic stability of the adsorbed reactants by means of quantum and molecular mechanics calculations, with special focus on the effect of the solvent. We have found that the impact of the solvent is significant for both ethanol and OH, contributing to a decrease in their adsorption free energies by a few dozen kcal mol-1 with respect to the adsorption energy under vacuum. Furthermore, we observe that hydrogen atom abstraction is enhanced for those simulation paths featuring large surface-reactant distances, namely, when the reactants weakly interact with the catalyst. The picture emerging from our study is therefore that of a catalyst whose coverage in an aqueous environment is largely dominated by OH with respect to ethanol. Nevertheless, only a small amount of them, specifically those weakly bound to the catalyst, is really active in the ethanol electro-oxidation reaction. These results open the idea of a rational design of co-catalysts based on the tuning of surface chemical properties to eventually enhance exchange current density.

Stochastic Modelling of 13C NMR Spin Relaxation Experiments in Oligosaccharides.

A framework for the stochastic description of relaxation processes in flexible macromolecules including dissipative effects has been recently introduced, starting from an atomistic view, describing the joint relaxation of internal coordinates and global degrees of freedom, and depending on parameters recoverable from classic force fields (energetics) and medium modelling at the continuum level (friction tensors). The new approach provides a rational context for the interpretation of magnetic resonance relaxation experiments. In its simplest formulation, the semi-flexible Brownian (SFB) model has been until now shown to reproduce correctly correlation functions and spectral densities related to orientational properties obtained by direct molecular dynamics simulations of peptides. Here, for the first time, we applied directly the SFB approach to the practical evaluation of high-quality 13C nuclear magnetic resonance relaxation parameters, T1 and T2, and the heteronuclear NOE of several oligosaccharides, which were previously interpreted on the basis of refined ad hoc modelling. The calculated NMR relaxation parameters were in agreement with the experimental data, showing that this general approach can be applied to diverse classes of molecular systems, with the minimal usage of adjustable parameters.

Multiscale modeling of reaction rates: application to archetypal SN2 nucleophilic substitutions.

We propose an approach to the evaluation of kinetic rates of elementary chemical reactions within Kramers' theory based on the definition of the reaction coordinate as a linear combination of natural, pseudo Z-matrix, internal coordinates of the system. The element of novelty is the possibility to evaluate the friction along the reaction coordinate, within a hydrodynamic framework developed recently [J. Campeggio et al., J. Comput. Chem. 2019, 40, 679-705]. This, in turn, allows to keep into account barrier recrossing, i.e. the transmission coefficient that is employed in correcting transition state theory evaluations. To test the capabilities and the flaws of the approach we use as case studies two archetypal SN2 reactions. First, we consider to the standard substitution of chloride ion to bromomethane. The rate constant at 295.15 K is evaluated to k/c⊖ = 2.7 × 10-6 s-1 (with c⊖ = 1 M), which compares well to the experimental value of 3.3 × 10-6 s-1 [R. H. Bathgate and E. A. Melwyn-Hughes, J. Chem. Soc 1959, 2642-2648]. Then, the method is applied to the SN2 reaction of methylthiolate to dimethyl disulfide in water. In biology, such an interconversion of thiols and disulfides is an important metabolic topic still not entirely rationalized. The predicted rate constant is k/c⊖ = 7.7 × 103 s-1. No experimental data is available for such a reaction, but it is in accord with the fact that the alkyl thiolates to dialkyl disulfides substitutions in water have been found to be fast reactions [S. M. Bachrach, J. M. Hayes, T. Dao and J. L. Mynar, Theor. Chem. Acc. 2002, 107, 266-271].

Glycosidic linkage flexibility: The ψ torsion angle has a bimodal distribution in α-L-Rhap-(1→2)-α-L-Rhap-OMe as deduced from 13C NMR spin relaxation.

The molecular dynamics (MD) computer simulation technique is powerful for the investigation of conformational equilibrium properties of biomolecules. In particular, free energy surfaces of the torsion angles (those degrees of freedom from which the geometry mostly depends) allow one to access conformational states, as well as kinetic information, i.e., if the transitions between conformational states occur by simple jumps between wells or if conformational regions close to these states also are populated. The information obtained from MD simulations may depend substantially on the force field employed, and thus, a validation procedure is essential. NMR relaxation data are expected to be highly sensitive to the details of the torsional free energy surface. As a case-study, we consider the disaccharide α-l-Rhap-(1 → 2)-α-l-Rhap-OMe that features only two important torsion angles, ϕ and ψ, which define the interglycosidic orientation of the sugar residues relative to each other, governed mainly by the exo-anomeric effect and steric interactions, respectively. In water, a ψ- state is preferred, whereas in DMSO, it is a ψ+ state, suggesting inherent flexibility at the torsion angle. MD simulations indicated that bistable potentials describe the conformational region well. To test whether a unimodal distribution suffices or if a bimodal distribution better represents molecular conformational preferences, we performed an alchemical morphing of the torsional free energy surface and computed T1, T2, and NOE13C NMR relaxation data that were compared to experimental data. All three NMR observables are substantially affected by the morphing procedure, and the results strongly support a bimodal Boltzmann equilibrium density with a major and a minor conformational state bisected at ψ ≈ 0°, in accord with MD simulations in an explicit solvent.

DiTe2: Calculating the diffusion tensor for flexible molecules.

We report on an extended hydrodynamic modeling of the friction tensorial properties of flexible molecules including all types of natural, Z-Matrix like, internal coordinates. We implement the new methodology by extending and updating the software DiTe [Barone et al. J. Comput. Chem. 30, 2 (2009)]. DiTe (DIffusion TEnsor) implements a hydrodynamic modeling of the generalized translational, rotational, and configurational friction and diffusion tensors of flexible molecules in which flexibility is described in terms of dihedral angles. The new tool, DiTe2, has been renewed to include also stretching and bending types of internal mobility. Furthermore, DiTe2 is able to calculate the friction and diffusion tensors along collective (or reaction) coordinates defined as linear combinations of the internal natural ones. A number of tests are reported to show the new features of DiTe2. As leitmotiv for the tests, the calmodulin protein is taken into consideration, described both at all-atom and coarse-grained levels. © 2018 Wiley Periodicals, Inc.

Evaluating rotation diffusion properties of molecules from short trajectories.

We show that under proper assumptions it is possible to estimate with good precision the principal values of the rotational diffusion tensor of proteins from the analysis of short (up to 2-3 ns) molecular dynamics trajectories. We apply this analysis to a few model cases: three polyalanine peptides (2, 5, and 10 aminoacids), the fragment B3 of protein G (GB3), the bovine pancreatic trypsin inhibitor (BPTI), the hen egg-white lysozyme (LYS), the B1 domain of plexin (PB1), and thrombin. The protocol is based on the analysis of the global angular momentum autocorrelation functions, complementing the standard approach based on rotational autocorrelation functions, which requires much longer trajectories. A comparison with values predicted by hydrodynamic modeling and available experimental data is presented.

Stochastic modeling of macromolecules in solution. II. Spectral densities.

In Paper I [Polimeno et al., J. Chem. Phys. 150, 184107 (2019)], we proposed a general approach for interpreting relaxation properties of a macromolecule in solution, derived from an atomistic description. A simple scheme (the semiflexible Brownian, SFB, model) has been defined for the case of limited internal flexibility, but retaining full coupling with external degrees of freedom, inclusion of all of the momenta, and dissipation. Here we discuss the application of the SFB model to the practical evaluation of orientation spectral densities, based on two complementary computational treatments.

Stochastic modeling of macromolecules in solution. I. Relaxation processes.

A framework for the stochastic description of relaxation processes in flexible macromolecules, including dissipative effects, is introduced from an atomistic point of view. Projection-operator techniques are employed to obtain multidimensional Fokker-Planck operators governing the relaxation of internal coordinates and global degrees of freedom and depending upon parameters fully recoverable from classic force fields (energetics) and continuum models (friction tensors). A hierarchy of approaches of different complexity is proposed in this unified context, aimed primarily at the interpretation of magnetic resonance relaxation experiments. In particular, a model based on a harmonic internal Hamiltonian is discussed as a test case.

Loop Electrostatics Asymmetry Modulates the Preexisting Conformational Equilibrium in Thrombin.

Thrombin exists as an ensemble of active (E) and inactive (E*) conformations that differ in their accessibility to the active site. Here we show that redistribution of the E*-E equilibrium can be achieved by perturbing the electrostatic properties of the enzyme. Removal of the negative charge of the catalytic Asp102 or Asp189 in the primary specificity site destabilizes the E form and causes a shift in the 215-217 segment that compromises substrate entrance. Solution studies and existing structures of D102N document stabilization of the E* form. A new high-resolution structure of D189A also reveals the mutant in the collapsed E* form. These findings establish a new paradigm for the control of the E*-E equilibrium in the trypsin fold.

Flexibility at a glycosidic linkage revealed by molecular dynamics, stochastic modeling, and (13)C NMR spin relaxation: conformational preferences of α-L-Rhap-α-(1 → 2)-α-L-Rhap-OMe in water and dimethyl sulfoxide solutions.

The monosaccharide L-rhamnose is common in bacterial polysaccharides and the disaccharide α-L-Rhap-α-(1 → 2)-α-L-Rhap-OMe represents a structural model for a part of Shigella flexneri O-antigen polysaccharides. Utilization of [1'-(13)C]-site-specific labeling in the anomeric position at the glycosidic linkage between the two sugar residues facilitated the determination of transglycosidic NMR (3)JCH and (3)JCC coupling constants. Based on these spin-spin couplings the major state and the conformational distribution could be determined with respect to the ψ torsion angle, which changed between water and dimethyl sulfoxide (DMSO) as solvents, a finding mirrored by molecular dynamics (MD) simulations with explicit solvent molecules. The (13)C NMR spin relaxation parameters T1, T2, and heteronuclear NOE of the probe were measured for the disaccharide in DMSO-d6 at two magnetic field strengths, with standard deviations ≤1%. The combination of MD simulation and a stochastic description based on the diffusive chain model resulted in excellent agreement between calculated and experimentally observed (13)C relaxation parameters, with an average error of <2%. The coupling between the global reorientation of the molecule and the local motion of the spin probe is deemed essential if reproduction of NMR relaxation parameters should succeed, since decoupling of the two modes of motion results in significantly worse agreement. Calculation of (13)C relaxation parameters based on the correlation functions obtained directly from the MD simulation of the solute molecule in DMSO as solvent showed satisfactory agreement with errors on the order of 10% or less.

Towards bulk thermodynamics via non-equilibrium methods: gaseous methane as a case study.

We illustrate how the Jarzynski equality (JE), which is the progenitor of non-equilibrium methods aimed at constructing free energy landscapes for molecular-sized fluctuating systems subjected to steered transformations, can be applied to derive equations of state for bulk systems. The key-step consists of physically framing the computational strategy of "total energy morphing", recently presented by us as an efficient implementation of the JE [M. Zerbetto, A. Piserchia, D. Frezzato, J. Comput. Chem., 2014, 35, 1865-1881], in terms of build-up of the real thermodynamic state of a bulk material from the corresponding ideal state, in which the particles are non-interacting. In this context, the JE machinery yields the excess free energy versus suitably chosen controlled state variables, whose thermodynamic derivatives eventually lead to the equation of state. As an explanatory case study, we apply the methodology to derive the equation of state of gaseous methane by constructing the Helmholtz free energy versus the particle density (at fixed temperature) and then evaluating the thermodynamic derivative with respect to the volume. In our intent, this "old-style" work on gaseous methane should open the way for the investigation of thermodynamics of extended systems via non-equilibrium methods.

Probing the conformational energetics of alkyl thiols on gold surfaces by means of a morphing/steering non-equilibrium tool.

In this work we show that a non-equilibrium statistical tool based on Jarzynski's equality (JE) can be applied to achieve a sufficiently accurate mapping of the torsion free energy, bond-by-bond, for an alkyl thiol ligand tethered to a gold surface and sensing the presence of the surrounding cluster of similar chains. The strength of our approach is the employment of a strategy to let grow the internal energetics of the whole system (namely, the "energy morphing" stage recently presented by us in J. Comput. Chem., 2014, 35, 1865-1881) before initiating the rotational steering, which yields accurate results in terms of statistical uncertainties and bias on the free energy profiles. The work is mainly methodological and illustrates the feasibility of this kind of inspection on nanoscale molecular clusters with conformational flexibility. The outcomes for the archetype of self-assembled-monolayers considered here, a regular pattern of 10-carbon alkyl thiols on an ideal gold surface, give information on the conformational mobility of the ligands. Notably, such information is unlikely to be obtained by means of standard equilibrium techniques or by conventional molecular dynamics simulations.

Looking for some free energy? Call JEFREE ( ).

In this communication, we present the Jarzynski's Equality FREe Energy (JEFREE) library, an efficient and easy-to-use C++ library targeted to the calculation of the free energy profile along a selected generalized coordinate of a system, within the framework of the nonequilibrium steered transformations as introduced by Jarzynski [Phys. Rev. E, 1997, 56, 5018]. JEFREE can be readily integrated into any code, since both C and FORTRAN wrappers have been developed, and easily customizable by a user thanks to the object-oriented programming paradigm offered by the C++ language. Also, JEFREE implements the novel idea of making a total "morphing" of the system energy landscape before initiating the proper steering stage. This proves to be an efficient mean to overtake the problematic sampling of the initial equilibrium state when the number of degrees of freedom is high and the landscape owns many local minima separated by large energy barriers. The calculation of the free energy profile for the rotation along torsion angles in alkyl chains is presented as an example of application of our tool.

Lifetime shortening and fast energy-tansfer processes upon dimerization of a A-π-D-π-A molecule.

Time-resolved fluorescence and transient absorption experiments uncover a distinct change in the relaxation dynamics of the homo-dimer formed by two 2,5-bis[1-(4-N-methylpyridinium)ethen-2-yl)]-N-methylpyrrole ditriflate (M) units linked by a short alkyl chain when compared to that of the monomer M. Fluorescence decay traces reveal characteristic decay times of 1.1 ns and 210 ps for M and the dimer, respectively. Transient absorption spectra in the spectral range of 425-1050 nm display similar spectral features for both systems, but strongly differ in the characteristic relaxation times gathered from a global fit of the experimental data. To rationalize the data we propose that after excitation of the dimer the energy localizes on one M branch and then decays to a dark state, peculiar only of the dimer. This dark state relaxes to the ground state within 210 ps through non-radiative relaxation. The nature of the dark state is discussed in relation to different possible photophysical processes such as excimer formation and charge transfer between the two M units. Anisotropy decay traces of the probe-beam differential transmittance of M and the dimer fall on complete different time scales as well. The anisotropy decay for M is satisfactorily ascribed to rotational diffusion in DMSO, whereas for the dimer it occurs on a faster time scale and is likely caused by energy-transfer processes between the two monomer M units.

Analysis of Velocity Autocorrelation Functions from Molecular Dynamics Simulations of a Small Peptide by the Generalized Langevin Equation with a Power-Law Kernel.

Internal motions play an essential role in the biological functions of proteins and have been the subject of numerous theoretical and spectroscopic studies. Such complex environments are associated with anomalous diffusion where, in contrast to the classical Brownian motion, the relevant correlation functions have power law decays with time. In this work, we investigate the presence of long memory stochastic processes through the analysis of atomic velocity autocorrelation functions. Analytical expressions of the velocity autocorrelation function spectrum obtained through a Mori-Zwanzig projection approach were shown to be compatible with molecular dynamics simulations of a small helical peptide (8-polyalanine).

Intrinsic Flexibility beyond the Highly Ordered DNA Tetrahedron: An Integrative Spectroscopic and Molecular Dynamics Approach.

Since the introduction of DNA-based architectures, in the past decade, DNA tetrahedrons have aroused great interest. Applications of such nanostructures require structural control, especially in the perspective of their possible functionalities. In this work, an integrated approach for structural characterization of a tetrahedron structure is proposed with a focus on the fundamental biophysical aspects driving the assembly process. To address such an issue, spin-labeled DNA sequences are chemically synthesized, self-assembled, and then analyzed by Continuous-Wave (CW) and pulsed Electron Paramagnetic Resonance (EPR) spectroscopy. Interspin distance measurements based on PELDOR/DEER techniques combined with molecular dynamics (MD) thus revealed unexpected dynamic heterogeneity and flexibility of the assembled structures. The observation of flexibility in these ordered 3D structures demonstrates the sensitivity of this approach and its effectiveness in accessing the main dynamic and structural features with unprecedented resolution.

Interpretation of cw-ESR spectra of p-methyl-thio-phenyl-nitronyl nitroxide in a nematic liquid crystalline phase.

In this paper we report on the characterization by continuous wave electron spin resonance spectroscopy (cw-ESR) of a nitronyl nitroxide radical in a nematic phase. A detailed analysis is performed by exploiting an innovative modeling strategy alternative to the usual spectral simulation approach: most of the molecular parameters needed to calculate the spectrum are evaluated a priori and the ESR spectrum is obtained by direct application of the stochastic Liouville equation. Allowing a limited set of fitting parameters it is possible to reproduce satisfactorily ESR spectra in the temperature range 260 K-340 K including the nematic-to-isotropic phase transition (325.1 K). Our results open the way to a more quantitative understanding of the ordering and mobility of nitronyl nitroxide radicals in nanostructured environments.