On the Statistical Regime, Coherence versus Incoherence and Ergodicity of Quantum Vibrational Trajectories in Soft Condensed Molecular Systems.

A theoretical-computational procedure, recently proposed for modelling Vibrational Energy Relaxation (VER) processes of a molecule (Quantum Center, QC) embedded in a complex atomic-molecular system, is extended and applied for analyzing in detail the features of the QC density matrix (DM) temporal evolution. The results, obtained using aqueous azide ion as a case study, show the total lack of coherence in the DM, when the system is prepared to be initially in a pure vibrational eigenstate. This finding is fully in line with the statistical interpretation of the process typically adopted also in the experimental studies where the relaxation processes are all described within the typical schemes of chemical kinetics. Consistently, when the initial vibrational state corresponds to an eigenstate mixture, although initially coherent, the DM relaxes to a fully incoherent condition with a mean lifetime related to the one of the diagonal elements relaxation. These specific DM features turn out to be essentially governed by the thermal equilibrium condition of the atomic-molecular classical coordinates which drive the ensemble of the quantum-trajectories toward the observed statistical regime. Finally, from the analysis of a single long timescale quantum vibrational trajectory it also clearly emerges its ergodic behaviour.

A Theoretical-Computational Study of Phosphodiester Bond Cleavage Kinetics as a Function of the Temperature.

The hydrolysis of the phosphodiester bond is an important chemical reaction involved in several biological processes. Here, we study the cleavage of this bond by means of a theoretical-computational method in a model system, the dineopentyl phosphate. By such an approach, we reconstructed the kinetics and related thermodynamics of this chemical reaction along an isochore. In particular, we evaluated the kinetic constants of all the reaction steps within a wide range of temperatures, mostly corresponding to conditions where no experimental measures are available due to the extremely slow kinetics. Our results, in good agreement with the experimental data, show the robustness of our theoretical-computational methodology which can be easily extended to more complex systems.

A statistical mechanical model of supercooled water based on minimal clusters of correlated molecules.

In this paper, we apply a theoretical model for fluid state thermodynamics to investigate simulated water in supercooled conditions. This model, which we recently proposed and applied to sub- and super-critical fluid water [Zanetti-Polzi et al., J. Chem. Phys. 156(4), 44506 (2022)], is based on a combination of the moment-generating functions of the enthalpy and volume fluctuations as provided by two gamma distributions and provides the free energy of the system as well as other relevant thermodynamic quantities. The application we make here provides a thermodynamic description of supercooled water fully consistent with that expected by crossing the liquid-liquid Widom line, indicating the presence of two distinct liquid states. In particular, the present model accurately reproduces the Widom line temperatures estimated with other two-state models and well describes the heat capacity anomalies. Differently from previous models, according to our description, a cluster of molecules that extends beyond the first hydration shell is necessary to discriminate between the statistical fluctuation regimes typical of the two liquid states.

Theoretical-Computational Modeling of CD Spectra of Aqueous Monosaccharides by Means of Molecular Dynamics Simulations and Perturbed Matrix Method.

The electronic circular dichroism (ECD) spectra of aqueous d-glucose and d-galactose were modeled using a theoretical-computational approach combining molecular dynamics (MD) simulations and perturbed matrix method (PMM) calculations, hereafter termed MD-PMM. The experimental spectra were reproduced with a satisfactory accuracy, confirming the good performances of MD-PMM in modeling different spectral features in complex atomic-molecular systems, as already reported in previous studies. The underlying strategy of the method was to perform a preliminary long timescale MD simulation of the chromophore followed by the extraction of the relevant conformations through essential dynamics analysis. On this (limited) number of relevant conformations, the ECD spectrum was calculated via the PMM approach. This study showed that MD-PMM was able to reproduce the essential features of the ECD spectrum (i.e., the position, the intensity, and the shape of the bands) of d-glucose and d-galactose while avoiding the rather computationally expensive aspects, which were demonstrated to be important for the final outcome, such as (i) the use of a large number of chromophore conformations; (ii) the inclusion of quantum vibronic coupling; and (iii) the inclusion of explicit solvent molecules interacting with the chromophore atoms within the chromophore itself (e.g., via hydrogen bonds).

Modelling Complex Bimolecular Reactions in a Condensed Phase: The Case of Phosphodiester Hydrolysis.

(1) Background: the theoretical modelling of reactions occurring in liquid phase is a research line of primary importance both in theoretical-computational chemistry and in the context of organic and biological chemistry. Here we present the modelling of the kinetics of the hydroxide-promoted hydrolysis of phosphoric diesters. (2) Method: the theoretical-computational procedure involves a hybrid quantum/classical approach based on the perturbed matrix method (PMM) in conjunction with molecular mechanics. (3) Results: the presented study reproduces the experimental data both in the rate constants and in the mechanistic aspects (C-O bond vs. O-P bond reactivity). The study suggests that the basic hydrolysis of phosphodiesters occurs through a concerted ANDN mechanism, with no formation of penta-coordinated species as reaction intermediates. (4) Conclusions: the presented approach, despite the approximations, is potentially applicable to a large number of bimolecular transformations in solution and therefore leads the way to a fast and general method to predict the rate constants and reactivities/selectivities in complex environments.

A general statistical mechanical model for fluid system thermodynamics: Application to sub- and super-critical water.

We propose in this paper a theoretical model for fluid state thermodynamics based on modeling the fluctuation distributions and, hence, the corresponding moment generating functions providing the free energy of the system. Using the relatively simple and physically coherent gamma model for the fluctuation distributions, we obtain a complete theoretical equation of state, also giving insight into the statistical/molecular organization and phase or pseudo-phase transitions occurring under the sub- and super-critical conditions, respectively. Application to sub- and super-critical fluid water and a comparison with the experimental data show that this model provides an accurate description of fluid water thermodynamics, except close to the critical point region where limited but significant deviations from the experimental data occur. We obtain quantitative evidence of the correspondence between the sub- and super-critical thermodynamic behaviors, with the super-critical water pseudo-liquid and pseudo-gas phases being the evolution of the sub-critical water liquid and gas phases, respectively. Remarkably, according to our model, we find that for fluid water the minimal subsystem corresponding to either the liquid-like or the gas-like condition includes an infinite number of molecules in the sub-critical regime (providing the expected singularities due to macroscopic phase transitions) but only five molecules in the super-critical regime (coinciding with the minimal possible hydrogen-bonding cluster), thus suggesting that the super-critical regime be characterized by the coexistence of nanoscopic subsystems in either the pseudo-liquid or the pseudo-gas phase with each subsystem fluctuating between forming and disrupting the minimal hydrogen-bonding network.

Modeling Charge Transfer Reactions by Hopping between Electronic Ground State Minima: Application to Hole Transfer between DNA Bases.

In this paper, we extend the previously described general model for charge transfer reactions, introducing specific changes to treat the hopping between energy minima of the electronic ground state (i.e., transitions between the corresponding vibrational ground states). We applied the theoretical-computational model to the charge transfer reactions in DNA molecules which still represent a challenge for a rational full understanding of their mechanism. Results show that the presented model can provide a valid, relatively simple, approach to quantitatively study such reactions shedding light on several important aspects of the reaction mechanism.

Theoretical Modeling of Redox Potentials of Biomolecules.

The estimation of the redox potentials of biologically relevant systems by means of theoretical-computational approaches still represents a challenge. In fact, the size of these systems typically does not allow a full quantum-mechanical treatment needed to describe electron loss/gain in such a complex environment, where the redox process takes place. Therefore, a number of different theoretical strategies have been developed so far to make the calculation of the redox free energy feasible with current computational resources. In this review, we provide a survey of such theoretical-computational approaches used in this context, highlighting their physical principles and discussing their advantages and limitations. Several examples of these approaches applied to the estimation of the redox potentials of both proteins and nucleic acids are described and critically discussed. Finally, general considerations on the most promising strategies are reported.

A Simplified Treatment for Efficiently Modeling the Spectral Signal of Vibronic Transitions: Application to Aqueous Indole.

In this paper, we introduce specific approximations to simplify the vibronic treatment in modeling absorption and emission spectra, allowing us to include a huge number of vibronic transitions in the calculations. Implementation of such a simplified vibronic treatment within our general approach for modelling vibronic spectra, based on molecular dynamics simulations and the perturbed matrix method, provided a quantitative reproduction of the absorption and emission spectra of aqueous indole with higher accuracy than the one obtained when using the existing vibronic treatment. Such results, showing the reliability of the approximations employed, indicate that the proposed method can be a very efficient and accurate tool for computational spectroscopy.

Theoretical-Computational Modeling of Gas-State Thermodynamics in Flexible Molecular Systems: Ionic Liquids in the Gas Phase as a Case Study.

A theoretical-computational procedure based on the quasi-Gaussian entropy (QGE) theory and molecular dynamics (MD) simulations is proposed for the calculation of thermodynamic properties for molecular and supra-molecular species in the gas phase. The peculiarity of the methodology reported in this study is its ability to construct an analytical model of all the most relevant thermodynamic properties, even within a wide temperature range, based on a practically automatic sampling of the entire conformational repertoire of highly flexible systems, thereby bypassing the need for an explicit search for all possible conformers/rotamers deemed relevant. In this respect, the reliability of the presented method mainly depends on the quality of the force field used in the MD simulations and on the ability to discriminate in a physically coherent way between semi-classical and quantum degrees of freedom. The method was tested on six model systems (n-butane, n-butane, n-octanol, octadecane, 1-butyl-3-methylimidazolium hexafluorophosphate and 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ionic pairs), which, being experimentally characterized and already addressed by other theoretical-computational methods, were considered as particularly suitable to allow us to evaluate the method's accuracy and efficiency, bringing out advantages and possible drawbacks. The results demonstrate that such a physically coherent yet relatively simple method can represent a further valid computational tool that is alternative and complementary to other extremely efficient computational methods, as it is particularly suited for addressing the thermodynamics of gaseous systems with a high conformational complexity over a large range of temperature.

Segregation on the nanoscale coupled to liquid water polyamorphism in supercooled aqueous ionic-liquid solution.

The most intriguing hypothesis explaining many water anomalies is a metastable liquid-liquid phase transition (LLPT) at high pressure and low temperatures, experimentally hidden by homogeneous nucleation. Recent infrared spectroscopic experiments showed that upon addition of hydrazinium trifluoroacetate to water, the supercooled ionic solution undergoes a sharp, reversible LLPT at ambient pressure, possible offspring of that in pure water. Here, we calculate the temperature-dependent signature of the OH-stretching band, reporting on the low/high density phase of water, in neat water and in the same experimentally investigated ionic solution. The comparison between the infrared signature of the pure liquid and that of the ionic solution can be achieved only computationally, providing insight into the nature of the experimentally observed phase transition and allowing us to investigate the effects of ionic compounds on the high to low density supercooled liquid water transition. We show that the experimentally observed crossover behavior in the ionic solution can be reproduced only if the phase transition between the low- and high-density liquid states of water is coupled to a mixing-unmixing transition between the water component and the ions: at low temperatures, water and ions are separated and the water component is a low density liquid. At high temperatures, water and ions get mixed and the water component is a high-density liquid. The separation at low temperatures into ion-rich and ion-poor regions allows unveiling the polyamorphic nature of liquid water, leading to a crossover behavior resembling that observed in supercooled neat water under high pressure.

Length-scale dependence of protein hydration-shell density.

Here we present a computational approach based on molecular dynamics (MD) simulation to study the dependence of the protein hydration-shell density on the size of the protein molecule. The hydration-shell density of eighteen different proteins, differing in size, shape and function (eight of them are antifreeze proteins), is calculated. The results obtained show that an increase in the hydration-shell density, relative to that of the bulk, is observed (in the range of 4-14%) for all studied proteins and that this increment strongly correlates with the protein size. In particular, a decrease in the density increment is observed for decreasing protein size. A simple model is proposed in which the basic idea is to approximate the protein molecule as an effective ellipsoid and to partition the relevant parameters, i.e. the solvent-accessible volume and the corresponding solvent density, into two regions: inside and outside the effective protein ellipsoid. It is found that, within the model developed here, almost all of the hydration-density increase is located inside the protein ellipsoid, basically corresponding to pockets within, or at the surface of the protein molecule. The observed decrease in the density increment is caused by the protein size only and no difference is found between antifreeze and non-antifreeze proteins.

Modeling amino-acid side chain infrared spectra: the case of carboxylic residues.

Infrared (IR) spectroscopy is commonly utilized for the investigation of protein structures and protein-mediated processes. While the amide I band provides information on protein secondary structures, amino acid side chains are used as IR probes for the investigation of protein reactions, such as proton pumping in rhodopsins. In this work, we calculate the IR spectra of the solvated aspartic acid, with both zwitterionic and protonated backbones, and of a capped form, i.e. mimicking the aspartic acid residue in proteins, by means of molecular dynamics (MD) simulations and the perturbed matrix method (PMM). This methodology has already proved its good modeling capabilities for the amide I mode and is here extended to the treatment of protein side chains. The computed side chain vibrational signal is in very good agreement with the experimental one, well reproducing both the peak frequency position and the bandwidth. In addition, the MD-PMM approach proposed here is able to reproduce the small frequency shift (5-10 cm-1) experimentally observed between the protonated and zwitterionic forms, showing that such a shift depends on the excitonic coupling between the modes localized on the side chain and on the backbone in the protonated form. The spectrum of the capped form, in which the amide I band is also calculated, agrees well with the corresponding experimental spectrum. The reliable calculation of the vibrational bands of carboxyl-containing side chains provides a useful tool for the interpretation of experimental spectra.

Evolutionary Modes in Protein Observable Space: The Case of Thioredoxins.

In this article, we investigated the structural and dynamical evolutionary behaviour of a set of ten thioredoxin proteins as formed by three extant forms and seven resurrected ones in laboratory. Starting from the crystallographic structures, we performed all-atom molecular dynamics simulations and compare the trajectories in terms of structural and dynamical properties. Interestingly, the structural properties related to the protein density (i.e. the number of residues divided by the excluded molecular volume) well describe the protein evolutionary behaviour. Our results also suggest that the changes in sequence as occurred during the evolution have affected the protein essential motions, allowing us to discriminate between ancient and extant proteins in terms of their dynamical behaviour. Such results are yet more evident when the bacterial, archaeal and eukaryotic thioredoxins are separately analysed.

Modelling vibrational relaxation in complex molecular systems.

In this paper we show how it is possible to treat the quantum vibrational relaxation of a chromophore, embedded in a complex atomic-molecular environment, via the explicit solution of the time-dependent Schroedinger equation once using a proper separation between quantum and semiclassical degrees of freedom. The rigorous theoretical framework derived, based on first principles and making use of well defined approximations/assumptions, is utilized to construct a general model for the kinetics of the vibrational relaxation as obtained by the direct evaluation of the density matrix for all the relevant quantum state transitions. Application to (deuterated) N-methylacetamide (the typical benchmark used as a model for the amino acids) shows that the obtained theoretical-computational approach captures the essential features of the experimental process, unveiling the basic relaxation mechanism involving several vibrational state transitions.

Theoretical-computational modelling of the temperature dependence of the folding-unfolding thermodynamics and kinetics: the case of a Trp-cage.

Here we present a theoretical-computational study of the thermodynamics and kinetics of an aqueous Trp-cage, a 20-residue long miniprotein. The combined use of accurate molecular dynamics simulations rigorously reconstructing the proper isobar of the system and a sound statistical-mechanical model provides a quantitative description of the temperature dependence of the relevant physical-chemical properties and insights into the detailed mechanisms regulating the folding-unfolding properties.

Tyrosine absorption spectroscopy: Backbone protonation effects on the side chain electronic properties.

The UV-vis spectrum of Tyrosine and its response to different backbone protonation states have been studied by applying the Perturbed Matrix Method (PMM) in conjunction with molecular dynamics (MD) simulations. Herein, we theoretically reproduce the UV-vis absorption spectrum of aqueous solution of Tyrosine in its zwitterionic, anionic and cationic forms, as well as of aqua-p-Cresol (i.e., the moiety that constitutes the side chain portion of Tyrosine). To achieve a better accuracy in the MD sampling, the Tyrosine Force Field (FF) parameters were derived de novo via quantum mechanical calculations. The UV-vis absorption spectra are computed considering the occurring electronic transitions in the vertical approximation for each of the chromophore configurations sampled by the classical MD simulations, thus including the effects of the chromophore semiclassical structural fluctuations. Finally, the explicit treatment of the perturbing effect of the embedding environment permits to fully model the inhomogeneous bandwidth of the electronic spectra. Comparison between our theoretical-computational results and experimental data shows that the used model captures the essential features of the spectroscopic process, thus allowing to perform further analysis on the strict relationship between the quantum properties of the chromophore and the different embedding environments. © 2018 Wiley Periodicals, Inc.

Extending the perturbed matrix method beyond the dipolar approximation: comparison of different levels of theory.

Some years ago we developed a theoretical-computational hybrid quantum/classical methodology, the Perturbed Matrix Method (PMM), to be used in conjunction with molecular dynamics simulations for the investigation of chemical processes in complex systems, that proved to be a valuable tool for the simulation of relevant experimental observables, e.g., spectroscopic signals, reduction potentials, kinetic constants. In typical PMM calculations the quantum sub-part of the system, the quantum centre, is embedded into an external perturbing field providing a perturbation operator explicitly calculated up to the dipolar terms. In this paper we further develop the PMM approach, beyond the dipolar terms in the perturbation operator expansion, by including explicitly the quadrupolar terms and/or by expanding the perturbation operator on each atom of the quantum centre. These different levels of the perturbation operator expansion, providing different levels of theory, have been tested by calculating three different spectroscopic observables: the spectral signal of liquid water and aqueous benzene due to the lowest energy electronic excitation and the infrared amide I band of aqueous trans-N-methylacetamide. All the systems tested show that, even though the previous PMM level of theory is already capable of reproducing the main features of the spectral signal, the higher levels of theory improve the quantitative reproduction of the spectral details.

The unfolding effects on the protein hydration shell and partial molar volume: a computational study.

In this paper we apply the computational analysis recently proposed by our group to characterize the solvation properties of a native protein in aqueous solution, and to four model aqueous solutions of globular proteins in their unfolded states thus characterizing the protein unfolded state hydration shell and quantitatively evaluating the protein unfolded state partial molar volumes. Moreover, by using both the native and unfolded protein partial molar volumes, we obtain the corresponding variations (unfolding partial molar volumes) to be compared with the available experimental estimates. We also reconstruct the temperature and pressure dependence of the unfolding partial molar volume of Myoglobin dissecting the structural and hydration effects involved in the process.

Extending the essential dynamics analysis to investigate molecular properties: application to the redox potential of proteins.

Here, a methodology is proposed to investigate the collective fluctuation modes of an arbitrary set of observables, maximally contributing to the fluctuation of another functionally relevant observable. The methodology, based on the analysis of fully classical molecular dynamics (MD) simulations, exploits the essential dynamics (ED) method, originally developed to analyse the collective motions in proteins. We apply this methodology to identify the residues that are more relevant for determining the reduction potential (E(0)) of a redox-active protein. To this aim, the fluctuation modes of the single-residue electrostatic potentials mostly contributing to the fluctuations of the total electrostatic potential (the main determinant of E(0)) are investigated for wild-type azurin and two of its mutants with a higher E(0). By comparing the results here obtained with a previous study on the same systems [Zanetti-Polzi et al., Org. Biomol. Chem., 2015, 13, 11003] we show that the proposed methodology is able to identify the key sites that determine E(0). This information can be used for a general deeper understanding of the molecular mechanisms on the basis of the redox properties of the proteins under investigation, as well as for the rational design of mutants with a higher or lower E(0). From the results of the present analysis we propose a new azurin mutant that, according to our calculations, shows a further increase of E(0).