Molecular Partition Function Research Articles

Chemical Protein Unfolding - A Simple Cooperative Model.

Chemical unfolding with guanidineHCl or urea is a common method to study the conformational stability of proteins. The analysis of unfolding isotherms is usually performed with an empirical linear extrapolation method (LEM). A large positive free energy is assigned to the native protein, which is usually considered to be a minimum of the free energy. The method thus contradicts common expectations. Here, we present a multistate cooperative model that addresses specifically the binding of the denaturant to the protein and the cooperativity of the protein unfolding equilibrium. The model is based on a molecular statistical-mechanical partition function of the ensemble, but simple solutions for the calculation of the binding isotherm and the associated free energy are presented. The model is applied to 23 published unfolding isotherms of small and large proteins. For a given denaturant, the binding constant depends on temperature and pH but shows little protein specificity. Chemical unfolding is less cooperative than thermal unfolding. The cooperativity parameter σ is at least 2 orders of magnitude larger than that of thermal unfolding. The multistate cooperative model predicts zero free energy for the native protein, which becomes strongly negative beyond the midpoint concentration of unfolding. The free energy to unfold a cooperative unit corresponds exactly to the diffusive energy of the denaturant concentration gradient necessary for unfolding. The temperature dependence of unfolding isotherms yields the denaturant-induced unfolding entropy and, in turn, the unfolding enthalpy. The multistate cooperative model provides molecular insight and is as simple to apply as the LEM but avoids the conceptual difficulties of the latter.

Mechanisms for gas-phase molecular formation of neutral formaldehyde (H2CO) in cold astrophysical regions

Context. Formaldehyde is a potential biogenic precursor involved in prebiotic chemical evolution. The cold conditions of the interstellar medium (ISM) allow H2CO to be reactive, playing a significant role as a chemical intermediate in formation pathways leading to interstellar complex organic molecules. However, gas-phase molecular formation mechanisms in cold regions of the ISM are poorly understood. Aims. We computationally determine the most favored gas-phase molecular formation mechanisms at local thermodynamic equilibrium conditions that can produce the detected amounts of H2CO in diffuse molecular clouds (DMCs), in dark, cold, and dense molecular clouds (DCDMCs), and in three regions of circumstellar envelopes of low-mass protostars (CELMPs). Methods. The potential energy surfaces, thermodynamic functions, and single-point energies for transition states were calculated at the CCSD(T)-F12/cc-pVTZ-F12 and MP2/aug-cc-pVDZ levels of theory and basis sets. Molecular thermodynamics and related partition functions were obtained by applying the Maxwell-Boltzmann quantum statistics theory from energies computed at CCSD(T)-F12/cc-pVTZ-F12 with corrections for zero-point energy. A literature review on detected abundances of reactants helped us to propose the most favorable formation routes. Results. The most probable reactions that produce H2CO in cold astrophysical regions are: 1CH2 + ⋅3O2 →1H2CO + O⋅(3P) in DMCs, ⋅3CH2 + ⋅3O2 →1H2CO + ⋅O(3P) in DCDMCs, and ⋅CH3 + ⋅O(3P) →1H2CO + ⋅H in region III, ⋅CH3 +⋅O(1D) →1H2CO + ⋅H in region II, and 1CH2 + ⋅3O2 →1H2CO + ⋅O(3P) in region I belonging to CELMPs. Conclusions. Quantum chemical calculations suggest that the principal carbonaceous precursors of H2CO in cold regions for the gas-phase are CH2(a1A1), and ⋅CH2(X3B1) combined with ⋅O2(3Σg) and ⋅CH3(2A”) + ⋅O(3P) / O(1D). Reactions based on more complex reagents yield less effective thermodynamics in the gas-phase H2CO molecular formation.

Prediction of thermodynamic properties of formation of HCl at temperatures 200–1000 K

In this work, the thermodynamic properties of the formation reaction of hydrogen chloride from the reactants chlorine and hydrogen are studied by employing the developed Manning-Rosen potential. A closed-form expression for the vibration part of the molecular partition function is used to calculate the vibrational contribution of the thermodynamic properties. The results obtained are used to investigate the change in the thermodynamic functions for the formation reaction of HCl at a high temperature range. The predicted results are compared with the ones obtained from the other theoretical approaches based on the harmonic and an-harmonic oscillator models.

High-temperature ideal-gas partition function of the molecule

The molecular partition functions of even the simplest molecules are uncertain at high temperatures. This paper presents an investigation of the existing discrepancies for the molecular ion. The specific aim, apart from a comparison of various results, is to include all interactions between atoms in the molecular partition function to obtain those values which can be called the ideal-gas partition function. A classical mechanics-based method is used, and to extend the results to temperatures below 7000 K the Wigner–Kirkwood quantum correction is applied. Analysis of the results of this paper and others shows significant differences from the results currently known in the field of molecular physics and those used in the field of astrophysics.

On the exclusion of the negative contribution to the molecular partition function

ABSTRACTThe article presents classical mechanics approach to thermodynamic properties of molecules at high temperatures which is based on a specific realisation of the idea of excluded volume and in the end avoids the use of second virial coefficient (in contrary to other approaches). The aim of this approach is to remove the negative contribution to the molecular partition function. The influence of the negative part on the partition function is investigated for the ground state of Na molecule. The full molecular partition functions (with bound, resonance and scattering contributions) are compared with the ones with negative contribution excluded. Additionally the applicability of the hard spheres repulsion is investigated and the comparison with an exact free (scattering) contribution is made.

Assessing the Molecular Basis of the Fuel Octane Scale: A Detailed Investigation on the Rate Controlling Steps of the Autoignition of Heptane and Isooctane.

N-Heptane and 2,2,4-trimethylpentane (isooctane) are the key species in the modeling of ignition of hydrocarbon-based fuel formulations. Isooctane is knock-resistant whereas n-heptane is a very knock-prone hydrocarbon. It has been suggested that interconversion of their associated alkylperoxy and hydroperoxyalkyl species via hydrogen-transfer isomerization reaction is the key step to understand their different knocking behavior. In this work, the kinetics of unimolecular hydrogen-transfer reactions of n-heptylperoxy and isooctylperoxy are determined using canonical variational transition-state theory and multidimensional small curvature tunneling. Internal rotation of involved molecules is taken explicitly into account in the molecular partition function. The rate coefficients are calculated in the temperature range 300-900 K, relevant to low-temperature autoignition. The concerted HO2 elimination is an important reaction that competes with some H-transfer and is associated with chain termination. Thus, the branching ratio between these reaction channels is analyzed. We show that variational and multidimensional tunneling effects cannot be neglected for the H-transfer reaction. In particular, the pre-exponential Arrhenius fitting parameter derived from our rate constants shows a strong dependence on the temperature, because tunneling increases quickly at temperatures below 500 K. On the basis of our results, the existing qualitative model for the reasons for different knock behavior observed for n-heptane and isooctane is quantitatively validated at the molecular level.

Dual-Level Method for Estimating Multistructural Partition Functions with Torsional Anharmonicity.

For molecules with multiple torsions, an accurate evaluation of the molecular partition function requires consideration of multiple structures and their torsional-potential anharmonicity. We previously developed a method called MS-T for this problem, and it requires an exhaustive conformational search with frequency calculations for all the distinguishable conformers; this can become expensive for molecules with a large number of torsions (and hence a large number of structures) if it is carried out with high-level methods. In the present work, we propose a cost-effective method to approximate the MS-T partition function when there are a large number of structures, and we test it on a transition state that has eight torsions. This new method is a dual-level method that combines an exhaustive conformer search carried out by a low-level electronic structure method (for instance, AM1, which is very inexpensive) and selected calculations with a higher-level electronic structure method (for example, density functional theory with a functional that is suitable for conformational analysis and thermochemistry). To provide a severe test of the new method, we consider a transition state structure that has 8 torsional degrees of freedom; this transition state structure is formed along one of the reaction pathways of the hydrogen abstraction reaction (at carbon-1) of ketohydroperoxide (KHP; its IUPAC name is 4-hydroperoxy-2-pentanone) by OH radical. We find that our proposed dual-level method is able to significantly reduce the computational cost for computing MS-T partition functions for this test case with a large number of torsions and with a large number of conformers because we carry out high-level calculations for only a fraction of the distinguishable conformers found by the low-level method. In the example studied here, the dual-level method with 40 high-level optimizations (1.8% of the number of optimizations in a coarse-grained full search and 0.13% of the number of optimizations in a fine-grained full search) reproduces the full calculation of the high-level partition function within a factor of 1.0 to 2.0 from 200 to 1000 K. The error in the dual-level method can be further reduced to factors of 0.6 to 1.1 over the whole temperature interval from 200 to 2400 K by optimizing 128 structures (5.9% of the number of optimizations in a fine-grained full search and 0.41% of the number of optimizations in a fine-grained full search). These factor-of-two or better errors are small compared to errors up to a factor of 1.0 × 103 if one neglects multistructural effects for the case under study.

A whole-path importance-sampling scheme for Feynman path integral calculations of absolute partition functions and free energies.

Using Feynman path integrals, a molecular partition function can be written as a double integral with the inner integral involving all closed paths centered at a given molecular configuration, and the outer integral involving all possible molecular configurations. In previous work employing Monte Carlo methods to evaluate such partition functions, we presented schemes for importance sampling and stratification in the molecular configurations that constitute the path centroids, but we relied on free-particle paths for sampling the path integrals. At low temperatures, the path sampling is expensive because the paths can travel far from the centroid configuration. We now present a scheme for importance sampling of whole Feynman paths based on harmonic information from an instantaneous normal mode calculation at the centroid configuration, which we refer to as harmonically guided whole-path importance sampling (WPIS). We obtain paths conforming to our chosen importance function by rejection sampling from a distribution of free-particle paths. Sample calculations on CH4 demonstrate that at a temperature of 200 K, about 99.9% of the free-particle paths can be rejected without integration, and at 300 K, about 98% can be rejected. We also show that it is typically possible to reduce the overhead associated with the WPIS scheme by sampling the paths using a significantly lower-order path discretization than that which is needed to converge the partition function.

The Physical Significance and the Role of the Molecular Partition Function

The Physical Significance and the Role of the Molecular Partition Function

Single-shot hyperspectral coherent Raman planar imaging in the range 0–4200 cm−1

We propose a technique for ultrabroadband planar coherent Raman spectroscopy that enables wideband chemically selective mapping of molecular partition functions in the gas-phase within a single-laser-shot. A spectral region spanning 0–4200 cm−1 is excited simultaneously, in principle allowing for coherent planar imaging of most all fundamental Raman-active modes. This unique instantaneous and spatially correlated assessment enables multiplexed studies of transient dynamical systems in a two-dimensional (2D) field. Here, we demonstrate single-laser-shot high temperature diagnostics of H2, with spatially resolved 2D measurement of transitions of both the pure-rotational H2 S-branch and the vibrational H2 Q-branch, analyzing the temperature contour of a reacting fuel-species as it evolves at a flame-front.

Binding polynomial in molecular self-assembly.

In the present work the concept of a binding polynomial is revisited for the most widely used case of self-assembly of identical molecular units and results in the re-construction of a link to the grand partition function of such a system. It is found that if the self-assembly process is not pronounced (i.e., the product of the equilibrium constant and the monomer concentration is close to zero), the binding polynomial has the meaning of a molecular partition function that is given by the summation over energy levels of any molecule in the system. In other cases the concept of a binding polynomial may be misleading.

On the Thermodynamics of Refrigerant + Heterogeneous Solid Surfaces Adsorption

This Article presents a theoretical framework for the understanding of pressurized adsorption systems using the statistical rate methodology. Utilizing results from the statistical rate theory, basic thermodynamic variables including enthalpy (h(a)), entropy (s(a)), and the specific heat capacity (c(p,a)) of the adsorbed phase are derived using the thermodynamic requirements of chemical equilibrium, Gibbs law, as well as Maxwell relations. A built-in constant (K) describes the adsorbed molecular partition function (q(s)), and it captures the heterogeneous properties of the adsorbent + adsorbate pair at equilibrium states. Improved adsorbed-phase volume considerations were incorporated in the formulations of these variables where they could be utilized with relative ease for analyzing the energetic performances of any practical adsorption system. In this Article, we have demonstrated how derived thermodynamic quantities can bridge the information gap with respect to the states of adsorbed phase, as well as resolved some theoretical inconsistencies that were found in previously derived quantities. Experimentally, the adsorption isotherms of propane (refrigerant) on activated carbon powder (Maxsorb III) for temperatures from 5 to 75 °C and pressures up to 8 bar are presented, and they are used to illustrate the behaviors of the adsorbed-phase during uptakes, temperatures, and pressure excursions or changes.

Computational study on the structure and properties of ternary complexes of Ln3+ (Ln = La, Ce, Nd AND Sm) with 5,7-dichloroquinoline-8-ol and 4-vinyl pyridine

In the present research, we have mainly concentrated on the survey of interactions in Ln3+ (Ln = La, Ce, Nd, and Sm) ternary complexes of 5,7-dichloroquinoline-8-ol (DCQ) and 4-vinyl pyridine (VP), [Ln(VP)2(DCQ)3]3+ by means of density functional theory, Hartree-Fock and Sparkle/PM3 semi-empirical computational methods. For VP and DCQ ligands, the cation binding energy sequence follows the order La3+ > Ce3+ > Nd3+ > Sm3+ as expected based on increasing in the hardness and decreasing in the ionic radius of this lanthanide cation series. A similar trend was observed in the calculated binding energy of the aforesaid ligands with the hydrated lanthanide cation series [Ln(H2O)9]3+, while the computed values of deformation energy of ligands upon complexation demonstrated an opposite order in the lanthanide cation series. Moreover, the solvent effects are considered via a polarized continuum model and provided a significant increase in the binding strength while the relative magnitude of binding energies is the same as that in the gas phase. Combining quantum and statistical mechanical calculations, we have also determined quantitatively a reliable estimate of the conformational distribution of the [Sm(VP)2(DCQ)3]3+ complex at various temperatures in the gas phase by computing the molecular partition functions and consequently the analysis of the conformational equilibrium constants.

Low-energy conformers of pamidronate and their intramolecular hydrogen bonds: a DFT and QTAIM study

Extensive DFT and ab initio calculations were performed to characterize the conformational space of pamidronate, a typical pharmaceutical for bone diseases. Mono-, di- and tri-protic states of molecule, relevant for physiological pH range, were investigated for both canonical and zwitterionic tautomers. Semiempirical PM6 method were used for prescreening of the single bond rotamers followed by geometry optimizations at the B3LYP/6-31++G(d,p) and B3LYP/6-311++G(d,p) levels. For numerous identified low energy conformers the final electronic energies were determined at the MP2/6-311++G(2df,2p) level and corrected for thermal effects at B3LYP level. Solvation effects were also considered via the COSMO and C-PCM implicit models. Reasonable agreement was found between bond lengths and angle values in comparison with X-ray crystal structures. Relative equilibrium populations of different conformers were determined from molecular partition functions and the role of electronic, vibrational and rotational degrees of freedom on the stability of conformers were analyzed. For no level of theory is a zwitterionic structure stable in the gas-phase while solvation makes them available depending on the protonation state. Geometrically identified intramolecular hydrogen bonds were analyzed by QTAIM approach. All conformers exhibit strong inter-phosphonate hydrogen bonds and in most of them the alkyl-amine side chain is folded on the P-C-P backbone for further hydrogen bond formation.

Use of Solution-Phase Vibrational Frequencies in Continuum Models for the Free Energy of Solvation

We find that vibrational contributions to a solute's free energy are in general insensitive to whether the solute vibrational frequencies are computed in the gas phase or in solution. In most cases, the difference is smaller than the intrinsic error in solvation free energies associated with the continuum approximation to solvation modeling, although care must be taken to avoid spurious results associated with limitations in the quantum-mechanical harmonic-oscillator approximation for very low-frequency molecular vibrations. We compute solute vibrational partition functions in aqueous and carbon tetrachloride solution and compare them to gas-phase molecular partition functions computed with the same level of theory and the same quasiharmonic approximation for the diverse and extensive set of molecules and ions included in the training set of the SMD continuum solvation model, and we find mean unsigned differences in vibrational contributions to the solute free energy of only about 0.2 kcal/mol. On the basis of these results and a review of the theory, we conclude, in contrast to previous work (Ho, J.; Klamt, H.; Coote, M. L. J. Phys. Chem. A 2010, 114, 13442), that using partition functions computed for molecules optimized in solution is a correct and useful approach for averaging over solute degrees of freedom when computing free energies of solutes in solution, and it is moreover recommended for cases where liquid and gas-phase solute structures differ appreciably or when stationary points present in liquid solution do not exist in the gas phase, for which we provide some examples. When gas-phase and solution-phase geometries and frequencies are similar, the use of gas-phase geometries and frequencies is a useful approximation.

First-principles simulation of molecular dissociation–recombination equilibrium

For the first time, the equilibrium composition of chemical dissociation-recombination reaction is simulated from first-principles, only. Furthermore, beyond the conventional ab initio Born-Oppenheimer quantum chemistry the effects from the thermal and quantum equilibrium dynamics of nuclei are consistently included, as well as, the nonadiabatic coupling between the electrons and the nuclei. This has been accomplished by the path integral Monte Carlo simulations for full NVT quantum statistics of the H(3)(+) ion. The molecular total energy, partition function, free energy, entropy, and heat capacity are evaluated in a large temperature range: from below room temperature to temperatures relevant for planetary atmospheric physics. Temperature and density dependent reaction balance of the molecular ion and its fragments above 4000 K is presented, and also the density dependence of thermal ionization above 10,000 K is demonstrated.

Free energy calculations for a flexible water model

In this work, we consider the problem of calculating the classical free energies of liquids and solids for molecular models with intramolecular flexibility. We show that thermodynamic integration from the fully-interacting solid of interest to a Debye crystal reference state, with anisotropic harmonic interactions derived from the Hessian of the original crystal, provides a straightforward route to calculating the Gibbs free energy of the solid. To calculate the molecular liquid free energy, it is essential to correctly account for contributions from both intermolecular and intramolecular motion; we employ thermodynamic integration to a Lennard-Jones reference fluid, coupled with direct evaluation of the molecular ro-vibrational partition function. These approaches are used to study the low-pressure classical phase diagram of the flexible q-TIP4P/F water model. We find that, while the experimental ice-I/liquid and ice-III/liquid coexistence lines are described reasonably well by this model, the ice-II phase is predicted to be metastable. In light of this finding, we go on to examine how the coupling between intramolecular flexibility and intermolecular interactions influences the computed phase diagram by comparing our results with those of the underlying rigid-body water model.

Symmetry Numbers for Rigid, Flexible, and Fluxional Molecules: Theory and Applications

The use of molecular simulations and ab initio calculations to predict thermodynamic properties of molecules has become routine. Such methods rely upon an accurate representation of the molecular partition function or configurational integral, which in turn often includes a rotational symmetry number. However, the reason for including the symmetry number is unclear to many practitioners, and there is also a need for a general prescription for evaluating the symmetry numbers of flexible molecules, i.e., for molecules with thermally active internal degrees of freedom, such as internal rotors. Surprisingly, we have been unable to find any complete and convincing explanations of these important issues in textbooks or the journal literature. The present paper aims to explain why symmetry numbers are needed and how their values should be determined. Both classical and quantum approaches are provided.

Fast Prediction of Protein Thermodynamics

A computational method for a Flexibility And Stability Test (FAST) on three-dimensional protein structure is described. A four-dimensional free energy landscape defined by temperature and three order parameters is calculated in matter of minutes. The order parameters characterize structure and solvent properties. Specifically, they depict the number of native vs. disordered residues that are within a particular solvent state. Thermodynamic properties are derived from the free energy landscape, including stability curves and heat capacity over all experimentally accessible temperatures. Protein flexibility and correlated motions for a given thermodynamic and solvent condition are also calculated to highlight structural mechanisms. A Free Energy Decomposition (FED) is employed to account for essential enthalpy-entropy compensation mechanisms that include: hydrogen bonding, chemical diversity among residues, atomic packing, strain and vibration energy, pH effects, solvation effects such as clathrate water interacting with residues, hydrophobic effects due to water transfer from buried regions to bulk solvent, and network rigidity. Network rigidity is a long-range underlying mechanical interaction that accounts for conformational entropy nonadditivity during Free Energy Reconstitution (FER). In contrast to molecular simulation, FAST is based on a free energy functional that is solved using self-consistent mean field theory. Individual free energy components come from molecular partition functions that are parameterized from a combination of long all-atom molecular dynamics simulations in explicit solvent, and empirical fitting to experimental data. FAST is a unified model that accounts for several modes of protein denaturation driven by extreme: temperature, pH, and concentration of co-solute. Pressure dependence will also be incorporated in future work. Because of its computational efficiency, FAST can be used in high throughput applications (i.e., design) to assess the consequences of all FED components on the free energy. This work is supported by NIH grant R01-GM073082.

A thermal self-consistent field theory for the calculation of molecular vibrational partition functions

A new approach for the calculation of anharmonic molecular vibrational partition functions is developed based on a separable ansatz to the thermal density matrix. The parameters appearing in the effective single particle Hamiltonians that generate the thermal density matrices are determined variationally. The resulting equations are the thermal analogs of the vibrational self-consistent field approximation. The method has the formal property that the free energy calculated by this approach is an upper bound to the exact free energy. Thermodynamic quantities calculated by this approach are generally in good agreement with the results of numerically converged calculations. This approach is more efficient than the standard sum over state approaches in that the computational resources scale with N(4) where N is the number of vibrational degrees of freedom. Thus it can be applied to fairly large systems.