Absolute free-energy calculations of liquids using a harmonic reference state

Absolute free-energy methods provide a potential solution to the overlap problem in free-energy calculations. In this paper, we report an extension of the previously published confinement method (J. Phys. Chem. B 2006, 110, 17212-20) to fluid simulations. Absolute free energies of liquid argon and liquid water are obtained accurately and compared with results from thermodynamic integration. The method works by transforming the liquid state into a harmonic, solid reference state. This is achieved using a special restraint potential that allows molecules to change their restraint position during the simulation, which circumvents the need for the molecules to sample the full extent of their translational freedom. The absolute free energy of the completely restrained reference state is obtained from a normal mode calculation. Because of the generic reference state used, the method is applicable to nonhomogeneous, diffusive systems and could provide an alternative method in situations in which solute annihilation fails due to the size of the solute. Potential applications include calculation of solvation energies of large molecules and free energies of peptide conformational changes in explicit solvent.     

Accurate calculations of free-energy differences by the distribution method

We employ the strategy used in the successive umbrella sampling method [P. Virnau and M. Muller, J. Chem. Phys. 120, 10925 (2004)] to obtain the energy-difference distribution over its desired range. This is very helpful in calculating free-energy differences, where the source of the error is well recognized as the insufficient sampling over the relevant tail region in the energy-difference distribution. The distribution method proposed here employs the idea of restricting the sampling within an appropriate energy range, as was presented by Shing and Gubbins in their restricted umbrella sampling method [Mol. Phys. 46, 1109 (1982)]. We demonstrate the efficiency of the distribution method by calculating the free-energy difference of a model of harmonic oscillators where the systems exhibit nonoverlap features in their important phase spaces through the original Metropolis sampling. For this particular case, we show that the distribution method outperforms the free-energy perturbation method and even the Bennett's acceptance ratio method [J. Comput. Phys. 22, 245 (1976)] with the fastest convergence and the smallest relative errors. We further demonstrate the application of the distribution method with a simple point charge water model.     

Phase-space overlap measures. II. Design and implementation of staging methods for free-energy calculations

We consider staged free-energy calculation methods in the context of phase-space overlap relations, and argue that the selection of work-based methods should be guided by consideration of the phase-space overlap of the systems of interest. Stages should always be constructed such that work is performed only into a system that has a phase-space subset relation with the starting system. Thus multiple stages are required if the systems of interest are not such that one forms a phase-space subset with the other. Three two-stage methods are possible, termed umbrella sampling, overlap sampling, and funnel sampling. The last is appropriate for cases in which the subset relation holds, but only in the extreme, meaning that one system's important phase space constitutes a very small portion of the others. Umbrella sampling is most suitable for nonoverlap systems, and overlap sampling is appropriate for systems exhibiting partial phase-space overlap. We review recently introduced metrics that characterize phase-space overlap, showing that the performance of the single- and two-stage methods is consistent with the phase-space picture. We also demonstrate that a recently introduced bias-detection measure is effective in identifying inaccuracy in single- and multistage calculations. The examples used are the chemical-potential calculation for a Lennard-Jones liquid at moderate and at high densities, the same for model water at ambient conditions, and a process of charging a neutral ion in water.     

Insights into the free-energy dependence of intramolecular dissociative electron transfers

To study the relationship between rate and driving force of intramolecular dissociative electron transfers, a series of donor-spacer-acceptor (D-Sp-A) systems has been devised and synthesized. cis-1,4-Cyclohexanedyil and a perester functional group were kept constant as the spacer and acceptor, respectively. By changing the aryl substituents of the phthalimide moiety, which served as the donor, the driving force could be varied by 0.74 eV. X-ray diffraction crystallography and ab initio conformational calculations pointed to D-Sp-A molecules having the cis-(cyclohexane) equatorial(phthalimido)-axial(perester) conformation and the same D/A orientation. The intramolecular dissociative electron-transfer process was studied by electrochemical means in N,N-dimethylformamide, in comparison with thermodynamic and kinetic information obtained with models of the acceptor and the donor. The intramolecular process consists of the electron transfer from the electrochemically generated phthalimide-moiety radical anion to the peroxide functional group. The electrochemical analysis provided clear evidence of a concerted dissociative electron-transfer mechanism, leading to the cleavage of the O-O bond. Support for this mechanism was obtained by ab initio MO calculations, which provided information about the LUMO of the acceptor and the SOMO of the donor. The intramolecular rate constants were determined and compared with the corresponding intermolecular values, the latter data being obtained by using the model molecules. As long as the effective location of the centroid of the donor SOMO does not vary significantly by changing the aryl substituent(s), the intramolecular dissociative electron transfer obeys the same main rules already highlighted for the corresponding intermolecular process. On the other hand, introduction of a nitro group drags the SOMO away from the acceptor, and consequently, the intramolecular rate drops by as much as 1.6 orders of magnitude from the expected value. Therefore, a larger solvent reorganization than for intermolecular electron transfers and the effective D/A distance and thus electronic coupling must be taken into account for quantitative predictions of intramolecular rates.     

Insights on the role of (dis)order from protein-protein interaction linear free-energy relationships

Protein-protein interactions (PPIs) are remarkably diverse and form the basis for various cellular functions. PPIs can be classified as ordered or disordered; the disordered ones do not have a well-defined structure prior to association, which is an exception to the conventional structure-function relationship. The occurrence of disordered proteins in functional roles is not explained by the conventional structure-function paradigm, and at present there is no clear understanding of the differences between the natures of these two PPIs. In this work, we studied the relationship between the kinetics and thermodynamics in PPIs to provide insights into the latter, with possible implications for the former. Analyzing the experimental data for various protein complexes, we found linear free-energy behavior with a striking kinetic difference between these two types of interactions. Binding affinities of (dis)ordered proteins are correlated with their (association) dissociation rates. Our observation, combined with the correspondence between biological activity and affinity, suggests that selection pressure on the dissociation or association kinetics in a functional context necessitates the presence of (dis)order in the structure.     

THE weighted histogram analysis method for free-energy calculations on biomolecules. I. The method

The Weighted Histogram Analysis Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomolecular Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free-energy and Potential of Mean Force calculations. This algorithm possesses the following advantages over methods that are currently employed: (1) It provides a built-in estimate of sampling errors thereby yielding objective estimates of the optimal location and length of additional simulations needed to achieve a desired level of precision; (2) it yields the “best” value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addition to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better estimates of the free-energy differences. By recasting the Ferrenberg–Swendsen Multiple Histogram equations in a form suitable for molecular mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine. © 1992 by John Wiley & Sons, Inc.     

Optimizing the driving function for nonequilibrium free-energy calculations in the linear regime: a variational approach

We consider the issue of optimizing linear-regime nonequilibrium simulations to estimate free-energy differences. In particular, we focus on the problem of finding the best-possible driving function lambda(t) that, for a given thermodynamic path, simulation algorithm, and amount of computational effort, minimizes dissipation. From the fluctuation-dissipation theorem it follows that, in the linear-response regime, the dissipation is controlled by the magnitude and characteristic correlation time of the equilibrium fluctuations in the driving force. As a result, the problem of finding the optimal switching scheme involves the solution of a standard problem in variational calculus: the minimization of a functional with respect to the switching function. In practice, the minimization involves solving the associated Euler-Lagrange equation subject to a set of boundary conditions. As a demonstration we apply the approach to the simple, yet illustrative problem of computing the free-energy difference between two classical harmonic oscillators with very different characteristic frequencies.     

Assessing the stability of free-energy perturbation calculations by performing variations in the method

We have calculated relative binding affinities for eight tetrafluorophenyl-triazole-thiogalactoside inhibitors of galectin-3 with the alchemical free-energy perturbation approach. We obtain a mean absolute deviation from experimental estimates of only 2–3 kJ/mol and a correlation coefficient (R²) of 0.5–0.8 for seven relative affinities spanning a range of up to 11 kJ/mol. We also studied the effect of using different methods to calculate the charges of the inhibitor and different sizes of the perturbed group (the atoms that are described by soft-core potentials and are allowed to have differing coordinates). However, the various approaches gave rather similar results and it is not possible to point out one approach as consistently and significantly better than the others. Instead, we suggest that such small and reasonable variations in the computational method can be used to check how stable the calculated results are and to obtain a more accurate estimate of the uncertainty than if performing only one calculation with a single computational setup.     

Reverse shuttling in a fullerene-stoppered rotaxane

The preparation and characterization of a solvent-switchable rotaxane that shuttles in the opposite direction to that expected are reported. The reverse shuttling is confirmed by NMR spectroscopy and can be monitored by cyclic voltammetry. The electrochemically generated anions on the fullerene moiety are stabilized by the closer proximity of the macrocycle.     

Formation of a rotaxane from the end-capping process of a pseudorotaxane. Effects of the solvent

The effects that the solvent exerts on the end-capping process of a pseudorotaxane formed by the [Ru(NH 3) 5(4,4'-bpy)]2+ complex and beta-cyclodextrin were studied. In this process the 4,4'-bpy ligand acts as rigid axle and the cyclodextrin as ring or macrocycle. The stopper used was the [Fe(CN) 5 H2O]3(-) complex. The solvents used were mixtures of ethyleneglycol-water and tert-butyl alcohol-water. Results showed similar, although strange, behavior in both media studied. Thus, a decrease of the observed rate constant was obtained when the concentration of cyclodextrin increases for all the media studied. However, at fixed cyclodextrin concentrations, an increase of k obs was obtained when small quantities of the cosolvent were added to the medium and, further, a decrease of k obs for the higher quantities of the organic solvents. This strange behavior could be explained by taking into account electrostatic and specific solvent (solvent-solvent and solvent-solute) effects.     

Nature of the excited state of triangulo silver N-heterocyclic carbenes. Insights from relativistic DFT calculations

Insights into the ground and excited states involved in the reported luminescent behavior of a complex involving the Ag(3) core stabilized by pyridil derivatives of N-heterocyclic carbenes has been achieved by means relativistic DFT calculations including scalar and spin-orbit coupling. The stabilization of the [Ag(3)](3+) core is enhanced by the population of a highly symmetric bonding Ag(3) orbital, composed of 75% from the 5s, 15% from 5p, and 10% from 4d. Thus, stabilization of the Ag(3) core involves a slightly bonded d(10) metallic core in addition to the pure nonbonding argentophilic interaction picture. It is suggested that the population of this highly bonding [Ag(3)](3+) orbital is responsible of the short Ag-Ag bond length observed in the studied compounds. The characterized electronic excitations allows to rule that the metal-ligand to ligand charge-transfer transitions account for the luminescent properties. The calculated Stokes shifts are in good agreement with the experimental data.     

Spatial arrangement of norbornanedicarboxylic acids N-arylimides

The effect of substituents on the spatial arrangement of norbornane compounds was investigated. Norbornanedicarboxylic acids N-arylimides with OCH₃, Cl, NO₂ groups in the ortho-position of the aromatic ring form as conformational isomers with hindered rotation. The specific spatial isomerism is confirmed by the quantum-chemical calculations.     

Insights into photodissociation dynamics of benzamide and formanilide from ab initio calculations

In the present study, the five lowest electronic states that control the UV photodissociation of formanilide and benzamide have been characterized using the complete active space self-consistent field theory. The mechanisms for the initial relaxation and subsequent dissociation processes have been determined on the basis of the calculated potential energy surfaces and their intersections. It was found that the S(1)/T(1)/T(2) three-surface intersection plays an important role in the photodissociation processes of benzamide. However, the dissociation behavior of formanilide and benzamide was found to be quite different from that for aliphatic amides. The present study provides several insights into the photodissociation dynamics of formanilide and benzamide.     

The construction of a supramolecular polymeric rotaxane from bipyridine-ruthenium and cyclodextrin

A luminescent supramolecular link is constructed by a very simple method using bipyridine-ruthenium and cyclodextrin, which displays not only a quasi-linear structure, but also a satisfactory fluorescence emission in both solution and the solid state.     

Selective photoinduced energy transfer from a thiophene rotaxane to acceptor

An energy transfer process was investigated using cyclodextrin-oligothiophene rotaxanes (2T-[2]rotaxane). The excited energy of 2T-[2]rotaxane is transferred to the sexithiophene derivative which is included in the cavity of β-CD stoppers of 2T-[2]rotaxane.