ALBERT

Notes: Inelastic neutron scattering spectra are calculated from harmonic and damped harmonic models of the internal dynamics of a small protein, the bovine pancreatic trypsin inhibitor (BPTI). Numerical Fourier transformation of the intermediate scattering function Fvibinc (q, t) is used to calculate the inelastic scattering. This permits the inclusion of multiphonon scattering and frictional damping effects. Although for a typical experimental configuration, the multiphonon contribution does not significantly alter the form of the scattering at frequencies below about 30 cm−1, it does have a significant effect on the scattering intensity at higher frequencies. Frictional damping is introduced into the harmonic model by assuming that each mode acts as an independent damped Langevin oscillator. With this model and the assumption that the lowest frequency modes are overdamped while the higher frequency modes are underdamped, improved agreement with the experimental BPTI powder results is obtained. The measured scattering from BPTI in solution shows increased intensity at frequencies below 50 cm−1 relative to the powder results. The solution scattering profile can be reproduced approximately by the addition of overdamped Langevin oscillator normal modes to the dynamic model in best agreement with the powder data. Several other aspects of neutron scattering from proteins are examined. Anisotropy in the harmonic resolution broadened scattering is demonstrated. Spectra calculated assuming classical equations of motion are shown to agree with those calculated with the full quantum-mechanical dynamical model. Translational diffusion broadening is found to be small compared to the instrumental resolution broadening for the range of scattering wave vectors of interest. The contribution of the coherent scattering to the measured intensity is calculated for the case of a partially hydrated protein. Under typical experimental conditions, the measured cross sections are dominated by the incoherent scattering and the self part of the coherent scattering, a result that justifies the comparison of experimental data with calculated incoherent scattering spectra.

Notes: The range of validity of diffusive algorithms is studied by comparing the isomerization rates and rotational reorientation times from a series of Langevin dynamics trajectories of butane with the results of a Brownian dynamics (diffusive) trajectory, as well as with analytic approximations. It is found that inertial effects may be ignored for rotation at most liquid densities and contribute approximately 10% to the isomerization rate at water viscosities. For neat butane, or other short-chain alkane solvents at room temperature, the effect of inertial terms on isomerization rates is significant.

Notes: A theoretical force field for in-plane vibrations of benzene has been determined from ab initio calculations at both the Hartree–Fock level with 4-21G, 6-31G, and 6-31G* basis sets and the MP2 level with 4-21G and 6-31G basis sets. The average error of the calculated frequencies at the MP2 level is between 2% and 3%. The reliability of the force field and vibrational frequency predictions of the calculations are analyzed. All diagonal stretching force constants obtained at the MP2 level are in quantitative agreement with Ozkabak–Goodman experimental force field, while the diagonal force constants involving ring deformation and CH rock are somewhat overestimated by the theory. Most of the off-diagonal force constants agree with the Ozkabak–Goodman results in sign but there are some significant quantitative differences in magnitude. Comparisons are made with other force fields, including results obtained by scaling ab initio calculations or introducing modified Hamiltonians. A simple extrapolation method for introducing correlation corrections into Hartree–Fock force constants gives excellent results for benzene.

Notes: The free energy, energy, and entropy of solvation, relative to the pure liquid, are analyzed. By a coupling parameter integration it is shown that only averages over the solute–solvent interaction energy contribute to the free energy and that the solvent–solvent interaction term, which contributes the so-called cavity (solvent reorganization) term to the energy, is cancelled exactly by a corresponding term in the entropy. These terms exist even in the infinite dilution limit since they arise from the derivative of the free energy with respect to the solute density. Following the approach of Garisto et al. [J. Chem. Phys. 79, 6294 (1983)], the site–site Ornstein–Zernike integral equations and HNC closures are used to determine the derivatives of the distribution functions with respect to the density. This makes it possible to calculate the energetic and entropic contributions to the solvation free energy in the infinite dilution limit. The method is applied to pure solvent and to infinitely dilute aqueous solutions of cations, anions and neutral Lennard-Jones particles. The results are in agreement with numerical calculations of the thermodynamic quantities by use of finite difference values for the temperature derivatives. A simple empirical relation for the charge dependence of the solvation free energy is observed; it is shown for the case of an ion in a dipolar solvent, as typified by aqueous electrolyte solutions, that the free energy of solvation varies quadratically with the charge and is very nearly equal to one-half the solute–solvent portion of the solvation energy. Some discussion of the relation of the present results to entropy–enthalpy compensation and to computer simulations is given.

Notes: The McLachlan variational principle for the time-dependent Schrödinger equation is utilized in conjunction with extant localized Guassian wave packet technology to deduce equations of motion for general multidimensional Gaussians. These equations of motion are characterized by the same simplicity as the local quadratic expansion results of Heller [J. Chem. Phys. 62, 1544 (1975)]. However, the resultant variational wave packet evolution is shown to be an improvement over its local quadratic analog as a tool for computing certain photodissociation spectra. Numerical examples drawn from the Beswick–Jortner model of ICN photodissociation [Chem. Phys. 24, 1 (1977)] are presented.

Notes: A Brownian dynamics simulation of a lipid chain is used to model the motional properties of a dipalmitoyl phosphatidylcholine bilayer. The effects of the bilayer environment on the chain are represented by a mean field derived from an extension of the Marcelja model. The simulation was run 44 million steps, the equivalent of approximately 0.66 μs for a viscosity of 2.2 cp. The results are compared with those of a 30 million step simulation of the chain in the absence of the mean field. Deuterium order parameters for the methylene groups along the chain and the average chain length calculated from the mean field trajectory are shown to converge to the experimentally determined values for DPPC with an appropriate choice of parameters. An analysis of the torsional dynamics of the chain, including transition rates and kink probabilities, is carried out. It is demonstrated that kink formation is sometimes, though not always, concerted. A comparison of the membrane and free chain simulations implies that the internal dynamics of a hydrocarbon chain in a bilayer is very similar to that of a neat alkane. Thus, the significant limits on the orientational freedom of the chain, as expected by the nonzero order parameters, are induced by a potential that has only a small effect on the local motions.

Notes: The results of the Brownian dynamics simulation of a hydrocarbon chain in a membrane bilayer described in the preceding paper are used to analyze the 13C NMR T1 relaxation in lipid bilayer vesicles. The analysis shows that the frequency dependence of the relaxation does not arise from gauche–trans isomerization or from axial rotation of the entire lipid molecule. However, a model in which fast axial rotation (D(parallel)≈2×1010 s−1) and slow noncollective diffusive director fluctuations (D⊥≈1–2×108 s−1) are superimposed on the internal motions quantitatively accounts for both the magnitude and frequency dependence of the T1 data. An effective viscosity for the interior of the bilayer in the range of 1 cp, and a director order parameter of 0.5–0.7 are required to fit the NMR data. Collective effects do not appear necessary for explaining the NMR T1 data in vesicles, although they may be important for multilamellar dispersions.

Notes: The potential energy function about the C–C single bond for the ground state 1,3-butadiene has been derived from ab initio calculations at both the Hartree–Fock (HF) level with 6-31G, 6-31G*, and 6-311G** basis sets and the second-order Møller–Plesset perturbation (MP2) level with 6-31G* basis set with the complete geometry optimizations at each of 15 fixed CCCC dihedral angles; the total energies and optimized geometries for the s-trans, gauche, and s-cis conformers were also determined at MP2 level with 6-311G* basis set and the third-order Møller–Plesset perturbation (MP3) level with 6-31G* basis set. The second stable conformer of the butadiene is predicted to be a gauche structure from all the calculations with a CCCC dihedral angle between 35° and 40° and a barrier of 0.5–1.0 kcal/mol to the s-cis transition state, and the theoretical torsional potentials are in good agreement with the experimental potential function of trans–gauche–gauche case derived by Durig et al.; by contrast, the theoretical torsional components differ significantly from the experimental results obtained from a trans–cis model. Vibrational frequencies and force field for s-trans and gauche conformers of 1,3-butadiene are determined at the Hartree–Fock and MP2 levels with 6-31G, 6-31G*, 6-311G, and 6-311G* basis sets. The mean absolute percentage deviations of the calculated frequencies from the experimental values (not corrected for anharmonicity) are ∼10%–13% and 3%–6% for the Hartree–Fock and MP2 methods, respectively. The effects of polarization functions and electron correlation on the force fields are studied, and the additivity of correlation and d function effects are discussed. Comparisons are made with other force fields, including experimental and previous ab initio results.

Notes: A method is presented that uses integral equation theory to determine analytic temperature derivatives of the radial distribution functions. It is illustrated by studying the solvation thermodynamics of monatomic solutes in aqueous solution. The results agree well with the density derivative method developed previously [Yu and Karplus, J. Chem. Phys. 89, 2366 (1988)]. An expression for the solvation enthalpy is derived which allows direct comparison with experimental and isobaric–isothermal (NPT) ensemble simulation data. Satisfactory agreement with experiment is found for pure water and for the aqueous solvation of monovalent ions. Simple equations that exploit the site–site HNC closures are given for the decomposition of the potential of mean force into its enthalpic (or energetic) and entropic components. Since the extended RISM (HNC-RISM) theory yields an incorrect (trivial) value of the dielectric constant, two different ways to correct for the asymptotic behavior of the solute–solute potential of mean force are compared. They lead to similar results but the method in which the solvent dielectric constant is modified from the outset can be applied more generally.The interactions between nonpolar and between polar solutes in water are decomposed into enthalpic and entropic contributions. This is difficult to do by computer simulations because of the lack of precision in such calculations. The association of nonpolar solutes in water is found to have comparable enthalpic and entropic contributions; this result disagrees with the usual description of an entropy-dominated hydrophobic interaction. For ions, the somewhat surprising result is that the association of like-charged species is enthalpy driven while for oppositely charged ions entropic effects are dominant. The process of bringing two like-charged ions together leads to higher local charge density; the more favorable solvation enthalpy arising from this increase in charge density (q2 dependence) more than compensates for the Coulombic repulsion. For oppositely charged ions, association leads to a partial charge neutralization in which the favorable Coulombic attraction is overwhelmed by the loss of stabilizing solvation enthalpy. The entropic increase is due to the greater freedom of the surrounding water molecules resulting from the partial charge neutralization.

Notes: A general theoretical framework for treating the kinetics of diffusion-influenced bimolecular reactions in solution is presented. It is based on a hierachy of phenomenological kinetic equations for the reduced distribution functions of reactant molecules. With this formalism, a perturbation series expression for the rate coefficient for irreversible reactions involving a long-ranged sink function is derived. For a delta-function sink, it reduces to that obtained previously by Northrup and Hynes [Chem. Phys. Lett. 54, 244 (1978)]. It is demonstrated that the correctness of the Smoluchowski's expression for the rate coefficient in the low concentration limit results from a cancellation of errors. For diffusion-influenced reversible reactions involving a delta-function sink, explicit expressions for the time-dependent forward and reverse rate coefficients are derived. Experimental data on the relaxation kinetics of the triiodide ion formation reaction are reinterpreted, and consideration of diffusion effects is found to be essential. It is shown that the rate coefficient for the diffusive encounter of reactant molecules is not the upper bound to the bimolecular rate coefficient for reversible reactions.