Hydrodynamic modeling of diffusion tensor properties of flexible molecules

We present a computationally efficient implementation of hydrodynamic modeling for the evaluation of diffusion tensors of molecules with internal degrees of freedom, adapted to take into account information from linear scaling computations of solvent accessible surfaces implemented in the framework of last generation continuum solvent models. Torsional angles are taken also explicitly into account, while retaining correct hydrodynamic interactions. A comparison with literature data is presented to prove the effectiveness of the approach for a wide range of molecular dimensions and solvent environments.     

Coupling of the overall molecular motion with the conformational transitions. II. The full rotational problem

The method proposed in the companion paper for analysing the coupling between overall and internal dynamics is applied to the study of the full rotational motion of a molecule with one internal degree of freedom. For systems characterized by a finite set of stable conformers determined by the minima of the intramolecular potential, a simplified time evolution operator of mixed type is derived, with the continuous diffusion equation and the generalized random walk operator representing the overall rotation and the internal dynamics, respectively. The dependence on the conformational state of the rotational diffusion tensor is one source of coupling between these two types of motion. Another source is represented by the recoil rotations acting on each subunit during a conformational transition. Both conformational-dependent rotational diffusion tensors and recoil rotations can be calculated from a model for the friction exerted by the solvent. Some applications of the theory are presented in relation to the butane molecule and the molecules having the structure of biphenyl, with particular emphasis on the calculation of the experimental observables in NMR and dielectric relaxation measurements.     

An efficient computational method for predicting rotational diffusion tensors of globular proteins using an ellipsoid representation

We propose a new computational method for predicting rotational diffusion properties of proteins in solution. The method is based on the idea of representing protein surface as an ellipsoid shell. In contrast to other existing approaches this method uses principal component analysis of protein surface coordinates, which results in a substantial increase in the computational efficiency of the method. Direct comparison with the experimental data as well as with the recent computational approach (Garcia de la Torre; et al. J. Magn. Reson. 2000, B147, 138-146), based on representation of protein surface as a set of small spherical friction elements, shows that the method proposed here reproduces experimental data with at least the same level of accuracy and precision as the other approach, while being approximately 500 times faster. Using the new method we investigated the effect of hydration layer and protein surface topography on the rotational diffusion properties of a protein. We found that a hydration layer constructed of approximately one monolayer of water molecules smoothens the protein surface and effectively doubles the overall tumbling time. We also calculated the rotational diffusion tensors for a set of 841 protein structures representing the known protein folds. Our analysis suggests that an anisotropic rotational diffusion model is generally required for NMR relaxation data analysis in single-domain proteins, and that the axially symmetric model could be sufficient for these purposes in approximately half of the proteins.     

Effective charge of polyelectrolytes as a function of the dielectric constant of a solution

The combination of diffusion and electrophoresis NMR is applied to determine the effective charge of poly(styrene sulfonate) in solution. While electrophoresis NMR yields the electrophoretic mobility of the molecules in solution, the hydrodynamic friction is determined from diffusion NMR. From the force balance between electrostatic force and hydrodynamic friction, the effective charge of the molecule is determined free of any model. In the present study poly(styrene sulfonate) has been investigated in mixtures of water and methanol of varying composition. The lower dielectric constant in the mixtures with high methanol content results in a drastically reduced effective charge of the polyelectrolytes. The reduced effective charge along the polymer chain is the reason for a much more compact conformation of the polyelectrolyte, which is seen in a smaller hydrodynamic size of the molecule.     

Size dependence of protein diffusion very close to membrane surfaces: measurement by total internal reflection with fluorescence correlation spectroscopy

The diffusion coefficients of nine fluorescently labeled antibodies, antibody fragments, and antibody complexes have been measured in solution very close to supported planar membranes by using total internal reflection with fluorescence correlation spectroscopy (TIR-FCS). The hydrodynamic radii (3-24 nm) of the nine antibody types were determined by comparing literature values with bulk diffusion coefficients measured by spot FCS. The diffusion coefficients very near membranes decreased significantly with molecular size, and the size dependence was greater than that predicted to occur in bulk solution. The observation that membrane surfaces slow the local diffusion coefficient of proteins in a size-dependent manner suggests that the primary effect is hydrodynamic as predicted for simple spheres diffusing close to planar walls. The TIR-FCS data are consistent with predictions derived from hydrodynamic theory. This work illustrates one factor that could contribute to previously observed nonideal ligand-receptor kinetics at model and natural cell membranes.     

Techniques for geometry optimization: A comparison of cartesian and natural internal coordinates

A comparison is made between geometry optimization in Cartesian coordinates, using an appropriate initial Hessian, and natural internal coordinates. Results on 33 different molecules covering a wide range of symmetries and structural types demonstrate that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internals to converge to global minima whereas Cartesian optimizations converge to the local minimum closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internals are to be preferred over Z-matrix coordinates. General optimization strategies using internal coordinates and/or Cartesians are discussed for both unconstrained and constrained optimization. © John Wiley & Sons, Inc.     

Two-particle friction in a mesoscopic solvent

The effects of hydrodynamic interactions on the friction tensors for two particles in solution are studied. The particles have linear dimensions on nanometer scales and are either simple spherical particles interacting with the solvent through repulsive Lennard-Jones forces or are composite cluster particles whose atomic components interact with the solvent through repulsive Lennard-Jones forces. The solvent dynamics is modeled at a mesoscopic level through multiparticle collisions that conserve mass, momentum, and energy. The dependence of the two-particle relative friction tensors on the interparticle separation indicates the importance of hydrodynamic interactions for these nanoparticles.     

Brownian motion model for collision phenomena: Translational energy distribution

A brownian motion model is assumed for the relative translational motion of colliding molecules with complex internal structures and the distribution function (probability) of the final relative velocity is obtained from a Fokker-Planck equation for conditional probability in this paper. Under certain further assumptions we are able to obtain a distribution function similar in form to the information theoretic result. A formula for angular distributions is also obtained, which shows a forward-backward or a slide-way peaking, depending on the ratio of “transversal and longitudinal temperatures” (friction tensors). The “friction tensors” are given in terms of interaction potentials and can be in principle calculated from their explicit formulas given in the paper.     

FCclasses3: Vibrationally-resolved spectra simulated at the edge of the harmonic approximation

We introduce FCclasses3, a code to carry out vibronic simulations of electronic spectra and nonradiative rates, based on the harmonic approximation. Key new features are: implementation of the full family of vertical and adiabatic harmonic models, vibrational analysis in curvilinear coordinates, extension to several electronic spectroscopies and implementation of time-dependent approaches. The use of curvilinear valence internal coordinates allows the adoption of quadratic model potential energy surfaces (PES) of the initial and final states expanded at arbitrary configurations. Moreover, the implementation of suitable projectors provides a robust framework for defining reduced-dimensionality models by sorting flexible coordinates out of the harmonic subset, so that they can then be treated at anharmonic level, or with mixed quantum classical approaches. A set of tools to facilitate input preparation and output analysis is also provided. We show the program at work in the simulation of different spectra (one and two-photon absorption, emission and resonance Raman) and internal conversion rate of a typical rigid molecule, anthracene. Then, we focus on absorption and emission spectra of a series of flexible polyphenyl molecules, highlighting the relevance of some of the newly implemented features. The code is freely available at http://www.iccom.cnr.it/en/fcclasses/ .     

Rovibrational molecular hamiltonian in mixed bond-angle and umbrella-like coordinates

A new exact quantum mechanical rovibrational Hamiltonian operator for molecules exhibiting large amplitude inversion and torsion motions is derived. The derivation is based on a division of a molecule into two parts: a frame and a top. The nuclei of the frame only are used to construct a molecular system of axes. The inversion motion of the frame is described in the umbrella-like coordinates, whereas the torsion motion of the top is described by the nonstandard torsion angle defined in terms of the nuclear vectors and one of the molecular axes. The internal coordinates chosen take into account the properties of the inversion and torsion motions. Vibrational s and rotational Omega vectors obtained for the introduced internal coordinates determine the rovibrational tensor G defined by simple scalar products of these vectors. The Jacobian of the transformation from the Cartesian to the internal coordinates considered and the G tensor specify the rovibrational Hamiltonian. As a result, the Hamiltonian for penta-atomic molecules like NH2OH with one inverter is presented and a complete set of the formulas necessary to write down the Hamiltonian of more complex molecules, like NH2NH2 with two inverters, is reported. The approach considered is essentially general and sufficiently simple, as demonstrated by derivation of a polyatomic molecule Hamiltonian in polyspherical coordinates, obtained by other methods with much greater efforts.     

Langevin dynamics for rigid bodies of arbitrary shape

We present an algorithm for carrying out Langevin dynamics simulations on complex rigid bodies by incorporating the hydrodynamic resistance tensors for arbitrary shapes into an advanced rotational integration scheme. The integrator gives quantitative agreement with both analytic and approximate hydrodynamic theories for a number of model rigid bodies and works well at reproducing the solute dynamical properties (diffusion constants and orientational relaxation times) obtained from explicitly solvated simulations.     

Long-time dynamics of Rouse-Zimm polymers in dilute solutions with hydrodynamic memory

The dynamics of flexible polymers in dilute solutions is studied taking into account the hydrodynamic memory, as a consequence of fluid inertia. As distinct from the Rouse-Zimm (RZ) theory, the Boussinesq friction force acts on the monomers (beads) instead of the Stokes force, and the motion of the solvent is governed by the nonstationary Navier-Stokes equations. The obtained generalized RZ equation is solved approximately using the preaveraging of the Oseen tensor. It is shown that the time correlation functions describing the polymer motion essentially differ from those in the RZ model. The mean-square displacement (MSD) of the polymer coil is at short times approximately t(2) (instead of approximately t). At long times the MSD contains additional (to the Einstein term) contributions, the leading of which is approximately t. The relaxation of the internal normal modes of the polymer differs from the traditional exponential decay. It is displayed in the long-time tails of their correlation functions, the longest lived being approximately t(-3/2) in the Rouse limit and t(-5/2) in the Zimm case, when the hydrodynamic interaction is strong. It is discussed that the found peculiarities, in particular, an effectively slower diffusion of the polymer coil, should be observable in dynamic scattering experiments.     

Development and validation of an integrated computational approach for the modeling of cw-ESR spectra of free radicals in solution: p-(methylthio)phenyl nitronylnitroxide in toluene as a case study

In this work we address the interpretation, via an ab initio integrated computational approach, of continuous wave electron spin resonance (cw-ESR) spectra of p-(methylthio)phenyl nitronylnitroxide (MTPNN) dissolved in toluene. Our approach is based on the determination of the spin Hamiltonian, averaged with respect to fast vibrational motions, with magnetic tensor parameters (Zeeman and hyperfine tensors) characterized by quantum mechanical density functional calculations. The system is then described by a stochastic Liouville equation, with inclusion of diffusive rotational dynamics. Parametrization of diffusion rotational tensor is provided by a hydrodynamic model. Cw-ESR spectra of MTPNN are simulated for a wide range of temperatures (155-292 K) with minimal resorting to fitting procedures, proving that the combination of sensitive ESR spectroscopy and sophisticated modeling can be highly helpful in providing structural and dynamic information on molecular systems.     

Explicit factorization of external coordinates in constrained statistical mechanics models

If a macromolecule is described by curvilinear coordinates or rigid constraints are imposed, the equilibrium probability density that must be sampled in Monte Carlo simulations includes the determinants of different mass-metric tensors. In this work, the authors explicitly write the determinant of the mass-metric tensor G and of the reduced mass-metric tensor g, for any molecule, general internal coordinates and arbitrary constraints, as a product of two functions; one depending only on the external coordinates that describe the overall translation and rotation of the system, and the other only on the internal coordinates. This work extends previous results in the literature, proving with full generality that one may integrate out the external coordinates and perform Monte Carlo simulations in the internal conformational space of macromolecules.     

Analytical algorithm for molecular modeling

A molecular modeling procedure, based on internal coordinates and strictly analytical even in the most intricated cases, is described. Internal coordinates, always nonredundant, become mutually independent and can be varied without constraints. Structural refinement from diffraction data (Least-square method, LS) can be done using the classical Gauss-Newton approach and avoiding Lagrange multipliers. A comparative test done using published data has shown that while the new method gives rise to a structural refinement in perfect agreement with the known structure, the traditional methods (z-matrix and constraints based) does not work.     

Effects of the Bead‐Solvent Interaction on the Dynamics of Macromolecules, 1

Summary: Hamiltonian dynamics and a chain model are used to study the dynamics of macromolecules immersed in a solution. From the Hamiltonian of the overall system, “macromolecule + solvent,” a master and a Fokker‐Planck equation are then derived for the phase‐space distribution of the macromolecule. In the Fokker‐Planck equation, all the information about the interaction among the beads of the macromolecule as well as the effects of the surrounding solvent is described by friction tensors, which are expressed in terms of the bead‐solvent interaction and the dynamic structure factor of the solvent. To explore the influence of the bead‐solvent potential on the dynamics of macromolecules, the friction tensors are calculated for a dumbbell molecule and for three choices of the interaction (Yukawa, Born‐Mayer, and Lennard‐Jones). Expressions are derived, in particular, for the friction tensor coefficients of the center‐of‐mass and the relative coordinates of the dumbbell. For the long‐term behaviour of the internal momentum autocorrelation function, moreover, an “algebraic decay” is found, in contrast to the (unphysical) exponential decay as known from phenomenological theory.

Yukawa, Born‐Mayer and Lennard‐Jones bead‐solvent interaction potentials.     

Gaussian approximation for Hookean dumbbells with hydrodynamic interaction: Translational diffusivity

The recently developed Gaussian approximation for the hydrodynamic interaction is used to discuss the translational diffusivity of polymers in dilute solutions undergoing homogeneous flows. For the Hookean dumbbell model, we derive the diffusion tensors associated with (i) the average polymer velocity caused by external forces; (ii) the mean-square displacement of a single polymer caused by the Brownian forces; and (iii) the polymer mass flux caused by concentration gradients. We discuss the model predictions for these diffusion tensors for steady shear flow in detail.This paper was presented at the Frühjahrstagung des Fachausschusses Polymerphysik der Deutschen Physikalischen Gesellschaft in Hamburg (West Germany), March 14–16,1988.     

Tork: Conformational analysis method for molecules and complexes

A conformational search method for organic molecules and bimolecular complexes is presented. The method, termed Tork, uses normal-mode analysis in bond-angle-torsion coordinates and focuses on a key subset of torsional coordinates to identify natural molecular motions that lead the initial conformation to new energy minima. New conformations are generated via distortion along these modes and their pairwise combinations, followed by energy minimization. For complexes, special treatment is accorded to the six coordinates that specify the position and orientation of one molecule relative to the other. Tests described here show that Tork is highly efficient for cyclic, acyclic, and mixed single molecules, as well as for host-guest complexes.     

On the use of optimal internal vibrational coordinates for symmetrical bent triatomic molecules

The use of generalized internal coordinates for the variational calculation of excited vibrational states of symmetrical bent triatomic molecules is considered with applications to the SO2, O3, NO2, and H2O molecules. These coordinates depend on two external parameters which can be properly optimized. We propose a simple analytical method to determine the optimal internal coordinates for this kind of molecules based on the minimization with respect to the external parameters of the zero-point energy, assuming only quadratic terms in the Hamiltonian and no quadratic coupling between the optimal coordinates. The optimal values of the parameters thus obtained are shown to agree quite well with those that minimize the sum of a number of unconverged energies of the lowest vibrational states, computed variationally using a small basis function set. The unconverged variational calculation uses a basis set consisting of the eigenfunctions of the uncoupled anharmonic internal coordinate Hamiltonian. Variational calculations of the excited vibrational states for the four molecules considered carried out with an increasing number of basis functions, also evidence the excellent convergence properties of the optimal internal coordinates versus those provided by other normal and local coordinate systems.