Dynamic models, flexible molecules

Throughout the realm of molecular modeling, the concept of molecular shape arises over and over in one form or another. Just what do scientists mean by a molecule s shape, and how can one use three-dimensional shape in modeling. In Chapter 5, Professor Gustavo A. Arteca examines these issues and delineates the hierarchical levels of molecular shape and shape descriptors. He explains molecular shape in terms of mathematical descriptors of nuclear geometry, connectivity, and molecular surfaces. Of special note are his comments on shape dynamics of flexible molecules and descriptors of relative shape. 

The density of states (DOS) of lattice phonons has been calculated by lattice dynamical methods [111]. The vibrational DOS of orthorhombic Ss up to about 500 cm has been determined by neutron scattering [121] and calculated by MD simulations of a flexible molecule model [118,122]. 

Here the vector rj represents the centre of mass position, and D is usually averaged over several time origins to to improve statistics. Values for D can be resolved parallel and perpendicular to the director to give two components (D//, Dj ), and actual values are summarised for a range of studies in Table 3 of [45]. Most studies have found diffusion coefficients in the 10 m s range with the ratio D///Dj between 1.59 and 3.73 for calamitic liquid crystals. Yakovenko and co-workers have carried out a detailed study of the reorientational motion in the molecule PCH5 [101]. Their results show that conformational molecular flexibility plays an important role in the dynamics of the molecule. They also show that cage models can be used to fit the reorientational correlation functions of the molecule. 

The theoretical prediction of these properties for branched molecules has to take into account the peculiar aspects of these chains. It is possible to obtain these properties as the low gradient Hmits of non-equilibrium averages, calculated from dynamic models. The basic approach to the dynamics of flexible chains is given by the Rouse or the Rouse-Zimm theories [12,13,15,21]. How-ever,both the friction coefficient and the intrinsic viscosity can also be evaluated from equilibrium averages that involve the forces acting on each one of the units. This description is known as the Kirkwood-Riseman (KR) theory [15,71 ]. Thus, the translational friction coefficient, fl, relates the force applied to the center of masses of the molecule and its velocity... 

The modeling of carbohydrates is undergoing rapid development. For example, the first comprehensive conformational mappings of disaccharides with flexible residues and the first molecular dynamics studies of carbohydrates have only recently been published. At the same time, interest in carbohydrates has been increasing dramatically, and there is a need for a publication that gently introduces the uninitiated and provides an overview of current research in the area. We feel that Computer Modeling ( Carbohydrate Molecules meets these needs. 

Molecular modeling using either Monte-Carlo simulations or molecular dynamics is used to apply molecular mechanics energy minimizations to very complex systems [348]. In complex flexible molecules such as proteins or nucleic adds, the number of variable parameters, i.e., bond torsion angles, is such that the global search for energy minima becomes impossible The same problem occurs with theoretical calculations of water structure in aqueous solutions or in heavily hydrated crystals. 

In main-chain LCPs, molecular flexibility can be distributed more-or-less uniformly along the chain, as is the case for PBLG, HPC, or Vectra A, or it can be concentrated in flexible spacers, as in OQO(phenylsulfonyl)lU (see Fig. 11-2). The former are called persistently flexible molecules, and are often modeled by the worm-like chain, with a uniform bending modulus, while for the latter, a reasonable model might be the freely jointed chain (see Fig. 11-3 and Section 2.2.3.2). For a recent discussion of the phase behavior and dynamics of worm-like chains, see Sato and Teramoto (1996). 

Most macro molecules in solution are neither as stiff as the rigid rod nor as flexible as the Gaussian coil. For particular systems of interest a dynamical model should be made and the corresponding spectrum (or time correlation function) calculated. Measurement of the spectrum and a fit to the theoretical form then allows extraction of the model dynamic constants. These dynamic constant may then be related to equilibrium structural properties of the molecule (end-to-end distances, backbone curvature, etc.). 

The earliest molecular dynamics simulations using realistic potentials were of atoms interacting under the Lennard-Jones potential. In such calculations the only forces on the atoms are those due to non-bonded interactions. It is rather more difficult to simulate molecules because the interaction between two non-spherical molecules depends upon their relative orientation as well as the distance between them If the molecules are flexible then there will also be intramolecular interactions, which give rise to changes in conformation. Clearly, the simplest model is to treat the species present as rigid bodies with no intramolecular conformational freedom. In such cases the dynamics of each molecule can often be considered in terms of translations of its centre of mass and rotations about its centre of mass. The force on the molecule equals the vector sum of all the forces acting at the... 

Dynamics of rodlike molecules is quite different from that of linear flexible chains. The rodlike molecule exhibits a well-defined rotational motion in addition to the center-of-mass motion (Fig. 3.58). The latter has two components parallel to the rod axis and perpendicular to the rod axis. The expressions for the translational diffusion coefficients D and in the directions parallel and perpendicular to the rod axis and the rotational diffusion coefficient were obtained by Kirkwood" for a model that consists of N beads in a straight line. 

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