Newton equations, molecular modelling

Nenitzescu reaction 100 Neumann principle, tensor properties 201 neural networks 810 neutron scattering 680-698 Newton equations, molecular modelling 74 nickel, X ray absorption curve 622 nitro, terminal substituents 147 nitrogen saturation, phase transitions 357... 

Molecular dynamics consists of the brute-force solution of Newton s equations of motion. It is necessary to encode in the program the potential energy and force law of interaction between molecules the equations of motion are solved numerically, by finite difference techniques. The system evolution corresponds closely to what happens in real life and allows us to calculate dynamical properties, as well as thennodynamic and structural fiinctions. For a range of molecular models, packaged routines are available, either connnercially or tlirough the academic conmuinity. 

The classical dynamics of molecular models is generated by Hamilton s (or Newton s) equations of motion. In the absence of external, time-dependent forces, and within the Born-Oppenheimer approximation, the dynamics of molecular vibrations, rotations, and reactions conserves the total energy . We therefore restrict our attention in the nonlinear dynamics literature to energy-conserving systems, which are technically referred to as Hamiltonian systems. For the purposes of the present discussion, we restrict our attention to Hamiltonian systems with two degrees of freedom ... 

Molecular simulations Computer modeling of the motion of an assembly of atoms or molecules. In molecular simulations only the motions of nuclei are considered, (i.e one assumes that the electronic Schrodinger equation has been solved providing intermolecular interaction potentials.) In practice empirical interaction potentials are utilized in most cases. Two main approaches are used the Monte Carlo (MC) method and molecular dynamics (MD). The former relies on statistical sampling of the configuration space of the systems, whereas the latter solves classical mechanics (Newton) equations to find trajectories of molecules. 

The strength of molecular mechanics is that by treating molecules as classical objects, fliUy described by Newton s equations of motion, quite large systems can be modeled. Computations involving enzymes with thousands of atoms are done routinely. As computational capabilities have advanced, so... 

Molecular dynamics, in contrast to MC simulations, is a typical model in which hydrodynamic effects are incorporated in the behavior of polymer solutions and may be properly accounted for. In the so-called nonequilibrium molecular dynamics method [54], Newton s equations of a (classical) many-particle problem are iteratively solved whereby quantities of both macroscopic and microscopic interest are expressed in terms of the configurational quantities such as the space coordinates or velocities of all particles. In addition, shear flow may be imposed by the homogeneous shear flow algorithm of Evans [56]. 

The molecular dynamics unit provides a good example with which to outline the basic approach. One of the most powerful applications of modem computational methods arises from their usefulness in visualizing dynamic molecular processes. Small molecules, solutions, and, more importantly, macromolecules are not static entities. A protein crystal structure or a model of a DNA helix actually provides relatively little information and insight into function as function is an intrinsically dynamic property. In this unit students are led through the basics of a molecular dynamics calculation, the implementation of methods integrating Newton s equations, the visualization of atomic motion controlled by potential energy functions or molecular force fields and onto the modeling and visualization of more complex systems. 

Molecular dynamics has been used to simulate water structures, wherein an accurate water potential function is used to enable solution of Newton s equations of motion for a small (e.g., 1000-10,000) number of molecules over time. In water and water structures, the SPC (Berendsen et al 1981) and the TIP4P (Jorgensen et al 1983) potential models are most often used. Reanalysis of extant diffraction data by Soper et al. (1997) has called both of these potentials into question. 

Molecular simulations are used most often for modeling proteins and nucleic acids. We mention the methods here only because they are methods for computing a free energy directly. However, they are rather complex calculations, so we will keep our comments brief. Molecular dynamics simulations give information about the variation in structure and energy of a molecule over an interval of time (78,79). In MD, each atom moves according to Newton s equations of motion for classical particles ... 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

The basic law of viscosity was formulated before an understanding or acceptance of the atomic and molecular structure of matter although just like Hooke s law for the elastic properties of solids the basic equation can be derived from a simple model, where a flnid is assumed to consist of hypothetical spherical molecules. Also like Hooke s law, this theory predicts linear behavior at low rates of strain and deviations at high strain rates. But we digress. The concept of viscosity was first introduced by Newton, who considered what we now call laminar flow and the frictional forces exerted between layers within a fluid. If we have a fluid placed between a stationary wall and a moving wall and we assume there is no slip at the walls (believe it or not, a very good assumption), then the velocity profile illustrated in Figure... 

In the case of rigid molecules, the force equations for the molecular center-of-mass motions are supplemented with an additional set of equations describing the torques on the molecules, the Newton-Euler equations. In this case more complex finite difference methods are used to generate the trajectories [32]. When simulating liquid water, a time step of about 10 s is used, whether a rigid molecule or flexible molecule model is under investigation. 

Motion in a protein may be modeled by computer using Isaac Newton s equation of motion, / = ma. This modeling requires the three-dimensional coordinates from an X-ray structure analysis as a starting point and some knowledge of interatomic potentials, so that only reasonable interatomic distances will be employed at all stages.Such molecular dynamics calculations lead to a prediction of where atoms will move in a short period of time, and result in the calculation of a time-dependent trajectory of all atoms. Initially each atom is moved in the direction of the force on it from other atoms and then, as each atom moves, its trajectory may change to accommodate this. In addition, this method aids in protein structure refinement,as was described in Chapter 10, although it is important to ensure that the model so refined still fits the electron density map. 

Molecular Simulations Molecular simulations are useful for predicting properties of bulk fluids and solids. Molecular dynamics (MD) simulations solve Newton s equations of motion for a small number (on the order of 10 ) of molecules to obtain the time evolution of the system. MD methods can be used for equilibrium and transport properties. Monte Carlo (MC) simulations use a model for the potential energy between molecules to simulate configurations of the molecules in proportion to their probability of occurrence. Statistical averages of MC configurations are useful for equilibrium properties, particularly for saturated densities, vapor pressures, etc. Property estimations using molecular simulation techniques are not illustrated in the remainder of this section as commercial software implementations are not generally available at this time. 

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