Holonomic constraint

An initial and desired final configuration of a system can be used by the targeted molecular dynamics (TMD) method (Schlitter et al., 1993) to establish a suitable pathway between the given configurations. The resulting pathway, can then be employed during further SMD simulations for choosing the direction of the applied force. TMD imposes time-dependent holonomic constraints which drive the system from one known state to another. This method is also discussed in the chapter by Helms and McCammon in this volume. 

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

The standard discretization for the equations (9) in molecular dynamics is the (explicit) Verlet method. Stability considerations imply that the Verlet method must be applied with a step-size restriction k < e = j2jK,. Various methods have been suggested to avoid this step-size barrier. The most popular is to replace the stiff spring by a holonomic constraint, as in (4). For our first model problem, this leads to the equations d... 

The concept of a symplectic method is easily extended to systems subject to holonomic constraints [22]. For example the RATTLE discretization is found to be a symplectic discretization. Since SHAKE is algebraically equiva lent to RATTLE, it, too, has the long-term stability of a symplectic method. 

I fall vertically downwards. Constraints are often categorised as holonomic or non-mic. Holonomic constraints can be expressed in the form... 

CE uses holonomic constraints. In a constrained system the coordinates of the particles 5t independent and the equations of motion in each of the coordinate directions are cted. A second difficulty is that the magnitude of the constraint forces is unknown, in the case of the box on the slope, the gravitational force acting on the box is in the ction whereas the motion is down the slope. The motion is thus not in the same direc-s the gravitational force. As such, the total force on the box can be considered to arise wo sources one due to gravity and the other a constraint force that is perpendicular to otion of the box (Figure 7.8). As there is no motion perpendicular to the surface of the the constraint force does no work. 

The first step of the structure refinement is the appHcation of distance geometry (DG) calculations which do not use an energy function but only experimentally derived distances and restraints which follow directly from the constitution, the so-caUed holonomic constraints. Those constraints are, for example, distances between geminal protons, which normally are in the range between 1.7 and 1.8 A, or the distance between vicinal protons, which can not exceed 3.1 A when protons are in anti-periplanar orientation. 

The first step in the DG calculations is the generation of the holonomic distance matrix for aU pairwise atom distances of a molecule [121]. Holonomic constraints are expressed in terms of equations which restrict the atom coordinates of a molecule. For example, hydrogen atoms bound to neighboring carbon atoms have a maximum distance of 3.1 A. As a result, parts of the coordinates become interdependent and the degrees of freedom of the molecular system are confined. The acquisition of these distance restraints is based on the topology of a model structure with an arbitrary, but energetically optimized conformation. 

Lipkowitz and D. B. Boyd, Eds., VCH, Weinheim, Germany, 1998, pp. 75-136. Molecular Dynamics with General Holonomic Constraints and Application to Internal Coordinate Constraints. 

One of the benefits of MC algorithms is the ability to naturally deal with constraints, that is, to set up a simulation in terms of arbitrary sets of degrees of freedom, which may well be different from the degrees of freedom over which the potential energy is evaluated. In MD, such functionality is introduced by holonomic constraints [23], for which a variety of popular algorithms have been... 

Kutteh, R., Straatsma, T.P. Molecular dynamics with general holonomic constraints and application to internal coordinate constraints. In Reviews in Computational Chemistry (eds... 

This section presents the notation for generalized coordinates, constraints, basis vectors, and tensors that is used throughout the paper. We consider a system consisting of N pointlike particles (beads) with positions R, ..., R with masses mi,..., mj. The positions of the beads are subject to K holonomic constraints, of the form... 

The differential form dXj = akjdqk + aktdt = 0 defines a general linear constraint condition. For an integrable or holonomic constraint, expressed by Xj( qk, t) = 0, the coefficients are partial derivatives of X such that akj = and cikt — Nonintegrable or nonholonomic constraints are defined by the differ-... 

Any conservative mechanical system which is either free or subject to holonomic constraints and whose potential does not depend on the generalized velocities is described by standard equations of motion (either Lagrangian or Hamiltonian). The kinetic energy of the iV-particIe system is ... 

Afh (q) is the current element of a matrix A (q) called the constraint-matrix. A (q) is a real symmetric matrix whose diagonal elements are positive it depends on the generalized coordinates as variables and parametrically on the constraints. This matrix possesses an inverse since det A(q) = 0 is not possible it would correspond to a supplementary relationship between the coordinates only, i.e a supplementary holonomic constraint. 

Implementing the shared-memory vector/parallel algorithms developed by Mertz et al. (fgr evaluation of the potential energies and forces, generation of the nonbonded neighbor list, and satisfaction of holonomic constraints) into CHARMM and AMBER resulted in near-linear speed-ups on eight processors of a Cray Y-MP for the forces and neighbor lists. For the holonomic constraints, speed-ups of 6.0 and 6.4 were obtained for the SHAKE and matrix inversion method, respectively. 

We finally note that is possible to use Gauss principle to obtain equations of motion when the system is subject to holonomic constraints such as bond length or bond angle constraints. In this case one obtains the same equations of motion as one would obtain by applying the Lagrange equation. 

R. Kutteh and T. P. Straatsma, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., Wiley-VCH, New York, 1998, Vol. 13, pp. 75-136. Molecular Dynamics with General Holonomic Constraints and Applications to Internal Coordinate Constraints. J. C. Shelley and D. R. Berard, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., Wiley-VCH, New York, 1998, Vol. 12, pp. 137-205. Computer Simulation of Water Physisorption at Metal-Water Interfaces. 

Lastly it is noted that an effective potential of a form similar to Eq. (2.19) appiears when use is made of holonomic constraints in Brownian dynamics simulations, that is, in the use of fixed bond lengths and bond angles in studies of dihedral angle conformational relaxation in polymer systems. In fact the two potentials represent different physical phenomena, but each acts so as to yield the correct equilibrium distribution in the coordinates of interest. The two approaches, of thermalization and constraint, have been compared for the case of the four-particle chain. ... 

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