Constraint SHAKE

Assuming a commonly used bond-stretch tolerance factor (sf) = 10 for a molecular model with specific geometrical parameters, Eq. [119] furnishes the angle tolerance factor needed for equal accuracy runs of triangulation and angle-constraint SHAKE. 

In Monte Carlo simulation, the choice of polymer model is governed by the choice of attempted moves. Typically kinetic energy is integrated analytically over all modes, and the partition function for the flexible model in the limit of infinite stiffness results [105]. In molecular dynamics, constraints (SHAKE, etc.) freeze kinetic energy contributions and the partition function for the rigid model results [105,197]. To achieve sampling from the desired partition function, it is necessary to add a pseudopotential based on the covariant metric tensor a [198]. 

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 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. 

Rotation matrices may be viewed as an alternative to particles. This approach is based directly on the orientational Lagrangian (1). Viewing the elements of the rotation matrix as the coordinates of the body, we directly enforce the constraint Q Q = E. Introducing the canonical momenta P in the usual manner, there results a constrained Hamiltonian formulation which is again treatable by SHAKE/RATTLE [25, 27, 20]. For a single rigid body we arrive at equations for the orientation of the form[25, 27]... 

The most commonly used method for applying constraints, particularly in molecula dynamics, is the SHAKE procedure of Ryckaert, Ciccotti and Berendsen [Ryckaert et a 1977]. In constraint dynamics the equations of motion are solved while simultaneous satisfying the imposed constraints. Constrained systems have been much studied in classics mechanics we shall illustrate the general principles using a simple system comprising a bo sliding down a frictionless slope in two dimensions (Figure 7.8). The box is constrained t remain on the slope and so the box s x and y coordinates must always satisfy the equatio of the slope (which we shall write as y = + c). If the slope were not present then the bo... 

Initial stmcture equUibration, wherein bad or close contacts are reheved this may be done with constraints on bonds, eg, to simplify the process (the premise of the SHAKE technique). 

The SHAKE method for bond constraints reduces the number of degrees of freedom during the initial stages of simulations it is good for minimizing solvent bath overhead. 

There are various ways to obtain the solutions to this problem. The most straightforward method is to solve the full problem by first computing the Lagrange multipliers from the time-differentiated constraint equations and then using the values obtained to solve the equations of motion [7,8,37]. This method, however, is not computationally cheap because it requires a matrix inversion at every iteration. In practice, therefore, the problem is solved by a simple iterative scheme to satisfy the constraints. This scheme is called SHAKE [6,14] (see Section V.B). Note that the computational advantage has to be balanced against the additional work required to solve the constraint equations. This approach allows a modest increase in speed by a factor of 2 or 3 if all bonds are constrained. 

In practice the equation of motion is solved first without considering the constraint force and in the next step the constraint forces are obtained by correcting the positions such that the molecule conserves its minimum structure, i.e., such that the constraints are fiilfilled. For small molecules direct inversion is possible, for large molecules iterative procedures are applied (60). This means that each constraint is corrected after the other until a certain convergence is reached. This algorithm is called Shake (65). Another important aspect of simulations concerns periodic boundary conditions. A virtual replication of the central box at each of its planes is carried out in order to avoid surface effects. A detailed description can be found in the excellent textbooks of Allen and Tildesley (60) and Frenkel and Smit (61). 

As an alternative, the SHAKE algorithm (21) is used if angle constraints are implemented or the LINCS algorithm otherwise fails. 

Many discussions of life on Titan have considered the possibility that water, normally frozen at the ambient temperature, might remain liquid following heating by impacts.22 Life in this aqueous environment would be subject to the same constraints and opportunities as life in water. Water droplets in hydrocarbon solvents are, in addition, convenient cellular compartments for evolution, as Tawfik and Griffiths have shown in the laboratory.23 An emulsion of water droplets in oil is obtainable by simple shaking. This could easily be a model for how life on Titan achieves the isolation necessary for Darwinian evolution, and it provides an interesting alternative for membranes, discussed in earlier chapters as a common feature of terran life. 

To achieve an evenly distributed thermal excitation, the nuclei were brought to a temperature of 300 K by applying a sequence of 30 sinusoidal pulses, each of which was chosen to raise the temperature by 10 K. Each of the excitation vectors was chosen to be orthogonal to the already excited modes. The warmed-up systems were equilibrated for the 10 000 timesteps. The time step of 7 au. was used. Constraints were maintained by SHAKE algorithm.36 A temperature of 300 K was controlled by a Nose thermostat.23,24 The fictitious kinetic energy of the electrons was controlled in a similar fashion by a Nose thermostat.52... 

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. 

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