Shaking

Although the yield of mitotic cells is poor (1-3%) it is increased by pooling many harvests taken over a 2-3-h period. 

Stubblefield et al. (1967) used dilute trypsin to release their accumulated metaphase cells and claim that the purity of the resulting suspension was better than with shaking. The details of their method 

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If two pure, immiscible liquids, such as benzene and water, are vigorously shaken together, they will form a dispersion, but it is doubtful that one phase or the other will be uniquely continuous or dispersed. On stopping the agitation, phase separation occurs so quickly that it is questionable whether the term emulsion really should be applied to the system. A surfactant component is generally needed to obtain a stable or reasonably stable emulsion. Thus, if a little soap is added to the benzene-water system, the result on shaking is a true emulsion that separates out only very slowly. Theories of... 

Andersen H C 1983 RATTLE a velocity version of the SHAKE algorithm for molecular dynamics calculations J. Comput. Phys. 52 24-34... 

Addition of dilute potassium dichromate(VI) solution, K2Cr207, to a solution of hydrogen peroxide produces chromium peroxide, CrOj, as an unstable blue coloration on adding a little ether and shaking this compound transfers to the organic layer in which it is rather more stable. 

In water pollution studies, the oxygen content can be measured by making the water alkaline and shaking a measured volume with an oxygen-free solution containing Mn- (aq). The solution is acidified with sulphuric acid, potassium iodide added and the liberated iodine titrated with sodium thiosulphate. 

Very recently, we have developed and incorporated into the CHARMM molecular mechanics program a version of LN that uses direct-force evaluation, rather than linearization, for the fast-force components [91]. The scheme can be used in combination with SHAKE (e.g., for freezing bond lengths) and with periodic boundary conditions. Results for solvated protein and nucleic-... 

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 constrained equations of motion in cartesian eoordinates can be solved by the SHAKE or (the essentially equivalent) RATTLE method (see [8]) which requires the solution of a non-linear system of equations in the Lagrange multiplier funetion A. The equivalent formulation in local coordinates ean still be integrated by using the explicit Verlet method. 

These various techniques were recently applied to molecular simulations [11, 20]. Both of these articles used the rotation matrix formulation, together with either the explicit reduction-based integrator or the SHAKE method to preserve orthogonality directly. In numerical experiments with realistic model problems, both of these symplectic schemes were shown to exhibit vastly superior long term stability and accuracy (measured in terms of energy error) compared to quaternionic schemes. 

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]... 

These equations reduce to a 3 x 3 matrix Ricatti equation in this case. In the appendix of [20], the efficient iterative solution of this nonlinear system is considered, as is the specialization of the method for linear and planar molecules. In the special case of linear molecules, the SHAKE-based method reduces to a scheme previously suggested by Fincham[14]. 

The standard numerical integrators for the constrained system (12) are the SHAKE scheme [23], which extends the Verlet method (2),... 

Both methods are time-reversible. For A = 0, they reduce to SHAKE and RATTLE. In contrast to SHAKE and RATTLE, the time step is not restricted by the largest eigenvalue of A. 

H. C. Andersen. Rattle A velocity version of the Shake algorithm for molecular dynamics calculations. J. Comp. Phys., 52 24-34, 1983. 

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... 

Our discussion so far has considered the use of SHAKE with the Verlet algorithm Versions have also been derived for other integration schemes, such as the leap-froj algorithm, the predictor-corrector methods and the velocity Verlet algorithm. In the cast of the velocity Verlet algorithm, the method has been named RATTLE [Anderson 1983]... 

Liquids are occasionally purified by removing impurities as constant-boiling mixtures, or by shaking with concentrated sulphuric acid and subsequently separating the dried liquid from the acid the second method is therefore limited to liquids which are insoluble in, and chemically unaffected by, the strong acid e.g., benzene, anhydrous chloral). 

If an alternative mixed solvent is required, shake thoroughly a mixture of 20 ml. of distilled water and 40 ml. of the -butanol, then... 

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