Long-time-scale motions

Protein dynamics occurs on very different time scales ([McCammon and Harvey 1987, Jardetzky 1996]). Here, we are most interested in long time scale motions such as relative motion between secondary structure elements, and inter-domain motion. 

Auerbach et al. (101) used a variant of the TST model of diffusion to characterize the motion of benzene in NaY zeolite. The computational efficiency of this method, as already discussed for the diffusion of Xe in NaY zeolite (72), means that long-time-scale motions such as intercage jumps can be investigated. Auerbach et al. used a zeolite-hydrocarbon potential energy surface that they recently developed themselves. A Si/Al ratio of 3.0 was assumed and the potential parameters were fitted to reproduce crystallographic and thermodynamic data for the benzene-NaY zeolite system. The functional form of the potential was similar to all others, including a Lennard-Jones function to describe the short-range interactions and a Coulombic repulsion term calculated by Ewald summation. 

To optimize force fields for long time scale motions Aliev et al. propose a new robust approach to use NMR spin-lattice relaxation times Ti of both backbone and sidechain carbons. This allows a selective determination of both overall molecular and intramolecular motional time scales. In addition they use motionally averaged experimental/ coupling constants for torsional FF parameters. The force constants in the FFs and the correlation times are fitted in an Arrhenius-type of equation. 

The first term represents the forces due to the electrostatic field, the second describes forces that occur at the boundary between solute and solvent regime due to the change of dielectric constant, and the third term describes ionic forces due to the tendency of the ions in solution to move into regions of lower dielectric. Applications of the so-called PBSD method on small model systems and for the interaction of a stretch of DNA with a protein model have been discussed recently ([Elcock et al. 1997]). This simulation technique guarantees equilibrated solvent at each state of the simulation and may therefore avoid some of the problems mentioned in the previous section. Due to the smaller number of particles, the method may also speed up simulations potentially. Still, to be able to simulate long time scale protein motion, the method might ideally be combined with non-equilibrium techniques to enforce conformational transitions. 

A variety of techniques have been introduced to increase the time step in molecular dynamics simulations in an attempt to surmount the strict time step limits in MD simulations so that long time scale simulations can be routinely undertaken. One such technique is to solve the equations of motion in the internal degree of freedom, so that bond stretching and angle bending can be treated as rigid. This technique is discussed in Chapter 6 of this book. Herein, a brief overview is presented of two approaches, constrained dynamics and multiple time step dynamics. 

The Eik/TDDM approximation can be computationally implemented with a procedure based on a local interaction picture for the density matrix, and on its propagation in a relax-and-drive perturbation treatment with a relaxing density matrix as the zeroth-order contribution and a correction due to the driving effect of nuclear motions. This allows for an efficient computational procedure for differential equations coupling functions with short and long time scales, and is of general applicability. 

While the idea of LF explained the H2 data quite well [28], we were surprised by the magnitude of the oscillations in our I2 data [16], as, unlike H2,12 is not vibrationally cold at room temperature - the conditions for our experiment. Generally, thermal motion is detrimental to observing coherent motion. Thus, we took a long time scale run to get a more accurate measurement of the frequency of the vibrations, shown in Fig. 1.5. These data also exhibit a vibrational revival, from which the anharmonicity of the potential well can be determined. Indeed, the vibrational frequency accurately matched that of the ground state. 

As it was mentioned previously the use of straightforward MD simulations to study slow processes is limited by the size of the time step that is required to obtain a stable trajectory. The SDEL algorithm allows the computation of atomically detailed trajectories connecting two known conformations of the molecule over long time scales. In contrast to normal MD, step sizes can be easily increased by factors of thousands (or more) without significative changes in many properties of the trajectory. The trade-off is that trajectories obtained with such a large step size are approximate molecular motions that occur on a scale shorter than the... 

The earliest molecular dynamics (MD) simulations on polyatomic molecular systems date back less than three decades. Since then, MD simulations have been applied to increasingly more complex molecules, such as n-alkanes" " and proteins. " In contrast to the earlier simulations of atomic systems, the simulation of polyatomic molecules is complicated by the existence of both intramolecular and intermolecular degrees of freedom. The presence of this variety of degrees of freedom leads to a wide range of time scales of molecular motions. Because the size of the time step used in MD simulations is limited by the shortest period of motion present, simulations of long time scale... 

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