Molecular dynamics

Brunger, A. T.; Adams, P. D.; Rice, L. M.

doi:10.1107/97809553602060000694

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 18.2, p. 378 | 1 | 2 |

Section 18.2.4.1. Molecular dynamics

A. T. Brunger,^a ^* P. D. Adams^b and L. M. Rice^c

^a The Howard Hughes Medical Institute, and Departments of Molecular and Cellular Physiology, Neurology and Neurological Sciences, and Stanford Synchrotron Radiation Laboratory, Stanford Universty, 1201 Welch Road, MSLS P210, Stanford, CA 94305-5489, USA,^bThe Howard Hughes Medical Institute and Department of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06511, USA, and ^cDepartment of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06511, USA
Correspondence e-mail: axel.brunger@stanford.edu

18.2.4.1. Molecular dynamics

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A suitably chosen set of atomic parameters can be viewed as generalized coordinates that are propagated in time by the classical equations of motion (Goldstein, 1980). If the generalized coordinates represent the x, y, z positions of the atoms of a molecule, the classical equations of motion reduce to the familiar Newton's second law: $[m_{i} {\partial^{2}{\bf r}_{i} \over \partial t^{2}} = -\nabla_{i}E. \eqno(18.2.4.1)]$ The quantities $[m_{i}]$ and $[{\bf r}_{i}]$ are, respectively, the mass and coordinates of atom i, and E is given by equation (18.2.3.1). The solution of the partial differential equations (18.2.4.1) can be achieved numerically using finite-difference methods (Verlet, 1967; Abramowitz & Stegun, 1968). This approach is referred to as molecular dynamics.

Initial velocities for the integration of equation (18.2.4.1) are usually assigned randomly from a Maxwell distribution at the appropriate temperature. Assignment of different initial velocities will generally produce a somewhat different structure after simulated annealing. By performing several refinements with different initial velocities, one can therefore improve the chances of success of simulated-annealing refinement. Furthermore, this improved sampling can be used to study discrete disorder and conformational variability, especially when using torsion-angle molecular dynamics (see below).

Although Cartesian (i.e. flexible bond lengths and bond angles) molecular dynamics places restraints on bond lengths and bond angles [through $[E_{\rm chem}]$ , equation (18.2.3.1)], one might want to implement these restrictions as constraints, i.e., fixed bond lengths and bond angles (Diamond, 1971). This is supported by the observation that the deviations from ideal bond lengths and bond angles are usually small in macromolecular X-ray crystal structures. Indeed, fixed-length constraints have been applied to crystallographic refinement by least-squares minimization (Diamond, 1971). It is only recently, however, that efficient and robust algorithms have become available for molecular dynamics in torsion-angle space (Bae & Haug, 1987, 1988; Jain et al., 1993; Rice & Brünger, 1994). We chose an approach that retains the Cartesian-coordinate formulation of the target function and its derivatives with respect to atomic coordinates, so that the calculation remains relatively straightforward and can be applied to any macromolecule or their complexes (Rice & Brünger, 1994). In this formulation, the expression for the acceleration becomes a function of positions and velocities. Iterative equations of motion for constrained dynamics in this formulation can be derived and solved by finite-difference methods (Abramowitz & Stegun, 1968). This method is numerically very robust and has a significantly increased radius of convergence in crystallographic refinement compared to Cartesian molecular dynamics (Rice & Brünger, 1994).

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International Tables for Crystallography (2006). Vol. F. ch. 18.2, p. 378