0.1 Representing proteins in silico and protein forward kinematics  (Page 7/7)

The only preprocessing step that is necessary to start working with this method, however, is to perform an initial pass on the protein to extract the initial values of the dihedral angles and the constant bond lengths and angles, from the Cartesian coordinates available from PDB files, if the intention is to start from the protein's native state. This is easily done; bond lengths can be obtained by computing the distance between the bonded atoms, and bond angles by computing the angle between the vectors formed by two consecutive bonds (recall that the dot product of two vectors yields the product of their lengths times the cosine of the angle between them). Next, we present the transformations required for the Denavit-Hartenberg method.

Consider three consecutive bonds as in the figure below. Suppose that a local coordinate frame is attached at the beginning of each bond. For example, local coordinate system $x_{i-1}$ , $y_{i-1}$ , $z_{i-1}$ is centered at atom $A_{i-1}$ . Therefore, imagine that the position of each atom in three-dimensional space is specified in terms ofa frame that is centered at the previous atom. Given the frames at atom $A_{i-2}$ , and atom $A_{i-1}$ , one can determine how the frames at atoms $A_i$ and atom $A_{i-1}$ will change in space as a consequence of a rotation around the bond that connects atoms $A_{i-1}$ and $A_i$ with the dihedral angle . The correct transformation can be computed in terms of three primitive operations: two rotations and one translation. The two rotations are a rotation around the dihedral bond by the dihedral angle and a rotation around an axis perpendicular to the bond angle, by the bond angle. The translation refers to the fact that the origins of the frames are on the respective centers of the atoms connected by the bond, thus separated by bond lengths.

Required reading

• Zhang-Kavraki 2002 [PDF] Zhang, M. and L. E. Kavraki, "A New Method for Fast and Accurate Derivation of Molecular Conformations". Journal of Chemical Information and Computer Sciences, 42:64-70, 2002.
• Stamati-Shehu-Kavraki. Computing Forward Kinematics for Protein-like linear systems using Denavit-Hartenberg Local Frames [PDF]

Resources

• VMD Visual Molecular Dynamics, is an excellent tool for visualization and scripted manipulation of protein structures that uses Tcl scripting - Humphrey, W., Dalke, A. and Schulten, K., "VMD - Visual Molecular Dynamics", J. Molec. Graphics, 1996, vol. 14, pp. 33-38.
• RasMol is mostly a viewer, but has some built-in tools. - Roger Sayle and E. James Milner-White. "RasMol: Biomolecular graphics for all", Trends in Biochemical Sciences (TIBS), September 1995, Vol. 20, No. 9, p. 374.
• Chimera is a very powerful visualizer that handles huge structures easily. - Pettersen, E.F., Goddard, T.D., Huang, C.C., Couch, G.S., Greenblatt, D.M., Meng, E.C., and Ferrin, T.E. "UCSF Chimera - A Visualization System for Exploratory Research and Analysis." J. Comput. Chem. 25(13):1605-1612 (2004).
• InsightII , Cerius2 and Catalyst are products for simulation, discovery and analysis that recently became commercial, and can be found here .
• CHARMM is a simulation package based on the CHARMM force field. Brooks BR, Bruccoleri RE, Olafson BD, States DJ, Swaminathan S, Karplus M (1983). "CHARMM: A program for macromolecular energy, minimization, and dynamics calculations". J Comp Chem 4: 187217.
• NAMD is another popular simulation package and can be obtained here . James C. Phillips, Rosemary Braun, Wei Wang, James Gumbart, Emad Tajkhorshid, Elizabeth Villa, Christophe Chipot, Robert D. Skeel, Laxmikant Kale, and Klaus Schulten. Scalable molecular dynamics with NAMD. Journal of Computational Chemistry, 26:1781-1802, 2005.
• Amber is one of the most widely used molecular dynamics simulators due to its speed. Duan et al. A point-charge force field for molecular mechanics simulations of proteins based on condensed-phase quantum mechanical calculations Journal of Computational Chemistry Vol. 24, Issue 16. Pages 1999-2012 (2003).