Constraint (computational chemistry)
In mechanics, a constraint algorithm is a method for satisfying hard constraints for rigid bodies that follow Newton's equations of motion.
Application to molecular dynamics
Constraint algorithms have been often applied to molecular dynamics simulations. Although such simulations are sometimes carried out in internal coordinates that automatically satisfy the bond-length and bond-angle constraints, they may also be carried out in Cartesian coordinates with explicit constraint forces for the bond lengths and bond angles. Unfortunately, such forces require a short time step (0.5-1.5 fs) and merely serve to simulate the rapid, physically irrelevant oscillations of the bond geometry. Therefore, it is better to integrate Newton's equations while constraining the bond geometry mathematically.
SETTLE
For three atoms satisfying three constraints, it is trivial to solve the constraint equations analytically. The SETTLE algorithm is commonly used to satisfy such sets of constraints, which commonly occur in simulations of three-atom molecules such as water.
SHAKE
The SHAKE algortihm was the first algorithm developed to satisfy bond geometry constraints during molecular dynamics simulations.[1]
The original SHAKE algorithm is limited to mechanical systems with a tree structure, i.e., no closed loops of constraints. A later extension of the method, QSHAKE (Quaternion SHAKE) was developed to amend this.[2] It works satisfactorily for rigid loops such as aromatic ring systems but fails for flexible loops, such as when a protein has a disulfide bond.
LINCS
An alternative constraint method, LINCS (Linear Constraint Solver) was published in 1997.[3]
Internal coordinate methods
The earliest and simplest approach to satisfying hard constraints in energy minimization and molecular dynamics is to represent the mechanical system in so-called internal coordinates corresponding to the true degrees of freedom of the system. For example, the dihedral angles of a protein are an independent set of coordinates that specify the positions of all the atoms without requiring any constraints. The difficulty of such internal-coordinate approaches is two-fold: the Newtonian equations of motion become much more complex and the internal coordinates may be difficult to define for cyclic systems of constraints, e.g., in aromatic ring systems or when a protein has a disulfide bond.
The original methods for efficient recursive energy minimization were developed by Gō and coworkers.[4][5]
Efficient recursive, internal-coordinate constraint solvers were extended to molecular dynamics.[6] Analagous methods were applied later to other systems.[7][8][9]
See also
References and footnotes
- ^ Ryckaert, J-P (1977). "Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of n-Alkanes". Journal of Computational Physics. 23: 327–341.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Forester, TR (1998). "SHAKE, Rattle, and Roll: Efficient Constraint Algorithms for Linked Rigid Bodies". Journal of Computational Chemistry. 19: 102–111.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Hess, B (1997). "LINCS: A Linear Constraint Solver for Molecular Simulations". Journal of Computational Chemistry. 18: 1463–1472.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Noguti T (1983). "Unknown". Journal of the Physical Society of Japan. 52: 3685–3690.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Abe, H (1984). "Unknown". Computational Chemistry. 8: 239–247.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Jain, A (1993). "A Fast recursive Algorithm for Molecular Dynamics Simulation". Journal of Computational Physics. 106: 258–268.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Rice, LM (1994). "Torsion Angle Dynamics: Reduced Variable Conformational Sampling Enhances Crystallographic Structure Refinement". Proteins: Structure, Function, and Genetics. 19: 277–290.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Mathiowetz, AM (1994). "Protein Simulations Using Techniques Suitable for Very Large Systems: The Cell Multipole Method for Nonbond Interactions and the Newton-Euler Inverse Mass Operator Method for Internal Coordinate Dynamics". Proteins: Structure, Function, and Genetics. 20: 227–247.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ Mazur, AK (1997). "Quasi-Hamiltonian Equations of Motion for Internal Coordinate Molecular Dynamics of Polymers". Journal of Computational Chemistry. 18: 1354–1364.