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## Constraint algorithmIn mechanics, a Constraint algorithms are 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 with explicit or implicit constraint forces for the bond lengths and bond angles. Explicit constraint forces typically shorten the time-step significantly, making the simulation less efficient computationally; in other words, more computer power is required to compute a trajectory of a given length. Therefore, internal coordinates and implicit-force constraint solvers are generally preferred. ## Additional recommended knowledge
## Mathematical backgroundThe motion of a set of where If where the index This problem was studied in detail by Joseph Louis Lagrange, who laid out most of the methods for solving it. A second approach is to introduce explicit forces that work to maintain the constraint; for example, one could introduce strong spring forces that enforce the distances among mass points within a "rigid" body. The two difficulties of this approach are that the constraints are not satisfied exactly, and the strong forces may require very short time-steps, making simulations inefficient computationally. A third approach is to use a method such as Lagrange multipliers or projection to the constraint manifold to determine the coordinate adjustments necessary to satisfy the constraints. Finally, there are various hybrid approaches in which different sets of constraints are satisfied by different methods, e.g., internal coordinates, explicit forces and implicit-force solutions. ## Internal coordinate methodsThe simplest approach to satisfying constraints in energy minimization and molecular dynamics is to represent the mechanical system in so-called The original methods for efficient recursive energy minimization in internal coordinates were developed by Gō and coworkers. Efficient recursive, internal-coordinate constraint solvers were extended to molecular dynamics. ## Lagrange multiplier-based methodsIn most molecular dynamics simulation, constraints are enforced using the method of Lagrange multipliers. Given a set of where and are the positions of the two particles involved in the These constraint equations, are added to the potential energy function in the equations of motion, resulting in, for each of the Adding the constraint equations to the potential does not change it, since all should, ideally, be zero. Integrating both sides of the equations of motion twice in time yields the constrained particle positions at the time where is the unconstrained (or uncorrected) position of the To satisfy the constraints in the next timestep, the Lagrange multipliers must be chosen such that This implies solving a systen of simultaneously for the This system of where is the Jacobian of the equations σ Since not all particles are involved in all constraints, is blockwise-diagonal and can be solved blockwise, i.e. molecule for molecule. Furthermore, instead of constantly updating the vector , the iteration is started with , resulting in simpler expressions for and . After each iteration, the unconstrained particle positions are updated using - .
The vector is then reset to This is repeated until the constraint equations are satisfied up to a prescribed tolerance. Although there are a number of algorithms to compute the Lagrange multipliers, they differ only in how they solve the system of equations, usually using Quasi-Newton methods. ## The SETTLE algorithmThe SETTLE algorithm ## The SHAKE algorithmThe SHAKE algorithm was the first algorithm developed to satisfy bond geometry constraints during molecular dynamics simulations. It solves the system of non-linear constraint equations using the Gauss-Seidel method to approximate the solution of the linear system of equations in the Newton iteration. This amounts to assuming that is diagonally dominant and solving the λ _{k}
for all iteratively until the constraint equations are solved to a given tolerance. Each iteration of the SHAKE algorithm costs operations and the iterations themselves converge linearly. A noniterative form of SHAKE was developed later. Several variants of the SHAKE algorithm exist. Although they differ in how they compute or apply the constraints themselves, the constraints are still modelled using Lagrange multipliers which are computed using the Gauss-Seidel method. 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. Further extensions include RATTLE A final modification is the P-SHAKE algorithm ## The M-SHAKE algorithmThe M-SHAKE algorithm is solved exactly using an LU decomposition. Each iteration costs operations, yet the solution converges quadratically, requiring much less iterations than SHAKE. This solution was first proposed in 1986 by Ciccotti and Ryckaert ## The LINCS algorithmAn alternative constraint method, LINCS (Linear Constraint Solver) was developed in 1997, LINCS applies Lagrange multipliers to the constraint forces and solves for the multipliers by using a series expansion to approximate the inverse of the Jacobian : in each step of the Newton iteration. The inversion only works for matrices with Eigenvalues smaller than 1, making the algorithm suitable only for molecules with low connectivity. ## Hybrid methodsHybrid methods have also been introduced in which the constraints are divided into two groups; the constraints of the first group are solved using internal coordinates whereas those of the second group are solved using constraint forces, e.g., by a Lagrange multiplier or projection method. ## See also## References and footnotes- ^
^{a}^{b}Laplace, PS (1788).*Mécanique analytique*. **^**Noguti T; Gō N (1983). "A Method of Rapid Calculation of a 2nd Derivative Matrix of Conformational Energy for Large Molecules".*Journal of the Physical Society of Japan***52**: 3685–3690.**^**Abe, H; Braun W, Noguti T, Gō N (1984). "Rapid Calculation of 1st and 2nd Derivatives of Conformational Energy with respect to Dihedral Angles for Proteins: General Recurrent Equations".*Computers and Chemistry***8**: 239–247.**^**Bae, D-S; Haug EJ (1988). "A Recursive Formulation for Constrained Mechanical System Dynamics: Part I. Open Loop Systems".*Mechanics of Structures and Machines***15**: 359–382.**^**Jain, A; Vaidehi N, Rodriguez G (1993). "A Fast Recursive Algorithm for Molecular Dynamics Simulation".*Journal of Computational Physics***106**: 258–268.**^**Rice, LM; Brünger AT (1994). "Torsion Angle Dynamics: Reduced Variable Conformational Sampling Enhances Crystallographic Structure Refinement".*Proteins: Structure, Function, and Genetics***19**: 277–290.**^**Mathiowetz, AM; Jain A, Karasawa N, Goddard III, WA (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.**^**Mazur, AK (1997). "Quasi-Hamiltonian Equations of Motion for Internal Coordinate Molecular Dynamics of Polymers".*Journal of Computational Chemistry***18**: 1354–1364.**^**Miyamoto, S; Kollman PA (1992). "SETTLE: An Analytical Version of the SHAKE and RATTLE Algorithm for Rigid Water Models".*Journal of Computational Chemistry***13**: 952–962.**^**Ryckaert, J-P; Ciccotti G, Berendsen HJC (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.**^**Yoneya, M; Berendsen HJC, Hirasawa K. "A Noniterative Matrix Method for Constraint Molecular-Dynamics Simulations".*Molecular Simulations***13**: 395–405.**^**Forester, TR; Smith W (1998). "SHAKE, Rattle, and Roll: Efficient Constraint Algorithms for Linked Rigid Bodies".*Journal of Computational Chemistry***19**: 102–111.**^**McBride, C; Wilson MR, Howard JAK (1998). "Molecular dynamics simulations of liquid crystal phases using atomistic potentials".*Molecular Physics***93**: 955–964.**^**Andersen, Hans C. (1983). "RATTLE: A "Velocity" Version of the SHAKE Algorithm for Molecular Dynamics Calculations".*Journal of Computational Physics***52**: 24-34. doi:10.1016/0021-9991(83)90014-1.**^**Lee, Sang-Ho; Kim Palmo, Samuel Krimm (2005). "WIGGLE: A new constrained molecular dynamics algorithm in Cartesian coordinates".*Journal of Computational Physics***210**: 171-182. doi:10.1016/j.jcp.2005.04.006.**^**Lambrakos, S. G.; J. P. Boris, E. S. Oran, I. Chandrasekhar, M. Nagumo (1989). "A Modified SHAKE algorithm for Maintaining Rigid Bonds in Molecular Dynamics Simulations of Large Molecules".*Journal of Computational Physics***85**: 473-486. doi:10.1016/0021-9991(89)90160-5.**^**Gonnet, Pedro (2007). "P-SHAKE: A quadratically convergent SHAKE in ".*Journal of Computational Physics***220**: 740-750. doi:10.1016/j.jcp.2006.05.032.**^**Kräutler, Vincent; W. F. van Gunsteren, P. H. Hünenberger (2001). "A Fast SHAKE Algorithm to Solve Distance Constraint Equations for Small Molecules in Molecular Dynamics Simulations".*Journal of Computational Chemistry***22**(5): 501-508.**^**Ciccotti, G.; J. P. Ryckaert (1986). "Molecular Dynamics Simulation of Rigid Molecules".*Computer Physics Reports***4**: 345-392.- ^
^{a}^{b}Hess, B; Bekker H, Berendsen HJC, Fraaije JGEM (1997). "LINCS: A Linear Constraint Solver for Molecular Simulations".*Journal of Computational Chemistry***18**: 1463–1472. **^**Edberg, R; Evans DJ, Morriss GP (1986). "Constrained Molecular-Dynamics Simulations of Liquid Alkanes with a New Algorithm".*Journal of Chemical Physics***84**: 6933–6939.**^**Baranyai, A; Evans DJ (1990). "New Algorithm for Constrained Molecular-Dynamics Simulation of Liquid Benzene and Naphthalene".*Molecular Physics***70**: 53–63.**^****^**Bae, D-S; Haug EJ (1988). "A Recursive Formulation for Constrained Mechanical System Dynamics: Part II. Closed Loop Systems".*Mechanics of Structures and Machines***15**: 481–506.**^**Rodriguez, G; Jain A, Kreutz-Delgado K (1991). "A Spatial Operator Algebra for Manipulator Modeling and Control".*The International Journal for Robotics Research***10**: 371–381.**^**Sommerfeld, Arnold (1952).*Lectures on Theoretical Physics, Vol. I: Mechanics*. New York: Academic Press. ISBN-10 0126546703.
Categories: Molecular dynamics | Computational chemistry | Molecular physics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Constraint_algorithm". A list of authors is available in Wikipedia. |