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Constraint Dynamics

Constraint Dynamics. Fortgeschrittene Methoden in der Moleküldynamik SS 05 Lars Petzold, Mattias Maneck. Overview. Introduction Theory SHAKE and RATTLE Summary Demonstration. Introduction. Simulation of biomolecules is complex High frequency motions (bonds)

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Constraint Dynamics

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  1. Constraint Dynamics Fortgeschrittene Methoden in der Moleküldynamik SS 05 Lars Petzold, Mattias Maneck

  2. Overview • Introduction • Theory • SHAKE and RATTLE • Summary • Demonstration Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  3. Introduction • Simulation of biomolecules is complex • High frequency motions (bonds) • Low frequency motions (torsions angles) • Simulation time step is dictated by the highest frequency Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  4. Introduction • Wish: Increase the time step without prejudicing the accuracy of the simulation. • High frequency motions are not needed • Fix high frequencyparts of the molecule. • Example: q2 q1 - q2 q1 Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  5. Theory – Constraints q2 • Two kinds of time independent (scleronom) constraints • holonomic • non-holonomic d q1 g q2 d q1 g Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  6. Theory – Constraints • a pendulum • length d • weight M • Newton Equation Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  7. Theory – Lagrange equation • Lagrange equation • equation of motion Force field Cartesian coordinates Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  8. Theory – Lagrange equation • Lagrange equation with constraints • equation of motion Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  9. Theory – Example • a pendulum • equation of motion Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  10. Theory – Example • a pendulum • three ways to solve the equation • coupled differential equations • polar coordinates • numerical integration Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  11. Theory – Example • Coupled differential equations • differentiate the constraint two times on the time • inserting Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  12. Theory – Example • polar coordinates Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  13. Theory – Example • polar coordinates Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  14. Theory – Example • polar coordinates • equation of motion Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  15. Theory • Numerical integration is shown by Lars • like Verlets algorithm • Constraints are included in the Lagrange equation by adding the sum of all constraints • Differentiating the Lagrange equation gives the equation of motion, equal to Newtons equation. • Sometimes constraints can be avoided by generalizing coordinates. Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  16. SHAKEby J.-P. Ryckaert, G. Ciccotti, H.J.C. Berendsen • Goal: • Use constraints to reduce number of degrees of freedom (df) example: N particles, with 3*N df, k constraints  3*N-k df remain • Compute constraints iteratively • Use Verlet algorithm to compute trajectory Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  17. Constraints on … • Positions ( distances): Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  18. Verlet + Constraints  SHAKE Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  19. Illustration of Fc Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  20. Compute without constraints: • check constraints (e.g. i<->j) to correct : Computation of Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  21. Computation of (cont‘) • Tolerance exceeded: Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  22. keeps bond-lengths rigid Avoids calculation of uninteresting motions increases timestep Speeds up computation Precision-factor: Adds „Δt 2“ to „ Δt 0“ terms No direct access to velocities Hard to implement NPT simulation Calculation of T over velocity SHAKE Pros & Cons Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  23. RATTLE by Hans Christian Andersen • Integration method: Velocity Verlet (VV) • VV involves velocities to the computation of the trajectory • extra hidden constraints on the velocities • Makes algorithm more accurate • NTP-ensemble simulation possible Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  24. Velocity Verlet (VV) Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  25. Constraints on … • Positions ( distances): • Time derivates of position constraints: Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  26. VV+Constraints  RATTLE Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  27. Computation of • Compute q without constraints: • check constraints (e.g. i<->j) to correct b: Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  28. Computation of (cont‘) • Tolerance exceeded: Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  29. Compute without constraints: Computation of • check hidden constraints (e.g. i<->j) to correct : Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  30. Computation of (cont‘) • ToleranceHC exceeded: Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  31. Example:Simulation of two water-molecules • Water model: SPC/E(Berendsen et al. ´87) • Used forces: Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  32. Simulation:without constraints Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  33. Simulation: SHAKE Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  34. Simulation:SHAKE + „constraints-application“ Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  35. Summary • Constraints are included in the Lagrange equation by adding the sum of all constraints • SHAKE computes trajectory by only checking distances • RATTLE additionally considers velocities • RATTLE is more flexible than SHAKE Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

  36. References Papers: Shake: Journal of Computational Physics 23, 327-341 (1977) Numerical Integration of the Cartesian Equations of Motion of a System with Constraints; Molecular Dynamics of n-Alkanes Jean-Paul Ryckaert, Giovanni Cicotti, and Herman J. C. Berendsen Rattle: Journal of Computational Physics 52, 24-34 (1983) Rattle: A "Velocity" Version of the Shake Algorithm for Molecular Dynamics Calculations Hans C. Andersen Books: Leach, Molecular Modelling-Principles and Applications Allen, Computer Simulation of Liquids Frenkel, Understanding Molecular Simulation Matthias Maneck, Lars Petzold: Fortgeschrittene Methoden in der Moleküldynamik SS 05

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