1. LINCS: A Linear Constraint Solver for Molecular Simulations Berk Hess, Hemk Bekker, Herman J.C.Berendsen, Johannes G.E.M.Fraaije Journal of Computational Chemistry, 1997 Ankur Dhanik

2. Outline Introduction to Molecular Dynamics Problem description Some solutions LINCS Results

3. Introduction to Molecular Dynamics A classical molecular simulations method Successive configurations of the system are generated by integrating Newton’s laws of motion Integration algorithms should strive to reduce computation and conserve energy Various integration algorithms –Standard Verlet algorithm –Leap-frog algorithm

4. Introduction to Molecular Dynamics Standard Verlet Algorithm –r(t+Δt) = 2r(t) - r (t-Δt) + Δt 2 a(t) –v(t) = [r(t+Δt) - r(t-Δt)]/2Δt –Velocities are not directly generated, and are one time step behind Leap-frog algorithm –r(t+Δt) = r(t) + Δtv(t+Δt/2) –v(t+Δt/2) = v(t-Δt/2) + Δta(t) –v(t) = [v(t+Δt/2) + v(t-Δt/2)]/2 –Velocities are half time step behind

5. Introduction to Molecular Dynamics Choosing the time step One order of magnitude smaller than the shortest motion (bond vibrations) Severe restriction as these high frequency motions have minimal effect on the overall behavior of the system Constrained dynamics

6. Introduction to Molecular Dynamics SystemType of motion present Suggested times step(s) Atomstranslation10 -14 Rigid moleculesTranslation and rotation 5 X 10 -15 Rigid molecules, rigid bonds Translation, rotation, torsion 2 X 10 -15 Rigid molecules, flexible bonds Translation, rotation, torsion, vibration 10 -15 or 5 X 10 -16 The different types of motion present in various systems together with suggested time steps

7. Introduction to Molecular Dynamics In constrained dynamics bonds and angles are forced to adopt specific values throughout a simulation Constraints are categorized as holonomic and non-holonomic. Suppose a particle on the surface of a sphere –r 2 – a 2 = 0 Holonomic –r 2 – a 2 >= 0 Non-holonomic In a constrained system –Particles are not independent –Magnitude of constraint forces are unknown FcFc g

8. Introduction to Molecular Dynamics Integration algorithms –Standard Verlet Algorithm –Leap-frog algorithm Choosing time step Constrained dynamics Holonomic and non-holonomic constraints

9. Outline Introduction to Molecular Dynamics Problem description Some solutions LINCS Results

10. Problem Description Design an algorithm for solving constrained molecular dynamics, the constraints being holonomic The algorithm should strive for following features: –Numerical stability –Energy conservation –Computational efficiency

11. Some solutions Reset coupled constraints after an unconstrained update –Non-linear problem –SETTLE Solves analytically Very fast, but unsuited for large molecules –SHAKE Iterative method Sequentially all the bonds are set to the correct length Simple and numerically stable No solutions may be found when displacements are large, difficult to parallelize

12. Some solutions EEM –The second derivatives of constraint equations are set to zero –All the constraints are dealt with simultaneously –Corrections are necessary to achieve accuracy and stability

13. LINCS Does an unconstrained update Sets the projection of the new bonds onto the old directions of the bonds to the prescribed lengths Similar to EEM, with some practical differences Implements –Efficient solver for the matrix equation –A velocity correction that prevents rotational lengthening –A length correction that improves accuracy and stability

14. LINCS Newton’s equation of motion Constraint equations Lagrangian formulation Gradient matrix of constraint equations, B

15. LINCS 1st and 2 nd derivatives of constraints are zero Constraint forces, where Newton’s equation of motion (I-TB) is projection matrix which sets the constrained coordinates to zero. T transforms motions in the constrained coordinates into motions in cartesian coordinates

16. LINCS Constrained leap-frog algorithm =0 is the projection matrix that sets the constrained coordinates to zero Since =0 If we setWe can use

17. LINCS Numerical errors can accumulate which leads to drifts Velocity correction Position correction Drawback: the projection of the new bonds onto the old directions rather than the new bonds are set to prescribed lengths

18. LINCS Correction for rotational lengthening –To correct rotation of bond i, the projection of the bond on the old direction is set to The corrected positions are

19. LINCS Constrained new position is given by Half of the CPU time goes to invert with diagonal element & A is symmetric, sparse and has zeros on the diagonal

20. LINCS The first power of A n gives the coupling effects of neighboring bonds The second power gives the coupling effect over a distance of two bonds The inversion through a series expansion makes parallelization easy In one timestep, the bonds influence each other when they are separated by fewer bonds than highest order in expansion

21. LINCS Parallelization –Consider a linear-bond constrained molecule of 100 atoms to be simulated on a dual processor computer –Uses rotation correction and an expansion to the second power of A n –Because order of expansion is two, bonds influence each other over a distance of 6 –Update of position and call of LINCS algorithm must be done for atom 1-56 and 44-100 on processors 1 and 2 respectively –1-50 update from processor 1 and 50-100 from processor 2

22. Results

23. Solves constrained molecular dynamics Numerically stable Conserves energy Three to four times faster than SHAKE Can be easily parallelized Problem Description Design an algorithm for solving constrained molecular dynamics, the constraints being holonomic The algorithm should strive for following features: –Numerical stability –Energy conservation –Computational efficiency