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LINCS: A Linear Constraint Solver for Molecular Simulations

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

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LINCS: A Linear Constraint Solver for Molecular Simulations

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  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) + Δt2a(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 The different types of motion present in various systems together with suggested time steps

  7. Fc g 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 • r2 – a2 = 0 Holonomic • r2 – a2 >= 0 Non-holonomic • In a constrained system • Particles are not independent • Magnitude of constraint forces are unknown

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

  15. 1st and 2nd derivatives of constraints are zero Constraint forces , where LINCS 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. Constrained leap-frog algorithm =0 is the projection matrix that sets the constrained coordinates to zero =0 Since If we set We can use LINCS

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

  20. LINCS • The first power of An 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 An • 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. 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 Results • Solves constrained molecular dynamics • Numerically stable • Conserves energy • Three to four times faster than SHAKE • Can be easily parallelized

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