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

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

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

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

Introduction to Molecular Dynamics

System Type of motion present

Suggested times step(s)

Atoms translation 10-14

Rigid molecules Translation 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

10-15or 5 X 10-16

The different types of motion present in various systems together with suggested time steps

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

Fc

g

Introduction to Molecular Dynamics

Integration algorithms– Standard Verlet Algorithm– Leap-frog algorithm

Choosing time stepConstrained dynamicsHolonomic and non-holonomic constraints

Outline

Introduction to Molecular Dynamics

• Problem description

• Some solutions

• LINCS

• Results

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

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

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

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

LINCS

Newton’s equation

of motion

Constraint equations

Lagrangian formulation

Gradient matrix of

constraint equations, B

LINCS

1st and 2nd derivatives

of constraints are zero

Constraint forces

, where

Newton’s equationof 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

LINCS

Constrained leap-frog algorithm

=0

is the projection matrix that sets the constrained coordinates to zero

Since =0

If we set We can use

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

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

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

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

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

Results

Results

• Solves constrained molecular dynamics

• Numerically stable

• Conserves energy

• Three to four times faster than SHAKE

• Can be easily parallelizedProblem 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