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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 Newtons 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?t) 2r(t) - r (t-?t) ?t2a(t)
- v(t) r(t?t) - r(t-?t)/2?t
- Velocities are not directly generated, and are

one time step behind - Leap-frog algorithm
- r(t?t) r(t) ?tv(t?t/2)
- v(t?t/2) v(t-?t/2) ?ta(t)
- v(t) v(t?t/2) v(t-?t/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 gt 0 Non-holonomic
- In a constrained system
- Particles are not independent
- Magnitude of constraint forces are unknown

Introduction to Molecular Dynamics

- Integration algorithms
- Standard Verlet Algorithm
- Leap-frog algorithm
- Choosing time step
- Constrained dynamics
- Holonomic 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

LINCS

Newtons 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

LINCS

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

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 parallelized

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