MOLECULAR SIMULATION TECHNIQUES
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Transcript of MOLECULAR SIMULATION TECHNIQUES
• Molecular dynamics (MD) is a computer simulation
technique where the time evolution of a set of
interacting atoms is followed by integrating their
equations of motion.
• We follow the laws of classical mechanics, and most
notably Newton's law:
Molecular dynamics - Introduction
A brief description of the molecular dynamics
method
Successive configuration of the molecular system can be obtained by
integrating Newton’s laws of motion. Positions and momenta of the particles
of the given molecular system are described by the trajectories obtained by
the successive integration of the Newton’s equations which are mathematical
description of the following natural rules:
1. A body continues to move in a straight line at a constant velocity
unless a force acts upon it;
2. Force equals the rate of change of momentum;
3. To every action there is an equal and opposite reaction;
The trajectories are obtained by solving the differential equations of the
Newton’s second law:
i
xi
m
F
dt
xdi
2
2
Simple models
Hard sphere potential
Square well potential
MOLECULAR DYNAMICS USING
SIMPLE METHODS
The steps involved in the hard-sphere calculation
as follows:
1. Identify the next pair of spheres to collide and calculate
when the collision will occur.
2. Calculate the positions of all the spheres at the collision
time.
3. Determine the new velocities of the two colliding
spheres after the collision.
4. Repeat from 1 until finished.
The new velocities of the colliding spheres are
calculated by applying the principle conservation
of linear momentum.
MOLECULAR DYNAMICS WITH
CONTINUOUS POTENTIALS
First MD with continuous potentials done in 1964 (simulation of argon
by Rahman).
Finite difference method: the integration is broken down into many small stages, each separated in time by a fixed time dt.
.........)()()(
.........)(2
1)(
. . . . . .)(6
1)(
2
1)()()(
. . . . . .)(24
1)(
6
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2
32
432
ttctbttb
tctttba(t)δt)a(t
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Verlet algorithm
The most widely used method in molecular dynamics programs is the Verlet algorithm. It uses the positions and accelerations at time t, and the positions from the previous step, r(t-δt) to calculate new positions at t+δt, r(t+δt). Relations between positions and velocities at those two moments in time can be written as:
Those two relations can be added to give:
The velocities do not explicitly appear in the Verlet algorithm. They can be calculated in several ways. A very simple approach is to divide the difference in positions at times t+δt and t-δt by 2δt, i.e.
Another approach calculates velocities at the half step :
Practical application of this algorithm is straightforward and memory requirements are modest, only positions at two time steps have to be recorded r(t), r(t-δt), and the acceleration a(t). The only drawback is that the new position r(t + δt) is obtained by adding small term δ2ta(t) to the difference of two much larger terms 2r(t) and r(t-δt), which requires high precision for r in the numerical calculation.
.....)(2
1)()()(
.....)(2
1)()()(
2
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tatttvtrttr
)()()(2)( 2 tatttrtrttr
tttrttrtv 2/)]()([)(
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Verlet algorithm
The leap-frog method is the variation of Verlet algorithm. It uses the following relations:
The name of this method comes from its nature, i.e., velocities make ‘leap-frog’ jumps over the positions to give their values at
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Verlet algorithm
The velocity Verlet algorithm gives positions, velocities and accelerations at the same time and does not compromise precision:
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1)()(
)(2
1)()()( 2
ttatattvttv
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Verlet algorithm
Beeman Algorithm
Better velocities, better energy conservation More expensive to calculate
)(6
1)(
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1)()(
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2)()()( 22
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General Predictor-Corrector
Algorithms
Predict the position x(t+dt) and velocity v(t+dt)
at the end of the next step.
Evaluate the forces at t+dt using the predicted
position.
Correct the predictions using some combination
of the predicted and previous values of position
and velocity.
Gear’s Predictor-Corrector methods
Predict ac(t+dt) from the Taylor expansion at the starting point
Begin with a simple prediction, as in any of the previous methods
Initially step to r(t+dt), v (t+dt), a(t+dt),b(t+dt) at that point.
The difference between the a(t+dt) and the predicted ac(t+dt):
a(t t) ac(t t) a(t t)
Estimates the error in the initial step, which is used to correct:
rc(t t) r(t t) c0 a(t t)
vc(t t) v(t t) c1 a(t t)
ac(t t) / 2 a(t t)/ 2 c2 a(t t)
bc(t t)/ 6 b(t t) / 6 c3 a(t t)
Predictor-corrector algorithms
1. Predictor: From the positions and their time derivatives up to a certain order q, all known at time t, one ``predicts'' the same quantities at time by means of a Taylor expansion. Among these quantities are, of course, accelerations .
2. Force evaluation: The force is computed taking the gradient of the potential at the predicted positions. The resulting acceleration will be in general different from the ``predicted acceleration''. The difference between the two constitutes an ``error signal''.
3. Corrector: This error signal is used to ``correct'' positions and their derivatives. All the corrections are proportional to the error signal, the coefficient of proportionality being a ``magic number'' determined to maximize the stability of the algorithm.
Evaluate integration methods
Fast, minimal memory, easy to program
Calculation of force is time consuming
Conservation of energy and momentum
Time-reversible
Long time step can be used
Which algorithm is appropriate
Cost effective
Energy conservation
Root-mean-square fluctuation
Total, 0.02 kcal/mol
KE and PE, 5 kcal/mol
Choosing the time step
Too small: covering small conformation space
Too large: instability
Suggested time steps
Translation, 10 fs
Flexible molecules and rigid bonds, 2fs
Flexible molecules and bonds, 1fs
Multiple time step dynamics
Reversible reference system propagation
algorithm (r-RESPA)
Forces within a system classified into a number of
groups according to how rapidly the force changes
Each group has its own time step, while maintaining
accuracy and numerical stability
Molecular dynamics setup
Initial configuration
Initial velocities (Maxwell-Boltzmann)
Force field
Cutoff: doesn’t save time by itself. But can
combine with neighbor list and speed-up the
simulation
Tk
vm
Tk
mvP
B
ixi
B
iix
22/1
2
1exp
2)(
Running molecular dynamics
Equilibration
Special care is needed for inhomogeneous system
Calculating the temperature
Nc is the number of constraints, so 3N – Nc is the total number of degrees of
freedom
Boundary conditions
No boundary
Periodic boundary condition
Non-periodic: reaction zone, harmonic constraint
boundary atoms
N
i
CB
i
i NNTk
m
pH
1
2
322
Constraint dynamics
High frequency modes takes all the computer time
Low frequency modes correspond to conformational
changes
Constraint: system is forced to satisfy certain
conditions
SHAKE: constraint the bond vibration
Molecular Modelling - Andrew R. Leach
(Principles and Applications)
http:/docjax.com/molecular dynamic simulation methods/
http:/google.com/molecular dynamics/ simulation simple methods/
http:/google.com/molecular dynamic simulation with continuous potential/