Transcript ng. ppt

Molecular Dynamics Simulations
An Introduction
N. Gautham
Department of Crystallography and Biophysics
University of Madras, Guindy Campus
Chennai 600 025
[email protected]
Molecular Dynamics
• Definitions, Motivations
• Force fields
• Algorithms and computations
• Water and Solvent
Molecular dynamics - Introduction
• 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
• Given an initial set of positions and velocities, the
subsequent time evolution is in principle completely
determined.
• Atoms and molecules will ‘move’ in the computer,
bumping into each other, vibrating about a mean
position (if constrained), or wandering around (if the
system is fluid), oscillating in waves in concert with
their neighbours, perhaps evaporating away from the
system if there is a free surface, and so on, in a way
similar to what real atoms and molecules would do.
Molecular dynamics -Motivation
• The computer experiment.
• In a computer experiment, a model is still provided
by theorists, but the calculations are carried out by
the machine by following a recipe (the algorithm,
implemented in a suitable programming language).
• In this way, complexity can be introduced (with
caution!) and more realistic systems can be
investigated, opening a road towards a better
understanding of real experiments.
Molecular dynamics -Motivation
• The computer calculates a trajectory of the system
• 6N-dimensional phase space (3N positions and 3N
momenta).
• A trajectory obtained by molecular dynamics provides
a set of conformations of the molecule,
• They are accessible without any great expenditure of
energy (e.g. breaking bonds)
• MD also used as an efficient tool for optimisation of
structures (simulated annealing).
Molecular dynamics - Motivation
• MD allows to study the dynamics of large
macromolecules
• Dynamical events control processes which affect
functional properties of the biomolecule (e.g. protein
folding).
• Drug design is used in the pharmaceutical industry to
test properties of a molecule at the computer without
the need to synthesize it.
Molecular dynamics - Introduction
• In molecular dynamics, atoms interact with each
other.
• These interactions are due to forces which act upon
every atom, and which originate from all other atoms
• Atoms move under the action of these instantaneous
forces.
• As the atoms move, their relative positions change
and forces change as well.
Molecular dynamics – Time Limitations
• Typical MD simulations are performed on systems
containing thousands of atoms
• Simulation times range from a few picoseconds to
hundreds of nanoseconds.
• A simulation is reliable when the simulation time is
much longer than the relaxation time of the quantities
we are interested in.
Molecular dynamics – The model
Molecular dynamics – Force Fields
Epot = SVbond + SVang + SVtorsion + SVvdW + SVele + …
Other terms (the ‘…’)
• Planarity constraints
• Hydrogen bonding potentials
• Interaction terms (between different types of motion
e.g. bond length stretch – bond angle bend)
Molecular dynamics – Force Fields
• The potential as specified by the above has an infinite
range.
• In practical applications, it is customary to establish a
cutoff radius Rc and disregard the interactions
between atoms separated by more than Rc
Molecular dynamics – Force Fields
• What should we do at the boundaries of our simulated
system?
• If nothing special is done, atoms near the boundary
would have less neighbours than atoms inside.
• This causes surface effects in the simulation to be
much more important than they are in the real
system.
Molecular dynamics – Force Fields
• A solution to this problem is to use periodic boundary
conditions (PBC).
-1,1
0,1
1,1
-1,0
0,0
1,0
Primary
Cell
-1,-1
0,-1
1,-1
• We use the minimum image criterion: among all
possible images of a particle j, select only the closest.
Molecular dynamics – Algorithms
• The engine of a molecular dynamics program is its
time integration algorithm.
• Time integration algorithms are based on finite
difference methods, where time is discretized on a
finite grid, the time step t being the distance
between consecutive points on the grid
• Knowing the positions and some of their time
derivatives at time t, the integration scheme gives the
same quantities at a later time t+t
• By iterating the procedure, the time evolution of the
system can be followed for long times.
Molecular dynamics – Algorithms
• These schemes are approximate and there are errors
associated with them
• Truncation errors are related to the accuracy of the
finite difference method with respect to the true
solution. These errors are intrinsic to the algorithm.
• Round-off errors are related to errors associated to a
particular implementation of the algorithm. For
instance, to the finite number of digits used in
computer arithmetic.
• Both errors can be reduced by decreasing t
Molecular dynamics – Algorithms
• Two popular integration methods for MD calculations
are the Verlet algorithm and predictor-corrector
algorithms
• The most commonly used time integration algorithm
is the Verlet algorithm
Molecular dynamics – Algorithms
• The predictor-corrector algorithm consists of three
steps
• Step 1: Predictor. From the positions and their time
derivatives at time t, one ‘predicts’ the same
quantities at time t+t by means of a Taylor
expansion. Among these quantities are, of course,
accelerations ‘a’
• Step 2: Force evaluation. The force is computed by
taking the gradient of the potential at the predicted
positions.
Molecular dynamics – Algorithms
• Step 2 (contd.): The difference between the resulting
acceleration and the predicted acceleration constitutes
an ‘error signal
• Step 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 determined to maximize the
stability of the algorithm.
Molecular dynamics – Algorithms
• To start the simulation we have to create a set of
initial positions and velocities for the atoms in the
molecule
• The initial positions usually correspond to a known
structure (from X-ray or NMR structures, or predicted
models)
• The initial velocities are assigned taking them from a
Maxwell distribution at a certain temperature T
• Another possibility is to take the initial positions and
velocities to be the final positions and velocities of a
previous MD run
Molecular dynamics – Water and solvent
• The molecule is positioned in a box of size
approximately twice the largest dimension of the
molecule
• The molecule is solvated by adding water (or other
solvent molecules) at random positions in the box –
no two atoms can be touching each other
Molecular dynamics – Algorithms
• Every time the state of the system changes (e.g.
when we start the simulation) the system will be out
of equilibrium for a while
• We usually want equilibrium to be reached before
starting performing measurements on the system
• A physical quantity A generally approaches its
equilibrium value exponentially with time:
•  may be a few hundred time steps, allowing us to
see A(t) converge to Ao
Molecular dynamics – Analyses
• The simplest way of analyzing the system during (or
after) its dynamic motion is looking at it.
• One can assign a radius to the atoms, represent the
atoms as balls having that radius, and have a
computer program construct a ‘photograph’ of the
system.
• We may also colour the atoms according to its
properties (charge, displacement, ‘temperature’…)
Molecular dynamics – Analyses
• We also can measure instantaneous and time
averages of various physically important quantities
• To measure time averages: If the instantaneous
values of some property A at time t is
then its average is
where NT is the number of steps in the trajectory
Molecular dynamics – Analyses
Slide
Shift
-3
3 -3
Tilt
3 -2
5
-30
Roll
30 -30
30 10
50
G 2- C 23
-0.8
0.4
3.5
-3.3
9.4
35.9
C 3- G 22
1.1
-0.7
3.5
6.0
-9.6
G 4- C 21
Analyses using ‘trajectories’
35.3
Molecular dynamics – Analyses
Molecular dynamics – Analyses

Molecular dynamics – Analyses







45
C
1
278
251
127
275
131
163
174
53
G
293
2
265
257
144
172
180
C
3
290
266
109
109
A
48
284
4
140 221
5
263
55
292
264
132
248
57
266
124
7
8
295
264
122
9
133
245
138
50
292
272
124
237
177
171
188
C
142
52
183
T
240
173
181
T
160
173
297
6
160
170
211
184
A
238
175
193
G
179
49
44
277
264
136
262
143
161
169
56
G
291
10
267
144
C
11
254
172
180
269
116
292
174
185
G
12
123
83 A
156 B
178
48
313 A
264 B
178 A
155 B
242
125
248
127
36 B
45 A
314 B
285 A
13 A
262 B
208 A
214 B
206 A
191 B
Molecular dynamics – Analyses
Molecular dynamics – Analyses
Molecular dynamics – Analyses
Molecular dynamics – Optimization tool
• Molecular Dynamics may also be used as an
optimization tool
• Traditional (optimization) minimization techniques
(steepest descent, conjugate gradient, etc.) do not
normally overcome energy barriers and tend to fall
into the nearest local minimum
energy
Global minimum
Conformational space
Molecular dynamics – Optimization tool
• Temperature in a molecular dynamics calculation
provides a way to fly over the barriers
• States with energy E are visited with a probability
exp(-E/kBT)
• By decreasing T slowly to 0, there is a good chance
that the system will be able to pick up the best
minimum and land into it
• This is the simulated annealing protocol, where the
system is equilibrated at a certain (high) temperature
and then slowly cooled down to T=0
Molecular dynamics – Optimization tool
Trajectory
energy
Conformational space
Molecular dynamics – Other Methods
• We have discussed so far the standard molecular
dynamics scheme, based on the time integration of
Newton's equations and leading to the conservation
of the total energy.
• In the statistical mechanics parlance, these
simulations are performed in the microcanonical
ensemble, or NVE ensemble
• The number of particles, the volume and the energy
are constant quantities.
Molecular dynamics – Other Methods
• There are other important alternatives to the NVE
ensemble
• A scheme for simulations in the isoenthalpic-isobaric
ensemble (NPH) has been developed
• The volume V of the box is variable. The enthalpy
H=(E + PV ) is a conserved quantity.
• Another very important ensemble is the canonical
ensemble (NVT).
• The temperature is kept constant
Molecular mechanics – References
• Molecular Modelling
A.R. Leach (2001) Prentice Hall.
• Understanding Molecular Simulation
D. Frenkel and B. Smit (1996) Academic Press
• Molecular Dynamics Simulation
J.M. Haile (1992) John Wiley
• http://www.fisica.uniud.it/~ercolessi/md/md/md.html