Molecular Dynamics Simulations

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Molecular Dynamics Simulations An
Introduction N. Gautham Department of
Crystallography and Biophysics University of
Madras, Guindy Campus Chennai 600
025 gautham_at_unom.ac.in
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• Molecular Dynamics
• Definitions, Motivations
• Force fields
• Algorithms and computations
• Water and Solvent

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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

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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.

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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.

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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).

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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.

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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.

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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.

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Molecular dynamics The model
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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)

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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

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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.

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Molecular dynamics Force Fields

• A solution to this problem is to use periodic
  boundary conditions (PBC).
• We use the minimum image criterion among all
  possible images of a particle j, select only the
  closest.

-1,1 0,1 1,1
-1,0 0,0 Primary Cell 1,0
-1,-1 0,-1 1,-1
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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.

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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

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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

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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.

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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.

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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

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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

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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

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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)

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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

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Molecular dynamics Analyses
Analyses using trajectories
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Molecular dynamics Analyses
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Molecular dynamics Analyses
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Molecular dynamics Analyses
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Molecular dynamics Analyses
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Molecular dynamics Analyses
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Molecular dynamics Analyses
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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

Global minimum
energy
Conformational space
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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 T0

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Molecular dynamics Optimization tool
Trajectory
energy
Conformational space
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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.

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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

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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.ht
  ml