1. Molecular Dynamics Simulations
An Introduction
N. Gautham
Department of Crystallography and Biophysics
University of Madras, Guindy Campus
Chennai

2. Molecular Dynamics Definitions, Motivations Force fields
Algorithms and computations
Water and Solvent

3. 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:

4. 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.

5. 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.

6. 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).

7. 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.

8. 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.

9. 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.

10. Molecular dynamics – The model

11. 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)

12. 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

13. 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.

14. 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

15. 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.

16. 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

17. 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

18. 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.

19. 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.

20. 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

21. 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

22. 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

23. 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’…)

24. 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

25. Molecular dynamics – Analyses
Analyses using ‘trajectories’

26. Molecular dynamics – Analyses

27. Molecular dynamics – Analyses

28. Molecular dynamics – Analyses

29. Molecular dynamics – Analyses

30. Molecular dynamics – Analyses

31. Molecular dynamics – Analyses

32. 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

33. 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

34. Molecular dynamics – Optimization tool
Trajectory
energy
Conformational space

35. 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.

36. 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

37. 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