Molecular Dynamics
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Molecular Dynamics
● is computer simulation of physical movements
by atoms and molecules
● is frequently used in the study of proteins and
biomolecules, as well as in materials science
● is a specialized discipline of molecular
modeling and computer simulation based on
statistical mechanics
Wikipedia
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Molecular Dynamics
Biological molecules exhibit a wide range of time scales over which
specific processes occur; for example
– Local Motions (0.01 to 5 Å, 10-15
to 10-1
s)
● Atomic fluctuations
● Sidechain Motions
● Loop Motions
– Rigid Body Motions (1 to 10Å, 10-9
to 1s)
● Helix Motions
● Domain Motions (hinge bending)
● Subunit motions
– Large-Scale Motions (> 5Å, 10-7
to 104
s)
● Helix coil transitions
● Dissociation/Association
● Folding and Unfolding
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Molecular Dynamics
● Molecular dynamics simulations permit the study of complex,
dynamic processes that occur in biological systems. These include,
for example,
● Protein stability
● Conformational changes
● Protein folding
● Molecular recognition: proteins, DNA, membranes,
complexes
● Ion transport in biological systems
● and provide the mean to carry out the following studies,
● Drug Design
● Structure determination: X-ray and NMR
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Molecular Dynamics
● First macromolecular MD simulation published (1977,
Size: 500 atoms, Simulation Time: 9.2 ps=0.0092 ns,
Program: CHARMM precursor) Protein: Bovine
Pancreatic Trypsine Inhibitor. This is one of the best
studied proteins in terms of folding and kinetics. Its
simulation published in Nature magazine paved the
way for understanding protein motion as essential in
function and not just accessory.
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Molecular Dynamics
● MD simulation of the complete satellite tobacco
mosaic virus (STMV) (2006, Size: 1 million atoms,
Simulation time: 50 ns, program: NAMD) This
virus is a small, icosahedral plant virus which
worsens the symptoms of infection by Tobacco
Mosaic Virus (TMV). Molecular dynamics
simulations were used to probe the mechanisms
of viral assembly. The entire STMV particle
consists of 60 identical copies of a single protein
that make up the viral capsid (coating), and a 1063
nucleotide single stranded RNA genome. One key
finding is that the capsid is very unstable when
there is no RNA inside. The simulation would take
a single 2006 desktop computer around 35 years
to complete. It was thus done in many processors
in parallel with continuous communication
between them
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MD ensembles
● Microcanonical ensemble (NVE) : The thermodynamic state
characterized by a fixed number of atoms, N, a fixed volume, V, and
a fixed energy, E. This corresponds to an isolated system.
● Canonical Ensemble (NVT): This is a collection of all systems
whose thermodynamic state is characterized by a fixed number of
atoms, N, a fixed volume, V, and a fixed temperature, T.
● Isobaric-Isothermal Ensemble (NPT): This ensemble is
characterized by a fixed number of atoms, N, a fixed pressure, P,
and a fixed temperature, T.
● Grand canonical Ensemble (mVT): The thermodynamic state for
this ensemble is characterized by a fixed chemical potential, m, a
fixed volume, V, and a fixed temperature, T.
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MD – how it works
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Molecular Dynamics
0 ps 1 ps Time2 ps
1 MD step = 0.002 ps = 2 fs
500 steps 500 steps
1 snapshot 1 snapshot
Saved each 500 step
...
...
..
.
MD trajectorie = set of snapshots
2 ns
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Molecular Dynamics - trajectory
MD trajectorie = set of snapshots
0 ps 200 ps 400 ps 600 ps
800 ps
1000 ps 1200 ps 1400 ps
1600 ps 1800 ps
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MD – average structure
..
.
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MD – structural changes
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MD – math behind
Newton’s equation of motion is given by
where Fi
is the force exerted on particle i, mi
is the mass of
particle i and ai
is the acceleration of particle i. The force can also
be expressed as the gradient of the potential energy,
Combining these two equations yields
where V is the potential energy of the system. Newton’s equation
of motion can then relate the derivative of the potential energy to
the changes in position as a function of time.
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MD – math behind
The potential energy is a function of the atomic positions
(3N) of all the atoms in the system. Due to the complicated
nature of this function, there is no analytical solution to the
equations of motion; they must be solved numerically.
Numerous numerical algorithms have been developed for
integrating the equations of motion.
* Verlet algorithm
* Leap-frog algorithm
* Velocity Verlet
* Beeman’s algorithm
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MD – Verlet algorithm
All the integration algorithms assume the
positions, velocities and accelerations can be
approximated by a Taylor series expansion:
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MD – Verlet algorithm
To derive the Verlet algorithm one can write
Summing these two equations, one obtains
The Verlet algorithm uses positions and accelerations at time t and the
positions from time t-δt to calculate new positions at time t+δt. The Verlet
algorithm uses no explicit velocities. The advantages of the Verlet algorithm
are, i) it is straightforward, and ii) the storage requirements are modest . The
disadvantage is that the algorithm is of moderate precision.
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MD - The Velocity Verlet algorithm
This algorithm yields positions, velocities and accelerations at
time t. There is no compromise on precision.
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MD – Beeman´s algorithm
This algorithm is closely related to the Verlet algorithm
The advantage of this algorithm is that it provides a more
accurate expression for the velocities and better energy
conservation. The disadvantage is that the more complex
expressions make the calculation more expensive.
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MD – The leap-frog algorithm
In this algorithm, the velocities are first calculated at time t+1/2δt; these are
used to calculate the positions, r, at time t+δt. In this way, the velocities leap
over the positions, then the positions leap over the velocities. The advantage
of this algorithm is that the velocities are explicitly calculated, however, the
disadvantage is that they are not calculated at the same time as the positions.
The velocities at time t can be approximated by the relationship:
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MD – Periodic boundary conditions
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MD – short range interactions
The pairwise interactions for a
single particle can be
computed by a) computing the
interaction to all other particles
or b) by dividing the domain
into cells with an edge length
of at least the cut-off radius of
the interaction potential and
computing the interaction
between the particle and all
particles in the same (red) and
in the adjacent (green) cells.
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MD – Long range interactions
● Particle Mesh Evald
Real space
Fourier space
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MD – Solvation model
● Implicit solvent
– a method of representing solvent as a
continuous medium instead of individual
“explicit” solvent molecules
– Poisson – Boltzmann model
– Generalized Born model
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MD – Solvation model
● Explicit solvent model
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Restraints x Constraints
r0 r0
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MD – post-simulation analysis
● MM-PBSA , MM-GBSA analysis
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MD – post-simulation analysis
● Secondary structure of proteins – DSSP
– DSSP recognizes eight types of secondary structure (each identified by
its own symbol), depending on the pattern of hydrogen bonds.The
310 helix, alpha helix and pi helix are symbolized as G, H and I,
respectively, and are recognized by having a repetitive sequence of
hydrogen bonds in which the donor residue is three, four, or five
residues later in the backbone. DSSP recognizes two types of
hydrogen-bond pairs in beta sheet structures, the parallel and
antiparallel bridge; a lone bridge is symbolized by B (a beta bridge),
whereas longer sets of H-bond pairs of the same type are symbolized
by E for extended, which is equivalent to a beta sheet. Sheets with
beta bulges are likewise symbolized by E. The remaining types are T
for turn (featuring a hydrogen bond typical of a helix), S for a region
of high curvature (if the angle between the vector from Cαi to Cαi+2
and the vector from Cαi-2
to Cαi is less than 70°, and a blank if no
other rule pertains. Additionally there is L meaning "loop" or "other"
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MD - software
● AMBER
● NAMD
● GROMOS
● GROMACS
● YASARA
● CHARMM