Molecular Simulation and Systems Biology - Systems Biology: Networks, Models, and Applications - page 70

all the atoms in the system, and the force on any particular atom is
the gradient of this function with respect to motion of that atom.

Types and Uses of Classical Molecular Simulations
Given a model for the interactions between atoms and molecules,
one can perform computations of various types to explore different
aspects of the behavior of the molecular system. These aspects usually
can be characterized as structural, thermodynamic, or kinetic. The first
relates to the determination of the most stable molecular conformation,
or structure, of the molecular system. Protein structure refinement falls
into this category. This problem is one of optimization: given an energy
expression, find the set of atomic coordinates that minimizes the
energy. This problem is nontrivial because of the high dimensionality
of the search space (a small protein with 2000 atoms has a search
space with 6000 dimensions). Furthermore, the energy function has
a large number of local minima, and so the global minimum can
be very difficult to find. Most energy functions can be analytically
differentiated, so one can use gradients and second, or even higher,
derivatives of the energy function to facilitate the search.

The Velocity Verlet Algorithm

Start with current atomic positions, velocities and forces:

{ r i ( t ), v i ( t ), F i ( t )}

Using the current atomic coordinates, velocities, and forces, update the
coordinates:

Ft
m

()

i

rt ( +∆ t ) = rt () + v t () ⋅ ∆ + t ⋅ ∆ t

2

i i i

2

Using the updated positions of all the atoms, update the forces:

− ∇ i U ( r 1 ( t +∆ t ), r 2 ( t +∆ t ), . . .) F i ( t + ∆ t ) =

Using the updated forces, update the velocities:

+ ∆

t
m

vt ( +∆ t ) = vt ( ) [ Ft ( +∆ t ) + Ft ( )]

i i i i

2

Finish with new atomic positions, velocities, and forces:

{ r i ( t + ∆ t ), v i ( t + ∆ t ), F i ( t + ∆ t )}