Molecular dynamics simulations with constrained roto-translational
motions: Theoretical basis and statistical mechanical consistency
A. Amadei,
a)
G. Chillemi,
b)
M. A. Ceruso, A. Grottesi, and A. Di Nola
Department of Chemistry, University of Rome ‘‘La Sapienza,’’ P.le A. Moro 5, 00185, Rome, Italy
Received 29 April 1999; accepted 4 October 1999
From a specific definition of the roto-translational external and intramolecular internal
coordinates, a constrained dynamics algorithm is derived for removing the roto-translational
motions during molecular dynamics simulations, within the leap-frog integration scheme. In the
paper the theoretical basis of this new method and its statistical mechanical consistency are reported,
together with two applications. © 2000 American Institute of Physics. S0021-96060050201-3
I. INTRODUCTION
Often in molecular simulations the interest is focused on
the structural properties of a molecule with internal degrees
of freedom. For simulations of a single molecule in vacuum
ideal gas condition as well as for simulations of a solute
molecule in its solvent infinite dilution condition, the roto-
translational motions are in general uninteresting, while the
behavior of the internal coordinates can be very important
for many different studies. In particular with large organic
molecules, and especially biomacromolecules, simulations
are usually performed to obtain a detailed investigation of
the conformational fluctuations, which can be studied only
after removing the overall translation and rotation of the
molecule. In these cases the ensemble of molecular configu-
rations obtained from the simulation is normally manipulated
to remove these roto-translational motions. In general the
roto-translation is eliminated by over-imposing the center of
mass of the actual configuration with that of a reference one,
and then least square fitting the atomic displacements be-
tween the two structures rotating the actual structure around
its center of mass.
1
This procedure, although usually efficient
and widely used, has one disadvantage. Its implicit definition
of external and internal coordinates is rather complicated,
especially for the definition of conjugated momenta, and
hence it is difficult to use this approach for theoretical me-
chanics or rigorous statistical mechanical studies, as well as
to derive ideal constraint forces to stop the molecular roto-
translational motions directly during the simulation. It should
be considered that a rigorous method to constrain the roto-
translational motions during a simulation can be advanta-
geous in the following cases:
1 For large and flexible molecules the removal of the dy-
namical coupling between the internal motions and mo-
lecular roto-translations can shorten the system’s relax-
ation time and hence provide a better configurational
sampling for the internal coordinates in simulations of
usual time lengths.
2 Simulations of a molecule in vacuum, with either a usual
force field or using a mean field, are usually performed
at zero angular momentum and this constraint can alter
the statistical mechanical consistency of the simulation.
On the contrary when ideal holonomic constraints are
used to stop the roto-translational motions, and the an-
gular momentum is not fixed anymore, the simulation
can provide the exact statistical mechanics of the system.
3 For simulations of large nonspherical molecules e.g.,
proteins in water, with the presence of the roto-
translational constraints we could use a simulation box
shaped on molecular geometry, reducing significantly
the number of necessary water molecules. The effect of
large solute rotations is negligible for short length simu-
lations hundreds of picoseconds but becomes relevant
for longer time simulations. In fact, from the nanosec-
onds range the solute has enough time to rotate signifi-
cantly and hence, without using a cubic simulation box,
to interact directly with its periodic images.
4 For the calculation of free energy differences due to
changes of the roto-translational configuration for inter-
acting molecules or for a molecule interacting with an
external field e.g., molecular docking the simulations
with the roto-translational constraints could be extremely
efficient.
In this paper we show that it is possible to use a defini-
tion of internal and external coordinates which is very suited
for theoretical derivations and that allows direct simulation
of only the molecular internal degrees of freedom. In fact,
this definition of the molecular coordinates standard in ana-
lytical mechanics allows the use of ideal constraint forces to
stop instantaneously the roto-translational motions during the
simulation. It is well known from theoretical mechanics
2,3
that the use of ideal holonomic constraints in a Hamiltonian
system still provides Hamiltonian dynamics in the con-
strained phase space constraint subspace, and so a con-
strained Hamiltonian system can still be described by the
microcanonical ensemble. We will show that more often in
general the holonomic constraints do not alter the basic type
of dynamics in the case of the usual molecular dynamics
MD equations of motion involving a frictional term, and
a
Author to whom correspondence should be addressed; electronic mail:
amadei@seurat.chem.uniroma1.it
b
Inter-University Computing Consortium CASPUR, University of Rome
‘‘La Sapienza,’’ P.le A. Moro 5, 00185, Rome, Italy.
JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 1 1 JANUARY 2000
9 0021-9606/2000/112(1)/9/15/$17.00 © 2000 American Institute of Physics
Downloaded 13 Jan 2003 to 151.100.52.54. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp