This module can be used to enforce the rotation of a group of atoms, as
e.g. a protein subunit. There are a variety of rotation potentials,
among them complex ones that allow flexible adaptations of both the
rotated subunit as well as the local rotation axis during the
simulation. An example application can be found in ref.
145.
In the fixed axis approach (see Fig. 45 B),
torque on a group of atoms with positions
(denoted “rotation group”) is applied
by rotating a reference set of atomic positions – usually their initial
positions – at a constant angular
velocity around an axis defined by a direction vector
and a pivot point
. To that aim, each atom with
position is attracted by a “virtual
spring” potential to its moving reference position
,
where is a matrix that describes the rotation
around the axis. In the simplest case, the “springs” are described by a
harmonic potential,
where , , and are the components of
the normalized rotation vector ,
and . As illustrated in
Fig. 46 A for a single atom ,
the rotation matrix operates on the initial
reference positions
of atom at . At a later time , the
reference position has rotated away from its initial place (along the
blue dashed line), resulting in the force
The forces generated by the isotropic potentials
(eqns. (361) and (365)) also contain components parallel to the
rotation axis and thereby restrain motions along the axis of either the
whole rotation group (in case of ) or within the
rotation group, in case of .
For cases where unrestrained motion along the axis is preferred, we have implemented a
“parallel motion” variant by eliminating all components parallel to the
rotation axis for the potential. This is achieved by projecting the
distance vectors between reference and actual positions
In the above variants, the minimum of the rotation potential is either a
single point at the reference position
(for the isotropic potentials) or a
single line through parallel to the
rotation axis (for the parallel motion potentials). As a result, radial
forces restrict radial motions of the atoms. The two subsequent types of
rotation potentials, and , drastically
reduce or even eliminate this effect. The first variant,
(Fig. 46 B), eliminates all force
components parallel to the vector connecting the reference atom and the
rotation axis,
As seen in Fig. 46 B, the force
resulting from still contains a small, second-order
radial component. In most cases, this perturbation is tolerable; if not,
the following alternative, , fully eliminates the
radial contribution to the force, as depicted in
Fig. 46 C,
where a small parameter has been introduced to avoid
singularities. For , the
equipotential planes are spanned by and ,
yielding a force perpendicular to
, thus not
contracting or expanding structural parts that moved away from or toward
the rotation axis.
Choosing a small positive (e.g.,
,
Fig. 46 D) in the denominator of
eqn. (380) yields a well-defined potential and
continuous forces also close to the rotation axis, which is not the case
for
(Fig. 46 C). With
As sketched in Fig. 45 A–B, the rigid body
behavior of the fixed axis rotation scheme is a drawback for many
applications. In particular, deformations of the rotation group are
suppressed when the equilibrium atom positions directly depend on the
reference positions. To avoid this limitation,
eqns. (378) and (383)
will now be generalized towards a “flexible axis” as sketched in
Fig. 45 C. This will be achieved by
subdividing the rotation group into a set of equidistant slabs
perpendicular to the rotation vector, and by applying a separate
rotation potential to each of these slabs.
Fig. 45 C shows the midplanes of the slabs
as dotted straight lines and the centers as thick black dots.
To avoid discontinuities in the potential and in the forces, we define
“soft slabs” by weighing the contributions of each slab to the
total potential function by a Gaussian function
with
.
This choice also implies that the individual contributions to the force
from the slabs add up to unity such that no further normalization is
required.
To each slab center , all atoms
contribute by their Gaussian-weighted (optionally also mass-weighted)
position vectors
.
The instantaneous slab centers are
calculated from the current positions
,
We consider two flexible axis variants. For the first variant, the slab
segmentation procedure with Gaussian weighting is applied to the radial
motion potential
(eqn. (378) / Fig. 46 B),
yielding as the contribution of slab
Note that for , as defined, the slabs are fixed in
space and so are the reference centers
. If during the simulation the
rotation group moves too far in
direction, it may enter a region where – due to the lack of nearby
reference positions – no reference slab centers are defined, rendering
the potential evaluation impossible. We therefore have included a
slightly modified version of this potential that avoids this problem by
attaching the midplane of slab to the center of mass of the
rotation group, yielding slabs that move with the rotation group. This
is achieved by subtracting the center of mass
of the group from the positions,
To simplify the force derivation, and for efficiency reasons, we here
assume to be constant, and thus
.
The resulting force error is small (of order or
if mass-weighting is applied) and can therefore be
tolerated. With this assumption, the forces
have the same form as eqn. (395).
In this second variant, slab segmentation is applied to
(eqn. (383)), resulting in
a flexible axis potential without radial force contributions
(Fig. 46 C),
Applying transformation (396) yields a
“translation-tolerant” version of the flexible2 potential,
. Again, assuming that
,
,
are small, the resulting equations for
and are similar
to those of and
.
To apply enforced rotation, the particles that are to be
subjected to one of the rotation potentials are defined via index groups
rot-group0, rot-group1, etc., in the
mdp input file. The reference positions
are read from a special
trr file provided to grompp. If no such
file is found, are used as
reference positions and written to trr such that they
can be used for subsequent setups. All parameters of the potentials such
as , , etc.
(Table 16) are provided as mdp
parameters; rot-type selects the type of the potential.
The option rot-massw allows to choose whether or not to
use mass-weighted averaging. For the flexible potentials, a cutoff value
(typically )
makes sure that only significant contributions to and
are evaluated, i.e. terms with
are omitted.
Table 17 summarizes observables that are
written to additional output files and which are described below.
Table 16 Parameters used by the various rotation potentials.
x indicate which parameter is actually used for a given potential#
Table 17 Quantities recorded in output files during enforced rotation simulations.
All slab-wise data is written every nstsout steps, other rotation data every nstrout steps.#
For fixed axis rotation, the average angle
of the group relative to the reference group is determined via the
distance-weighted angular deviation of all rotation group atoms from
their reference positions,
Here, is the distance of the reference position to the
rotation axis, and the difference angles are determined
from the atomic positions, projected onto a plane perpendicular to the
rotation axis through pivot point (see
eqn. (368) for the definition of
),
For flexible axis rotation, two outputs are provided, the angle of the
entire rotation group, and separate angles for the segments in the
slabs. The angle of the entire rotation group is determined by an RMSD
fit of to the reference positions
at , yielding
as the angle by which the reference has to
be rotated around for the optimal
fit,
To determine the local angle for each slab , both reference
and actual positions are weighted with the Gaussian function of slab
, and is calculated as in
eqn. (403) from the Gaussian-weighted
positions.
For all angles, the mdp input option
rot-fit-method controls whether a normal RMSD fit is
performed or whether for the fit each position
is put at the same distance to the
rotation axis as its reference counterpart
. In the latter case, the RMSD
measures only angular differences, not radial ones.
Angle Determination by Searching the Energy Minimum#
Alternatively, for rot-fit-method=potential, the angle
of the rotation group is determined as the angle for which the rotation
potential energy is minimal. Therefore, the used rotation potential is
additionally evaluated for a set of angles around the current reference
angle. In this case, the rotangles.log output file
contains the values of the rotation potential at the chosen set of
angles, while rotation.xvg lists the angle with minimal
potential energy.
where is the distance vector from
the rotation axis to and
is the force component
perpendicular to and
. For flexible axis rotation,
torques are calculated for
each slab using the local rotation axis of the slab and the
Gaussian-weighted positions.