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r"""Bond-Angle-Torsion coordinates analysis --- :mod:`MDAnalysis.analysis.bat`
===========================================================================
:Author: Soohaeng Yoo Willow and David Minh
:Year: 2020
:Copyright: GNU Public License, v2 or any higher version
.. versionadded:: 2.0.0
This module contains classes for interconverting between Cartesian and an
internal coordinate system, Bond-Angle-Torsion (BAT) coordinates [Chang2003]_,
for a given set of atoms or residues. This coordinate system is designed
to be complete, non-redundant, and minimize correlations between degrees
of freedom. Complete and non-redundant means that for N atoms there will
be 3N Cartesian coordinates and 3N BAT coordinates. Correlations are
minimized by using improper torsions, as described in [Hikiri2016]_.
More specifically, bond refers to the bond length, or distance between
a pair of bonded atoms. Angle refers to the bond angle, the angle between
a pair of bonds to a central atom. Torsion refers to the torsion angle.
For a set of four atoms a, b, c, and d, a torsion requires bonds between
a and b, b and c, and c and d. The torsion is the angle between a plane
containing atoms a, b, and c and another plane containing b, c, and d.
For a set of torsions that share atoms b and c, one torsion is defined as
the primary torsion. The others are defined as improper torsions, differences
between the raw torsion angle and the primary torsion. This definition reduces
the correlation between the torsion angles.
Each molecule also has six external coordinates that define its translation and
rotation in space. The three Cartesian coordinates of the first atom are the
molecule's translational degrees of freedom. Rotational degrees of freedom are
specified by the axis-angle convention. The rotation axis is a normalized vector
pointing from the first to second atom. It is described by the polar angle,
:math:`\phi`, and azimuthal angle, :math:`\theta`. :math:`\omega` is a third angle
that describes the rotation of the third atom about the axis.
This module was adapted from AlGDock [Minh2020]_.
See Also
--------
:class:`~MDAnalysis.analysis.dihedrals.Dihedral`
class to calculate dihedral angles for a given set of atoms or residues
:func:`MDAnalysis.lib.distances.calc_dihedrals()`
function to calculate dihedral angles from atom positions
Example applications
--------------------
The :class:`~MDAnalysis.analysis.bat.BAT` class defines bond-angle-torsion
coordinates based on the topology of an atom group and interconverts between
Cartesian and BAT coordinate systems.
For example, we can determine internal coordinates for residues 5-10
of adenylate kinase (AdK). The trajectory is included within the test data files::
import MDAnalysis as mda
from MDAnalysisTests.datafiles import PSF, DCD
import numpy as np
u = mda.Universe(PSF, DCD)
# selection of atomgroups
selected_residues = u.select_atoms("resid 5-10")
from MDAnalysis.analysis.bat import BAT
R = BAT(selected_residues)
# Calculate BAT coordinates for a trajectory
R.run()
After :meth:`R.run()<BAT.run>`, the coordinates can be accessed with
:attr:`R.results.bat<BAT.bat>`. The following code snippets assume that the
previous snippet has been executed.
Reconstruct Cartesian coordinates for the first frame::
# Reconstruct Cartesian coordinates from BAT coordinates
# of the first frame
XYZ = R.Cartesian(R.results.bat[0,:])
# The original and reconstructed Cartesian coordinates should all be close
print(np.allclose(XYZ, selected_residues.positions, atol=1e-6))
Change a single torsion angle by :math:`\pi`::
bat = R.results.bat[0,:]
bat[bat.shape[0]-12] += np.pi
XYZ = R.Cartesian(bat)
# A good number of Cartesian coordinates should have been modified
np.sum((XYZ - selected_residues.positions)>1E-5)
Store data to the disk and load it again::
# BAT coordinates can be saved to disk in the numpy binary format
R.save('test.npy')
# The BAT coordinates in a new BAT instance can be loaded from disk
# instead of using the run() method.
Rnew = BAT(selected_residues, filename='test.npy')
# The BAT coordinates before and after disk I/O should be close
print(np.allclose(Rnew.results.bat, R.results.bat))
Analysis classes
----------------
.. autoclass:: BAT
:members:
:inherited-members:
.. attribute:: results.bat
Contains the time series of the Bond-Angle-Torsion coordinates as a
(nframes, 3N) :class:`numpy.ndarray` array. Each row corresponds to
a frame in the trajectory. In each column, the first six elements
describe external degrees of freedom. The first three are the center
of mass of the initial atom. The next three specify the external angles
according to the axis-angle convention: :math:`\phi`, the polar angle,
:math:`\theta`, the azimuthal angle, and :math:`\omega`, a third angle
that describes the rotation of the third atom about the axis. The next
three degrees of freedom are internal degrees of freedom for the root
atoms: :math:`r_{01}`, the distance between atoms 0 and 1,
:math:`r_{12}`, the distance between atoms 1 and 2,
and :math:`a_{012}`, the angle between the three atoms.
The rest of the array consists of all the other bond distances,
all the other bond angles, and then all the other torsion angles.
References
----------
.. [Chang2003] Chang, Chia-En, Michael J Potter, and Michael K Gilson (2003).
"Calculation of Molecular Configuration Integrals". *Journal of Physical
Chemistry B* 107(4): 1048–55. doi:`10.1021/jp027149c
<https://doi.org/10.1021/jp027149c>`_
.. [Hikiri2016] Hikiri, Simon, Takashi Yoshidome, and Mitsunori Ikeguchi (2016).
"Computational Methods for Configurational Entropy Using Internal and
Cartesian Coordinates." *Journal of Chemical Theory and Computation*
12(12): 5990–6000. doi:`10.1021/acs.jctc.6b00563
<https://doi.org/10.1021/acs.jctc.6b00563>`_
.. [Minh2020] Minh, David D L (2020). "Alchemical Grid Dock (AlGDock): Binding
Free Energy Calculations between Flexible Ligands and Rigid Receptors."
*Journal of Computational Chemistry* 41(7): 715–30.
doi:`10.1002/jcc.26036 <https://doi.org/10.1002/jcc.26036>`_
"""
import logging
import warnings
import numpy as np
import copy
import MDAnalysis as mda
from .base import AnalysisBase
from MDAnalysis.lib.distances import calc_bonds, calc_angles, calc_dihedrals
from MDAnalysis.lib.mdamath import make_whole
from ..due import due, Doi
logger = logging.getLogger(__name__)
def _sort_atoms_by_mass(atoms, reverse=False):
r"""Sorts a list of atoms by name and then by index
The atom index is used as a tiebreaker so that the ordering is reproducible.
Parameters
----------
ag_o : list of Atoms
List to sort
reverse : bool
Atoms will be in descending order
Returns
-------
ag_n : list of Atoms
Sorted list
"""
return sorted(atoms, key=lambda a: (a.mass, a.index), reverse=reverse)
def _find_torsions(root, atoms):
"""Constructs a list of torsion angles
Parameters
----------
root : AtomGroup
First three atoms in the coordinate system
atoms : AtomGroup
Atoms that are allowed to be part of the torsion angle
Returns
-------
torsions : list of AtomGroup
list of AtomGroup objects that define torsion angles
"""
torsions = []
selected_atoms = list(root)
while len(selected_atoms) < len(atoms):
torsionAdded = False
for a1 in selected_atoms:
# Find a0, which is a new atom connected to the selected atom
a0_list = _sort_atoms_by_mass(a for a in a1.bonded_atoms \
if (a in atoms) and (a not in selected_atoms))
for a0 in a0_list:
# Find a2, which is connected to a1, is not a terminal atom,
# and has been selected
a2_list = _sort_atoms_by_mass(a for a in a1.bonded_atoms \
if (a!=a0) and len(a.bonded_atoms)>1 and \
(a in atoms) and (a in selected_atoms))
for a2 in a2_list:
# Find a3, which is
# connected to a2, has been selected, and is not a1
a3_list = _sort_atoms_by_mass(a for a in a2.bonded_atoms \
if (a!=a1) and \
(a in atoms) and (a in selected_atoms))
for a3 in a3_list:
# Add the torsion to the list of torsions
torsions.append(mda.AtomGroup([a0, a1, a2, a3]))
# Add the new atom to selected_atoms
# which extends the loop
selected_atoms.append(a0)
torsionAdded = True
break # out of the a3 loop
break # out of the a2 loop
if torsionAdded is False:
print('Selected atoms:')
print([a.index + 1 for a in selected_atoms])
print('Torsions found:')
print([list(t.indices + 1) for t in torsions])
raise ValueError('Additional torsions not found.')
return torsions
[docs]class BAT(AnalysisBase):
"""Calculate BAT coordinates for the specified AtomGroup.
Bond-Angle-Torsions (BAT) internal coordinates will be computed for
the group of atoms and all frame in the trajectory belonging to `ag`.
"""
@due.dcite(Doi("10.1002/jcc.26036"),
description="Bond-Angle-Torsions Coordinate Transformation",
path="MDAnalysis.analysis.bat.BAT")
def __init__(self, ag, initial_atom=None, filename=None, **kwargs):
r"""Parameters
----------
ag : AtomGroup or Universe
Group of atoms for which the BAT coordinates are calculated.
`ag` must have a bonds attribute.
If unavailable, bonds may be guessed using
:meth:`AtomGroup.guess_bonds <MDAnalysis.core.groups.AtomGroup.guess_bonds>`.
`ag` must only include one molecule.
If a trajectory is associated with the atoms, then the computation
iterates over the trajectory.
initial_atom : :class:`Atom <MDAnalysis.core.groups.Atom>`
The atom whose Cartesian coordinates define the translation
of the molecule. If not specified, the heaviest terminal atom
will be selected.
filename : str
Name of a numpy binary file containing a saved bat array.
If filename is not ``None``, the data will be loaded from this file
instead of being recalculated using the run() method.
Raises
------
AttributeError
If `ag` does not contain a bonds attribute
ValueError
If `ag` contains more than one molecule
"""
super(BAT, self).__init__(ag.universe.trajectory, **kwargs)
self._ag = ag
# Check that the ag contains bonds
if not hasattr(self._ag, 'bonds'):
raise AttributeError('AtomGroup has no attribute bonds')
if len(self._ag.fragments) > 1:
raise ValueError('AtomGroup has more than one molecule')
# Determine the root
# The initial atom must be a terminal atom
terminal_atoms = _sort_atoms_by_mass(\
[a for a in self._ag.atoms if len(a.bonds)==1], reverse=True)
if (initial_atom is None):
# Select the heaviest root atoms from the heaviest terminal atom
initial_atom = terminal_atoms[0]
elif (not initial_atom in terminal_atoms):
raise ValueError('Initial atom is not a terminal atom')
# The next atom in the root is bonded to the initial atom
# Since the initial atom is a terminal atom, there is only
# one bonded atom
second_atom = initial_atom.bonded_atoms[0]
# The last atom in the root is the heaviest atom
# bonded to the second atom
# If there are more than three atoms,
# then the last atom cannot be a terminal atom.
if self._ag.n_atoms != 3:
third_atom = _sort_atoms_by_mass(\
[a for a in second_atom.bonded_atoms \
if (a in self._ag) and (a!=initial_atom) \
and (a not in terminal_atoms)], \
reverse=True)[0]
else:
third_atom = _sort_atoms_by_mass(\
[a for a in second_atom.bonded_atoms \
if (a in self._ag) and (a!=initial_atom)], \
reverse=True)[0]
self._root = mda.AtomGroup([initial_atom, second_atom, third_atom])
# Construct a list of torsion angles
self._torsions = _find_torsions(self._root, self._ag)
# Get indices of the root and torsion atoms
# in a Cartesian positions array that matches the AtomGroup
self._root_XYZ_inds = [(self._ag.indices==a.index).nonzero()[0][0] \
for a in self._root]
self._torsion_XYZ_inds = [[(self._ag.indices==a.index).nonzero()[0][0] \
for a in t] for t in self._torsions]
# The primary torsion is the first torsion on the list
# with the same central atoms
prior_atoms = [sorted([a1, a2]) for (a0, a1, a2, a3) in self._torsions]
self._primary_torsion_indices = [prior_atoms.index(prior_atoms[n]) \
for n in range(len(prior_atoms))]
self._unique_primary_torsion_indices = \
list(set(self._primary_torsion_indices))
self._ag1 = mda.AtomGroup([ag[0] for ag in self._torsions])
self._ag2 = mda.AtomGroup([ag[1] for ag in self._torsions])
self._ag3 = mda.AtomGroup([ag[2] for ag in self._torsions])
self._ag4 = mda.AtomGroup([ag[3] for ag in self._torsions])
if filename is not None:
self.load(filename)
def _prepare(self):
self.results.bat = np.zeros(
(self.n_frames, 3*self._ag.n_atoms), dtype=np.float64)
def _single_frame(self):
# Calculate coordinates based on the root atoms
# The rotation axis is a normalized vector pointing from atom 0 to 1
# It is described in two degrees of freedom
# by the polar angle and azimuth
if self._root.dimensions is None:
(p0, p1, p2) = self._root.positions
else:
(p0, p1, p2) = make_whole(self._root, inplace=False)
v01 = p1 - p0
v21 = p1 - p2
# Internal coordinates
r01 = np.sqrt(np.sum(v01 *
v01)) # Distance between first two root atoms
r12 = np.sqrt(np.sum(v21 *
v21)) # Distance between second two root atoms
# Angle between root atoms
a012 = np.arccos(max(-1.,min(1.,np.sum(v01*v21)/\
np.sqrt(np.sum(v01*v01)*np.sum(v21*v21)))))
# External coordinates
e = v01 / r01
phi = np.arctan2(e[1], e[0]) # Polar angle
theta = np.arccos(e[2]) # Azimuthal angle
# Rotation to the z axis
cp = np.cos(phi)
sp = np.sin(phi)
ct = np.cos(theta)
st = np.sin(theta)
Rz = np.array([[cp * ct, ct * sp, -st], [-sp, cp, 0],
[cp * st, sp * st, ct]])
pos2 = Rz.dot(p2 - p1)
# Angle about the rotation axis
omega = np.arctan2(pos2[1], pos2[0])
root_based = np.concatenate((p0, [phi, theta, omega, r01, r12, a012]))
# Calculate internal coordinates from the torsion list
bonds = calc_bonds(self._ag1.positions,
self._ag2.positions,
box=self._ag1.dimensions)
angles = calc_angles(self._ag1.positions,
self._ag2.positions,
self._ag3.positions,
box=self._ag1.dimensions)
torsions = calc_dihedrals(self._ag1.positions,
self._ag2.positions,
self._ag3.positions,
self._ag4.positions,
box=self._ag1.dimensions)
# When appropriate, calculate improper torsions
shift = torsions[self._primary_torsion_indices]
shift[self._unique_primary_torsion_indices] = 0.
torsions -= shift
# Wrap torsions to between -np.pi and np.pi
torsions = ((torsions + np.pi) % (2 * np.pi)) - np.pi
self.results.bat[self._frame_index, :] = np.concatenate(
(root_based, bonds, angles, torsions))
[docs] def load(self, filename, start=None, stop=None, step=None):
"""Loads the bat trajectory from a file in numpy binary format
Parameters
----------
filename : str
name of numpy binary file
start : int, optional
start frame of analysis
stop : int, optional
stop frame of analysis
step : int, optional
number of frames to skip between each analysed frame
See Also
--------
save: Saves the bat trajectory in a file in numpy binary format
"""
logger.info("Choosing frames")
self._setup_frames(self._trajectory, start, stop, step)
logger.info("Loading file")
self.results.bat = np.load(filename)
# Check array dimensions
if self.results.bat.shape != (self.n_frames, 3*self._ag.n_atoms):
errmsg = ('Dimensions of array in loaded file, '
f'({self.results.bat.shape[0]},'
f'{self.results.bat.shape[1]}), differ from required '
f'dimensions of ({self.n_frames, 3*self._ag.n_atoms})')
raise ValueError(errmsg)
# Check position of initial atom
if (self.results.bat[0, :3] != self._root[0].position).any():
raise ValueError('Position of initial atom in file ' + \
'inconsistent with current trajectory in starting frame.')
return self
[docs] def save(self, filename):
"""Saves the bat trajectory in a file in numpy binary format
See Also
--------
load: Loads the bat trajectory from a file in numpy binary format
"""
np.save(filename, self.results.bat)
[docs] def Cartesian(self, bat_frame):
"""Conversion of a single frame from BAT to Cartesian coordinates
One application of this function is to determine the new
Cartesian coordinates after modifying a specific torsion angle.
Parameters
----------
bat_frame : numpy.ndarray
an array with dimensions (3N,) with external then internal
degrees of freedom based on the root atoms, followed by the bond,
angle, and (proper and improper) torsion coordinates.
Returns
-------
XYZ : numpy.ndarray
an array with dimensions (N,3) with Cartesian coordinates. The first
dimension has the same ordering as the AtomGroup used to initialize
the class. The molecule will be whole opposed to wrapped around a
periodic boundary.
"""
# Split the bat vector into more convenient variables
origin = bat_frame[:3]
(phi, theta, omega) = bat_frame[3:6]
(r01, r12, a012) = bat_frame[6:9]
n_torsions = (self._ag.n_atoms - 3)
bonds = bat_frame[9:n_torsions + 9]
angles = bat_frame[n_torsions + 9:2 * n_torsions + 9]
torsions = copy.deepcopy(bat_frame[2 * n_torsions + 9:])
# When appropriate, convert improper to proper torsions
shift = torsions[self._primary_torsion_indices]
shift[self._unique_primary_torsion_indices] = 0.
torsions += shift
# Wrap torsions to between -np.pi and np.pi
torsions = ((torsions + np.pi) % (2 * np.pi)) - np.pi
# Set initial root atom positions based on internal coordinates
p0 = np.array([0., 0., 0.])
p1 = np.array([0., 0., r01])
p2 = np.array([r12 * np.sin(a012), 0., r01 - r12 * np.cos(a012)])
# Rotate the third atom by the appropriate value
co = np.cos(omega)
so = np.sin(omega)
# $R_Z(\omega)$
Romega = np.array([[co, -so, 0], [so, co, 0], [0, 0, 1]])
p2 = Romega.dot(p2)
# Rotate the second two atoms to point in the right direction
cp = np.cos(phi)
sp = np.sin(phi)
ct = np.cos(theta)
st = np.sin(theta)
# $R_Z(\phi) R_Y(\theta)$
Re = np.array([[cp * ct, -sp, cp * st], [ct * sp, cp, sp * st],
[-st, 0, ct]])
p1 = Re.dot(p1)
p2 = Re.dot(p2)
# Translate the first three atoms by the origin
p0 += origin
p1 += origin
p2 += origin
XYZ = np.zeros((self._ag.n_atoms, 3))
XYZ[self._root_XYZ_inds[0]] = p0
XYZ[self._root_XYZ_inds[1]] = p1
XYZ[self._root_XYZ_inds[2]] = p2
# Place the remaining atoms
for ((a0,a1,a2,a3), r01, angle, torsion) \
in zip(self._torsion_XYZ_inds, bonds, angles, torsions):
p1 = XYZ[a1]
p3 = XYZ[a3]
p2 = XYZ[a2]
sn_ang = np.sin(angle)
cs_ang = np.cos(angle)
sn_tor = np.sin(torsion)
cs_tor = np.cos(torsion)
v21 = p1 - p2
v21 /= np.sqrt(np.sum(v21 * v21))
v32 = p2 - p3
v32 /= np.sqrt(np.sum(v32 * v32))
vp = np.cross(v32, v21)
cs = np.sum(v21 * v32)
sn = max(np.sqrt(1.0 - cs * cs), 0.0000000001)
vp = vp / sn
vu = np.cross(vp, v21)
XYZ[a0] = p1 + \
r01*(vu*sn_ang*cs_tor + vp*sn_ang*sn_tor - v21*cs_ang)
return XYZ
@property
def atoms(self):
"""The atomgroup for which BAT are computed (read-only property)"""
return self._ag