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# MDAnalysis --- https://www.mdanalysis.org
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# Please cite your use of MDAnalysis in published work:
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# R. J. Gowers, M. Linke, J. Barnoud, T. J. E. Reddy, M. N. Melo, S. L. Seyler,
# D. L. Dotson, J. Domanski, S. Buchoux, I. M. Kenney, and O. Beckstein.
# MDAnalysis: A Python package for the rapid analysis of molecular dynamics
# simulations. In S. Benthall and S. Rostrup editors, Proceedings of the 15th
# Python in Science Conference, pages 102-109, Austin, TX, 2016. SciPy.
# doi: 10.25080/majora-629e541a-00e
#
# N. Michaud-Agrawal, E. J. Denning, T. B. Woolf, and O. Beckstein.
# MDAnalysis: A Toolkit for the Analysis of Molecular Dynamics Simulations.
# J. Comput. Chem. 32 (2011), 2319--2327, doi:10.1002/jcc.21787
#
"""
HELANAL --- analysis of protein helices
=======================================
:Author: Lily Wang
:Year: 2020
:Copyright: Lesser GNU Public License v2.1+
.. versionadded:: 2.0.0
This module contains code to analyse protein helices using the
HELANAL_ algorithm
([Bansal2000]_ , [Sugeta1967]_ ).
HELANAL_ quantifies the geometry of helices in proteins on the basis of their
Cα atoms. It can determine local structural features such as the local
helical twist and rise, virtual torsion angle, local helix origins and
bending angles between successive local helix axes.
.. _HELANAL: https://pubmed.ncbi.nlm.nih.gov/10798526/
.. [Sugeta1967] Sugeta, H. and Miyazawa, T. 1967. General method for
calculating helical parameters of polymer chains from bond lengths, bond
angles and internal rotation angles. *Biopolymers* 5 673 - 679
.. [Bansal2000] Bansal M, Kumar S, Velavan R. 2000.
HELANAL - A program to characterise helix geometry in proteins.
*J Biomol Struct Dyn.* 17(5):811-819.
Example use
-----------
You can pass in a single selection::
import MDAnalysis as mda
from MDAnalysis.tests.datafiles import PSF, DCD
from MDAnalysis.analysis import helix_analysis as hel
u = mda.Universe(PSF, DCD)
helanal = hel.HELANAL(u, select='name CA and resnum 161-187')
helanal.run()
All computed properties are available in ``.results``::
print(helanal.results.summary)
Alternatively, you can analyse several helices at once by passing
in multiple selection strings::
helanal2 = hel.HELANAL(u, select=('name CA and resnum 100-160',
'name CA and resnum 200-230'))
The :func:`helix_analysis` function will carry out helix analysis on
atom positions, treating each row of coordinates as an alpha-carbon
equivalent::
hel_xyz = hel.helix_analysis(u.atoms.positions, ref_axis=[0, 0, 1])
Classes
-------
.. autoclass:: HELANAL
Functions
---------
.. autofunction:: helix_analysis
.. autofunction:: vector_of_best_fit
.. autofunction:: local_screw_angles
"""
import warnings
import numpy as np
import MDAnalysis as mda
from ..lib import util, mdamath
from .base import AnalysisBase
[docs]
def vector_of_best_fit(coordinates):
"""Fit vector through the centered coordinates,
pointing to the first coordinate (i.e. upside-down).
Parameters
----------
coordinates : :class:`numpy.ndarray` of shape (N, 3)
Returns
-------
:class:`numpy.ndarray` of shape (3,)
Vector of best fit.
"""
centered = coordinates - coordinates.mean(axis=0)
Mt_M = np.matmul(centered.T, centered)
u, s, vh = np.linalg.svd(Mt_M)
vector = vh[0]
# does vector face first local helix origin?
angle = mdamath.angle(centered[0], vector)
if angle > np.pi / 2:
vector *= -1
return vector
[docs]
def local_screw_angles(global_axis, ref_axis, helix_directions):
"""
Cylindrical azimuth angles between the local direction vectors,
as projected onto the cross-section of the helix, from (-pi, pi].
The origin (angle=0) is set to the plane of global_axis and ref_axis.
Parameters
----------
global_axis : :class:`numpy.ndarray` of shape (3,)
Vector of best fit. Screw angles are calculated perpendicular to
this axis.
ref_axis : :class:`numpy.ndarray` of shape (3,)
Reference length-wise axis. One of the reference vectors is
orthogonal to this axis.
helix_directions : :class:`numpy.ndarray` of shape (N, 3)
array of vectors representing the local direction of each
helix window.
Returns
-------
:class:`numpy.ndarray` of shape (N,)
Array of screw angles.
"""
global_axis = np.asarray(global_axis)
# normal to the plane of `ref_axis` & `global_axis`
perp = np.cross(ref_axis, global_axis)
if not np.any(perp): # zero when ref_axis, global_axis parallel
# use random orthogonal vector
new_ref = [[1, 0, 0], [0, 0, 1]]
while not np.any(perp) and new_ref:
perp = np.cross(new_ref.pop(), global_axis)
# normal for angle to plane of perp and global_axis
ortho = np.cross(-perp, global_axis)
# project helix_directions onto global to remove contribution
norm_global_sq = np.dot(global_axis, global_axis)
mag_g = np.matmul(global_axis, helix_directions.T) / norm_global_sq
# projection onto global_axis
proj_g = mag_g.reshape(-1, 1) @ global_axis.reshape(1, -1)
# projection onto plane w/o global_axis contribution
proj_plane = helix_directions - proj_g
# angles from projection to perp
refs = np.array([perp, ortho]) # (2, 3)
norms = _, ortho_norm = np.outer(
mdamath.pnorm(refs), mdamath.pnorm(proj_plane)
)
cos = cos_perp, cos_ortho = np.matmul(refs, proj_plane.T) / norms
to_perp, to_ortho = np.arccos(np.clip(cos, -1, 1)) # (2, n_vec)
to_ortho[ortho_norm == 0] = 0 # ?
to_ortho[cos_perp < 0] *= -1
to_ortho[to_ortho == -np.pi] = np.pi # leave 180 alone
return np.rad2deg(to_ortho)
[docs]
def helix_analysis(positions, ref_axis=(0, 0, 1)):
r"""
Calculate helix properties from atomic coordinates.
Each property is calculated from a sliding window of 4 atoms,
from i to i+3. Any property whose name begins with 'local' is a
property of a sliding window.
Parameters
----------
positions : :class:`numpy.ndarray` of shape (N, 3)
Atomic coordinates.
ref_axis : array-like of length 3, optional
The reference axis used to calculate the tilt of the vector
of best fit, and the local screw angles.
Returns
-------
dict with the following keys:
local_twists : array, shape (N-3,)
local twist angle from atom i+1 to i+2
local_nres_per_turn : array, shape (N-3,)
number of residues per turn, based on local_twist
local_axes : array, shape (N-3, 3)
the length-wise helix axis of the local window
local_bends : array, shape (N-6,)
the angles between local helix angles, 3 windows apart
local_heights : array, shape (N-3,)
the rise of each local helix
local_helix_directions : array, shape (N-2, 3)
the unit vector from each local origin to atom i+1
local_origins : array, shape (N-2, 3)
the projected origin for each helix
all_bends : array, shape (N-3, N-3)
angles between each local axis
global_axis : array, shape (3,)
vector of best fit through origins, pointing at the first origin.
local_screw_angles : array, shape (N-2,)
cylindrical azimuth angle to plane of global_axis and ref_axis
"""
# ^ ^
# \ / bi
# \ /
# CA_i+2 <----- CA_i+1
# / \ / ^
# / r \ / \
# V / \ θ / \
# / \ / CA_i
# v origin
# CA_i+3
#
# V: vectors
# bi: approximate "bisectors" in plane of screen
# Note: not real bisectors, as the vectors aren't normalised
# θ: local_twists
# origin: origins
# local_axes: perpendicular to plane of screen. Orthogonal to "bisectors"
vectors = positions[1:] - positions[:-1] # (n_res-1, 3)
bisectors = vectors[:-1] - vectors[1:] # (n_res-2, 3)
bimags = mdamath.pnorm(bisectors) # (n_res-2,)
adjacent_mag = bimags[:-1] * bimags[1:] # (n_res-3,)
# find angle between bisectors for twist and n_residue/turn
cos_theta = mdamath.pdot(bisectors[:-1], bisectors[1:]) / adjacent_mag
cos_theta = np.clip(cos_theta, -1, 1)
twists = np.arccos(cos_theta) # (n_res-3,)
local_twists = np.rad2deg(twists)
local_nres_per_turn = 2 * np.pi / twists
# find normal to bisectors for local axes
cross_bi = np.cross(bisectors[:-1], bisectors[1:]) # (n_res-3, 3)
local_axes = (cross_bi.T / mdamath.pnorm(cross_bi)).T # (n_res-3, 3)
local_axes = np.nan_to_num(local_axes)
zero_vectors = np.tile(np.any(local_axes, axis=1), (len(local_axes), 1)).T
# find angles between axes for bends
bend_theta = np.matmul(local_axes, local_axes.T) # (n_res-3, n_res-3)
# set angles to 0 between zero-vectors
bend_theta = np.where(
zero_vectors + zero_vectors.T, bend_theta, 1 # (n_res-3, n_res-3)
)
bend_matrix = np.rad2deg(np.arccos(np.clip(bend_theta, -1, 1)))
# local bends are between axes 3 windows apart
local_bends = np.diagonal(bend_matrix, offset=3) # (n_res-6,)
# radius of local cylinder
radii = (adjacent_mag**0.5) / (2 * (1.0 - cos_theta)) # (n_res-3,)
# special case: angle b/w bisectors is 0 (should virtually never happen)
# guesstimate radius = half bisector magnitude
radii = np.where(cos_theta != 1, radii, (adjacent_mag**0.5) / 2)
# height of local cylinder
heights = np.abs(mdamath.pdot(vectors[1:-1], local_axes)) # (n_res-3,)
local_helix_directions = (bisectors.T / bimags).T # (n_res-2, 3)
# get origins by subtracting radius from atom i+1
origins = positions[1:-1].copy() # (n_res-2, 3)
origins[:-1] -= (radii * local_helix_directions[:-1].T).T
# subtract radius from atom i+2 in last one
origins[-1] -= radii[-1] * local_helix_directions[-1]
helix_axes = vector_of_best_fit(origins)
screw = local_screw_angles(
helix_axes, np.asarray(ref_axis), local_helix_directions
)
results = {
"local_twists": local_twists,
"local_nres_per_turn": local_nres_per_turn,
"local_axes": local_axes,
"local_bends": local_bends,
"local_heights": heights,
"local_helix_directions": local_helix_directions,
"local_origins": origins,
"all_bends": bend_matrix,
"global_axis": helix_axes,
"local_screw_angles": screw,
}
return results
[docs]
class HELANAL(AnalysisBase):
r"""
Perform HELANAL helix analysis on your trajectory.
Parameters
----------
universe : Universe or AtomGroup
The Universe or AtomGroup to apply the analysis to.
select : str or iterable of str, optional
The selection string to create an atom selection that the HELANAL
analysis is applied to. Note that HELANAL is designed to work on the
alpha-carbon atoms of protein residues. If you pass in multiple
selections, the selections will be analysed separately.
ref_axis : array-like of length 3, optional
The reference axis used to calculate the tilt of the vector
of best fit, and the local screw angles.
flatten_single_helix : bool, optional
Whether to flatten results if only one selection is passed.
split_residue_sequences : bool, optional
Whether to split the residue sequence into individual helices.
This keyword only applies if a residue gap is present in the
AtomGroup generated by a ``select`` string.
If ``False``, the residues will be analysed as a single helix.
If ``True``, each group of consecutive residues will be treated
as a separate helix.
verbose : bool, optional
Turn on more logging and debugging.
Attributes
----------
results.local_twists : array or list of arrays
The local twist angle from atom i+1 to i+2.
Each array has shape (n_frames, n_residues-3)
results.local_nres_per_turn : array or list of arrays
Number of residues per turn, based on local_twist.
Each array has shape (n_frames, n_residues-3)
results.local_axes : array or list of arrays
The length-wise helix axis of the local window.
Each array has shape (n_frames, n_residues-3, 3)
results.local_heights : array or list of arrays
The rise of each local helix.
Each array has shape (n_frames, n_residues-3)
results.local_helix_directions : array or list of arrays
The unit vector from each local origin to atom i+1.
Each array has shape (n_frames, n_residues-2, 3)
results.local_origins :array or list of arrays
The projected origin for each helix.
Each array has shape (n_frames, n_residues-2, 3)
results.local_screw_angles : array or list of arrays
The local screw angle for each helix.
Each array has shape (n_frames, n_residues-2)
results.local_bends : array or list of arrays
The angles between local helix axes, 3 windows apart.
Each array has shape (n_frames, n_residues-6)
results.all_bends : array or list of arrays
The angles between local helix axes.
Each array has shape (n_frames, n_residues-3, n_residues-3)
results.global_axis : array or list of arrays
The length-wise axis for the overall helix. This points at
the first helix window in the helix, so it runs opposite to
the direction of the residue numbers.
Each array has shape (n_frames, 3)
results.global_tilts : array or list of arrays
The angle between the global axis and the reference axis.
Each array has shape (n_frames,)
results.summary : dict or list of dicts
Summary of stats for each property: the mean, the sample
standard deviation, and the mean absolute deviation.
"""
# shapes of properties from each frame, relative to n_residues
attr_shapes = {
"local_twists": (-3,),
"local_bends": (-6,),
"local_heights": (-3,),
"local_nres_per_turn": (-3,),
"local_origins": (-2, 3),
"local_axes": (-3, 3),
"local_helix_directions": (-2, 3),
"local_screw_angles": (-2,),
}
def __init__(
self,
universe,
select="name CA",
ref_axis=(0, 0, 1),
verbose=False,
flatten_single_helix=True,
split_residue_sequences=True,
):
super(HELANAL, self).__init__(
universe.universe.trajectory, verbose=verbose
)
selections = util.asiterable(select)
atomgroups = [universe.select_atoms(s) for s in selections]
consecutive = []
# check that residues are consecutive and long enough sequence
for s, ag in zip(selections, atomgroups):
groups = util.group_same_or_consecutive_integers(ag.resindices)
counter = 0
if len(groups) > 1:
msg = "Your selection {} has gaps in the residues.".format(s)
if split_residue_sequences:
msg += " Splitting into {} helices.".format(len(groups))
else:
groups = [ag.resindices]
warnings.warn(msg)
for g in groups:
ng = len(g)
counter += ng
if ng < 9:
warnings.warn(
"Fewer than 9 atoms found for helix in "
"selection {} with these resindices: {}. "
"This sequence will be skipped. HELANAL "
"is designed to work on at sequences of "
"≥9 residues.".format(s, g)
)
continue
ids, counts = np.unique(g, return_counts=True)
if np.any(counts > 1):
dup = ", ".join(map(str, ids[counts > 1]))
warnings.warn(
"Your selection {} includes multiple atoms "
"for residues with these resindices: {}."
"HELANAL is designed to work on one alpha-"
"carbon per residue.".format(s, dup)
)
consecutive.append(ag[counter - ng : counter])
self.atomgroups = consecutive
self.ref_axis = np.asarray(ref_axis)
self._flatten = flatten_single_helix
def _zeros_per_frame(self, dims, n_positions=0):
"""Create zero arrays where first 2 dims are n_frames, n_values"""
first = dims[0] + n_positions
npdims = (
self.n_frames,
first,
) + dims[
1:
] # py27 workaround
return np.zeros(npdims, dtype=np.float64)
def _prepare(self):
n_res = [len(ag) for ag in self.atomgroups]
for key, dims in self.attr_shapes.items():
empty = [self._zeros_per_frame(dims, n_positions=n) for n in n_res]
self.results[key] = empty
self.results.global_axis = [self._zeros_per_frame((3,)) for n in n_res]
self.results.all_bends = [
self._zeros_per_frame((n - 3, n - 3)) for n in n_res
]
def _single_frame(self):
_f = self._frame_index
for i, ag in enumerate(self.atomgroups):
results = helix_analysis(ag.positions, ref_axis=self.ref_axis)
for key, value in results.items():
attr = self.results[key]
attr[i][_f] = value
def _conclude(self):
# compute tilt of global axes
self.results.global_tilts = tilts = []
norm_ref = (self.ref_axis**2).sum() ** 0.5
for axes in self.results.global_axis:
cos = np.matmul(self.ref_axis, axes.T) / (
mdamath.pnorm(axes) * norm_ref
)
cos = np.clip(cos, -1.0, 1.0)
tilts.append(np.rad2deg(np.arccos(cos)))
global_attrs = ["global_axis", "global_tilts", "all_bends"]
attrnames = list(self.attr_shapes.keys()) + global_attrs
# summarise
self.results.summary = []
for i in range(len(self.atomgroups)):
stats = {}
for name in attrnames:
attr = self.results[name]
mean = attr[i].mean(axis=0)
dev = np.abs(attr[i] - mean)
stats[name] = {
"mean": mean,
"sample_sd": attr[i].std(axis=0, ddof=1),
"abs_dev": dev.mean(axis=0),
}
self.results.summary.append(stats)
# flatten?
if len(self.atomgroups) == 1 and self._flatten:
for name in attrnames + ["summary"]:
attr = self.results[name]
self.results[name] = attr[0]
def universe_from_origins(self):
"""
Create MDAnalysis Universe from the local origins.
Returns
-------
Universe or list of Universes
"""
try:
origins = self.results.local_origins
except AttributeError:
raise ValueError("Call run() before universe_from_origins")
if not isinstance(origins, list):
origins = [origins]
universe = []
for xyz in origins:
n_res = xyz.shape[1]
u = mda.Universe.empty(
n_res,
n_residues=n_res,
atom_resindex=np.arange(n_res),
trajectory=True,
).load_new(xyz)
universe.append(u)
if not isinstance(self.results.local_origins, list):
universe = universe[0]
return universe