Source code for MDAnalysis.analysis.helix_analysis

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HELANAL --- analysis of protein helices

:Author: Lily Wang
:Year: 2020
:Copyright: GNU Public License v3

.. 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.


.. [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')

All computed properties are available in ``.results``::


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

    hel_xyz = hel.helix_analysis(u.atoms.positions, ref_axis=[0, 0, 1])


.. autoclass:: HELANAL


.. 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 =, 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, # (n_res-3, n_res-3) bend_theta, 1) 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