4.2.1. Coordinate fitting and alignment — `MDAnalysis.analysis.align`

Author: Oliver Beckstein, Joshua Adelman
Year: 2010–2013
Copyright: GNU Public License v3

The module contains functions to fit a target structure to a reference structure. They use the fast QCP algorithm to calculate the root mean square distance (RMSD) between two coordinate sets [Theobald2005] and the rotation matrix R that minimizes the RMSD [Liu2010]. (Please cite these references when using this module.).

Typically, one selects a group of atoms (such as the C-alphas), calculates the RMSD and transformation matrix, and applys the transformation to the current frame of a trajectory to obtain the rotated structure. The alignto() and AlignTraj functions can be used to do this for individual frames and trajectories respectively.

The RMS-fitting tutorial shows how to do the individual steps manually and explains the intermediate steps.

See also

MDAnalysis.analysis.rms: contains functions to compute RMSD (when structural alignment is not required)
MDAnalysis.lib.qcprot: implements the fast RMSD algorithm.

4.2.1.1. RMS-fitting tutorial

The example uses files provided as part of the MDAnalysis test suite (in the variables PSF, DCD, and PDB_small). For all further examples execute first

>>> import MDAnalysis as mda
>>> from MDAnalysis.analysis import align
>>> from MDAnalysis.analysis.rms import rmsd
>>> from MDAnalysis.tests.datafiles import PSF, DCD, PDB_small

In the simplest case, we can simply calculate the C-alpha RMSD between two structures, using rmsd():

>>> ref = mda.Universe(PDB_small)
>>> mobile = mda.Universe(PSF,DCD)
>>> rmsd(mobile.select_atoms('name CA').positions, ref.select_atoms('name CA').positions)
28.20178579474479

Note that in this example translations have not been removed. In order to look at the pure rotation one needs to superimpose the centres of mass (or geometry) first:

>>> rmsd(mobile.select_atoms('name CA').positions, ref.select_atoms('name CA').positions, center=True)
21.892591663632704

This has only done a translational superposition. If you want to also do a rotational superposition use the superposition keyword. This will calculate a minimized RMSD between the reference and mobile structure.

>>> rmsd(mobile.select_atoms('name CA').positions, ref.select_atoms('name CA').positions,
>>>      superposition=True)
6.809396586471815

The rotation matrix that superimposes mobile on ref while minimizing the CA-RMSD is obtained with the rotation_matrix() function

>>> mobile0 = mobile.select_atoms('name CA').positions - mobile.select_atoms('name CA').center_of_mass()
>>> ref0 = ref.select_atoms('name CA').positions - ref.select_atoms('name CA').center_of_mass()
>>> R, rmsd = align.rotation_matrix(mobile0, ref0)
>>> print rmsd
6.809396586471805
>>> print R
[[ 0.14514539 -0.27259113  0.95111876]
 [ 0.88652593  0.46267112 -0.00268642]
 [-0.43932289  0.84358136  0.30881368]]

Putting all this together one can superimpose all of mobile onto ref:

>>> mobile.atoms.translate(-mobile.select_atoms('name CA').center_of_mass())
>>> mobile.atoms.rotate(R)
>>> mobile.atoms.translate(ref.select_atoms('name CA').center_of_mass())
>>> mobile.atoms.write("mobile_on_ref.pdb")

4.2.1.2. Common usage

To fit a single structure with alignto():

>>> ref = mda.Universe(PSF, PDB_small)
>>> mobile = mda.Universe(PSF, DCD)     # we use the first frame
>>> align.alignto(mobile, ref, select="protein and name CA", weights="mass")

This will change all coordinates in mobile so that the protein C-alpha atoms are optimally superimposed (translation and rotation).

To fit a whole trajectory to a reference structure with the AlignTraj class:

>>> ref = mda.Universe(PSF, PDB_small)   # reference structure 1AKE
>>> trj = mda.Universe(PSF, DCD)         # trajectory of change 1AKE->4AKE
>>> alignment = align.AlignTraj(trj, ref, filename='rmsfit.dcd')
>>> alignment.run()

It is also possible to align two arbitrary structures by providing a mapping between atoms based on a sequence alignment. This allows fitting of structural homologs or wild type and mutant.

If a alignment was provided as “sequences.aln” one would first produce the appropriate MDAnalysis selections with the fasta2select() function and then feed the resulting dictionary to AlignTraj:

>>> seldict = align.fasta2select('sequences.aln')
>>> alignment = align.AlignTraj(trj, ref, filename='rmsfit.dcd', select=seldict)
>>> alignment.run()

(See the documentation of the functions for this advanced usage.)

4.2.1.3. Functions and Classes

Changed in version 0.10.0: Function rmsd() was removed from this module and is now exclusively accessible as rmsd().

Changed in version 0.16.0: Function rms_fit_trj() deprecated in favor of AlignTraj class.

Changed in version 0.17.0: removed deprecated rms_fit_trj()

MDAnalysis.analysis.align.alignto(mobile, reference, select=None, weights=None, subselection=None, tol_mass=0.1, strict=False, match_atoms=True)[source]

Perform a spatial superposition by minimizing the RMSD.

Spatially align the group of atoms mobile to reference by doing a RMSD fit on select atoms.

The superposition is done in the following way:

A rotation matrix is computed that minimizes the RMSD between the coordinates of mobile.select_atoms(sel1) and reference.select_atoms(sel2); before the rotation, mobile is translated so that its center of geometry (or center of mass) coincides with the one of reference. (See below for explanation of how sel1 and sel2 are derived from select.)
All atoms in Universe that contain mobile are shifted and rotated. (See below for how to change this behavior through the subselection keyword.)

The mobile and reference atom groups can be constructed so that they already match atom by atom. In this case, select should be set to “all” (or None) so that no further selections are applied to mobile and reference, therefore preserving the exact atom ordering (see Ordered selections).

Warning

The atom order for mobile and reference is only preserved when select is either “all” or None. In any other case, a new selection will be made that will sort the resulting AtomGroup by index and therefore destroy the correspondence between the two groups. It is safest not to mix ordered AtomGroups with selection strings.

Parameters

mobile (Universe or AtomGroup) – structure to be aligned, a AtomGroup or a whole Universe
reference (Universe or AtomGroup) – reference structure, a AtomGroup or a whole Universe
select (str or dict or tuple (optional)) –
The selection to operate on; can be one of:
1. any valid selection string for select_atoms() that produces identical selections in mobile and reference; or
2. a dictionary {'mobile': sel1, 'reference': sel2} where sel1 and sel2 are valid selection strings that are applied to mobile and reference respectively (the MDAnalysis.analysis.align.fasta2select() function returns such a dictionary based on a ClustalW or STAMP sequence alignment); or
3. a tuple (sel1, sel2)
When using 2. or 3. with sel1 and sel2 then these selection strings are applied to atomgroup and reference respectively and should generate groups of equivalent atoms. sel1 and sel2 can each also be a list of selection strings to generate a AtomGroup with defined atom order as described under Ordered selections).
match_atoms (bool (optional)) – Whether to match the mobile and reference atom-by-atom. Default True.
weights ({“mass”, None} or array_like (optional)) – choose weights. With "mass" uses masses as weights; with None weigh each atom equally. If a float array of the same length as mobile is provided, use each element of the array_like as a weight for the corresponding atom in mobile.
tol_mass (float (optional)) – Reject match if the atomic masses for matched atoms differ by more than tol_mass, default [0.1]
strict (bool (optional)) –

True
Will raise SelectionError if a single atom does not match between the two selections.

False [default]
Will try to prepare a matching selection by dropping residues with non-matching atoms. See get_matching_atoms() for details.
subselection (str or AtomGroup or None (optional)) –
Apply the transformation only to this selection.

None [default]
Apply to mobile.universe.atoms (i.e., all atoms in the context of the selection from mobile such as the rest of a protein, ligands and the surrounding water)

selection-string
Apply to mobile.select_atoms(selection-string)

AtomGroup
Apply to the arbitrary group of atoms

Returns

old_rmsd (float) – RMSD before spatial alignment
new_rmsd (float) – RMSD after spatial alignment

See also

AlignTraj: More efficient method for RMSD-fitting trajectories.

Changed in version 1.0.0: Added match_atoms keyword to toggle atom matching.

Changed in version 0.8: Added check that the two groups describe the same atoms including the new tol_mass keyword.

Changed in version 0.10.0: Uses get_matching_atoms() to work with incomplete selections and new strict keyword. The new default is to be lenient whereas the old behavior was the equivalent of strict = True.

Changed in version 0.16.0: new general ‘weights’ kwarg replace mass_weighted, deprecated mass_weighted

Deprecated since version 0.16.0: Instead of mass_weighted=True use new weights='mass'

Changed in version 0.17.0: Deprecated keyword mass_weighted was removed.

class MDAnalysis.analysis.align.AlignTraj(mobile, reference, select='all', filename=None, prefix='rmsfit_', weights=None, tol_mass=0.1, match_atoms=True, strict=False, force=True, in_memory=False, **kwargs)[source]

RMS-align trajectory to a reference structure using a selection.

Both the reference reference and the trajectory mobile must be MDAnalysis.Universe instances. If they contain a trajectory then it is used. The output file format is determined by the file extension of filename. One can also use the same universe if one wants to fit to the current frame.

Changed in version 1.0.0: save() has now been removed, as an alternative use np.savetxt() on results.rmsd.

Parameters

mobile (Universe) – Universe containing trajectory to be fitted to reference
reference (Universe) – Universe containing trajectory frame to be used as reference
select (str (optional)) – Set as default to all, is used for Universe.select_atoms to choose subdomain to be fitted against
filename (str (optional)) – Provide a filename for results to be written to
prefix (str (optional)) – Provide a string to prepend to filename for results to be written to
weights ({“mass”, None} or array_like (optional)) – choose weights. With "mass" uses masses of reference as weights; with None weigh each atom equally. If a float array of the same length as the selection is provided, use each element of the array_like as a weight for the corresponding atom in the selection.
tol_mass (float (optional)) – Tolerance given to get_matching_atoms to find appropriate atoms
match_atoms (bool (optional)) – Whether to match the mobile and reference atom-by-atom. Default True.
strict (bool (optional)) – Force get_matching_atoms to fail if atoms can’t be found using exact methods
force (bool (optional)) – Force overwrite of filename for rmsd-fitting
in_memory (bool (optional)) – Permanently switch mobile to an in-memory trajectory so that alignment can be done in-place, which can improve performance substantially in some cases. In this case, no file is written out (filename and prefix are ignored) and only the coordinates of mobile are changed in memory.
verbose (bool (optional)) – Set logger to show more information and show detailed progress of the calculation if set to True; the default is False.

reference_atoms

Atoms of the reference structure to be aligned against

Type: AtomGroup

mobile_atoms

Atoms inside each trajectory frame to be rmsd_aligned

Type: AtomGroup

results.rmsd

Array of the rmsd values of the least rmsd between the mobile_atoms and reference_atoms after superposition and minimimization of rmsd

New in version 2.0.0.

Type: numpy.ndarray

rmsd

Alias to the results.rmsd attribute.

Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use results.rmsd instead.

Type: numpy.ndarray

filename

String reflecting the filename of the file where the aligned positions will be written to upon running RMSD alignment

Type: str

Notes

If set to verbose=False, it is recommended to wrap the statement in a try ... finally to guarantee restoring of the log level in the case of an exception.
The in_memory option changes the mobile universe to an in-memory representation (see MDAnalysis.coordinates.memory) for the remainder of the Python session. If mobile.trajectory is already a MemoryReader then it is always treated as if in_memory had been set to True.

Changed in version 1.0.0: Default filename has now been changed to the current directory.

Deprecated since version 0.19.1: Default filename directory will change in 1.0 to the current directory.

Changed in version 0.16.0: new general weights kwarg replace mass_weights

Deprecated since version 0.16.0: Instead of mass_weighted=True use new weights='mass'

Changed in version 0.17.0: removed deprecated mass_weighted keyword

Changed in version 1.0.0: Support for the start, stop, and step keywords has been removed. These should instead be passed to AlignTraj.run().

Changed in version 2.0.0: rmsd results are now stored in a MDAnalysis.analysis.base.Results instance.

class MDAnalysis.analysis.align.AverageStructure(mobile, reference=None, select='all', filename=None, weights=None, tol_mass=0.1, match_atoms=True, strict=False, force=True, in_memory=False, ref_frame=0, **kwargs)[source]

RMS-align trajectory to a reference structure using a selection, and calculate the average coordinates of the trajectory.

Both the reference reference and the trajectory mobile must be MDAnalysis.Universe instances. If they contain a trajectory, then it is used. You can also use the same universe if you want to fit to the current frame.

The output file format is determined by the file extension of filename.

Example

import MDAnalysis as mda
from MDAnalysis.tests.datafiles import PSF, DCD
from MDAnalysis.analysis import align

u = mda.Universe(PSF, DCD)

# align to the third frame and average structure
av = align.AverageStructure(u, ref_frame=3).run()
averaged_universe = av.results.universe

Parameters

mobile (Universe) – Universe containing trajectory to be fitted to reference
reference (Universe (optional)) – Universe containing trajectory frame to be used as reference
select (str (optional)) – Set as default to all, is used for Universe.select_atoms to choose subdomain to be fitted against
filename (str (optional)) – Provide a filename for results to be written to
weights ({“mass”, None} or array_like (optional)) – choose weights. With "mass" uses masses of reference as weights; with None weigh each atom equally. If a float array of the same length as the selection is provided, use each element of the array_like as a weight for the corresponding atom in the selection.
tol_mass (float (optional)) – Tolerance given to get_matching_atoms to find appropriate atoms
match_atoms (bool (optional)) – Whether to match the mobile and reference atom-by-atom. Default True.
strict (bool (optional)) – Force get_matching_atoms to fail if atoms can’t be found using exact methods
force (bool (optional)) – Force overwrite of filename for rmsd-fitting
in_memory (bool (optional)) – Permanently switch mobile to an in-memory trajectory so that alignment can be done in-place, which can improve performance substantially in some cases. In this case, no file is written out (filename and prefix are ignored) and only the coordinates of mobile are changed in memory.
ref_frame (int (optional)) – frame index to select frame from reference
verbose (bool (optional)) – Set logger to show more information and show detailed progress of the calculation if set to True; the default is False.

reference_atoms

Atoms of the reference structure to be aligned against

Type: AtomGroup

mobile_atoms

Atoms inside each trajectory frame to be rmsd_aligned

Type: AtomGroup

results.universe

New Universe with average positions

New in version 2.0.0.

Type: MDAnalysis.Universe

universe

Alias to the results.universe attribute.

Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use results.universe instead.

Type: MDAnalysis.Universe

results.positions

Average positions

New in version 2.0.0.

Type: np.ndarray(dtype=float)

positions

Alias to the results.positions attribute.

Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use results.positions instead.

Type: np.ndarray(dtype=float)

results.rmsd

Average RMSD per frame

New in version 2.0.0.

Type: float

rmsd

Alias to the results.rmsd attribute.

Deprecated since version 2.0.0: Will be removed in MDAnalysis 3.0.0. Please use results.rmsd instead.

Type: float

filename

String reflecting the filename of the file where the average structure is written

Type: str

Notes

If set to verbose=False, it is recommended to wrap the statement in a try ... finally to guarantee restoring of the log level in the case of an exception.
The in_memory option changes the mobile universe to an in-memory representation (see MDAnalysis.coordinates.memory) for the remainder of the Python session. If mobile.trajectory is already a MemoryReader then it is always treated as if in_memory had been set to True.

New in version 1.0.0.

Changed in version 2.0.0: universe, positions, and rmsd are now stored in a MDAnalysis.analysis.base.Results instance.

MDAnalysis.analysis.align.rotation_matrix(a, b, weights=None)[source]

Returns the 3x3 rotation matrix R for RMSD fitting coordinate sets a and b.

The rotation matrix R transforms vector a to overlap with vector b (i.e., b is the reference structure):

\[\mathbf{b} = \mathsf{R} \cdot \mathbf{a}\]

Parameters

a (array_like) – coordinates that are to be rotated (“mobile set”); array of N atoms of shape N*3 as generated by, e.g., MDAnalysis.core.groups.AtomGroup.positions.
b (array_like) – reference coordinates; array of N atoms of shape N*3 as generated by, e.g., MDAnalysis.core.groups.AtomGroup.positions.
weights (array_like (optional)) – array of floats of size N for doing weighted RMSD fitting (e.g. the masses of the atoms)

Returns

R (ndarray) – rotation matrix
rmsd (float) – RMSD between a and b before rotation
(R, rmsd) rmsd and rotation matrix R

Example

R can be used as an argument for MDAnalysis.core.groups.AtomGroup.rotate() to generate a rotated selection, e.g.

>>> R = rotation_matrix(A.select_atoms('backbone').positions,
>>>                     B.select_atoms('backbone').positions)[0]
>>> A.atoms.rotate(R)
>>> A.atoms.write("rotated.pdb")

Notes

The function does not shift the centers of mass or geometry; this needs to be done by the user.

See also

MDAnalysis.analysis.rms.rmsd: Calculates the RMSD between a and b.
alignto: A complete fit of two structures.
AlignTraj: Fit a whole trajectory.

4.2.1.4. Helper functions

The following functions are used by the other functions in this module. They are probably of more interest to developers than to normal users.

MDAnalysis.analysis.align._fit_to(mobile_coordinates, ref_coordinates, mobile_atoms, mobile_com, ref_com, weights=None)[source]

Perform an rmsd-fitting to determine rotation matrix and align atoms

Parameters

mobile_coordinates (ndarray) – Coordinates of atoms to be aligned
ref_coordinates (ndarray) – Coordinates of atoms to be fit against
mobile_atoms (AtomGroup) – Atoms to be translated
mobile_com (ndarray) – array of xyz coordinate of mobile center of mass
ref_com (ndarray) – array of xyz coordinate of reference center of mass
weights (array_like (optional)) – choose weights. With None weigh each atom equally. If a float array of the same length as mobile_coordinates is provided, use each element of the array_like as a weight for the corresponding atom in mobile_coordinates.

Returns

mobile_atoms (AtomGroup) – AtomGroup of translated and rotated atoms
min_rmsd (float) – Minimum rmsd of coordinates

Notes

This function assumes that mobile_coordinates and ref_coordinates have already been shifted so that their centers of geometry (or centers of mass, depending on weights) coincide at the origin. mobile_com and ref_com are the centers before this shift.

The rotation matrix \(\mathsf{R}\) is determined with rotation_matrix() directly from mobile_coordinates and ref_coordinates.
mobile_atoms \(X\) is rotated according to the rotation matrix and the centers according to

\[X' = \mathsf{R}(X - \bar{X}) + \bar{X}_{\text{ref}}\]

where \(\bar{X}\) is the center.

MDAnalysis.analysis.align.fasta2select(fastafilename, is_aligned=False, ref_resids=None, target_resids=None, ref_offset=0, target_offset=0, verbosity=3, alnfilename=None, treefilename=None, clustalw='clustalw2')[source]

Return selection strings that will select equivalent residues.

The function aligns two sequences provided in a FASTA file and constructs MDAnalysis selection strings of the common atoms. When these two strings are applied to the two different proteins they will generate AtomGroups of the aligned residues.

fastafilename contains the two un-aligned sequences in FASTA format. The reference is assumed to be the first sequence, the target the second. ClustalW produces a pairwise alignment (which is written to a file with suffix .aln). The output contains atom selection strings that select the same atoms in the two structures.

Unless ref_offset and/or target_offset are specified, the resids in the structure are assumed to correspond to the positions in the un-aligned sequence, namely the first residue has resid == 1.

In more complicated cases (e.g., when the resid numbering in the input structure has gaps due to missing parts), simply provide the sequence of resids as they appear in the topology in ref_resids or target_resids, e.g.

target_resids = [a.resid for a in trj.select_atoms('name CA')]

(This translation table is combined with any value for ref_offset or target_offset!)

Parameters

fastafilename (str, path to filename) – FASTA file with first sequence as reference and second the one to be aligned (ORDER IS IMPORTANT!)
is_aligned (bool (optional)) –

False (default)
run clustalw for sequence alignment;

True
use the alignment in the file (e.g. from STAMP) [False]
ref_offset (int (optional)) – add this number to the column number in the FASTA file to get the original residue number, default: 0
target_offset (int (optional)) – add this number to the column number in the FASTA file to get the original residue number, default: 0
ref_resids (str (optional)) – sequence of resids as they appear in the reference structure
target_resids (str (optional)) – sequence of resids as they appear in the target
alnfilename (str (optional)) – filename of ClustalW alignment (clustal format) that is produced by clustalw when is_aligned = False. default None uses the name and path of fastafilename and substitutes the suffix with ‘.aln’.
treefilename (str (optional)) – filename of ClustalW guide tree (Newick format); if default None the the filename is generated from alnfilename with the suffix ‘.dnd’ instead of ‘.aln’
clustalw (str (optional)) – path to the ClustalW (or ClustalW2) binary; only needed for is_aligned = False, default: “ClustalW2”

Returns

select_dict – dictionary with ‘reference’ and ‘mobile’ selection string that can be used immediately in AlignTraj as select=select_dict.

Return type

dict

See also

sequence_alignment(), which, programs.

Changed in version 1.0.0: Passing alnfilename or treefilename as None will create a file in the current working directory.

MDAnalysis.analysis.align.sequence_alignment(mobile, reference, match_score=2, mismatch_penalty=-1, gap_penalty=-2, gapextension_penalty=-0.1)[source]

Generate a global sequence alignment between two residue groups.

The residues in reference and mobile will be globally aligned. The global alignment uses the Needleman-Wunsch algorithm as implemented in Bio.pairwise2. The parameters of the dynamic programming algorithm can be tuned with the keywords. The defaults should be suitable for two similar sequences. For sequences with low sequence identity, more specialized tools such as clustalw, muscle, tcoffee, or similar should be used.

Parameters

mobile (AtomGroup) – Atom group to be aligned
reference (AtomGroup) – Atom group to be aligned against
match_score (float (optional), default 2) – score for matching residues, default 2
mismatch_penalty (float (optional), default -1) – penalty for residues that do not match , default : -1
gap_penalty (float (optional), default -2) – penalty for opening a gap; the high default value creates compact alignments for highly identical sequences but might not be suitable for sequences with low identity, default : -2
gapextension_penalty (float (optional), default -0.1) – penalty for extending a gap, default: -0.1

Returns

alignment – Tuple of top sequence matching output (‘Sequence A’, ‘Sequence B’, score, begin, end)

Return type

tuple

See also

BioPython documentation for pairwise2. Alternatively, use fasta2select() with clustalw2 and the option is_aligned=False.

New in version 0.10.0.

MDAnalysis.analysis.align.get_matching_atoms(ag1, ag2, tol_mass=0.1, strict=False, match_atoms=True)[source]

Return two atom groups with one-to-one matched atoms.

The function takes two AtomGroup instances ag1 and ag2 and returns two atom groups g1 and g2 that consist of atoms so that the mass of atom g1[0] is the same as the mass of atom g2[0], g1[1] and g2[1] etc.

The current implementation is very simplistic and works on a per-residue basis:

The two groups must contain the same number of residues.
Any residues in each group that have differing number of atoms are discarded.
The masses of corresponding atoms are compared. and if any masses differ by more than tol_mass the test is considered failed and a SelectionError is raised.

The log file (see MDAnalysis.start_logging()) will contain detailed information about mismatches.

Parameters

ag1 (AtomGroup) – First AtomGroup instance that is compared
ag2 (AtomGroup) – Second AtomGroup instance that is compared
tol_mass (float (optional)) – Reject if the atomic masses for matched atoms differ by more than tol_mass [0.1]
strict (bool (optional)) –

True
Will raise SelectionError if a single atom does not match between the two selections.

False [default]
Will try to prepare a matching selection by dropping residues with non-matching atoms. See get_matching_atoms() for details.
match_atoms (bool (optional)) –

True
Will attempt to match atoms based on mass

False
Will not attempt to match atoms based on mass

Returns

(g1, g2) – Tuple with AtomGroup instances that match, atom by atom. The groups are either the original groups if all matched or slices of the original groups.

Return type

tuple

Raises

SelectionError – Error raised if the number of residues does not match or if in the final matching masses differ by more than tol.

Notes

The algorithm could be improved by using e.g. the Needleman-Wunsch algorithm in Bio.profile2 to align atoms in each residue (doing a global alignment is too expensive).

New in version 0.8.

Changed in version 0.10.0: Renamed from check_same_atoms() to get_matching_atoms() and now returns matching atomgroups (possibly with residues removed)

4.2.1. Coordinate fitting and alignment — MDAnalysis.analysis.align

4.2.1.1. RMS-fitting tutorial

4.2.1.2. Common usage

4.2.1.3. Functions and Classes

4.2.1.4. Helper functions

4.2.1. Coordinate fitting and alignment — `MDAnalysis.analysis.align`