# Source code for MDAnalysis.analysis.msd

```
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# J. Comput. Chem. 32 (2011), 2319--2327, doi:10.1002/jcc.21787
#
r"""
Mean Squared Displacement --- :mod:`MDAnalysis.analysis.msd`
==============================================================
:Authors: Hugo MacDermott-Opeskin
:Year: 2020
:Copyright: GNU Public License v2
This module implements the calculation of Mean Squared Displacements (MSDs)
by the Einstein relation. MSDs can be used to characterize the speed at
which particles move and has its roots in the study of Brownian motion.
For a full explanation of the theory behind MSDs and the subsequent calculation
of self-diffusivities the reader is directed to :cite:p:`Maginn2019`.
MSDs can be computed from the following expression, known as the
**Einstein formula**:
.. math::
MSD(r_{d}) = \bigg{\langle} \frac{1}{N} \sum_{i=1}^{N} |r_{d}
- r_{d}(t_0)|^2 \bigg{\rangle}_{t_{0}}
where :math:`N` is the number of equivalent particles the MSD is calculated
over, :math:`r` are their coordinates and :math:`d` the desired dimensionality
of the MSD. Note that while the definition of the MSD is universal, there are
many practical considerations to computing the MSD that vary between
implementations. In this module, we compute a "windowed" MSD, where the MSD
is averaged over all possible lag-times :math:`\tau \le \tau_{max}`,
where :math:`\tau_{max}` is the length of the trajectory, thereby maximizing
the number of samples.
The computation of the MSD in this way can be computationally intensive due to
its :math:`N^2` scaling with respect to :math:`\tau_{max}`. An algorithm to
compute the MSD with :math:`N log(N)` scaling based on a Fast Fourier
Transform is known and can be accessed by setting ``fft=True`` [Calandri2011]_
[Buyl2018]_. The FFT-based approach requires that the
`tidynamics <https://github.com/pdebuyl-lab/tidynamics>`_ package is
installed; otherwise the code will raise an :exc:`ImportError`.
Please cite [Calandri2011]_ [Buyl2018]_ if you use this module in addition to
the normal MDAnalysis citations.
.. warning::
To correctly compute the MSD using this analysis module, you must supply
coordinates in the **unwrapped** convention. That is, when atoms pass
the periodic boundary, they must not be **wrapped** back into the primary
simulation cell. MDAnalysis does not currently offer this functionality in
the ``MDAnalysis.transformations`` API despite having functions with
similar names. We plan to implement the appropriate transformations in the
future. In the meantime, various simulation packages provide utilities to
convert coordinates to the unwrapped convention. In GROMACS for example,
this can be done using ``gmx trjconv`` with the ``-pbc nojump`` flag.
Computing an MSD
----------------
This example computes a 3D MSD for the movement of 100 particles undergoing a
random walk. Files provided as part of the MDAnalysis test suite are used
(in the variables :data:`~MDAnalysis.tests.datafiles.RANDOM_WALK` and
:data:`~MDAnalysis.tests.datafiles.RANDOM_WALK_TOPO`)
First load all modules and test data
.. code-block:: python
import MDAnalysis as mda
import MDAnalysis.analysis.msd as msd
from MDAnalysis.tests.datafiles import RANDOM_WALK_TOPO, RANDOM_WALK
Given a universe containing trajectory data we can extract the MSD
analysis by using the class :class:`EinsteinMSD`
.. code-block:: python
u = mda.Universe(RANDOM_WALK_TOPO, RANDOM_WALK)
MSD = msd.EinsteinMSD(u, select='all', msd_type='xyz', fft=True)
MSD.run()
The MSD can then be accessed as
.. code-block:: python
msd = MSD.results.timeseries
Visual inspection of the MSD is important, so let's take a look at it with a
simple plot.
.. code-block:: python
import matplotlib.pyplot as plt
nframes = MSD.n_frames
timestep = 1 # this needs to be the actual time between frames
lagtimes = np.arange(nframes)*timestep # make the lag-time axis
fig = plt.figure()
ax = plt.axes()
# plot the actual MSD
ax.plot(lagtimes, msd, lc="black", ls="-", label=r'3D random walk')
exact = lagtimes*6
# plot the exact result
ax.plot(lagtimes, exact, lc="black", ls="--", label=r'$y=2 D\tau$')
plt.show()
This gives us the plot of the MSD with respect to lag-time (:math:`\tau`).
We can see that the MSD is approximately linear with respect to :math:`\tau`.
This is a numerical example of a known theoretical result that the MSD of a
random walk is linear with respect to lag-time, with a slope of :math:`2d`.
In this expression :math:`d` is the dimensionality of the MSD. For our 3D MSD,
this is 3. For comparison we have plotted the line :math:`y=6\tau` to which an
ensemble of 3D random walks should converge.
.. _figure-msd:
.. figure:: /images/msd_demo_plot.png
:scale: 100 %
:alt: MSD plot
Note that a segment of the MSD is required to be linear to accurately
determine self-diffusivity. This linear segment represents the so called
"middle" of the MSD plot, where ballistic trajectories at short time-lags are
excluded along with poorly averaged data at long time-lags. We can select the
"middle" of the MSD by indexing the MSD and the time-lags. Appropriately
linear segments of the MSD can be confirmed with a log-log plot as is often
reccomended :cite:p:`Maginn2019` where the "middle" segment can be identified
as having a slope of 1.
.. code-block:: python
plt.loglog(lagtimes, msd)
plt.show()
Now that we have identified what segment of our MSD to analyse, let's compute
a self-diffusivity.
Computing Self-Diffusivity
--------------------------------
Self-diffusivity is closely related to the MSD.
.. math::
D_d = \frac{1}{2d} \lim_{t \to \infty} \frac{d}{dt} MSD(r_{d})
From the MSD, self-diffusivities :math:`D` with the desired dimensionality
:math:`d` can be computed by fitting the MSD with respect to the lag-time to
a linear model. An example of this is shown below, using the MSD computed in
the example above. The segment between :math:`\tau = 20` and :math:`\tau = 60`
is used to demonstrate selection of a MSD segment.
.. code-block:: python
from scipy.stats import linregress
start_time = 20
start_index = int(start_time/timestep)
end_time = 60
linear_model = linregress(lagtimes[start_index:end_index],
msd[start_index:end_index])
slope = linear_model.slope
error = linear_model.stderr
# dim_fac is 3 as we computed a 3D msd with 'xyz'
D = slope * 1/(2*MSD.dim_fac)
We have now computed a self-diffusivity!
Combining Multiple Replicates
--------------------------------
It is common practice to combine replicates when calculating MSDs. An example
of this is shown below using MSD1 and MSD2.
.. code-block:: python
u1 = mda.Universe(RANDOM_WALK_TOPO, RANDOM_WALK)
MSD1 = msd.EinsteinMSD(u1, select='all', msd_type='xyz', fft=True)
MSD1.run()
u2 = mda.Universe(RANDOM_WALK_TOPO, RANDOM_WALK)
MSD2 = msd.EinsteinMSD(u2, select='all', msd_type='xyz', fft=True)
MSD2.run()
combined_msds = np.concatenate((MSD1.results.msds_by_particle,
MSD2.results.msds_by_particle), axis=1)
average_msd = np.mean(combined_msds, axis=1)
The same cannot be achieved by concatenating the replicas in a single run as
the jump between the last frame of the first trajectory and frame 0 of the
next trajectory will lead to an artificial inflation of the MSD and hence
any subsequent diffusion coefficient calculated.
Notes
_____
There are several factors that must be taken into account when setting up and
processing trajectories for computation of self-diffusivities.
These include specific instructions around simulation settings, using
unwrapped trajectories and maintaining a relatively small elapsed time between
saved frames. Additionally, corrections for finite size effects are sometimes
employed along with various means of estimating errors
:cite:p:`Yeh2004,Bulow2020` The reader is directed to the following review,
which describes many of the common pitfalls :cite:p:`Maginn2019`. There are
other ways to compute self-diffusivity, such as from a Green-Kubo integral. At
this point in time, these methods are beyond the scope of this module.
Note also that computation of MSDs is highly memory intensive. If this is
proving a problem, judicious use of the ``start``, ``stop``, ``step`` keywords
to control which frames are incorporated may be required.
References
----------
.. bibliography::
:filter: False
:style: MDA
Maginn2019
Yeh2004
Bulow2020
Classes
-------
.. autoclass:: EinsteinMSD
:members:
:inherited-members:
"""
import numpy as np
import logging
from ..due import due, Doi
from .base import AnalysisBase
from ..core import groups
from tqdm import tqdm
logger = logging.getLogger('MDAnalysis.analysis.msd')
due.cite(Doi("10.21105/joss.00877"),
description="Mean Squared Displacements with tidynamics",
path="MDAnalysis.analysis.msd",
cite_module=True)
due.cite(Doi("10.1051/sfn/201112010"),
description="FCA fast correlation algorithm",
path="MDAnalysis.analysis.msd",
cite_module=True)
del Doi
[docs]class EinsteinMSD(AnalysisBase):
r"""Class to calculate Mean Squared Displacement by the Einstein relation.
Parameters
----------
u : Universe or AtomGroup
An MDAnalysis :class:`Universe` or :class:`AtomGroup`.
Note that :class:`UpdatingAtomGroup` instances are not accepted.
select : str
A selection string. Defaults to "all" in which case
all atoms are selected.
msd_type : {'xyz', 'xy', 'yz', 'xz', 'x', 'y', 'z'}
Desired dimensions to be included in the MSD. Defaults to 'xyz'.
fft : bool
If ``True``, uses a fast FFT based algorithm for computation of
the MSD. Otherwise, use the simple "windowed" algorithm.
The tidynamics package is required for `fft=True`.
Defaults to ``True``.
Attributes
----------
dim_fac : int
Dimensionality :math:`d` of the MSD.
results.timeseries : :class:`numpy.ndarray`
The averaged MSD over all the particles with respect to lag-time.
results.msds_by_particle : :class:`numpy.ndarray`
The MSD of each individual particle with respect to lag-time.
ag : :class:`AtomGroup`
The :class:`AtomGroup` resulting from your selection
n_frames : int
Number of frames included in the analysis.
n_particles : int
Number of particles MSD was calculated over.
.. versionadded:: 2.0.0
"""
def __init__(self, u, select='all', msd_type='xyz', fft=True, **kwargs):
r"""
Parameters
----------
u : Universe or AtomGroup
An MDAnalysis :class:`Universe` or :class:`AtomGroup`.
select : str
A selection string. Defaults to "all" in which case
all atoms are selected.
msd_type : {'xyz', 'xy', 'yz', 'xz', 'x', 'y', 'z'}
Desired dimensions to be included in the MSD.
fft : bool
If ``True``, uses a fast FFT based algorithm for computation of
the MSD. Otherwise, use the simple "windowed" algorithm.
The tidynamics package is required for `fft=True`.
"""
if isinstance(u, groups.UpdatingAtomGroup):
raise TypeError("UpdatingAtomGroups are not valid for MSD "
"computation")
super(EinsteinMSD, self).__init__(u.universe.trajectory, **kwargs)
# args
self.select = select
self.msd_type = msd_type
self._parse_msd_type()
self.fft = fft
# local
self.ag = u.select_atoms(self.select)
self.n_particles = len(self.ag)
self._position_array = None
# result
self.results.msds_by_particle = None
self.results.timeseries = None
def _prepare(self):
# self.n_frames only available here
# these need to be zeroed prior to each run() call
self.results.msds_by_particle = np.zeros((self.n_frames,
self.n_particles))
self._position_array = np.zeros(
(self.n_frames, self.n_particles, self.dim_fac))
# self.results.timeseries not set here
def _parse_msd_type(self):
r""" Sets up the desired dimensionality of the MSD.
"""
keys = {'x': [0], 'y': [1], 'z': [2], 'xy': [0, 1],
'xz': [0, 2], 'yz': [1, 2], 'xyz': [0, 1, 2]}
self.msd_type = self.msd_type.lower()
try:
self._dim = keys[self.msd_type]
except KeyError:
raise ValueError(
'invalid msd_type: {} specified, please specify one of xyz, '
'xy, xz, yz, x, y, z'.format(self.msd_type))
self.dim_fac = len(self._dim)
def _single_frame(self):
r""" Constructs array of positions for MSD calculation.
"""
# shape of position array set here, use span in last dimension
# from this point on
self._position_array[self._frame_index] = (
self.ag.positions[:, self._dim])
def _conclude(self):
if self.fft:
self._conclude_fft()
else:
self._conclude_simple()
def _conclude_simple(self):
r""" Calculates the MSD via the simple "windowed" algorithm.
"""
lagtimes = np.arange(1, self.n_frames)
positions = self._position_array.astype(np.float64)
for lag in tqdm(lagtimes):
disp = positions[:-lag, :, :] - positions[lag:, :, :]
sqdist = np.square(disp).sum(axis=-1)
self.results.msds_by_particle[lag, :] = np.mean(sqdist, axis=0)
self.results.timeseries = self.results.msds_by_particle.mean(axis=1)
def _conclude_fft(self): # with FFT, np.float64 bit prescision required.
r""" Calculates the MSD via the FCA fast correlation algorithm.
"""
try:
import tidynamics
except ImportError:
raise ImportError("""ERROR --- tidynamics was not found!
tidynamics is required to compute an FFT based MSD (default)
try installing it using pip eg:
pip install tidynamics
or set fft=False""")
positions = self._position_array.astype(np.float64)
for n in tqdm(range(self.n_particles)):
self.results.msds_by_particle[:, n] = tidynamics.msd(
positions[:, n, :])
self.results.timeseries = self.results.msds_by_particle.mean(axis=1)
```