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# D. L. Dotson, J. Domanski, S. Buchoux, I. M. Kenney, and O. Beckstein.
# MDAnalysis: A Python package for the rapid analysis of molecular dynamics
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# doi: 10.25080/majora-629e541a-00e
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# MDAnalysis: A Toolkit for the Analysis of Molecular Dynamics Simulations.
# J. Comput. Chem. 32 (2011), 2319--2327, doi:10.1002/jcc.21787
#
"""Water dynamics analysis --- :mod:`MDAnalysis.analysis.waterdynamics`
=======================================================================
:Author: Alejandro Bernardin
:Year: 2014-2015
:Copyright: GNU Public License v3
.. versionadded:: 0.11.0
This module provides functions to analyze water dynamics trajectories and water
interactions with other molecules. The functions in this module are: water
orientational relaxation (WOR) :footcite:p:`Yeh1999`, hydrogen bond lifetimes (HBL)
:footcite:p:`Rapaport1983`, angular distribution (AD) :footcite:p:`Grigera1995`, mean
square displacement (MSD) :footcite:p:`Brodka1994` and survival probability (SP)
:footcite:p:`Liu2004`.
For more information about this type of analysis please refer to
:footcite:p:`ArayaSecchi2014` (water in a protein cavity) and
:footcite:p:`Milischuk2011` (water in a nanopore).
.. rubric:: References
.. footbibliography::
Example use of the analysis classes
-----------------------------------
HydrogenBondLifetimes
~~~~~~~~~~~~~~~~~~~~~
To analyse hydrogen bond lifetime, use
:meth:`MDAnalysis.analysis.hydrogenbonds.hbond_analysis.HydrogenBondAnalysis.lifetime`.
See Also
--------
:mod:`MDAnalysis.analysis.hydrogenbonds.hbond_analysis`
WaterOrientationalRelaxation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Analyzing water orientational relaxation (WOR)
:class:`WaterOrientationalRelaxation`. In this case we are analyzing "how fast"
water molecules are rotating/changing direction. If WOR is very stable we can
assume that water molecules are rotating/changing direction very slow, on the
other hand, if WOR decay very fast, we can assume that water molecules are
rotating/changing direction very fast::
import MDAnalysis
from MDAnalysis.analysis.waterdynamics import WaterOrientationalRelaxation as WOR
u = MDAnalysis.Universe(pdb, trajectory)
select = "byres name OH2 and sphzone 6.0 protein and resid 42"
WOR_analysis = WOR(universe, select, 0, 1000, 20)
WOR_analysis.run()
time = 0
#now we print the data ready to plot. The first two columns are WOR_OH vs t plot,
#the second two columns are WOR_HH vs t graph and the third two columns are WOR_dip vs t graph
for WOR_OH, WOR_HH, WOR_dip in WOR_analysis.timeseries:
print("{time} {WOR_OH} {time} {WOR_HH} {time} {WOR_dip}".format(time=time, WOR_OH=WOR_OH, WOR_HH=WOR_HH,WOR_dip=WOR_dip))
time += 1
#now, if we want, we can plot our data
plt.figure(1,figsize=(18, 6))
#WOR OH
plt.subplot(131)
plt.xlabel('time')
plt.ylabel('WOR')
plt.title('WOR OH')
plt.plot(range(0,time),[column[0] for column in WOR_analysis.timeseries])
#WOR HH
plt.subplot(132)
plt.xlabel('time')
plt.ylabel('WOR')
plt.title('WOR HH')
plt.plot(range(0,time),[column[1] for column in WOR_analysis.timeseries])
#WOR dip
plt.subplot(133)
plt.xlabel('time')
plt.ylabel('WOR')
plt.title('WOR dip')
plt.plot(range(0,time),[column[2] for column in WOR_analysis.timeseries])
plt.show()
where t0 = 0, tf = 1000 and dtmax = 20. In this way we create 20 windows
timesteps (20 values in the x axis), the first window is created with 1000
timestep average (1000/1), the second window is created with 500 timestep
average(1000/2), the third window is created with 333 timestep average (1000/3)
and so on.
AngularDistribution
~~~~~~~~~~~~~~~~~~~
Analyzing angular distribution (AD) :class:`AngularDistribution` for OH vector,
HH vector and dipole vector. It returns a line histogram with vector
orientation preference. A straight line in the output plot means no
preferential orientation in water molecules. In this case we are analyzing if
water molecules have some orientational preference, in this way we can see if
water molecules are under an electric field or if they are interacting with
something (residue, protein, etc)::
import MDAnalysis
from MDAnalysis.analysis.waterdynamics import AngularDistribution as AD
u = MDAnalysis.Universe(pdb, trajectory)
selection = "byres name OH2 and sphzone 6.0 (protein and (resid 42 or resid 26) )"
bins = 30
AD_analysis = AD(universe,selection,bins)
AD_analysis.run()
#now we print data ready to graph. The first two columns are P(cos(theta)) vs cos(theta) for OH vector ,
#the seconds two columns are P(cos(theta)) vs cos(theta) for HH vector and thirds two columns
#are P(cos(theta)) vs cos(theta) for dipole vector
for bin in range(bins):
print("{AD_analysisOH} {AD_analysisHH} {AD_analysisDip}".format(AD_analysis.graph0=AD_analysis.graph[0][bin], AD_analysis.graph1=AD_analysis.graph[1][bin],AD_analysis.graph2=AD_analysis.graph[2][bin]))
#and if we want to graph our results
plt.figure(1,figsize=(18, 6))
#AD OH
plt.subplot(131)
plt.xlabel('cos theta')
plt.ylabel('P(cos theta)')
plt.title('PDF cos theta for OH')
plt.plot([float(column.split()[0]) for column in AD_analysis.graph[0][:-1]],[float(column.split()[1]) for column in AD_analysis.graph[0][:-1]])
#AD HH
plt.subplot(132)
plt.xlabel('cos theta')
plt.ylabel('P(cos theta)')
plt.title('PDF cos theta for HH')
plt.plot([float(column.split()[0]) for column in AD_analysis.graph[1][:-1]],[float(column.split()[1]) for column in AD_analysis.graph[1][:-1]])
#AD dip
plt.subplot(133)
plt.xlabel('cos theta')
plt.ylabel('P(cos theta)')
plt.title('PDF cos theta for dipole')
plt.plot([float(column.split()[0]) for column in AD_analysis.graph[2][:-1]],[float(column.split()[1]) for column in AD_analysis.graph[2][:-1]])
plt.show()
where `P(cos(theta))` is the angular distribution or angular probabilities.
MeanSquareDisplacement
~~~~~~~~~~~~~~~~~~~~~~
Analyzing mean square displacement (MSD) :class:`MeanSquareDisplacement` for
water molecules. In this case we are analyzing the average distance that water
molecules travels inside protein in XYZ direction (cylindric zone of radius
11[nm], Zmax 4.0[nm] and Zmin -8.0[nm]). A strong rise mean a fast movement of
water molecules, a weak rise mean slow movement of particles::
import MDAnalysis
from MDAnalysis.analysis.waterdynamics import MeanSquareDisplacement as MSD
u = MDAnalysis.Universe(pdb, trajectory)
select = "byres name OH2 and cyzone 11.0 4.0 -8.0 protein"
MSD_analysis = MSD(universe, select, 0, 1000, 20)
MSD_analysis.run()
#now we print data ready to graph. The graph
#represents MSD vs t
time = 0
for msd in MSD_analysis.timeseries:
print("{time} {msd}".format(time=time, msd=msd))
time += 1
#Plot
plt.xlabel('time')
plt.ylabel('MSD')
plt.title('MSD')
plt.plot(range(0,time),MSD_analysis.timeseries)
plt.show()
.. _SP-examples:
SurvivalProbability
~~~~~~~~~~~~~~~~~~~
Analyzing survival probability (SP) :class:`SurvivalProbability` of molecules.
In this case we are analyzing how long water molecules remain in a
sphere of radius 12.3 centered in the geometrical center of resid 42 and 26.
A slow decay of SP means a long permanence time of water molecules in
the zone, on the other hand, a fast decay means a short permanence time::
import MDAnalysis
from MDAnalysis.analysis.waterdynamics import SurvivalProbability as SP
import matplotlib.pyplot as plt
universe = MDAnalysis.Universe(pdb, trajectory)
select = "byres name OH2 and sphzone 12.3 (resid 42 or resid 26) "
sp = SP(universe, select, verbose=True)
sp.run(start=0, stop=101, tau_max=20)
tau_timeseries = sp.tau_timeseries
sp_timeseries = sp.sp_timeseries
# print in console
for tau, sp in zip(tau_timeseries, sp_timeseries):
print("{time} {sp}".format(time=tau, sp=sp))
# plot
plt.xlabel('Time')
plt.ylabel('SP')
plt.title('Survival Probability')
plt.plot(tau_timeseries, sp_timeseries)
plt.show()
One should note that the `stop` keyword as used in the above example has an
`exclusive` behaviour, i.e. here the final frame used will be 100 not 101.
This behaviour is aligned with :class:`AnalysisBase` but currently differs from
other :mod:`MDAnalysis.analysis.waterdynamics` classes, which all exhibit
`inclusive` behaviour for their final frame selections.
Another example applies to the situation where you work with many different "residues".
Here we calculate the SP of a potassium ion around each lipid in a membrane and
average the results. In this example, if the SP analysis were run without treating each lipid
separately, potassium ions may hop from one lipid to another and still be counted as remaining
in the specified region. That is, the survival probability of the potassium ion around the
entire membrane will be calculated.
Note, for this example, it is advisable to use `Universe(in_memory=True)` to ensure that the
simulation is not being reloaded into memory for each lipid::
import MDAnalysis as mda
from MDAnalysis.analysis.waterdynamics import SurvivalProbability as SP
import numpy as np
u = mda.Universe("md.gro", "md100ns.xtc", in_memory=True)
lipids = u.select_atoms('resname LIPIDS')
joined_sp_timeseries = [[] for _ in range(20)]
for lipid in lipids.residues:
print("Lipid ID: %d" % lipid.resid)
select = "resname POTASSIUM and around 3.5 (resid %d and name O13 O14) " % lipid.resid
sp = SP(u, select, verbose=True)
sp.run(tau_max=20)
# Raw SP points for each tau:
for sps, new_sps in zip(joined_sp_timeseries, sp.sp_timeseries_data):
sps.extend(new_sps)
# average all SP datapoints
sp_data = [np.mean(sp) for sp in joined_sp_timeseries]
for tau, sp in zip(range(1, tau_max + 1), sp_data):
print("{time} {sp}".format(time=tau, sp=sp))
.. _Output:
Output
------
WaterOrientationalRelaxation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Water orientational relaxation (WOR) data is returned per window timestep,
which is stored in :attr:`WaterOrientationalRelaxation.timeseries`::
results = [
[ # time t0
<WOR_OH>, <WOR_HH>, <WOR_dip>
],
[ # time t1
<WOR_OH>, <WOR_HH>, <WOR_dip>
],
...
]
AngularDistribution
~~~~~~~~~~~~~~~~~~~
Angular distribution (AD) data is returned per vector, which is stored in
:attr:`AngularDistribution.graph`. In fact, AngularDistribution returns a
histogram::
results = [
[ # OH vector values
# the values are order in this way: <x_axis y_axis>
<cos_theta0 ang_distr0>, <cos_theta1 ang_distr1>, ...
],
[ # HH vector values
<cos_theta0 ang_distr0>, <cos_theta1 ang_distr1>, ...
],
[ # dip vector values
<cos_theta0 ang_distr0>, <cos_theta1 ang_distr1>, ...
],
]
MeanSquareDisplacement
~~~~~~~~~~~~~~~~~~~~~~
Mean Square Displacement (MSD) data is returned in a list, which each element
represents a MSD value in its respective window timestep. Data is stored in
:attr:`MeanSquareDisplacement.timeseries`::
results = [
#MSD values orders by window timestep
<MSD_t0>, <MSD_t1>, ...
]
SurvivalProbability
~~~~~~~~~~~~~~~~~~~
Survival Probability (SP) computes two lists: a list of taus (:attr:`SurvivalProbability.tau_timeseries`) and a list of
the corresponding survival probabilities (:attr:`SurvivalProbability.sp_timeseries`).
results = [ tau1, tau2, ..., tau_n ], [ sp_tau1, sp_tau2, ..., sp_tau_n]
Additionally, a list :attr:`SurvivalProbability.sp_timeseries_data`, is provided which contains
a list of all SPs calculated for each tau. This can be used to compute the distribution or time dependence of SP, etc.
Classes
--------
.. autoclass:: WaterOrientationalRelaxation
:members:
:inherited-members:
.. autoclass:: AngularDistribution
:members:
:inherited-members:
.. autoclass:: MeanSquareDisplacement
:members:
:inherited-members:
.. autoclass:: SurvivalProbability
:members:
:inherited-members:
"""
from MDAnalysis.lib.correlations import autocorrelation, correct_intermittency
import MDAnalysis.analysis.hbonds
from itertools import zip_longest
import logging
import warnings
import numpy as np
logger = logging.getLogger('MDAnalysis.analysis.waterdynamics')
from MDAnalysis.lib.log import ProgressBar
[docs]
class WaterOrientationalRelaxation(object):
r"""Water orientation relaxation analysis
Function to evaluate the Water Orientational Relaxation proposed by Yu-ling
Yeh and Chung-Yuan Mou :footcite:p:`Yeh1999`. WaterOrientationalRelaxation
indicates "how fast" water molecules are rotating or changing direction.
This is a time correlation function given by:
.. math::
C_{\hat u}(\tau)=\langle \mathit{P}_2[\mathbf{\hat{u}}(t_0)\cdot\mathbf{\hat{u}}(t_0+\tau)]\rangle
where :math:`P_2=(3x^2-1)/2` is the second-order Legendre polynomial and :math:`\hat{u}` is
a unit vector along HH, OH or dipole vector.
Parameters
----------
universe : Universe
Universe object
selection : str
Selection string for water [‘byres name OH2’].
t0 : int
frame where analysis begins
tf : int
frame where analysis ends
dtmax : int
Maximum dt size, `dtmax` < `tf` or it will crash.
.. versionadded:: 0.11.0
.. versionchanged:: 1.0.0
Changed `selection` keyword to `select`
"""
def __init__(self, universe, select, t0, tf, dtmax, nproc=1):
self.universe = universe
self.selection = select
self.t0 = t0
self.tf = tf
self.dtmax = dtmax
self.nproc = nproc
self.timeseries = None
def _repeatedIndex(self, selection, dt, totalFrames):
"""
Indicates the comparation between all the t+dt.
The results is a list of list with all the repeated index per frame
(or time).
Ex: dt=1, so compare frames (1,2),(2,3),(3,4)...
Ex: dt=2, so compare frames (1,3),(3,5),(5,7)...
Ex: dt=3, so compare frames (1,4),(4,7),(7,10)...
"""
rep = []
for i in range(int(round((totalFrames - 1) / float(dt)))):
if (dt * i + dt < totalFrames):
rep.append(self._sameMolecTandDT(
selection, dt * i, (dt * i) + dt))
return rep
def _getOneDeltaPoint(self, universe, repInd, i, t0, dt):
"""
Gives one point to calculate the mean and gets one point of the plot
C_vect vs t.
Ex: t0=1 and tau=1 so calculate the t0-tau=1-2 intervale.
Ex: t0=5 and tau=3 so calcultate the t0-tau=5-8 intervale.
i = come from getMeanOnePoint (named j) (int)
"""
valOH = 0
valHH = 0
valdip = 0
n = 0
for j in range(len(repInd[i]) // 3):
begj = 3 * j
universe.trajectory[t0]
Ot0 = repInd[i][begj]
H1t0 = repInd[i][begj + 1]
H2t0 = repInd[i][begj + 2]
OHVector0 = H1t0.position - Ot0.position
HHVector0 = H1t0.position - H2t0.position
dipVector0 = ((H1t0.position + H2t0.position) * 0.5) - Ot0.position
universe.trajectory[t0 + dt]
Otp = repInd[i][begj]
H1tp = repInd[i][begj + 1]
H2tp = repInd[i][begj + 2]
OHVectorp = H1tp.position - Otp.position
HHVectorp = H1tp.position - H2tp.position
dipVectorp = ((H1tp.position + H2tp.position) * 0.5) - Otp.position
normOHVector0 = np.linalg.norm(OHVector0)
normOHVectorp = np.linalg.norm(OHVectorp)
normHHVector0 = np.linalg.norm(HHVector0)
normHHVectorp = np.linalg.norm(HHVectorp)
normdipVector0 = np.linalg.norm(dipVector0)
normdipVectorp = np.linalg.norm(dipVectorp)
unitOHVector0 = [OHVector0[0] / normOHVector0,
OHVector0[1] / normOHVector0,
OHVector0[2] / normOHVector0]
unitOHVectorp = [OHVectorp[0] / normOHVectorp,
OHVectorp[1] / normOHVectorp,
OHVectorp[2] / normOHVectorp]
unitHHVector0 = [HHVector0[0] / normHHVector0,
HHVector0[1] / normHHVector0,
HHVector0[2] / normHHVector0]
unitHHVectorp = [HHVectorp[0] / normHHVectorp,
HHVectorp[1] / normHHVectorp,
HHVectorp[2] / normHHVectorp]
unitdipVector0 = [dipVector0[0] / normdipVector0,
dipVector0[1] / normdipVector0,
dipVector0[2] / normdipVector0]
unitdipVectorp = [dipVectorp[0] / normdipVectorp,
dipVectorp[1] / normdipVectorp,
dipVectorp[2] / normdipVectorp]
valOH += self.lg2(np.dot(unitOHVector0, unitOHVectorp))
valHH += self.lg2(np.dot(unitHHVector0, unitHHVectorp))
valdip += self.lg2(np.dot(unitdipVector0, unitdipVectorp))
n += 1
return (valOH/n, valHH/n, valdip/n) if n > 0 else (0, 0, 0)
def _getMeanOnePoint(self, universe, selection1, selection_str, dt,
totalFrames):
"""
This function gets one point of the plot C_vec vs t. It uses the
_getOneDeltaPoint() function to calculate the average.
"""
repInd = self._repeatedIndex(selection1, dt, totalFrames)
sumsdt = 0
n = 0.0
sumDeltaOH = 0.0
sumDeltaHH = 0.0
sumDeltadip = 0.0
for j in range(totalFrames // dt - 1):
a = self._getOneDeltaPoint(universe, repInd, j, sumsdt, dt)
sumDeltaOH += a[0]
sumDeltaHH += a[1]
sumDeltadip += a[2]
sumsdt += dt
n += 1
# if no water molecules remain in selection, there is nothing to get
# the mean, so n = 0.
return (sumDeltaOH / n, sumDeltaHH / n, sumDeltadip / n) if n > 0 else (0, 0, 0)
def _sameMolecTandDT(self, selection, t0d, tf):
"""
Compare the molecules in the t0d selection and the t0d+dt selection and
select only the particles that are repeated in both frame. This is to
consider only the molecules that remains in the selection after the dt
time has elapsed.
The result is a list with the indices of the atoms.
"""
a = set(selection[t0d])
b = set(selection[tf])
sort = sorted(list(a.intersection(b)))
return sort
def _selection_serial(self, universe, selection_str):
selection = []
for ts in ProgressBar(universe.trajectory, verbose=True,
total=universe.trajectory.n_frames):
selection.append(universe.select_atoms(selection_str))
return selection
[docs]
@staticmethod
def lg2(x):
"""Second Legendre polynomial"""
return (3*x*x - 1)/2
[docs]
def run(self, **kwargs):
"""Analyze trajectory and produce timeseries"""
# All the selection to an array, this way is faster than selecting
# later.
if self.nproc == 1:
selection_out = self._selection_serial(
self.universe, self.selection)
else:
# selection_out = self._selection_parallel(self.universe,
# self.selection, self.nproc)
# parallel selection to be implemented
selection_out = self._selection_serial(
self.universe, self.selection)
self.timeseries = []
for dt in list(range(1, self.dtmax + 1)):
output = self._getMeanOnePoint(
self.universe, selection_out, self.selection, dt, self.tf)
self.timeseries.append(output)
[docs]
class AngularDistribution(object):
r"""Angular distribution function analysis
The angular distribution function (AD) is defined as the distribution
probability of the cosine of the :math:`\theta` angle formed by the OH
vector, HH vector or dipolar vector of water molecules and a vector
:math:`\hat n` parallel to chosen axis (z is the default value). The cosine
is define as :math:`\cos \theta = \hat u \cdot \hat n`, where :math:`\hat
u` is OH, HH or dipole vector. It creates a histogram and returns a list
of lists, see Output_. The AD is also know as Angular Probability (AP).
Parameters
----------
universe : Universe
Universe object
select : str
Selection string to evaluate its angular distribution ['byres name OH2']
bins : int (optional)
Number of bins to create the histogram by means of :func:`numpy.histogram`
axis : {'x', 'y', 'z'} (optional)
Axis to create angle with the vector (HH, OH or dipole) and calculate
cosine theta ['z'].
.. versionadded:: 0.11.0
.. versionchanged:: 1.0.0
Changed `selection` keyword to `select`
"""
def __init__(self, universe, select, bins=40, nproc=1, axis="z"):
self.universe = universe
self.selection_str = select
self.bins = bins
self.nproc = nproc
self.axis = axis
self.graph = None
def _getCosTheta(self, universe, selection, axis):
valOH = []
valHH = []
valdip = []
i = 0
while i <= (len(selection) - 1):
universe.trajectory[i]
line = selection[i].positions
Ot0 = line[::3]
H1t0 = line[1::3]
H2t0 = line[2::3]
OHVector0 = H1t0 - Ot0
HHVector0 = H1t0 - H2t0
dipVector0 = (H1t0 + H2t0) * 0.5 - Ot0
unitOHVector0 = OHVector0 / \
np.linalg.norm(OHVector0, axis=1)[:, None]
unitHHVector0 = HHVector0 / \
np.linalg.norm(HHVector0, axis=1)[:, None]
unitdipVector0 = dipVector0 / \
np.linalg.norm(dipVector0, axis=1)[:, None]
j = 0
while j < len(line) / 3:
if axis == "z":
valOH.append(unitOHVector0[j][2])
valHH.append(unitHHVector0[j][2])
valdip.append(unitdipVector0[j][2])
elif axis == "x":
valOH.append(unitOHVector0[j][0])
valHH.append(unitHHVector0[j][0])
valdip.append(unitdipVector0[j][0])
elif axis == "y":
valOH.append(unitOHVector0[j][1])
valHH.append(unitHHVector0[j][1])
valdip.append(unitdipVector0[j][1])
j += 1
i += 1
return (valOH, valHH, valdip)
def _getHistogram(self, universe, selection, bins, axis):
"""
This function gets a normalized histogram of the cos(theta) values. It
return a list of list.
"""
a = self._getCosTheta(universe, selection, axis)
cosThetaOH = a[0]
cosThetaHH = a[1]
cosThetadip = a[2]
lencosThetaOH = len(cosThetaOH)
lencosThetaHH = len(cosThetaHH)
lencosThetadip = len(cosThetadip)
histInterval = bins
histcosThetaOH = np.histogram(cosThetaOH, histInterval, density=True)
histcosThetaHH = np.histogram(cosThetaHH, histInterval, density=True)
histcosThetadip = np.histogram(cosThetadip, histInterval, density=True)
return (histcosThetaOH, histcosThetaHH, histcosThetadip)
def _hist2column(self, aList):
"""
This function transform from the histogram format
to a column format.
"""
a = []
for x in zip_longest(*aList, fillvalue="."):
a.append(" ".join(str(i) for i in x))
return a
[docs]
def run(self, **kwargs):
"""Function to evaluate the angular distribution of cos(theta)"""
if self.nproc == 1:
selection = self._selection_serial(
self.universe, self.selection_str)
else:
# not implemented yet
# selection = self._selection_parallel(self.universe,
# self.selection_str,self.nproc)
selection = self._selection_serial(
self.universe, self.selection_str)
self.graph = []
output = self._getHistogram(
self.universe, selection, self.bins, self.axis)
# this is to format the exit of the file
# maybe this output could be improved
listOH = [list(output[0][1]), list(output[0][0])]
listHH = [list(output[1][1]), list(output[1][0])]
listdip = [list(output[2][1]), list(output[2][0])]
self.graph.append(self._hist2column(listOH))
self.graph.append(self._hist2column(listHH))
self.graph.append(self._hist2column(listdip))
def _selection_serial(self, universe, selection_str):
selection = []
for ts in ProgressBar(universe.trajectory, verbose=True,
total=universe.trajectory.n_frames):
selection.append(universe.select_atoms(selection_str))
return selection
[docs]
class MeanSquareDisplacement(object):
r"""Mean square displacement analysis
Function to evaluate the Mean Square Displacement (MSD_). The MSD gives the
average distance that particles travels. The MSD is given by:
.. math::
\langle\Delta r(t)^2\rangle = 2nDt
where :math:`r(t)` is the position of particle in time :math:`t`,
:math:`\Delta r(t)` is the displacement after time lag :math:`t`,
:math:`n` is the dimensionality, in this case :math:`n=3`,
:math:`D` is the diffusion coefficient and :math:`t` is the time.
.. _MSD: http://en.wikipedia.org/wiki/Mean_squared_displacement
Parameters
----------
universe : Universe
Universe object
select : str
Selection string for water [‘byres name OH2’].
t0 : int
frame where analysis begins
tf : int
frame where analysis ends
dtmax : int
Maximum dt size, `dtmax` < `tf` or it will crash.
.. versionadded:: 0.11.0
.. versionchanged:: 1.0.0
Changed `selection` keyword to `select`
"""
def __init__(self, universe, select, t0, tf, dtmax, nproc=1):
self.universe = universe
self.selection = select
self.t0 = t0
self.tf = tf
self.dtmax = dtmax
self.nproc = nproc
self.timeseries = None
def _repeatedIndex(self, selection, dt, totalFrames):
"""
Indicate the comparation between all the t+dt.
The results is a list of list with all the repeated index per frame
(or time).
- Ex: dt=1, so compare frames (1,2),(2,3),(3,4)...
- Ex: dt=2, so compare frames (1,3),(3,5),(5,7)...
- Ex: dt=3, so compare frames (1,4),(4,7),(7,10)...
"""
rep = []
for i in range(int(round((totalFrames - 1) / float(dt)))):
if (dt * i + dt < totalFrames):
rep.append(self._sameMolecTandDT(
selection, dt * i, (dt * i) + dt))
return rep
def _getOneDeltaPoint(self, universe, repInd, i, t0, dt):
"""
Gives one point to calculate the mean and gets one point of the plot
C_vect vs t.
- Ex: t0=1 and dt=1 so calculate the t0-dt=1-2 interval.
- Ex: t0=5 and dt=3 so calcultate the t0-dt=5-8 interval
i = come from getMeanOnePoint (named j) (int)
"""
valO = 0
n = 0
for j in range(len(repInd[i]) // 3):
begj = 3 * j
universe.trajectory[t0]
# Plus zero is to avoid 0to be equal to 0tp
Ot0 = repInd[i][begj].position + 0
universe.trajectory[t0 + dt]
# Plus zero is to avoid 0to be equal to 0tp
Otp = repInd[i][begj].position + 0
# position oxygen
OVector = Ot0 - Otp
# here it is the difference with
# waterdynamics.WaterOrientationalRelaxation
valO += np.dot(OVector, OVector)
n += 1
# if no water molecules remain in selection, there is nothing to get
# the mean, so n = 0.
return valO/n if n > 0 else 0
def _getMeanOnePoint(self, universe, selection1, selection_str, dt,
totalFrames):
"""
This function gets one point of the plot C_vec vs t. It's uses the
_getOneDeltaPoint() function to calculate the average.
"""
repInd = self._repeatedIndex(selection1, dt, totalFrames)
sumsdt = 0
n = 0.0
sumDeltaO = 0.0
valOList = []
for j in range(totalFrames // dt - 1):
a = self._getOneDeltaPoint(universe, repInd, j, sumsdt, dt)
sumDeltaO += a
valOList.append(a)
sumsdt += dt
n += 1
# if no water molecules remain in selection, there is nothing to get
# the mean, so n = 0.
return sumDeltaO/n if n > 0 else 0
def _sameMolecTandDT(self, selection, t0d, tf):
"""
Compare the molecules in the t0d selection and the t0d+dt selection and
select only the particles that are repeated in both frame. This is to
consider only the molecules that remains in the selection after the dt
time has elapsed. The result is a list with the indexs of the atoms.
"""
a = set(selection[t0d])
b = set(selection[tf])
sort = sorted(list(a.intersection(b)))
return sort
def _selection_serial(self, universe, selection_str):
selection = []
for ts in ProgressBar(universe.trajectory, verbose=True,
total=universe.trajectory.n_frames):
selection.append(universe.select_atoms(selection_str))
return selection
[docs]
def run(self, **kwargs):
"""Analyze trajectory and produce timeseries"""
# All the selection to an array, this way is faster than selecting
# later.
if self.nproc == 1:
selection_out = self._selection_serial(
self.universe, self.selection)
else:
# parallel not yet implemented
# selection = selection_parallel(universe, selection_str, nproc)
selection_out = self._selection_serial(
self.universe, self.selection)
self.timeseries = []
for dt in list(range(1, self.dtmax + 1)):
output = self._getMeanOnePoint(
self.universe, selection_out, self.selection, dt, self.tf)
self.timeseries.append(output)
[docs]
class SurvivalProbability(object):
r"""
Survival Probability (SP) gives the probability for a group of particles to remain
in a certain region. The SP is given by:
.. math::
P(\tau) = \langle \frac{ N(t, t + \tau )} { N(t) }\rangle
where :math:`\tau` is the timestep, :math:`N(t)` the number of particles at time
:math:`t`, and :math:`N(t, t+\tau)` is the number of particles at every frame from
:math:`t` to :math:`t + \tau`. The angular brackets represent an average over all time
origins, :math:`t`. See :func:`MDAnalysis.lib.correlations.autocorrelation` for
technical details.
Parameters
----------
universe : Universe
Universe object
select : str
Selection string; any selection is allowed. With this selection you
define the region/zone where to analyze, e.g.: "resname SOL and around 5
(resid 10)". See `SP-examples`_.
verbose : Boolean, optional
When True, prints progress and comments to the console.
Notes
-----
Currently :class:`SurvivalProbability` is the only on in
:mod:`MDAnalysis.analysis.waterdynamics` to support an `exclusive`
behaviour (i.e. similar to the current behaviour of :class:`AnalysisBase`
to the `stop` keyword passed to :meth:`SurvivalProbability.run`. Unlike
other :mod:`MDAnalysis.analysis.waterdynamics` final frame definitions
which are `inclusive`.
.. versionadded:: 0.11.0
.. versionchanged:: 1.0.0
Using the MDAnalysis.lib.correlations.py to carry out the intermittency
and autocorrelation calculations.
Changed `selection` keyword to `select`.
Removed support for the deprecated `t0`, `tf`, and `dtmax` keywords.
These should instead be passed to :meth:`SurvivalProbability.run` as
the `start`, `stop`, and `tau_max` keywords respectively.
The `stop` keyword as passed to :meth:`SurvivalProbability.run` has now
changed behaviour and will act in an `exclusive` manner (instead of it's
previous `inclusive` behaviour),
.. versionchanged:: 2.7.0
Updated docs to align with discrete autocorrelation function.
"""
def __init__(self, universe, select, verbose=False):
self.universe = universe
self.selection = select
self.verbose = verbose
[docs]
def run(self, tau_max=20, start=None, stop=None, step=None, residues=False,
intermittency=0, verbose=False):
"""
Computes and returns the Survival Probability (SP) timeseries
Parameters
----------
start : int, optional
Zero-based index of the first frame to be analysed, Default: None
(first frame).
stop : int, optional
Zero-based index of the last frame to be analysed (exclusive),
Default: None (last frame).
step : int, optional
Jump every `step`-th frame. This is compatible but independant of
the taus used, and it is good to consider using the `step` equal
to `tau_max` to remove the overlap. Note that `step` and `tau_max`
work consistently with intermittency. Default: None
(use every frame).
tau_max : int, optional
Survival probability is calculated for the range
1 <= `tau` <= `tau_max`.
residues : Boolean, optional
If true, the analysis will be carried out on the residues
(.resids) rather than on atom (.ids). A single atom is sufficient
to classify the residue as within the distance.
intermittency : int, optional
The maximum number of consecutive frames for which an atom can
leave but be counted as present if it returns at the next frame.
An intermittency of `0` is equivalent to a continuous survival
probability, which does not allow for the leaving and returning of
atoms. For example, for `intermittency=2`, any given atom may leave
a region of interest for up to two consecutive frames yet be
treated as being present at all frames. The default is continuous
(0).
verbose : Boolean, optional
Print the progress to the console.
Returns
-------
tau_timeseries : list
tau from 1 to `tau_max`. Saved in the field tau_timeseries.
sp_timeseries : list
survival probability for each value of `tau`. Saved in the field
sp_timeseries.
sp_timeseries_data: list
raw datapoints from which the average is taken (sp_timeseries).
Time dependancy and distribution can be extracted.
.. versionchanged:: 1.0.0
To math other analysis methods, the `stop` keyword is now exclusive
rather than inclusive.
"""
start, stop, step = self.universe.trajectory.check_slice_indices(
start,
stop,
step
)
if tau_max > (stop - start):
raise ValueError("Too few frames selected for given tau_max.")
# preload the frames (atom IDs) to a list of sets
self._selected_ids = []
# fixme - to parallise: the section should be rewritten so that this loop only creates a list of indices,
# on which the parallel _single_frame can be applied.
# skip frames that will not be used in order to improve performance
# because AtomGroup.select_atoms is the most expensive part of this calculation
# Example: step 5 and tau 2: LLLSS LLLSS, ... where L = Load, and S = Skip
# Intermittency means that we have to load the extra frames to know if the atom is actually missing.
# Say step=5 and tau=1, intermittency=0: LLSSS LLSSS
# Say step=5 and tau=1, intermittency=1: LLLSL LLLSL
frame_loaded_counter = 0
# only for the first window (frames before t are not used)
frames_per_window = tau_max + 1 + intermittency
# This number will apply after the first windows was loaded
frames_per_window_subsequent = (tau_max + 1) + (2 * intermittency)
num_frames_to_skip = max(step - frames_per_window_subsequent, 0)
frame_no = start
while frame_no < stop: # we have already added 1 to stop, therefore <
if num_frames_to_skip != 0 and frame_loaded_counter == frames_per_window:
logger.info("Skipping the next %d frames:", num_frames_to_skip)
frame_no += num_frames_to_skip
frame_loaded_counter = 0
# Correct the number of frames to be loaded after the first window (which starts at t=0, and
# intermittency does not apply to the frames before)
frames_per_window = frames_per_window_subsequent
continue
# update the frame number
self.universe.trajectory[frame_no]
logger.info("Loading frame: %d", self.universe.trajectory.frame)
atoms = self.universe.select_atoms(self.selection)
# SP of residues or of atoms
ids = atoms.residues.resids if residues else atoms.ids
self._selected_ids.append(set(ids))
frame_no += 1
frame_loaded_counter += 1
# adjust for the frames that were not loaded (step>tau_max + 1),
# and for extra frames that were loaded (intermittency)
window_jump = step - num_frames_to_skip
self._intermittent_selected_ids = correct_intermittency(self._selected_ids, intermittency=intermittency)
tau_timeseries, sp_timeseries, sp_timeseries_data = autocorrelation(self._intermittent_selected_ids,
tau_max, window_jump)
# warn the user if the NaN are found
if all(np.isnan(sp_timeseries[1:])):
logger.warning('NaN Error: Most likely data was not found. Check your atom selections. ')
# user can investigate the distribution and sample size
self.sp_timeseries_data = sp_timeseries_data
self.tau_timeseries = tau_timeseries
self.sp_timeseries = sp_timeseries
return self