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# R. J. Gowers, M. Linke, J. Barnoud, T. J. E. Reddy, M. N. Melo, S. L. Seyler,
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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
#
"""
Hydrogen bond autocorrelation --- :mod:`MDAnalysis.analysis.hbonds.hbond_autocorrel`
====================================================================================
:Author: Richard J. Gowers
:Year: 2014
:Copyright: GNU Public License v3
.. versionadded:: 0.9.0
Description
-----------
Calculates the time autocorrelation function, :math:`C_x(t)`, for the hydrogen
bonds in the selections passed to it. The population of hydrogen bonds at a
given startpoint, :math:`t_0`, is evaluated based on geometric criteria and
then the lifetime of these bonds is monitored over time. Multiple passes
through the trajectory are used to build an average of the behaviour.
.. math::
C_x(t) = \\left \\langle \\frac{h_{ij}(t_0) h_{ij}(t_0 + t)}{h_{ij}(t_0)^2} \\right\\rangle
The subscript :math:`x` refers to the definition of lifetime being used, either
continuous or intermittent. The continuous definition measures the time that
a particular hydrogen bond remains continuously attached, whilst the
intermittent definition allows a bond to break and then subsequently reform and
be counted again. The relevent lifetime, :math:`\\tau_x`, can then be found
via integration of this function
.. math::
\\tau_x = \\int_0^\\infty C_x(t) dt`
For this, the observed behaviour is fitted to a multi exponential function,
using 2 exponents for the continuous lifetime and 3 for the intermittent
lifetime.
:math:`C_x(t) = A_1 \\exp( - t / \\tau_1)
+ A_2 \\exp( - t / \\tau_2)
[+ A_3 \\exp( - t / \\tau_3)]`
Where the final pre expoential factor :math:`A_n` is subject to the condition:
:math:`A_n = 1 - \\sum\\limits_{i=1}^{n-1} A_i`
For further details see [Gowers2015]_.
.. rubric:: References
.. [Gowers2015] Richard J. Gowers and Paola Carbone,
A multiscale approach to model hydrogen bonding: The case of polyamide
The Journal of Chemical Physics, 142, 224907 (2015),
DOI:http://dx.doi.org/10.1063/1.4922445
Input
-----
Three AtomGroup selections representing the **hydrogens**, **donors** and
**acceptors** that you wish to analyse. Note that the **hydrogens** and
**donors** selections must be aligned, that is **hydrogens[0]** and
**donors[0]** must represent a bonded pair. For systems such as water,
this will mean that each oxygen appears twice in the **donors** AtomGroup.
The function :func:`find_hydrogen_donors` can be used to construct the donor
AtomGroup
::
import MDAnalysis as mda
from MDAnalysis.analysis import hbonds
from MDAnalysis.tests.datafiles import waterPSF, waterDCD
u = mda.Universe(waterPSF, waterDCD)
hydrogens = u.select_atoms('name H*')
donors = hbonds.find_hydrogen_donors(hydrogens)
Note that this requires the Universe to have bond information. If this isn't
present in the topology file, the
:meth:`MDAnalysis.core.groups.AtomGroup.guess_bonds` method can be used
as so
::
import MDAnalysis as mda
from MDAnalysis.analysis import hbonds
from MDAnalysis.tests.datafiles import GRO
# we could load the Universe with guess_bonds=True
# but this would guess **all** bonds
u = mda.Universe(GRO)
water = u.select_atoms('resname SOL and not type DUMMY')
# guess bonds only within our water atoms
# this adds the bond information directly to the Universe
water.guess_bonds()
hydrogens = water.select_atoms('type H')
# this is now possible as we guessed the bonds
donors = hbonds.find_hydrogen_donors(hydrogens)
The keyword **exclusions** allows a tuple of array addresses to be provided,
(Hidx, Aidx), these pairs of hydrogen-acceptor are then not permitted to be
counted as part of the analysis. This could be used to exclude the
consideration of hydrogen bonds within the same functional group, or to perform
analysis on strictly intermolecular hydrogen bonding.
Hydrogen bonds are defined on the basis of geometric criteria; a
Hydrogen-Acceptor distance of less then **dist_crit** and a
Donor-Hydrogen-Acceptor angle of greater than **angle_crit**.
The length of trajectory to analyse in ps, **sample_time**, is used to choose
what length to analyse.
Multiple passes, controlled by the keyword **nruns**, through the trajectory
are performed and an average calculated. For each pass, **nsamples** number
of points along the run are calculated.
Output
------
All results of the analysis are available through the *solution* attribute.
This is a dictionary with the following keys
- *results* The raw results of the time autocorrelation function.
- *time* Time axis, in ps, for the results.
- *fit* Results of the exponential curve fitting procedure. For the
*continuous* lifetime these are (A1, tau1, tau2), for the
*intermittent* lifetime these are (A1, A2, tau1, tau2, tau3).
- *tau* Calculated time constant from the fit.
- *estimate* Estimated values generated by the calculated fit.
The *results* and *time* values are only filled after the :meth:`run` method,
*fit*, *tau* and *estimate* are filled after the :meth:`solve` method has been
used.
Worked Example for Polyamide
----------------------------
This example finds the continuous hydrogen bond lifetime between N-H..O in a
polyamide system. This will use the default geometric definition for hydrogen
bonds of length 3.0 Å and angle of 130 degrees.
It will observe a window of 2.0 ps (`sample_time`) and try to gather 1000
sample point within this time window (this relies upon the trajectory being
sampled frequently enough). This process is repeated for 20 different start
points to build a better average.
::
import MDAnalysis as mda
from MDAnalysis.analysis import hbonds
from MDAnalysis.tests.datafiles import TRZ_psf, TRZ
import matplotlib.pyplot as plt
# load system
u = mda.Universe(TRZ_psf, TRZ)
# select atoms of interest into AtomGroups
H = u.select_atoms('name Hn')
N = u.select_atoms('name N')
O = u.select_atoms('name O')
# create analysis object
hb_ac = hbonds.HydrogenBondAutoCorrel(u,
acceptors=O, hydrogens=H, donors=N,
bond_type='continuous',
sample_time=2.0, nsamples=1000, nruns=20)
# call run to gather results
hb_ac.run()
# attempt to fit results to exponential equation
hb_ac.solve()
# grab results from inside object
tau = hb_ac.solution['tau']
time = hb_ac.solution['time']
results = hb_ac.solution['results']
estimate = hb_ac.solution['estimate']
# plot to check!
plt.plot(time, results, 'ro')
plt.plot(time, estimate)
plt.show()
Classes
-------
.. autofunction:: find_hydrogen_donors
.. autoclass:: HydrogenBondAutoCorrel
:members:
"""
from __future__ import division, absolute_import
from six.moves import zip
from six import raise_from
import numpy as np
import scipy.optimize
import warnings
from MDAnalysis.lib.log import ProgressBar
from MDAnalysis.lib.distances import capped_distance, calc_angles, calc_bonds
from MDAnalysis.lib._cutil import _in2d
from MDAnalysis.core.groups import requires
from MDAnalysis.due import due, Doi
due.cite(Doi("10.1063/1.4922445"),
description="Hydrogen bonding autocorrelation time",
path='MDAnalysis.analysis.hbonds.hbond_autocorrel',
)
del Doi
[docs]@requires('bonds')
def find_hydrogen_donors(hydrogens):
"""Returns the donor atom for each hydrogen
Parameters
----------
hydrogens : AtomGroup
the hydrogens that will form hydrogen bonds
Returns
-------
donors : AtomGroup
the donor atom for each hydrogen, found via bond information
.. versionadded:: 0.20.0
"""
return sum(h.bonded_atoms[0] for h in hydrogens)
[docs]class HydrogenBondAutoCorrel(object):
"""Perform a time autocorrelation of the hydrogen bonds in the system.
Parameters
----------
universe : Universe
MDAnalysis Universe that all selections belong to
hydrogens : AtomGroup
AtomGroup of Hydrogens which can form hydrogen bonds
acceptors : AtomGroup
AtomGroup of all Acceptor atoms
donors : AtomGroup
The atoms which are connected to the hydrogens. This group
must be identical in length to the hydrogen group and matched,
ie hydrogens[0] is bonded to donors[0].
For water, this will mean a donor appears twice in this
group, once for each hydrogen.
bond_type : str
Which definition of hydrogen bond lifetime to consider, either
'continuous' or 'intermittent'.
exclusions : ndarray, optional
Indices of Hydrogen-Acceptor pairs to be excluded.
With nH and nA Hydrogens and Acceptors, a (nH x nA) array of distances
is calculated, *exclusions* is used as a mask on this array to exclude
some pairs.
angle_crit : float, optional
The angle (in degrees) which all bonds must be greater than [130.0]
dist_crit : float, optional
The maximum distance (in Angstroms) for a hydrogen bond [3.0]
sample_time : float, optional
The amount of time, in ps, that you wish to observe hydrogen
bonds for [100]
nruns : int, optional
The number of different start points within the trajectory
to use [1]
nsamples : int, optional
Within each run, the number of frames to analyse [50]
pbc : bool, optional
Whether to consider periodic boundaries in calculations [``True``]
..versionchanged: 1.0.0
``save_results()`` method was removed. You can instead use ``np.savez()``
on :attr:`HydrogenBondAutoCorrel.solution['time']` and
:attr:`HydrogenBondAutoCorrel.solution['results']` instead.
"""
def __init__(self, universe,
hydrogens=None, acceptors=None, donors=None,
bond_type=None,
exclusions=None,
angle_crit=130.0, dist_crit=3.0, # geometric criteria
sample_time=100, # expected length of the decay in ps
time_cut=None, # cutoff time for intermittent hbonds
nruns=1, # number of times to iterate through the trajectory
nsamples=50, # number of different points to sample in a run
pbc=True):
#warnings.warn("This class is deprecated, use analysis.hbonds.HydrogenBondAnalysis "
# "which has .autocorrelation function",
# category=DeprecationWarning)
self.u = universe
# check that slicing is possible
try:
self.u.trajectory[0]
except Exception:
raise_from(ValueError("Trajectory must support slicing"), None)
self.h = hydrogens
self.a = acceptors
self.d = donors
if not len(self.h) == len(self.d):
raise ValueError("Donors and Hydrogen groups must be identical "
"length. Try using `find_hydrogen_donors`.")
if exclusions is not None:
if len(exclusions[0]) != len(exclusions[1]):
raise ValueError(
"'exclusion' must be two arrays of identical length")
self.exclusions = np.column_stack((exclusions[0], exclusions[1])).astype(np.intp)
else:
self.exclusions = None
self.bond_type = bond_type
if self.bond_type not in ['continuous', 'intermittent']:
raise ValueError(
"bond_type must be either 'continuous' or 'intermittent'")
self.a_crit = np.deg2rad(angle_crit)
self.d_crit = dist_crit
self.pbc = pbc
self.sample_time = sample_time
self.nruns = nruns
self.nsamples = nsamples
self._slice_traj(sample_time)
self.time_cut = time_cut
self.solution = {
'results': None, # Raw results
'time': None, # Time axis of raw results
'fit': None, # coefficients for fit
'tau': None, # integral of exponential fit
'estimate': None # y values of fit against time
}
def _slice_traj(self, sample_time):
"""Set up start and end points in the trajectory for the
different passes
"""
dt = self.u.trajectory.dt # frame step size in time
req_frames = int(sample_time / dt) # the number of frames required
n_frames = len(self.u.trajectory)
if req_frames > n_frames:
warnings.warn("Number of required frames ({}) greater than the"
" number of frames in trajectory ({})"
.format(req_frames, n_frames), RuntimeWarning)
numruns = self.nruns
if numruns > n_frames:
numruns = n_frames
warnings.warn("Number of runs ({}) greater than the number of"
" frames in trajectory ({})"
.format(self.nruns, n_frames), RuntimeWarning)
self._starts = np.arange(0, n_frames, n_frames / numruns, dtype=int)
# limit stop points using clip
self._stops = np.clip(self._starts + req_frames, 0, n_frames)
self._skip = req_frames // self.nsamples
if self._skip == 0: # If nsamples > req_frames
warnings.warn("Desired number of sample points too high, using {0}"
.format(req_frames), RuntimeWarning)
self._skip = 1
[docs] def run(self, force=False):
"""Run all the required passes
Parameters
----------
force : bool, optional
Will overwrite previous results if they exist
"""
# if results exist, don't waste any time
if self.solution['results'] is not None and not force:
return
master_results = np.zeros_like(np.arange(self._starts[0],
self._stops[0],
self._skip),
dtype=np.float32)
# for normalising later
counter = np.zeros_like(master_results, dtype=np.float32)
for i, (start, stop) in ProgressBar(enumerate(zip(self._starts,
self._stops)), total=self.nruns,
desc="Performing run"):
# needed else trj seek thinks a np.int64 isn't an int?
results = self._single_run(int(start), int(stop))
nresults = len(results)
if nresults == len(master_results):
master_results += results
counter += 1.0
else:
master_results[:nresults] += results
counter[:nresults] += 1.0
master_results /= counter
self.solution['time'] = np.arange(
len(master_results),
dtype=np.float32) * self.u.trajectory.dt * self._skip
self.solution['results'] = master_results
def _single_run(self, start, stop):
"""Perform a single pass of the trajectory"""
self.u.trajectory[start]
# Calculate partners at t=0
box = self.u.dimensions if self.pbc else None
# 2d array of all distances
pair = capped_distance(self.h.positions, self.a.positions, max_cutoff=self.d_crit, box=box,
return_distances=False)
if self.exclusions is not None:
pair = pair[~ _in2d(pair, self.exclusions)]
hidx, aidx = np.transpose(pair)
a = calc_angles(self.d.positions[hidx], self.h.positions[hidx],
self.a.positions[aidx], box=box)
# from amongst those, who also satisfiess angle crit
idx2 = np.where(a > self.a_crit)
hidx = hidx[idx2]
aidx = aidx[idx2]
nbonds = len(hidx) # number of hbonds at t=0
results = np.zeros_like(np.arange(start, stop, self._skip),
dtype=np.float32)
if self.time_cut:
# counter for time criteria
count = np.zeros(nbonds, dtype=np.float64)
for i, ts in enumerate(self.u.trajectory[start:stop:self._skip]):
box = self.u.dimensions if self.pbc else None
d = calc_bonds(self.h.positions[hidx], self.a.positions[aidx],
box=box)
a = calc_angles(self.d.positions[hidx], self.h.positions[hidx],
self.a.positions[aidx], box=box)
winners = (d < self.d_crit) & (a > self.a_crit)
results[i] = winners.sum()
if self.bond_type == 'continuous':
# Remove losers for continuous definition
hidx = hidx[np.where(winners)]
aidx = aidx[np.where(winners)]
elif self.bond_type == 'intermittent':
if self.time_cut:
# Add to counter of where losers are
count[~ winners] += self._skip * self.u.trajectory.dt
count[winners] = 0 # Reset timer for winners
# Remove if you've lost too many times
# New arrays contain everything but removals
hidx = hidx[count < self.time_cut]
aidx = aidx[count < self.time_cut]
count = count[count < self.time_cut]
else:
pass
if len(hidx) == 0: # Once everyone has lost, the fun stops
break
results /= nbonds
return results
[docs] def solve(self, p_guess=None):
"""Fit results to an multi exponential decay and integrate to find
characteristic time
Parameters
----------
p_guess : tuple of floats, optional
Initial guess for the leastsq fit, must match the shape of the
expected coefficients
Continuous defition results are fitted to a double exponential with
:func:`scipy.optimize.leastsq`, intermittent definition are fit to a
triple exponential.
The results of this fitting procedure are saved into the *fit*,
*tau* and *estimate* keywords in the solution dict.
- *fit* contains the coefficients, (A1, tau1, tau2) or
(A1, A2, tau1, tau2, tau3)
- *tau* contains the calculated lifetime in ps for the hydrogen
bonding
- *estimate* contains the estimate provided by the fit of the time
autocorrelation function
In addition, the output of the :func:`~scipy.optimize.leastsq` function
is saved into the solution dict
- *infodict*
- *mesg*
- *ier*
"""
if self.solution['results'] is None:
raise ValueError(
"Results have not been generated use, the run method first")
# Prevents an odd bug with leastsq where it expects
# double precision data sometimes...
time = self.solution['time'].astype(np.float64)
results = self.solution['results'].astype(np.float64)
def within_bounds(p):
"""Returns True/False if boundary conditions are met or not.
Uses length of p to detect whether it's handling continuous /
intermittent
Boundary conditions are:
0 < A_x < 1
sum(A_x) < 1
0 < tau_x
"""
if len(p) == 3:
A1, tau1, tau2 = p
return (A1 > 0.0) & (A1 < 1.0) & \
(tau1 > 0.0) & (tau2 > 0.0)
elif len(p) == 5:
A1, A2, tau1, tau2, tau3 = p
return (A1 > 0.0) & (A1 < 1.0) & (A2 > 0.0) & \
(A2 < 1.0) & ((A1 + A2) < 1.0) & \
(tau1 > 0.0) & (tau2 > 0.0) & (tau3 > 0.0)
def err(p, x, y):
"""Custom residual function, returns real residual if all
boundaries are met, else returns a large number to trick the
leastsq algorithm
"""
if within_bounds(p):
return y - self._my_solve(x, *p)
else:
return np.full_like(y, 100000)
def double(x, A1, tau1, tau2):
""" Sum of two exponential functions """
A2 = 1 - A1
return A1 * np.exp(-x / tau1) + A2 * np.exp(-x / tau2)
def triple(x, A1, A2, tau1, tau2, tau3):
""" Sum of three exponential functions """
A3 = 1 - (A1 + A2)
return A1 * np.exp(-x / tau1) + A2 * np.exp(-x / tau2) + A3 * np.exp(-x / tau3)
if self.bond_type == 'continuous':
self._my_solve = double
if p_guess is None:
p_guess = (0.5, 10 * self.sample_time, self.sample_time)
p, cov, infodict, mesg, ier = scipy.optimize.leastsq(
err, p_guess, args=(time, results), full_output=True)
self.solution['fit'] = p
A1, tau1, tau2 = p
A2 = 1 - A1
self.solution['tau'] = A1 * tau1 + A2 * tau2
else:
self._my_solve = triple
if p_guess is None:
p_guess = (0.33, 0.33, 10 * self.sample_time,
self.sample_time, 0.1 * self.sample_time)
p, cov, infodict, mesg, ier = scipy.optimize.leastsq(
err, p_guess, args=(time, results), full_output=True)
self.solution['fit'] = p
A1, A2, tau1, tau2, tau3 = p
A3 = 1 - A1 - A2
self.solution['tau'] = A1 * tau1 + A2 * tau2 + A3 * tau3
self.solution['infodict'] = infodict
self.solution['mesg'] = mesg
self.solution['ier'] = ier
if ier in [1, 2, 3, 4]: # solution found if ier is one of these values
self.solution['estimate'] = self._my_solve(
self.solution['time'], *p)
else:
warnings.warn("Solution to results not found", RuntimeWarning)
def __repr__(self):
return ("<MDAnalysis HydrogenBondAutoCorrel analysis measuring the "
"{btype} lifetime of {n} different hydrogens>"
"".format(btype=self.bond_type, n=len(self.h)))