Source code for MDAnalysis.analysis.hbonds.hbond_autocorrel

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Hydrogen bond autocorrelation --- :mod:`MDAnalysis.analysis.hbonds.hbond_autocorrel`

:Author: Richard J. Gowers
:Year: 2014
:Copyright: GNU Public License v3

.. versionadded:: 0.9.0


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

    :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),


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

  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
  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.


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

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,
              sample_time=2.0, nsamples=1000, nruns=20)
  # call run to gather results
  # attempt to fit results to exponential equation
  # 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)


.. autofunction:: find_hydrogen_donors

.. autoclass:: HydrogenBondAutoCorrel

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.core.groups import requires

from MDAnalysis.due import due, Doi
         description="Hydrogen bonding autocorrelation time",
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`.") self.exclusions = exclusions if self.exclusions: if not len(self.exclusions[0]) == len(self.exclusions[1]): raise ValueError( "'exclusion' must be two arrays of identical length") 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, d = capped_distance(self.h.positions, self.a.positions, max_cutoff=self.d_crit, box=box) if self.exclusions: # set to above dist crit to exclude exclude = np.column_stack((self.exclusions[0], self.exclusions[1])) pair = np.delete(pair, np.where(pair==exclude), 0) 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)))