# 4.4.2. Hydrogen bond autocorrelation — `MDAnalysis.analysis.hydrogenbonds.hbond_autocorrel`

Author:

  Richard J. Gowers

Year:

  2014

Copyright:

  Lesser GNU Public License v2.1+

Added in version 0.9.0.

Changed in version 2.0.0: Module moved from `MDAnalysis.analysis.hbonds.hbond_autocorrel` to `MDAnalysis.analysis.hydrogenbonds.hbond_autocorrel`.

## 4.4.2.1. Description

Calculates the time autocorrelation function, \(C_x(t)\), for the hydrogen bonds in the selections passed to it. The population of hydrogen bonds at a given startpoint, \(t_0\), 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.

\[C_x(t) = \left \langle \frac{h_{ij}(t_0) h_{ij}(t_0 + t)}{h_{ij}(t_0)^2} \right\rangle\]

The subscript \(x\) 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, \(\tau_x\), can then be found via integration of this function

\[\tau_x = \int_0^\infty C_x(t) dt`\]

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.

\[C_x(t) = A_1 \exp( - t / \tau_1) + A_2 \exp( - t / \tau_2) [+ A_3 \exp( - t / \tau_3)]\]

Where the final pre expoential factor \(A_n\) is subject to the condition:

\[A_n = 1 - \sum\limits_{i=1}^{n-1} A_i\]

For further details see [1].

## 4.4.2.2. 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 `find_hydrogen_donors()` can be used to construct the donor AtomGroup

    import MDAnalysis as mda
    from MDAnalysis.analysis import hydrogenbonds
    from MDAnalysis.tests.datafiles import waterPSF, waterDCD
    u = mda.Universe(waterPSF, waterDCD)
    hydrogens = u.select_atoms('name H*')
    donors = hydrogenbonds.find_hydrogen_donors(hydrogens)

Note that this requires the Universe to have bond information. If this isn’t present in the topology file, the `MDAnalysis.core.groups.AtomGroup.guess_bonds()` method can be used as so

    import MDAnalysis as mda
    from MDAnalysis.analysis import hydrogenbonds
    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 = hydrogenbonds.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.

## 4.4.2.3. 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 `run()` method, fit, tau and estimate are filled after the `solve()` method has been used.

## 4.4.2.4. 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 hydrogenbonds
    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 = hydrogenbonds.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()

## 4.4.2.5. Functions and Classes

MDAnalysis.analysis.hydrogenbonds.hbond_autocorrel.find_hydrogen_donors(hydrogens)

  Returns the donor atom for each hydrogen

  Parameters:

    hydrogens (AtomGroup) – the hydrogens that will form hydrogen bonds

  Returns:

    donors – the donor atom for each hydrogen, found via bond information

  Return type:

    AtomGroup

  Added in version 0.20.0.

class MDAnalysis.analysis.hydrogenbonds.hbond_autocorrel.HydrogenBondAutoCorrel(universe, hydrogens=None, acceptors=None, donors=None, bond_type=None, exclusions=None, angle_crit=130.0, dist_crit=3.0, sample_time=100, time_cut=None, nruns=1, nsamples=50, pbc=True)

  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 `HydrogenBondAutoCorrel.solution['time']` and `HydrogenBondAutoCorrel.solution['results']` instead.

  run(force=False)

    Run all the required passes

    Parameters:

      force (bool, optional) – Will overwrite previous results if they exist

  solve(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 `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 `leastsq()` function is saved into the solution dict

    > - infodict

    > - mesg

    > - ier