1 Dihedral principal component analysis
2 -------------------------------------
4 | :ref:`gmx angle <gmx angle>`, :ref:`gmx covar <gmx covar>`,
5 :ref:`gmx anaeig <gmx anaeig>`
6 | Principal component analysis can be performed in dihedral
7 space \ :ref:`172 <refMu2005a>` using |Gromacs|. You start by defining the
8 dihedral angles of interest in an index file, either using
9 :ref:`gmx mk_angndx <gmx mk_angndx>` or otherwise. Then you use the
10 :ref:`gmx angle <gmx angle>` program with the ``-or`` flag to
11 produce a new :ref:`trr` file containing the cosine and sine
12 of each dihedral angle in two coordinates, respectively. That is, in
13 the :ref:`trr` file you will have a series of numbers
14 corresponding to: cos(\ :math:`\phi_1`), sin(\ :math:`\phi_1`),
15 cos(\ :math:`\phi_2`), sin(\ :math:`\phi_2`), ...,
16 cos(\ :math:`\phi_n`), sin(\ :math:`\phi_n`), and the array is padded
17 with zeros, if necessary. Then you can use this :ref:`trr`
18 file as input for the :ref:`gmx covar <gmx covar>` program and perform
19 principal component analysis as usual. For this to work you will need
20 to generate a reference file (:ref:`tpr`,
21 :ref:`gro`, :ref:`pdb` etc.) containing the same
22 number of “atoms” as the new :ref:`trr` file, that is for
23 :math:`n` dihedrals you need 2\ :math:`n`/3 atoms (rounded up if not
24 an integer number). You should use the ``-nofit`` option
25 for :ref:`gmx covar <gmx covar>` since the coordinates in the dummy
26 reference file do not correspond in any way to the information in the
27 :ref:`trr` file. Analysis of the results is done using
28 :ref:`gmx anaeig <gmx anaeig>`.
33 | :ref:`gmx hbond <gmx hbond>`
34 | The program :ref:`gmx hbond <gmx hbond>`
35 analyzes the *hydrogen bonds* (H-bonds) between all possible donors D
36 and acceptors A. To determine if an H-bond exists, a geometrical
37 criterion is used, see also :numref:`Fig. %s <fig-hbond>`:
39 .. math:: \begin{array}{rclcl}
40 r & \leq & r_{HB} & = & 0.35~\mbox{nm} \\
41 \alpha & \leq & \alpha_{HB} & = & 30^o \\
43 :label: eqnhbondgeomtric
47 .. figure:: plots/hbond.*
50 Geometrical Hydrogen bond criterion.
52 The value of :math:`r_{HB} = 0.35 \mathrm{nm}` corresponds to the first minimum
53 of the RDF of SPC water (see also :numref:`Fig. %s <fig-hbondinsert>`).
55 The program :ref:`gmx hbond <gmx hbond>` analyzes all hydrogen bonds
56 existing between two groups of atoms (which must be either identical or
57 non-overlapping) or in specified donor-hydrogen-acceptor triplets, in
62 .. figure:: plots/hbond-insert.*
65 Insertion of water into an H-bond. (1) Normal H-bond between two
66 residues. (2) H-bonding bridge via a water molecule.
68 - Donor-Acceptor distance (:math:`r`) distribution of all H-bonds
70 - Hydrogen-Donor-Acceptor angle (:math:`\alpha`) distribution of all
73 - The total number of H-bonds in each time frame
75 - The number of H-bonds in time between residues, divided into groups
76 :math:`n`-:math:`n`\ +\ :math:`i` where :math:`n` and
77 :math:`n`\ +\ :math:`i` stand for residue numbers and :math:`i` goes
78 from 0 to 6. The group for :math:`i=6` also includes all H-bonds for
79 :math:`i>6`. These groups include the
80 :math:`n`-:math:`n`\ +\ :math:`3`, :math:`n`-:math:`n`\ +\ :math:`4`
81 and :math:`n`-:math:`n`\ +\ :math:`5` H-bonds, which provide a
82 measure for the formation of :math:`\alpha`-helices or
83 :math:`\beta`-turns or strands.
85 - The lifetime of the H-bonds is calculated from the average over all
86 autocorrelation functions of the existence functions (either 0 or 1)
89 .. math:: C(\tau) ~=~ \langle s_i(t)~s_i (t + \tau) \rangle
92 - with :math:`s_i(t) = \{0,1\}` for H-bond :math:`i` at time
93 :math:`t`. The integral of :math:`C(\tau)` gives a rough estimate of
94 the average H-bond lifetime :math:`\tau_{HB}`:
96 .. math:: \tau_{HB} ~=~ \int_{0}^{\infty} C(\tau) d\tau
99 - Both the integral and the complete autocorrelation function
100 :math:`C(\tau)` will be output, so that more sophisticated analysis
101 (*e.g.* using multi-exponential fits) can be used to get better
102 estimates for :math:`\tau_{HB}`. A more complete analysis is given in
103 ref. \ :ref:`173 <refSpoel2006b>`; one of the more fancy option is the Luzar
104 and Chandler analysis of hydrogen bond kinetics \ :ref:`174 <refLuzar96b>`, :ref:`175 <refLuzar2000a>`.
106 - An H-bond existence map can be generated of dimensions
107 *# H-bonds*\ :math:`\times`\ *# frames*. The ordering is identical to
108 the index file (see below), but reversed, meaning that the last
109 triplet in the index file corresponds to the first row of the
112 - Index groups are output containing the analyzed groups, all
113 donor-hydrogen atom pairs and acceptor atoms in these groups,
114 donor-hydrogen-acceptor triplets involved in hydrogen bonds between
115 the analyzed groups and all solvent atoms involved in insertion.