Order parameters and statistics#

The mean order parameter of bond \(CH_j\) (i.e. the \(j^{th}\) C-H bond) is calculated using the standard formula:

\[\overline{S_{CH_j}} = \frac{1}{2} \left \langle 3cos^2(\theta) -1 \right \rangle\]

where \(\theta\) is the angle between the \(CH_j\) bond and the normal to the membrane (usually the z axis), \(\langle ... \rangle\) means averaging over molecules and frames. \(S_{CH}\) can be measured by NMR which is useful to validate simulation results, as largely described in the NMRlipids project.

The order parameter output of buildH (default name OP_buildH.out) looks like this:

# OP_name            resname atom1 atom2  OP_mean OP_stddev OP_stem
#--------------------------------------------------------------------
gamma1_1             POPC    C1    H11    0.01304  0.12090  0.01069
gamma1_2             POPC    C1    H12    0.00666  0.09279  0.00820
gamma1_3             POPC    C1    H13   -0.01531  0.09141  0.00808
[...]

Each line corresponds to a given CH. The 4 first columns contain the generic name, residue name, carbon and hydrogen names respectively. The other columns contains different statistics on order parameters (OP):

  • OP_mean, also written \(\overline{S_{CH_j}}\) as described above, is the mean OP of bond \(CH_j\) averaged over all lipids and all frames of the trajectory.

  • OP_stddev is the standard deviation of the OP over residues, we shall write it \(\sigma(S_{CH_j})\); first we average each OP of bond \(CH_j\) (e.g. the C-H of beta1) of residue \(i\) (i.e. lipid \(i\)) over the whole trajectory:

\[ \overline{S_{CH_j}(i)} = \frac{1}{nframes} \sum_{t=0}^{t=nframes} S_{CH_j}(i)(t) \]

where \(nframes\) is the total number of frames, then we calculate the standard deviation of those means over all residues:

\[ \sigma(S_{CH_j}) = \sqrt{ \frac{1}{nres} \sum_{i=1}^{i=nres} (\overline{S_{CH_j}(i)} - \overline{S_{CH_j}})^2 }\]

where \(nres\) is the total number of residues (i.e. lipids).

  • OP_stem is the standard error of the mean averaged in the same spirit, let’s call it \(err(S_{CH_j})\):

\[err(S_{CH_j}) = \frac{\sigma(S_{CH_j})}{\sqrt{nres}}\]