7.1.4. Radius of gyration¶

The radius of gyration measures the compactness of a protein structure.

\[R_\mathrm{gyr}^2 = \frac{1}{M}\sum_{i=1}^{N} m_i(\mathbf{r}_i - \mathbf{R})^2\]

where \(M = \sum_{i=1}^{N} m_i\) is the total mass and \(\mathbf{R} = N^{-1}\sum_{i=1}^{N} \mathbf{r}_i\) is the center of mass of the protein consisting of \(N\) atoms.

The Gromacs tool gmx gyrate can be used to compute the radius of gyration for the whole protein (using the pre-defined “Protein” index group)

mkdir analysis/rgyr && cd analysis/rgyr
echo Protein | gmx gyrate -s ../../MD/md.tpr -f ../../MD/md.xtc -o gyrate.xvg

and the resulting time series in file gyrate.xvg can be plotted [1].

Timeseries of the radius of gyration computed for the whole protein.

(In principle, \(R_\mathrm{gyr}\) can indicate changes in conformation but the simulation time in our test trajectory is too short to reveal the large conformational transition that AdK can undergo [Beckstein2009].)

Footnotes

[1]

To plot in Python, make sure to not write xvg legend information to the output file (using the -xvg none flag)

echo Protein | \
    gmx gyrate -s ../../MD/md.tpr -f ../../MD/md.xtc -o gyrate.xvg -xvg none

so that you can easily read the data with numpy.loadtxt():

import matplotlib.pyplot as plt
import numpy

t,data,x,y,z = numpy.loadtxt("gyrate.xvg", unpack=True)

fig = plt.figure(figsize=(5,2.5))
ax = fig.add_subplot(111)
fig.subplots_adjust(bottom=0.2)

ax.fill_between(t,data, color="magenta", linestyle="-", alpha=0.1)
ax.plot(t,data, color="magenta", linestyle="-")

ax.set_xlabel("time $t$ (ps)")
ax.set_ylabel(r"protein $R_\mathrm{gyr}$ (nm)")

fig.savefig("rgyr.png", dpi=300)
fig.savefig("rgyr.svg")
fig.savefig("rgyr.pdf")