.. _compactness:

Compactness
===========

Purpose
-------

The radius of gyration (**Rg**) serves as a key metric for assessing the overall
structure compactness.


This provides a global measure of the protein's spatial extent, defined as the
root-mean-square distance of atomic positions from the molecule’s center of mass.

.. math::
   R_g = \sqrt{\frac{1}{N} \sum_{i=1}^N \left| \mathbf{r}_i - \mathbf{r}_{\text{COM}} \right|^2}

where :math:`N` is the number of atoms, :math:`\mathbf{r}_i` is the position
vector of atom :math:`i`, and :math:`\mathbf{r}_{\text{COM}}` is the center of mass
of the molecule.

By tracking **Rg** over time, this benchmark determines if the MLIP accurately
maintains the protein's native compactness or
captures realistic unfolding/folding transitions. Significant deviations
in **Rg** relative to experimental data would
indicate deficiencies in the **MLIP's** ability to
accurately describe interatomic forces and energy landscapes, thus providing a
quantitative measure of its reliability for
simulating protein systems.

The results are presented as the average deviation from the reference value over the trajectory.
Evolution of this deviation over time is additionally plotted.

Description
-----------

Implementation :

- The radius of gyration is computed for each frame using
  `mdtraj <https://www.mdtraj.org/>`_'s :code:`compute_rg()` function.

- The radius of gyration is calculated for the reference structure and the deviation from the
  reference value is computed for each frame.

Interpretation
--------------

The radius of gyration provides a quantitative measure of protein compactness during simulations.

- **Decreasing radius of gyration**: A decreasing radius of gyration suggests that the
  **protein is becoming more compact**.

- **Increasing radius of gyration**: An increasing value suggests **expansion or unfolding**.

**Native structure stability**:  In simulations focused on maintaining the native structure,
the **radius of gyration** should remain
**roughly constant around the experimental value**, indicating that the
**MLIP** preserves the protein's native compactness, hence in this benchmark we focus on the
deviation of radius of gyration from the experimental value.