# Computing the Gromov-Wasserstein Distance between Two Surface Meshes Using Optimal Transport
## 5 Computational Experiments
### Shape Similarity: Morphodynamics of Protein Structure Surfaces  
Different experimental techniques lead to different representations of protein structures. For example, high-resolution X-ray crystallography and NMR techniques derive models of proteins that include all their atoms and that are accurate at the Angstrom level. Recent progress on cryo electron microscopy (EM) can now often also reach atomic resolution. However, the difficulty of maintaining the integrity of stable complexes of interest on the various types of grids necessary to mount the sample in thin ice makes it often desirable to resort to negative staining techniques, in which case the resolution is much lower (typically 10-25 Angstroms). A low-resolution model derived from EM techniques is often represented as a density map, namely a shape characterized by its surface. Such a model is often available long before its high-resolution counterpart, as EM techniques are usually easier, faster, and cheaper to implement. It is therefore of interest to develop methods that can analyze the geometry of a protein directly from its EM density map. Such methods should generate information similar to those derived from methods that work directly on the high-resolution model of the protein structure. We consider here the problem of comparing the geometry of two protein structures using the GW distance between the surfaces of their density maps. To assess if this distance obtained from low resolution models of the protein structures mimic what could be found from high-resolution models, we compared it with the cRMS distance between atomistic models of the same proteins. We performed these tests on the protein calmodulin. Results are shown in Figure 5.  
Calmodulin is a calcium binding protein that is found in all eukaryotic cells. Its structure looks like a dumbbell, with two small domains separated by a linker region. It is the flexibility of this linker that defines the ability of calmodulin to bind to a wide range of ligands [45]. We considered two conformations for calmodulin, a conformation in the absence of a ligand (referred to as the apo or ligand-free conformation) and a conformation in the presence of a substrate (referred to as holo ligand-bound conformation, where we use interchangeably the terms ligand and substrate to indicate a molecule that binds to calmodulin). Those conformations were found in the database of protein structure, the PDB [46], with codes 1 CLL and 1A29, respectively. We built a trajectory between these two conformations. This trajectory is designed to mimic the structural transition that results from the binding of the ligand. We used the program MinActionPath who is designed to generate the most probable trajectory between the two conformers, namely the one with minimal action (for details, see [44]). The trajectory was sampled over 51 conformations, each represented with all the atoms of calmodulin. We then computed the distances between any two of these conformations in two different ways. First, we  
Figure 5: **Analyzing the dynamics of the conformational transition of calmodulin using coarse and high resolution models of the protein. ** We built a trajectory including 51 conformations between the apo (i.e., ligand-free) structure and an holo (i.e., with a ligand-bound) structure of the protein calmodulin using the program MinActionPath [44]. The transition between those two conformations leads to significant changes in the structure, as illustrated with the models of the structures shown below the horizontal axis. For all those 51 conformations, we computed their cRMS distances to the apo structure (structure number 0). These cRMS values are plotted versus the conformation number as a red solid line. In parallel, we plot the GW distance between the surfaces representing the same conformers and the surface of the apo protein as blue dots. The cRMS values and corresponding GW values exhibit a high correlation (0.985).