2.1.1. Computation of the SWIM descriptor

The Solvent Excluded Surface (SES) of the all atoms protein structure (Fig. 1a) retrieved from the PDB [3] is computed using EDTSurf [53] (Fig. 1b). The Wave Kernel Signature (WKS) is then computed on the resulting 3D mesh M [2] (Fig. 1c). For each point of the surface of the 3D mesh, a vector of size N, representing the WKS descriptor is computed. This descriptor, based on the eigenvalues of the Laplace-Beltrami operator, has the property of invariance to isometry and is robust to perturbations [2].

Diagram for the generation of a SWIM. The Solvent Excluded Surface is computed using EDTSurf [53] (a-b). In a second step, the Wave Kernel Signature (WKS) [2] is calculated for each point of the surface (c). The surface is then projected on a unit sphere [1] (d). The sphere is mapped onto the 2D plane [9] (e). The points of the map are interpolated (f) to form the final descriptor called SWIM (g).

On a second step, the 3D mesh is projected on a unit sphere S using the ITK algorithm [13] based on the method described by Angenent et al. [1] (Fig. 1d). The frame of the unit sphere is defined using a reference point on its surface called the pole. The pole of the unit sphere is selected arbitrarily, and the triangles of the mesh are projected while preserving their angles. This transformation is conformal and bijective. It is to note that, even if the distances and surface areas are not preserved in the projection, they are only modified by a scaling factor.

Then, the unit sphere is transformed onto the 2D plane using the two spherical coordinates of the angles (θ, ϕ) [8], [9] (Fig. 1e). A map of size (θmax−θmin)/δ,(ϕmax−ϕmin)/δ is created. θmax and θmin are the maximum and minimum values of θ and ϕmax and ϕmin are the maximum and minimum values of ϕ. δ is the step for dividing the sphere in areas, where each area is represented by a point on the map in the discrete plan. The value associated to each point on the map is the WKS descriptor represented by a vector (Fig. 2). These maps are called Surface Wave Interpolated Maps (SWIM).

WKS on the surface of one conformation of ubiquitin (5xbo_A_1) (top) and its corresponding SWIM (bottom) for the 1st (a), 50th (b) and 100th (c) value of the WKS.

The final step is the interpolation of each point of the map (Fig. 1f). The map is encoded as an image and the projection on the 2D plane is not filling each pixel with a value. This can create an imbalance if a map has large areas with no value, or if the neighborhood of a point of the map is considered for comparison. Therefore, for each pixel with no value, the three points on the map defining the triangle with the smallest area containing the pixel are used to interpolate the value of this pixel.

The main issue with this representation is the deformation in the neighborhood of the poles while passing from the unit sphere to the 2D plane. To handle this issue, we use a multiview approach where the pole axis is rotated by an angle α in the planes perpendicular to each of the three Cartesian axes of the unit sphere (Fig. 3). Then, a SWIM is created as mentioned above (Fig. 1g). We used a multiview approach to generate a set of seven SWIMs by using seven projections with a 2π3 rotation (Fig. 3). This approach offers an optimal balance between minimizing the impact of the initial arbitrary pole selection and maintaining the computational efficiency of the method. As the SWIMs are compared locally (see section 2.1.2 below), the multiview approach helps us avoid the arbitrary choice of the pole of the unit sphere at the second step of the workflow. This set of seven SWIMs is the final descriptor used for comparing protein surfaces.

Illustration of the multiview approach with a representation of one of the three rotations of the pole axis achieved in the planes perpendicular to the Cartesian axis of the Unit sphere (left panel). The corresponding SWIMs for the 50th value of the WKS with a rotation of 2π3 around a pole of the unit sphere of Ubiquitin (5xbo_A_1) are shown in the right panel.