# Fractal Aspects of Human S100 Protein Structures
## 2 Materials and Methods  
In this study, human S100 proteins containing the EF-Hand motif and possessing three-dimensional dimer structures determined using crystallographic and/or NMR techniques were considered: S100A1, S100A2, S100A3, S100A4, S100A5, S100A6, S100A7, S100A8, S100A9, D100A10, S100A11, S100A12, S100A13, S100A14, S100A16, S100B, and S100P (Supplementary Table S1). As dimerization is essential for the functional roles of S100 proteins, only structures containing dimers have been considered in the present study. For S100A13, structures having the PDB IDs 2 KI4 and 2 KI6 are perfectly similar (Supplementary Table S2) and we only considered the structural file 2 KI4 in our analysis. The sequences of these proteins were extracted from the UniProt database [40] and their spatial structures were extracted from the Protein Data Bank [11]. In the case of NMR structures, only model 1 was analyzed for each dimer structure, as this model is often considered more accurate because it has the lowest overall energy or best satisfies the experimental constraints. The UniProt and PDB IDs of the investigated proteins are shown in Supplementary Table S1.  
The Clustal Omega tool [41] was considered to perform multiple sequence alignment of the investigated proteins and the Chimera tool [12] was used for structure analysis and superimposition.  
The mass fractal dimension, Dm, and surface fractal dimension, Ds, associated with each protein considered were calculated using the methods described by [PERSON] [21]. The mass of a protein (M), or its amino acid number (N) scales with its mean radius, R, as  
\[M\,\sim\,N\,\sim\,R^{D_{m}}\]  
with Dm being the mass fractal dimension. Consequently, the mass fractal dimension may be computed from the slope of the line obtained when plotting the number of amino acids or the molecular mass versus the mean radius in a double logarithmical plot. To compute the mean radius of every investigated protein, the Protein Dipole Moments Server ([[https://dipole.proteopedia.org/--accessed](https://dipole.proteopedia.org/--accessed)]([https://dipole.proteopedia.org/--accessed](https://dipole.proteopedia.org/--accessed)) on 2 August 2024) [42] was considered. This server has been also used to assess the overall conformation of the investigated structures.  
Computation of the accessible surface area (ASA) of the protein involves the use of a sphere with a given radius, R, which is rolled onto the surface of the protein maintaining contact with the van der Waals surface and gives  
\[ASA=N(R)R^{2}\]where N(R) is the number of spheres required to cover the surface. As the radius is higher, fewer spheres are required to cover the surface  
\[N(R)\,\sim R^{-D_{s}}\]  
resulting in  
\[ASA=R^{2-D_{s}}\]  
where D\({}_{\text{s}}\) is the fractal dimension of the protein surface that can be further computed from the slope of the line obtained when plotting ASA versus R in a double logarithmical plot [21]. For the investigated proteins, the accessible surface areas for spheres with radii of 1.0, 1.4, 1.8, 2.2, 2.6, and 3.0 A, respectively, were computed using GetArea online tool ([[https://curie.utmb.edu/getarea.html--accessed](https://curie.utmb.edu/getarea.html--accessed)]([https://curie.utmb.edu/getarea.html--accessed](https://curie.utmb.edu/getarea.html--accessed)) on 26 July 2024) [43]. This tool has been also used to compute the apolar surface areas of investigated proteins, the sphere radius used in this case being 1.4 A.