# Fractal Aspects of Human S100 Protein Structures
## 4 Discussion  
The mass fractal dimension obtained from the investigated S100 proteins is D\({}_{\rm m}\) = 1.54, this value being obtained for both solution and crystal structures. This result is not unexpected because the fractal dimension primarily reflects how mass is distributed at different scales, being less sensitive to the fine details of flexibility or the ordered nature of crystals. Solution structures capture a more dynamic and flexible state, but they maintain a fractal-like distribution of mass at larger scales. Crystal structures capture a more rigid and static form of the proteins, but the internal fold complexity is similar and the overall fractal nature of molecular mass distribution in proteins remains similar. It is known that for fully extended polymers D\({}_{\rm m}\) = 1.67, and for ideal compact polymers, D\({}_{\rm m}\) = 2 [21]. Proteins are complex macromolecules that fold into specific three-dimensional structures dictated by their amino acid sequences. Due to this folding process, proteins do not behave as ideal polymers and their scaling properties differ from those of ideal polymers. The value of the mass fractal dimension obtained for the dimers of S100 proteins suggests that they have extended global conformations in good correlation with outcomes of the Protein Dipole Moments Server for the great majority of investigated proteins, as presented in Table 1. This extended conformation results from their EF-hand calcium-binding motifs, dimerization, and their ability to undergo calcium-induced conformational changes [44, 45]. These structural properties allow S100 proteins to engage in diverse biological processes such as calcium signaling, immune response, and protein regulation, making the extended shape critical for their functional versatility. In another study regarding the mass fractal dimensions of CaBPs containing EF-hand regions, the mass fractal dimensions for CaBPs with extended structures were calculated as D\({}_{\rm m}\) = 1.38, and D\({}_{\rm m}\) = 1.43 for CaBPs with compact structures [25]. The difference between the value obtained in this study when compared to that obtained in the previous one could be explained by the fact that in the present study, the dimers of S100 proteins are considered, whereas the previous study involved only monomers of other families of CaBPs. The values of the surface fractal dimensions of S100 proteins obtained in this study, D\({}_{\rm s}\) = 2.35 \(\pm\) 0.09 for solution structures and D\({}_{\rm s}\) = 2.23 \(\pm\) 0.05 for crystal structures, are quite similar to the values of surface fractal dimension obtained for the other extended CaBPs presented in the study of [PERSON] and coworkers [25]. The surface fractal dimension is related to the roughness or complexity of the surface of a structure, and a higher surface fractal dimension indicates a more irregular and complex surface. When comparing protein structures obtained from NMR and X-ray crystallography, NMR structures generally have larger surface fractal dimensions than crystallographic structures, due to the nature of the environments in which these structures are determined [46]. In solution, proteins are in a dynamic, flexible state, leading to more irregular surface topologies.