# Fractal Aspects of Human S100 Protein Structures
## 4 Discussion
A cysteine-to-serine substitution results in the loss of the thiol group, which is critical for forming disulfide bonds. The absence of these bonds can lead to changes in the protein's tertiary or quaternary structure, making it less stable or altering its conformation. Also, these mutations in S100A4 could have profound effects on the structural integrity, calcium-binding properties, and protein-protein interactions of S100A4. These changes could result in altered cellular functions, particularly in processes like cell motility, invasion, and metastasis, making them potentially relevant for understanding cancer progression. For the S100A9 protein, the surface fractal dimension obtained for the CYSSSER mutant (PDB ID 5l8N), 2.282, decreases to 2.2510 for the CYS3 SER and MET67 PHE mutant structure (PDB ID 7U15). Methionine is a flexible, non-polar amino acid with a sulfur-containing side chain, while phenylalanine is a bulkier, aromatic amino acid. MET67 PHE substitution, due to the aromatic ring of phenylalanine, introduces a more rigid structure and could have a significant impact on the protein's structure, dimerization with S100A8, and its role in calcium and zinc binding [54]. These changes are likely to affect its inflammatory signaling and antimicrobial functions, with potential consequences for diseases involving chronic inflammation, autoimmune disorders, and cancer. It may be noticed from Table 1 that, in the case of the S100B protein, the change in the pH also affects the surface characteristics as the surface fractal dimension slightly decreases from 2.296 at pH 6.5 (PDB ID 3 DOY) to 2.234 at pH 10 (PDB ID 3d10). Protein surface charges are determined by the ionizable amino acid residues that are exposed on the protein's surface. These charges depend on the pH of the environment, which affects the protonation or deprotonation of these residues. This highlights that the surface fractal dimension of proteins can be used to assess the effects of the characteristics of the environment in which the protein is found, particularly pH, as pH affects protein conformation, interactions, and aggregation behavior, which in turn modifies the surface complexity of the protein structures. Changes in the values of the surface fractal dimensions are usually accompanied by changes in polar and apolar surface areas. There is no direct correlation between polar/apolar surface area and surface fractal dimension, but it is reasonable to expect that increased surface complexity and polar character can influence each other in specific contexts. A protein with a complex surface may have a larger number of polar residues exposed to the solvent, potentially increasing the polar surface area, but a complex surface could also result in increased exposure of hydrophobic residues, small pockets on the surface might expose regions that would otherwise be buried in a smoother structure. Changes in polar/apolar surface area may affect protein folding, as hydrophobic residues tend to be buried inside the protein to minimize exposure to the aqueous environment, and are also important for understanding how proteins interact with hydrophobic/hydrophilic environments. This underlines that the polar/apolar surface area and surface fractal dimension of proteins are measures that provide insights into protein structure, surface complexity, and behavior in biological systems, but they describe different aspects of protein surfaces. Changes in surface fractal dimension can directly affect protein function. These structural changes of S100 proteins are crucial for their role as calcium sensors and signal transducers, modulating interactions with other proteins based on calcium levels. By analyzing how the surface fractal dimension changes due to different factors, details can be obtained about the nature of the binding interaction, such as its strength, the degree of induced conformational change, and potential biological implications.