# Fractal Aspects of Human S100 Protein Structures
## 1 Introduction  
In living organisms, calcium is a cation that has both a structural and a functional role in most cellular processes. One of the most important roles of calcium is as a second messenger in a cell, a role that is dependent on the existence of calcium-binding proteins (CaBPs), proteins that have the ability to form a complex with calcium under various conditions. CaBPs have different cellular and tissue distributions and are involved in specific functions. These proteins typically contain specific domains or motifs that have a high affinity for calcium, allowing them to modulate intracellular calcium levels and mediate calcium signaling [1]. CaBPs can have both intracellular and extracellular localization. Intracellular CaBPs contain or lack the EF-hand structural domain (a helix-loop-helix structure), which is the most common calcium-binding (sometimes magnesium-binding) domain and binds ions with high affinity and specificity [2]. In the case of EF-hand CaBPs, their structures comprise two domains, with each domain containing two EF-hand motifs [3]. Intracellular CaBPs containing the EF-hand motif have been grouped into two functional categories [4]: (i) calcium buffer, proteins that bind calcium to transport or regulate its concentration; and (ii) calcium sensors, proteins that bind calcium to decode its signal by interacting with other target proteins. The two functional categories of EF-hand motif-containing CaBPs also differ in their overall structure, with calcium buffers typically presenting a compact spatial structure and calcium sensors typically exhibiting an extended dumbbell-like spatial structure [4, 5]. Data from the literature indicate that the overall extended or compact structures of EF-hand CaBPs depend on the protonation of several acidic residues [6] and the hydrophilicity of the residues linking the two domains [7]. S100 proteins are a large family of CaBPs containing EF-hand motifs and are involved in a wide range of intracellular and extracellular functions, playing critical roles in regulating cellular processes and disease pathology. Importantly, S100 proteins are expressed exclusively in vertebrates and have other functions complemented by their calcium-binding ability [8]. S100 proteins typically contain two EF-hand calcium-binding motifs. The first is called the pseudo-EF-hand motif, a loop that contains 14 residues that chelate calcium via backbone carbonyls, resulting in a weaker calcium-binding affinity. The second EF-hand calcium-binding motif contains a loop of 12 residues that bind calcium via sidechain carboxylates or carbonyls, resulting in a higher calcium-binding affinity [9]. Most S100 proteins form dimers, the dimerization being essential for their functional roles and interactions with target proteins [10]. Figure 1 presents the spatial structure of the dimer of S100A1 protein (model 1), retrieved from Protein Data Bank (PDB) [11] (PDB ID 2 LLS) and visualized using Chimera 1.16 software [12]. Fractals are geometric shapes exhibiting self-similarity across different scales, and this concept can be extended to biological macromolecules such as proteins. The concepts of fractal geometry can be applied to protein sequences and structures leading to a deeper understanding of their complexity and self-similar patterns. Data from the literature show that proteins have fractal aspects of their sequences and structures. Protein sequences can exhibit patterns in which certain motifs or domains repeat at different scales that can be considered self-similar structures because they occur at different locations in the sequence and contribute to the overall function and stability of the protein. There have been intensive studies on the correlation properties of protein sequences that have used both statistical methods (such as correlation function, random walk, and power spectrum) and nonlinear dynamics theory, which is a powerful method for identifying determinism or randomness of irregular systems.