# Finite Element Modeling of Cells Adhering to a Substrate: An Overview
## 1 Introduction  
### Theoretical Background (Principles) of Cell Adhesion and Mechanotransduction Mechanism  
The interaction between the cell and environment originating from cell-substrate contact is mediated by cell adhesion phenomena entailed in crucial cell mechanisms (e.g., motility and cellular life cycle phases from growth to differentiation) [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. The assembly of protein complexes called focal adhesions (FAs), deriving from early protein agglomerate at the lamellipodia-environment interface, allows cells to adhere to the underlying surface and carry on cellular activities triggered by cell-environment interaction [20; 21; 22; 23; 24; 25; 26; 27]. FA architecture consists of (i) transmembrane proteins (i.e., integrins), which link to extra-cellular matrix ligands, thus ensuring cell-substrate connection; and (ii) an intracellular protein structure composed of vinculin, paxillin, talin, adaptor protein p130 Cas, zyxin, vasodilitor-stimulated phosphoprotein, and focal adhesion kinase, which binds integrins to stress fibers, thus enabling the interaction between cytoskeleton and cellular membrane [28; 29; 30; 31; 32; 33; 34] (Figure 1). The cytoskeleton serves as a framework, offering mechanical support to the cell to uphold specific morphology and internal organization. Cytoskeleton structural components (i.e., F-actin bundles (F-abs), microtubules (MTs), and intermediate filaments (IFs)) are, in turn, connected to internal nuclear architecture through resprin and Sad1-UNC-84 homology (SUN) protein complexes (i.e., the linker of nucleoside and cytoskeleton (LINC) complexes) [35; 36]. These protein complexes are constituted by SUN proteins that span over the inner nuclear membrane (INM) and bind the perinuclear space (PS) to resprin isoforms that pass through the outer nuclear membrane (ONM) and link directly with F-actin and IFs and MTs via plectin and kinesin-1,respectively. On the nucleoskeleton side, SUN proteins connect to the nuclear lamina and interact with chromatin (Figure 2). Therefore, the interconnection between the cellular adhesion region and the nuclear domain allows environment mechanical cues to be detected by FAs at the contact interface and to be transmitted to the cell nucleus through the load-carrying cytoskeletal components in the form of stress waves traveling along cytoskeleton structures at a speed approximately 15 times faster than chemical diffusion and 28 times faster than motor-driven transport [20; 22]. This architecture ensures that mechanical information can be swiftly transferred to multiple locations within the cell, resulting in a cascade of biochemical signals. Force transmission towards the nuclear region leads to altered cellular behavior and functions in response to environment cues by determining the rearrangement of chromatin, the opening of nuclear pores, and inducing genetic and transcription programs [35]. In order to prevent the dissipation of mechanical energy caused by the transmission across totally soft cytoskeletal material, the information contained in mechanical stimuli must be carried to the nucleus along cytoskeletal preferential transferring channels (i.e., actin bundles). These channels are characterized by a prestressed state that results in stiffened  
Figure 1: Schematization of the focal adhesion structure (FAs). As shown, FAs have a multilayer architecture, allowing cells to interact with the extracellular matrix (ECM). Starting from the cell-substrate interface and proceeding towards the inner part of the cell, FA structure can be schematized into two main regions: a transmembrane domain and an intracellular one. The FA transmembrane domain consists of heterodimeric proteins (i.e., integrins (IT)), which span the plasma membrane and act as receptors to ensure a high affinity bind with ECM ligands.