# Migration of the 3T3 Cell with a Lamellipodium on Various Stiffness Substrates--Tensegrity Model
## 2 Materials and Methods  
### Tensegirty Model  
A tensegirty structure comprises of a network of connected elements each supporting compression or tension. They are arranged to provide a stable volume and shape in the space while loaded by external loads. The tensegirty structure (Figure 2) is a useful tool to describe the cell motion and morphology changes because it is based on the mechanical integrity and equilibrium through the coordination of the tensile actin filaments and compressive microtubules [42; 43]. The tensegirty model is constructed from the set of compression components inside of a net of tension components that keep the compressed members from each other. The tensile elements provide a load-supporting capability to the structure. Tensegirty models were used to describe the mechanical responses of various cells, like viruses and erythrocytes [36].  
The basic tensegirty model (Figure 2) consists of 30 elements. Six pre-compressed struts represent microtubule members and 24 pre-tensed cables represent the microfilaments.  
The current tensegirty structure (Figure 3) represents the cytoskeleton of a 3T3 cell, its nucleus, and lamellipodia. 3T3 cells are mouse fibroblast cells [43]. The 3T3 cells originated from a 1962 cell  
Figure 1: Lamellipodia and filopodia at the cell’s leading edge. [41]. The green line shows the cell’s adherent area, while the red lines represent a network of actin filaments.  
Figure 2: Cytoskeleton tensegirty model. (Solid bars represent pre-compressed microtubules; thin lines represent pre-tensed microfilaments).  
line [44]. In this model, a spring element is added between node 46 and 49 that will simulate the connection of a cell to the substrate. The spring element's stiffness varies to simulate the substrate stiffness so that the strain energy of the model can be found at each stiffness value.  
The intermediate filaments have been ignored since their influence on the cell's mechanical behavior becomes significant only at the high deformations of the cell (over 20%) [45; 46], but at the small deformation of the cell, there is only a slight increase in the reaction force when the intermediate filaments are added.  
A recent study found that the elastic energy of the cytoskeleton of a 3T3 cell is a dependent on the substrate stiffness it is attached to [47]. The current study aims to find the internal elastic strain energy of a cytoskeleton, its nucleus, and the lamellipodium of a 3T3 cell for a range of the substrate stiffness values.  
Node 3 is located at the coordinate system's origin and represents the point of focal adhesion. This links the substrate and cell and it is fixed in the current model. All other nodes can move in any direction to simulate the cell spreading; however, no rotation is allowed. The length of the cables and struts of this model are able to change due to the response to the applied prestress and deformation.  
The distance between the \"superior plane\" (the top X-Y plane) and the \"inferior plane\" (the bottom X-Y plane) represents the initial cell height of 8.7 \(\upmu\)m. The superior and interior planes were set to be parallel to the X-Y plane before the application of deformation.  
Three tensegrity structures are shown in Figure 3. The furthest to the left with the largest structure is the tensegrity model of the cytoskeleton and the nucleus of a 3T3 cell. The two smaller structures to the right represent the lamellipodium of the 3T3 cell. The spring element connecting nodes 46 and 49 represent substrate stiffness.