# Migration of the 3T3 Cell with a Lamellipodium on Various Stiffness Substrates--Tensegrity Model
## 1 Introduction
A prestressed tensegrity network model was successfully used to describe the deformation of the adherent cell and compared with the values measured on cells in vivo and on isolated actin filaments [31]. The tensegrity-based architecture was introduced by [PERSON] in 1961 [35]. The approach of tensegrity is often used to describe how tissues and cells are constructed and respond to external excitation [36]. The tissues may act as a continuous structure at the macroscopic scale, however, they should be treated as discrete structural entities when viewed at the microscale [37]. Cells form attachments to the substrate and neighboring cells that distribute over the cell's surfaces discretely [38]. A tensegrity structure can easily represent these features. As it was mentioned above [1; 2; 3; 4; 5; 6], there are number of factors influencing a cell migration, the current work only examines the effect of the substrate's mechanical properties (stiffness) on cell migration. In this paper, a tensegrity structure was used to model the cell-substrate interaction to explain the dependency of the cell's motility on the stiffness of the substrate. During the cell's attachment to a substrate, the geometry of the cell changes similarly to the effect of the application of external loads to the cell membrane. The movement of the cell along a substrate is dependent on its internal elastic strain and interfacial energies. It is hypothesized here that a cell probes its environment and guides its directional movement depending on the underlying substrate stiffness. It is also hypothesized that a cell senses the stiffness of the substrate in a number of directions by extending the lamellipodium. When the cell finds a direction that would decrease its internal elastic strain energy, it chooses it as the preferred direction of the movement. It is also hypothesized that this direction corresponds to the direction of the higher stiffness of the substrate. Since the cell prefers to stay in the lowest possible energetic state [39; 40], it will select the migration direction that will lower its internal elastic energy. In other words, giving the choice of substrate stiffness, the cell will move and adhere to the stiffer area of the substrate since its internal elastic energy would be lower. One of the mechanisms that allows a cell to migrate is to use lamellipodia. The lamellipodia and filopodia are the structures extending from the leading edge of a cell (Figure 1). A thin protrusion filled with a network of actin is a lamellipodium. Filopodia may be described as'sensors' that cells use to probe their environment [41]. When a cell probes any substrate, the lamellipodia expands. When a cell encounters a soft substrate, the lamellipodia retracts causing the cell to change directions [1]. The experimental observation of the cell-substrate interaction was used to develop a tensegirty-based model. This allows to model the cell probing of the stiffness of a substrate. During this process, the internal elastic strain energy of the cell is changed. The cytoskeleton and lamellipodium of a cell were represented as a tensegirty structure with the prestress. Its internal elastic strain energy was evaluated by a finite element method for an array of the substrate's stiffness value.