# Migration of the 3T3 Cell with a Lamellipodium on Various Stiffness Substrates--Tensegrity Model
## 1 Introduction  
A single cell migration is an important component of many biological events. It may be initiated by number of various factors, such as mechanical [1] (mechanotaxis), electrical [2] (galvanotaxis), thermal [3] (therotaxis), topological [4, 5] (topotaxis) and chemical [6] (chemotaxis). Cell migration is dependent on many factors, among these the mechanical interaction between the adherent cell and the substrate. The particular mechanisms that govern how the various environmental factors influence the cell's biological responses are not well known. However, it is obvious that physical environmental changes in the cell's motility and morphology are influenced by the cell's internal mechanical balance [7]. It was shown that an adherent cell senses mechanical stimulus [8, 9, 10] and responds to it by keeping the cell's intracellular and extracellular forces in balance [11, 12]. Many studies have focused on developing a mechanism called \"mechanotransduction\". It is a process that allows a cell to sense mechanical loads and to transduce them into a biochemical signal. It has been shown that the motility of a cell is dependent on the substrate's stiffness [13, 14, 15]. [PERSON] [13] called this process a \"durotaxis\". This process is dependent on the assumption that a cell is capable of probing the stiffness of the substrate it is adherent to and respond to it by applying contractile forces. A number of theoretical [16] and numerical [17, 18, 19] models were developed to describe and understand the mechanical cellular responses. Numerical mechanical models based on the finite element analysis of tensegrity structures [20, 21] are often utilized to describe the cell's response to environmental mechanical stimuli. Modifications in the morphology and structure of the cell due to the external loads are important for various cell functions, as growth, proliferation, differentiation and motility [22, 23]. It was shown that the directional motility of a cell is a significant part of tumor metastasis and wound healing [24, 25]. The formation of focal adhesion and cell migration [26, 27] is dependent on the forces generated within the actin cytoskeleton, as these forces are dependent on the cell's interaction with the extracellular matrix. Through probing the environment, i.e., the substrate's stiffness, cells can guide their activities that will lead to changes in the morphologies and direction of migration. A recent study [28] showed that cells are capable of sensing and reacting to the substrate's stiffness. Conversely, the opposite results were obtained [29] for epithelial cells. They were more often extending into branches while on a low stiffness substrate than on a stiffer substrate. Models have been introduced to understand and describe cell behavior on the substrates with different stiffnesses [30; 31; 32; 33; 34; 35; 36; 37; 38]. These models provide an understanding of the effects of the substrate's physical properties on adherent cell morphologies and motility. Two main approaches are used to investigate the reaction of the cells to environmental changes: continuum [31; 32; 33; 34] and discrete approaches [31; 36; 37; 38]. In continuum mechanical models, the cells are treated as a continuum material. The continuum approach was successfully utilized to present a blood cell as a viscous liquid, while the cortex as a membrane [32]. It was used to study the small deformation characteristics of leukocytes [34]. It was also used to model the interaction of a single chondrocytes with the extracellular matrix [33]. The discrete models are based on the micro and nano-structural approach. They comprise a tensegrity model, an open-cell foam model, tensed cable network model, or a spectrin network model, to name a few. A cytoskeleton is a main structural component in micro and nano structural approaches when modeling the adherent cell [31].