**4. Conclusions**

In this chapter, we introduced our novel finite element cellular model to explore the mechanism behind collective cell migration using a simplified tissue model. This model includes a detailed mechano-chemical feedback loop, which takes into account of formation of focal adhesion and cell protrusion initiated by Rac signaling. In addition, our model incorporates the mechanical cue transmitting between the follower cell and the leader cell. We further examined the effects of cell-substrate contact and intercellular adhesions on collective cell migration.

An important result of this study is that we find the cell-substrate mechanics plays crucial role in guiding collective cell migration with higher persistence, more accurate direction, and better coordination between cell pairs (**Figure 5**). Previous *in vitro* study has shown that cells tend to have elongated shape on stiffer substrate while cells tend to have spherical shape on softer substrate [33]. This is compatible with our simulation (**Figure 4a** and **c**). We anticipate that our finite element cellular model can be applied to a broad of studies of cellular tissue problems.

#### **Appendix A: cell migration model**

In our model, cell migration is initiated and maintained by the protrusion force on the leading edge and the cell migration speed varies with the cell-substrate friction following [16].

#### **A.1 Cell-substrate depending on substrate stiffness**

The adhesion coefficient *Y*ð Þ *x*, *t* of a cell vertex *x* at time *t* and set to be proportional to the strength of focal adhesions [18]: *<sup>Y</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>n</sup>x*,*<sup>t</sup> <sup>n</sup>*<sup>0</sup> *EstYa*, where *nx*,*<sup>t</sup>* is the number of binding integrins at location *x* at *t*, *n*<sup>0</sup> is a normalizing constant number, *Est* is the stiffness of the substrate, *Ya* is the basic adhesion constant taken from [18]. In this way, the cell-substrate friction is related with the stiffness of the substrate.

#### **A.2 Cell protrusion depends on substrate stiffness**

In our model, there is a mechano-chemical pathway dictating the cell protrusion. The bound integrin initiates the activation of Rac which regulates the cell protrusion. At time *t*, the migration direction of the cell *C* is sampled from all the boundary vertice according to their Rac concentration. One vertex *v<sup>i</sup>* is stochastically selected with the probability P *Rac*ð Þ *v<sup>i</sup> i Rac*ð Þ *v<sup>i</sup>* . The outward unit normal vector *n v*ð Þ*<sup>i</sup>* of *v<sup>i</sup>* is chosen as the cell migration direction. Any vertex *v <sup>j</sup>* whose outward unit normal vector *n v <sup>j</sup>* � � is positively aligned with *n v*ð Þ*<sup>i</sup>* , is treated as leading edge vertex. The protrusion force *f <sup>a</sup>* is then applied on each leading edge vertex *v<sup>i</sup>* as *f <sup>a</sup>*ð Þ¼ *v<sup>i</sup> f <sup>a</sup>R*ð Þ *v<sup>i</sup> n v*ð Þ*<sup>i</sup>* , where *f <sup>a</sup>* is a constant, *R*ð Þ *v<sup>i</sup>* is the normalized Rac concentration at *vi*.

#### **A.3 Calibrating the cell protrusion parameter**

As shown in **Figure 6a**–**c**, the cell leading edge has higher level of bound integrin, along with higher level of Rac due to the effect of positive feedback loop. The Merlin expression is also mechano-dependent. As shown of the pair of cells in the green box of **Figure 6b**, after the right cell migrates, the stretch force on the cadherin spring between them make the Merlin delocate from the left cell. As a result, the left cell can express Rac to protrude following the right cell. If the pair of static cell are simply in contact (**Figure 6b**), the Merlin is expressed on both of them. Therefore, the Rac expression is inhibited. Both of the two cells do not

*A Dynamic Finite Element Cellular Model and Its Application on Cell Migration DOI: http://dx.doi.org/10.5772/intechopen.94181*

**Figure 6.**

*The cell protrusion depends on mechano-chemical process. (a–c) the spatial distribution of the normalized concentration of bound integrin, Merlin, and Rac. The black arrows indicate the migration direction. The pattern of Merlin expression depends on cell status. Green box in (b): The left cell follows the right one. Merlin is expressed only on the right cell; Orange box in (b): The two static cells are in contact. Merlin is expressed on both of them. (d) Cell migration speed of our simulation is consistent with the experimental observation [33, 34].*

protrude against each other. To fit our cell protrusion model to the in vitro data, we calibrate the parameter of *Est*0: when *Est*<sup>0</sup> ¼ 40*kPa*, the cell migration speed of our simulation has the best match with the in vitro studies [33, 34] (**Figure 6d**).
