**3. Results**

#### **3.1 Mechanics of cell-substrate is crucial to regulate collective cell migration**

#### *3.1.1 Morphology and migration pattern under different substrate stiffness*

We fist studied collective cell migration under the mechano-chemical mechanism. The trajectories of our simulation showed that cells migrate faster and more persistently on stiffer substrate (**Figure 3a** and **b**). This is compatible with the observed pattern from the *in vitro* study of collective cell migration (**Figure 3c** and **d**). Furthermore, the shape of cell also changes with different substrate stiffness. Cells

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

#### **Figure 3.**

*Cell migrating trajectories. (a–b) the migrating trajectory in our simulation using two substrate stiffness: 3 and 65 kPa. (c–d) the migrating trajectory from the in vitro study using the same substrate stiffness: 3 and 65 kPa [10]. The scale bar is 100 μm.*

adopted a more spherical shape on softer substrate (**Figure 4a**) while cells were more elongated on stiffer substrate (**Figure 4c**). The same pattern of cell morphology was observed in [10], where cell extended its protrusions in all directions on softer substrate (**Figure 4b**) while cell protruded only on the leading edge with a long tail on stiffer substrate (**Figure 4d**).

Overall, the patterns of cell trajectory and cell morphology of our simulation are consistent with that from *in vitro* study. This indicated that our mechano-chemical model is valid.

#### *3.1.2 The mechanical signal has long-distance impacts on collective cell migration*

We then quantified the cell migration to explore the role of mechanics of cell-substrate and cell–cell mechanics on collective cell migration using the three measurements: persistent ratio *p t*ð Þ*<sup>n</sup>* , normalized separation distance *di*,*<sup>j</sup>*ð Þ *tn* and direction angle *α*ð Þ *tn* .

**Figure 4.**

*Cell morphology. (a, c) the cell morphology in our simulation using two substrate stiffness: 3 and 65 kPa. (b, d) the cell morphology from the in vitro study [10] using the same substrate stiffness: 3 and 65 kPa. The cell boundary is highlighted in black.*

**Figure 5.**

*Measurements of the collective cell migration. (a–c) the cell migration speed, persistence ratio and normalized separation distance of our simulation and in the in vitro study [10]. (d–g) the migration direction angle of the cells on the leading edge and more than 500 μm from the wound edge in our simulation and that in the in vitro study [10] on the substrate with stiffness 65 kPa (d–e) and 3 kPa (f–g). The colors indicating simulation and experiment are shown in (a). The error bars of our simulation depict the standard deviations of four runs of simulation.*

We first examined the migrating speed of the cell. In general, cells migrate with higher speed on stiffer substrate (**Figure 5a**, more details of cell migration speed can be found in Appendix). In addition, cells close to the wound edge migrated with higher speed on both stiffer and softer substrate. This speed decreased gradually as the distance to the wound edge increased. On substrate with stiffness of 65 kPa, the migration speed decreased from 0*:*69 0*:*01*μm= min* in Region I to 0*:*49 0*:*02*μm= min* in Region IV, while on substrate with stiffness of 3 kPa, the migration

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

speed decreased from 0*:*38 0*:*02*μm= min* in Region I to 0*:*25 0*:*02*μm= min* in Region IV (**Figure 5a**). The cell migration speed of our simulation was consistent with that from the *in vitro* study [10]. It is easy to interpret such pattern of cell migration speed. For cells in Region I, especially on the wound edge, there are fewer or even no cells ahead. As the distance to the wound edge increased, it was more crowded and more difficult for cells to migrate forward.

We next examined the migration persistence of the cells. As shown in **Figure 5b**, cells migrate more persistently on stiffer substrate. In addition, cells close to the wound edge migrated with higher migration persistence. For cells on substrate with stiffness of 65 kPa, the persistence ratio decreased from 82 2% in Region I to 58 3% in Region IV, while for cells on substrate with stiffness of 3 kPa, the persistence ratio decreased from 71 1% in Region I to 55 3% in Region IV (**Figure 5b**). As shown in **Figure 5c**, collective cell migration was coordinated better on stiffer substrate.

In addition, we examined the normalized separation distance of the pairs of migrating cells. As shown in **Figure 5c**, the normalized separation distance increased as the distance to the wound edge increased. In our simulation, for cells on substrate at stiffness of 65 kPa, the separation distance decreased from 0*:*15 0*:*02 in Region I to 0*:*11 0*:*02 in Region II and then increased to 0*:*21 0*:*03 in Region IV, while for cells on substrate at stiffness of 3 kPa, the separation distance decreased from 0*:*22 0*:*02 in Region I to 0*:*17 0*:*02 in Region II and then increased to 0*:*19 0*:*04 in Region IV (**Figure 5c**). This pattern of separation distance in our simulation was also observed in the *in vitro* study [10].

Furthermore, we examined the migration direction angle. We compared this angle for cells on the leading edge of the tissue and cells 500 *μm* away. Since the cell migration direction is usually along the cell polarity direction [32], we also compared this direction angle to the cell polarization direction reported in [10]. As shown in **Figure 5d**–**g**, cells exhibit more accurate migration direction towards the wound on stiffer substrate (65 kPa). Only about 10 % of the cells on the leading edge had migration direction opposite to the wound (**Figure 5d**, 90°–270°). For cells > 500 *μm* away from the wound edge, 30 % of them had migration direction opposite to the wound (**Figure 5f**, 90°–270°). However, for cells on softer substrate (3 kPa), cell migration deviated more from the direction towards the wound where 35 % of the cells on leading edge had migration direction opposite to the wound direction (**Figure 5e**, 90°–270°), while for cells > 500 *μm* away from the wound edge, this fraction increased to 45 % (**Figure 5g**, 90°–270°).

These measurements implied that substrate stiffness is important to guide collective cell migration. Cells on stiffer substrate can migrate with high persistence, good coordination between cell pairs, and accurate migration direction. Our simulation suggests that the mechano-chemical feedback loop in each cell ensured it to dictate its migration direction. Furthermore, the individual cell movements were organized into a global migrative wave through intercellular adhesions.
