**2. Methods**

#### **2.1 Cell geometry**

In our model, a cell in 2D Ω ⊂ <sup>2</sup> is represented as an oriented polygon including a number of boundary vertices *<sup>V</sup>*<sup>∂</sup><sup>Ω</sup> � *vi* ∈ ∂<sup>Ω</sup> <sup>⊂</sup> <sup>2</sup> , where the location of the vertex *v<sup>i</sup>* is denoted as *xi*. The set of boundary vertices *V*<sup>∂</sup>Ω, together with a set of internal vertices *VInt* and a set of triangular elements *<sup>T</sup>*<sup>Ω</sup> � *<sup>τ</sup><sup>i</sup>*,*j*,*<sup>k</sup>* : *<sup>v</sup>i*, *<sup>v</sup> <sup>j</sup>*, *<sup>v</sup><sup>k</sup>* <sup>∈</sup> *V*<sup>∂</sup><sup>Ω</sup> ∪*V*Intg define the geometry of cell Ω (**Figure 1a**). If two cells are closely in contact, a set of adhesive springs are generated between them (**Figure 1a**, red bars in the dashed blue box). There are several interior vertice on each cell boundary edge. They are evenly distributed along that edge. These interior vertice are the potential locations for newly generated adhesive spring to attach on. Any force applied on that interior vertex through the attached adhesive spring will be mapped onto its nearest end-node vertex of the corresponding boundary edge.

#### **2.2 Viscoelasticity of the cell**

Previous researches have demonstrated that the cell cytoskeleton exhibits viscoelastic characteristic [14, 15]. Following the studies of [16, 17], we assume that, during cell deformation and cell migration, linear viscoelasticity is adequate to describe the mechanical properties of the cell.

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

#### **Figure 1.**

*The cell geometry and the chemical pathway between cell-substrate and intercellular adhesion. (a) the cell in our model is represented as following: The cell boundary is defined by an oriented polygon including a number of boundary vertices. A triangular mesh tiling up a cell is generated based on the method of farthest sampling [11]. The E-cadhesion type of intercellular adhesions between two neighboring cells are represented as elastic springs (red bars in the blue box, the dashed blue box is for a closer view). (b) each triangular element exhibits viscoelastic characteristic using a generalized Maxwell model following [5, 12, 13]. (c) the positive feedback loop between focal adhesion and cell protrusion is built up in each vertex of the triangular mesh following [8]. Such network includes the proteins of integrin, Paxillin, Rac, and PAK. The protein Merlin on the cadherin is also included to count the effects of intercellular adhesion on cell migration [9].*

#### *2.2.1 Strain and stress tensors*

We use the strain tensor *ε*ð Þ *x*, *t* to describe the local cell deformation at *x* at time *<sup>t</sup>*. *<sup>ε</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* takes the form of *<sup>ε</sup>*1,1 <sup>¼</sup> *<sup>∂</sup>u*1*=∂x*1, *<sup>ε</sup>*2,2 <sup>¼</sup> *<sup>∂</sup>u*2*=∂x*2, and *<sup>ε</sup>*1,2 <sup>¼</sup> *<sup>ε</sup>*2,1 <sup>¼</sup> 1 <sup>2</sup> ð Þ *<sup>∂</sup>u*1*=∂x*<sup>2</sup> <sup>þ</sup> *<sup>∂</sup>u*2*=∂x*<sup>1</sup> , where *u x*ð Þ , *<sup>t</sup>* defined as ð Þ *<sup>u</sup>*1ð Þ *<sup>x</sup>*, *<sup>t</sup>* , *<sup>u</sup>*2ð Þ *<sup>x</sup>*, *<sup>t</sup> <sup>T</sup>* <sup>⊂</sup> <sup>2</sup> is the displacement of *x* at time *t*. We use the stress tensor *σ* ð Þ *x*, *t* to describe the local forces at *x* at time *t*. Here *σ* is correlated with *ε* by a generalized Maxwell model: *σ* ð Þ¼ *x*, *t σ*∞ð Þþ *x*, *t σm*ð Þ *x*, *t* [5, 12, 13], where *σ*∞ð Þ *x*, *t* is the stress of the long-term elastic element and *σm*ð Þ *x*, *t* is the stress of the Maxwell elastic element. *E*∞, *Em*, and *η<sup>m</sup>* denote the long-term elastic modulus, elastic modulus of the Maxwell elastic element, and viscous coefficient of the Maxwell viscous element, respectively (**Figure 1b**). The strain of the Maxwell elastic element *ε*1ð Þ *x*, *t* and the strain of the viscous element *ε*2ð Þ *x*, *t* sum up to the strain tensor *ε*ð Þ *x*, *t* : *ε*1ð Þþ *x*, *t ε*2ð Þ¼ *x*, *t ε*ð Þ *x*, *t* .

We assume that the total free energy of a cell is the summation of its elastic energy, its adhesion energy due to the contact with the substrate, its elastic energy due to the intercellular adhesions with neighboring cells, and its energy due to the forces exerting on the boundary.

#### *2.2.2 Cell elastic energy*

The elastic energy due to the deformation of the cell Ω is given by

$$E\_{\Omega}(t) = \frac{1}{2} \int\_{\Omega} \left(\sigma\_{\Leftrightarrow}(\mathfrak{x}, t) + \sigma\_{d}\mathfrak{d}\_{\vec{\eta}}(\mathfrak{x})\right)^{T} \mathfrak{e}\left(\mathfrak{x}, t\right) d\mathfrak{x} + \frac{1}{2} \int\_{\Omega} \sigma\_{m}(\mathfrak{x}, t)^{T} \mathfrak{e}\_{1}(\mathfrak{x}, t) d\mathfrak{x},\tag{1}$$

where *σ<sup>a</sup>* is a homogeneous contractile pressure following [6].

#### *2.2.3 Cell adhesion energy due to the contact with the substrate*

The energy due to the adhesion between the cell and the substrate is given by [6].

$$\frac{Y(\mathbf{x},t)}{2}\int\_{\Omega} \mathfrak{u}\left(\mathbf{x},t\right)^{2}d\mathbf{x},\tag{2}$$

where *Y*ð Þ *x*, *t* is the adhesion coefficient at time *t* and is proportional to the strength of local 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 *n<sup>x</sup>*,*<sup>t</sup>* is the number of bound integrins at location *x* at *t* (more details of calculating *n<sup>x</sup>*,*<sup>t</sup>* in **Model of focal adhesion**), *n*<sup>0</sup> is a normalized constant number, *Est* is the stiffness of the substrate and *Ya* is the basic adhesion constant following [18].

#### *2.2.4 Cell adhesion energy due to intercellular adhesion*

The energy due to the intercellular adhesions, which are modeled as elastic springs, is given by <sup>1</sup> 2 P *l kl <sup>u</sup>l*ð Þ*<sup>t</sup>* <sup>2</sup> , where *kl* is the spring constant of the spring *l*. Its orientation angle at time *t* is denoted as *θl*ð Þ*t* . Its transformation vector *T*ð Þ*θ* is denoted as cosð Þ *θ<sup>l</sup>* , sin ð Þ *θ<sup>l</sup>* , � cosð Þ *θ<sup>l</sup>* ð Þ , � sin ð Þ *θ<sup>l</sup>* . So *ul*ð Þ*t* can be written as *ul*ðÞ¼ *t T*ð Þ *θ<sup>l</sup>* ð Þ *u*11ð Þ*t* , *u<sup>l</sup>*2ð Þ*t* , where *u<sup>l</sup>*1ð Þ*t* and *u<sup>l</sup>*2ð Þ*t* are the displacements of the two endnode vertice *x*<sup>1</sup> and *x*<sup>2</sup> of *l* at time *t*. The elastic force of *l* due to displacement of Δ*l* is applied on *x<sup>i</sup>* and *x <sup>j</sup>* as *f<sup>l</sup>* ¼ *f*ð Þ Δ*l e<sup>l</sup>* and -*f*ð Þ Δ*l el*, respectively, where *f*ð Þ Δ*l* is the magnitude of *f<sup>l</sup>* , *e<sup>l</sup>* is the unit vector of the orientation of *l*.

#### *2.2.5 Boundary and protrusion forces*

Furthermore, the local forces applied on the cell boundary also contribute to the energy. Following [6], the tension force along the cell boundary is considered. In addition, we also incorporate the protrusion force on the leading edge of migrating cell. The contribution of these two forces can be written as

$$\int\_{\partial\Omega} (\lambda(\mathfrak{x},t) + f(\mathfrak{x},t)) \mathfrak{u}(\mathfrak{x},t) d\mathfrak{x},\tag{3}$$

where *λ*ð Þ *x*, *t* is the line tension force and *f x*ð Þ , *t* is the protrusion force. Line tension force is written as *λ*ð Þ¼� *x*, *t f <sup>m</sup>κ*ð Þ *x*, *t n x*ð Þ , *t* , where *f <sup>m</sup>* is a contractile force per unit length, *κ*ð Þ *x*, *t* is the curvature, and *n x*ð Þ , *t* is the outward unit normal at *x* at time *t* [6]. Protrusion force is denoted as *f* ð Þ¼� *x*, *t f <sup>a</sup>n x*ð Þ , *t* , where *f <sup>a</sup>* is the protrusion force per unit length.

#### *2.2.6 The total free energy and its dissipation*

In summary, the total free energy of cell Ω at time *t* is given by

$$E\_{\Omega}(t) = \frac{1}{2} \int\_{\Omega} \left( \left( \sigma\_{\alpha}(\mathbf{x}, t) + \sigma\_{a} \delta\_{\overline{\eta}} \right)^{T} \mathbf{e} \left( \mathbf{x}, t \right) + \sigma\_{m}^{T}(\mathbf{x}, t) \mathbf{e}\_{1} (\mathbf{x}, t) \right) d\mathbf{x} + \frac{1}{2} \int\_{\Omega} \left( \sigma\_{a}, \sigma\_{a}, 0 \right) \mathbf{e} \left( \mathbf{x}, t \right) d\mathbf{x} \tag{4}$$

$$+ \frac{Y(\mathbf{x}, t)}{2} \int\_{\Omega} \mathbf{u} \left( \mathbf{x}, t \right)^{2} d\mathbf{x} + \frac{1}{2} \sum\_{l} k\_{l} \mathbf{u}\_{l}(t)^{2} + \int\_{\partial \Omega} \left( \lambda(\mathbf{x}, t) + f(\mathbf{x}, t) + f\_{l}(\mathbf{x}, t) \right) \mathbf{u}(\mathbf{x}, t) d\mathbf{x} \right). \tag{4}$$

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

The energy dissipation of *E*Ωð Þ*t* due to cell viscosity is determined by the viscous coefficient *η<sup>m</sup>* and the strain of the viscous element *ε*2ð Þ *x*, *t* [19]: � Ð <sup>Ω</sup>*η<sup>m</sup> <sup>∂</sup> <sup>ε</sup>*<sup>2</sup> *∂t* � �<sup>2</sup> *dx*. The dissipation of the total free energy of the cell can be written as

$$\begin{split} \frac{\partial}{\partial t} E\_{\Omega}(t) &= \frac{\partial}{\partial t} \Big[ \frac{1}{2} \Big]\_{\Omega} \Big( \left( \sigma\_{\text{ov}}(\mathbf{x}, t) + \sigma\_{d} \delta\_{\overline{\eta}} \right)^{T} \mathbf{e} \left( \mathbf{x}, t \right) + \sigma\_{m}^{T}(\mathbf{x}, t) \mathbf{e}\_{1} (\mathbf{x}, t) \Big) d\mathbf{x} + \frac{1}{2} \Big[ \int\_{\Omega} \left( \sigma\_{\text{av}}, \sigma\_{\text{a}} \right) \mathbf{e} \left( \mathbf{x}, t \right) d\mathbf{x} \Big] \\ &+ \frac{Y(\mathbf{x}, t)}{2} \int\_{\Omega} \mathbf{u} \left( \mathbf{x}, t \right)^{2} d\mathbf{x} + \frac{1}{2} \sum\_{l} k\_{l} \mathbf{u}\_{l} (t)^{2} + \int\_{\partial \Omega} \left( \lambda(\mathbf{x}, t) + f(\mathbf{x}, t) + f\_{l}(\mathbf{x}, t) \right) \mathbf{u} \left( \mathbf{x}, t \right) d\mathbf{x} \Big] \\ &= - \int\_{\Omega} \eta\_{m} \Big( \frac{\partial \mathbf{e}\_{2}(\mathbf{x}, t)}{\partial t} \Big)^{2} d\mathbf{x}. \end{split} \tag{5}$$

Since *<sup>η</sup><sup>m</sup> <sup>∂</sup> <sup>ε</sup>*<sup>2</sup> *<sup>∂</sup><sup>t</sup>* ¼ *Emε*<sup>1</sup> and *σ*<sup>∞</sup> ¼ *E*∞*ε*, and *ul*ðÞ¼ *t T*ð Þ *θ<sup>l</sup> u<sup>l</sup>*<sup>1</sup> ð Þ , *u<sup>l</sup>*<sup>2</sup> . Eq. (5) can be rewritten as

$$\begin{split} & \int\_{\Omega} E\_{\omega} \mathbf{e}(\mathbf{x}, t) \frac{\partial \mathbf{e}(\mathbf{x}, t)}{\partial t} d\mathbf{x} + \int\_{\Omega} \boldsymbol{\sigma}\_{m}^{T}(\mathbf{x}, t) \frac{\partial \mathbf{e}(\mathbf{x}, t)}{\partial t} d\mathbf{x} + \int\_{\Omega} (\boldsymbol{\sigma}\_{a}, \boldsymbol{\sigma}\_{a}, \mathbf{0}) \frac{\partial \mathbf{e}(\mathbf{x}, t)}{\partial t} d\mathbf{x} \\ & \quad + Y \int\_{\Omega} \mathbf{u}(\mathbf{x}, t) \frac{\partial \mathbf{u}(\mathbf{x}, t)}{\partial t} d\mathbf{x} + \sum\_{l} T(\theta\_{l})^{T} T(\theta\_{l}) k\_{l}(\mathbf{u}\_{l1}(t), \mathbf{u}\_{l2}(t)) \frac{\partial \mathbf{u}(\mathbf{x}, t)}{\partial t} \\ & \quad + \int\_{\partial \Omega} \left( \lambda(\mathbf{x}, t) + f(\mathbf{x}, t) + f\_{l}(\mathbf{x}, t) \right) \frac{\partial \mathbf{u}(\mathbf{x}, t)}{\partial t} d\mathbf{x} = \mathbf{0}. \end{split} \tag{6}$$
 
$$\text{Denoting } \mathbf{B} = \begin{pmatrix} \partial / \partial \mathbf{x}\_{1} & \mathbf{0} \\ \mathbf{0} & \partial / \partial \mathbf{x}\_{2} \end{pmatrix}, \text{then } \mathbf{e}(\mathbf{x}, t) = \mathbf{B} \mathbf{u}(\mathbf{x}, t). \text{ According to Gauss's}$$

*∂=∂x*<sup>2</sup> *∂=∂x*<sup>1</sup> divergence theorem, we rewrote Ð <sup>Ω</sup>ð Þ *<sup>σ</sup>a*, *<sup>σ</sup>a*, 0 *<sup>B</sup> <sup>∂</sup>u x*ð Þ , *<sup>t</sup> <sup>∂</sup><sup>t</sup>* as <sup>Ð</sup> <sup>Ω</sup>*σa*<sup>∇</sup> � *<sup>∂</sup>u x*ð Þ , *<sup>t</sup> <sup>∂</sup><sup>t</sup> dx*, which leads to Ð <sup>∂</sup>Ω*σan x*ð Þ , *<sup>t</sup> <sup>∂</sup> u x*ð Þ , *<sup>t</sup> <sup>∂</sup><sup>t</sup> dx*.

$$\text{Denoting } \mathbf{A}\_{l} = \begin{pmatrix} c\_{l}^{2} & c\_{l}s\_{l} & -c\_{l}^{2} & -c\_{l}s\_{l} \\ c\_{l}s\_{l} & s\_{l}^{2} & -c\_{l}s\_{l} & -s\_{l}^{2} \\ -c\_{l}^{2} & -c\_{l}s\_{l} & c\_{l}^{2} & c\_{l}s\_{l} \\ -c\_{l}s\_{l} & -s\_{l}^{2} & c\_{l}s\_{l} & s\_{l}^{2} \end{pmatrix}, \text{ where } c\_{l} = \cos\left(\theta\_{l}\right) \text{ and }$$
 
$$\text{i.e. } (0) \text{ E- (6) are the constraints in.}$$

*sl* ¼ *sin* ð Þ *θ<sup>l</sup>* . Eq. (6) can be rewritten as

$$\begin{aligned} \int\_{\Omega} \mathbf{B}^T \boldsymbol{\sigma}(\mathbf{x}, t)^T d\mathbf{x} + Y \int\_{\Omega} \boldsymbol{\mathfrak{u}}(\mathbf{x}, t) d\mathbf{x} + \sum\_{l} \mathbf{A}\_l k\_l(\boldsymbol{\mathfrak{u}}\_{l1}(t), \boldsymbol{\mathfrak{u}}\_{l2}(t)) &= \\ - \int\_{\partial\Omega} \boldsymbol{\sigma}\_d \boldsymbol{\mathfrak{u}}(\mathbf{x}, t) + \boldsymbol{\lambda}(\mathbf{x}, t) + f(\mathbf{x}, t) + f\_l(\mathbf{x}, t) d\mathbf{x} \end{aligned} \tag{7}$$

#### *2.2.7 Stress of viscoelastic cell and its update*

By using general Maxwell model, the stress *σ* ð Þ *x*, *t* can be written as [12, 13, 19]:

$$E\_{\approx} \mathfrak{e}(\mathfrak{x}, t) + \int\_{0}^{t} E\_{m} \mathfrak{e}\_{m \stackrel{t \to}{\mathbb{R}}} \frac{\partial \mathfrak{e}(\mathfrak{x}, s)}{\partial s} ds = \mathfrak{e}\_{\approx}(\mathfrak{x}, t) + \mathfrak{e}\_{m}(\mathfrak{x}, t). \tag{8}$$

During the time interval <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> *tn*þ<sup>1</sup> � *tn*, where *tn* is the *<sup>n</sup>*-th time step, *<sup>σ</sup><sup>n</sup>*þ<sup>1</sup> *<sup>m</sup>* ð Þ *<sup>x</sup>* can be written as [19]:

$$\left[e^{-\frac{\Delta t}{\eta\_m/E\_{\infty}}}\sigma\_m^n(\mathfrak{x}) + \frac{E\_m}{E\_{\infty}}\right]\_{t\_n}^{t\_{n+1}} e^{-\frac{t\_{n+1} - \varepsilon}{\eta\_m/E\_{\infty}}} ds \frac{\sigma\_{\infty}^{n+1}(\mathfrak{x}) - \sigma\_{\infty}^n(\mathfrak{x})}{\Delta t}.\tag{9}$$

Therefore, the stress *<sup>σ</sup>n*ð Þ *<sup>x</sup>* at *tn* can be written as

$$
\sigma^n(\mathfrak{x}) = \sigma^n\_{\infty}(\mathfrak{x}) + \sigma^n\_m(\mathfrak{x}) \tag{10}
$$

#### *2.2.8 Force balance equation for discretized time step*

For each triangular element *τi*,*j*,*k*, Eq. (7) at time step *tn*þ<sup>1</sup> can be rewritten using Eq. (10) as

$$\begin{split} & \int\_{\mathfrak{r}\_{ij,k}} \mathbf{B}^T \left( E\_{\alpha} \mathbf{B} \mathbf{u}^{n+1} (\boldsymbol{\omega}) + e^{-\frac{\Delta t}{\mathfrak{r}\_m \boxplus \boldsymbol{\omega}\_m}} \ \sigma\_m^n + \gamma\_m A\_m \left( E\_{\alpha} \mathbf{B} \mathbf{u}^{n+1} (\boldsymbol{\omega}) - E\_{\alpha} \mathbf{B} \mathbf{u}^n (\boldsymbol{\omega}) \right) \right) d\mathbf{x} \\ & + Y \Big[\_{\mathfrak{r}\_{ijk}} \mathbf{u}^{n+1} d\mathbf{x} + \sum\_{l \in \mathfrak{r}\_{ij,k}} A\_l k\_l \left( \mathfrak{u}\_{l1}^{n+1}, \mathfrak{u}\_{l2}^{n+1} \right) = F^{n+1} (\boldsymbol{\omega}), \end{split} \tag{11}$$

where *<sup>γ</sup><sup>m</sup>* <sup>¼</sup> *Em=E*∞, *Am* <sup>¼</sup> <sup>1</sup>�*<sup>e</sup>* � <sup>Δ</sup>*<sup>t</sup> ηm=E*∞ Δ*t ηm=E*∞ , and *<sup>F</sup><sup>n</sup>*þ<sup>1</sup> ¼ �<sup>Ð</sup> <sup>∂</sup>Ω*σ<sup>a</sup> n x*ð Þþ , *t λ*ð Þþ *x*, *t f x*ð Þþ , *t f<sup>l</sup>* ð Þ *x*, *t dx*. Eq. (11) leads to the following linear force-balance equation

$$\mathbf{K}^{n+1}\_{\mathbf{r}\_{i\neq k}} \mathbf{u}^{n+1}\_{\mathbf{r}\_{i\neq k}} = \mathbf{f}^{n+1}\_{\mathbf{r}\_{i\neq k}},\tag{12}$$

where *<sup>K</sup><sup>n</sup>*þ<sup>1</sup> *<sup>τ</sup>i*,*j*,*<sup>k</sup>* , *<sup>u</sup><sup>n</sup>*þ<sup>1</sup> *<sup>τ</sup>i*,*j*,*<sup>k</sup>* , and *<sup>f</sup> <sup>n</sup>*þ<sup>1</sup> *<sup>τ</sup>i*,*j*,*<sup>k</sup>* are the stiffness matrix, displacement vector, and integrated force vector of *τ<sup>i</sup>*,*j*,*<sup>k</sup>* at time step *tn*þ<sup>1</sup> (see more details of derivation of (Eq. 12) in [11]).

We can then assemble the element stiffness matrices of all triangular elements into one big global stiffness matrix *K<sup>n</sup>*þ<sup>1</sup> . Therefore, the linear relationship between the concatenated displacement vector *u<sup>n</sup>*þ<sup>1</sup> of all cell vertice and the force vector *f <sup>n</sup>*þ<sup>1</sup> on them is given by

$$\mathbf{K}^{n+1} \mathbf{u}^{n+1} = \mathbf{f}^{n+1}.\tag{13}$$

Changes in the cell shape at time step *tn*þ<sup>1</sup> can be obtained by solving Eq. (13). For vertex *v<sup>i</sup>* at *xi*, its new location at next time step is then updated as *x<sup>n</sup>*þ<sup>1</sup> *<sup>i</sup>* <sup>¼</sup> *<sup>x</sup><sup>n</sup> <sup>i</sup>* <sup>þ</sup> *<sup>u</sup><sup>n</sup>*þ<sup>1</sup>ð Þ *<sup>v</sup><sup>i</sup>* .

#### **2.3 Mechano-chemical pathway in the cell**

Upon contact with the environment, cells can transfer the mechanical cues into biochemical signals, which can trigger the initiation of further cellular behaviors [20]. In our model, we considered a mechano-chemical pathway consisting of two parts, where one is to regulate the feedback loop between focal adhesion and cell protrusion and the other is to regulate the transmission of mechanical signal between adjacent cells through intercellular adhesions.

#### *2.3.1 Model of focal adhesion*

For each vertex *v<sup>i</sup>* in cell Ω, we assign a constant number of integrin ligand on it. These integrin molecules can bind or unbind with fibronectin molecules on the

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

substrate underneath. Following [21], the numbers of bound and unbound integrin ligand molecules are determined by

$$\frac{dR\_b}{dt} = k\_f n\_s R\_u - k\_r R\_b,\tag{14}$$

where *Ru* and *Rb* are the numbers of unbound and bound integrin ligand, respectively; *k <sup>f</sup>* is the binding rate coefficient; *ns* is the concentration of fibronectin per cell vertex; *kr* is the unbinding rate coefficient. *kr* depends on the magnitude of the traction force *f<sup>r</sup>* applied on *vi*. Traction force *fr*ð Þ *x*, *t* on *x* at time *t* is given by *<sup>Y</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup> u x*ð Þ , *<sup>t</sup>* following [6]. *kr* is determined by *kr* <sup>¼</sup> *kr*<sup>0</sup> *<sup>e</sup>*�0*:*<sup>04</sup> *fr* <sup>þ</sup> <sup>4</sup>*<sup>e</sup>* � <sup>7</sup>*e*0*:*<sup>2</sup> *fr* � � following [22], where *kr*<sup>0</sup> is a constant. *k <sup>f</sup>* is related with the substrate stiffness by *<sup>k</sup> <sup>f</sup>* <sup>¼</sup> *kf*0*E*<sup>2</sup> *st= E*<sup>2</sup> *st* <sup>þ</sup> *<sup>E</sup>*<sup>2</sup> *st*0 � � [16, 23], where *kf*<sup>0</sup> and *Est*<sup>0</sup> are constants (see Appendix for choosing *Est*0).

#### *2.3.2 Model of feedback loop between focal adhesion and cell protrusion*

We introduced a simplified model of a positive feedback loop to control the spatial distribution of the focal adhesions, which governs the direction of cell protrusion [8, 24]. In our model, this feedback loop involves proteins of Paxillin, Rac, and PAK (**Figure 1c**). Upon formation of focal adhesion, Paxillin is activated by active PAK. The active Paxillin then activates Rac, which in turn triggers the activation of PAK. The activated Rac is responsible for protruding cells [25]. Since the protein Merlin on the intercellular cadherin complex also plays a role in activating Rac [9], we include Merlin in our feedback loop. The protein concentration over time is updated through a set of differential equations following [8]:

$$k\frac{dr}{dt} = k\_{\mathbf{x},r} \left( k\_m \frac{C\_r^2}{C\_r^2 + m^2} \mathbf{x} - r \right) \tag{15}$$

$$\frac{dp}{dt} = k\_{r,p}(r-p) \tag{16}$$

$$\frac{d\mathbf{x}}{dt} = k\_{p,\mathbf{x}} \left( k\_{\mathbf{x}} \frac{p^2}{p^2 + C\_{\mathbf{x}}^2} n - \mathbf{x} \right) \tag{17}$$

where *x*, *r*, *p*, *m* and *n* are the concentrations of activated Paxillin, activated Rac, activated PAK, Merlin and bound integrins, respectively. *kx*,*<sup>r</sup>*, *kr*,*<sup>p</sup>*, *kp*,*<sup>x</sup>*, *km*, *kx*, *Cr*, *Cx* are the parameters of corresponding rates. The level of activated Rac was used to determine the protrusion force on the leading edge of the migrating cell (see details of cell protrusion model in Appendix).

#### *2.3.3 Model of mechanosensing through intercellular adhesion*

We added Merlin in the feedback loop (**Figure 1c**) following a previous study reporting that Merlin on the intercellular cadherin complex regulates the Rac activity [26]. As illustrated in **Figure 2a**, for two adjacent cells *C*<sup>1</sup> and *C*<sup>2</sup> where *C*<sup>1</sup> is the leader cell and *C*<sup>2</sup> is the follower cell, if both cells are at static state, Merlin molecules only locate on the cadherin spring (**Figure 2b** top). As reported by [9], Merlin suppressed the binding of integrin. Due to such suppression, Rac turns to inactivated on the Merlin-expressed site. Once cell *C*<sup>1</sup> starts to migrate, tension force is generated on the cadherin spring between *C*<sup>1</sup> and *C*2. Merlin is therefore

#### **Figure 2.**

*Mechanosensing through intercellular adhesion and tissue model for collective cell migration. (a) the intercellular adhesion of the cadherin spring (red springs in the blue box) are responsible for transmitting mechanical stimulus from leader cell (C*1*) to its follower cells (C*<sup>2</sup> *and C*3*). (b) When cells are static, Merlin which inhibits the Rac activation is bound on the cadherin spring. Once leader cell migrates, stretch is generated on the cadherin spring, Merlin on the follower cell is delocated. Therefore, Rac is activated on the follower cell. (c) the size of the wound epithelial tissue is 720 μm* � *240 μm. The right boundary is set as the wound edge (yellow line). Cells can migrate towards the open space on the right. Three measurements are introduced to measure the collective cell migration: (1) migration persistence p t*ð Þ*<sup>n</sup> , the ratio of the distance from the current position at time tn to its initial position (green line), over the length of the traversed path (red curve); (2) normalized pair separation distance di*,*<sup>j</sup>*ð Þ *tn is the separation distance between a pair of cells at time tn (green lines) divided by the average length of the two cells' traversed path (red curves); (3) migration direction angle α*ð Þ *tn is the angle between the migration direction (red arrow) and the direction towards the wound (green arrow).*

delocalized from cadherin-attached site in response to the generated tension force. As a consequence, Rac is activated there to generate protrusion force to follow the leader cell (**Figure 2b** bottom [26]). For simplicity, we introduced the inactive Merlin phenotype along with the active Merlin phenotype on the two end vertice of one intercellular cadherin adhesion. The negative feedback loop of Merlin-Rac is modeled through a set of differential equations following [9, 26], where active Merlin and inactive Merlin can switch their phenotype, but only active Merlin can suppress the Rac activity (**Figure 1c**). The delocalization of Merlin was simply modeled as Merlin switching to inactive phenotype:

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

$$\frac{dm}{dt} = e - \delta\left(f\_t > f\_{t-thr}\right)k\_{m,\epsilon}\left(k\_p\left(\frac{p^2}{p^2 + C\_\epsilon^2} + k\_\epsilon\right)m\right) \tag{18}$$

$$\frac{de}{dt} = \delta(\left.f\_t > f\_{t-thr}\right)k\_{m,\epsilon}\left(k\_p\left(\frac{p^2}{p^2 + C\_\epsilon^2} + k\_\epsilon\right)m\right) - e\tag{19}$$

where *m*, *e* and *p* are the concentrations of Merlin, inactive Merlin, and PAK, respectively. *km*,*e*, *kp*, *ke*, *Ce* are the corresponding rate parameters. *δ*ð Þ *x* is a kronecker function that *δ*ð Þ¼ *TRUE* 1 and *δ*ð Þ¼ *FALSE* 0. *ft* is the tension force through the cadherin spring and *ft*�*thr* is a force threshold.

#### **2.4 Cellular tissue for collective cell migration**

In our model, the collective cell migration was modeled using a wound tissue of epithelial cells. The tissue size is 720 *μm* � 240 *μm*. The epithelial cell type is set to MCF-10A, which is used in the *in vitro* study [10]. The corresponding epithelialspecific parameters can be found in **Table 1**. We arbitrarily set the right boundary of the tissue as the wound edge, and cells can migrate towards the open space to the right of the wound edge (**Figure 2c**). The mechano-chemical pathway was initiated first in the cells on the wound edge after they migrate. We followed a previous study [10] to divide the location of cells into four sub-regions according to their distance to the wound edge: Regions I, II, III, and IV whose distance to the wound edge is 0–160 *μm*, 160–320 *μm*, 320–480 *μm*, and 480–640 *μm*, respectively (**Figure 2c**). We ran the simulation for 12 biological hours, the same experimental duration time in the *in vitro* study [10].

#### **2.5 Measurements of the collective cell migration**

In our model, the collective cell migration are measured using four measurements following [10]:

The migration persistence. At time step *tn*, the length of the straight line between cell positions at *tn* and initial time step *t*<sup>0</sup> over the length of the migrating trajectory:

$$p(t\_n) = \frac{|\mathbf{x}(t\_n) - \mathbf{x}(t\_0)|}{\sum\_{k=0}^{n-1} |\mathbf{x}(t\_{k+1}) - \mathbf{x}(t\_k)|},\tag{20}$$

where *t*<sup>0</sup> is the initial time, *x*ð Þ*ti* is the position at time step *ti* (**Figure 2c.1**).

The normalized separation distance. At time step *tn*, the separation distance of a pair of two adjacent cells 1-2, divided by the average length of their migrating trajectories:

$$d\_{1,2}(t\_i) = \frac{||\mathbf{x}\_1(t\_i) - \mathbf{x}\_2(t\_i)| - |\mathbf{x}\_1(t\_0) - \mathbf{x}\_2(t\_0)||}{\frac{1}{2} \left(\sum\_{k=0}^{i-1} |\mathbf{x}\_1(t\_{k+1}) - \mathbf{x}\_1(t\_k)| + \sum\_{k=0}^{i-1} |\mathbf{x}\_2(t\_{k+1}) - \mathbf{x}\_2(t\_k)|\right)},\tag{21}$$

where the numerator is the separation distance between cells *i* and *j* at time *tn*, and the denominator is the average path length of cells *i* and *j* at time *tn* (**Figure 2c.2**).

The direction angle. The angle between the cell migration direction and the direction towards the wound.

