Open access peer-reviewed chapter

A Dynamic Finite Element Cellular Model and Its Application on Cell Migration

Written By

Jieling Zhao

Submitted: 15 June 2020 Reviewed: 24 September 2020 Published: 31 October 2020

DOI: 10.5772/intechopen.94181

From the Edited Volume

Finite Element Methods and Their Applications

Edited by Mahboub Baccouch

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While the tissue is formed or regenerated, cells migrate collectively and remained adherent. However, it is still unclear what are the roles of cell-substrate and intercellular interactions in regulating collective cell migration. In this chapter, we introduce our newly developed finite element cellular model to simulate the collective cell migration and explore the effects of mechanical feedback between cells and between cell and substrate. Our viscoelastic model represents one cell with many triangular elements. Intercellular adhesions between cells are represented as linear springs. Furthermore, we include a mechano-chemical feedback loop between cell-substrate mechanics and cell migration. Our results reproduce a set of experimental observation of patterns of collective cell migration during epithelial wound healing. In addition, we demonstrate that cell-substrate determined mechanics play an important role in regulating persistent and oriented collective cell migration. This chapter illustrates that our finite element cellular model can be applied to study a number of tissue related problems regarding cellular dynamic changes at subcellular level.


  • finite element model
  • collective cell migration
  • cell-substrate mechanics
  • intercellular adhesion
  • model developing

1. Introduction

Thanks to the accurate description of changes in material mechanics, finite element method has been widely used in the field of bioengineering to study cellular tissue related problems such as neurulation and epithelial mechanics [1, 2]. However, majority of current finite element models are only restricted on tissues undergoing changes of shapes and displacements at small scale. In addition, during the simulation, the cellular tissue is required to be remained as one integrity. These limitations restrict the traditional finite element method to be applied to study the essential physiological processes such as morphogenesis, tissue regeneration, tumor metastasis, and cancer invasion, where cells often migrate collectively as large coherent strands or tubes. Such large scale of collective cell movement is recognized as the hallmark of tissue-remodeling events. During the past decade, to overcome the limitation of traditional finite element method, dynamic finite element method such as PFEM has been developed to extend the traditional FEM to study mechanics of materials with more flexibility or undergoing larger scale of motility. The object domain (either fluid or solid) is represented as nodes tessellated by triangular mesh. The mathematical equations governing the physical rules of the mechanical property of the discretized domain defined by the mesh connecting nodes are subsequentially solved in the standard FEM. Under the analysis using dynamic finite element method, the motion of sub-domain of the object can freely move and even separate from the main domain [3]. The advancement of dynamic finite element in achieving both accurate description of material mechanics and large scale of geometric and topological changes makes it suitable to simulate the physiological processes such as wound healing and cancer invasion. During these physiological processes, cells move in collective fashion and respond with chemical and mechanical signals through cell–cell junctions and interactions between cells and their micro-environment.

In this chapter, we introduced our newly developed dynamic finite element cellular model and its application to study the influence of cell-substrate mechanics and intercellular adhesions on collective cell migration. Our model represents each cell as a mesh of triangular elements at sub-cellular level [4]. Each triangular element exhibits viscoelastic characteristic using a Maxwellian model [5]. The effects of line tension forces along the cell boundary according to the local curvature is incorporated [6]. The intercellular adhesions are modeled as elastic springs at sub-cellular scale [7]. In addition, a mechano-chemical feedback pathway including focal adhesion, proteins of Paxillin, Rac, PAK, and Merlin, which are all responsible for cell protrusion [8] is embedded in individual cell. This pathway is collaborated with another mechano-chemical pathway, which is responsible for transmitting mechanical cue through intercellular adhesions [9]. Our model is used to study collective cell migration using a simplified wound tissue. We then compare our simulation results to an in vitro study [10]. Finally, we discussed and made the conclusion that the mechanics between cell-substrate play a crucial role in guiding highly efficient collective cell migration. This guidance cue is well maintained and transmitted between cells through the intercellular adhesions.


2. Methods

2.1 Cell geometry

In our model, a cell in 2D ΩR2 is represented as an oriented polygon including a number of boundary vertices V∂Ωvi∂ΩR2, where the location of the vertex vi is denoted as xi. The set of boundary vertices V∂Ω, together with a set of internal vertices VInt and a set of triangular elements TΩτi,j,k:vivjvkV∂ΩVInt 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.

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 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.

2.2.1 Strain and stress tensors

We use the strain tensor εxt to describe the local cell deformation at x at time t. εxt takes the form of ε1,1=u1/x1, ε2,2=u2/x2, and ε1,2=ε2,1=12u1/x2+u2/x1, where uxt defined as u1xtu2xtTR2 is the displacement of x at time t. We use the stress tensor σxt to describe the local forces at x at time t. Here σ is correlated with ε by a generalized Maxwell model: σxt=σxt+σmxt [5, 12, 13], where σxt is the stress of the long-term elastic element and σmxt is the stress of the Maxwell elastic element. E, Em, and ηm 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 ε1xt and the strain of the viscous element ε2xt sum up to the strain tensor εxt: ε1xt+ε2xt=εxt.

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


where σa 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].


where Yxt is the adhesion coefficient at time t and is proportional to the strength of local focal adhesions [18]: Yxt=nx,tn0EstYa, where nx,t is the number of bound integrins at location x at t (more details of calculating nx,t in Model of focal adhesion), n0 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 12lklult2, where kl is the spring constant of the spring l. Its orientation angle at time t is denoted as θlt. Its transformation vector Tθ is denoted as cosθlsinθlcosθlsinθl. So ult can be written as ult=Tθlu11tul2t, where ul1t and ul2t are the displacements of the two end-node vertice x1 and x2 of l at time t. The elastic force of l due to displacement of Δl is applied on xi and xj as fl=fΔlel and -fΔlel, respectively, where fΔl is the magnitude of fl, el 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


where λxt is the line tension force and fxt is the protrusion force. Line tension force is written as λxt=fmκxtnxt, where fm is a contractile force per unit length, κxt is the curvature, and nxt is the outward unit normal at x at time t [6]. Protrusion force is denoted as fxt=fanxt, where fa 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


The energy dissipation of EΩt due to cell viscosity is determined by the viscous coefficient ηm and the strain of the viscous element ε2xt [19]: Ωηmε2t2dx. The dissipation of the total free energy of the cell can be written as


Since ηmε2t=Emε1 and σ=Eε, and ult=Tθlul1ul2. Eq. (5) can be rewritten as


Denoting B=/x100/x2/x2/x1, then εxt=Buxt. According to Gauss’ divergence theorem, we rewrote Ωσaσa0Buxtt as Ωσauxttdx, which leads to ∂Ωσanxtuxttdx.

Denoting Al=cl2clslcl2clslclslsl2clslsl2cl2clslcl2clslclslsl2clslsl2, where cl=cosθl and sl=sinθl. Eq. (6) can be rewritten as


2.2.7 Stress of viscoelastic cell and its update

By using general Maxwell model, the stress σxt can be written as [12, 13, 19]:


During the time interval Δt=tn+1tn, where tn is the n-th time step, σmn+1x can be written as [19]:


Therefore, the stress σnx at tn can be written as


2.2.8 Force balance equation for discretized time step

For each triangular element τi,j,k, Eq. (7) at time step tn+1 can be rewritten using Eq. (10) as


where γm=Em/E, Am=1eΔtηm/EΔtηm/E, and Fn+1=∂Ωσanxt+λxt+fxt+flxtdx. Eq. (11) leads to the following linear force-balance equation


where Kτi,j,kn+1, uτi,j,kn+1, and fτi,j,kn+1 are the stiffness matrix, displacement vector, and integrated force vector of τi,j,k at time step tn+1 (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 Kn+1. Therefore, the linear relationship between the concatenated displacement vector un+1 of all cell vertice and the force vector fn+1 on them is given by


Changes in the cell shape at time step tn+1 can be obtained by solving Eq. (13). For vertex vi at xi, its new location at next time step is then updated as xin+1=xin+un+1vi.

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


where Ru and Rb are the numbers of unbound and bound integrin ligand, respectively; kf 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 fr applied on vi. Traction force frxt on x at time t is given by Yxtuxt following [6]. kr is determined by kr=kr0e0.04fr+4e7e0.2fr following [22], where kr0 is a constant. kf is related with the substrate stiffness by kf=kf0Est2/Est2+Est02 [16, 23], where kf0 and Est0 are constants (see Appendix for choosing Est0).

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]:


where x, r, p, m and n are the concentrations of activated Paxillin, activated Rac, activated PAK, Merlin and bound integrins, respectively. kx,r, kr,p, kp,x, 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 C1 and C2 where C1 is the leader cell and C2 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 C1 starts to migrate, tension force is generated on the cadherin spring between C1 and C2. Merlin is therefore 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:

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 (C1) to its follower cells (C2 and C3). (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 ptn, 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,jtn 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 αtn is the angle between the migration direction (red arrow) and the direction towards the wound (green arrow).


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. δx is a kronecker function that δTRUE=1 and δFALSE=0. ft is the tension force through the cadherin spring and ftthr 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 epithelial-specific 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].

Time step lapse0.1 secNAa
Cell radius10 μm[27]
Young’s modulus of cell5 kPa[28]
Poisson ratio of cell0.40[29]
Contractile pressure σa2 kPa[6]
Adhesion energy constant Ya0.9 /μm[6]
Spring constant of cadherin spring3.0 nN/μmNA
Default length of cadherin spring100 nm[30]
Maximum length of cadherin spring400 nm[31]
Protrusion force constant fa2.0 nN/μm[11]
Integrin bound rate kf00.5[22]
Integrin unbound rate kr00.4[22]
Reference substrate stiffness Est040 kPaNA
Rac deactivation rate kx,r4/min[8]
PAK deactivation rate kr,p10/min[8]
Paxillin dephosphorylation rate kp,x10/min[8]
Saturation of phosphorylated Paxillin kx1[8]
Saturation of PAK activation kp1[8]
Saturation of Merlin km1NA
Merlin phosphorylation rate km,e10/minNA
Saturation of phosphorylated Merlin ke1 e3NA
Force threshold of delocating Merlin ftthr0.15 nNNA

Table 1.

Parameters used in the model.

aEstimated value marked as NA.

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 t0 over the length of the migrating trajectory:


where t0 is the initial time, xti 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:


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.


where uc is the unit vector of the cell migration direction, uw is the unit vector of direction from the cell mass center towards the wound, sgnx is the sign of x (Figure 2c.3).


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 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).

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.

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.

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 ptn, normalized separation distance di,jtn and direction angle αtn.

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±0.01μm/min in Region I to 0.49±0.02μm/min in Region IV, while on substrate with stiffness of 3 kPa, the migration speed decreased from 0.38±0.02μm/min in Region I to 0.25±0.02μm/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.

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 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 5dg, 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.


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.


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 Yxt of a cell vertex x at time t and set to be proportional to the strength of focal adhesions [18]: Yxt=nx,tn0EstYa, where nx,t is the number of binding integrins at location x at t, n0 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 vi is stochastically selected with the probability RacviiRacvi. The outward unit normal vector nvi of vi is chosen as the cell migration direction. Any vertex vj whose outward unit normal vector nvj is positively aligned with nvi, is treated as leading edge vertex. The protrusion force fa is then applied on each leading edge vertex vi as favi=faRvinvi, where fa is a constant, Rvi is the normalized Rac concentration at vi.


A.3 Calibrating the cell protrusion parameter

As shown in Figure 6ac, 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 protrude against each other. To fit our cell protrusion model to the in vitro data, we calibrate the parameter of Est0: when Est0=40kPa, the cell migration speed of our simulation has the best match with the in vitro studies [33, 34] (Figure 6d).

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].


  1. 1. Chen X., Brodland G.W. Multi-scale finite element modeling allows the mechanics of amphibian neurulation to be elucidated. Physical Biology. 2008;5: 015003
  2. 2. Hutson M.S., Veldhuis J., Ma X., Lynch H.E., Cranston P.G., Brodland G.W. Combining laser microsurgery and finite element modeling to assess cell-level epithelial mechanics. Biophysical Journal. 2009;97:3075–3085
  3. 3. Oñate E., Idelsohn S., Del Pin F., Aubry R. The particle finite element methodan overview. International Journal of Computational Methods. 2004;1:267–307
  4. 4. Zhao J., Manuchehrfar F., Liang J. Cell-substrate mechanics guide collective cell migration through intercellular adhesion: a dynamic finite element cellular model. Biomechanics and Modeling in Mechanobiology. 2020;19(5):1781–1796
  5. 5. Karcher H., Lammerding J., Huang H., Lee R.T., Kamm R.D., Kaazempur-Mofrad M.R. A three-dimensional viscoelastic model for cell deformation with experimental verification. Biophysical Journal. 2003;85(5):3336–3349
  6. 6. Oakes P., Banerje S., Marchetti M. Gardel M. Geometry regulates traction stresses in adherent cells. Biophysical Journal. 2014;107:825–833
  7. 7. Jamali Y., Azimi M., Mofrad M.R. A sub-cellular viscoelastic model for cell population mechanics. PLoS One. 2010;5(8):e12097
  8. 8. Cirit M., Krajcovic M., Choi C.K., Welf E.S., Horwitz A.F., Haugh J.M. Stochastic model of integrin-mediated signaling and adhesion dynamics at the leading edges of migrating cells. PLoS Comput. Biol. 2010;6(2):e1000688
  9. 9. Okada T., Lopez-Lago M., Giancotti F.G. Merlin/nf-2 mediates contact inhibition of growth by suppressing recruitment of Rac to the plasma membrane. J. Cell Biol. 2005;171(2):361–371
  10. 10. Ng M.R., Besser A., Danuser G., Brugge J.S. Substrate stiffness regulates cadherin-dependent collective migration through myosin-II contractility. The Journal of Cell Biology. 2012;199(3):545–563
  11. 11. Zhao J., Cao Y., DiPietro L.A., Liang J. Dynamic cellular finite-element method for modelling large-scale cell migration and proliferation under the control of mechanical and biochemical cues: a study of re-epithelialization. Journal of The Royal Society Interface. 2017;14(129):20160959
  12. 12. Schoner J.L., Lang J., Seidel H.P. Measurement-based interactive simulation of viscoelastic solids. Computer Graphics Forum. 2004;23:547–556
  13. 13. Schwartz J.M., Denninger M., Rancourt D., Moisan C. Laurendeau D. Modelling liver tissue properties using a non-linear visco-elastic model for surgery simulation. Medical Image Analysis. 2005;9(2):103–112
  14. 14. Rubinstein B., Fournier M.F., Jacobson K., Verkhovsky A.B., Mogilner A. Actin-myosin viscoelastic flow in the keratocyte lamellipod. Biophysical Journal. 2009;97(7):1853–1863
  15. 15. Ladoux B., Mège R.M., Trepat X. Front–rear polarization by mechanical cues: From single cells to tissues. Trends in Cell Biology. 2016;26(6):420–433
  16. 16. Dokukina I.V. Gracheva M.E. A model of fibroblast motility on substrates with different rigidities. Biophysical Journal. 2010;98(12):2794–2803
  17. 17. Barnhart E., Lee K.C., Keren K., Mogilner A., Theriot J. An adhesion-dependent switch between mechanisms that determine motile cell shape. PLoS Biol. 2011;9(5):e1001059
  18. 18. Banerjee S., Marchetti M.C. Contractile stresses in cohesive cell layers on finite-thickness substrates. Physical Review Letters. 2012;109(10):108101
  19. 19. Sedef M., Samur E., Basdogan C. Real-time finite-element simulation of linear viscoelastic tissue behavior based on experimental data. Computer Graphics and Applications. 2006;26(6):58–68
  20. 20. Holle A.W., Engler A.J. More than a feeling: discovering, understanding, and influencing mechanosensing pathways. Current Opinion in Biotechnology. 2011;22(5):648–654
  21. 21. DiMilla P., Barbee K., Lauffenburger D. Mathematical model for the effects of adhesion and mechanics on cell migration speed. Biophysical Journal. 1991;60(1):15
  22. 22. Li Y., Bhimalapuram P., Dinner A.R. Model for how retrograde actin flow regulates adhesion traction stresses. Journal of Physics: Condensed Matter. 2010;22(19):194113
  23. 23. Yeh Y.C., Ling J.Y., Chen W.C., Lin H.H., Tang M.J. Mechanotransduction of matrix stiffness in regulation of focal adhesion size and number: reciprocal regulation of caveolin-1 and β1 integrin. Scientific reports. 2017;7(1):15008
  24. 24. Stéphanou A., Mylona E., Chaplain M., Tracqui P. A computational model of cell migration coupling the growth of focal adhesions with oscillatory cell protrusions. Journal of Theoretical Biology. 2008;253(4):701–716
  25. 25. Wu Y.I., Frey D., Lungu O.I., Jaehrig A., Schlichting I., Kuhlman B., Hahn K.M. A genetically encoded photoactivatable rac controls the motility of living cells. Nature. 2009;461(7260):104–108
  26. 26. Das T., Safferling K., Rausch S., Grabe N., Boehm H., Spatz J.P. A molecular mechanotransduction pathway regulates collective migration of epithelial cells. Nature Cell Biology. 2015;17(3):276
  27. 27. Watt F.M., Green H. Involucrin synthesis is correlated with cell size in human epidermal cultures. The Journal of Cell Biology. 1981;90:738–742
  28. 28. Hu S., Wang R., Tsang C.M., Tsao S.W., Sun D., et al. Revealing elasticity of largely deformed cells flowing along confining microchannels. RSC Advances. 2018;8:1030–1038
  29. 29. Zielinski R., Mihai C., Kniss D., Ghadiali S.N. Finite element analysis of traction force microscopy: Influence of cell mechanics, adhesion, and morphology. Journal of Biomechanical Engineering. 2013;135:071009
  30. 30. Changede R., Sheetz M. Integrin and cadherin clusters: A robust way to organize adhesions for cell mechanics. BioEssays. 2017;39:1–12
  31. 31. Nematbakhsh A., Sun W., Brodskiy P.A., Amiri A., Narciso C., et al. Multi-scale computational study of the mechanical regulation of cell mitotic rounding in epithelia. PLoS Computational Biology. 2017;13:e1005533
  32. 32. Rappel W.J., Edelstein-Keshet L. Mechanisms of cell polarization. Current Opinion in Systems Biology. 2017;3:43–53
  33. 33. Ansardamavandi A., Tafazzoli-Shadpour M., Shokrgozar M.A. Behavioral remodeling of normal and cancerous epithelial cell lines with differing invasion potential induced by substrate elastic modulus. Cell Adhesion and Migration. 2018;12(5):472–488
  34. 34. Cai P., Layani M., Leow W.R., Amini S., Liu Z., et al. Bio-inspired mechanotactic hybrids for orchestrating traction-mediated epithelial migration. Advanced Materials. 2016;28:3102–3110

Written By

Jieling Zhao

Submitted: 15 June 2020 Reviewed: 24 September 2020 Published: 31 October 2020