Transmission of cell internal stresses to the substrate

Recent investigations have demonstrated that active (actin filaments and AM machinery) and passive (microtubules and cell membrane) cellular elements play a key role in generating the cell contractile stress which is transmitted to the substrate through integrins. The former, which generates active cell stress, basically depends on the minimum, ϵ_min, and maximum, ϵ_max, internal strains, which is zero outside of ϵ_max-ϵ_min range, while the latter, which generates passive cell stress, is directly proportional to stiffness of passive cellular elements and internal strains. Therefore, the mean contractile stress arisen due to incorporation of the active and passive cellular elements can be presented by [66–69] (1) where K_pas, K_act, ϵ_cell and σ_max represent the stiffness of the passive and active cellular elements, the internal strain of the cell and the maximum contractile stress exerted by the actin-myosin machinery, respectively, while $\widetilde{ϵ}={\sigma }_{\mathrm{\text{max}}}/K_{\mathrm{\text{act}}}$.

Effective mechanical forces

A cell extends protrusions in leading edges in the direction of migration and adheres to its substrate pulling itself forward in direction of the most effective signal. The cell membrane area is as tiny as to produce strong traction force due to cell internal stress, consequently, adhesion is thought to compensate this shortage by providing the sufficient traction required for efficient cell translocation [3]. The equilibrium of forces exerted on the cell body should be satisfied by cell migration and cell shape changes [70, 71]. In the meantime, two main mechanical forces act on a cell body: traction force and drag force. The former is exerted due to the contraction of the actin-myosin apparatus which is proportional to the stress transmitted by the cell to the ECM by means of integrins and adhesion. Representing the cell by a connected group of finite elements, the nodal traction force exerted by the cell to the surrounding substrate at each finite element node of the cell membrane can be expressed as [69] (2) where σ_i is the cell internal stress in ith node of the cell membrane and e_i represents a unit vector passing from the ith node of the cell membrane towards the cell centroid. S(t) is the cell membrane area which varies with time. During cell migration, it is assumed that the cell volume is constant [72–74], however the cell shape and cell membrane area change. ζ is the adhesivity which is a dimensionless parameter proportional to the binding constant of the cell integrins, k, the total number of available receptors, n_r, and the concentration of the ligands at the leading edge of the cell, ψ. Therefore, it can be defined as [66–68] (3) ζ depends on the cell type and can be different in the anterior and posterior parts of the cell. Its definition is given in the following sections. Thereby, the net traction force affecting on the whole cell because of cell-substrate interaction can be calculated by [69] (4) where n is the number of the cell membrane nodes. During migration, nodal traction forces (contraction forces) exerted on cell membrane towards its centroid compressing the cell. Consequently, each finite element node on the cell membrane, which has less internal deformation, will have a higher traction force [69]. On the contrary, the drag force opposes the cell motion through the substrate that depends on the relative velocity and the linear viscoelastic character of the cell substrate. At micro-scale the viscous resistance dominates the inertial resistance of a viscose fluid [75]. Assuming ECM as a viscoelastic medium and considering negligible convection, Stokes’ drag force around a sphere can be described as [76] (5) where v is the relative velocity and r is the spherical object radius. η(E_sub) is the effective medium viscosity. Within a substrate with a linear stiffness gradient, we assume that effective viscosity is linearly proportional to the medium stiffness, E_sub, at each point. Therefore it can be calculated as (6) where λ is the proportionality coefficient and η_min is the viscosity of the medium corresponding to minimum stiffness. Although, the viscosity coefficient may be finally saturated with higher substrate stiffness, this saturation occurs outside the substrate stiffness range that is proper for some cells [58].

Equation 5 was developed by Stoke to calculate the drag force around a spherical shape object with radius r. This typical equation was employed in our previous works for cell migration with constant spherical shape [66, 69]. In the present work, according to Equations 17–19, an inaccurate calculation of the drag force may affect considerably the calculation accuracy of the cell velocity and polarization direction. So that, according to [77, 78], a shape factor is appreciated to moderate the Stokes’ drag expression to be suitable for irregular cell shape. The drag of irregular solid objects depends on the degree of non-sphericity and their relative orientation to the flow. Therefore for an irregular object shape the drag is basically anisotropic compared to movement direction. Since here the objective is to investigate cell migration while cell morphology changes, calculation of the drag force using Equation 5 will not be precise enough. Due to the randomness of the cell shapes and dynamics, description of drag force for objects with irregular shape is extremely complicated. It is thought that only probabilistic and approximate predictions can be reasonable and useful to describe drag force for highly irregular particles [77, 78]. Therefore, referring to experimental observations, an appropriate shape factor, f_shape, is appreciated to moderate the Stokes’ drag expression for highly irregularly-shaped objects which is accurate enough [68, 77, 78] (7)

A wide variety of shape-characterizing parameters has been suggested for irregular particles. Here we have employed Corey Shape Factor (CSF) which is the most common and accurate shape factor. It appreciates three main lengths of an object that are mutually perpendicularly to each other as (8) where l_max, l_med and l_min are the cell’s longest, intermediate and the shortest dimensions, respectively, which are representative of cell surface area changes [77]. In the case of a spherical cell shape, this shape factor delivers 1. Although other shape factors have been proposed to characterize the shape irregularity, using the max-med-min length factor leads to reliable results [77, 79].

Protrusion force

To migrate, cells extend local protrusions to probe their environment. This is the duty of protrusion force generated by actin polymerization which has a stochastic nature during cell migration [80]. It should be distinguished from the cytoskeletal contractile force [68, 75]. The order of the protrusion force magnitude is the same as that of the traction force but with lower amplitude [69, 75, 81–83]. Therefore, we randomly estimate it as (9) where e_rand is a random unit vector and $F_{\mathrm{\text{net}}}^{\mathrm{\text{trac}}}$ is the magnitude of the net traction force while κ is a random number, such that 0 ≤ κ < 1, [66, 68].

Electrical force in presence of electrotactic cue

Exogenous EFs imposed to a cell have been proposed as a directional cue that directs the cells to migrate in cell therapy. Besides, studies in the last decade have provided convincing evidence that there is a role for EFs in wound healing [6]. Significantly, this role is highlighted more than expected due to overriding other cues in guiding cell migration during wound healing [6, 31]. Experimental works demonstrate that Ca²⁺ influx into cell plays a significant role in the electrotactic cell response [25, 26, 28]. Although this is still a controversial open question, Ca²⁺ dependence of electrotaxis has been observed in many cells such as neural crest cells, embryo mouse fibroblasts, fish and human keratocytes [23, 25, 27, 30, 40]. On the other hand, Ca²⁺ independent electrotaxis has been observed in mouse fibroblasts [32]. The precise mechanism behind intracellular Ca²⁺ influx during electrotaxis is not well-known. A simple cell at resting state maintain a negative membrane potential [25] so that exposing it to a dcEF causes that the side of the plasma membrane near the cathode depolarizes while the the other side hyperpolarizes [23, 25, 30]. For a cell with trivial voltage-gated conductance, the membrane side which is hyperpolarized attracts Ca²⁺ due to passive electrochemical diffusion. Therefore, this side of the cell contracts and propels the cell towards the cathode which causes to open the voltage-gated Ca²⁺ channels (VGCCs) near the cathode (depolarised) and allows intracellular Ca²⁺ influx (Fig 1). So, on both anodal and cathodal sides of the cell, intracellular Ca²⁺ level enhances. Balance between the opposing magnetic forces defines the resultant electrical force affecting the cell body [25]. That is the reason that some cells tend to reorient towards the anode, like metastatic human breast cancer cells [84], human granulocytes [85], while some others do towards the cathode, such as human keratinocytes [26, 86], embryo fibroblasts [27], human retinal pigment epithelial cells [87] and fish epidermal cells [40].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Response of a cell to a dcEFs.

A simple cell in the resting state has a negative membrane potential [25]. When a cell with a negligible voltage-gated conductance is exposed to a dcEF, it is hyperpolarised membrane near the anode attracts Ca²⁺ due to passive electrochemical diffusion. Consequently, this side of the cell contracts, propelling the cell towards the cathode. Therefore, voltage-gated Ca²⁺ channels (VGCCs) near cathode (depolarised side) open and a Ca²⁺ influx occurs. In such a cell, intracellular Ca²⁺ level rises in both sides. The direction of cell movement, then, depends on the difference of the opposing magnetic contractile forces, which are exerted by cathode and anode [25].

https://doi.org/10.1371/journal.pone.0122094.g001

A single cell embedded within a uniform EF will be ionized and charged. Therefore the electrical force experienced by this individual cell can be obtained by (10) where E is uniform dcEF strength and Ω(E) stands for the surface charge density of the cell. e_EF is a unit vector in the direction of the dcEF toward the cathode or anode, depending on the cell type. The time course of the translocation response during exposing a cell to a dcEF demonstrates that the cell velocity versus translocation varies depending on the dcEF strength. Experiments of Nishimura et al. [26] on human keratinocytes indicate that the net migration velocity raises by increase the dcEF strength to about 100 mV/mm while further increase the dcEF strength does not affect the cell net migration velocity. Since the Ca²⁺ influx into intracellular may play a role in this process [25, 26, 28, 88–90], it is thought that the imposed dcEF regulates the concentration of intracellular Ca²⁺. Therefore, it can be deduced that the cell surface charge is directly proportional to the imposed dcEF strength [25, 26]. Consequently, we assume a linear relationship between the cell surface charge and the applied dcEF strength as (11) where Ω_satur is the saturation value of the surface charge and E_satur is the maximum dcEF strength that causes Ca²⁺ influx into intracellular.

Deformation and reorientation of the cell

Solid line in Fig 2 shows a spherical cell configuration which is initially considered. It is assumed that the cell first exerts mechano-sensing forces on the membrane to probe its surrounding micro-environment which is named mechano-sensing process. Thus, the cell internal strain at each finite element node of the cell membrane along e_i can be calculated by (12) where ϵ_i is the strain tensor of ith node located on cell membrane due to mechano-sensing process.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Calculation of the cell reorientation.

a- A initially spherical cell (solid line) is deformed (dashed line) during mechano-sensing process. e_mech is mechanotaxis reorientation of the cell. b- A cell is reoriented due to exposing to chemotaxis, thermotaxis and electrotaxis where e_ch, e_th and e_EF denote the unit vector in the direction of each cue, respectively. The coefficients μ_mech, μ_ch, and μ_th are effective factors of mechanotactic, chemotactic and thermotactic cues, respectively. $F_{\mathrm{\text{net}}}^{\mathrm{\text{trac}}}$ is the magnitude of the net traction force, F_prot is the random protrusion force, F_EF represents the electrical force that is exerted by dcEF and F_drag stands for drag force. e_pol represents the net polarisation direction of a cell in a multi-signaling environment.

https://doi.org/10.1371/journal.pone.0122094.g002

A cell exerts contraction forces towards its centroid compressing itself so that the cell internal deformation, ϵ_cell, created by these forces on each finite element node of the cell membrane is negative. Hence, according to Equations 1 and 2 nodes with a less internal deformation experience a higher internal stress and traction force. Therefore, the net traction forces, ${\mathbf{\text{F}}}_{\mathrm{\text{net}}}^{\mathrm{\text{trac}}}$, points towards the direction of minimum cell internal deformation (Equation 4), presenting the mechanotaxis reorientation of the cell [69]. Consequently, the unit vector of the mechanotactic reorientation of the cell, e_mech, reads (13)

In presence of thermotaxis or chemotaxis, the cell polarisation direction will be controlled by all the existent stimuli. It is assumed that the presence of both additional cues does not affect either the physical or the mechanical properties of a typical cell, nor its surrounding ECM. Traction forces exerted by a typical cell depend on the mechanical apparatus of the cell and the mechanical properties of the substrate [21]. Therefore, the mechanotactic tool practically drives the cell body forward while the presence of chemotaxis and/or thermotaxis cues only changes the cell polarisation direction such that a part of the net traction force is guided by mechanotaxis and the rest is guided by these stimuli (Fig 2). Consequently, under chemical and/or thermal gradients, the unit vectors associated to the chemotactic and thermotactic stimuli can be represented, respectively, as [66, 67] (14) (15) where ∇ denotes the gradient operator while C and T represent the chemoattractant concentration and the temperature, respectively. As mentioned above, the realignment of the net traction force under these cues is affected by the direction of chemical and thermal gradients, so that the effective force, F_eff, which incorporates mechanotactic, chemotactic and thermotactic effects can be defined as (16) where μ_mech, μ_ch and μ_th are the effective factors of mechanotaxis, chemotaxis, and thermotaxis cues respectively, μ_mech + μ_ch + μ_th = 1. It is assumed that there is neither degradation nor remodeling of the ECM during cell motility. Having in account that the inertial force is negligible, the cell motion equation delivers drag force as (17) Thereby, using Equation 7, the instantaneous velocity of the cell is defined as (18) with the net polarisation direction (19)

Cell morphology and cell remodeling during cell migration

Cell migration composed of several coordinated cyclic cellular processes. At the light microscope level, many authors summarize this process into several steps such as leading-edge protrusion, formation of new adhesions near the front, contraction, releasing old adhesions and rear retraction [11, 91]. At the trailing end the cortical tension squeezes or presses the cytoplasm in the direction of migration while at the leading edge, the tension generated due to protrusions drives the cells forward [3, 92].

Guided by the aforementioned experimental observations, the regulatory process behind the cell shape during cell migration is here simplified to analyze cell shape changes coupled with the cell traction forces. Therefore, we model the dominant modes of cell morphological changes considering the cell body retraction at the rear and extension at the front. Referring to Fig 3, the initial domain of the cell, which is located within the working space of Λ ⊂ R³ with the global coordinates of X, may be described as (20) where X′ denotes the local cell coordinates located in the cell centroid. Accordingly the cell membrane can be represented by ∂Ω′. Thereby, the substrate domain can be defined as (21) During cell migration, both domains Ω′ and Ω vary such that Ω′ ∪ Ω = Λ and Ω′ ∩ Ω = ∅.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Definition of extension and retraction points as well as anterior and posterior parts of the cell at each time step.

Λ ⊂ R³, Ω and Ω′ represent the 3D working space, matrix and cell domains, respectively. X stands for the global coordinates and X′ represents the local cell coordinates located in the cell centroid, O. χ is a plane passing by the cell centroid with unit normal vector n parallel to the cell polarisation direction, e_pol. P denotes a finite element node located on the cell membrane, ∂Ω. ∂Ω′⁺ and ∂Ω′⁻ are the finite element nodes located on the front and rear of the cell membrane, respectively.

https://doi.org/10.1371/journal.pone.0122094.g003

To correctly incorporate adhesivity, ζ, of cell in the cell front and rear, it is essential to define the cell anterior and posterior during cell motility. Assuming χ is a plane passing by the cell centroid, O, with unit normal vector n, parallel to e_pol, and s(X′) is a position vector of an arbitrary node located on ∂Ω′ (Fig 3), projection of s on n can be defined as (22) Consequently, nodes with positive δ are located on the cell membrane at the front, ∂Ω′⁺, while nodes with negative δ belong to the cell membrane at the cell rear, ∂Ω′⁻, where ∂Ω′ = ∂Ω′⁺ ∪ ∂Ω′⁻ should be satisfied.

We assume that the cell extends the protrusion from the membrane vertex whose position vector is approximately in the direction of cell polarisation, on the contrary, it retracts the trailing end from the membrane vertex whose position vector is totally in the opposite direction of cell polarisation. Thus, the maximum value of δ delivers the membrane node located on ∂Ω′⁺ from which the cell must be extended while the minimum value of δ represents the membrane node located on ∂Ω′⁻ from which the cell must be retracted. Assume e^ex ∈ Ω is the finite element that the membrane node with the maximum value of δ belongs to its space and e^re ∈ Ω′ is the finite element that the membrane node with the minimum value of δ belongs to its space. To integrate cell shape changes and cell migration, simply, e^re is moved from the Ω′ domain to the Ω domain, in contrast, e^ex is eliminated from the Ω domain and is included in the Ω′ domain [68].

In the present model the cell is not allowed to obtain infinitely thin shape during migration. Therefore, consistent with the experimental observation of Wessels et al. [93, 94], it is considered that the cell can extend approximately 10% of its whole volume as pseudopodia.

Cell behavior in a 3D matrix with a pure mechanotaxis

Experimental investigations demonstrate that cells located within 3D matrix actively migrate in direction of stiffness gradient towards stiffer regions [103]. In addition, it has been observed that during cell migration towards stiffer regions, the cell elongates and subsequently the cell membrane area increases [13, 96].

To consider the effect of mechanotaxis on cell behavior, it is assumed that there is a linear stiffness gradient in x direction which changes from 1 kPa at x = 0 to 100 kPa at x = 400 μm. The cell is initially located at a corner of the matrix near the boundary surface with lowest stiffness. Fig 5 and Fig 6 show the cell configuration and the trajectory tracked by the cell centroid within a matrix with stiffness gradient, respectively. As expected, independent from the initial position of the cell, when the cell is placed within a substrate with pure stiffness gradient it tends to migrate in direction of the stiffness gradient towards the stiffer region and it becomes gradually elongated. The cell experiences a maximum elongation in the intermediate region of the substrate since it is far from unconstrained boundary surface which is discussed in the previously presented work [66]. As the cell approaches the end of the substrate the cell elongation and CMI decrease (see Fig 7). Despite the boundary surface at x = 400 μm has maximum elastic modulus, due to unconstrained boundary, the cell does not tend to move towards it and maintains at a certain distance from it. The cell may extend random protrusions to the end of the substrate but it retracts again and maintains its centroid around an imaginary equilibrium plane (IEP) located far from the end of the substrate at x = 351 ± 5 μm (see Fig 8) [69]. Therefore, the cell never spread on the surface with the maximum stiffness. It is worth noting that the deviation of the obtained IEP coordinates is due to the stochastic nature of cell migration (random protrusion force). Fig 8 represents cell RI for the imposed stiffness gradient slope. The simulation was repeated for several initial positions of the cell and several values of the gradient slope, all the obtained results were consistent. However, change in the gradient slope can change the cell random movement and slightly displace the IEP position (results of different gradient slopes are not shown here). Cell behavior within the substrate with stiffness gradient is in agreement with experimental observations [13, 96, 103] and the results of the previous works presented by the same authors in which a constant spherical configuration has been considered for the cell [67, 69]. It is worth mentioning that the net cell traction force and velocity curves are not presented here since they roughly follow the same trend as the previous work [67].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Shape changes during cell migration within a substrate with a linear stiffness gradient.

The substrate stiffness changes linearly in x direction from 1 kPa at x = 0 to 100 kPa at x = 400 μm. At the beginning the cell is located at the corner of the substrate near the soft region. The results demonstrate that the cell migrates in the direction of stiffness gradient and the cell centroid finally moves around an IEP located at x = 351 ± 5 μm. a- The cell at the middle of the substrate, b- the cell final position (see also S1 Video).

https://doi.org/10.1371/journal.pone.0122094.g005

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Trajectory of the cell centroid within a substrate with stiffness gradient in presence of different stimuli.

Examples are run 10 times in order to check consistency of the results. The slop of the cell centroid trajectory reflects the attractivity of every cue to the cell.

https://doi.org/10.1371/journal.pone.0122094.g006

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Cell elongation, ϵ_elong (left axis), and CMI (right axis) versus the cell centroid translocation within a substrate with a pure stiffness gradient.

As the cell approaches the intermediate regions of the substrate (rigid regions) both the ϵ_elong and CMI increase. On the contrary, they decrease near the surface with maximum stiffness because the cell retracts protrusions due to unconstrained boundary surface.

https://doi.org/10.1371/journal.pone.0122094.g007

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Mean RI (left axis) and IEP position (right axis) of cell in the presence of different cues.

The error bars represent mean standard deviation among different runs. Adding a new stimulus to the substrate with stiffness gradient decreases the cell random alignment (increases mean RI) and moves the cell towards the end of the substrate.

https://doi.org/10.1371/journal.pone.0122094.g008

Cell behavior in presence of thermotaxis

Several experimental studies [18, 19] have demonstrated that, in vivo, different cell types are affected by thermal gradient. Here, employing the present model, we investigate that how the cell can sense and respond to the presence of thermal gradient in its substrate. To do so, a thermal gradient is added to the aforementioned substrate with stiffness gradient. It is assumed that the temperature at x = 0 is equal to 36°C and at x = 400 μm is 39°C [19], while μ_th = 0.2. This creates a linear thermal gradient throughout the substrate along x axis. At the beginning, the cell is located at one of the corners of the substrate near the boundary surface with minimum temperature. The results indicate that the cell gradually elongates and migrates towards warmer zone in direction of the thermal gradient by means of thermotaxis (Fig 9). Fig 6 demonstrates the trajectory that is tracked by the cell centroid. In this case also there is an IEP located at x = 359 ± 3 μm (Fig 8) that the cell centroid finally move around it. Comparing the trajectory of the cell centroid in the presence of thermotaxis with that of pure mechanotaxis indicates that the cell centroid slightly moves towards the end of the substrate with greater temperature. Once the cell achieves IEP, it extends protrusions randomly in different directions maintaining the position of the cell centroid near the IEP. These findings are independent from the initial cell position and are consistent with experimental findings of Higazi et al. [19] who demonstrated that trophoblasts migrate towards warmer locations due to thermal gradient. Comparing RIs of mechanotaxis and thermotaxis cases in Fig 8 illustrates that adding thermotaxis cue to the substrate with stiffness gradient causes decrease in cell random motility (increase in RI). Because mechanical and thermal gradients, which are in the same directions, contribute with each other to more directionally guide the cell. Both the cell elongation and the CMI follow the same trend as mechanotaxis example but in average there is an increase in their amount, which means the contribution of mechanotaxis and thermotaxis increases the cell elongation and the CMI (Fig 10). The thermal gradient imposed here may be considered as the maximum biological gradient, which is applicable in cell environment. We have repeated the simulation for mild thermal gradients but there is no considerable deviation in results (results not shown). Therefore, it can be deduced that the variation of gradient slope in thermotaxis do not dramatically affect the final results because in biological ranges, sharp thermal gradients are not applicable. However, it is noticeable that the cell does not exhibit significant thermotactic response to a very mild thermal gradients (when difference between maximum and minimum temperatures is less than 0.2°C in the substrate).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Shape changes during cell migration within a substrate with conjugate linear stiffness and thermal gradients.

It is assumed that there is a linear thermal gradient in x direction (as stiffness gradient) which changes from 36°C at x = 0 to 39°C at x = 400 μm. At the beginning the cell is located at a corner of the substrate near the surface with lower temperature. The results demonstrate that the cell migrates along the thermal gradient towards warmer region. Finally, the cell centroid moves around an IEP located at x = 359 ± 3 μm. When the cell centroid is near the IEP the cell may send out and retract protrusions but it maintains the position around IEP. a- The cell at the middle of the substrate, b- the cell final position (see also S2 Video).

https://doi.org/10.1371/journal.pone.0122094.g009

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 10. Cell elongation, ϵ_elong (left axis), and CMI (right axis) versus the cell centroid translocation in the presence of thermotaxis.

The cell elongation and CMI are maximum in the intermediate regions of the substrate and decreases as the cell approaches the unconstrained surface with higher temperature.

https://doi.org/10.1371/journal.pone.0122094.g010

Cell behavior in the presence of chemotaxis

Many experimental investigations have demonstrated that the cell has a directional migratory capability in presence of a shallow chemoattractant gradient within 3D surrounding substrates [16, 104]. In vitro, observations indicate that cells include a strong basal pseudopod cycle by which pseudopod extension occurs along chemical gradient at the close side of the cell to the higher chemical concentration [20]. This means that the cell elongates its body in direction of chemical gradient towards the higher concentration of chemoattractant substance.

Here, to consider effect of chemotaxis on cell behavior, a chemical gradient is added into the same substrate with stiffness gradient. It is assumed that a chemoattractant substance with concentration of 5×10⁻⁵ M exists at x = 400 μm while chemoattractant concentration at x = 0 μm is null. This creates a linear chemical gradient along the x axis. The evolution of shape changes during cell migration in the presence of chemotaxis is presented in Fig 11 for two different chemotaxis effective factors, μ_ch = 0.35 and μ_ch = 0.4. In Fig 6, the trajectory, which is tracked by the cell centroid, is compared with that of the previous experiments. It implies that the cell centroid ultimately moves around an IEP located at x = 368 ± 3 μm and x = 374 ± 4 μm for μ_ch = 0.35 and μ_ch = 0.4, respectively, (Fig 8). Therefore, it can be deduced that adding a chemotactic stimulus to the substrate moves the final position of the cell centroid towards the chemoattractant source, of course depending on the employed chemotactic effective factor. Similar behavior of cell motility has been observed in the previously presented work by the same authors in which the cell has been represented by a constant spherical shape [67]. In both cases, when the cell is near to the chemoattractant source, it may extend or retract protrusions in random directions, no cell tendency to leave the IEP. It is clear from Fig 12 that for both cases the cell follows the same trend as that of the previous examples in terms of the cell elongation and CMI. However, here, the peak of the cell elongation and CMI slightly increases in comparison with mechanotaxis and/or thermotaxis. In the presence of chemotaxis, the cell tends to spread on the surface on which chemoattractant source is located. It causes cell elongation and CMI increase in perpendicular direction to the imposed chemical gradient, which is considerable in case of greater chemotaxis effective factor (see Figs 11d and 12a). Because of the higher chemotaxis effective factor, the cell receives stronger chemotactic signal to spread more on the surface with chemoattractant source. Besides, the cell random movement relatively decreases for both cases in comparison with either mechanotaxis or thermotaxis example (Fig 8).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 11. Shape changes during cell migration in presence of chemotaxis within a substrate with stiffness gradient.

It is assumed that there is a chemoattractant substance with concentration of 5×10⁻⁵ M at x = 400 μm, which creates a linear chemical gradient across x direction. At the beginning the cell is located at one of the corners of the substrate near the surface of null chemoattractant substance. Two chemotaxis effective factors are considered; μ_ch = 0.35 (a and b) and μ_ch = 0.4 (c and d). The results demonstrate that, for both cases, the cell migrates along the chemical gradient towards the higher chemoattractant concentration. Depending on chemical effective factor, the ultimate position of the cell centroid will be different, for μ_ch = 0.35 the cell centroid keeps moving around an IEP located at x = 368 ± 3 μm (b) while for higher chemical effective factor, μ_ch = 0.4, the position of the IEP moves towards chemoattractant source to locate at x = 374 ± 4 μm (d). It is remarkable that in both cases the IEP displaces further towards the end of substrate in comparison with thermotaxis case (see also S3 and S4 Videos for low and high chemical effective factors, respectively).

https://doi.org/10.1371/journal.pone.0122094.g011

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 12. Cell elongation, ϵ_elong (left axis), and CMI (right axis) versus the cell centroid translocation in the presence of chemotaxis as well as mechanotaxis.

a- μ_ch = 0.35 and b- μ_ch = 0.40. For both cases, the cell elongation and CMI are maximum in the intermediate regions of the substrate and decreases as the cell approaches the unconstrained surface with chemoattractant source. Because, when the cell reaches the surface with maximum chemoattractant concentration, it tends to adhere to and spread over that surface. However, in the case of chemotaxis cue with higher effective factor the cell again elongates in perpendicular direction to the imposed chemical gradient.

https://doi.org/10.1371/journal.pone.0122094.g012

Cell migration towards chemoattractant source is qualitatively consistent with many experimental [20, 105, 106] and numerical [17, 51, 107] studies. Besides, cell elongation and shape change during migration is consistent with finding of Maeda et al. [108] implying that gradient sensing and polarization direction of the cell are linked to the cell shape changes and accompanied with motility length of pseudopods.

Cell behavior in presence of electrotaxis

As mentioned above, endogenous EF is developed around wounds during tissues injury, causing cell migration towards wound cites. Experiments show that in a Guinea pig skin injury just 3 mm away from wound, lateral potential drops to 0 from 140 mV/mm at the wound edge [6, 109–111]. Besides, in cornea ulcer, an EF equal to 42 mV/mm is measured [6, 112]. The cell movement can be also directed and accelerated via exposing it to an exogenous dcEF depending on cell phenotype. In this process, both calcium ion release from and influx into intracellular are generally associated with cell polarisation direction. For instance, human granulocytes [85], rabbit corneal endothelial cells [113], metastatic human breast cancer cells [84] are attracted by anode. Unlike metastatic rat prostate cancer cells [114], embryo fibroblasts [27], human keratinocytes [86], fish epidermal cells [40], human retinal pigment epithelial cells [87], epidermal and human skin cells [30] that move towards cathode. Therefore, altogether, different cell phenotypes may present different electrotactic behavior.

To consider the influence of the electrotaxis on cell behavior, it is considered that the cell is exposed to a dcEF through which the anode is located at x = 0 μm and the cathode at x = 400 μm. It is assumed that the cell phenotype is such that to be attracted by the cathode, such as human keratinocytes [86] or embryo fibroblasts [27]. First, the cell is located near the anode at x = 0. To demonstrate effect of dcEF strength on cell behavior the simulation is repeated for two different dcEF strength, E = 10 mV/mm and E = 10 100 mV/mm. Cell migration and shape change in the presence of both weak and strong EF are presented in Fig 13. In response of an EF, the cell re-organizes its side that is facing the cathode, and migrates directionally towards the cathode. The presence of the EF can dominate mechanotaxis effect and move the cell to the end of the substrate even more than previous cases where the cell centroid locates around IEP at x = 379 ± 3 μm and x = 383 ± 2 μm for the weak and strong EF strengths, respectively, (Fig 6 and Fig 8). Besides, the presence of the EF decreases considerably the random movement of the cell (see Fig 8). Near the cathode pole in the presence of weak EF the cell may extend many protrusions in different directions but the change of the cell centroid position is trivial. This is not the case in presence of strong EF, the position of the cell centroid remains constant due to the domination EF role. The cell is even unable to send out any protrusion. This takes place because the strong EF provides a dominant directional signal to guide the migrating cell towards the cathode, dominating the effect of other forces. This is consistent with previous work presented by the same authors assuming constant spherical cell shape [67] where the cell became immobile when it reaches the cathode in the presence of stronger EF strength. EF induces morphological change in the migrating cell where for both cases the average cell elongation and CMI are higher than those of all the previous cases (Fig 14). In presence of electrotaxis the cell achieves the maximum elongation sooner than the other cases and it maintains the maximum amounts until it reaches the end of substrate. Therefore, a flat region can be seen in the fitted elongation and CMI curves (Fig 14). For both cases, near the cathode, the cell elongation and CMI decrease, because in the presence of dcEF the cell tends to spread on the surface where the cathodal pole is located. However, in case of strong EF the cell elongation and CMI again increases because the electrical force acting on the cell body is strong enough to cause the cell elongation perpendicularly to dcEF direction, leading increase in the cell elongation and CMI. It is noteworthy mentioning that for both cases the ultimate cell elongation and CMI are greater that all previous studied cases.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 13. Shape changes during cell migration in presence of electrotaxis within a substrate with stiffness gradient.

A cell is exposed to a dcEF where the anode is located at x = 0 and the cathode at x = 400 μm. It is supposed that the cell is attracted to the cathode pole. At the beginning, the cell is placed in one of the corners of the substrate near the anode and far from the cathode pole. Two EF strength are considered; E = 10 mV/mm (a and b) and E = 100 mV/mm (c and d). For both cases, the cell migrates along the dcEF towards the surface in which the cathode pole is located. Depending on EF strength, the ultimate location of the cell centroid will be different so that for E = 10 mV/mm the cell centroid keeps moving around an IEP located at x = 379 ± 3 μm (b) while for saturation EF strength, E = 100 mV/mm, the position of the IEP moves further to the cathode pole to locate at x = 383 ± 2 μm (d). In the case of saturation EF strength (E = 100 mV/mm) the cell perfectly elongates on the surface of cathode pole without extending any protrusion (see also S5 and S6 Videos for low and high EF strengths, respectively).

https://doi.org/10.1371/journal.pone.0122094.g013

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 14. Cell elongation, ϵ_elong (left axis), and CMI (right axis) versus the cell centroid translocation in the presence of electotaxis as well as mechanotaxis.

a- E = 10 mV/mm and b- E = 100 mV/mm. The cell elongation and CMI reaches a maximum amount sooner than previous cases and are aproximately constant until the cell reaches the cathode pole. The cell elongation and CMI decrease when the cell reaches the surface on which the cathode pole is located but they never diminish less than those of other stimuli. However, in the case of higher EF strength the cell elongation and CMI again increase. The cell elongation and CMI are maximum in this case compared to the other previous cases.

https://doi.org/10.1371/journal.pone.0122094.g014

Cell shape change in Multi-signalling substrate

Finally, to simultaneously evaluate the effect of different stimuli on cell shape change during cell migration, we have designed 30 different cases through which different thermotaxis and chemotaxis effective factors as well as different EF strengths are applied. The maximum cell elongation, ϵ_elong, and CMI versus the combination of stimuli, which occur in the intermediate area of the substrate, are summarized in Figs 15 and 16, respectively. Our findings indicate that the increase of each stimulus effect increases both the cell elongation and CMI. Obviously, Figs 15 and 16 illustrate that the rate of changes in the cell elongation and CMI is greater in the direction of the electrotactic axis (E.Ω) than that of other cues ($\frac{{\mu }_{\mathrm{\text{ch}}}+{\mu }_{\mathrm{\text{th}}}}{{\mu }_{\mathrm{\text{mech}}}}$), indicating dominant role of electrotaxis. Moreover, increasing the EF strength more than the saturation value does not remarkably affect the cell elongation and CMI. It should be mentioned that, generally, the greater the cell elongation and CMI the less cell random movement. The dominant role of the electrotaxis on cell directional movement is already discussed in the previous work in which a constant spherical cell shape was considered [67].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 15. Variation of the maximum cell elongation, ϵ_elong, versus thermotaxis, chemotaxis and electrotaxis stimuli.

https://doi.org/10.1371/journal.pone.0122094.g015

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 16. Variation of the maximum CMI versus thermotaxis, chemotaxis and electrotaxis stimuli.

https://doi.org/10.1371/journal.pone.0122094.g016