Prediction of Cell Alignment on Cyclically Strained Grooved Substrates

Abstract

Cells respond to both mechanical and topographical stimuli by reorienting and reorganizing their cytoskeleton. Under certain conditions, such as for cells on cyclically stretched grooved substrates, the effects of these stimuli can be antagonistic. The biophysical processes that lead to the cellular reorientation resulting from such a competition are not clear yet. In this study, we hypothesized that mechanical cues and the diffusion of the intracellular signal produced by focal adhesions are determinants of the final cellular alignment. This hypothesis was investigated by means of a computational model, with the aim to simulate the (re)orientation of cells cultured on cyclically stretched grooved substrates. The computational results qualitatively agree with previous experimental studies, thereby supporting our hypothesis. Furthermore, cellular behavior resulting from experimental conditions different from the ones reported in the literature was simulated, which can contribute to the development of new experimental designs.

Introduction

In vitro studies have demonstrated that cells react to both mechanical and topographical stimuli rearranging their orientation and internal cytoskeleton. For instance, cells orient perpendicular to cyclic strain when cultured on stiff substrates or on biaxially constrained collagen gels (1, 2, 3, 4, 5, 6, 7). This phenomenon, called strain avoidance (SA), depends on the profile of the cyclic strain because the degree of cellular (re)orientation is proportional to the amplitude and frequency of the strain (1, 2). Cellular orientation is also influenced by topographical features, a phenomenon known as contact guidance (CG). For example, experimental studies have demonstrated that, when seeded on substrates having nano/microgrooves, cells tend to orient themselves along the direction of these patterns (8, 9, 10, 11, 12, 13, 14, 15, 16). A similar behavior can be observed when cells are cultured on substrates having linear patterns created with microcontact printing (17, 18, 19, 20), or on top of microposts with an ellipsoidal base (21).

Understanding the biochemical processes that determine the effects of the interaction between mechanical and topographical stimuli on cells is fundamental to shed light on their behavior in vivo, where multiple sources provide such types of stimuli. In fact, load-bearing tissues such as cardiovascular tissues, skeletal muscle, and ligaments, are continuously subjected to mechanical strains. At the same time, as experimental evidence has revealed, cells receive topographical cues from collagen, one of the most abundant proteins inside the human body.

Lamers et al. (9) demonstrated that cells on grooved surfaces do not respond to these topographical stimuli when their groove width is lower than a certain threshold. In particular, in their experiments, they cultured osteoblast-like cells on surfaces with grooves of different sizes and they showed that these cells were influenced by the patterns only when the groove width was >75 nm. In fact, for smaller widths, cells oriented themselves isotropically, while for larger values, cells aligned along the direction of the grooves, with the strength of alignment being proportional to the width size.

It is interesting to observe that, when cells are cultured on substrates that are cyclically stretched along the direction of topographical patterns, the effects of SA and CG are in competition. Many different experimental studies have been performed to investigate the final outcome of this competition (22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33). The final cellular organization in these studies appeared to depend on several factors, such as the amplitude of the applied cyclic strain, the size of the used topographical features, and the type of cells investigated.

The experiments of Prodanov et al. (28) are particularly interesting. In their study, the authors used the same technique for the fabrication of grooves and a similar cell type as reported in Lamers et al. (9) and, in addition, they mechanically stimulated cells along the direction of the patterns. Even though only two pitch sizes were investigated (600 nm and 2 μm), the similarities with the work of Lamers et al. (9) facilitate the identification of the effects of both strain and topographical features on the process of cellular (re)orientation. In particular, it was observed that SA can overrule the effects of CG when the size of the pitches is 600 nm. On the other hand, for a pitch size of 2 μm, cells did not respond to mechanical stimuli and aligned along the grooves, following the effects of CG. Therefore, it appears that only relatively large grooves can inhibit the biological mechanisms responsible for the remodeling potential of cells in response to cyclic strain. However, further research is needed to investigate these phenomena and understand the biological processes involved.

In the context of cellular (re)orientation and mechanotransduction, a fundamental role is played by actin stress fibers (SFs) and focal adhesions (FAs). In fact, cellular alignment generally corresponds to the main SF direction (2), and experiments have shown that these acto-myosin bundles are essential for the reorientation of cells in response to cyclic strain (3, 5). On the other hand, FAs are crucial to sense and interact with topographical stimuli, because they form the link between cells and their surroundings (34). Understanding the behavior and interaction of SFs and FAs is therefore key to understand cellular (re)orientation. However, the processes involved are very complex and depend on many factors. Numerical simulations are a valuable tool to test hypotheses and investigate the effects of single physical phenomena, and thereby increase our understanding of the underlying processes.

Several mathematical models have been proposed to explain the remodeling of SFs and predict their interaction with the cellular surroundings (35, 36, 37, 38). In 2006, Deshpande et al. (35) introduced a bio-chemo-mechanical model to predict the effects of cyclic strain on SFs, and therefore give a possible explanation for the phenomenon of SA (39). In their study, they proposed that SFs tend to establish a homeostatic equilibrium with their surroundings, and this leads to the final SF organization. This model has been successively coupled with a thermodynamic model for FA maturation and signal production and diffusion (40, 41). Recently, Vigliotti et al. (42) used this computational framework to explain the phenomenon of CG on grooved surfaces observed in the experiments of Lamers et al. (9). In the proposed theory, CG is caused by the inability of FAs to form above grooves. When the grooves are too large, the signal that originates at the location of FAs on the ridges does not reach the middle of the grooves due to the limited diffusion distance. Therefore, the cascade of intracellular signal that leads to SF formation cannot initiate in this region, and SFs polymerize only on (or in proximity to) the ridges. This spatial heterogeneity affects the orientation of SFs and cells, which consequently align along the direction of the patterns.

After considering their numerical predictions, we can affirm that the mathematical models previously discussed are based on hypotheses that can explain both SA and CG individually. However, the mechanisms responsible for the potential competition between these two phenomena are not yet clearly understood. In this study, we hypothesized that when both mechanical and topographical stimuli are present, SA and CG can both be explained by the diffusion of the FA intracellular signal, and hence this single factor could be able to explain the competition between both phenomena. To investigate the validity of our hypothesis, the numerical model of Vigliotti et al. (42) was extended to analyze the behavior of SFs and FAs of cells cultured on cyclically stretched grooved surfaces (or linear topographical patterns), as cells in this context are subjected to stimuli that induce both SA (stretch) and CG (grooves/patterns). The experiments of Prodanov et al. (28) were chosen as a reference to qualitatively assess our numerical predictions, due to the similarities with the study of Lamers et al. (9) previously simulated by Vigliotti et al. (42). In addition, cellular behaviors resulting from experimental conditions different from the ones reported in the literature were predicted, which could be helpful for the development of new experimental designs.

Materials and Methods

In this study, we simulated the remodeling of cells seeded on grooved surfaces subjected to cyclic strain by adapting the computational framework proposed by Vigliotti et al. (42). The focus was on the homeostatic behavior of a cell in response to cyclic strain, achieved a long time after the moment of seeding; thus, the transient dynamics of FA and SF after seeding were neglected. For these conditions, the following loop of interdependencies between FAs and SFs is proposed (Fig. 1): 1) the applied cyclic stretch causes a periodic increase of high-affinity integrins on a small portion of the ridges present in the surface, where integrins and ligands are in contact; 2) this phenomenon induces FA intracellular signaling cascades involving several proteins (such as IP₃, Rho, Ca²⁺, and ROCK) that diffuse in the cytosol and activate SFs; and 3) SFs exert traction upon their surroundings, causing relative displacement between cell and ridges, which in turn induces more FA formation. The FA dynamics 1) and FA intracellular signaling 2) could be described with equations previously introduced by Deshpande et al. (40) and Pathak et al. (41). However, due to excessive computational costs, the numerical resolution of these equations is not feasible when considering the effects of cyclic stretch. Therefore, we developed some biophysical and mathematical approximations to enable the computational simulations. In the following, the approximated mathematical model is introduced. Readers are referred to Vigliotti et al. (42) for a more comprehensive overview of the original model, while the biological details motivating the equations can be found at the references quoted in each case. For completeness, a brief description of the original model is also provided in the Supporting Material.

Figure 1.

Scheme of the 1D substrate (with ridges and grooves) and cell (with FAs and SFs). To see this figure in color, go online.

Geometry and boundary conditions

The case modeled was a grooved surface parallel to the plane $x_1-x_2$ (Fig. 1) featuring grooves (along x₂) as wide as half of the pitch length $L_0$ (along x₁). We assumed that a cell, with homogeneous thickness b along x₃, was lying on top of the surface. The cell could adhere only on the ridges, without sinking on the grooves. The focus was on the FA and SF dynamics in a central portion of this cell, far from its periphery. In this case, because typical eukaryotic cells are considerably larger than the pitches considered in this study, it is reasonable to assume that the quantities under analysis are spatially periodic along x₁ (with period L₀) and uniform along x₂. Due to this assumption, it was sufficient to model a one-dimensional (1D) unit of the cell along x₁, with the length of this unit equal to L₀ centered at the middle of the groove. In particular, the modeled 1D portion of the cell started from the middle of a ridge $\left(x_1=0\right)$ and ended in the middle of the next one $\left(x_1=L_0\right)$, while the points $x_1=L_0/4$ and $x_1=3L_0/4$ corresponded to the transitions between the ridges and a groove (Fig. 1).

By assuming that the cell is attached to the ridges of a very stiff substrate, we can consider the displacements along the direction x₁ as negligible. In particular, following from the spatial periodicity and symmetries along x₁, the spatial boundary conditions $u_{\text{cell},1}\left(0\right)=u_{\text{cell},1}\left(L_0\right)=u_{\text{sub},1}\left(0\right)=u_{\text{sub},1}\left(L_0\right)=0$ were prescribed, where $u_{\text{cell},1}$ and $u_{\text{sub},1}$ denote the cellular and substrate displacements along x₁. Furthermore, additional boundary conditions were added along x₂ to include the cyclic strain, with a strain amplitude ${\epsilon }_{22,\max }$ and frequency f.

Approximation of the mathematical model for FA dynamics

The computational model developed by Deshpande et al. (40) analyzes the flux of integrins in a cell and the change of state of integrins. Experimental studies have demonstrated that integrins can have two conformational states, defined as low-affinity (or bent) state and high-affinity (i.e., straight) state. Due to their geometry, the bent integrins do not interact with the extracellular molecules and can move within the plasma membrane. Only high-affinity integrins, which can bind with the extracellular ligands, form FAs. Relative displacements between cell and substrate induce the conversion of integrins from the bent to the straight state, a phenomenon regulated by thermodynamic laws. Specifically, chemical potentials can be associated to both states. The chemical potential ${\chi }_L$ of low-affinity integrins is given by the summation of internal energy and entropy of mixing:

$${\chi }_L\left(x,t\right)={\mu }_L+kT\text{ln}\left[{\xi }_L\left(x,t\right)/{\xi }_R\right],$$ (1)

with ${\xi }_L$ as the concentration of low-affinity integrins, μ_L as their reference potential, k as the Boltzmann’s constant, T as the absolute temperature, and ξ_R as the reference binder concentration. The chemical potential χ_H of high-affinity integrins was similar, with additional terms representing the mechanical interaction with the extracellular ligands, as follows:

$${\chi }_H\left(x,t\right)={\mu }_H+kT\text{ln}\left[{\xi }_H\left(x,t\right)/{\xi }_R\right]+\Phi (\Delta )-F(\Delta )\Delta ,$$ (2)

where ξ_H labels the concentration of high-affinity integrins, μ_H is their reference potential, Φ(Δ) is the bond energy stored in a single integrin-ligand complex due to a relative displacement Δ between the integrin and the substrate, and F is the work-conjugate force. This last term relates to the bond energy, and is classically expressed as $F(\Delta )=\left(\partial \Phi (\Delta )/\partial \Delta \right)$ (43). The function describing the relationship between Φ and Δ was defined as a piecewise quadratic potential:

$$\Phi (\Delta )=\{\begin{array}{c}\left(1/2\right)k_s{\Delta }^2\\ k_s\left(2{\Delta }_n\Delta -{\Delta }_n^2-{\Delta }^2/2\right)\\ k_s{\Delta }_n^2\end{array}\begin{array}{c}\Delta <{\Delta }_n\\ {\Delta }_n\le \Delta \le 2{\Delta }_n\\ \Delta >2{\Delta }_n\end{array}.$$ (3)

This function was derived by treating each high-affinity integrin as a nonlinear spring to include the effects of stability of the integrin-ligand complex and by considering the possibility of a rupture. Although simple, this function captures the essential characteristics of FA formation and satisfies the requirement that integrin-ligand complexes should have a finite energy of rupture: $\gamma =\Phi \left(\Delta \to +\infty \right)<+\infty$. In the definition, k_s represents the stiffness of the integrin-ligand bonds, while Δ_n is the stretch such that F is maximum.

The conversion from low- to high-affinity integrins (and vice versa) is typically fast compared to the biological processes considered in our study. Therefore, the two chemical potentials of Eqs. 1 and 2 were assumed to be always in equilibrium, implying ${\chi }_L={\chi }_H$. Thus, given the total integrin conservation statement $\xi ={\xi }_H+{\xi }_L$ and the relative displacement Δ, it would be possible to determine ${\xi }_H$ and ${\xi }_L$. In the original model, this is achieved by numerically solving a diffusion equation for the trafficking of low-affinity integrins along the cellular membrane (Eq. S9 in the Supporting Material) and a model for SF contractility. However, this is not feasible when considering the effects of cyclic strain, due to excessive computational costs, thus the value for ${\xi }_H$ had to be determined with an alternative method. First, the trafficking of low-affinity integrins was neglected by assuming that, because of homeostasis, the concentration of integrins is constantly homogeneous. In particular, we assumed ${\xi }_H+{\xi }_L\equiv {\xi }_0$, where ${\xi }_0$ is the homeostatic value of the total integrin concentration at any location on the cell membrane. Second, the relative displacement Δ was approximated using biophysical rules. In what follows, we first approximate the region where cyclic strain causes a nonzero relative displacement Δ between cell and substrate; then, an evolution over time for the magnitude of Δ is derived; and finally, the increase of high-affinity integrins caused by this displacement is identified.

Recall that the cell is assumed homogenous in the x₂ direction. Moreover, we assumed that in the homeostatic state the cell is almost completely attached to the substrate on the ridges (where FAs are formed) except for a transitional region that we denote U_S. This transitional region has length D_S and is situated immediately adjacent to the groove as indicated in Fig. 1. Higher shear tractions are expected to persist in this transitional region; so, it is reasonable to assume that there exists a nonzero relative displacement of the cell and the substrate in the x₁ direction over this transitional region. Then, shear-lag theory and mechanical equilibrium dictate that the length D_S is given as

$$b{\sigma }_{\text{max}}=k_s{\Delta }_n{\xi }_H{D}_S,$$ (4)

where $b{\sigma }_{\max }$ is the isometric traction force exerted by SFs over the grooves, and k_sΔ_n is the maximum force per integrin-ligand complex (from $F(\Delta )=\left(\partial \Phi (\Delta )/\partial \Delta \right)$ and Eq. 3). Consequently, $k_s{\Delta }_n{\xi }_H$ is the force per unit area, while D_S represents the length over which this force acts to give the required mechanical equilibrium. Eq. 4 cannot be solved directly because D_S and ${\xi }_H$ are both unknown. Rather, we approximate Eq. 4 as $b{\sigma }_{\text{max}}=k_s{\Delta }_n{D}_S/{\beta }_1$, where we have replaced ${\xi }_H$ by a material parameter 1/β₁, which is representative of the maximum density of high-affinity integrins that can form on the cell surface (this maximum density is expected to be attained at the edges of the ridges). Then, by taking into account the inability of FAs to form on a region larger than the size of a ridge, we write

$$D_S=\text{min}\left[\left({\beta }_{1}b{\sigma }_{\max }\right)/\left(k_s{\Delta }_n\right),L_0/4\right],$$ (5)

and it follows that $U_S$ is the region of the pitch from $x=L_0/4-D_S$ until $L_0/4$, and from $3L_0/4$ until $3L_0/4+D_S$.

After approximating the region where $\Delta \left(x,t\right)\ne 0$, the evolution over time of this displacement and its magnitude was determined. First, for simplicity, we assumed that the relative displacement caused by SFs in U_S is instantaneous, nonzero only at the middle of every stretching period (when the amplitude of the stretch is maximal; Fig. 1). Second, at the middle of every stretching period, we hypothesized a linear relationship between the magnitude of the relative displacement and the pitch length, such that

$$\Delta \left(x,t\right)=\{\begin{array}{cc}{\beta }_2{L}_0& x\in U_S,tf=n+1/2,n\in \mathbb{N}\\ 0& \text{otherwise,}\end{array}$$ (6)

with β₂ as a positive parameter. This linear proportion is motivated by the observation that in Vigliotti et al. (42), where FA dynamics was computed by direct numerical integration for static conditions, the magnitude of Δ increased with L₀.

From Eq. 6, ${\xi }_H+{\xi }_L\equiv {\xi }_0$, and ${\chi }_L={\chi }_H$, the concentration of high-affinity integrins can be obtained:

$${\xi }_H\left(x,t\right)=\{\begin{array}{cc}{\xi }_0{\left(\text{exp}\left(\left({\mu }_H-{\mu }_L+\Phi \left({\beta }_2{L}_0\right)-F\left({\beta }_2{L}_0\right){\beta }_2{L}_0\right)/kT\right)+1\right)}^{-1}& x\in U_S,tf=n+1/2,n\in \mathbb{N},\\ {\xi }_0{\left(\text{exp}\left(\left({\mu }_H-{\mu }_L\right)/kT\right)+1\right)}^{-1}& \text{otherwise}.\end{array}$$ (7)

Consequently, at the middle of every stretching cycle, cyclic strain causes an increase ${\hat{\xi }}_H$ of the high-affinity integrin concentration equal to

$${\hat{\xi }}_H={\xi }_0\left|{\left(\text{exp}\left(\left({\mu }_H-{\mu }_L+\Phi \left({\beta }_2{L}_0\right)-F\left({\beta }_2{L}_0\right){\beta }_2{L}_0\right)/kT\right)+1\right)}^{-1}-{\left(\text{exp}\left(\left({\mu }_H-{\mu }_L\right)/kT\right)+1\right)}^{-1}\right|.$$ (8)

FA intracellular signaling cascade

Similar to Vigliotti et al. (42), in this study we reinterpret the computational model of Pathak et al. (41) as a mathematical description of the Rho-ROCK signaling pathway. As presented above, cyclic strain induces conversion of low-affinity integrins to their straight counterpart. These “new” high-affinity integrins bind with extracellular ligands and recruit proteins to form FAs. In turn, this biological process induces activation of Rho. Subsequently, the activated Rho diffuses through the cytosol while it is being dephosphorylated. These phenomena can be modeled by a diffusion equation with additional terms that take into account both the dephosphorylation and activation of Rho:

$$\dot{S}\left(x,t\right)=m_{s}kT\frac{{\partial }^{2}S\left(x,t\right)}{\partial x^2}-k_{d}S\left(x,t\right)+S_{\text{add}}\left(x,t\right),$$ (9)

where S is the concentration of Rho per unit of volume and the overdot indicates the time derivative. In this evolution law we have that: the first term on the right side corresponds to the diffusion of Rho with mobility m_s; the second term models the dephosphorylation of Rho as a first-order reaction with a forward reaction rate equal to k_d and a negligible reverse reaction rate; and the third term corresponds to the increase of Rho in response to every increase of ${\xi }_H$. Due to Eq. 8, this last term can be defined as:

$$S_{\text{add}}\left(x,t\right)=\{\begin{array}{cc}\left(\alpha /b\right){\hat{\xi }}_H& x\in U_S,tf=n+1/2,n\in \mathbb{N},\\ 0& \text{otherwise},\end{array}$$ (10)

such that every increase of concentration of high-affinity integrins corresponds to an increase of α of activated Rho, distributed over the cellular thickness b.

We observe that Eq. 9 was obtained by modifying the evolution law previously proposed by Pathak et al. (41) (and see Eq. S1), to enable the computation of the effects of cyclic strain on the FA intracellular signaling. This equation could be solved independently, without numerically computing the FA remodeling and, in particular, its solution could be found analytically by writing the signal as a Fourier series multiplied by an exponential. The resulting intracellular signal was periodic, due to the periodic nature of $S_{\text{add}}\left(x,t\right)$ and the rate of signal dephosphorylation given by k_d, which is known to be very high. For more information, readers are referred to the section of the Supporting Material that focuses on the analytical resolution of Eq. 9.

The activated Rho, while diffusing and before being dephosphorylated, unfolds ROCK, which is later unfolded by ROCK-specific inhibitors. This can be modeled as follows:

$$\dot{C}\left(x,t\right)={\lambda }_f\left[S\left(x,t\right)/S_0\right]\left[1-C\left(x,t\right)\right]-{\lambda }_{b}C\left(x,t\right),$$ (11)

where C is the ratio between the concentration of activated ROCK and the maximum possible value; S₀ is the reference concentration of Rho; and ${\lambda }_f$ and ${\lambda }_b$ are the rate parameters of the unfolding and folding of ROCK, respectively.

Due to the temporal periodicity of the intracellular signal S and the slow remodeling of C, it was possible to approximate the solution of Eq. 11 by using a similar approach as reported in Ristori et al. (44), thereby further decreasing the computational costs:

$$C_{\infty }\left(x\right)={\left[1+\left({\lambda }_b{S}_0\right)/({\lambda }_f\overline{S}\left(x\right))\right]}^{-1},$$ (12)

where C_∞ is the asymptotic value of ROCK, and $\overline{S}$ is the average value of Rho over its period. Further information about the approximation and an analytical expression for C_∞ are present in the Supporting Material.

SF dynamics

The main components of SFs are F-actin and myosin II proteins, in addition to cross linkers. In the resting state, myosin II and F-actin float into the cytosol, until a number of parallel intracellular signaling pathways induce the formation and contraction of SFs. For example, it is well established that the Rho-ROCK pathway triggers SF activation. We thus coupled Eq. 12 with an evolution law for the activation level of SFs η, where we define this term as the ratio of the concentration of polymerized actin and phosphorylated myosin within a SF bundle to the maximum concentration permitted by biochemistry.

The evolution law for η proposed by Deshpande et al. (35) was derived assuming that η increases proportionally with the concentration of ROCK, while SF dissociation is initiated when SF stress differs from an isometric value:

$$\dot{\eta }\left(x,\phi ,t\right)=k_f{C}_{\infty }\left(x\right)\left[1-\eta \left(x,\phi ,t\right)\right]-k_b\left[1-\sigma \left(x,\phi ,t\right)/{\sigma }_0\left(x,\phi ,t\right)\right]\eta \left(x,\phi ,t\right),$$ (13)

where φ is the angle between the SF direction and x₁ (Fig. 1); k_f and k_b are parameters for the forward and backward rates, respectively; σ is the current SF stress; and ${\sigma }_0={\sigma }_{\text{max}}\eta$ is the isometric SF stress (with σ_max maximum SF stress).

Because SFs exert tension by means of a biological mechanism similar to sarcomeres in muscular cells, the ratio between the current SF stress and its isometric value could be approximated with a simplified version of the Hill equation (45), which has been frequently adopted to model this kind of behavior:

$$\sigma \left(x,\phi ,t\right)/{\sigma }_0\left(x,\phi ,t\right)=\{\begin{array}{c}0\\ 1-\left(k_v\dot{\epsilon }\left(x,\phi ,t\right)\right)/\left(\eta \left(x,\phi ,t\right){\dot{\epsilon }}_0\right)\\ 1\end{array}\begin{array}{c}\dot{\epsilon }\left(x,\phi ,t\right)/{\dot{\epsilon }}_0<-\eta \left(x,\phi ,t\right)/k_v\\ -\eta \left(x,\phi ,t\right)/k_v\le \dot{\epsilon }\left(x,\phi ,t\right)/{\dot{\epsilon }}_0\le 0\\ \dot{\epsilon }\left(x,\phi ,t\right)/{\dot{\epsilon }}_0>0\end{array},$$ (14)

with k_v nondimensional parameter regulating the decrease of the ratio when the SF strain rate $\dot{\epsilon }\left(x,\phi ,t\right)$ differs from a reference value ${\dot{\epsilon }}_0$. The SF strain rate is related to the material strain rate with equation $\dot{\epsilon }\left(x,\phi ,t\right)={\dot{\epsilon }}_{11}{\text{cos}}^2\phi +{\dot{\epsilon }}_{22}{\text{sin}}^2\phi$ (having assumed ${\dot{\epsilon }}_{12}=0$), and the average stress exerted by SFs can be found with a homogenization analysis:

$$\left(\begin{array}{cc}{\sigma }_{11}& {\sigma }_{12}\\ {\sigma }_{12}& {\sigma }_{22}\end{array}\right)=\left(1/\pi \right)\underset{-\pi /2}{\overset{\pi /2}{\int }}\left(\begin{array}{cc}\sigma {\text{cos}}^2\phi & \left(\sigma /2\right)\text{sin}2\phi \\ \left(\sigma /2\right)\text{sin}2\phi & \sigma {\text{sin}}^2\phi \end{array}\right)d\phi .$$ (15)

In addition to the active stress, we considered the passive elasticity of cells, which was modeled with a Neo-Hookean constitutive equation:

$${\sigma }_i^P=E/\left[4\left(1+\upsilon \right)\right]J^{-2}\left(2{\lambda }_i^2-I_1\right)+E/\left[2\left(1-\upsilon \right)\right]\left(J-1\right),$$ (16)

where ${\sigma }_i^p$ and ${\lambda }_i$ are the principal passive stress and the principal stretch values along the direction x_i; E denotes the Young’s modulus; υ is the Poisson’s ratio; and J and I₁ are defined as $J={\lambda }_1{\lambda }_2$ and $I_1={\lambda }_1^2+{\lambda }_2^2$, respectively. As a consequence of Eqs. 15 and 16, the total cellular stress Σ_i (along direction x_i) can be obtained adding the active and passive stresses:

$${\Sigma }_i={\sigma }_{ii}+{\sigma }_i^P.$$ (17)

Finally, the equation that regulates the mechanical equilibrium in the cell was specified as:

$$b\frac{\partial {\Sigma }_i}{\partial x_i}={\xi }_{H}F.$$ (18)

Summary of the equations and computational approach

In this study, an approximation was derived for the mathematical model of the FA intracellular signal used in the study of Vigliotti et al. (42). In particular, the trafficking of low-affinity integrins was neglected, while integrin thermodynamics (Eqs. 1–3) was incorporated into the model by approximating the effects of cyclic strain on the increase of high-affinity integrins (Eq. 8). Subsequently, an equation for the diffusion of Rho at every stretching cycle was identified (Eq. 9). This equation was coupled with the boundary conditions $\left(\partial S/\partial x\right)\left(0,t\right)=\left(\partial S/\partial x\right)\left(L_0,t\right)=0$, resulting from the symmetries of the problem, and was analytically solved, which considerably decreased the computational time. Then, the evolution law for ROCK (Eq. 11) was analytically approximated with Eq. 12, thereby further decreasing the computational cost. Readers are referred to the Supporting Material for more details about the resolution and approximation of Eqs. 9 and 11. The remaining Eqs. 13–18 for SF remodeling and mechanical equilibrium were numerically solved until the achievement of a homeostatic state. The parameters used for the computational simulations were very similar to the ones chosen in Vigliotti et al. (42) and Wei et al. (39), and are reported in Table 1. The computational simulations were repeated for several pitch sizes $\left(0.05\mu \text{m}\le L_0\le 1\mu \text{m}\right)$ and strain amplitudes $\left(0.01\le {\epsilon }_{22,\max }\le 0.08\right)$, consistent with Vigliotti et al. (42) and Prodanov et al. (28). Similar to this last study, the cyclic strain frequency was chosen equal to 1 Hz, to mimic physiological conditions.

Table 1.

Parameters Used in the Computational Simulation

Parameters	Values
b	1 μm
α	7 × 10−2
λb	1.5 × 10−2 s−1
μH − μL	5 kT
E	4 kPa
kv	2.5 × 10−1
kb	3 × 10−3 s−1
T	3.1 × 102K
ms	104 s(mg)−1
kd	7.5 × 102 s−1
ks	15 pN (μm)s−1
υ	0.45
${\dot{\epsilon }}_0$	2.8 s−1
β1	2 × 10−2μm2
S0	103 molecules × μm−3
λf	2.5 × 105 s−1
ξ0	103 integrins × μm−2
Δn	1.3 × 10−1μm
σmax	2.5 kPa
kf	3 × 10−2 s−1
β2	5 × 10−3

Results

In accordance with the literature, the model predicted that small values of L₀ corresponded to a SA behavior of cells, while cells showed a CG behavior for large pitch sizes. These differences can be explained by the hypothesized influence of L₀ on the FA intracellular signal and, ultimately, on the ROCK profile and SF distributions. For brevity, the results of these last quantities will be shown only for SFs oriented parallel and perpendicular to the grooves, as we observed that the results of the remaining SF orientations varied monotonically from $\eta \left(x,0\right)$ to $\eta \left(x,90\right)$. Consequently, the extreme values of SF activation provide all the information relevant to interpreting the results.

The periodic intracellular signal diffusion

In the mathematical model, we assumed that the amplitude of the cyclic strain ε_22,max does not significantly affect the magnitude of S; therefore, Fig. 2 can be considered as representative for every value of ε_22,max.

Figure 2.

(a and b) Profile of the signal S on a pitch for a full period of 1 s, for different groove sizes. (c and d) Signal S in the middle of the groove for different groove sizes just after the initiation of the signal. The value t = 0 s corresponds to the moment of initiation of the signal. To see this figure in color, go online.

Several physical phenomena and parameters determine whether a significant amount of intracellular signal can diffuse all over the pitch. For t = 0 s, which corresponds to the moment of initiation of the signal, S was initially nonzero only on a small part of the ridges with a magnitude equal to $\max \left(S\left(x,0\right)\right)=\left(\alpha /b\right){\hat{\xi }}_H$ (Fig. 2, a and b), with this quantity increasing with L₀ (Eqs. 7–10). From this region, the signal diffused toward the middle of the grooves and ridges present in the substrate while, in addition, it decreased homogenously over time due to signal dephosphorylation (Fig. 2, a and b). Therefore, in the case of slow diffusion compared to dephosphorylation and for large groove sizes, the amount of intracellular signal that reached the middle of the grooves was negligible (Fig. 2 d).

The asymptotic profile of ROCK C_∞

As determined by Eqs. 11 and 12, the asymptotic profile of C_∞ is largely dependent on the intracellular signal S and correspondences between their graphs can be identified (Fig. 3). First, because the intracellular signal started diffusing from the ridges, the maximum value of C_∞ was located in that region. Second, similar to $\max \left(S\left(x,0\right)\right)$, max(C_∞) increased with L₀ and was independent of the magnitude of ε_22,max. Finally, similar to $S\left(L_0/2,t\right)$, the value of C_∞ in the middle of the grooves decreased when L₀ increased.

Figure 3.

(a–d) Representation of values of the SF activation level for the direction parallel (${\eta }_{90}$, green) and perpendicular (${\eta }_0$, blue) to the grooves and spatial distribution of C_∞ (red) for different pitch sizes and strain amplitudes. To see this figure in color, go online.

The activation level of SFs perpendicular to the strain and grooves η₀

As shown in Fig. 3, ${\eta }_0$ appeared to be scarcely dependent on ε_22,max but strongly dependent on L₀. This behavior is a consequence of Eqs. 13 and 14, which describe the dependence of the SF activation level on C and the strain rate. In particular, Eq. 13 determines that ${\eta }_0$ increases proportionally with C_∞, which is in turn largely dependent on L₀. On the other hand, Eqs. 13 and 14 together dictate that a decrease in SF activation level is caused only by compressions along the specific SF direction. Because the applied stretch ε_22,max was perpendicular to the direction of ${\eta }_0$ and resulted only in a low strain along x₁, the amplitude of the applied stretch ε_22,max hardly affected the value of ${\eta }_0$.

Consequently, ${\eta }_0$ was almost maximal on every point where $C_{\infty }>>0$. In particular, it was uniformly high for $L_0=0.05\mu \text{m}$ (Fig. 3, a and b) while, for $L_0=1\mu \text{m}$, the value of ${\eta }_0$ was approximately zero in the middle of the pitch (Fig. 3, c and d), similar to C_∞. It is worth noticing that in the middle of the ridges ${\eta }_0=1$, although we do not have $C_{\infty }>>0$. This occurs because in this region ${\dot{\epsilon }}_{11}=0$, so there was no SF depolymerization perpendicular to the grooves in this region, and even a very small value of C_∞ asymptotically induced the maximal value of ${\eta }_0$ (Eqs. 13 and 14).

The activation level of SFs parallel to the strain and grooves η₉₀

The activation level of SFs parallel to the strain and grooves (labeled with ${\eta }_{90}$) was always lower than ${\eta }_0$ (Fig. 3), because the cyclic strain was applied along the associated direction. Consequently, the difference between ${\eta }_0$ and ${\eta }_{90}$ increased with ε_22,max. It can also be noticed that the decrease in ${\eta }_{90}$ caused by the applied stretch was lower when L₀ increased. This occurred because C_∞ was larger above the ridges in those simulations, and thus cells tended to polymerize more SFs in that region, thereby counteracting the depolymerization induced by the applied stretch (Fig. 3, b and d).

Interpretation of the computational results

Fig. 4 shows the SF activation levels ${\eta }_0$ and ${\eta }_{90}$ on a square with dimensions $2L_0\times 2L_0$ to describe how the profiles of the activation levels may lead to a specific cellular alignment as a consequence of the mechanisms described in the previous sections. In particular, to interpret the SF activation levels in terms of global cellular orientation, we hypothesized that cells prefer to orient toward the direction(s) along which they can form long SF bundles, which correspond to the directions for which the SF activation levels are continuously nonzero.

Figure 4.

(a–h) Representation of the SF activation level on a square of dimension $2L_0\times 2L_0$. The model is explicitly 1D, in x₁, and assumes constant levels of SFs and FAs in x₂. The x₂ dimension is added here for clarity. The values of the SF activation level for the direction perpendicular (${\eta }_0$, first and third columns) and parallel (${\eta }_{90}$, second and fourth columns) to the grooves are shown for different groove sizes and strain amplitudes. (Dashed lines) Direction of the grooves; (solid lines) direction of the represented SF activation level. To see this figure in color, go online.

Starting from the smallest pitch size (0.05 μm), Fig. 4, a and b, shows that the SF distribution was isotropic for low values of ε_22,max, because the predicted SF activation levels lead to long SF bundles in both directions. In fact, both ${\eta }_0$ and ${\eta }_{90}$ were continuously nonzero along their associated directions (x₁ and x₂, respectively), indicating the formation of long SFs perpendicular and parallel to the grooves. On the other hand, for ${\epsilon }_{22,\max }=0.08$ (Fig. 4, c and d), the SF distribution is likely to be anisotropic due to the different responses of ${\eta }_0$ and ${\eta }_{90}$ to the applied stretch. Specifically, while ${\eta }_0$ was still continuous and maximal along its direction, ${\eta }_{90}$ decreased considerably due to the applied strain. Thus, for this combination of pitch size and strain magnitude, the phenomenon of SA was predicted to be dominant over CG.

For larger pitch sizes, CG was predicted to become dominant over SA. In fact, for $L_0=1\mu \text{m}$, the increase of ${\epsilon }_{22,\max }$ did not cause significant changes in the SF activation levels (see Fig. 4 e versus Fig. 4 g, and Fig. 4 f versus Fig. 4 h). In the middle of the grooves, for both pitch sizes and all strain magnitudes, ${\eta }_0$ and ${\eta }_{90}$ were both equal to zero, while their values were almost maximal above the ridges. Consequently, while ${\eta }_{90}$ was continuous along its direction and maximal above the ridges, ${\eta }_0$ was not continuously nonzero along the direction of the pitch. For these reasons, cells on substrates with large grooves are more likely to form long SF bundles only along the direction of these patterns, independently of the applied strain amplitude, implying that CG is dominating SA under these conditions.

SA for small grooves and CG for large grooves

When $\text{min}\left({\eta }_0\right)\ne 0$, then ${\eta }_0$ is continuously nonzero along the pitch. On the other hand, when $\max \left({\eta }_{90}\right)>0$, then ${\eta }_{90}>0$ above the ridges in proximity of the grooves and it is continuous due to the hypotheses of the mathematical model. Therefore, due to the previous observations, determining $\max \left({\eta }_{90}\right)$ is sufficient to identify when we have long parallel SF bundles, while $\text{min}\left({\eta }_0\right)$ is important for perpendicular SFs.

Due to these observations, a representation of the interpretation of the results (Fig. 5) can be obtained drawing two lines for every computational simulation: the vertical lines are proportional to the average of ${\eta }_{90}$ along the direction of the grooves calculated in $x_1=L_0/4$, as a reference value, while the horizontal lines are proportional to the average of ${\eta }_0$ along the pitch. This last quantity was set equal to zero when ${\eta }_0=0$ in the middle of the pitch, due to the fact that ${\eta }_0$ has to be continuous to induce long SFs along that direction.

Figure 5.

Overview of the results for different pitch sizes and strain amplitudes. The vertical and horizontal lines are proportional to the average of, respectively, ${\eta }_{90}$ and ${\eta }_0$ along their particular direction, where $x_1=L_0/4$ was chosen as a representative location for ${\eta }_{90}$. In addition, the average of ${\eta }_0$ was set to zero if ${\eta }_0=0$ in the middle of the grooves, as SFs are not able to form when the activation level is discontinuous.

Fig. 5 shows that the numerical simulations predicted SA for pitch sizes $L_0\le 0.15\mu \text{m}$, with the degree of cell (re)orientation proportional to the strain amplitude, and CG for values $L_0\ge 0.25\mu \text{m}$.

The transition phase

Interestingly, for the pitch size $L_0=0.2\mu \text{m}$, an enhancement of the effects of CG by the cyclic strain was predicted. For the pitch size $L_0=0.2\mu \text{m}$ and ${\epsilon }_{22,\max }=0.01$, Fig. 6 a shows that $\text{min}\left({\eta }_0\right)\ne 0$ and $\max \left({\eta }_{90}\right)>0$, and thus long SF bundles can form along both x₁ and x₂. Because the average values of ${\eta }_0$ and ${\eta }_{90}$ are comparable (Fig. 5), in this case the cellular orientation was predicted as isotropic. Conversely, as shown in Fig. 6 b, when ${\epsilon }_{22,\max }=0.08$, the values of ${\eta }_0$ were zero in the middle of the pitch; consequently, ${\eta }_0$ was not continuously nonzero along its specific direction. Therefore, according to the interpretation that we proposed, cells are likely to align parallel to the grooves. In conclusion, in the case of $L_0=0.2\mu \text{m}$, our simulations predicted an isotropic alignment for ${\epsilon }_{22,\max }=0.01$ and an alignment with the grooves for ${\epsilon }_{22,\max }=0.08$ (and therefore CG), which is opposite to the behavior observed for smaller grooves, for which strain overruled CG in the case of large amplitudes.

Figure 6.

(a and b) Representation of the values of the SF activation level for the direction parallel (${\eta }_{90}$, green) and perpendicular (${\eta }_0$, blue) to the grooves and the spatial distribution of C_∞ (red) for $L_0=0.2\mu \text{m}$ and for different strain amplitudes. To see this figure in color, go online.

Discussion

The aim of this study was to enhance our understanding of the mechanisms that determine the (re)orientation of cells in response to both mechanical and topographical stimuli. Our main hypothesis was that, under these conditions, the final cellular orientation is determined by a combination of FA intracellular signal diffusion and mechanical cues that affect the polymerization and depolymerization of SFs, respectively. To confirm our hypothesis, we adapted the computational framework proposed by Vigliotti et al. (42) to study the remodeling of SFs and FAs in cells cultured on cyclically stretched grooved surfaces. In addition, we physically approximated some of the equations present in the original mathematical model to avoid excessive computational costs.

The computational simulations predicted that cells mainly respond to mechanical stimuli when cultured on surfaces having pitch sizes $L_0\le 0.15\mu \text{m}$, while they show a CG behavior for pitch sizes $L_0\ge 0.25\mu \text{m}$. In particular, when $L_0\le 0.15\mu \text{m}$, the results showed that cells tend to orient perpendicular to the topographical patterns and the applied cyclic strain, with the strength of this alignment proportional to the cyclic strain amplitude. On the other hand, for $L_0\ge 0.25\mu \text{m}$, the simulations predicted a cellular alignment parallel to the direction of the grooves and the mechanical stimuli, regardless of the strain amplitude, due to the effects of CG.

The computational results can be explained considering the hypothesized characteristics of the FA intracellular signal, which was produced on a small portion of the ridges and then diffused through the whole pitch. When the pitch size was small, significant signal could reach the middle of the grooves through diffusion. As a consequence, SFs polymerized in a spatially homogenous fashion, without exhibiting significant effects due to the topographical patterns. In these conditions, the only asymmetry in SF formation was caused by the compression along the grooves, which led to SA in the case of high strain magnitudes. In contrast, when the pitch size was large, a significant amount of FA intracellular signal was dephosphorylated before reaching the middle of the grooves, and thus SF polymerization was not induced in this region. Thus, the activation level of SFs perpendicular to the grooves was predicted to be discontinuous along the corresponding direction. Due to this discontinuity, cells cannot form long SFs along direction x₁. On the other hand, in the case of large pitch sizes, despite the presence of compression, long SFs parallel to the ridges were predicted to form due to the high FA intracellular signal production, and thereby SF polymerization, above the ridges, which dominated the SF depolymerization due to the cyclic strain. In conclusion, in this case, cells only formed long SFs that were parallel to the grooves, and responded to CG by aligning along these patterns.

Verifying the computational results is challenging, due to the lack of experimental data on the behavior of a single cell type with multiple groove sizes and strain amplitudes. Nevertheless, we can observe that the results obtained for very small and large patterns are qualitatively in agreement with the experiments of Prodanov et al. (28), who observed that osteoblast-like cells exhibit SA behavior when cultured on relatively small grooves, and CG behavior in case of large grooves. On the other hand, when quantitatively compared, the model predictions do not match perfectly with the experimental observations. For example, the threshold between SA and CG is predicted for a different size than the one expected analyzing the results of Prodanov et al. (28). These differences might be smaller in case of a different set of model parameters, but quantitatively matching one set of experimental results was not the aim of our study, which was focused on the general cellular behavior and not on a specific cell type. Moreover, Prodanov et al. (28) applied a different cyclic strain profile than the one used in this study, which might have affected the results. In particular, they applied an intermittent cyclic strain instead of a continuous one, and this difference is known to affect the behavior of cells (46). In addition, changing the model parameters to match specific experimental results might be challenging, due to the large number and high level of interdependency of the parameters. The mathematical model adopted in this study considers several complex biological phenomena, which was essential to capture the effects of both mechanical and topographical stimuli but, at the same time, implied that many model parameters were necessary. Due to these considerations, the original parameter values were used, also to maintain the similarities with the studies of Vigliotti et al. (42) and Wei et al. (39).

Interestingly, for a pitch size equal to 0.2 μm, the simulation results suggested that the applied cyclic strain contributes to CG together with the topographical cues, rather than inducing SA. In fact, for this pitch size and low levels of strain, cells were predicted to be slightly aligned along the direction perpendicular to the strain and grooves (SA), while for larger strain amplitudes, only SFs parallel to the grooves were able to form (CG). This phenomenon was a consequence of the characteristics of the FA intracellular signal diffusion and the mechanical properties of the cellular cytoskeleton. For this pitch size, the intracellular signal was able to reach the middle of the grooves sufficiently to induce a significant SF polymerization in this region. At the same time, due to the cytoskeleton elasticity, a small portion of the cyclic strain applied along the direction of the grooves was transmitted perpendicularly, causing SF depolymerization in this direction. For small levels of strain, this depolymerization was not sufficient to cause the disassembly of SFs perpendicular to the grooves, and thus cells aligned perpendicularly to the direction of stretch (SA). On the other hand, for larger strain amplitudes, the perpendicular SFs were completely depolymerized in the middle of the grooves, and therefore they could no longer form because of the discontinuity of the SF activation level along their direction. As a result, for large strain amplitudes, cells were predicted to be oriented parallel to the topographical patterns (CG).

The enhancement of CG due to cyclic strain suggested by the numerical predictions has never been directly observed in experiments, and its experimental verification may be complex. It is, however, interesting to notice that Prodanov et al. (28) reported that cyclic strain applied to substrates with 1 μm-wide grooves diminished the number of cells aligned perpendicular to the strain and topographical patterns compared to the one observed under static conditions. Therefore, this result may support our prediction that cyclic strain enhances the effects of CG for specific conditions. On the other hand, the experimental results of Wang and Grood (22) suggest that a further increase of strain amplitude might stop the enhancement of CG by mechanical stimuli and cause realignment of cells in response to strain. In their experiments, they observed that human skin fibroblasts cultured on 1 μm-wide grooves did not respond to strain with strain amplitudes <8%, while they realigned in response to mechanical stimuli when cyclically strained with 12% strain amplitude. With the mathematical model proposed in this study, predicting this kind of behavior is not possible. This qualitative difference might be explained by the relationship between the amplitude of the strain and the FA intracellular signal hypothesized in the mathematical model. In our study, we assumed that the dependency between strain amplitude and FA intracellular signal magnitude is negligible, while this might not be the case when very large strains are applied. The exact relationship between the amplitude of the strain and the FA intracellular signal magnitude could be investigated in future experimental studies. This would then enable the refinement of our model assumption and numerical predictions. In addition, although some interaction between the different orientations of stress fibers is likely to occur, we ignored these interactions in this study, as this phenomenon was assumed negligible in the original model of Deshpande et al. (35). Considering the capability of this model in capturing many cellular behaviors, we decided not to modify its assumptions in our study, although it could still be a subject of future work.

In the Materials and Methods, to motivate the assumptions that led to the equations, we associated the FA intracellular signal S to the concentration of Rho, while C was related to the concentration of ROCK, with consequent SF activation. On the other hand, we can also interpret the model as a mathematical abstraction of the signaling pathway that leads to the opening of stress-activated calcium channels (SACCs) and subsequent SF formation. Thodeti et al. (47) showed that, during the remodeling of endothelial cells in response to cyclic strain, an increase of calcium influx through SACCs is induced by mechanical strain to bound integrins. This occurs in a similar fashion as in our model. Here, once having identified the variables S and C as IP₃ ions and Ca²⁺, respectively, we have that: 1) mechanical stimuli of high-affinity integrins induce an increase of IP₃ ions; 2) IP₃ ions diffuse through the cytosol and stimulate the opening of SACCs and an increase of Ca²⁺ influx; and 3) a higher concentration of Ca²⁺ stimulates the activation of SF contraction. Thus, it may be possible that our simulations match with the experimental results because they approximate the function of SACCs. However, experimental studies aimed at unraveling whether SACCs are a determinant of cellular (re)orientation have shown contradictory results. For instance, while Standley et al. (48) observed reorientation of rat vascular smooth muscle cells in response to cyclic stretch even when SACCs were inhibited, this was not the case for Hayakawa et al. (49), who showed that inhibition of SACCs with gadolinium can restrain cellular orientation. Nevertheless, in this last study, reorientation of SFs was obtained, a phenomenon known to be the first step toward cellular realignment. Due to the uncertainty of the exact biochemical processes involved in the interplay between mechanical and topographical stimuli, we decided to focus first on whether or not signal diffusion would at least be able to describe experimental findings, regardless of the specific biochemical pathway involved. Of course, a more detailed investigation of the underlying processes would be very worthwhile for future studies.

In this study, the computational method and assumptions were focused on substrates with parallel grooves or lines created with a microcontact printing technique. These substrates are characterized by a spatial homogeneity along the direction of the patterns, which was used to prevent excessive computational costs. The competition between SA and CG has also been observed with different experimental conditions, e.g., with cells positioned on top of elliptic microposts stretched along their long axis (21). With those experiments, it was demonstrated that cap and basal actin fibers respond differently to mechanical and topographical stimuli. Unfortunately, simulating these experiments with the computational framework used in this study is challenging due to excessive computational costs associated with the three-dimensional nature of the problem. Similarly, simulating experiments where cells are surrounded by extracellular matrix or collagen gels is not currently feasible with this computational method, due to the difference of spatial scales between the FAs and the whole tissue. For these situations, the development of continuum or multiscale approaches might be necessary. Nevertheless, our study can be useful to gain the necessary information to develop these continuum or multiscale computational frameworks.

Conclusions

This study suggests that the interaction between FA intracellular signal diffusion and mechanical cues is important in the context of cellular realignment in response to both mechanical and topographical stimuli. The computational results demonstrated that the mechanisms previously suggested by Vigliotti et al. (42) represent a possible explanation not only for solely CG, but also for its competition with SA. In addition, the computational simulations suggested that, under certain conditions, mechanical stimuli can actually enhance the effects of topographical stimuli, and this behavior could be verified by means of new experimental protocols.

Author Contributions

T.R., A.V., F.P.T.B., S.L., and V.S.D. designed the research; T.R., A.V., and V.S.D. performed the research; T.R., F.P.T.B., S.L., and V.S.D. analyzed the data; and T.R. and S.L. wrote the article.

Editor: Charles Wolgemuth.