A theoretical model of collective cell polarization and alignment

Abstract

Collective cell polarization and alignment play important roles in tissue morphogenesis, wound healing and cancer metastasis. How cells sense the direction and position in these processes, however, has not been fully understood. Here we construct a theoretical model based on describing cell layer as a nemato-elastic medium, by which the cell polarization, cell alignment and cell active contraction are explicitly expressed as functions of components of the nematic order parameter. To determine the order parameter we derive two sets of governing equations, one for the force equilibrium of the system, and the other for the minimization of the system’s free energy including the energy of cell polarization and alignment. By solving these coupled governing equations, we can predict the effects of substrate stiffness, geometries of cell layers, external forces and myosin activity on the direction- and position-dependent cell aspect ratio and cell orientation. Moreover, the axisymmetric problem with cells on a ring-like pattern is solved analytically, and the analytical solution for cell aspect ratio are governed by parameter groups which include the stiffness of the cell and the substrate, the strength of myosin activity and the external forces. Our predictions of the cell aspect ratio and orientation are generally comparable to experimental observations. These results show that the pattern of cell polarization is determined by the anisotropic degree of active contractile stress, and suggest a stress-driven polarization mechanism that enables cells to sense their spatial positions to develop direction- and position-dependent behavior. This, in turn, sheds light on the ways to control pattern formation in tissue engineering for potential biomedical applications.

1. Introduction

Cells often change their shape and become anisotropic while adapting to inhomogeneities of an extracellular matrix or responding to chemical or mechanical stimuli. The transition from an isotropic to an anisotropic state is known as cell polarization. Pattern formation caused by cell polarization has been observed in various studies of collective cell migration and associated tissue development processes, indicating that a specific arrangement of cell polarization and alignment are indispensable for processes of directional migration (Vedula et al., 2012), tissue development (Leptin and Grunewald, 1990), wound healing (Brugues et al., 2014), and cancer metastasis (Friedl et al., 2012). For instance, during embryonic development the cell collective grown from a single cell, the zygote, are shaped into different organs via the direction- and position-dependent behaviors (Keller, 2002; Kimmel et al., 1995). It has been extensively studied that spatial gradients of chemical signals, such as morphogen and chemokines, can induce direction- or position-dependent behaviors of collective cells (Driever and Nüsslein-Volhard, 1988; Nelson, 2009). However, recent studies suggest that besides the chemical signals, the mechanical factors also play critical roles in the direction- or position-dependent collective cell behaviors (Heisenberg and Bellaïche, 2013).

Given the intrinsic active contraction of cytoskeleton, throughout the collective cells the tension forces are transmitted among cells via intercellular adhesion, while the associated cell-matrix interaction, cell traction forces, are simultaneously produced. As thus collective cells with their substrates constitute a complex multi-body mechanical system in which the transduction of the tension and the traction force is intrinsic. Recent studies suggest that the direction- or position-dependent collective behaviors of cells can be regulated by the spatial distribution of traction and tension forces controlled by the mechanical factors, including geometric constraints, substrate stiffness and external forces (He et al., 2019b). For example, cell proliferation and differentiation are correlated with the spatial gradient of traction forces. Regions of high traction result in fast proliferation and stem cell’s differentiation towards osteogenesis, whereas the regions of low traction suppress cell proliferation and cause stem cells to differentiate to adipocytes (Nelson et al., 2005; Ruiz and Chen, 2008). High expression of cancer stem cell markers is located in the regions of large von Mises stress in cell layer, while low expression of the markers is located in the regions of small von Mises stress (Lee et al., 2016). Moreover, driven by the in-plane maximum shear stress, collective cells polarize and migrate along the direction of the maximum principal stress (He et al., 2015; Liu et al., 2016, 2018; Tambe et al., 2011), and larger shear stress causes a higher cell aspect ratio (AR) (He et al., 2015). In addition, strong cellular cortical contraction and weak cell-cell adhesion restrict cell collectives in a jammed solid-like state, however, to the opposite, the weak cellular cortical contraction and strong cell-cell adhesion cause cell collectives to unjam and behave like fluid (Bi et al., 2015; Park et al., 2015). Counterintuitively, p53 which is usually thought to function as a tumor suppressor and suppress cell migration, can promote the motility of cancer cells in confluent cell layers by decreasing cell-cell adhesion and increasing the traction forces (He et al., 2019a). Strikingly, during embryogenesis the direction- and position-dependent rearrangement of collective cells can be predicted by the pattern of the tension/stress in collective cells (He et al., 2019b; Streichan et al., 2018).

The important roles of mechanical forces in cell polarization and alignment have been confirmed by theoretical studies through various models. For example, continuum-based models simulating active contractile cells as prestressed elastic layers can reproduce the distribution of the in-plane tension/stresses in cell layers and traction forces between a cell layer and a matrix, and thereby predict the position-dependent distribution of focal adhesion, alignment of cytoskeleton, and cell polarization and migration (Deshpande et al., 2008; Edwards and Schwarz, 2011; Feng et al., 2018; Gao and Gao, 2016; He et al., 2019b, 2015, 2014; Notbohm et al., 2016; Rosakis et al., 2015). The vertex model, modeling cells as polygons and considering the elasticity of cell area, active contraction of cell membrane cortex and intercellular interaction energy, predicted that the cell polarization plays a critical role in collective cell migration and proliferation (Bi et al., 2015; Farhadifar et al., 2007), and the tension among cells can determine the patterns of collective cell oscillation in the Drosophila amnioserosa (Lin et al., 2017, 2018). The phase-field model (e.g. cellular Potts model), in which the phase index reflecting cell location and shape are calculated by minimizing system’s free energy, can capture the main features of cell polarization and alignment regulated by cell-cell and cell-matrix interactions, and predict different patterns of collective cell migration (Löber et al., 2015) and cancer cell invasion (Palmieri et al., 2015).

As briefly reviewed above, it has been shown both experimentally and theoretically that the spatial distribution of mechanical forces drives the direction- or position-dependent collective cell behaviors. However, the underpinning mechanism has not been fully clarified. As opposed to passive materials, cells actively modulate the ordering of their cytoskeleton and their shapes to fit external environments such as substrate stiffness (Gupta et al., 2015; He et al., 2015; Liu et al., 2016), cell-substrate interfacial geometries and curvature (He et al., 2015; Liu et al., 2018), and stimuli of external forces (Kong et al., 2008; Zhong et al., 2011). Moreover it has been found that cells are stretched along the maximum principal stress direction in cell layer due to the intrinsic contractility of cytoskeleton, and this elastic deformation of cells causes their shape, as well as actin-myosin cytoskeleton, to polarize along the same direction (He et al., 2015; Kong et al., 2010; Liu et al., 2016). Due to cell/cytoskeleton’s polarization, the cell’s active contraction becomes anisotropic, which in turn changes the stress state in the cell layer. This series of responses forms an active mechanical feedback between the intrinsic active contraction and the polarization of cell shape (He et al., 2015; Ladoux et al., 2016; Liu et al., 2016). This feedback is reminiscent of the responding behaviors of active gel or nematic material (Duclos et al., 2014; Prost et al., 2015) which has been used to simulate cell arrangement and its effect on cell motility or apoptotic cell extrusion (Duclos et al., 2017; Kawaguchi et al., 2017; Saw et al., 2017), and cell polarization caused by substrate stiffening (Gupta et al., 2015). However, it remains unknown how this active mechanical feedback contribute to the direction- or position-dependent behaviors in collective cells.

In this work we developed a mechanical model of cell polarization and alignment along the lines of active gel theory (Pismen and Koepf, 2014; Prost et al., 2015). Our model introduces the nematic tensor order parameter to characterize the intrinsic active contraction of cytoskeleton. In this way, the cell AR and cell angle are explicitly expressed as functions of the components of the order parameter, which are determined by solving two sets of governing equations. Moreover, the model is used to predict the distribution of cell AR and orientation angle in different contexts, such as varying substrate stiffness, different size/shape of geometric constraints, different types of mechanical forces, and varying myosin activity. Our predictions are generally comparable with experimental observations. These results shed light on the way active mechanical feedback regulates the direction- and position-dependent cell polarization and alignment under different conditions, and suggest the underlying mechanism of the stress-driven cell polarization in cell collectives. In particular, an analytical solution of cell polarization on a ring-like pattern is derived for the first time, which enhances the insight into the effects of the different physical factors on cell polarization and alignment.

2. The model

2.1. Governing equations

The nematic tensor order parameter Q is introduced to characterize the cellular active contraction which is coupled to cell polarization and orientation, as shown in Fig. 1. Cell polarization has different meanings, and here it refers to the cell aspect ratio (AR). The tensor Q is expressed in Cartesian coordinates through two scalars p and q (Koepf and Pismen, 2015; Pismen and Koepf, 2014)

$$Q=\left[\begin{array}{cc}p& q\\ q& -p\end{array}\right]$$ (1)

Fig. 1.

Schematic illustration of cell polarization and orientation using the nematic model. The cell aspect ratio (AR) is represented by AR = (1 + S)/(1 − S). The angle between the long axis of a polarized cell and the x axis ϕ_cell coincides with the direction of the maximum principal active stress of ${\sigma }_{ij}^{ac}$.

The order parameter S that quantifies the anisotropic degrees of the cellular active contraction can be calculated as

$$S=\sqrt{\sum _{i,j}\left(Q_{ij}^2/2\right)}\hspace{0.17em}i,j=1,2.$$ (2)

Specifically, p = S cos 2ϕ_cell and q = S sin 2ϕ_cell, where ϕ_cell is the angle between the long axis of a polarized cell and the x axis.

The cellular active contractile stress can be expressed as function of the order parameter

$${\sigma }_{ij}^{ac}=\gamma \left({\delta }_{ij}+Q_{ij}\right),$$ (3)

where the parameter γ quantifies the strength of myosin activity within cells and δ_ij is the Kronecker delta. As the cellular active stress is contractile, γ should be positive. For a non-polarized cell, S = 0 and ${\sigma }_{ij}^{ac}=\gamma {\delta }_{ij}$, indicating isotropic active contraction of a non-polarized cell. For a polarized cell, S varies between zero and unity. According to the traction-distance law (He et al., 2014), the active contraction force increases with the distance from cell center (He et al., 2014; Lemmon and Romer, 2010). One can assume therefore that the maximum and minimum principal stresses of ${\sigma }_{ij}^{ac}$ align with the long and short axes of cells, respectively. If the coordinate system is rotated to align the x axis with the long axis of a cell, the principle active stress, denoted with a prime, can be expressed as

$${\sigma }_{ij}^{ac}\prime =\gamma \left[\begin{array}{cc}1+S& 0\\ 0& 1-S\end{array}\right],$$ (4)

so that the two principal stresses, γ(1 + S) and γ(1 − S), are active contractile stresses along the long and short axes of the cells, respectively (Fig. 1). As thus, S quantifies the anisotropic degree of the active stress. According to the traction-distance law (He et al., 2014), cells contracts stronger along their long axis than along their short axis. Therefore, the cell AR can be estimated by the ratio between the two principal stresses (Fig. 1)

$$AR=\left(1+S\right)/\left(1-S\right)$$ (5)

The cell aligns with the direction of the maximum principal stress that its angle is given by 2ϕ_cell = arctan(q/p). In this way, the nematic tensor order parameter Q couples the cellular active contraction ${\sigma }_{ij}^{ac}$ with the cell AR and orientation angle ϕ_cell, and reflects the active mechanical feedback, by positively correlating cell AR with the anisotropic degree of the cell active contraction.

The cell layer is modeled as a 2D continuous layer with a constant thickness h_c. The elastic stress ${\sigma }_{ij}^{el}$ in the layer is

$${\sigma }_{ij}^{el}=2\mu {\epsilon }_{ij}+\left(K-\mu \right){\delta }_{ij}{\epsilon }_{kk},$$ (6)

where μ is the shear modulus and K is the bulk modulus; ε_ij = (u_i,j + u_j,i)/2 is the strain tensor in which u_i is the displacement of the cell layer, and the commas denote partial derivatives. The equilibrium equation of the layer is

$$h_c\left({\sigma }_{ij,j}^{el}+{\sigma }_{ij,j}^{ac}\right)-\rho k_{eff}{u}_i=0.$$ (7)

The last term in the left-hand side of this equation represents the traction exerted by adhesive bonds; ρ is the number density of the bonds, and ρk_eff is the effective stiffness per unit area of the substrate. Substituting Eqs. (3), (6) into Eq. (7) gives

$$\mu {\nabla }^2{u}_i+K{u}_{j,ij}+\gamma Q_{ij,j}-\frac{\rho k_{eff}}{h_c}{u}_i=0.$$ (8)

The corresponding boundary condition (BC) is

$$\left({\sigma }_{ij}^{el}+{\sigma }_{ij}^{ac}\right)n_j=F_i$$ (9)

where n_j is the outward normal to the boundary of the cell layer, and F_i is the external force.

Collective cells resembling the nematic phase display an orientational order with polarized geometry (Ladoux et al., 2016; Prost et al., 2015). Based on the nematic theory, the equilibrium configuration of the cell layer is assumed to minimize the free energy functional (Koepf and Pismen, 2015; Lubensky et al., 2002)

$$F=h_c{\iint }_{\Omega }\left[\frac{\alpha }{2}{S}^2+\frac{k}{2}{Q}_{ij,j}^2-\eta Q_{ij}\cdot {\epsilon }_{ij}+f_{el}\left({\epsilon }_{ij}\right)\right]dA$$ (10)

where α characterizes the cell polarization energy, k represents the strength of cell alignment. η is the coupling coefficient between the nematic order and the elastic deformation, considering the energy of the cell alignment induced by strain/stress in cell layer (Gupta et al., 2015; He et al., 2015). The term $f_{el}=\frac{1}{2}{\sigma }_{ij}^{el}{\epsilon }_{ij}+\frac{1}{2}\rho k_{eff}{u}_i^2$ denotes the sum of the elastic energy of the cell layer and the cell-substrate adhesion energy.

The free energy Eq. (10) can be expressed as a function of p and q as

$$F=h_c{\iint }_{\Omega }\left\{\frac{\alpha }{2}\left(p^2+q^2\right)+\frac{k}{2}\left[{\left(\frac{\partial p}{\partial x}+\frac{\partial q}{\partial y}\right)}^2+{\left(\frac{\partial q}{\partial x}-\frac{\partial p}{\partial y}\right)}^2\right]-\eta \left(p{\epsilon }_{xx}+2q{\epsilon }_{xy}-p{\epsilon }_{yy}\right)+f_{el}\left({\epsilon }_{ij}\right)\right\}dxdy$$ (11)

Varying Eq. (11) with respect to p gives

$$\delta F=h_c{\iint }_{\Omega }\left\{k\left[\left(\frac{\partial p}{\partial x}+\frac{\partial q}{\partial y}\right)\frac{\partial \delta p}{\partial x}+\left(\frac{\partial p}{\partial y}-\frac{\partial q}{\partial x}\right)\frac{\partial \delta p}{\partial y}\right]+\left[\alpha p-\eta \left({\epsilon }_{xx}-{\epsilon }_{yy}\right)\right]\delta p\right\}dxdy$$ (12)

Using Green’s theorem Eq. (12) can be transformed to

$$\delta F=h_c{\iint }_{\Omega }\left\{-k\left(\frac{{\partial }^{2}p}{\partial x^2}+\frac{{\partial }^{2}p}{\partial y^2}\right)+\left[\alpha p-\eta \left({\epsilon }_{xx}-{\epsilon }_{yy}\right)\right]\right\}\delta pdxdy+h_c{\int }_{S}k\left[\left(\frac{\partial p}{\partial x}+\frac{\partial q}{\partial y}\right)n_i+\left(\frac{\partial p}{\partial y}-\frac{\partial q}{\partial x}\right)n_j\right]\delta pds$$ (13)

Minimizing the free energy, i.e., δF = 0 gives

$$\frac{k}{\alpha }\left(\frac{{\partial }^{2}p}{\partial x^2}+\frac{{\partial }^{2}p}{\partial y^2}\right)-p+\frac{\eta }{\alpha }\left(u_{x,x}-u_{y,y}\right)=0$$ (14)

$$\left(\frac{\partial p}{\partial x}+\frac{\partial q}{\partial y}\right)n_i+\left(\frac{\partial p}{\partial y}-\frac{\partial q}{\partial x}\right)n_j=0$$ (15)

Similarly, varying Eq. (11) with respect to q gives

$$\delta F=h_c{\iint }_{\Omega }\left\{k\left[\left(\frac{\partial p}{\partial x}+\frac{\partial q}{\partial y}\right)\frac{\partial \delta q}{\partial y}+\left(\frac{\partial q}{\partial x}-\frac{\partial p}{\partial y}\right)\frac{\partial \delta q}{\partial x}\right]+\left(\alpha q-2\eta {\epsilon }_{xy}\right)\delta q\right\}dxdy$$ (16)

Using Green’s theorem Eq. (16) can be transformed to

$$\delta F=h_c{\iint }_{\Omega }\left[-k\left(\frac{{\partial }^{2}q}{\partial x^2}+\frac{{\partial }^{2}q}{\partial y^2}\right)+\left(\alpha q-2\eta {\epsilon }_{xy}\right)\right]\delta qdxdy+h_c{\int }_{S}k\left[\left(\frac{\partial p}{\partial x}+\frac{\partial q}{\partial y}\right)n_j+\left(\frac{\partial q}{\partial x}-\frac{\partial p}{\partial y}\right)n_i\right]\delta qds$$ (17)

Minimizing the free energy, i.e., δF = 0 gives

$$\frac{k}{\alpha }\left(\frac{{\partial }^{2}q}{\partial x^2}+\frac{{\partial }^{2}q}{\partial y^2}\right)-q+\frac{\eta }{\alpha }\left(u_{x,y}-u_{y,x}\right)=0$$ (18)

$$\left(\frac{\partial p}{\partial x}+\frac{\partial q}{\partial y}\right)n_j+\left(\frac{\partial p}{\partial x}-\frac{\partial q}{\partial y}\right)n_i=0$$ (19)

Therefore, Eqs. (14 and 18) yield the governing equations,

$$\begin{gathered} L^2{\nabla }^{2}p-p+c\left(u_{x,x}-u_{y,y}\right)=0 \\ {L}^2{\nabla }^{2}q-q+c\left(u_{x,y}+u_{y,x}\right)=0, \end{gathered}$$ (20)

where $L=\sqrt{k/\alpha }$ represents a characteristic length which should be of the order of cell size, and c = η/α. The effects of L on cell AR and alignment are shown in Fig. S1 in Supplementary materials.

Eqs. (15) and (19) lead to the corresponding boundary conditions

$$Q_{ij,j}=0$$ (21)

The governing equations, (8) and (20) are coupled partial differential equations. Solving them typically requires resorting to numerical evaluation. But for axisymmetric problems the equations can be simplified to give an analytical solution, as derived in the following.

2.2. Analytical solution for an axisymmetric ring pattern

Assuming the cell polarization and alignment to be axisymmetric, according to Eq. (3) the intercellular active stresses in the polar coordinates r − θ are

$${\sigma }_{rr}^{ac}=\gamma \left(1-S\right),\hspace{1em}{\sigma }_{\theta \theta }^{ac}=\gamma \left(1+s\right),\hspace{0.17em}\hspace{1em}{\sigma }_{r\theta }^{ac}=0,$$ (22)

When s>0, the maximum principal direction of the active stress is along the circumferential direction, but when s<0, it is aligned with the radial direction. According to Eq. (2), S = |s|. And according to Eq. (6), the intercellular elastic stresses in the polar coordinates are

$${\sigma }_{rr}^{el}=\left(K+\mu \right)\frac{d{u}_r}{dr}+\left(K-\mu \right)\frac{u_r}{r},\hspace{1em}{\sigma }_{\theta \theta }^{el}=\left(K-\mu \right)\frac{d{u}_r}{dr}+\left(K+\mu \right)\frac{u_r}{r}.$$ (23)

The force balance equation along the radial coordinate r is

$$\frac{d\left({\sigma }_{rr}^{el}+{\sigma }_{rr}^{ac}\right)}{dr}+\frac{{\sigma }_{rr}^{el}+{\sigma }_{rr}^{ac}-\left({\sigma }_{\theta \theta }^{el}+{\sigma }_{\theta \theta }^{ac}\right)}{r}-\frac{\rho k_{eff}{u}_r}{h_c}=0.$$ (24)

Substituting Eqs. (22) and (23) into Eq. (24) yields

$$\left(K+\mu \right)\left(\frac{d^2{u}_r}{d{r}^2}+\frac{d{u}_r}{rdr}-\frac{u_r}{r^2}\right)-\gamma \left(\frac{ds}{dr}+\frac{2s}{r}\right)-\frac{\rho k_{eff}{u}_r}{h_c}=0.$$ (25)

The BC Eq. (9) becomes

$$\left(K+\mu \right)\frac{d{u}_r}{dr}+\left(K-\mu \right)\frac{u}{r}+\gamma \left(1-s\right)=F_{\alpha }\hspace{0.17em}\text{at}\hspace{0.17em}r=r_0,r_1.$$ (26)

where r₀ and r₁ are the inner and outer radii of the ring, respectively, and the subscript α = 0, 1 corresponds to the inner and outer edges, respectively. The free energy functional can be expressed in the polar coordinates as

$$F=h_c{\iint }_{\Omega }\left[\frac{\alpha }{2}{s}^2+\frac{k}{2}{\left(\frac{ds}{dr}+\frac{s}{r}\right)}^2+\eta s\left(\frac{d{u}_r}{dr}-\frac{ur}{r}\right)+f_{el}\left({\epsilon }_{ij}\right)\right]rd\theta dr,$$ (27)

Varying Eq. (27) with respect to s yields

$$\frac{k}{\alpha }\left(\frac{d^{2}s}{d{r}^2}-\frac{2s}{r^2}\right)-s-\frac{\eta }{\alpha }\left(\frac{d{u}_r}{dr}-\frac{u_r}{r}\right)=0.$$ (28)

The corresponding boundary conditions of a ring pattern are

$$\frac{ds}{dr}+\frac{s}{r}=0\hspace{0.17em}\text{at}\hspace{0.17em}r=r_0,r_1.$$ (29)

We non-dimensionalize the governing Eqs. (25) and (28) by setting $L=\sqrt{k/\alpha }$, ${\overline{u}}_r=u_r/L$, $\overline{r}=r/L$,

$$\left(\frac{d^2{\overline{u}}_r}{d{\overline{r}}^2}+\frac{d{\overline{u}}_r}{\overline{r}d\overline{r}}-\frac{{\overline{u}}_r}{{\overline{r}}^2}\right)-a\left(\frac{ds}{d\overline{r}}+\frac{2s}{\overline{r}}\right)-b{\overline{u}}_r=0,$$ (30)

$$\frac{d^{2}s}{d{\overline{r}}^2}-\frac{2s}{{\overline{r}}^2}-s-c\left(\frac{d{\overline{u}}_r}{d\overline{r}}-\frac{{\overline{u}}_r}{\overline{r}}\right)=0.$$ (31)

The solution depends on the parameter groups a = γ/(K + μ), b = L²ρk_eff/[(K + μ)h_c], c = η/α. BC Eq. (29) remains unchanged, while BC Eq. (26) takes the form

$$\frac{d{\overline{u}}_r}{d\overline{r}}+d\frac{{\overline{u}}_r}{\overline{r}}+a\left(1-s\right)={\overline{F}}_{\alpha }\hspace{0.17em}\text{at}\hspace{0.17em}\overline{r}={\overline{r}}_0,{\overline{r}}_1$$ (32)

where d = (K − μ)/(K + μ) and ${\overline{F}}_{\alpha }=\frac{F_{\alpha }}{K+\mu }$. Since $\overline{r}>>1$ in the ring patterns studied here and 0 < |s| < 1, the first two terms in Eq. (31) are negligible, as justified in Fig. S3 in Supplementary materials. Then, solving Eqs. (30) and (31) we obtain

$${\overline{u}}_r=C_1{I}_1\left(\beta \overline{r}\right)+C_2{K}_1\left(\beta \overline{r}\right),$$ (33)

$$s=c\beta \left[-C_1{I}_2\left(\beta \overline{r}\right)+C_2{K}_2\left(\beta \overline{r}\right)\right]$$ (34)

where $\beta =\sqrt{b/\zeta }$ and ζ = ac + 1; I_n(x) and K_n(x) are the modified Bessel functions of the first and second kind, respectively. While BC Eq. (29) results in a trivial solution, the BC Eq. (32) yields

$$C_1=C_{1,num}/C_{denom},$$ (35)

$$C_2=C_{2,num}/C_{denom},$$ (36)

where

$$\begin{gathered} C_{1,num}={\overline{r}}_0{\overline{r}}_1\beta \zeta \left[\left(a-{\overline{F}}_1\right)K_0\left({\overline{r}}_0\beta \right)-\left(a-{\overline{F}}_0\right)K_0\left({\overline{r}}_1\beta \right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}-\left(1+d-2\zeta \right)\left[\left(a-{\overline{F}}_1\right){\overline{r}}_1{K}_1\left({\overline{r}}_0\beta \right)-\left(a-{\overline{F}}_0\right){\overline{r}}_0{K}_1\left({\overline{r}}_1\beta \right)\right], \end{gathered}$$ (37)

$$\begin{gathered} C_{2,num}={\overline{r}}_0{\overline{r}}_1\beta \zeta \left[\left(a-{\overline{F}}_1\right)I_0\left({\overline{r}}_0\beta \right)-\left(a-{\overline{F}}_0\right)I_0\left({\overline{r}}_1\beta \right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}+\left(1+d-2\zeta \right)\left[\left(a-{\overline{F}}_1\right){\overline{r}}_1{I}_1\left({\overline{r}}_0\beta \right)-\left(a-{\overline{F}}_0\right){\overline{r}}_0{I}_1\left({\overline{r}}_1\beta \right)\right], \end{gathered}$$ (38)

and

$$\begin{gathered} C_{denom}=\left[\left(1+d\right)I_1\left({\overline{r}}_1\beta \right)+{\overline{r}}_1\beta \zeta I_2\left({\overline{r}}_1\beta \right)\right]\left[\left(1+d\right)K_1\left({\overline{r}}_0\beta \right)-{\overline{r}}_0\beta \zeta K_2\left({\overline{r}}_0\beta \right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}+\left[\left(1+d\right)I_1\left({\overline{r}}_0\beta \right)+{\overline{r}}_0\beta \zeta I_2\left({\overline{r}}_0\beta \right)\right]\left[{\overline{r}}_1\beta \zeta K_2\left({\overline{r}}_1\beta \right)-\left(1+d\right)K_1\left({\overline{r}}_1\beta \right)\right]. \end{gathered}$$ (39)

2.3. Numerical solution

For the general case when an analytical solution is unavailable, a numerical solver is needed. The ‘2D- Mathematics-Coefficient form PDE’ module from the finite element program Comsol Multiphysics (Comsol, 2019) is used. We calculated the fields u_i, p and q by solving the governing equations, Eqs. (8) and (20), with the BCs, Eqs. (9) and (21). Once the fields p and q are known, one can analyze the polarization and orientation of cells. In addition, considering the discrete features of individual cells, we define $\langle •\rangle =\iint •dA/A$ as the area average of ‘•’ in the region of interest (ROI), where ‘•’ could be AR or ϕ_cell, while A is the area of the region. In the calculation, the elastic constants for cells are μ = 17.2kPa and K = 45.5kPa, and the cell layer thickness is h_c = 2μm (He et al., 2015, 2014; Mertz et al., 2012). Assuming the active contractile strain of cytoskeleton under physiological conditions is 0.1 (Deguchi et al., 2006; Lu et al., 2008; Zhong et al., 2011), we estimate γ = 9kPa. The two-spring model (Schwarz et al., 2006) gives k_eff = k_sk_b/(k_s + k_b) with k_s and k_b being the spring constants of the substrate and molecular bond, respectively. The substrate spring constant is $k_s=E_s{d}_m/\left(1-v_s^2\right)$, where E_s and ν_s are Young’s modulus and Poisson’s ratio of the substrate respectively, and d_m is the diameter of adhesion molecules (Kendall, 1971). Additionally, we supplement the density of adhesion molecules $\rho ={\alpha }_1{e}^{E_s/{\alpha }_2}$ as a function of the substrate stiffness to consider the effect of substrate stiffness on focal adhesions (Goffin et al., 2006). E_s varies from 1 kPa to 70 kPa in Fig. 2. Unless otherwise noted, the following values are kept as constant in the calculations: d_m = 1nm, k_b = 0.005nN/μm (Kong et al., 2008), ν_s = 0.4, c = 100, L = 10μm, α₁ = 0.7μm⁻², α₂ = 14.5kPa, T = 200nN/μm.

Fig. 2.

Effects of substrate stiffness on cell AR in the ring patterns. (a) The model prediction. The color map illustrates the cell AR and white spindles indicate the cell orientation. (b) Cell AR as function of the radial coordinate for different substrate stiffness (analytical- solid line, Eq. (34); numerical-dashed line, Eqs. (8) and (20) solved by Comsol; experimental data are shown by markers). (c) The mean value of cell aspect ratio <AR> (averaged over the region 150 μm < r < 300 μm) biphasically depends on the substrate stiffness. An improved analytical solution specifically for soft substrates is derived in Supplementary materials, which shows a better agreement with the numerical solutions when E_s < 20 kPa (Fig. S4 in Supplementary materials).

3. Results

3.1. Effects of the substrate stiffness

Fig. 2 shows the predicted cell orientation and AR as functions of the radial coordinate for a ring-like cell layer (Fig. 1) on three different substrates, stiff (60 kPa), medium (40 kPa), and soft (10 kPa). The analytical and numerical solutions are in agreement with each other, and capture the main features of the experimental results of MC3T3 cells (He et al., 2015) (see Figs. 2a &b, and Figs. S2a &S2b). For example, the cells align along the direction of maximum principal stress, i.e. the circular direction of the ring pattern, and the cell AR near the inner and outer boundary of the ring is larger than in the middle region. More strikingly, the cell AR is the highest on the substrate of the intermediate stiffness, compared to both the softest and the stiffest substrates, indicating a biphasic behavior. Fig. 2c plots the cell AR as the function of the substrate stiffness, and shows the predicted biphasic dependence of AR on E_s consistent with the experimental results (He et al., 2015). In addition, the distributions of the cell AR in Fig. 2a are in accordance with the distributions of the in-plane maximum shear stress in Fig. S2c. This result confirms that the role of the in-plane maximum shear stress as the driving force of cell polarization suggested by previous studies (He et al., 2015; Liu et al., 2016).

3.2. Effects of the geometry of substrate patterns

Our model is also used to predict the effects of varying size/shape of geometric constraints on the cell polarization and alignment. Fig. 3a shows that the width of the strip-like patterns has a significant effect on the cell AR and ϕ_cell. The narrower is the strip, the larger is the cell AR and the higher is the orientation index along the strip where the orientation index is defined by sin ϕ_cell (Fig. 3b). These trends of the cell AR and ϕ_cell predicted by the model are consistent with the experimental measurements by Vedula et al. (2012). This result suggests that the edge constraints influence the stress field and polarization energy of the cell layer, and become stronger as the width of the strip decreases.

Fig. 3.

Cell polarization and orientation on three strips of varying width. (a) The model prediction. The color map illustrates the cell AR and white spindles indicate the cell orientation. (b) The comparison between the simulation and the experimental results from Vedula et al. (2012). $\langle AR\rangle$ and $\langle \sin {\varphi }_{cell}\rangle$ are averaged AR and orientation index over the area of each strip protruded above the bottom reservoir.

Using a cross pattern, Duclos et al. (2014) reported an unexpected effect of geometric size on cell alignment. Our model reproduces this effect, as shown in Fig. 4. The central part of the cross pattern is a particularly interesting region where ϕ_cell is controlled by competition between the nematic orientation in the two arms of the pattern. As seen in Fig. 4(a–c), the nematic orientation is horizontal when the vertical branches are narrow, i.e., w_v/w_h = 0.5, and gradually rotates with the increasing width of the vertical branches, i.e., w_v/w_h = 1, 2. When the width of the branches is larger than a certain critical value, nematic orientation is no longer aligned with the direction of the branch, as seen in Fig. 4d. These results are consistent with experimental observations by Duclos et al. (2014). Besides the cell alignment the cell AR in the cross pattern can also be predicted by our model (Fig. 5). As we see, when the width of the vertical branch is smaller than the horizontal one, i.e., w_v/w_h = 0.5, the cell AR is larger in the vertical branch than in the horizontal one (Fig. 5a), but when w_v/w_h = 2, 3, the cell AR becomes smaller in the vertical branch than in the horizontal one (Figs. 5c and d). This result is consistent with that in Fig. 3, increasing the width of the strips decreases the cell AR. Nevertheless, the width has less influence at the center region of the cross pattern.

Fig. 4.

Cell alignment in the center region of a cross pattern. The color maps show cell orientation angle ϕ_cell, while the white spindles represent cell direction. (a–c) The cell orientation ϕ_cell in the cross center rotates gradually from horizontal to vertical as w_v/w_h increases from 0.5 to 2; (d) When w_v/w_h = 3, ϕ_cell in the cross center becomes parallel to the horizontal direction again.

Fig. 5.

The cell aspect ratio (AR) on cross patterns corresponding to Fig. 4. (a) The cell AR is larger in vertical branch than in the horizontal one when w_v/w_h = 0.5. (b) The AR in the vertical branch is the same as in the horizontal branch when w_v/w_h = 1. (c–d) When increasing the width of the vertical strip tow_v/w_h = 2, 3, the AR becomes smaller in the vertical branch than in the horizontal one.

Besides the size effect shown in Figs. 3–5 our model can also be used to simulate the effect of different substrate shapes, such as ellipse, indented square, and rectangle, on cell polarization and alignment (Fig. 6). Fig. 6a shows the color map of AR with cell alignment indicated by white spindle marks. Figs. 6b and c compare the experimental results (He et al., 2015) with the simulations for the cell AR and orientation, respectively. The aspect ratio in box 1 is significantly larger than in boxes 2 and 3, while that in box 2 is larger than in box 3. Although the experimental results are noisy, the averages in different areas fit theoretical predictions reasonably well. In addition, the mean angle difference ϕ between experimental and theoretical results shows a reverse trend, indicating that the cells with larger AR more preferentially align with the maximum principal stress direction (Fig. 6c).

Fig. 6.

Cell polarization and orientation on patterns of different geometries: an ellipse, an indented square, and a rectangle. (a) The color maps of cell aspect ratios predicted by the theoretical model; white spindles show the cell orientation. (b) Comparison of the model prediction of cell aspect ratios with experimental results in the boxes marked in panel (a). (c) The difference in orientation angles between the experimental measurement and theoretical predictions in the marked boxes. Symbols * and # represent that there are statistical differences when comparing with Region 1 and Region 2, respectively (p < 0.05).

3.3. Effects of external forces

Our model is further used to study the effects of various external forces, e.g. pulling or shearing forces applied at the wound edge, on the cell polarization and orientation (Fig. 7). The pulling force P = T κ is caused by the line tension T, dependent on the curvature κ of the wound, where the tension T is applied by the transcellular actomyosin ring around the wound (Abreu-Blanco et al., 2012; Brugues et al., 2014; Danjo and Gipson, 1998). Using the BC Eq. (9) to take into account these forces applied on the edges, our model predicts that the pulling force P induces polarization of cells perpendicularly to the wound edge to form a rosette-like pattern (Brugues et al., 2014). The closer the cells are to the wound edge, the higher is the AR (Figs. 7a and b). Moreover, the cell AR depends on the curvature of the wound. For instance, the cells in box 2 near the long axis (of a higher curvature) have a larger AR compared to box 1 near the short axis (of a lower curvature) of the oval wound, while for the circular wound the AR is identical in the corresponding two boxes. To verify these predictions, we performed the wound healing experiments (Figs. 7d and e, see the experimental method in SM), and analyzed the averaged AR of the cells around the circular and oval wounds. Our model predictions of the cell AR and orientation are generally in agreement with the experiments (Fig. 7f, Movies. S1 and S2). In contrast to the normal pulling force, when applying a shearing force τ at the wound edge, our model predicts that cells are aligned to form a swirl-like pattern, which is similar to the morphology of cells in the rat subcutaneous tissue when an acupuncture needle is rotated in the tissue (Langevin et al., 2002) (Figs. 7c and g).

Fig. 7.

Cell polarization and orientation at the wound edge under pulling and shearing forces. (a) The circular wound under a pulling force perpendicular to the wound edge, (b) the oval wound under a pulling force, and (c) the circular wound under a shearing force along the wound edge. The color maps illustrate the cell AR, and white spindles illustrate the cell direction predicted by our model. The black arrows illustrate the forces. The phase contrast images show the cell polarization and orientation around circular (d) and oval (e) wounds that had healed for 80 min. (f) The comparison of the predicted AR at the circular and oval wound edges with the corresponding experiments. (g) The histology image of a deformed rat subcutaneous tissue under the shearing force by the rotation of an acupuncture needle (Langevin et al., 2002). τ = 70kPa. The scale bar is 50μm.

3.4. Effects of myosin activity

It is interesting to study the effect of myosin activity as it endows the intrinsic active contraction of cells. In our model the strength/degree of myosin activity is quantified by the parameter γ as shown in Eq. (3). By varying the value of γ we test the effects of myosin activity on cell polarization and alignment in two cases: with or without external forces around the inner edge of the ring pattern (Fig. 8). The problems are solved not only numerically via governing Eqs. (8) and (20) in Comsol, but also analytically via Eq. (34).

Fig. 8.

Effects of the myosin activity γ on cell AR and alignment in a ring pattern with r₀ =100 μm and r₁ =700 μm. The color map represents the cell AR, and the white spindles represent the cell orientation. (a–c) Varying the value of γ does not change much the patterns of the cell AR and alignment in the absence of external forces. (d) The analytical solutions of AR are consistent with the numerical calculations for the cell layers without the pulling forces. Cells always polarize along the circular direction, i.e., s > 0 in Eq. (20). (e–g) The pulling forces rotate cells to polarize along the direction perpendicular to the edge, but increasing γ decreases the cell polarization along the direction perpendicular to the edge, and realigns cells back to the direction parallel to the edge. F₀ = 15kPa. (h) The analytical solutions of AR are consistent with the numerical calculations for the cell layers with the pulling forces. The cells near the inner edge polarize along the radial direction, i.e., s < 0, when γ = 9 kPa and 15 kPa. When γ = 45 kPa, all the cells polarize along the circumferential direction, i.e., s > 0.

The effects of γ depends on whether there is an external force. In the absence of external forces, the cell orientation is not sensitive to the change of γ; nevertheless, the cell AR increases with increasing γ, in accordance to both numerical and analytical solutions (Fig. 8a–d). However, in the presence of the external forces, the effects of γ on cell orientations are significant. For instance, when there are pulling forces F₀ around the inner edge, by changing γ from 9 kPa to 45 kPa, the orientation of cells around the edge turns from the direction perpendicular to the edge to the direction parallel to the edge (Fig. 8e–h). Meanwhile, the cell AR around the edge changes non-monotonically: first decreases before turning to the direction parallel to the edge, but then increases after becoming parallel to the edge (Fig. 8e–h).

The analytical solutions for cell AR and orientation are consistent with numerical solutions (Figs. 8d and h). In the analytical solution, the manner of cell polarization is determined by the sign of s, i.e., s>0 and s<0 respectively represent polarizing along the circular direction and the radial direction (Eq. (22)). As we see, when γ =9 kPa and 15 kPa, both the numerical and analytical solutions at a given pulling forces F₀ consistently show the cell orientation transition with varying radial position: from the outer to the inner edges cell polarization direction changes from the circular direction to the radial direction (Figs. 8e,f and h).

Strikingly, increasing the myosin activity γ can diminish the effects of external forces. As we see, increasing γ to 45 kPa changes the pattern of cell AR and alignment under the pulling forces (Fig. 8g) to be similar with the patterns in the absence of the pulling forces (Fig. 8a–c), i.e., cell AR are larger around the edges than the interior, and all the cells polarize along the circular direction. In addition, as shown in Fig. S5 in Supplementary materials, increasing γ prevents the formation of swirl-like pattern induced by the shearing forces in Fig. 7c. These results suggest that increasing the strength of myosin activity can stabilize the cell polarization and alignment. Therefore, γ is an important parameter that quantifies the capability of cells resisting the effects of the external forces for keeping their homeostatic state.

4. Discussion

4.1. The nematic cell model

In this study, the nematic cell model is used to predict the direction- and position- dependent cell polarization and alignment. In the model, cell AR, angle and active contractile stress are explicitly expressed as the functions of the components of the nematic order tensor parameter Q (Fig. 1 and Eq. (2)). To determine the order parameter Q, the governing Eqs. (8) and (20) are derived, respectively, from the force equilibrium (Eq. (7)) and by minimization of the free energy functional of the system, including the energy of cell polarization and alignment and of their coupling with elastic deformation (Eq. (10)). Then the spatial distribution of nematic order parameter Q is used to quantify the direction- and position-dependent cell AR and cell orientation. Importantly, for an axisymmetric ring-like cell layer, an analytical solution depending on key dimensionless combinations of model parameters is achieved (Eq. (34)), providing an explicit relationship between cell polarization and main biophysical parameters, such as cell elasticity K and μ, substrate stiffness k_eff, and myosin activity γ.

4.2. The polarization-contraction relation

Consistently with our previous studies, this study shows that cells tend to polarize along the maximum principal stress in a cell layer driven by the in-plane maximum shear stress. Accordingly, the larger is the in-plane maximum shear stress, the higher is the cell AR (He et al., 2015; Liu et al., 2016). This response causes the cell active contraction exerted by the cytoskeleton to be anisotropic, i.e., cells contract stronger along their long axis than their short axis (the ‘traction-distance law’ (He et al., 2014)), which, in turn, alters the stress state in the cell layer. Correspondingly, the cell and cytoskeleton need to align with the new maximum principal stresses in the updated stress state. As thus the relation between the polarized cell shape and the anisotropy of the cell’s active contractile stress forms an active mechanical feedback which suggests a polarization-contraction relation. This relation allows cells to sense the spatial distribution of mechanical forces and, in turn, to develop direction- and position-dependent cell polarization and alignment. In our model, the cell AR and the cell’s active contractile stress, linked by the nematic order parameter, are positively correlated, which satisfies the ‘traction-distance law’. Moreover, the coupling between the nematic order and the elastic deformation of cells, i.e. the third term in Eq. (10), indicates that the cell’s active contractile stress tends to align with the maximum principal stresses. Therefore the polarization-contraction relation built in our model enables the prediction of the stress-driven cell polarization and alignment.

4.3. The stress-driven cell polarization in different contexts

This nematic cell model can be generally used for predicting the direction- and position- dependent cell AR and cell angle in different contexts, such as varying substrate stiffness (Fig. 2), size/shape of geometrical constraint (Figs. 3–6), and external forces (Fig. 7). For example, the model reproduces the biphasic effect of substrate stiffness on the cell AR. Compared to the soft and stiff substrate, cells acquire the largest AR on medium-stiff substrate (Fig. 2). In addition, the model captures the size-dependent behaviors of cell polarization observed by Duclos et al. (2014). The cells in the central part of the cross pattern polarize along the horizontal direction when the width of the vertical arm is small, but they tend to polarize along the horizontal direction when the width of the arms becomes larger (Fig. 4a–c). Our model also predicts the effect of physical forces on cell polarization. The pulling forces at the wound edge cause cells to polarize along the direction perpendicular to the wound edge, and the cells near the long axis of the oval wound polarize more than those near the short axis (Figs. 7a–b and d–f). In contrast, in the presence of shearing forces at the wound edge, the model predicts the formation of a swirl-like pattern of cell polarization (Figs. 7c and g). Our model also predicts the effect of the strength of myosin activity on cell polarization, suggesting that increasing myosin activity can stabilize the cell alignment by resisting the effect of external forces (Fig. 8). These predictions are generally consistent with experimental observation. Taken together, this model reveals the underpinning mechanisms of the stress-driven cell polarization.

4.4. Edge effects versus the stress-driven mechanism of cell polarization

Our computations show that in the absence of external forces, if cells are located around the edges of the cell layer, they tend to polarize along the edges, regardless of the geometry of the cell layer, as shown in Figs. 2–4 and 6. This effect has been noticed in many experimental studies (Luo et al., 2008; Thery, 2010; Wan et al., 2011). Since the underlying mechanisms were not understood, it was called the edge effect or boundary effect by assuming that the edge plays a role in cell alignment due to chemical interactions. This might introduce an interesting debate: can the patterns of cell polarization and alignment be explained by the so-called edge effects? It is true that, compared to the cells at the center of the patterns, the cells at the edges seem different. For example, the cells lack cell-cell interaction on the sides facing the edges, and are restricted by the edges rather than by other cells. It is, however, impossible to use these edge effects to explain or predict the spatial distribution of cell polarization and alignment in an entire collective of cells in different geometries. In addition, when external forces are present, the cells may not align with the edge anymore. For example, cells preferentially align in the radial direction of the wound because of the contractile force exerted by the actomyosin ring, which cannot be explained by the edge effect (Fig. 7). It is, therefore, impossible that edge effects can serve as a general mechanism explaining the complex patterns of cell polarization and alignment. In fact, the edge effect is a special case of stress-driven cell behaviors in the absence of external forces. Once the cell layer has a stress-free boundary condition at the edge, the direction of the maximum principal stress, which is also the cell orientation predicted by our model, should precisely align with the edge, exhibiting the stress-driven cell polarity.

4.5. Outlook

Although we studied here the stress-driven cell polarization using the nematic cell model, it remains unclear how this behavior might impact collective cell migration. In an epithelial monolayer expanding into free space, cells migrate along the direction of the maximum principal stress, a phenomenon that is termed “plithotaxis” (Tambe et al., 2011). The cell orientation predicted by our model roughly coincides with the orientation of cell migration as described by plithotaxis. Other mechanisms besides plithotaxis, such as unjamming transition, are also involved in collective cell migration. Moreover, the unjamming transition is associated with changes of the cell shape, i.e., cell AR becomes progressively smaller and less variable as the collective cells become progressively more jammed (Atia et al., 2018). It would be interesting to develop computational models by using the nematic order parameter to take into account the polarization-contraction relation and the related changes of collective cell migration (Atia et al., 2018; Kim et al., 2013; Park et al., 2015).

5. Conclusions

In this work we constructed a theoretical model to describe the mechanics of cell polarization and alignment by introducing the polarization-contraction relation of cells. This relation is characterized by the nematic tensor order parameter, i.e., cell AR is positively correlated with the anisotropic degree of cell active contraction. Moreover, cell AR and orientation can be explicitly expressed as the functions of the nematic order parameter. The nematic order parameter is also used in different context to study the effects of substrate stiffness, the geometries of cell layers, pulling or shearing forces, and myosin activity. In these different contexts, the predicted cell AR and orientation are generally consistent with the experimental observations. These results suggest that the polarization-contraction relation characterized by the nematic order parameter can function as a stress-driven mechanism by which cells are able to sense the spatial distribution of mechanical forces to develop direction- and position-dependent behaviors. Our model also provides a possible tool to predict the direction-and position-dependent cell polarization and alignment in tissue morphogenesis, cancer metastasis, and pattern formation in tissue engineering for potential biomedical applications.