Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Koen
Schakenraad
^{ab},
Jeremy
Ernst
^{a},
Wim
Pomp
^{cd},
Erik H. J.
Danen
^{e},
Roeland M. H.
Merks
^{bf},
Thomas
Schmidt
^{c} and
Luca
Giomi
*^{a}
^{a}Instituut-Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands. E-mail: giomi@lorentz.leidenuniv.nl
^{b}Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
^{c}Kamerlingh Onnes-Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands
^{d}Division of Gene Regulation, The Netherlands Cancer Institute, P.O. Box 90203, 1006 BE Amsterdam, The Netherlands
^{e}Leiden Academic Center for Drug Research, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands
^{f}Institute of Biology, Leiden University, P.O. Box 9505, 2300 RA Leiden, The Netherlands

Received
20th March 2020
, Accepted 19th May 2020

First published on 21st May 2020

We investigate the mechanical interplay between the spatial organization of the actin cytoskeleton and the shape of animal cells adhering on micropillar arrays. Using a combination of analytical work, computer simulations and in vitro experiments, we demonstrate that the orientation of the stress fibers strongly influences the geometry of the cell edge. In the presence of a uniformly aligned cytoskeleton, the cell edge can be well approximated by elliptical arcs, whose eccentricity reflects the degree of anisotropy of the cell's internal stresses. Upon modeling the actin cytoskeleton as a nematic liquid crystal, we further show that the geometry of the cell edge feeds back on the organization of the stress fibers by altering the length scale at which these are confined. This feedback mechanism is controlled by a dimensionless number, the anchoring number, representing the relative weight of surface-anchoring and bulk-aligning torques. Our model allows to predict both cellular shape and the internal structure of the actin cytoskeleton and is in good quantitative agreement with experiments on fibroblastoid (GDβ1, GDβ3) and epithelioid (GEβ1, GEβ3) cells.

By adjusting their shape, cells can sense the mechanical properties of their microenvironment and regulate traction forces,^{8–10} with prominent consequences on bio-mechanical processes such as cell division, differentiation, growth, death, nuclear deformation and gene expression.^{11–16} On the other hand, the cellular shape itself depends on the mechanical properties of the environment. Experiments on adherent cells have shown that the stiffness of the substrate strongly affects cell morphology^{17,18} and triggers the formation of stress fibers.^{19,20} The cell spreading area increases with the substrate stiffness for several cell types, including cardiac myocytes,^{17} myoblasts,^{18} endothelial cells and fibroblasts,^{19} and mesenchymal stem cells.^{21}

In our previous work^{22} we have investigated the shape and traction forces of concave cells, adhering to a limited number of discrete adhesion sites and characterized by highly anisotropic actin cytoskeletons. Using a contour model of cellular adhesion,^{8,23–26} we demonstrated that the edge of these cells can be accurately approximated by a collection of elliptical arcs obtained from a unique ellipse, whose eccentricity depends on the degree of anisotropy of the contractile stresses arising from the actin cytoskeleton. Furthermore, our model quantitatively predicts how the anisotropy of the actin cytoskeleton determines the magnitudes and directions of traction forces. Both predictions were tested in experiments on highly anisotropic fibroblastoid and epithelioid cells^{27} supported by microfabricated elastomeric pillar arrays,^{28–30} finding good quantitative agreement.

Whereas these findings shed light on how cytoskeletal anisotropy controls the geometry and forces of adherent cells, they raise questions on how anisotropy emerges from the three-fold interplay between isotropic and directed stresses arising within the cytoskeleton, the shape of the cell envelope and the geometrical constraints introduced by focal adhesions. It is well known that the orientation of the stress fibers in elongated cells strongly correlates with the polarization direction of the cell.^{31–34} Consistently, our results indicate that, in highly anisotropic cells, stress fibers align with each other and with the cell's longitudinal direction (see, e.g., Fig. 1A).^{22} However, the physical origin of these alignment mechanisms is less clear and inevitably leads to a chicken-and-egg causality dilemma: do the stress fibers align along the cell's axis or does the cell elongate in the direction of the stress fibers?

Fig. 1 (A) A fibroblastoid cell with an anisotropic actin cytoskeleton cultured on a microfabricated elastomeric pillar array.^{22} The color scale indicates the measured orientation of the actin stress fibers. (B) Schematic representation of a contour model for the cell in (A). The cell contour consists of a collection of concave cellular arcs (red lines) that connect pairs of adhesion sites (blue dots). These arcs are parameterized as curves spanned counterclockwise around the cell by the arclength s, and are entirely described via the tangent unit vector T = (cosθ,sinθ) and the normal vector N = (−sinθ,cosθ), with θ the turning angle. The unit vector n = (cosθ_{SF},sinθ_{SF}) describes the local orientation θ_{SF} of the stress fibers. |

The biophysical literature is not scarce of cellular processes that might possibly contribute to alignment of stress fibers with each other and with the cell edge. Mechanical feedback between the cell and the extracellular matrix or substrate, for instance, has been shown to play an important role in the orientation and alignment of stress fibers.^{21,35–38} Molecular motors have also been demonstrated to produce an aligning effect on cytoskeletal filaments, both in mesenchymal stem cells^{39} and in purified cytoskeletal extracts,^{40} where the observation is further corroborated by agent-based simulations.^{41} A similar mechanism has been theoretically proposed for microtubules–kinesin mixtures.^{42} Studies in microchambers demonstrated that steric interactions can also drive alignment of actin filaments with each other and with the microchamber walls.^{43–45} Theoretical studies have highlighted the importance of the stress fibers' assembly and dissociation dynamics,^{35,46} the dynamics of focal adhesion complexes,^{47,48} or both.^{49,50} Also the geometry of actin nucleation sites has been shown to affect the growth direction of actin filaments, thus promoting alignment.^{51,52} Finally, mechanical coupling between the actin cytoskeleton and the plasma membrane has been theoretically shown to lead to fiber alignment, as bending moments arising in the membrane result into torques that reduce the amount of splay within the filaments.^{53} Despite such a wealth of possible mechanisms, it is presently unclear which one or which combination is ultimately responsible for the observed alignment of stress fibers between each other and with the cell's longitudinal direction. Analogously, it is unclear to what extent these mechanisms are sensitive to the specific mechanical and topographic properties of the environment, although some mechanisms rely on specific environmental conditions that are evidently absent in certain circumstances (e.g., the mechanical feedback between the cell and the substrate discussed in ref. 35, 37, 48 and 54 relies on deformations of the substrate and is unlikely to play an important role in experiments performed on micro-pillar arrays).

In this paper we investigate the interplay between the anisotropy of the actin cytoskeleton and the shape of cells adhering to microfabricated elastomeric pillar arrays.^{28–30} Rather than pinpointing a specific alignment mechanism, among those reviewed above, we focus on the interplay between cell shape and the spatial organization of the actin cytoskeleton. This is achieved by means of a phenomenological treatment of the stress fiber orientation based on the continuum description of nematic liquid crystals, coupled with a contour model of the cell edge.^{22} The paper is organized as follows: in Section II we introduce our contour model for cells with anisotropic cytoskeleton. We first review the key theoretical results, already reported in ref. 22, followed by an in-depth and previously unreported analysis of the model. In Section III we further generalize this approach by studying the mechanical interplay between the shape of the cell, described by our contour model, and the orientation of the actin cytoskeleton, modeled as a nematic liquid crystal confined by the cell edge, and we compare our results to experimental data on highly anisotropic cells. In both sections we assume that the coordinates of the adhesion sites along the cell contour are constant in time and known. A theoretical description of the dynamics of these adhesion sites, as a result of focal adhesion dynamics, is beyond the scope of this study and can be found, for example, in ref. 47 and 48.

ξ_{t}∂_{t}r = ∂_{s}F_{cortex} + (_{out} − _{in})·N, | (1) |

The stress tensor can be modeled upon taking into account isotropic and directed stresses. The latter are constructed by treating the stress fibers as contractile force dipoles, whose average orientation θ_{SF} is parallel to the unit vector n = (cosθ_{SF},sinθ_{SF}) (see Fig. 1B). This gives rise to an overall contractile bulk stress of the form:^{58,59}

_{out} − _{in} = σÎ + αnn, | (2) |

ξ_{t}∂_{t}r = ∂_{s}(λT) + σN + α(n·N)n. | (3) |

0 = (∂_{s}λ)T + (λκ + σ)N + α(n·N)n, | (4) |

Finally, all the material parameters appearing in eqn (4) depend, in principle, on the density of actin. Here we focus on the orientational structure of the cytoskeleton and, for sake of simplicity, we assume the density of actin to be uniform throughout the cell. Therefore our model does not account for local density variations that have been found experimentally on several cell types, where stress fibers occur most prominently along concave cell edges.^{60–63} A complementary approach, where the density rather than the orientation of the actin fibers is explicitly modeled, can be found in ref. 64.

For α = 0, eqn (4) describes the special case of a cell endowed with a purely isotropic cytoskeleton.^{8,23,24} Force balance requires λ to be constant along a single cellular arc (i.e. ∂_{s}λ = 0), whereas the bulk and cortical tension compromise along an arc of constant curvature, i.e. κ = −σ/λ, with the negative sign of κ indicating that the arcs are curved inwards. The cell edge is then described by a sequence of circular arcs, whose radius R = 1/|κ| = λ/σ depends on the local cortical tension λ of the arc. This model successfully describes the shape of adherent cells in the presence of strictly isotropic forces. However, as we showed in ref. 22, isotropic models are not suited for describing cells whose anisotropic cytoskeleton develops strong directed forces originating from actin stress fibers.^{65,66}

In the presence of an anisotropic cytoskeleton, α > 0 and the cell contour is no longer subject to purely normal forces. As a consequence, the cortical tension λ varies along a given cellular arc to balance the tangential component of the contractile forces arising from the actin cytoskeleton. In order to highlight the physical mechanisms described, in this case, by eqn (4), we introduce a number of simplifications that make the problem analytically tractable. These will be lifted in Section III, where we will consider the most general scenario. First, because the orientation of the stress fibers typically varies only slightly along a single arc, we assume the orientation of the stress fibers, θ_{SF}, to be constant along a single cellular arc, but different from arc to arc. Furthermore, without loss of generality, we orient the reference frame such that the stress fibers are parallel to the y-axis. Thus, θ_{SF} = π/2 and n = ŷ. Then, solving eqn (4) with respect to λ yields:

(5) |

The shape of a cellular arc is given by a segment of an ellipse, which is given by:^{22}

(6) |

Fig. 2 Schematic representation of a cellular arc, described by eqn (6), for n = (cosθ_{SF},sinθ_{SF}) = ŷ, hence θ_{SF} = π/2. A force balance between isotropic stress, directed stress and line tension results in the description of each cell edge segment (red curve) as part of an ellipse of aspect ratio . The cell exerts forces F_{0} and F_{1} on the adhesion sites (blue). The vector d = d(cosϕ,sinϕ) describes the relative position of the two adhesion sites, d^{⊥} = d(−sinϕ,cosϕ) is a vector perpendicular to d, and θ is the turning angle of the cellular arc. The coordinates of the ellipse center (x_{c},y_{c}) and the angular coordinates of the adhesion sites along the ellipse ψ_{0} and ψ_{1} are given in the ESI.† |

Fig. 3A shows an example of a fibroblastoid cell with ellipses fitted to its arcs. Because ellipse fitting is very sensitive to noise on the cell shape, only the longer arcs are considered here (see Materials and methods). We stress that, as long as the contractile stresses arising from the actin cytoskeleton are roughly uniform across the cell (i.e., α, σ and λ_{min} are constant), all cellular arcs of sufficient length are approximated by a unique ellipse (see Fig. 3A). The parameters that describe this ellipse are, in general, different for each individual cell. A survey of these parameters over a sample of 285 fibroblastoid and epithelioid cells yields the distributions displayed in Fig. 3B–D for the parameters λ_{min}, α and σ. The corresponding population averages are: λ_{min} = 7.6 ± 5.6 nN, α = 1.7 ± 1.7 nN μm^{−1}, σ = 0.87 ± 0.70 nN μm^{−1} and γ = 0.33 ± 0.20. Evidently, the variance in the parameter values is in part due to the natural variations across the cell population, and in part to possible statistical fluctuations in the experiments. Further insight about the distribution of material parameters can be addressed in the future by combining our model with experiments of cells adhering to micropatterned substrates, which impose reproducible cell shapes.^{67} Finally, we note that some of the smaller cellular arcs, such as those in the bottom left corner of Fig. 3, cannot be approximated by the same ellipse as the longer arcs. This may be due to local fluctuations in the density and orientation of stress fibers at the small scale or to other effects that are not captured by our model. For a description of the selection of the fitted arcs and of the endpoints of the arcs, see Materials and methods. For more experimental data on the elliptical fits, see ref. 22.

Fig. 3 (A) A fibroblastoid cell with an anisotropic actin cytoskeleton on a microfabricated elastomeric pillar array^{22} (same cell as in Fig. 1A), with a unique ellipse (white) fitted to its arcs of sufficient length (see Materials and methods). The actin, cell edge, and micropillar tops are in the red, green, and blue channels respectively. The endpoints of the arcs (cyan) are identified based on the forces that the cell exerts on the pillars (Materials and methods). Scale bar is 10 μm. (B–D) Histograms of the parameters λ_{min}, α and σ estimated from a survey of these parameters over a sample of 285 fibroblastoid and epithelioid cells. |

(7) |

(8a) |

(8b) |

(9a) |

(9b) |

(10) |

Another interesting quantity is obtained by adding the forces F_{0} and F_{1} from the same arc. Although these two forces act on two different adhesion sites, their sum represents the total net force that a single cellular arc exerts on the substrate. This is given by

(11) |

F_{⊥}^{2} + γF_{‖}^{2} = const., | (12) |

λ^{3}κ = −λ_{min}^{2}(α + σ) = const. | (13) |

(14) |

As mentioned in the Introduction, experimental observations, by us^{22} and others,^{31–34} have indicated that stress fibers tend to align with each other and with the cell's longitudinal direction. As we discussed, several cellular processes might contribute to these alignment mechanisms, such as mechanical cell–matrix feedback,^{21,35–38} motor-mediated alignment,^{39–42} steric interactions,^{43–45} stress fiber formation and dissociation,^{35,46,49,50} focal adhesion dynamics,^{47–50} the geometry of actin nucleation sites,^{51,52} or membrane-mediated alignment,^{53} but it is presently unclear which combination of mechanisms is ultimately responsible for the orientational correlation observed in experiments. Our phenomenological description of the actin cytoskeleton allows us to focus on the interplay between cellular shape and the orientation of the stress fibers, without the loss of generality that would inevitably result from selecting a specific alignment mechanism among those listed above.

This phenomenological description necessitates a number of simplifying assumptions that can be addressed in future work. First, we again assume the typical timescale associated with the equilibration of the forces (hence the reorientation of the actin filaments) to be much shorter than that associated with cell motility (see also Section II.A). Consequently, experiments on migrating cells^{76} or cells subject to cyclic mechanical loading^{77,78} are outside of the scope of the present paper. Moreover, our model does not take into account dynamical effects, such as actin filament turnover and the viscoelasticity of stress fibers.^{79,80} Second, as we did with in Section II, we restrict our model to effectively two-dimensional cells. This is not unreasonable, as cells adhering to a stiff surface have a largely flat shape,^{25} but it does imply that our model cannot capture three-dimensional stress fiber structures around the nucleus, such as the actin cap,^{81} or distinguish between the orientations of apical and basal stress fibers.^{82} Third, we do not model signalling pathways, thus our approach cannot account for variations of myosin activity (thus contractile stress) in response to the substrate stiffness and other mechanical cues, but, as already discussed in Section II, it can describe the modulation in spread-area and traction force originating from the mechanical coupling between the cell and the substrate^{63,68,69,71} (see Section II.C). Fourth, our model describes the overall cell-scale structure of the actin cytoskeleton and does not include local effects such as the interactions of individual stress fibers with focal adhesions in the cell interior.^{28,71,72}

(15) |

(16) |

(17) |

The second integral, which is extended over the cell contour, is the Nobili–Durand anchoring energy^{83} and determines the orientation of the stress fibers along the edge of the cell, with the tensor _{0} representing their preferential orientation. Experimental evidence form our (Fig. 3 and ref. 22) and other's work (e.g., ref. 31–34), suggests that, in highly anisotropic cells, peripheral stress fibers are preferentially parallel to the cell edge. The same trend is recovered in experiments with purified actin bundles confined in microchambers.^{43,44} In the language of Landau–de Gennes theory, this effect can be straightforwardly reproduced by setting

(18) |

In order to generate stationary configurations of the actin cytoskeleton, we numerically integrate the following overdamped equation:

(19) |

KN·∇Q_{ij} − 2W(Q_{ij} − Q_{0,ij}) = 0. | (20) |

(21) |

Next, we decompose eqn (3) along the normal and tangent directions of the cell contour. Since the cells considered here are pinned at the adhesion sites, which we again assume to be rigid, and the density of actin along the cell contour is assumed to be constant, tangential motion is suppressed, i.e., T·∂_{t}r = 0. Together with eqn (21) this yields:

0 = ∂_{s}λ + α_{0}T··N, | (22a) |

(22b) |

Integrating eqn (22a) then yields the cortical tension along an arc:

(23) |

Combining the dynamics of the cell contour and that of the cell bulk, we obtain the following coupled differential equations:

(24a) |

(24b) |

To highlight the physical meaning of our numerical results, we introduce two dimensionless numbers, namely the contractility number, Co, and the anchoring number, An. Co is defined as the ratio between the typical distance between two adhesion sites d and the major semi-axis of the ellipse approximating the corresponding cellular arc (b = λ_{min}/σ, see Section II.A):

(25) |

(26) |

To get insight on the effect of the combinations of these dimensionless parameters on the spatial organization of the cell, we first consider the simple case in which the adhesion sites are located at the corners of squares and rectangles (Section IV.A). In Section IV.B we consider more realistic adhesion geometries and compare our numerical results with experimental observations on highly anisotropic cells adhering to a small number of discrete adhesions.

Fig. 4 Configurations of cells whose adhesion sites are located at the vertices of a square. The thick black line represents the cell boundary, the black lines in the interior of the cells represent the orientation field n = (cosθ_{SF},sinθ_{SF}) of the stress fibers and the background color indicates the local nematic order parameter S. The spatial averages of the order parameter S are given, from left to right, by: 0.80; 0.80; 0.77 (top row), 0.94; 0.92; 0.92 (middle row), and 1.0; 1.0; 1.0 (bottom row). The vertical axis corresponds to the anchoring number An = WR/K and the horizontal axis to the contractility number Co = σd/λ_{min}. The cells shown correspond to values of An = 0, 1, 10 and Co = 0, 0.125, 0.25, where we take both d and R equal to the length of the square side. The ratios σ/(σ + α_{0}) = 1/9, λ_{min}Δt/(ξ_{t}R^{2}) = 2.8 × 10^{−6}, and KΔt/(ξ_{r}R^{2}) = 2.8 × 10^{−6}, and the parameters δ = 0.15R, N_{arc} = 20, and Δx = R/19 are the same for all cells. The number of iterations is 5.5 × 10^{5}. For definitions of Δt, Δx, and N_{arc}, see the ESI.† |

As expected, for low Co values (left column), cells with large An exhibit better parallel anchoring than cells with small An values, but lower nematic order parameter S in the cell interior (spatial average of S decreases from 1.0 at the bottom left to 0.80 at the top left, see Fig. 4). Interestingly, the alignment of stress fibers with the walls in the configuration with large An value (top left) resembles the configurations found by Deshpande et al.,^{35,46} who specifically accounted for the assembly and dissociation dynamics of the stress fibers. More strikingly, the structure reported in the top left of Fig. 4 appears very similar to those found in experiments of dense suspensions of pure actin in cell-sized square microchambers,^{43,44} simulations of hard rods in quasi-2D confinement,^{43} and results based on Frank elasticity,^{85} even though these systems are very different from living cells. As is the case in our simulations, in these studies the tendency of the filaments to align along the square edges competes with that of aligning along the diagonal.

For large Co values (right column of Fig. 4), the cell deviates from the square shape. Interestingly, although the contractile stresses (σ and α_{0}) do not directly affect the configuration of the cytoskeleton, they do it indirectly by influencing the shape of the cell. This results into an intricate interplay between shape and orientation, controlled by the numbers An and Co. In particular, for constant Co, i.e., for fixed stress fiber contractility, increasing An leads to higher tangential alignment of the stress fibers with the cell edge, thus increasing An decreases the contractile force experienced by the cell edge, which is proportional to (n·N)^{2} [eqn (24a)]. Conversely, for constant An, increasing Co leads to a more concave cell shape which forces the stress fibers to bend more. Consequently, the average order parameter in the cell decreases with increasing Co (see Fig. 4).

Finally, we note that all configurations in Fig. 4 display a broken rotational and/or chiral symmetry. For An = 0 the stress fibers are uniformly oriented, but any direction is equally likely. For non-zero An, the stress fibers tend to align along either of the diagonals (with the same probability) to reduce the amount of distortion. Upon increasing Co, chirality emerges in the cytoskeleton and in the cell contour (see, e.g., the cell in the middle of the right column in Fig. 4). In light of the recent interest in chiral symmetry breaking in single cells^{86} and in multicellular environments,^{87} we find it particularly interesting that chiral symmetry breaking can originate from the natural interplay between the orientation of the stress fibers and the shape of the cell.

To conclude this section, we focus on four-sided cells whose adhesion sites are located at the vertices of a rectangle and explore the effect of the cell aspect ratio (i.e., height/width). Fig. 5 displays three configurations having fixed maximal width and aspect ratio varying from 1 to 2. Fig. S3 in the ESI† shows the effect of increasing the aspect ratio while keeping instead the area of the rectangle fixed. Upon increasing the cell aspect ratio, the mean orientation of the stress fibers switches from the diagonal (aspect ratio 1) to longitudinal (aspect ratio 2), along with an increase in the order parameter in the cell bulk, as can be seen in Fig. 5 by the slightly more red-shifted cell interior (spatial average of S increases from 0.92 to 0.96). This behavior originates from the competition between bulk and boundary effects. Whereas the bulk energy favors longitudinal alignment, as this reduces the amount of bending of the nematic director, the anchoring energy favors alignment along all four edges alike, thus favoring highly bent configurations at the expense of the bulk elastic energy. When the aspect ratio increases, the bending energy of the bulk in the diagonal configuration increases, whereas the longitudinal state only pays the anchoring energy at the short edges, hence independently on the aspect ratio. Therefore, elongating the cell causes the stress fibers to transition from tangential to longitudinal alignment, with a consequent increase of the nematic order parameter. Interestingly, similar observations were made in experiments on pure actin filaments in cell-sized microchambers.^{43,44} More importantly, the longitudinal orientation of the stress fibers in cells of aspect ratio 2 is consistent with several experimental studies of cells adhering on adhesive stripes and elongated adhesive micropatterns.^{33,34,62,63,88} Fig. S4 and S5 in the ESI† show the effect of the anchoring number An, the contractility number Co, and the ratio between σ and α_{0} on a cell with aspect ratio 2.

Fig. 5 Effect of the aspect ratio of the cell, ranging from 1 to 2, on cytoskeletal organization for cells whose four adhesion sites are located at the vertices of a rectangle. The thick black line represents the cell boundary, the black lines in the interior of the cells represent the orientation field n = (cosθ_{SF},sinθ_{SF}) of the stress fibers and the background color indicates the local nematic order parameter S. The spatial averages of the order parameter S are given, from left to right, by: 0.92; 0.95; 0.96. The simulations shown are performed with An = WR/K = 1 where R is equal to the short side of the rectangle, and Co = σd/λ_{min} equal to 0.125, 0.1875, and 0.25 respectively, where d is equal to the long side of the rectangle. The ratios σ/(σ + α_{0}) = 1/9, λ_{min}Δt/(ξ_{t}R^{2}) = 2.8 × 10^{−6}, and KΔt/(ξ_{r}R^{2}) = 2.8 × 10^{−6}, and the parameters δ = 0.15R and Δx = R/19 are the same for all cells. N_{arc} = 20, 30, 40 from left to right and the number of iterations is 5.5 × 10^{5}. For definitions of Δt, Δx, and N_{arc}, see the ESI.† |

Fig. 6 Validation of our model to experimental data. (A) Optical micrograph (TRITC–Phalloidin) of a fibroblastoid cell (same cell as in Fig. 1 and 3).^{22} The adhesions (cyan circles) are determined by selecting micropillars that are close to the cell edge and experience a significant force (Materials and methods). (B) Experimental data of cell shape and coarse-grained cytoskeletal structure of this cell. The white line represents the cell boundary, black lines in the interior of the cells represent the orientation field n = (cosθ_{SF},sinθ_{SF}) of the stress fibers and the background color indicates the local nematic order parameter S. The spatial average of the order parameter is S = 0.54. (C–E) Configurations obtained from a numerical solution of eqn (24) using the adhesion sites of the experimental data (cyan circles) as input, and with various anchoring number (An) values. This is calculated from eqn (26), with R = 23.6 μm the square root of the cell area. The corresponding values of the length scale K/W are 71 μm (C), 14 μm (D), and 2.9 μm (E) respectively. The spatial averages of the order parameter S are given by: 0.85 (C), 0.60 (D), and 0.56 (E) respectively. The values for λ_{min}/σ = 14.7 μm and σ/(σ + α_{0}) = 0.40 are found by an analysis of the elliptical shape of this cell.^{22} The ratios λ_{min}Δt/ξ_{t} = 1.2 × 10^{−3} μm^{2} and KΔt/ξ_{r} = 1.2 × 10^{−3} μm^{2}, and the parameters δ = 11 μm, N_{arc} = 20, and Δx = 1.1 μm are the same for figures (C–E). The number of iterations is 2.1 × 10^{6}. For definitions of Δt, Δx, and N_{arc}, see the ESI.† |

Consistent with our results on rectangular cells (Fig. 5), the stress fibers align parallel to the cell's longitudinal direction and perpendicularly to the cell's shorter edges. Furthermore, the nematic order parameter is close to unity in proximity of the cell contour, indicating strong orientational order near the cell edge, but drops in the interior. This behavior is in part originating from the lower density of stress fibers around the center of mass of the cell, and in part from the presence of ±1/2 nematic disclinations away from the cell edge. These topological defects naturally arise from the tangential orientation along the boundary. Albeit not uniform throughout the whole cell contour, thus not sufficient to impose hard topological constraints on the configuration of the director in the bulk (i.e., Poincaré–Hopf theorem), this forces a non-zero winding of the stress fibers, which in turn is accommodated via the formation of one or more disclinations. As a consequence of the concave shape of the cell boundary, these defects have most commonly strength −1/2. The average order parameter in the cell is S = 0.54.

To compare our theoretical and experimental results, we extract the locations of the adhesion sites from the experimental data by selecting micropillars that are close to the cell edge and experience a significant force (for details, see Materials and methods), and use them as input parameters for the simulations. In Fig. 6C–E we show results of simulations of the cell in Fig. 6A and B for increasing An values, thus decreasing magnitude of the length scale K/W. Here, we take the length scale R = 23.6 μm to be the square root of the cell area and we use constant values for the ratios λ_{min}/σ = 14.7 μm and σ/(σ + α_{0}) = 0.40 as found by an analysis of the elliptical shape of this cell.^{22}Fig. 6C shows the results of a simulation where bulk alignment dominates over boundary alignment (An = 0.33, K/W = 71 μm), resulting in an approximately uniform cytoskeleton oriented along the cell's longitudinal direction. The nematic order parameter is also approximatively uniform and close to unity (spatial average of the order parameter is S = 0.85). For larger An values (Fig. 6D, An = 1.7 and K/W = 14 μm), anchoring effects become more prominent, resulting in distortions of the bulk nematic director, a lower nematic order parameter (spatial average S = 0.60), and the emergence of a −1/2 disclination in the bottom left side of the cell. Upon further increasing An (Fig. 6E, An = 8.0 and K/W = 2.9 μm), the −1/2 topological defect moves towards the interior, as a consequence of the increased nematic order along the boundary. This results in a decrease in nematic order parameter in the bulk of the cell, consistent with our experimental data. The spatial average is S = 0.56, close to the experimental average of S = 0.54.

A qualitative comparison between our in vitro (Fig. 6B) and in silico cells (Fig. 6E) highlights a number of striking similarities, such as the overall structure of the nematic director, the large value of the order parameter along the cell edge and in the thin neck at the bottom-right of the cell and the occurrence of a −1/2 disclination on the bottom-left side. The main difference is the order parameter away from the cell edges, which is lower in the experimental data than in the numerical prediction. The lower order parameter also results in an additional −1/2 disclination at the top-left of the cell in Fig. 6B which is absent in Fig. 6E. We hypothesize that this discrepancy is caused by a lower actin density in the cell interior, as observed before in many other experimental studies.^{60–63} As a consequence of the actin depletion, the nematic order parameter can decrease, and potentially vanish, in a way that cannot be described by our model, where the density of the actin fibers is, by contrast, assumed to be uniform across the cell.

In order to make this comparison quantitative and infer the value of the phenomenological parameters introduced in this section, we have further analyzed the residual function

(27) |

Fig. 7 Residual function Δ^{2}, defined in eqn (27), as a function of the anchoring number An (eqn (26) with R = 23.6 μm) for the cell displayed in Fig. 6. The error bars display the standard deviation obtained using jackknife resampling. For large An values the residual flattens, indicating that the actual value of An becomes unimportant once the anchoring torques (with magnitude W), which determine the tangential alignment of the stress fibers in the cell's periphery, outcompete the bulk elastic torques (with magnitude K). The minimum (Δ^{2} = 0.027) is found for An = 8.0. |

The same analysis presented above has been repeated for five other cells (Fig. 8). The first column shows the raw experimental data, the second column shows the coarse-grained experimental data, and the third column shows the simulations. For these we used the values of λ_{min}/σ and σ/(σ + α_{0}) obtained from a previous analysis of the cell morphology^{22} and the An values found by a numerical minimization of Δ^{2} (see Fig. S6 in the ESI†). Also for these cells Δ^{2} flattens for large An values, and we estimate An ≳ 3 and K/W ≲ 7 μm. The minima of Δ^{2} are given, from top to bottom, by 0.016, 0.058, 0.057, 0.034, and 0.037. This indicates reasonable quantitative agreement between experiment and simulation for all cells, even though the agreement is significantly better for the cell in Fig. 8F than for those in Fig. 8G and H. Similar to the cell in Fig. 6, we observe that the main discrepancies are the order parameter in the cell interior, which is smaller in the experimental data than in the numerical results, and a number of topological defects in this low nematic order region of the experimental data that are absent in the numerical data. The cell in Fig. 8F shows good agreement between the average order parameter in the experimental (S = 0.54) and numerical (S = 0.52) data, but for the other cells the average order parameter is overestimated by the simulations. We again attribute this discrepancy to actin density variations in the experiments that are not captured by the theory. On the other hand, we note that the overall structure of the stress fiber orientation, including the emergence of a number of −1/2 topological defects (see, e.g., Fig. 8F and K), is captured well by our approach. By contrast, because of the overall convexity of the cells and the lack of strong anchoring at the boundary, + 1/2 disclinations are less prominent in both our in vitro and in silico cells and are mainly localized at the actin-depleted regions.

Fig. 8 Comparison of experimental data on five anisotropic cells with the results of computer simulations. (A–E) Optical micrographs (TRITC–Phalloidin) of epithelioid (A, B and E) and fibroblastoid (C and D) cells on a microfabricated elastomeric pillar array.^{22} The adhesions (cyan circles) are determined by selecting micropillars that are close to the cell edge and experience a significant force (Materials and methods). (F–J) Experimental data of cell shape and coarse-grained cytoskeletal structure on a square lattice of these cells. The white line represents the cell boundary, the black lines in the interior of the cells represent the orientation field n = (cosθ_{SF},sinθ_{SF}) of the stress fibers and the background color indicates the local nematic order parameter S. The spatial averages of the order parameter S are given, from top to bottom, by: 0.54; 0.44; 0.45; 0.46; 0.37. (K–O) Simulations with the adhesion sites of the experimental data as input. The spatial averages of the order parameter S are given, from top to bottom, by: 0.52; 0.68; 0.61; 0.59; 0.53. The values for λ_{min}/σ = 12.6; 15.7; 18.0; 10.8; 13.4 μm and σ/(σ + α_{0}) = 0.75; 0.25; 0.46; 0.95; 0.52 are found by an analysis of the elliptical shape of these cells.^{22} The values of An = 4.4; 4.1; 19; 4.6; 4.7, where R = 17.3; 24.4; 39.9; 24.9; 25.3 μm is defined as the square root of the cell area, are determined by minimizing Δ^{2}, with the minima given by Δ^{2} = 0.016; 0.058; 0.057; 0.034; 0.037. These An values correspond to K/W = 3.9; 5.9; 2.1; 5.4; 5.4 μm. The ratios λ_{min}Δt/ξ_{t} = 1.2 × 10^{−3} μm^{2} and KΔt/ξ_{r} = 1.2 × 10^{−3} μm^{2}, and the parameters δ = 11 μm, N_{arc} = 20, and Δx = 1.1 μm are the same for all cells. The number of iterations is 2.1 × 10^{6}. For definitions of Δt, Δx, and N_{arc}, see the ESI.† |

Finally, in nematic liquid crystals, anchoring, namely the orientation of the nematogens by a surface, originates at the molecular scale as consequence of steric, van der Waals and dipolar interactions, and, similarly to epitaxy in solids, can be controlled via the surface chemistry (see e.g.ref. 89). Whereas the biological role of anchoring in the actin cytoskeleton is yet to be explored and understood, the prevalence of parallel anchoring (i.e. the stress fibers are tangent to the cell contour) highlighted by our comparative analysis, suggests that steric interactions may be instrumental in the organization of the actin cytoskeleton, consistently with studies of actin assemblies in microchambers.^{43–45}

Lifting the assumption that the stress fibers are uniformly oriented along individual cellular arcs allows one to describe the mechanical interplay between cellular shape and the configuration of the actin cytoskeleton. Using numerical simulations and inputs from experiments on fibroblastoid and epithelioid cells plated on micropillar arrays, we identified a feedback mechanism rooted in the competition between the tendency of stress fibers to align uniformly in the bulk of the cell, but tangentially with respect to the cell edge. Our approach enables us to predict both the shape of the cell and the orientation of the stress fibers and can account for the emergence of topological defects and other distinctive morphological features. The predicted stress fiber orientations are in good agreement with the experimental data and further offer an indirect way to estimate quantities that are generally precluded to direct measurement, such as the cell's internal stresses and the orientational stiffness of the actin cytoskeleton. The main discrepancy between our predictions and the experimental data is the overestimation of the nematic order parameter in the cell interior, which should be addressed in future work by explicitely accounting for actin density variations.

The success of this relatively simple approach is remarkable given the enormous complexity of the cytoskeleton and the many physical, chemical, and biological mechanisms associated with stress fiber dynamics and alignment.^{21,35–53} Yet, the agreement between our theoretical and experimental results suggests that, on the scale of the whole cell, the myriad of complex mechanisms that govern the dynamics of the stress fibers in adherent cells can be effectively described in terms of simple entropic mechanisms, as those at the heart of the physics of liquid crystals. Moreover, this quantitative agreement further establishes the fact that the dynamics and alignment of stress fibers in cells cannot be understood from dynamics at the sub-cellular scale alone, and highlights the crucial role of the boundary conditions inferred by cellular shape.^{60,61}

In addition, our analysis demonstrates that chiral symmetry breaking can originate from the natural interplay between the orientation of the stress fibers and the shape of the cell. A more detailed investigation of this mechanism is beyond the scope of this study, but will represent a challenge in the near future with the goal of shedding light on the fascinating examples of chiral symmetry breaking observed both in single cells^{86} and tissues.^{87}

In the future, we plan to use our model to investigate the mechanics of cells adhering to micropatterned substrates that impose reproducible cell shapes,^{67} with special emphasis to the interplay between cytoskeletal anisotropy and the geometry of the adhesive patches. These systems are not new to theoretical research, but previous studies have focused on either the cytoskeleton^{49} or on cell shape,^{90} rather then on their interaction. This will enable us to more rigorously compare our model predictions with existing experimental data on stress fiber orientation in various adhesive geometries,^{16,60–62,91} including convex shapes such as circles or spherocylinders.^{63,86,92} Additionally, our model could be further extended to account for the mechanical feedback on myosin activity (see e.g.ref. 63, 68, 69 and 71) as well as the interactions of the stress fibers with the micropillars in the bulk of the cell body.^{28,71,72} Furthermore, our framework could be extended to study the role of cytoskeletal anisotropy in cell motility, for instance by taking into account the dynamics of focal adhesions,^{47,48} biochemical pathways in the actin cytoskeleton,^{93} actin filament turnover and the viscoelasticity of stress fibers,^{79,80} or cellular protrusions and retractions.^{94} Finally, our approach could be extended to computational frameworks such as vertex models, Cellular Potts Models, or phase field models,^{95} and could provide a starting point for exploring the role of anisotropy in multicellular environments such as tissues.^{96–103}

For all pixels that are inside the cell, the configuration of the stress fibers was analyzed by calculating the nematic tensor using ImageJ with the OrientationJ plugin,^{105} see the ESI.† The data were further coarse-grained in blocks of 8 × 8 pixels corresponding to regions of size 1.104 × 1.104 μm^{2} in real space. This results in a new 64 × 64 lattice. The value of the nematic tensor in the new coarse-grained pixels was obtained from an average over those of the original 8 × 8 pixels located inside the cell. In turn, the coarse-grained pixels were considered inside the cell if more than half of the original pixels were inside the cell.

- C. M. Lo, H. B. Wang, M. Dembo and Y. L. Wang, Biophys. J., 2000, 79, 144–152 CrossRef CAS PubMed.
- R. D. Sochol, A. T. Higa, R. R. R. Janairo, S. Li and L. Lin, Soft Matter, 2011, 7, 4606 RSC.
- C. A. Reinhart-King, M. Dembo and D. A. Hammer, Biophys. J., 2008, 95, 6044–6051 CrossRef CAS PubMed.
- Y. Sawada, M. Tamada, B. J. Dubin-Thaler, O. Cherniavskaya, R. Sakai, S. Tanaka and M. P. Sheetz, Cell, 2006, 127, 1015–1026 CrossRef CAS PubMed.
- A. J. Engler, S. Sen, H. L. Sweeney and D. E. Discher, Cell, 2006, 126, 677–689 CrossRef CAS PubMed.
- B. Trappmann, J. E. Gautrot, J. T. Connelly, D. G. T. Strange, Y. Li, M. L. Oyen, M. A. Cohen Stuart, H. Boehm, B. Li, V. Vogel, J. P. Spatz, F. M. Watt and W. T. S. Huck, Nat. Mater., 2012, 11, 642–649 CrossRef CAS PubMed.
- T. Panciera, L. Azzolin, M. Cordenonsi and S. Piccolo, Nat. Rev. Mol. Cell Biol., 2017, 18, 758 CrossRef CAS PubMed.
- I. Bischofs, S. Schmidt and U. Schwarz, Phys. Rev. Lett., 2009, 103, 1–4 CrossRef PubMed.
- M. Ghibaudo, L. Trichet, J. Le Digabel, A. Richert, P. Hersen and B. Ladoux, Biophys. J., 2009, 97, 357–368 CrossRef CAS PubMed.
- D. A. Fletcher and R. D. Mullins, Nature, 2010, 463, 485–492 CrossRef CAS PubMed.
- N. Minc, D. Burgess and F. Chang, Cell, 2011, 144, 414–426 CrossRef CAS PubMed.
- M. Versaevel, T. Grevesse and S. Gabriele, Nat. Commun., 2012, 3, 671 CrossRef PubMed.
- C. S. Chen, M. Mrksich, S. Huang, G. M. Whitesides and D. E. Ingber, Science, 1997, 276, 1425–1428 CrossRef CAS PubMed.
- N. Jain, K. V. Iyer, A. Kumar and G. V. Shivashankar, Proc. Natl. Acad. Sci. U. S. A., 2013, 110, 11349–11354 CrossRef CAS PubMed.
- R. McBeath, D. M. Pirone, C. M. Nelson, K. Bhadriraju and C. S. Chen, Dev. Cell, 2004, 6, 483–495 CrossRef CAS PubMed.
- K. A. Kilian, B. Bugarija, B. T. Lahn and M. Mrksich, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 4872–4877 CrossRef CAS PubMed.
- A. Chopra, E. Tabdanov, H. Patel, P. A. Janmey and J. Y. Kresh, Am. J. Physiol.: Heart Circ. Physiol., 2011, 300, H1252–H1266 CrossRef CAS PubMed.
- A. J. Engler, M. A. Griffin, S. Sen, C. G. Bönnemann, H. L. Sweeney and D. E. Discher, J. Cell Biol., 2004, 166, 877–887 CrossRef CAS PubMed.
- T. Yeung, P. C. Georges, L. A. Flanagan, B. Marg, M. Ortiz, M. Funaki, N. Zahir, W. Ming, V. Weaver and P. A. Janmey, Cell Motil., 2005, 60, 24–34 CrossRef PubMed.
- F. Grinnell, Trends Cell Biol., 2000, 10, 362–365 CrossRef CAS.
- A. Zemel, F. Rehfeldt, A. E. X. Brown, D. E. Discher and S. A. Safran, J. Phys.: Condens. Matter, 2010, 22, 194110 CrossRef CAS.
- W. Pomp, K. Schakenraad, H. E. Balcıoğlu, H. van Hoorn, E. H. J. Danen, R. M. H. Merks, T. Schmidt and L. Giomi, Phys. Rev. Lett., 2018, 121, 178101 CrossRef CAS PubMed.
- R. Bar-Ziv, T. Tlusty, E. Moses, S. A. Safran and A. Bershadsky, Proc. Natl. Acad. Sci. U. S. A., 1999, 96, 10140–10145 CrossRef CAS PubMed.
- I. B. Bischofs, F. Klein, D. Lehnert, M. Bastmeyer and U. S. Schwarz, Biophys. J., 2008, 95, 3488–3496 CrossRef CAS PubMed.
- U. S. Schwarz and S. A. Safran, Rev. Mod. Phys., 2013, 85, 1327–1381 CrossRef CAS.
- L. Giomi, Advances in Experimental Medicine and Biology, in Cell migrations: causes and function, ed. S. Zapperi and C. A. M. La Porta, Springer International Publishing, 2019, vol. 1146, DOI:10.1007/978-3-030-17593-1.
- E. H. J. Danen, P. Sonneveld, C. Brakebusch, R. Fässler and A. Sonnenberg, J. Cell Biol., 2002, 159, 1071–1086 CrossRef CAS PubMed.
- J. L. Tan, J. Tien, D. M. Pirone, D. S. Gray, K. Bhadriraju and C. S. Chen, Proc. Natl. Acad. Sci. U. S. A., 2003, 100, 1484–1489 CrossRef CAS.
- L. Trichet, J. Le Digabel, R. J. Hawkins, S. R. K. Vedula, M. Gupta, C. Ribrault, P. Hersen, R. Voituriez and B. Ladoux, Proc. Natl. Acad. Sci. U. S. A., 2012, 109, 6933–6938 CrossRef CAS PubMed.
- H. Van Hoorn, R. Harkes, E. M. Spiesz, C. Storm, D. Van Noort, B. Ladoux and T. Schmidt, Nano Lett., 2014, 14, 4257–4262 CrossRef CAS PubMed.
- T. Vignaud, L. Blanchoin and M. Théry, Trends Cell Biol., 2012, 22, 671–682 CrossRef CAS PubMed.
- B. Ladoux, R.-M. Mège and X. Trepat, Trends Cell Biol., 2016, 26, 420–433 CrossRef PubMed.
- N. T. Lam, T. J. Muldoon, K. P. Quinn, N. Rajaram and K. Balachandran, Integr. Biol., 2016, 8, 1079–1089 CrossRef CAS PubMed.
- S. K. Gupta, Y. Li and M. Guo, Soft Matter, 2019, 15, 190–199 RSC.
- V. S. Deshpande, R. M. McMeeking and A. G. Evans, Proc. R. Soc. London, Ser. A, 2007, 463, 787–815 CrossRef CAS.
- S. Walcott and S. X. Sun, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 7757–7762 CrossRef CAS PubMed.
- A. Zemel, F. Rehfeldt, A. E. X. Brown, D. E. Discher and S. A. Safran, Nat. Phys., 2010, 6, 468 Search PubMed.
- N. Nisenholz, M. Botton and A. Zemel, Soft Matter, 2014, 10, 2453–2462 RSC.
- M. Raab, J. Swift, P. D. P. Dingal, P. Shah, J.-W. Shin and D. E. Discher, J. Cell Biol., 2012, 199, 669–683 CrossRef CAS.
- V. Schaller, C. Weber, C. Semmrich, E. Frey and A. R. Bausch, Nature, 2010, 467, 73 CrossRef CAS.
- P. Kraikivski, R. Lipowsky and J. Kierfeld, Phys. Rev. Lett., 2006, 96, 258103 CrossRef PubMed.
- S. Swaminathan, F. Ziebert, D. Karpeev and I. S. Aranson, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 79, 036207 CrossRef PubMed.
- M. Soares e Silva, J. Alvarado, J. Nguyen, N. Georgoulia, B. M. Mulder and G. H. Koenderink, Soft Matter, 2011, 7, 10631–10641 RSC.
- J. Alvarado, B. M. Mulder and G. H. Koenderink, Soft Matter, 2014, 10, 2354–2364 RSC.
- S. Deshpande and T. Pfohl, Biomicrofluidics, 2012, 6, 034120 CrossRef PubMed.
- V. S. Deshpande, R. M. McMeeking and A. G. Evans, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 14015–14020 CrossRef CAS PubMed.
- V. S. Deshpande, M. Mrksich, R. M. McMeeking and A. G. Evans, J. Mech. Phys. Solids, 2008, 56, 1484–1510 CrossRef CAS.
- E. G. Rens and R. M. H. Merks, 2019, arXiv e-prints, arXiv:1906.08962.
- A. Pathak, V. S. Deshpande, R. M. McMeeking and A. G. Evans, J. R. Soc., Interface, 2008, 5, 507–524 CrossRef.
- W. Ronan, V. S. Deshpande, R. M. McMeeking and J. P. McGarry, Biomech. Model. Mechanobiol., 2014, 13, 417–435 CrossRef.
- A.-C. Reymann, J.-L. Martiel, T. Cambier, L. Blanchoin, R. Boujemaa-Paterski and M. Théry, Nat. Mater., 2010, 9, 827 CrossRef CAS PubMed.
- G. Letort, A. Z. Politi, H. Ennomani, M. Théry, F. Nedelec and L. Blanchoin, PLoS Comput. Biol., 2015, 11, 1–21 CrossRef PubMed.
- A. P. Liu, D. L. Richmond, L. Maibaum, S. Pronk, P. L. Geissler and D. A. Fletcher, Nat. Phys., 2008, 4, 789 Search PubMed.
- S. Thomopoulos, G. M. Fomovsky and J. W. Holmes, J. Biomech. Eng., 2005, 127, 742–750 CrossRef PubMed.
- K. Burridge and M. Chrzanowska-Wodnicka, Annu. Rev. Cell Dev. Biol., 1996, 12, 463–519 CrossRef CAS PubMed.
- S. Banerjee and L. Giomi, Soft Matter, 2013, 9, 5251–5260 RSC.
- L. Giomi, Soft Matter, 2013, 9, 8121 RSC.
- T. J. Pedley and J. O. Kessler, Annu. Rev. Fluid Mech., 1992, 24, 313–358 CrossRef.
- R. A. Simha and S. Ramaswamy, Phys. Rev. Lett., 2002, 89, 058101 CrossRef PubMed.
- M. Théry, A. Pépin, E. Dressaire, Y. Chen and M. Bornens, Cell Motil., 2006, 63, 341–355 CrossRef PubMed.
- M. Théry, V. Racine, M. Piel, A. Pépin, A. Dimitrov, Y. Chen, J.-B. Sibarita and M. Bornens, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 19771–19776 CrossRef PubMed.
- J. James, E. D. Goluch, H. Hu, C. Liu and M. Mrksich, Cell Motil., 2008, 65, 841–852 CrossRef PubMed.
- P. W. Oakes, S. Banerjee, M. C. Marchetti and M. L. Gardel, Biophys. J., 2014, 107, 825–833 CrossRef CAS PubMed.
- J. P. McGarry, J. Fu, M. T. Yang, C. S. Chen, R. M. McMeeking, A. G. Evans and V. S. Deshpande, Philos. Trans. R. Soc., A, 2009, 367, 3477–3497 CrossRef CAS PubMed.
- S. Pellegrin and H. Mellor, J. Cell Sci., 2007, 120, 3491–3499 CrossRef CAS PubMed.
- K. Burridge and E. S. Wittchen, J. Cell Biol., 2013, 200, 9–19 CrossRef CAS PubMed.
- M. Théry, J. Cell Sci., 2010, 123, 4201–4213 CrossRef PubMed.
- A. D. Rape, W.-H. Guo and Y.-L. Wang, Biomaterials, 2011, 32, 2043–2051 CrossRef CAS.
- J. Hanke, D. Probst, A. Zemel, U. S. Schwarz and S. Köster, Soft Matter, 2018, 14, 6571–6581 RSC.
- P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, 1995 Search PubMed.
- J. Fu, Y.-K. Wang, M. T. Yang, R. A. Desai, X. Yu, Z. Liu and C. S. Chen, Nat. Methods, 2010, 7, 733–736 CrossRef CAS.
- G. L. Lin, D. M. Cohen, R. A. Desai, M. T. Breckenridge, L. Gao, M. J. Humphries and C. S. Chen, FEBS Lett., 2013, 587, 763–769 CrossRef CAS.
- C. Roux, A. Duperray, V. M. Laurent, R. Michel, V. Peschetola, C. Verdier and J. Étienne, Interface Focus, 2016, 6, 20160042 CrossRef.
- D. Riveline, E. Zamir, N. Q. Balaban, U. S. Schwarz, T. Ishizaki, S. Narumiya, Z. Kam, B. Geiger and A. D. Bershadsky, J. Cell Biol., 2001, 153, 1175–1186 CrossRef CAS PubMed.
- N. Q. Balaban, U. S. Schwarz, D. Riveline, P. Goichberg, G. Tzur, I. Sabanay, D. Mahalu, S. Safran, A. Bershadsky, L. Addadi and B. Geiger, Nat. Cell Biol., 2001, 3, 466–472 CrossRef CAS.
- B. Wehrle-Haller and B. A. Imhof, Int. J. Biochem. Cell Biol., 2003, 35, 39–50 CrossRef CAS.
- R. De, A. Zemel and S. A. Safran, Nat. Phys., 2007, 3, 655 Search PubMed.
- A. Livne, E. Bouchbinder and B. Geiger, Nat. Commun., 2014, 5, 3938 Search PubMed.
- J. Étienne, J. Fouchard, D. Mitrossilis, N. Bufi, P. Durand-Smet and A. Asnacios, Proc. Natl. Acad. Sci. U. S. A., 2015, 112, 2740–2745 CrossRef PubMed.
- P. W. Oakes, E. Wagner, C. A. Brand, D. Probst, M. Linke, U. S. Schwarz, M. Glotzer and M. L. Gardel, Nat. Commun., 2017, 8, 15817 CrossRef CAS PubMed.
- S. B. Khatau, C. M. Hale, P. J. Stewart-Hutchinson, M. S. Patel, C. L. Stewart, P. C. Searson, D. Hodzic and D. Wirtz, Proc. Natl. Acad. Sci. U. S. A., 2009, 106, 19017–19022 CrossRef CAS PubMed.
- K. Nagayama, Y. Yahiro and T. Matsumoto, Cell. Mol. Bioeng., 2013, 6, 473–481 CrossRef CAS.
- M. Nobili and G. Durand, Phys. Rev. A: At., Mol., Opt. Phys., 1992, 46, R6174–R6177 CrossRef CAS.
- Our code is available for download on GitHub: https://github.com/rmerks/CellQTensor.
- J. Galanis, D. Harries, D. L. Sackett, W. Losert and R. Nossal, Phys. Rev. Lett., 2006, 96, 028002 CrossRef.
- Y. H. Tee, T. Shemesh, V. Thiagarajan, R. F. Hariadi, K. L. Anderson, C. Page, N. Volkmann, D. Hanein, S. Sivaramakrishnan, M. M. Kozlov and A. D. Bershadsky, Nat. Cell Biol., 2015, 17, 445 CrossRef CAS.
- G. Duclos, C. Blanch-Mercader, V. Yashunsky, G. Salbreux, J.-F. Joanny, J. Prost and P. Silberzan, Nat. Phys., 2018, 14, 728–732 Search PubMed.
- P. Roca-Cusachs, J. Alcaraz, R. Sunyer, J. Samitier, R. Farré and D. Navajas, Biophys. J., 2008, 94, 4984–4995 CrossRef CAS PubMed.
- B. Jerome, Rep. Prog. Phys., 1991, 54, 391–451 CrossRef CAS.
- P. J. Albert and U. S. Schwarz, Biophys. J., 2014, 106, 2340–2352 CrossRef CAS PubMed.
- Q. Tseng, I. Wang, E. Duchemin-Pelletier, A. Azioune, N. Carpi, J. Gao, O. Filhol, M. Piel, M. Théry and M. Balland, Lab Chip, 2011, 11, 2231–2240 RSC.
- S. Jalal, S. Shi, V. Acharya, R. Y.-J. Huang, V. Viasnoff, A. D. Bershadsky and Y. H. Tee, J. Cell Sci., 2019, 132, jcs220780 CrossRef CAS PubMed.
- A. F. M. Marée, V. A. Grieneisen and L. Edelstein-Keshet, PLoS Comput. Biol., 2012, 8, 1–20 CrossRef PubMed.
- F. J. Segerer, F. Thüroff, A. Piera Alberola, E. Frey and J. O. Rädler, Phys. Rev. Lett., 2015, 114, 228102 CrossRef PubMed.
- P. J. Albert and U. S. Schwarz, Cell Adhes. Migr., 2016, 10, 516–528 CrossRef CAS PubMed.
- M. Eastwood, V. C. Mudera, D. A. McGrouther and R. A. Brown, Cell Motil. Cytoskeleton, 1998, 40, 13–21 CrossRef CAS PubMed.
- D. W. J. Van der Schaft, A. C. C. Van Spreeuwel, H. C. Van Assen and F. P. T. Baaijens, Tissue Eng., Part A, 2011, 17, 2857–2865 CrossRef CAS PubMed.
- O. Wartlick, P. Mumcu, F. Jülicher and M. Gonzalez-Gaitan, Nat. Rev. Mol. Cell Biol., 2011, 12, 594–604 CrossRef CAS PubMed.
- R. F. M. Van Oers, E. G. Rens, D. J. LaValley, C. A. Reinhart-King and R. M. H. Merks, PLoS Comput. Biol., 2014, 10, e1003774 CrossRef PubMed.
- P. Santos-Oliveira, A. Correia, T. Rodrigues, T. M. Ribeiro-Rodrigues, P. Matafome, J. C. Rodríguez-Manzaneque, R. Seiça, H. Girão and R. D. M. Travasso, PLoS Comput. Biol., 2015, 11, 1–20 CrossRef PubMed.
- D. S. Vijayraghavan and L. A. Davidson, Birth Defects Res., 2017, 109, 153–168 CrossRef CAS PubMed.
- T. B. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. T. Lim, J. M. Yeomans and B. Ladoux, Nature, 2017, 544, 212 CrossRef CAS PubMed.
- D. L. Barton, S. Henkes, C. J. Weijer and R. Sknepnek, PLoS Comput. Biol., 2017, 13, 1–34 CrossRef PubMed.
- H. E. Balcioglu, H. van Hoorn, D. M. Donato, T. Schmidt and E. H. J. Danen, J. Cell Sci., 2015, 128, 1316–1326 CrossRef CAS PubMed.
- OrientationJ, http://bigwww.epfl.ch/demo/orientation.

## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0sm00492h |

This journal is © The Royal Society of Chemistry 2020 |