Filamentous network mechanics and active contractility determine cell and tissue shape

Abstract

For both cells and tissues, shape is closely correlated with function presumably via geometry-dependent distribution of tension. In this study, we identify common shape determinants spanning cell and tissue scales. For cells whose sites of adhesion are restricted to small adhesive islands on a micropatterned substrate, shape resembles a sequence of inward-curved circular arcs. The same shape is observed for fibroblast-populated collagen gels that are pinned to a flat substrate. Quantitative image analysis reveals that, in both cases, arc radii increase with the spanning distance between the pinning points. Although the Laplace law for interfaces under tension predicts circular arcs, it cannot explain the observed dependence on the spanning distance. Computer simulations and theoretical modeling demonstrate that filamentous network mechanics and contractility give rise to a modified Laplace law that quantitatively explains our experimental findings on both cell and tissue scales. Our model in conjunction with actomyosin inhibition experiments further suggests that cell shape is regulated by two different control modes related to motor contractility and structural changes in the actin cytoskeleton.

FIGURE 1

Cell shape on micropatterned substrates. (A–C) Arc-like contours composed of actin fibers characterize the shape of BRL (A and B) and B16 cells (C) cultured on substrates of micropatterned fibronectin dots. Cultures were labeled for actin (green), paxillin (red), and fibronectin (blue). Scale bars 10 μm. (A′–C′) For all cases, arc-like contours fit well to circles determined by custom-made software. (B and C) The circles spanning diagonal distances show larger radii than the circles spanning the shorter distances between neighboring adhesions.

FIGURE 2

Simple tissue model. (A) Like for cells on micropatterned substrates, arc-like contours also mark the shape of a fibroblast-populated rat tail collagen gel pinned to the substrate at discrete sites by steel needles. Space bar is 5 mm. (B) The arcs are circular as revealed by automated fitting. (C) Detail of the upper right corner of the tissue model in (A) taken with widefield fluorescence microscopy (actin in green; nuclei in red). Note the circular hole where the steel needle has been removed and an elongated hole toward the gel center, which resulted from gel rupture close to the steel needles due to cellular contractility. (C′) Single optical section through the gel's rim taken with an ApoTome microscope (Carl Zeiss) reveals parallel-aligned cells in the periphery. (D and D′) In the bulk of the collagen gel, cells have a random orientation.

FIGURE 3

R-d relation. (A) For each arc, radius R and spanning distance d are recorded. (B) Simple tension model: Line tension λ straightens the contour between neighboring adhesions, whereas surface tension σ pulls the envelope inward toward the bulk. In mechanical equilibrium, the local radius of curvature is R = λ/σ and does not depend on the spanning distance d. (C) For both, BRL (blue) and B16 (green) cells, arc radius R increases with spanning distance d. (D) R and d also correlate for the simple tissue model based on rat tail (rt, blue crosses) and bovine dermis (bd, black pluses) collagen. The linear fits serve as guides to the eye.

FIGURE 4

Computer simulations of mechanical networks under isometric tension. (A) Tension-free configuration of a regular network of mechanical links pinned to a square pattern. (B and C) In mechanical equilibrium with links simulated as harmonic springs (B) or cables (C), only cable networks show circular arcs. In contrast to harmonic springs, cables show no resistance to compression and so have an asymmetric force-extension relation. (D) In mechanical equilibrium for a disordered cable network, circular arcs form regardless of the details of the adhesion geometry and the network topology.

FIGURE 5

In contrast to the simple tension model, the tension-elasticity model predicts an R(d)-dependence due to the boundary conditions imposed by the adhesion constraints. (A) Elasticity control (EC): R(d) from the effective contour model with varying ratios of fiber rigidity and surface tension l_f = EA/σ (l_f = 500, 200, 100, 50 from top to bottom; rest-length parameter α = 1). The agreement with the computer simulations of cable networks, as shown in Fig. 4, C and D, (circles) is excellent. (B) Tension control (TC): R(d) from the effective contour model with varying rest-length parameter (α = 1.01, 1.05, 1.1, 1.2, and 1.3 from top to bottom, infinite l_f). (C and D) Fits of experimental data for BRL cells (C) and bd-collagen gels (D) to the EC model (green) and the TC model (red). A bootstrap analysis shows that both fits are statistically significant. The predictions of the simple tension model (average R without dependence d) are shown in black.

FIGURE 6

Effect of arc strength. (A) BRL cell labeled for actin (green) and fibronectin (blue) indicates that arcs with stronger actin fluorescence (defined as arc strength S) might have a larger arc radius R than weak arcs. (B) Correlation between arc strength S and radius R for a fixed adhesion distance d between 7 and 8 μm. Despite the large data spread, the correlation is statistically significant.

FIGURE 7

Effects of actomyosin contractility. (A) BRL cell labeled for actin (green), paxillin (red), and fibronectin (blue) treated with blebbistatin, a myosin II inhibitor. (B) Quantitative analysis revealed that blebbistatin treatment caused a shift in the probability distributions P of arc strength S to smaller values (light green) compared to control cells (blue, overlap in dark green). Inhibition of contractility also caused a reduction in arc radius R. (Inset: R(d) for inhibited cells (green) compared to control cells (blue). Lines are fits to the EC model). (C) BRL cell treated with Y-27632, an inhibitor of the Rho-kinase pathway. (D) Y-27632 treatment caused similar effects as blebbistatin. (E) Evidence for TC control: the average of R/d as a function of arc strength S for control (blue), blebbistatin-treated (red), and Y-27632-treated (green) cells. The two inhibition experiments are statistically identical and lead to smaller arc radii than predicted from variations of arc strength S alone.