Geometry and network connectivity govern the mechanics of stress fibers

Significance

Actomyosin stress fiber (SF) networks transmit tension to the microenvironment, which contributes to cell shape and tissue assembly. However, virtually nothing is known about how SF geometry and network structure control the ability of an SF to generate tension. We normalized cell shape and SF length with micropatterning and used laser ablation to probe the viscoelastic properties of the resulting standardized SFs. We find that the retraction dynamics of a cut SF are strongly regulated by SF length, and this relationship may be amplified or offset by the orientation of the attached SFs in single cells as well as in a monolayer. Thus, SF force generation is controlled both locally via adhesive geometry and globally via connections to the actin network.

Abstract

Actomyosin stress fibers (SFs) play key roles in driving polarized motility and generating traction forces, yet little is known about how tension borne by an individual SF is governed by SF geometry and its connectivity to other cytoskeletal elements. We now address this question by combining single-cell micropatterning with subcellular laser ablation to probe the mechanics of single, geometrically defined SFs. The retraction length of geometrically isolated SFs after cutting depends strongly on SF length, demonstrating that longer SFs dissipate more energy upon incision. Furthermore, when cell geometry and adhesive spacing are fixed, cell-to-cell heterogeneities in SF dissipated elastic energy can be predicted from varying degrees of physical integration with the surrounding network. We apply genetic, pharmacological, and computational approaches to demonstrate a causal and quantitative relationship between SF connectivity and mechanics for patterned cells and show that similar relationships hold for nonpatterned cells allowed to form cell–cell contacts in monolayer culture. Remarkably, dissipation of a single SF within a monolayer induces cytoskeletal rearrangements in cells long distances away. Finally, stimulation of cell migration leads to characteristic changes in network connectivity that promote SF bundling at the cell rear. Our findings demonstrate that SFs influence and are influenced by the networks in which they reside. Such higher order network interactions contribute in unexpected ways to cell mechanics and motility.

Actomyosin stress fibers (SFs) enable mammalian cells to generate traction forces against the extracellular matrix (ECM) (1–3). These forces are increasingly recognized to play central roles in the regulation of cell shape, migration, and stem cell fate decisions (1, 4). At the multicellular level, these forces also contribute significantly to tissue morphogenesis, wound healing, and neoplasia (5, 6). SFs are composed of F-actin; structural proteins such as α-actinin; and frequently, nonmuscle myosin II (NMMII) (2, 7). When NMMII filaments are present, these structures generate tension that is transmitted along the length of the SF to ECM adhesions, surrounding SFs, and other connected structural elements such as the actin cortex, microtubules, intermediate filaments, and the nucleus (8). As such, SFs are capable of acting both locally and globally through their networked interactions with other cellular structures to impose tensile loads within the cell (9). This notion is a critical yet largely unexplored linchpin in current models of cell motility, where specific SF subsets are thought to coordinate tensile activities to direct remodeling of ECM adhesions and sculpt migratory processes (1, 10, 11).

The increasing appreciation of SFs as important players in cell mechanics and motility has stimulated great interest in measuring the contractile properties of individual SFs. Several techniques have been used over the past 2 decades, spanning two broad categories. The first and more reductionist category includes characterization of SFs extracted from cells (12) or reconstituted from molecular components (13). The second category includes characterization of SFs in living cells using tools such as subcellular laser ablation (SLA), which allows for the mechanical interrogation of single SFs (3). We and others have applied SLA to determine the viscoelastic contraction of sarcomeric structures within SFs (14–16) and tension distribution after SF ablation to focal adhesions (FAs) throughout the cell (9). Moreover, SLA has been used to spatially map SF viscoelastic properties within the cell and associate activities of specific NMMII activators and isoforms to these subpopulations (17, 18).

An important, emergent theme from these studies is that the effective mechanical contributions of an SF depend strongly on its geometry and structural context, although these relationships remain largely uninvestigated and controversial. Cellular heterogeneity represents a critical barrier to clarifying this, because cultured cells adopt a variety of morphologies giving rise to a poorly controlled diversity of SF geometries. Thus, it is challenging to understand fundamental material properties of SFs, including how length and ECM adhesivity regulate elastic recoil. This heterogeneity has also frustrated efforts to understand how single SF mechanics are related to the network properties in which they reside, even though this interconnectedness is broadly understood to play essential roles in cellular structure and motility.

Results

Single-cell ECM micropatterning represents a valuable tool for accomplishing this goal. The ability of this technology to arbitrarily standardize cell geometry has dramatically improved the field’s understanding of how cell shape controls traction force, cytoskeletal architecture, proliferation/apoptosis, and stem cell fate (8, 19, 20). Micropatterning has also been used to prescribe the geometry of peripheral SFs (21–23). We reasoned that we could productively exploit micropatterning approaches to control cell and SF geometry in combination with SLA and thereby gain new insights into SF mechanics. We were first interested in determining how SF length regulates viscoelastic retraction. To accomplish this, we designed three fibronectin (FN)-coated U-shaped patterns, i.e., patterns consisting of a rectangular frame of matrix with one long edge missing, of aspect ratios 1.5, 1.9, and 3.0 using a UV photopatterning strategy (Fig. 1A and Fig. 1 B and C, fourth column) (24). This strategy produced SFs of similar thickness with two terminal FAs and a length closely conforming to that of the pattern edge across all aspect ratios (Fig. S1 and Fig. 1 B and C, fourth column). We then applied SLA on SFs along the FN-free edge and observed their retraction kinetics. As in our previous studies, we fit each retraction curve to a Kelvin–Voigt viscoelastic cable model described by a time constant (τ), which reflects the SF’s effective viscosity/elasticity ratio, and a plateau retraction distance (L_o), which correlates to the elastic energy dissipated by half of the severed SF (3, 17, 18) (Fig. 1D) (SI Text and Fig. S2 for model choice). The retraction kinetics of these SFs exhibited a clear length dependence, with both L_o and τ (P < 0.01, Dunn test for nonparametric multiple comparison) increasing with length (Fig. S1). On the contrary, SFs produced on FN-filled rectangular patterns showed no statistical variation in dissipated elastic energy (L_o) or τ with SF length (Fig. S3) due to the presence of vinculin-positive FAs along the SF length, which pin the SF and prevent it from freely retracting (14). Thus, dissipated SF elastic energy and viscoelastic properties depend strongly on adhesive spacing, with longer SFs storing more elastic energy. These results validate indirect predictions from earlier micropattern-based studies on the elastic nature of SFs (21–23).

Fig. 1.

Dissipation of elastic energy in severed SFs depends on fiber length. To elucidate SF mechanics and SF length relationships for fixed cell geometry, we created spacing patterns in which cells are cultured on patterns consisting of a rectangular frame that contains a variable-length gap. (A) Schematic of pattern fabrication. (B) (Top) FN distribution on patterns of aspect ratio 1.9 with gap lengths ranging from 6 to 38 μm. (Bottom) Distribution of F-actin (magenta) and vinculin (green) in U2OS cells seeded on the corresponding patterns. A length-defined SF is formed across the gap with focal adhesions formed at the ends. Gap ends depicted by white arrows. (C) (Top) FN stain on patterns of aspect ratio 3.0. (Bottom) F-actin and vinculin distributions. (D) SF retraction analysis. D_a, SF material destroyed by ablation; 2L, distance between fiber ends over time (L is the retraction distance of a severed SF fragment). Length L vs. time t is fit to the Kelvin–Voigt model to determine L_o, whose magnitude correlates with the SF’s dissipated elastic energy, and τ, the viscoelastic time constant, which is the ratio of viscosity/elasticity. (E) Average L_o values for each pattern. A, B, and C statistical families show differences P < 0.05 determined using Dunn test for multiple comparisons of nonnormally distributed data. (F) Average τ values for each pattern. Statistical differences of *P < 0.05 using Kruskal–Wallis followed by Dunn test (N = 27, 58, 89, 72, and 126 for each spacing of aspect ratio 1.9, and n = 13, 21, 38, 40, and 120 for each spacing of aspect ratio 3.0). Data points at 19 μm (aspect ratio 1.9) and 25 μm (aspect ratio 3.0) correspond to the U-shaped patterns (replotted from Fig. S1). Note that for the Kelvin–Voigt model, we measure the retraction of one end of the cut fiber, and as such, SF length is halved. (Scale bars, 10 μm.)

Fig. S1.

(A) Distribution of matrix, actin, and focal adhesions on U-shaped patterns. First row shows fibronectin staining of patterns of three aspect ratios (1.5, 1.9, and 3.0) and three SF lengths (30, 38, and 50 μm). Second row shows distribution of SFs and focal adhesions in U2OS cells as visualized by F-actin (red) and vinculin (green). Third and fourth rows show higher-magnification images show the absence of focal adhesions along the defined-length SF formed at the fibronectin-free side and the presence of adhesions along the pattern’s bottom edge. (B) SF thickness does not vary based on aspect ratio. (C) Length vs. time curves for retracting SFs on U-shaped patterns for varying aspect ratio. (D) L_o increases with SF length as a function of aspect ratio. (E) τ statistically increases with SF length (n = 84, 126, and 120). Statistical differences determined using Dunn test for multiple comparisons of nonnormally distributed data (*P < 0.01). Boxes represent 25th and 75th percentiles; whiskers represent 10th and 90th percentiles. (Scale bars, 10 μm.)

Fig. S2.

(A) An active Kelvin–Voigt model consisting of an elastic, a motor, and a viscous element acting in parallel. $L$ is the retraction length, and it increases after laser cutting due to motor activity and prestress. (B) A beam of length $l$ contracts under the action of force dipoles $P$. The retraction length can be identified as $L=\left|u\left(l\right)\right|$, where $u\left(x\right)$ is the displacement field. The beam retracts if the connection to the wall is soft or if the connection is cut.

Fig. S3.

Viscoelastic properties of SFs of cells grown on fibronectin-coated rectangular patterns of variable-aspect ratio do not depend on SF length due to the presence of vinculin-positive focal adhesions. (A) L_o obtained from the Kelvin–Voigt fit does not vary with SF length as a function of aspect ratio. (B) τ does not vary with SF length (n = 22, 24, and 26 for each aspect ratio). Statistical differences were determined using Dunn test for multiple comparisons of nonnormally distributed data. Boxes represent 25th and 75th percentiles; whiskers represent 10th and 90th percentiles. (C) Distribution of SFs and focal adhesions of U2OS RFP-LifeAct cells seeded in varying-aspect ratio patterns as visualized by F-actin (RFP-LifeAct signal; red) and vinculin (immunofluorescence; green). Second row shows higher-magnification images of cells seeded on patterns of aspect ratio 1.9 showing the presence of focal adhesions along the length of the fiber. (Scale bars, 10 μm.)

Earlier studies have shown that when area is conserved, cellular prestress increases with aspect ratio, raising concerns that these differences could contribute to the observed length dependence (19, 25). However, when we measured whole-cell RMS traction and strain energy on the varying aspect ratio U-shaped patterns, we did not observe differences (Fig. S4) (26). Nevertheless, to directly probe for SF mechanics rather than whole-cell prestress, we designed patterns in which the two parallel arms of the U are symmetrically but incompletely connected with matrix (spacing patterns), leaving a gap of defined length (6, 12, 24, and 30 μm for aspect ratio 1.9 and 6, 12, 24, and 36 μm for aspect ratio 3.0). Cells formed single SFs across the gap, with FAs present at the edge of each SF (Fig. 1 B and C). Dissipated elastic energy released by an SF (corresponding to L_o) did indeed scale with length, whereas τ remained relatively constant (Fig. 1 E and F). The length sensitivity of L_o decreased with increased SF length, which may be due to subtle differences in matrix geometry, SF connectivity, or prestress across these patterns. Additionally, at higher SF lengths, we observed differences in the elastic energy dissipated by SFs of equal length within cells of different aspect ratios (Fig. S5 for primary data).

Fig. S4.

At the whole-cell level, U2OS RFP-LifeAct cells produce similar traction and strain energy on all variable-aspect ratio patterns. (A) Changes in bead displacement and ECM substrate strain distribution after removal of U2OS RFP-LifeAct cells seeded on fibronectin-coated U-shaped patterns of varying aspect ratio made on polyacrylamide gels (9.22 kPa). (B) Map of traction forces relaxed into the ECM substrate after cell removal. (C) No statistical differences observed in average whole-cell traction of cells seeded on U-shaped patterns on varying aspect ratio. (D) Average strain energy of U2OS RFP-LifeAct cells seeded on U-shaped patterns of varying aspect ratio is not dependent on aspect ratio. Bars are mean ± SEM. Statistical differences were determined using ANOVA followed by Tukey (N = 8, 22, and 17 cells for each aspect ratio).

Fig. S5.

Length vs. time curve for retracting SFs on (A) spacing patterns of aspect ratio 1.9 and (B) spacing patterns of aspect ratio 3.0.

As noted earlier, a key motivation for using single-cell micropatterning was to standardize SF geometry and facilitate the development of relationships between SF length and viscoelasticity. Despite this effort, we still observed experimental variation in SF retraction. Identifying remaining sources of heterogeneity could potentially yield valuable and unappreciated regulatory principles underlying SF mechanics. Indeed, when we examined our retraction distributions more closely, we noted that the L_o values were highly skewed on U-shaped patterns of all aspect ratios (Fig. 2A). The distributions closely follow lognormal distributions, with most cells falling under the peak and some cells falling under the long, right-sided tail of the fit (Fig. 2A and Table S1). The appearance of a lognormal distribution suggests that SF mechanical properties depend on their growth history. When we more closely inspected preablation and postablation RFP-LifeAct images for cells found under the peak (average retracting) and under the long tail (highly retracting) (Fig. 2 B and C and Fig. S6), we noted broad, cell-to-cell structural heterogeneities in the SF networks surrounding the target fiber. This in turn led us to hypothesize that variations in network structure might contribute to heterogeneities in SF viscoelasticity. We counted the number of SF connections to the length-defined SF (SF formed across the pattern gap) and found no statistical difference in the number of connections as a function of aspect ratio (Fig. 2D). We then asked whether the orientation of these connected SFs varies across cells. We hypothesized that connecting SFs apply a force on the length-defined SF whose y component (F_y; red arrow) is always downward and whose x component (F_x; green arrow) depends on the intersecting angle and location relative to the ablation site (Fig. 2E). Differences in angular distributions can presumably determine whether these connections and their corresponding forces are enhancing or impeding retraction. We calculated the average angle distribution per cell and binned the L_o values based on the average angle measurements (Fig. S7 for examples of angle analysis). We first observed that for average angles >90°, L_o values are smaller compared with angles <90°, possibly due to the presence of an F_x value that is in the opposite direction of retraction (Fig. 2F). As the average angle distribution increases from 0° to 90°, we observed an increase in L_o that peaks around 20°– 40°. The presence of a peak suggests that retraction kinetics are affected by both the F_x and F_y imposed by connecting SFs. The angle of the connecting SF therefore contributes to dissipated elastic energy (L_o).

Fig. 2.

Analysis of L_o distributions of U-shaped patterns reveals that elastic energy dissipation is heterogeneous and depends on network connectivity. (A) Probability density histograms of L_o on each U-shaped pattern fit to a lognormal distribution. (B) (Top) Preablation F-actin distribution (RFP-LifeAct) of average retracting (A.R.) cells whose L_o values fall under the peak. (Bottom) Overlay of F-actin distribution before (red) and after (green) ablation. (C) (Top) Preablation F-actin distribution of highly retracting (H.R.) cells whose L_o values fall under the tail of the fit. (Bottom) Overlay of actin network before (red) and after (green) ablation. (D) Dependence of number of connecting fibers to length-defined SF does not reveal statistical differences (N = 66, 94, and 103 for each aspect ratio, ANOVA followed by Tukey). Error bars represent SEM. (E) Schematic depicting angles with which the connecting fibers intersect the severed length-defined fiber. If the angle is between 0° and 90°, its cosine is positive and corresponds to an x component of the force (F_x) (green arrow) parallel to the direction of retraction (blue arrow). If the angle is between 90° and 180°, its cosine is negative resulting in an F_x that is antiparallel to the direction of retraction. Each cell is assigned an average angle value. (F) Correlation between observed L_o and average angle for a given cell. Bars are mean ± SEM, and lines portray statistical differences determined using ANOVA followed by Tukey (P < 0.05). (Scale bars, 10 μm.)

Table S1.

Bayesian information criterion and Akaike information criterion values of top three distribution fits of the L_o values of cells seeded on U-shaped patterns of varying aspect ratios

Distribution	Aspect ratio	BIC	AIC
Inverse Gaussian	1.5	187.61	182.75
Lognormal	1.5	187.66	182.80
Bimbaumsaunders	1.5	187.72	182.86
Inverse Gaussian	1.9	330.18	324.50
Bimbaumsaunders	1.9	330.31	324.64
Lognormal	1.9	330.84	325.17
Inverse Gaussian	3.0	427.25	421.68
Bimbaumsaunders	3.0	427.27	421.70
Lognormal	3.0	427.94	422.37

Fits were calculated based on histograms shown in Fig. 2A using the allfitdist command in MATLAB. The command returns all possible distributions that fit the data (normal and nonnormal) including the distribution parameters and the respective Bayesian information criterion (BIC) and Akaike information criterion (AIC) values. We list the top three distribution fits for each aspect ratio (distribution fits are also drawn in Fig. 2A). All fits provided have an average peak and a long tail showing the highly heterogeneous nature of the L_o collected.

Fig. S6.

Despite micropatterning attempts, cells exhibit heterogeneity in retraction kinetics. Kymographs of SF ends of two cells retracting away from each other over time for each aspect ratio. (Top) Cells that undergo small retraction and (Bottom) cells that undergo large retractions. Each frame is 1.96 s. (Scale bar, 10 µm.)

Fig. S7.

Examples of angle analysis and SF segmentation for U2OS RFP-LifeAct cells seeded on U-shaped patterns of aspect ratios (A) 1.9 and (B) 3.0, respectively. First, the image orientation is adjusted in order for the cell horizontal axis to be at 0°. Second, the intensity of RFP-LifeAct images of U2OS cells before ablation was adjusted using ImageJ. Connecting SFs were then visualized by RFP-LifeAct and manually traced. The angle between the connecting fiber and the horizontal axis is measured and recorded in degrees in ImageJ. Angle is recorded as discussed in the model presented in Fig. 2E. We then calculate the cosine and sine of each angle and finally add up all of the values to determine the sum of cos (α) and the sum of sin (α) per cell.

To experimentally test a causal role for SF connectivity in driving retraction, we performed gain- and loss-of-function studies with the myosin activator Rho-associated kinase (ROCK), which has been shown to govern SF assembly and contractility within the cellular interior, including the connecting SFs seen here. By contrast, peripherally located SFs such as the ones severed in our experiments are primarily regulated by myosin light chain kinase and are thus expected to be minimally perturbed by ROCK manipulation (17, 18). To suppress ROCK activity and dissipate central connections, we treated cells with 5 μM Y-27632. To enhance ROCK activity and strengthen connections, we stably overexpressed a constitutively active mutant of ROCK that is induced by doxycycline (CA-ROCK) (27) (Fig. S8). Immunostaining of FAs and SFs confirmed the increase of FA size in the presence of CA-ROCK and reduction in SF and FA formation in the presence of Y-27632 (Fig. 3A). In the absence of doxycyline, CA-ROCK cells have similar FAs to naive cells. Doxycycline addition induced the assembly of numerous SFs connected to the length-defined SF, whereas Y-27632–treated cells exhibited fewer connections (Fig. 3 B and C). The average number of connections per cell increased with CA-ROCK expression compared with naive and Y-27632–treated cells (Fig. 3D). To understand how these manipulations might influence the total F_y and F_x, we quantified the cell-to-cell distribution of the sums of the sines (${\displaystyle \sum }\sin \alpha$ for F_y) and cosines (${\displaystyle \sum }\cos \alpha$ for F_x) of the angles of intersection as defined earlier. Compared with naive cells, Y-27632 treatment shifted the ${\displaystyle \sum }\sin \alpha$ distribution to smaller values for all aspect ratios, whereas CA-ROCK expression shifted the distribution to larger values (Fig. 3E; *P < 0.05 and **P < 0.001). Similarly, Y-27632 treatment produced shifts to slightly more negative values for ${\displaystyle \sum }\cos \alpha$, and CA-ROCK overexpression led to more positive values (Fig. 3F; *P < 0.05 and **P < 0.001). When we performed SLA, we found that Y-27632 treatment reduced elastic energy dissipation (L_o) for all aspect ratios (relative to naive) and muted pattern to pattern differences. Conversely, CA-ROCK overexpression increased dissipated elastic energy compared with naive (P < 0.05 for 1.9 and 3) and enhanced pattern-to-pattern differences (Fig. 3G). With no doxycycline, CA-ROCK cells exhibited similar L_o values to naive U2OS RFP-LifeAct (Fig. S8). In parallel experiments, we found that wholesale inhibition of NMMII with blebbistatin reduced retraction distance while also producing an inward deflection of the length-defined SF (Fig. S9). This finding, together with our previous results, is consistent with our hypothesis that the viscoelastic properties of an SF are governed both by its own axial geometry and that of the network to which it connects.

Fig. S8.

Characterization of pSLIK CA-ROCK cells and 5 μM Y-27632–treated cells. (A) Schematic of the pSLIK system used for expression of CA-ROCK. Briefly, a constitutively active form of ROCK is placed under the control of a doxycycline-inducible promoter. The construct is then placed in a lentiviral vector, which is used to transduce cells and form stable cell lines. (B) Westerns were carried out to verify the effectiveness of the construct and Y-27632 incubation. In the absence of doxycyline (lane 2), the construct remains silent with no observable change in myosin light chain phosphorylation compared with naive cells. In the presence of doxycycline (lane 3), CA-ROCK is activated, leading to increases in myosin light chain phosphorylation. For Y-27632–treated cells, we observe a decrease in myosin light chain phosphorylation after 1 h of incubation. (C) Quantification of Western blot bands normalized to GAPDH and to the expression of phosphorylated myosin light chain of naive cells (n = 4 blots, and bars represent means ± SEM). (D) L_o of U2OS RFP-LifeAct cells and U2OS RFP-LifeAct CA-ROCK cells with no doxycycline are statistically similar, showing that the transduction did not affect SF viscoelastic properties (Student’s t test comparing conditions at each aspect ratio: for CA-Rock, no doxycycline cells, n = 23, 26, and 26 for each aspect ratio, and for U2OS RFP-LifeAct cells, n = 78, 127, and 116 for each aspect ratio). Error bars are SEM.

Fig. 3.

Network control of SF retraction is regulated by Rho-associated kinase-mediated assembly of connecting fibers. (A) Vinculin (green) and SF (red) distributions in naive U2OS, U2OS pSLIK CA ROCK ± doxycycline, and Y-27632–treated U2OS cells (U-shaped patterns, aspect ratio 3.0). (B and C) Effect of CA ROCK and Y-27632 on SF architecture. (Top) SF distributions before (red) and after (green) ablation. (Bottom) High-magnification images of connecting fibers to length-defined SF. (D) Quantification of the average number of connecting fibers per cell (N for Y-27632 = 21, 27, and 28, and N for induced CA-ROCK = 20, 33, and 30 for each aspect ratio). Data points represent mean ± SEM. Statistical comparisons performed using ANOVA followed by Tukey (*P < 0.05, **P < 0.001). Data points for U2OS are transposed from Fig. 2D. (E and F) Histograms of ${\displaystyle \sum }\sin \alpha$ and ${\displaystyle \sum }\cos \alpha$ values per cell obtained for Y-27632, naive, and induced CA-ROCK cells per aspect ratio. Black lines depict the mean value. Statistical comparisons of cosine and sine distributions within same aspect ratio were performed using ANOVA followed by Tukey (*P < 0.05, **P < 0.001). (G) L_o for cells treated with Y-27632 (rectangles), induced CA-ROCK (triangles), and naive (diamonds) on each aspect ratio. N for Y-27632 = 21, 26, and 45, and N for induced CA-ROCK = 18, 33, and 30 for each aspect ratio. Data points for U2OS are transposed from Fig. S1. Data points are mean ± SEM. Statistical differences calculated using Dunn test for multiple comparison of nonnormal data (*P < 0.01). (Scale bars, 10 μm.)

Fig. S9.

Blebbistatin-treated cells undergo smaller retraction but greater inward movement. (A) Quantification of cellular area of cells seeded on U-shaped patterns of aspect ratio 1.9 in the absence and presence of 10 µM blebbistatin (n = 106 and 61, respectively). (B) Length vs. time curves for retracting SFs in cells seeded on U-shaped patterns in the presence and absence of blebbistatin (n = 54 and 126 for each condition). Both follow an exponential retraction. (C) L_o decreases in the presence of blebbistatin. (D) τ is unchanged with treatment of blebbistatin. (E) Overlay of U2OS RFP-LifeAct cells before (magenta) and after (green) ablation. Length-defined SFs of blebbistatin-treated cells undergo a larger inward movement. (F) Quantification of percentage area change after ablation for each population shows that there is a statistically larger change in area in blebbistatin-treated cells due to a larger inward movement. Statistical differences determined using a Student’s t test (**P < 0.0001). Boxes represent 25th and 75th percentiles; whiskers represent 10th and 90th percentiles (N = 61 for and 106). Data for control cells transposed from Fig. 1 E and F (aspect ratio 1.9). (Scale bars, 10 μm.)

To investigate causal relationships between network architecture and SF mechanics more precisely, we developed a simple mechanical model of networked elastic cables subjected to external force (Fig. 4A) (28, 29). Shapes and forces of peripheral SFs in the absence of connecting SFs have been previously described with the tension-elasticity model (TEM) (21). Peripheral SFs show a circularly invaginated shape due to the force balance of an elastic line tension λ and a homogenous surface tension σ (Fig. S10). We extended this model by including internal SFs that connect to the peripheral one and exert a force $F_o$ at a prescribed angle α (SI Text). To simplify the model, we first considered a symmetric cell with one connecting SF at each side. Numerical solution of the system produces a distribution of F_center values with a peak at α ∼70°, reminiscent of our experimental observations. We explored the sensitivity of this solution to model parameters by varying the geometry of the system (connection point as determined by $l_{1,o}$), the active force ($F_o$), and the fiber stiffness EA (Fig. 4B). If the angle α is close to 0° or 180°, moving the connection points closer to the center (large $l_{1,o}$) does not affect the force calculated. For intermediate values of α, however, the force increases with increased $l_{1,o}$ (Fig. 4B, Top). In contrast, varying the active force $F_o$ changes both the basal and maximal force values (Fig. 4B, Middle). Changing the SF stiffness leaves the force levels unchanged for both large and small α (Fig. 4B, Bottom). Based on this simple symmetric model, we conclude that the magnitude of $F_o$ is not the sole determinant for predicting the dissipated elastic energy of a length-defined SF. Instead, the force $F_o$ has both an x component in the direction of retraction and a y component in the normal direction that pulls on the severed SF. This pull distends the retracting SF, introducing an additional elastic component that further enhances retraction. Without this y component force and the additional spring, SFs would retract maximally at α = 0, a prediction not consistent with our experimental observations. The presence of this elastic force is further confirmed by our experimental observation that the length-defined SF moves outward following SLA of the internal connecting fibers showing that connecting fibers contribute through prestress (Fig. S11).

Fig. 4.

An active cable network (ACN) model recapitulates key experimental results. (A) Symmetric model in which connecting SFs are modeled as exerting an active force $F_o$ and defined-length SFs are modeled as active elastic cables. (B) Parametric studies relating the magnitude of F_center to changes in geometry ($l_{1,o}$), SF active tension ($F_o$), and SF stiffness (EA). All curves exhibit the same shape and show that intermediate angles (60°–100°) support the greatest force, which is due to the additional elastic component of the length-defined SF from the pull of the internal $F_o$ that stretches the spring of the length-defined SF. (C–E) Segmentation process of the ACN model. U2OS RFP-LifeAct images obtained before SLA were manually segmented and SF network simulations are created as discussed in SI Text. (F) Plot of simulated retraction lengths $L_f$ (y axis) against experimentally derived retraction lengths L_o (x axis) (n = 8 and 5 for aspect ratios 1.9 and 3.0, respectively). Parameters used for ACN are as follows: $F_o$ = 4 nN, EA = 40 nN at aspect ratio 1.9 and $F_o$ = 7 nN, and EA = 80 nN at aspect ratio 3.0.

Fig. S10.

Schematics of the symmetric tension–elasticity model extended with a connecting fiber. The figure shows the left half of the symmetrical situation described in Fig. 4A in more detail. The connecting SF (blue) separates the length-defined SF (red) in two segments, which we label by indices 1 (segment A–C) and 2 (segment C–B). Each segment forms a circular arc of radius $R_i=\raisebox{1ex}{${\text{λ}}_i$}\!\left/ \!\raisebox{-1ex}{$\sigma $}\right.$, where ${\text{λ}}_i$ is the tension along the segment and $\sigma$ is a homogeneous surface tension. Note that F_center pulls in the horizontal direction for symmetry reasons. The geometric relations and derived model equations are explained in SI Text.

Fig. S11.

Removal of connections causes length-defined SF to shift away from the cell body showing that the SF is under prestress from the connecting network. (Left) Overlay of RFP-LifeAct images of cells before (green) and after (red) ablation of connecting fibers. Ablation point highlighted by red circle. (Scale bar, 10 µm.) (Middle) Normalized fluorescent intensity plot of length-defined SF over highlighted white box. Blue curve shows fluorescent intensity before ablation, and orange curve shows fluorescent intensity after ablation and with the removal of connections. All curves shift to more positive values, suggesting that the SF moves to the right (away from the cell body). (Right) Normalized fluorescent intensity plot of length-defined SF over highlighted gray box. Blue curve shows fluorescent intensity before ablation, and orange curve shows fluorescent intensity after ablation and with the removal of connections. No visible shift is observed, suggesting that the shift shown in Middle is indeed due to the removal of connections.

To further understand how connecting fibers contribute to local retraction within a single SF, we photobleached fiduciary markers in a single SF, photoablated the SF, and followed the retraction of each intermarker segment. When an SF fragment was highly connected at acute angles, SF segments within that fragment retracted more than those in the nonconnected SF fragment (Fig. S12 A and C). However, when an SF fragment was connected at large angles (>90°), SF segments underwent smaller retraction than the nonconnected SF fragment (Fig. S12B). Based on our results and our simple active cable model, we conclude that differences in the location and angle of connecting fibers can explain cell to cell heterogeneities in the elastic energy dissipated by SFs after SLA.

Fig. S12.

Retraction within an SF is heterogeneous and correlates with the geometry of the internal SF network. U2OS RFP-LifeAct cells exhibiting asymmetric connections [(A) top SF fragment highly connected with acute angles, (B) top SF fragment connected at angles >90°, and (C) bottom SF fragment highly connected with acute angles) were photobleached to create rectangular segments within the length-defined SF. The SF was then ablated, and retraction of each segment was traced (third and fourth columns). In A and C, segments within the SF fragment that is highly connected undergo greater retraction than segments in the nonconnected SF fragment. In B, segments in the SF fragment that is connected to the internal SF at an angle >90° undergo smaller retraction due to an opposing force that impedes recoil.

The symmetry inherent in our model (Fig. 4A) represents a major simplification of cellular SF networks. To test whether this reasoning could be extended to more complex experimentally observed SF distributions, we manually traced the location and angle of connecting SFs in a population of cells and generated a triangular mesh network that embeds the SFs as marked edges (Fig. 4 C–E). For illustration, we calculated the equilibrium force distribution for the length-defined SF using active force $F_o$= 5 nN and EA = 500 nN. Although a connecting SF adds only 5 nN of active force, the total force in the length-defined SF is as high as 25 nN at the center where SLA occurs and exhibits a heterogeneous energy distribution depending on the connecting SFs (Fig. 4E). To find optimal parameters for both active force and stiffness, we carried out active cable network simulations based on two data sets (n = 8 and n = 5 cells cultured on U-shaped patterns with aspect ratios of 1.9 and 3.0, respectively). Within each data set, we used the same parameters for all cells. Thus, the differences in the predicted forces between cells on the same pattern arise solely from the geometry of the internal SF distribution. To convert forces retrieved from the model to retraction distances L_f using a simple spring model (k = 3 nN/µm), we adjusted $F_o$ and EA for all cells in the data set to achieve optimal agreement with experiments. We found that connecting SF geometries are sufficient to explain the differences in retraction within each data set (Fig. 4F), although not with the same parameters for both aspect ratios. In contrast, we observed that SFs are both stiffer (EA_{r = 1.9} = 40 nN compared with EA_{r = 3.0} = 80 nN) and more contractile ($F_o$_{,r = 1.9} = 4 nN and $F_o$_{,r = 3.0} = 7 nN) at aspect ratio 3.0. Thus, a simple model that considers SF networks as a series of connected elastic cables enables us to dissect differences in the mechanical properties of SFs and accurately predict the retracted distance of a peripheral SF based only on its connections to other SFs.

To test whether the concepts developed with patterned single cells can also be extended to multicellular structures in which SF networks may be physically integrated across several cells, we performed SLA on cells grown in a monolayer (Fig. 5 A and B). Quantification of retraction kinetics showed that the relationship between SF length and retraction distance observed in patterned single cells (Fig. 1E) holds true in a monolayer setting: increasing SF length leads to increased L_o, reaching an eventual plateau (Fig. 5C). Remarkably, the effect of connectivity is amplified in the presence of cell–cell adhesions, as evidenced by the transmission of tension release from the targeted cell to its nearest neighbors. In some instances, this tension release is felt within distant cells tens of micrometers away (Fig. 5B and Movies S1 and S2).

Fig. 5.

Dissipation of elastic energy in a monolayer setting is related to SF length and connectivity. To elucidate whether the relationships uncovered using patterned single cells hold true in nonpatterned cells, we performed SLA of SFs in cells grown as a monolayer. (A and B) Examples of an ablated SF within cells that are growing in a monolayer. Overlay of cells before (green) and after ablation (magenta). Solid red boxes show overlay of images before (green) and after (magenta) ablation. Dashed red boxes show overlay of images taken at different times with no ablation to control for any SF movement during imaging. Arrowheads within solid red boxes point to regions of increased SF network rearrangement. In B, the ablated SF is highly connected and therefore undergoes large retraction, which leads to SF network rearrangements within cells farther away from the site of ablation (arrowheads within solid red boxes). (C) Overlay of patterned single-cell retraction kinetics (data transposed from Fig. 1E) and monolayer ablation retraction kinetics (gray diamonds). Both single-cell and monolayer ablation results exhibit similar relationships between SF length and dissipated elastic energy after SLA. (Scale bars are 10 μm for all images except zoomed-in images in A, where scale bars are 5 μm.)

Finally, we wondered whether changes in SF connectivity might contribute to broader cell-scale functions such as directional motility. To study the effect of connectivity on migration, we stimulated patterned cells using epidermal growth factor (EGF) to initiate the formation of migratory processes (20). When cells grown on the spacing patterns of aspect ratio 1.9 were EGF-stimulated, migratory processes were formed in ways that depended on the gap length (Fig. S13): for small gaps, migratory processes were formed at either the spacing side or the FN side, whereas for larger gaps, migratory processes only formed at the FN side, leading to a polarized cell with a rear SF. Moreover, the rear SFs of EGF-stimulated cells on U-shaped patterns (aspect ratio 1.5 and 1.9) exhibited increased connectivity with more internal SFs connecting to them (Fig. 6 A and B). When comparing the average angle distribution of naive versus EGF-treated cells, we observed a decrease in the average angle, which suggests that under EGF treatment, cells rearrange their SF networks to both increase the number of connections and align these connecting SFs at more acute angles (Fig. S14). Connectivity analysis revealed increases in both ${\displaystyle \sum }\cos \text{α}$ and ${\displaystyle \sum }\sin \text{α}$ distributions for both aspect ratios (Fig. 6C; **P < 0.001) due to both the change in the angle of connecting fibers and the increase in the number of connections. Based on the broader force distributions, we hypothesized and observed that the rear SFs of EGF-treated cells dissipate increased elastic energy after ablation (Fig. 6D; **P < 0.001). Our results suggest that as the cell is induced to migrate, SF connectivity evolves to suit the tensile/contractile needs of the rear SF, resulting in changes in SF mechanical properties as measured by SLA.

Fig. S13.

Treatment with EGF leads to a formation of a lamella-like structure whose orientation depends on the spacing of the pattern. (A) Actin distributions (phalloidin stains) within U2OS cells seeded on spacing patterns of aspect ratio 1.9. On all patterns, cells form four peripheral SFs across the boundaries of the pattern. (B) Actin distributions within U2OS cells seeded on spacing patterns and treated with EGF for 4 h. EGF induces a motile phenotype where the peripheral SFs at the pattern boundaries dissociate to form dorsal SFs and transverse arcs. When the spacing is small (6 and 12 µm), the SF formed across the gap may also dissociate, and migratory-like structures can form at either the top or the bottom part of the pattern. As the spacing increases, the SF formed across the gap remains and becomes the rear of the cell. For all images, the spacing is at the top. (Scale bar, 10 µm.)

Fig. 6.

U2OS RFP-LifeAct cells treated with EGF acquire a motile phenotype where the rear SF exhibits increased connectivity and increased elastic energy. (A) U2OS RFP-LifeAct cells seeded on U patterns treated with EGF. (B) Quantification of the average number of connecting fibers per cell (N for EGF treatment = 27 and 31 for each aspect ratio). Data points represent mean ± SEM. Data for non-EGF–treated cells are transposed from Fig. 3D. Statistical comparisons performed using two-way Student t test (**P < 0.001). (C) Histograms of (Left) ${\displaystyle \sum }\cos \text{α}$ and (Right) ${\displaystyle \sum }\sin \text{α}$ values per cell obtained for EGF-treated cells grown on U patterns of aspect ratio 1.5 (Top) and aspect ratio 1.9 (Bottom). Black lines depict the mean value of EGF-treated cells, and red lines depict the mean value of non-EGF–treated cells (averages transposed from Fig. 3 E and F). Statistical comparisons of cosine and sine distributions within same aspect ratio were performed using two-way Student t test (**P < 0.001). (D) L_o for cells treated with EGF on each aspect ratio. n = 44 and 48 for EGF-treated cells for each aspect ratio. Data points are mean ± SEM. Statistical differences calculated using two-way Student t test (**P < 0.001). Data for non-EGF–treated cells are transposed from Fig. S1. (Scale bars, 10 μm.)

Fig. S14.

Treatment with EGF leads to a shift in the average angle distribution to smaller (more acute) angles for both aspect ratios 1.5 and 1.9. (Top) Average angle distribution of naive U2OS RFP-LifeAct cells seeded on U patterns of varying aspect ratios. (Bottom) Average angle distribution of cells seeded on the respective patterns after EGF stimulation. Black lines depict the mean of the distributions and shift to smaller angles with treatment of EGF. *P < 0.05 using Student's t test.

Discussion

In this study, we have combined single-cell micropatterning and SLA to elucidate how SF geometry and connections within a complex network alter the elastic energy dissipated by an SF. Previous studies have produced several models that suggest that retraction length scales with SF length but make no predictions on how SF contractility and stiffness vary with SF length (14–16). Our results show that the relationship between SF length and dissipated elastic energy after SLA is more complex than previously appreciated and offers insights into this intricate relationship.

First, simulations demonstrate that longer SFs are stiffer and more contractile, a result that would not be predicted by existing SF models. Second, we do not find a simple linear increase, as suggested by simple mathematical models for SF retraction (SI Text), but a decrease in slope with increasing SF length. This is an insight that could be due to the interconnectedness of the actin SF network. Our study shows that the presence of internal connections leads to inhomogeneous retraction within the same SF as well as heterogeneities across standardized cells. In addition to producing prestress on the length-defined SF, differences in connectivity may also affect τ due to differences in drag force, in that a highly connected SF pulls a greater subset of the actin cytoskeleton as it retracts, effectively increasing viscous drag (Fig. S15). Our results directly show that connecting fibers, rather than the viscosity of the cytosol, dominate the viscous component of the retraction, thereby resolving an important open-ended question in the field of SF mechanics (15).

Fig. S15.

Overlay of cells before (green) and after (magenta) ablation to show that connecting fibers move along with the cut fiber [(A) observed green and purple SFs in overlay as shown by the white arrows]. The time constant τ is (A) 10.93 s and (B) 8.19 s.

Furthermore, when we cut single SFs in a monolayer setting, we observed tension release and SF rearrangement in distant cells. Recent work has suggested that transmission of contractile forces between cells critically regulates collective migration and durotaxis. Much of the evidence for this concept has been indirectly inferred from whole-cell traction force and genetic and pharmacological manipulations of cadherins and myosin (30–33). Our findings now provide direct evidence that single cytoskeletal structures in cells mediate tensile forces over long distances across a monolayer. Our results also provide mechanistic insight into how aggregates of cells can cooperatively generate outsize traction forces, which is important in collective cell migration and matrix remodeling (30).

Although our cable network model was quite minimalistic and only incorporated a small portion of the cytoskeletal network, it successfully predicted key experimental results and achieved remarkable quantitative agreement with the experimental results for specific cells. This suggests that cellular geometry and SF network properties are the primary drivers of SF viscolelastic retraction, whatever other cell-to-cell variations in structure or mechanics may exist. Precisely how SF network heterogeneities arise and evolve represents an important open question, and it is possible that this may be a function of the cell’s adhesive and spreading history. Indeed, our findings complement the recent, elegant demonstration that patterning reconstituted actomyosin network architecture can modulate contractility (34). Development of analogous micropatterning strategies for living cells may make it possible to test the hypotheses raised in our study by engineering the details of the internal SF network through strategic placement of adhesions.

Materials and Methods

Cell Lines and Reagents.

Viral particles of the pFUG-RFP-LifeAct vector were packaged in 293T cells and used to infect U2OS cells (ATCC HBT-96) (35). Cells were cultured in DMEM with 10% (vol/vol) FBS (JR Scientific), 1% penicillin/strep (Thermo Fisher Scientific), and 1% nonessential amino acids (Life Technologies).

Confocal Imaging of Immunofluorescence and SF Photodisruption.

Immunostaining imaging and SF SLA experiments were performed on a Zeis LSM 510 Meta Confocal microscope equipped with a MaiTai Ti:sapphire femtosecond laser (Spectra Physics) (9, 17, 18).

Deep UV-Based Pattern Fabrication.

Patterns were made as described elsewhere (24) and as shown in Fig. 1A.

SI Materials and Methods for details on all methods and data analysis.

SI Text

Theory of Retraction.

Active Kelvin–Voigt model.

Stress fibers (SFs) are defined as actin-based bundles with periodic structure (2). They can be classified into different subclasses (dorsal, ventral, transverse arcs, and peripheral) (1, 2, 7), but all of them have a periodic arrangement of its constituents, in particular the actin crosslinker α-actinin. Here we consider contractile SFs, such as ventral SFs, that also have a periodic arrangement of myosin II minifilaments. We conclude that such SFs have sarcomeric structures, similar to striated muscle (36). Because of the one-dimensional nature of the system, each sarcomere must sustain the same force and for many purposes, it is sufficient to consider one such element. Because SFs can maintain force over a long period, they seem to be dominated by the elastic element, and we model them as a Kelvin–Voigt model for an elastic solid, supplemented by a contractile element (Fig. S2A) (15, 23, 37–39). This model has also been used before to evaluate laser cutting experiments of the actin cortex (40).

Fig. S2A shows that in an active Kelvin–Voigt model, the elastic, viscous, and contractile elements are arranged in parallel; thus, the force balance simply reads

$$\text{ξ}\dot{L}+kL-F_m=0,$$ [S1]

where $\text{ξ}$ is the friction coefficient, $k$ the spring constant, and $F_m$ is the motor force. If we use a linearized force–velocity relation for the motor activity

$$F_m=F_s(1-\frac{v}{v_o})$$ [S2]

with stall force $F_s$, free velocity $v_o$, and velocity $v=\dot{L}$, we see that the second part adds a constant contribution to the friction coefficient, $\text{ξ}$ → $\text{ξ}+\frac{F_s}{v_o}$. We define the relaxation time $\tau =\frac{\text{ξ}}{k}$ to write the force balance as

$$\tau \dot{L}+L-\frac{F_s}{k}=0.$$ [S3]

With the appropriate initial conditions, this is solved by

$$L=D_a+\frac{F_s}{k}(1-e^{-\frac{t}{\tau }}).$$ [S4]

Because all terms in the force balance are linear, we obtain an exponential relaxation. Eq. S3 is commonly used for evaluating retraction after laser cutting and usually gives an excellent fit to the experimental data (3).

From Eq. S4 we see that the final retraction length is simply

$$L_o=\frac{F_s}{k}.$$ [S5]

$F_s$ scales linearly with the number of motors pulling in parallel in the actin bundle. The fact that retraction is caused by motor force $F_s$ corresponds to the experimental observation that the myosin II inhibitor blebbistatin strongly reduces retraction (compare Fig. S9). The larger the motor force $F_s$, the larger is the retraction length $L_o$, as is the initial speed, $\dot{L}\left(t=0\right)=\frac{F_s}{\text{ξ}}$. Because the spring constant $k$ has been measured in SF-pulling experiments to be of the order of 3 $\frac{\text{nN}}{\text{μm}}$ (23) and because the typical final retraction length is $L_o=2\hspace{0.17em}\text{μm}$, we can estimate the typical motor stall force as $F_s=k{L}_o=6\hspace{0.17em}\text{nN}$, in agreement with earlier results (23, 41). Note that in our simple model, the system comes to a halt because compressive energy is built up in the spring. Alternatively, one might argue that it comes to a halt because the myosin motors can only pull out a finite length because they eventually reach the barbed ends of the actin filaments (23). Another potential mechanism is that the contracted SF is held by connections in its environment, e.g., the substrate or the surrounding cytoskeleton (see below).

Until now, we have assumed that before cutting, the motors are stalled and that the elastic spring is relaxed. However, it is very likely that the system is prestressed, as suggested, e.g., by analysis of cell shape on micropatterned substrates (21, 22) and by initial jumps observed in the retraction (16). We therefore now include a prestress in the equations:

$$\text{ξ}\dot{L}+k(L-L_1)-F_s=0;$$ [S6]

thus, both the motor force and the prestressed spring now pull on an SF right after cutting $(L=0)$. We immediately note that the equations remain linear, and we thus get the same exponential relaxation, with the only difference being that $F_s\to F_s+k{L}_1$. For the final retraction we thus get a larger value:

$$L_o=\frac{F_s+k{L}_1}{k}.$$ [S7]

If we assume that the retraction after cutting is completely determined by the prestress $(F_s=0)$, then $L_o=L_1$. The above estimate $F_s=6\hspace{0.17em}\text{nN}$ for the motor force now becomes an estimate for the force generated by the prestress before cutting. This would also be the force $F_a$ transmitted to a focal adhesion holding the SF. In general, we expect the force

$$F_a=F_s+k{L}_1$$ [S8]

transmitted to the focal adhesion to contain contributions from both motors and prestress. We conclude that prestress might be a substantial part of the force measured at the SF endpoint (i.e., at the focal adhesion), but this cannot be deduced from the retraction kinetics, because it has the same shape both without and with prestress.

SF as contractile beam.

Until now, we did not explicitly consider the spatial extent of the SF. We now consider the equations for a spatially extended beam of length $l$ contracted locally by force dipoles $P$ (corresponding to the bipolar myosin II minifilaments) (Fig. S2B). We introduce the spatial coordinate $x(0\le x\le l)$ along the beam of length $l$. The 1D displacement is called $u\left(x\right)$ (we do not treat the dynamic case, so there is no time $t$ here). The final retraction length can now be identified as $L_O=\left|u\left(l\right)\right|$ [we take the modulus because $u\left(l\right)$ will be negative]. Assuming a linear constitutive law, the internal tension $T$ (stress times cross-sectional area) in the beam is

$$T\left(x\right)=Cu\prime \left(x\right)+P\left(x\right),$$ [S9]

where $C=EA$ is the 1D modulus. In the following, we assume homogeneous contractility, $P\left(x\right)=\text{const}.$ Force balance dictates $T\left(x\right)\prime =0$ (no external forces), and therefore,

$$Cu\prime \prime =0.$$ [S10]

With the boundary condition $u\left(0\right)=0$ and $T\left(l\right)=0$ (no displacement at the left and free stress boundary at the right), this is solved by

$$u\left(x\right)=-\frac{P}{C}x.$$ [S11]

Thus, the displacement increases linearly along the beam. It is driven by internal contractility $P$, which is balanced by compressive energy being stored in the beam. For the final retraction length, we get

$$L_O=\left|u\left(l\right)\right|=\frac{P}{C}l,$$ [S12]

which corresponds to Eq. S5 with $k=\frac{C}{l}=\frac{\text{EA}}{l}$. We conclude that for the simplest possible SF model, retraction length $L_o$ should scale linearly with SF length $l$. We also note that with the estimate $k=3\frac{\text{nN}}{\text{μm}}$ from above and the typical length $l=10\text{μm}$ of the cut segment, we get $C=\text{EA}=kl=30\text{nN}$ as a typical value for the 1D modulus.

Because we assume contractility $P$ to be constant, it enters the calculation only through the boundary condition, exactly in the way an elastic prestress would enter, because prestress would be generated by an external force $F$ pulling the beam to a certain extension. Thus, like for the active Kelvin–Voigt model, we again find that it is difficult to differ between the effect of motor force and prestress. If we go back to Eq. S7 and again use $k=\frac{\text{EA}}{l}$, we find that the linear scaling now only follows if we assume that $L_1=cl$ with a constant $c$, meaning that the resting length exactly scales with SF length $l$. Then we would get

$$L_o=\frac{F_s+\text{EA}c}{\text{EA}}l,$$ [S13]

and thus, the final retraction length $L_o$ should scale again linearly with SF length $l$. Of course these results only hold when both motor force $F_s$ and SF rigidity $k$ do not show some unexpected scaling with $l$. Thus, the standard model for SFs suggests a linear scaling between retraction length and SF length, even for a prestressed fiber. However, this result is based on many strong assumptions, and experimentally, there might be several reasons why this might not be the case. In particular, it has been argued earlier that nonlinear geometric effects for peripheral SFs break this kind of linear scaling (21), as also confirmed here by our computer simulations.

We next consider the force that should be exerted by an SF that does not have a free end but is attached at the end to a focal adhesion. Then we have to implement the boundary condition $T\left(l\right)=-Ku\left(l\right)$ with an external spring constant $K$. The solution is

$$u\left(x\right)=-\frac{P}{C+Kl}x,$$ [S14]

and the force on the boundary is

$$T\left(l\right)=\frac{Kl}{C+Kl}P=\frac{K}{k+K}P,$$ [S15]

which corresponds to two springs in series. In the limit of a stiff focal adhesion, $K\gg k$, the force on the substrate is simply $P$, independent of the length of the SF. Thus, for a contractile beam, $P$ has the same function as $F_s$ for the active Kelvin–Voigt model.

We also consider the elastic energy stored in the contractile beam. Because constant contractility $P$ only enters through the boundary condition, the elastic energy can be written as

$$U=\underset{0}{\overset{l}{{\displaystyle \int }}}\frac{1}{2}Cu\prime {\left(x\right)}^{2}dx=\frac{1}{2}Cl{\left(\frac{P}{C}\right)}^2=\frac{P^2}{2C}l,$$ [S16]

and, thus, a longer SF stores more energy. With $k=\frac{C}{l}$ and $F_s=P$, the same results follow from the active Kelvin–Voigt model with $U=\frac{F^s}{2k}$.

We finally consider the case that the beam is attached along its length to the environment through springs with spring constant per length $\rho$. Now the force balance reads $T\prime =\rho u$ and, thus,

$$Cu\prime \prime =\rho u.$$ [S17]

Defining the decay length $\text{λ}=\sqrt{\frac{C}{\rho }}$, we can write

$${\text{λ}}^{2}u\prime \prime -u=0.$$ [S18]

With the same boundary conditions as above this is solved by

$$u\left(x\right)=-\frac{P\text{λ}}{C}\frac{\sinh \left(\frac{x}{\text{λ}}\right)}{\cosh \left(\frac{l}{\text{λ}}\right)}.$$ [S19]

For $\rho \to 0$, $\text{λ}\to \infty$, we recover the result from Eq. S11 through a Taylor expansion. For the tension in the beam, we find

$$T\left(x\right)=P\left[1-\frac{\cosh \left(\frac{x}{\text{λ}}\right)}{\cosh \left(\frac{l}{\text{λ}}\right)}\right];$$ [S20]

thus, it is maximal at the left end and vanishes at the right end. For small $\text{λ}$, the tension at the left reaches the maximal possible value of $P$. In general, this calculation shows that stress will localize to the cut end if there is some connection to the environment. For the final retraction length, we find

$$L_o=\left|u\left(l\right)\right|=\frac{P\text{λ}}{C}\tanh \left(\frac{l}{\text{λ}}\right).$$ [S21]

Thus, it initially increases with beam length $l$ but then plateaus for large $l$ because for a long beam, only the end parts retract, as indeed observed experimentally when the SF is attached to the substrate (14). Here we consider SFs that are isolated from the substrate but do have some connections to the surrounding network. Then one would also expect that the final retraction length starts to plateau with increased SF length, in agreement with our experimental findings.

Symmetric Model Introduced in Fig. 4.

Mathematical description.

To understand the effect of internal SFs pulling a length-defined SF inward, we develop a simple mechanical model of connected cables subjected to external force. The connecting SFs are modeled by an active force $F_o$ at an angle α with the horizontal axis (Fig. S10). The length-defined SF is modeled by an active cable intersected by the connection points of the connecting SF. Further, a homogenous surface tension σ exerts a constant inward pull along the two segments of the peripheral SF, which results in a circular shape as described previously with the tension-elasticity model (TEM) (21). Note that the curvature along the peripheral SF is discontinuous at the connection point to satisfy force balance. For comparison with experiments, we are interested in the force F_central, which can be assessed if the rest lengths of the two fiber segments $l_{\mathrm{1,0}}$ and $l_{\mathrm{2,0}}$, the fiber stiffness $EA$, and the active force level $F_o$ are known parameters.

We first identify the length of the first segment, $l_1$; the length of the second segment, $l_2$; and the angle $\beta$ of the tangent of the first segment with the horizontal line at the intersection point as free variables and express all other variables in terms of these three. This leads to

$${\delta }_i=\raisebox{1ex}{$l_i$}\!\left/ \!\raisebox{-1ex}{$2\pi R_i$}\right.,\hspace{1em}{R}_i=\raisebox{1ex}{${\text{λ}}_i$}\!\left/ \!\raisebox{-1ex}{$\sigma $}\right.,\hspace{1em}{\text{λ}}_i=\text{EA}\frac{l_i-l_{i,0}}{l_{i,0}},\hspace{1em}{d}_i=2R_i\sin \left(\frac{{\delta }_i}{2}\right).$$

Further, we establish the geometric properties

$$\beta \prime =\beta +\frac{{\delta }_1}{2},\hspace{1em}\gamma \prime +\frac{{\delta }_2}{2}=\gamma \hspace{0.17em},\hspace{1em}\gamma ={\delta }_2.$$

Note that by the last geometric property, γ is not a free variable but depends on l₂. To solve the problem, we need to find three relations between the free variables. The first two are given by the force balance equation at the connection point,

$${\overrightarrow{F}}_0+{\text{λ}}_1{\overrightarrow{t}}_1+{\text{λ}}_2{\overrightarrow{t}}_2=\overrightarrow{0}-F_0\sin \alpha +{\text{λ}}_1\sin \beta -{\text{λ}}_2\sin \gamma =0-F_0\cos \alpha -{\text{λ}}_1\cos \beta +{\text{λ}}_2\cos \gamma =0.$$

Here ${\overrightarrow{t}}_1$, ${\overrightarrow{t}}_2$ denote the tangents of the two circle segments at the connection point C. The third relation follows from the geometrical consideration that the horizontal distance between points A and B in Fig. S10 equals the sum of the two rest lengths, $l_{\mathrm{1,0}}+l_{\mathrm{2,0}}$, which leads to

$$\cos \beta \prime =\frac{l_{\mathrm{1,0}}+l_{\mathrm{2,0}}-d_2\cos \gamma \prime }{d_1}.$$

We can further simplify this relation if we assume that $l_f=EA/\sigma \gg 1\hspace{0.17em}\text{µm}$, which implies approximately straight segments and seems to be a good approximation for the experimental observations (as depicted in Fig. 4A). In this limit, $R_i=\infty ,\hspace{1em}{\delta }_i=0,\hspace{1em}\gamma =0,\hspace{1em}{l}_i=d_i,$ and consequently,

$$F_o\times \sin \left(\alpha \right)-\left[F_o+k\left(l-l_o\right)\right]\times \sin \left(\beta \right)=0$$

$$F_o\times \cos \left(\alpha \right)+\left[F_o+k\left(l-l_o\right)\right]\times \cos \left(\beta \right)=F_o+k_2(l_2-l_{2,o})$$

$$\cos \left(\beta \right)=\frac{l_o+l_{2,o}-l_2}{l}$$

$$0\le \beta \le \frac{\pi }{2}.$$

Note that here and before the elastic parts are only evaluated if $l-l_o>0$, which represents the cable asymmetry between stretch and compression. We numerically solved this system of equations with the commercial software package Mathematica using $F_o$ = 5 nN, $k=\frac{\text{EA}}{l_o}$, $k_2=\frac{\text{EA}}{l_2}$, $\text{EA}$ = 500 nN, $l_o$ = 26 µm, $l_{2,o}$ = 4 µm, which are typical values for U2OS cells (29). Rather than exhibiting a simple force decay with increasing α, the force distribution shows a peak at $\alpha$ ∼ 70°, and the maximal force is considerably higher than the active forces involved (Fig. 4B). The force decreases for higher α until it reaches the active force level at $\alpha$ ∼180°. The plateaus on the left and right end of the curve correspond to conditions in which either the outer (left) or central (right) cable is in the compression regime.

Interpretation of results.

This simple model shows that the exact magnitude of active forces might not be the only determinant for the retraction length. As the connecting SFs pull the contour inward, an elastic stress occurs in the length-defined SF that might be higher than the active force of the connecting SF and is released when the length-defined SF is severed. Experiments confirm that the contour not only retracts along itself after ablation but also moves inward in this process. This indicates that there is indeed a stress caused by connecting SFs that pull inward. This stress is primarily determined by the connection location of the connecting SF to the length-defined SF, and by the connection angle. We conclude that configurations with either only small or only large α values are well suited to figure out the scale of active forces in experiments.

Active Cable Network Simulations.

Description of workflow.

Unlike the symmetric model, cells exhibit asymmetric SF distributions. To determine whether the principles underlying the model may be extended to these more complex experimental networks, we conducted an additional set of simulations based on active cable networks (ACN). The advantages of ACN simulations are that they are very flexible regarding the geometry of the fiber distribution, and they allow for the introduction of background network tension.

The workflow is illustrated in Fig. 4 C–E and is as follows: We take the actin fluorescence image and enhance the contrast in ImageJ such that the SFs are clearly visible. We then use our plugin for the Segmentation of Focal Adhesions and Stress Fibers (29) where the user can manually segment SFs by marking them with segmented lines. The plugin also allows us to segment cell area from the actin image and generate a triangular network that covers the whole cell area and embeds the SFs as marked edges. Due to the difficulty of segmenting a cell in such a way that a curved SF lies exactly at the cell rim, we implemented a mesh alteration routine, which removes the network part on the outside of the peripheral SF. We then perform an optimization with respect to the ACN energy using the same active force and one-dimensional Young’s modulus as for the symmetric model, $F_o$ = 5 nN and $EA$ = 500 nN for all SFs. Because we segment the invaginated contour of the cell, we set the rest length of the length-defined SF to 98% of its segmented length when comparing simulations and experiments. The background network was chosen to be softer by an order of magnitude and with a very small active component, which was added for numerical reasons. An example of a network after energy optimization with the custom-written software SurfaceMaster is shown in Fig. 4E, where the force magnitude is shown in color code. Fig. 4E shows that the qualitative behavior of the symmetric model is broadly preserved in the ACN simulation. Although the individual connecting SFs add only 5 nN of active force, the total force in the fiber is as high as 25 nN at the center. The other segments show lower forces depending on the connectivity of the SFs. These force predictions enable comparison of model predictions to experiments. To perform this comparison, we investigated two datasets, where either n = 8 or n = 5 cells where cultured on U-shaped patterns with aspect ratio 1.9 and 3.0, respectively. Because we used the same parameters for all cell simulations within one dataset, any differences observed in force magnitude at point of ablation are attributed only to the geometry of the internal SF network, which was based on segmentations of the RFP-LifeAct images of individual cells. We then recorded the predicted force at the location of ablation in the corresponding experiment. We converted these forces to lengths $L_f$ for comparison with the retraction length $L_o$ using a simple spring model, $F_{simulation}=k\times L_f$, where k = 3 nN/µm was chosen.

Interpretation of results.

The retraction lengths and the force lengths $L_o$ and $L_f$ match surprisingly well. Apart from one cell in each dataset, the difference $\mathrm{\Delta }L=\text{abs}\left(L_o-L_f\right)$ is below 1 µm. Cells that show large retraction lengths show high forces at the point of ablation in the simulations, and conversely, SFs with low retraction lengths are predicted to be under low force. We emphasize that information about the experimental retraction length did not enter the force prediction process and that the parameter settings were the same for all simulations within each dataset. This means that the active force was assumed to be the same for all SFs. This is obviously a very strong assumption and underlines the claim from the symmetric model that the connectivity determines peripheral SF forces more than force magnitudes. We conclude that ACN simulations are a fast and reliable way of estimating the force distribution along the contour.

SI Materials and Methods

Cell Lines and Reagents.

Viral particles of the pFUG-RFP-LifeAct vector were packaged in 293T cells and used to infect U2OS cells (ATCC HBT-96) at an MOI of 1.5 IU/cell (33). Cells expressing the pFUG-RFP-LifeAct vector were sorted on a DAKO-Cytomation MoFlo High-Speed Sorter based on RFP fluorescence (33). Myc-tagged p160ROCK Δ3 (kindly provided by Shuh Narumiya, Kyoto University, Kyoto) was subcloned into the lentiviral vector pSLIK containing the TRE tight doxycycline-inducible promoter (27). U2OS RFP-LifeAct cells were further stably transduced with the pSLIK vector at an MOI of 0.5 IU/cell. Cells were sorted based on RFP and Venus fluorescence. Cells were cultured in DMEM with 10% (vol/vol) FBS (JR Scientific), 1% penicillin/strep (Thermo Fisher Scientific), and 1% nonessential amino acids (NEAA; Life Technologies). Y-27632 2HCL was from Selleck Chemicals, and doxycycline and blebbistatin were purchased from Sigma Aldrich. EGF was obtained from R&D Systems and was used at 100 ng/mL. Cells were stimulated with EGF for at least 2 h and 30 min before ablation.

Deep UV-Based Pattern Fabrication.

Chrome quartz photomasks were designed using CleWin 4.0 and were printed from aBeam Technologies. Patterns were made as described elsewhere (24). Briefly, glass coverslips were coated with 0.01 mg/mL PLL-PEG (Surface Solutions) in 10 mM Hepes, pH = 7.4, for 1 h. After drying with air, coverslips were exposed to deep UV (UVO cleaner; Jelight) through a photomask. A drop of water was used to hold the coverslips on the photomask, and care was taken to remove any excess liquid and air bubbles. After deep UV incubation, coverslips were placed in DI water and left to incubate for at least 30 min. Coverslips were then incubated with 34.25 μg/mL of fibronectin (EMD Millipore Corporation) in 10 mM Hepes, pH = 8.5, for 1 h at 37° (Fig. 1A). Patterns were washed 3× for 5 min in PBS before cells were plated.

Confocal Imaging of Immunofluorescence and Photodisruption of Single SFs.

U2OS RFP-LifeAct cells were seeded on patterns and allowed to spread for 3 h. Media was changed to Live Cell Imaging Solution (Invitrogen) before experiment. For SF photodisruption, the femtosecond laser was used at 770 nm resulting in an energy deposition of 1–2 nJ on a single SF (9, 17, 18). SLA experiments were performed between 3 and 6 h after cell seeding. All images were acquired with a 40× water-dipping objective (N.A = 0.8). For Y-27632 experiments, 5 μM of Y-27632 was added 2 h after seeding, and for blebbistatin, 10 μM was added 3 h after seeding for 1 h and 30 min. CA-ROCK cells were cultured in doxycycline for 2 days before experiments.

Data Analysis of SF Retraction.

SF retraction distance was recorded every 1.96 s for 43 s following SLA. Retraction dynamics were obtained through tracking of the two ends in ImageJ. Results were then fit to a Kelvin–Voigt model,

$$L=D_a+L_o\left[1-\exp \left(-\frac{t}{\tau }\right)\right],$$ [S22]

where L is defined as half the distance between the two severed ends, $D_a$ is the length of SF destroyed by ablation, $L_o$ is the retraction plateau distance, and τ is the viscoelastic time constant determined using CurveFit, MATLAB. Note that this equation is equivalent to Eq. S4, discussed above. $L_o$ is proportional to the ratio of prestress to elasticity of the fiber, whereas τ is the ratio of viscosity to elasticity (9, 17). Connections to the length-defined SF were manually traced using confocal images of U2OS RFP-LifeAct cells and angles were obtained using ImageJ (Fig. S9).

Monolayer Laser Ablation.

A 35-mm glass bottom dish (MaTek Corporations) was coated with 25 µg/mL fibronectin diluted in PBS and incubated at 37° for at least 30 min. Dishes were washed with PBS before use. 200,000 U2OS RFP-LifeAct cells were seeded and allowed to grow until the dish surface was fully covered. Laser ablation was performed and analyzed as described before.

Western Blots.

U2OS RFP-LifeAct pSLIK CA-ROCK cells were cultured in 150 ng/mL doxycycline for 2 days before lysis. U2OS RFP LifeAct cells were incubated in 5 μM Y-27632 for 1 and 3 h before lysis. As described previously, cells were lysed in RIPA buffer with phosphatase and protease inhibitors (EMD Millipore) (27). Protein content was measured by BCA and used to normalize samples before loading. Lysates were boiled and run on 4–12% Bis-Tris gels and transferred onto a PVDF membrane. The following primary antibodies were used: anti-phosphorylated myosin light chain 2 (Thr18/Ser19) (Cell Signaling Technology) and anti-GAPDH (Sigma-Aldrich). HRP-conjugated secondary antibodies (Life Technologies) and ECL reagent (Thermo Fisher Scientific) were used for detection. Bands were quantified and normalized to GAPDH with the built-in gel analyzer tool in ImageJ.

Traction Force Microscopy.

Polyacrylamide gel micropatterning was performed as described previously (25). Briefly, gels were synthesized with 7.7% acrylamide, 0.29% bis-acrylamide, 1% ammonium persulfate, 0.1% TEMED, and 0.5% far-red fluorescent beads [0.2-μm diameter (Invitrogen)] and left to polymerize for 30 min on a chrome quartz photomask, treated with hexane to prevent gel sticking. After polymerization, the gel was exposed to deep UV (UVO cleaner; Jelight) for 80 s. The gels were then incubated with 10 mg/mL EDC (Thermo Fisher Scientific) and 17 mg/mL NHS (Thermo Fisher Scientific) for 15 min. After EDC/NHS incubation, gels were incubated with 34.25 μg/mL of fibronectin (EMD Millipore Corporation) in 10 mM Hepes, pH = 8.5, for 1 h at 37°. Gels were washed 3× for 5 min in PBS and 2× for 5 min in cell media. Cells were seeded on the patterns and left to spread for 6 h before imaging. We computed maps of cellular traction stresses from bead positions before and after cell detachment using constrained Fourier transform traction cytometry by manually drawing the cell area (3, 9).

Immunofluorescence Staining.

Briefly, patterned cells were fixed using 4% (vol/vol) paraformaldehyde for 10 min and washed using PBS. Cells were permeabilized in 0.5% Triton-X for 15 min and blocked in 5% (vol/vol) goat serum. We then incubated the cells with the primary antibody for 1 h at room temperature, followed by extensive washing in 1% goat serum and secondary incubation for 1 h at room temperature. We used the following antibodies: anti-fibronectin clone HFN 7.1 (Thermo Scientific), anti-vinculin (Sigma-Aldrich), and Alexa-Fluor 647 mouse. F-actin was stained with 546-phalloidin. Immunostaining images were obtained by one-photon confocal imaging. For presentation purposes, the contrast and brightness of fluorescence images were optimized using Image J.