Unexpected softening of a fibrous matrix by contracting inclusions

Abstract

Cells respond to the stiffness of their surrounding environment, but quantifying the stiffness of a fibrous matrix at the scale of a cell is complicated, due to the effects of nonlinearity and complex force transmission pathways resulting from randomness in fiber density and connections. While it is known that forces produced by individual contractile cells can stiffen the matrix, it remains unclear how simultaneous contraction of multiple cells in a fibrous matrix alters the stiffness at the scale of a cell. Here, we used computational modeling and experiments to quantify the stiffness of a random fibrous matrix embedded with multiple contracting inclusions, which mimicked the contractile forces of a cell. The results showed that when the matrix was free to contract as a result of the forces produced by the inclusions, the matrix softened rather than stiffened, which was surprising given that the contracting inclusions applied tensile forces to the matrix. Using the computational model, we identified that the underlying cause of the softening was that the majority of the fibers were under a local state of axial compression, causing buckling. We verified that this buckling-induced matrix softening was sufficient for cells to sense and respond by altering their morphology and force generation. Our findings reveal that the localized forces induced by cells do not always stiffen the matrix; rather, softening can occur in instances wherein the matrix can contract in response to the cell-generated forces. This study opens up new possibilities to investigate whether cell-induced softening contributes to maintenance of homeostatic conditions or progression of disease.

Graphical Abstract

1. Introduction

Many tissues are composed of cells encapsulated in a three-dimensional fibrous extracellular matrix [1]. Cells remodel the matrix, either biochemically by synthesis and degradation of the matrix, or mechanically by virtue of their contractility [2–5]. The cell-matrix interaction is bidirectional [6], meaning that cellular contractility is tuned by the stiffness of the matrix surrounding the cell [7–10], while cell forces can also modify matrix stiffness, for example, by mechanically stiffening the matrix at the scale of the bulk material [11, 12]. Cells sample the matrix not at the bulk scale, but rather at scales of tens of microns, which renders knowledge of the bulk matrix stiffness insufficient to predict cell sensing at the microscale. Several studies deciphered that individual contracting cells can mechanically stiffen the matrix at these small length scales [13–15], but tissues rarely contain solitary cells; rather, it is common for multiple contracting cells to be interspersed within the same matrix. Due to the effects of nonlinearity and complex force transmission pathways within the matrix, it remains unclear how collective contraction by multiple cells affects the mechanics and whether those collective effects are sufficiently large for cells to sense and respond.

We focus here on the question of how multiple contracting cells within a random fibrous matrix alter the stiffness. Despite the common understanding that an individual contractile cell stiffens the matrix, a few recent reports studying multi-cellular matrices, wherein cells were either seeded on decellularized matrices [16] or embedded inside collagen matrices [17, 18], described a contradicting observation, that contracting cells soften the matrix at the scale of tens of microns. This softening is often thought to be a consequence of matrix degradation due to cell-secreted metalloproteinases [17, 18]. The role of mechanical cell-matrix interactions, if any, in instrumenting such softening is not yet clear. In this light, prior studies on multi-cellular interactions in fibrous matrices observed the existence of complex mechanical coupling between contracting cells [19, 20], which results in cell-mediated global contraction of the matrix [2, 21], however, the current literature lacks a clear quantification of the changes in fiber structure and mechanical stiffness resulting from the contraction of multiple cells.

Here, we combined experiments and simulations to study how multiple contracting cells mechanically alter the stiffness of the matrix. Focusing solely on the mechanical interactions between the cells and the matrix is challenging given the potential of cells to biochemically remodel the matrix. To mitigate this challenge, we studied an analogous system wherein cells were replaced with contracting inclusions made of poly(N-isopropylacrylamide) (PNIPAAm), which could not degrade or synthesize the matrix as cells do. In parallel, we used mechanics-based computational models of inclusion-matrix systems, wherein the inclusions contracted, and we studied the resulting matrix fiber deformations and mapped them to the matrix stiffness. We validated the model results by performing microscopic indentation experiments on PNIPAAm-embedded collagen matrices. Finally, we verified that the changes in matrix stiffness caused by local contraction of PNIPAAm microspheres could be sensed by cells by measuring cellular morphology and cell-induced matrix displacements.

2. Methods

2.1. Preparation of Collagen Matrix with Embedded Contracting Microspheres

Thermosensitive microspheres of poly(N-isopropylacrylamide) (PNIPAAm) with average diameter ≈ 65 m were prepared as described in our prior work [22]. These microspheres were treated to covalently bind to the collagen fibers using sulfo-SANPAH as previously described [22]. Treated microspheres were mixed with neutralized rat tail collagen I (Corning, Inc.) to make matrices with random fibrous structure having a final collagen concentration of 3 mg/mL, as previously described [22]. As these collagen matrices were heated from 22°C to 39°C, the embedded PNIPAAm microspheres contracted by ≈ 40% [23], thereby applying microscale displacements in the matrices mimicking the displacements induced by contracting cells.

2.2. Rheometry of the Bulk Matrix

A commercial rheometer (Kinexus Ultra+, Malvern Pananlytical) was used to quantify the global shear modulus of the PNIPAAm-embedded collagen matrices. To grip the collagen matrices to the rheometer, two functionalized glass coverslips (treated with 0.5% (3-aminopropyl)triethoxysilane and 0.5% gluteraldehyde) were adhered to the flat plate geometries of the rheometer by double sided acrylic adhesives (3M Inc.). Cylindrical collagen matrices with a radius of 9 mm, and a height of 1.77 mm were polymerized between these two coverslips on the rheometer at 22°C for 90 min. A schematic of the test setup is depicted in Fig. 1a. Small shear strains were induced by twisting the matrix about its axis. The maximum shear strain applied was 0.36% to remain in the linear regime of deformation. To ensure that the loading was quasi-static, we kept the maximum strain rate below 0.02% /s. The angular acceleration was below 1.17 × 10⁻⁵ rad/s², meaning that inertial loads were negligible. To measure the shear modulus (G₀), we fitted a line to the data of torque versus angle and applied the standard equation for torsion. For each matrix, the shear modulus was first measured at 22°C, and then the temperature of the matrix was increased to 39°C (chosen to be substantially above the phase transition temperature of PNIPAAm, to induce maximal contraction). We waited for 1 hr for the temperature of the matrix to equilibrate, and the shear modulus at 39°C was measured.

Figure 1:

Experiments and simulations characterizing the global modulus of fibrous matrix with contracting inclusions. (a) Diagram of experiments with the rheometer. A collagen matrix with embedded PNIPAAm microspheres was polymerized between the two coverslips attached to the parallel plates of the rheometer, and the top plate was axially rotated to twist the sample. (b) Small strain shear modulus (G₀) of nine independent PNIPAAm-embedded matrices measured before and after the contraction of embedded PNIPAAm microspheres, respectively at temperatures 22°C and 39°C. The increase in the shear modulus from 22°C to 39°C is statistically significant (p = 0.0039, sign test). (c) Small strain shear modulus (G₀) of eight independent control collagen matrices at temperatures 22°C and 39°C. The temperature change had no significant effect on the modulus (p = 1, sign test). (d) Simulation on an equivalent 2D matrix model with multiple inclusions and simple shear boundary conditions. Prior to the contraction of these inclusions, the shear modulus at 0.5% shear strain was quantified and defined as G₁. (e) The deformed model after the contraction of inclusions by 40%, with the shear modulus at 0.5% shear strain called G₂. (f) Ratios G₂/G₁ for six independent models were always greater than unity, indicating global stiffening of the matrix due to contracted inclusions.

2.3. Experiments Quantifying the Mechanics of Collagen Matrices at Scales of a Cell

2.3.1. Matrix Preparation

Treated PNIPAAm microspheres were mixed with neutralized collagen labeled with Alexa Fluor 488, as described in our prior work [22]. The collagen concentration was 3 mg/mL, and collagen matrices were polymerized into disks of 20 mm diameter and 1.6 mm thickness in glass-bottom dishes at 26°C for 45 min, resulting in a random network of fibers (Supplemental Fig. S10). To ensure that the collagen fibers did not adhere to the glass, the surfaces of the glass bottom dishes were pre-treated with a solution of 4% pluronic acid (Sigma-Aldrich) for at least 24 hr. The PNIPAAm microspheres were observed to sink to the bottom of the collagen matrices during polymerization. Upon heating to 37°C, the collagen matrices contracted globally. The concentration of PNIPAAm microspheres in the collagen matrices produced a global matrix contraction of ≈ 27% at 37°C (Supplemental Fig. S10b). Additional collagen matrices were made without the PNIPAAm microspheres; these matrices were unaffected by temperature changes and served as controls.

2.3.2. Microscopic Indentation

We measured the stiffness of the top surfaces of the collagen matrices using microscopic indentation experiments, as elaborated in the following three steps.

First, glass microspheres with nominal diameter 138 m and density 2.5 g/cm³ (Cospheric) were used as spherical indenters. These micro-indenters were mixed with 1× phosphate buffered saline (PBS) and sparsely spread on the top surfaces of the collagen matrices just before imaging. Since the actual diameters of these indenters were not constant and varied between 120 m and 160 m, the diameter of each indenter was measured by manual image analysis.

Second, the matrices were fully submerged in 1× PBS, and image stacks of their top surfaces were collected on a Nikon upright FN1 confocal microscope. These stacks contained images of Alexa-488 labelled collagen fibers which were excited by the 488 nm laser, with emission collected by high sensitivity GaAsP detectors and recorded in Nikon NIS-Elements software. A 25× water immersion objective with numerical aperture 1.1 (Nikon) was used, and the lens of the objective remained submerged inside the PBS and above the surface of the matrix as depicted schematically in Supplemental Fig. S11a. Image stacks were collected with a step size of 1 μm. Images were scanned at 1024 × 1024 pixels with a pixel size of 0.499 × 0.499 μm².

Third, we analyzed the images to determine the actual diameter of each indenter and the corresponding indentation depth. Briefly, the z cross-section of the image stack through an indenter was imported into a vector graphics editor (Adobe Illustrator) and a circle was manually fitted to match the profile of the indented regime. The diameter of this circle was the actual diameter of the spherical indenter, and the indentation depth (δ) was determined by measuring the depth of penetration in a direction perpendicular from the surface of the matrix. Knowing the diameter of the indenter and considering its buoyancy in PBS, we calculated the component of the effective weight of the indenter along the direction of the indentation depth. The ratio of the effective weight of the indenter and the indentation depth was computed to determine the secant stiffness of the matrix at the location of each indenter.

2.3.3. Experiments with Cells

MDA-MB-231 cells and MDA-MB-231 cells expressing Lifeact-GFP were cultured in high glucose Dulbecco’s Modified Eagle’s Medium (DMEM) with 10% fetal bovine serum at 37°C and 5% CO₂.Medium from cell culture was collected, centrifuged, and the supernatant was used as conditioned medium in later experiments.

Both control and PNIPAAm-embedded collagen matrices were floated in 1× PBS at 37°C for 45 min, during which time the PNIPAAm-embedded matrices radially contracted. After warming the matrices, the PBS was removed and the matrices were washed with DMEM, which was pre-warmed at 37°C. Then the medium was removed and ≈ 20,000 MDA-MB-231 cells were seeded near the center of the matrices. The cells were allowed to adhere to the collagen matrix for 45 min at 37°C before adding DMEM supplemented with conditioned medium at a 1:5 ratio. The cells were allowed to spread for 18 hr. For assessing morphological features, we used matrices with unlabelled collagen fibers; these matrices were fixed with paraformaldehyde, and the cells were stained with Alexa-488 phalloidin (Invitrogen, R37110). For assessing the cell-induced displacements, we used Alexa-594 labelled collagen and cells expressing Lifeact-GFP.

The cells and the matrix fibers were imaged on a Bruker Ultima Multiphoton microscope equipped with a Coherent Charmeleon Ti-Sapphire laser and Hamamatsu multi-alkali photomultipliers. The matrices were excited at 890 nm and a 20× air objective with numerical aperture 0.45 (Nikon) was used for imaging. A 445/40 nm filter was used to capture the second harmonic generation (SHG), while a 525/70 nm and 565 nm long-pass filters were used to capture the green and red fluorescence, respectively. For assessing cellular morphology, stacks of images were collected with a step size of 5 m and the images were scanned at 1024 × 1024 pixels with a pixel size of 0.347 × 0.347 μm². For studying cell-mediated matrix deformation, time lapse image-stacks were collected over a period of 48 min and the stacks had a step size of 2.5 m; these images were scanned at 1024 × 1024 pixels with a pixel size of 0.293 × 0.293 μm². During the time lapse acquisitions, the matrices with live cells were placed in a commercial stage top incubator (Okolab), which was maintained at 37°C and 5% CO₂.

For evaluating cell morphology, images were analyzed in ImageJ to identify the boundaries of each cell. For each cell, we quantified shape by measuring the length of its longest axis (feret’s diameter) and the ratio of the square of the cell perimeter to the cell area. For evaluating cell-induced matrix displacements, displacement fields were computed with Fast Iterative Digital Image Correlation [24] using a subset size of 64 × 64 pixels (18.75 × 18.75 μm²) and a subset spacing of 16 pixels (4.69 μm), to correlate pairs of images separated by 48 min, similar to our prior work [22, 25–28]. Considering each cell to act like a dipole, we identified the center and the alignment of this analogous dipole by inspecting the displacement fields in the matrix. To identify the regions in the matrix most affected by cellular forces, we used the continuum mechanics solution for the displacement field due to a force dipole, which defined a dumbell-shaped region of large cell-induced displacements. Finally, for each cell, the 80th percentile of the magnitude of the matrix displacement inside the dumbell-shaped region was computed as a measure of displacement induced by that cell.

2.4. Computational Modeling

2.4.1. Finite Element Model of Inclusion-Matrix Systems

To model fibrous matrices with fully embedded inclusions, we constructed two-dimensional (2D) networks of randomly oriented fibers; see Supplemental Note 1 for justification of the use of 2D models. The fiber networks were designed to match the architecture of type I collagen using a modified version of a previously developed algorithm [29–31]. Firstly, this algorithm randomly deposits a set of nodes in a 2D domain and connects them with fibers. Then it uses a simulated annealing-based optimization technique to move the nodes and swap the fibers connecting at different nodes until a desired average fiber length (L_f) and connectivity (defined as the number of fibers connecting each node) is achieved. Noting that collagen matrices have nodal connectivity less than Maxwell’s isostatic threshold (which is twice the system dimension) [29, 32], we developed sub-isostatic 2D matrices with a connectivity of 3.4, which matches that of collagen [29, 30] and is below the 2D isostatic threshold of 4. Our matrix models had regions where fibers crossed but did not connect, similar to observations from images of collagen matrices [26, 33]. The matrix generation algorithm is available in a public repository at https://github.com/jknotbohm/fiber_network_model.

The fibers in the matrix were modeled as linear elastic Timoshenko beams whose shearing stiffness was half of the axial stiffness. The fibers were soft in bending and had the ratio of bending stiffness to axial stiffness, k = 1 × 10⁻⁴, which is a value typically reported for type I collagen fibers [34–37]. In the finite element software, we used two three-node quadratic beam elements for each fiber, and the fibers were “welded” at the connections to transmit both forces and moments. Circular inclusions were introduced inside the domain of the matrix, and their diameters matched the relative sizes of PNIPAAm microspheres in experimental matrices (diameters ≥ 2L_f). To add the inclusions to the fiber network, fibers with end nodes inside the region of an inclusion were trimmed at their intersection with the periphery of the inclusion. Fibers crossing the inclusions were kept. The circular inclusions were filled with three-node continuum triangle elements which underwent isotropic thermal contraction. Additional cross-links connected the matrix fibers to the inclusions to obtain a fully connected system. These cross-links were two-node linear beams which transferred forces and moments between the matrix and the inclusions. These cross-links and inclusions had the same Young’s modulus as that of the matrix fibers. The size of the matrix models was always ≥ 30L_f, and the models contained two or more embedded inclusions.

2.4.2. Boundary Conditions and Finite Element Simulations

All simulations were performed with multiple load steps, wherein one load step was used to induce contraction of the inclusions and separate load steps were used to measure the matrix stiffness. When inducing contraction of the inclusions, both free and fixed external boundaries were used, as described in the Results section. To characterize how the contracting inclusions influenced the global shear modulus of the matrices, two load steps were used, the first being contraction of the inclusions, and the second being a simple shear boundary condition. To measure the stiffness of the matrix at length scales that would be sensed by cells, stiffness was measured by contracting dipoles of different orientations introduced in the matrix at various locations as described in the Results. The dipoles were two-node truss elements of length 2L_f, and their end points were connected to nodes in the matrix. Each dipole underwent thermal contraction such that the dipole was under a state of local equilibrium. The Young’s modulus of the dipoles was six orders of magnitude larger than that of the matrix fibers to ensure that the thermal strain of the dipoles was the same as the actual strain. The internal force in the dipole at 10% axial contraction was extracted. The ratio of this internal force and the dipole displacement gave the secant stiffness of the matrix microenvironment sensed by the dipole. All finite element simulations were performed on Abaqus (Dassault Systèmes). Given that large local deformations were produced inside the matrix, we used the implicit dynamic quasistatic solver with the option of nonlinear geometry, as in our prior work [27, 38, 39]. During post-processing, we characterized the local matrix mechanics at the level of the fibers by identifying force chains and quantifying fiber excess lengths (difference between the fiber’s contour length and the node-to-node distance), similar to our prior work [39].

2.5. Statistical analysis

Statistical analysis was performed in MATLAB R2020a. Statistical tests used were the sign test, Wilcoxon rank sum test, and Kruskal-Wallis test for, respectively, one, two, more than two groups of data, as indicated in the figure captions. To define the level of significance of the correlation between two different parameters, a Pearson’s test was used. Values of p < 0.05 were considered statistically significant. For binary events, we used a bootstrapping approach (with 10,000 iterations) to obtain the 95% confidence interval.

3. Results

3.1. PNIPAAm–Matrix System Behaves Analogously to the Cell–Matrix System

To begin our study on the effects of multiple contracting cells on the mechanics of fibrous matrices, we considered the global shear modulus, which has been commonly studied. For example, the presence of uniaxial global pretension increases the global shear modulus of the fibrous extracellular matrix [35, 37, 40, 41]. Similarly, contracting cells or contracting inclusions induce local pretensions inside the matrix resulting in an increase in the global shear modulus of the matrix [11, 12, 42]. Here, we began by verifying these findings using contracting PNIPAAm microspheres, which produce well-controlled forces and do not degrade or synthesize matrix as cells do. We cast collagen matrices embedded with PNIPAAm microspheres and, using a shear rheometer, measured their shear moduli (G₀) at the global scale at low strains before and after contraction of the embedded microspheres (Fig. 1a), expecting to observe stiffening caused by contraction, similar to a recent study [42]. For nine independent PNIPAAm–matrix systems, the average value of G₀ was 65.7 Pa when the microspheres were in the uncontracted state (at 22°C, Fig. 1b). When the system temperature was raised from 22°C to 39°C to induce contraction of the embedded microspheres, the modulus G₀ increased to an average value of 206 Pa (Fig. 1b), which was 3.1 times the global shear modulus at 22°C. To verify that the contracting microspheres alone altered modulus, and that the change of temperature had no mechanical effect on the collagen matrix, we conducted control experiments on collagen matrices in the absence of PNIPAAm microspheres. Results from eight independent control collagen matrices showed no clear effect of the temperature change on the global shear modulus of the control matrices (Fig. 1c), similar to our prior studies [22, 28]. The factor of 3.1 stiffening caused by contraction of the PNIPAAm microspheres is similar to the typical extent of the macroscale stiffening exhibited by cell–matrix systems due to cellular contractions [11, 12].

Given that the contracting inclusions only interact mechanically with the surrounding collagen matrix, the experimentally observed macroscale stiffening of the system is a mechanical phenomenon, which we further validated by simulating these boundary conditions with mechanics-based models of the fibrous matrices (Fig. 1d). These models contained a cluster of inclusions embedded in a fiber matrix and, the inclusions were radially contracted by 40%. Next, we imitated the boundary condition of the shear rheometer (Fig. 1d,e) and computed the small-strain global shear modulus of the models before and after the contraction of inclusions, G₁ and G₂, respectively. For six independent inclusion-matrix models, contracted inclusions increased the global shear modulus by an average factor of G₂/G₁ = 3.1 (Fig. 1f), similar to the bulk stiffening exhibited by PNIPAAm-matrix systems.

3.2. Simulations on Fiber Matrix with Contracting Inclusions

We observed in the previous section that the contracting microspheres stiffened the matrix under conditions of simple shear. These contracting microspheres mimicked cell contraction in cell-matrix systems studied in prior work, wherein cell contraction induced similar stiffening of the matrix under the same conditions of simple shear [11, 12]. Similarly, single contracting cells can increase the matrix stiffness at scales of tens of microns [13–15]. However, for the case of multiple cells contracting in a random fibrous matrix, it is not clear that the changes in matrix stiffness could be sensed by a cell, because at this scale, the matrix exhibits spatial heterogeneity in which the stiffness can vary by more than an order of magnitude [23, 28, 43–46]. Thus, the influence of multiple contracting cells on the matrix stiffness at the length scale of a cell remains an open question.

To determine how multiple contracting cells could alter the matrix stiffness at scales sensed by a cell, we began with computational modeling of the contracting inclusions and fibrous matrix. The models consisted of multiple inclusions, each of diameter D = 2L_f, randomly embedded inside the fibrous matrix at an average inter-inclusion spacing ≈ 1.5L_f (Fig. 2a). Without constraining the outer boundaries of the matrix, the embedded inclusions were radially contracted by 40%, causing them to locally pull on the surrounding matrix fibers, mimicking forces produced by cells. In response to these tensile forces, fibers oriented perpendicular to the periphery of each inclusion were axially stretched, constituting force chains (Fig. 2b).These tensile force chains are expected to be associated with stiffening of the matrix [39, 47, 48]. To determine how the contracting inclusions altered the matrix stiffness, we measured the stiffness at different locations and in different directions, using contracting dipoles, each of which mimicked the contraction of a cell, as shown in Figs. 2c,d. The dipoles contracted and sensed the stiffness, similar to the way a cell senses its microenvironment. Each dipole axially contracted by 10%, and the ratio of internal force and the contracted displacement was computed to quantify the secant stiffness. Surprisingly, the values of stiffness measured after the contraction of the inclusions, k_s, were always less than those measured before the contraction of the inclusions, k_s,ref, meaning that the contracting inclusions mechanically changed the microenvironment of the dipole. This mechanical change, as indicated by the values of k_s/k_s,ref < 1 indicated matrix softening (Fig. 2e). Dipoles aligned in various different directions across six independent fiber network models all showed softening, indicating the softening was consistent and occurred in all directions (Fig. 2e). To investigate why the contracting inclusions caused the matrix to soften, we investigated the deformed microstructure of the matrix at the level of fibers. Despite the fact that the contracting inclusions induced tensile force chains (Fig. 2b), the majority of the matrix fibers (≈ 60%) were not stretched, but rather in a state of axial compression, potentially causing buckling. To quantify buckling of the fibers, we computed and mapped each fiber’s excess length, defined as the difference between the contour length of the fiber and its node-to-node distance [49] (Fig. 2c,d). Fibers circumferential to the periphery of the inclusions had large excess lengths, indicating buckling.

Figure 2:

Effect of a group of contracting inclusions on stiffness at the scale of a cell. (a) A representative undeformed state of the matrix containing a cluster of inclusions. The inclusions had a diameter of D = 2L_f and were randomly embedded inside the matrix at an average inter-inclusion spacing of ≈ 1.5L_f. (b) After inclusions contracted by 40%, the network globally contracted (uncontracted matrix size shown by the dashed box). Force transmission is shown by force chains constituting the top 20% highly stretched fibers in the matrix (red). These force chains emanated radially from the periphery of the inclusions. (c, d) The deformed matrix (of panel b) with a contracting dipole in the inter-inclusion space as depicted by the black line. Panels c and d depict two different representative orientations of the dipole. Insets depict the enlarged views of the respective dipole. The fibers of the deformed matrix are color coded based on the normalized values of their excess lengths (ξ/L_f). The dipoles contracted and registered the stiffness of the matrix inside the inter-inclusion space. (e) For differently aligned dipoles and for six independent matrices, the ratio of the values of matrix stiffness sensed by the dipole after and before the contraction of inclusions (k_s/k_s,ref) were recorded. k_s/k_s,ref < 1 indicates matrix softening due to contracting inclusions. Each marker indicates a different dipole; each marker shape indicates a different independent fiber network.

To establish if the fiber excess lengths were the cause of the softening, we focused in detail on the excess lengths of the fibers near to each contracting dipole. Considering that the fibers closest to each dipole’s end points have the largest effect on the mechanics sensed by that dipole, we defined circular regions of radius 0.25L_f surrounding the end points of each dipole; all fibers passing through these circular regions defined the microenvironment of that dipole (e.g., Fig. 3a, I and Fig. 3b, I). In the microenvironment of the dipole, the initially undeformed fibers (Fig. 3a, I and Fig. 3b, I) buckled after the contraction of the inclusions, and accumulated excess lengths (Fig. 3a, II and Fig. 3b, II). These observations remained consistent across multiple independent inclusion-matrix models, and an additional representative example is depicted in Supplemental Fig. S1. Measurements from several independent dipoles in six different models revealed a statistically significant correlation between the extent of matrix softening and the average excess length of the fibers in the microenvironment of the dipoles (p = 0.0076, Pearson) (Fig. 3c). Therefore, the multiple contracting inclusions buckled the majority of the matrix fibers, thereby triggering softening in the matrix in all directions.

Figure 3:

Simulations on inclusion-matrix models relate fiber buckling to matrix softening. (a, b) Enlarged views of the dipoles of Fig. 2c and d, respectively, along with nearby matrix fibers before (I) and after (II) the contraction of the inclusions. Fibers are color coded by normalized excess length ξ/L_f. After contraction of the inclusions, the fibers are highly buckled, as indicated by the large excess lengths. The fibers constituting the microenvironment of the dipole were defined using circles of radius r = 0.25L_f at the end points of the dipole. (c) Scatter plot of normalized stiffness (k_s/k_s,ref) sensed by each dipole against average excess length (ξ/L_f) in the vicinity of that dipole. The values of k_s/k_s,ref were always less than one, indicating that the matrix softened at the scale of the dipoles. The correlation between the reduction in matrix stiffness and excess lengths was statistically significant (p = 0.0076, Pearson). Each marker type indicates a dipole from an independent model.

To consider other loading conditions, we repeated the simulations after replacing the contracting circular inclusions with contracting dipoles, which might more closely resemble contracting cells. As above, the dipoles contracted in the first load step with free boundaries, and another dipole was used to measure the stiffness before and after contraction. In these simulations, the trends were the same, namely that the matrix softened in all directions and the degree of softening was correlated to the amount of fiber buckling present (Supplemental Fig. S2). Next, we designed a simulation to measure stiffness at a different length scale, namely at the scale of the global material. To this end, we embedded inclusions in the fibrous matrix, and, after inducing contraction of the inclusions under free boundary conditions, we applied simple shear, observing global softening caused by the contracting inclusions (Supplemental Fig. S3). Lastly, to verify that fiber buckling is essential for the softening, we repeated the simulations with the contracting circular inclusions in matrices having fixed boundaries. Compared to free boundaries, the fixed boundaries had greatly reduced excess lengths, indicating the fixed boundaries suppressed fiber buckling (Supplemental Fig. S4a–h). In turn, the matrix softening was largely eliminated, with some dipoles sensing a stiffer local environment caused by the contracting inclusions and some sensing approximately the same stiffness as before contraction of the inclusions (Supplemental Fig. S4i). Together, these findings indicate that when the boundary conditions allow the matrix to contract, the localized forces produced by contracting inclusions induce fiber buckling that leads to matrix softening.

3.3. Simulations on Fiber Matrix with Two Contracting Inclusions

We noted in the prior section that multiple contracting inclusions softened the matrix. To better understand the mechanics and the role of underlying system length scales that tuned this unexpected softening, we studied simplified inclusion-matrix models with only two embedded inclusions. To this end, two inclusions, each of diameter D = 2L_f and separated by a variable inter-inclusion spacing d, were embedded inside matrices with free boundaries (Fig. 4a). Prior models showed substantial fiber alignment and densification between the pairs of contracting inclusions in fibrous materials [40, 50–53], but such alignment and densification was not present here, which is due to the fact that the inclusions were simulated by elastic elements that are able to move closer together upon contraction, which is required to maintain force equilibrium (see Supplemental Fig. S5 and Supplemental Note 2 for more details).

Figure 4:

Simulations on matrix models containing two contracting inclusions. (a) A representative model containing two inclusions of diameter D = 2L_f, separated by inter-inclusion spacing d. The central dipole of length 2L_f is transverse to the imaginary line joining the centers of the inclusions. The inclusions were radially contracted by 40%. (b) Enlarged view of the central dipole which samples fibers passing through the circular regions of radius 0.25L_f centered at the dipole’s end points. (c) For four independent matrices, the ratio of the values of matrix stiffness sensed by the dipole after and before the contraction of inclusions (k_s/k_s,ref) were recorded, with k_s/k_s,ref < 1 indicating softening. Each marker color indicates an independent model. (d) A representative deformed state of the matrix with d = 8.5D, where the force chains consisting of top 5% most stretched fibers are shown in red. (e) The deformed matrix of panel d, where the top 5% most buckled fibers are shown blue. (f) The dipole of the system described in panels d and e. The matrix fibers in its microenvironment are colored based on the values of their excess lengths. (g, h, i) A representative deformed state of the matrix with d = 4.5D, where the force chains, buckled fibers, and dipole are shown, similar to panels d–f. (j, k, l) A representative deformed state of the matrix with d = 0.5D, where the force chains, buckled fibers, and dipole are shown, similar to panels d–f. (m) Plot showing the dependence of the average excess length of fibers in the microenvironment of the dipole to the model parameter d/D. (n) The normalized matrix stiffness in the vicinity of the transverse dipole (k_s/k_s,ref < 1) was negatively correlated to the average excess length of fibers in the microenvironment of the dipole. (o) The normalized matrix stiffness (k_s/k_s,ref) against the model parameter d/D. Panels a–o considered the transversely aligned dipole of panel a. (p) The matrix stiffness in the direction longitudinal to the imaginary center-line of the inclusions (inset) was plotted against the model parameter d/D. More than half (18 out of 28 instances, 95% confidence interval [13, 23]) of the data points exhibited k_s/k_s,ref < 1, indicating softening. In panels m–p, each marker color indicates an independent model. In panels c & n–p, the black dashed line at k_s/k_s,ref = 1 indicates no matrix softening.

A dipole of length 2L_f was introduced in the matrix between the two inclusions (Fig. 4a), and the dipole was initially oriented perpendicular to the common center line of the inclusions. As in the previous section, we identified the microenvironment of the dipole by defining threshold circles of radius 0.25L_f at the end points of the dipole, and the excess lengths of all the fibers passing through these circular regions were quantified (Fig. 4b). The contracting dipoles were again used to measure the matrix stiffness. Similar to the results with the multi-inclusion systems, the values of stiffness after the contraction of the inclusions, k_s, were smaller than those measured prior to the contraction of the inclusions, k_s,ref for all inter-inclusion spacing d, in four independent models, again indicating softening (Fig. 4c). To investigate how the contracting inclusions softened the matrix, we again characterized force chains and excess lengths. Each contracting inclusion pulled the radially aligned fibers, and similar to Fig. 2b, we mapped the force chains, which were perpendicular to the periphery of the inclusions (Fig. 4d,g,j). Again similar to Fig. 2c,d, matrix fibers aligned parallel to the periphery of the contracting inclusions had large excess lengths (Fig. 4e,h,k). We verified that the excess lengths appropriately indicated fiber buckling by noting the positive correlation between excess length and normalized curvature of the fibers (Supplemental Fig. S6). We also tested the effect of magnitude of contraction on matrix softening, observing that both excess lengths and the extent of softening were correlated to the amount of radial contraction (Supplemental Fig. S7).

Next, we studied how the ratio of the two system length scales, the inter-inclusion spacing (d) and the inclusion diameter (D), tuned the geometric proximity of the dipole from the contracted inclusions to influence the matrix in the dipole microenvironment. To this end, the excess lengths of the fibers (ξ) in the vicinity of the dipole, as identified in Fig. 4b, were calculated and mapped (Fig. 4f). We conducted a parametric study by varying the inter-inclusion spacing (d) alone (e.g., Figs. 4g,h,j,k), and mapped the fibers in the dipole microenvironment based on their excess lengths (e.g., Figs. 4i,l), similar to Fig. 4f. From a qualitative evaluation of Figs. 4f,i,l, when inter-inclusion spacing d decreased, there was an increase in the number of highly buckled fibers in the dipole microenvironment. At each value of the parameter d/D, we quantified the average excess length of fibers in the microenvironment of the dipole (ξ/L_f). Results from four independent matrix models consistently revealed a negative correlation between ξ/L_f and d/D (Fig. 4m). A second parametric study considering variable inclusion diameter D (Supplemental Fig. S8a–k) exhibited similar dependence between ξ/L_f and d/D (Supplemental Fig. S8l).

Interestingly, as the average excess length of the fibers in the microenvironment of the dipole increased, so did the extent of softening in the matrix (Fig. 4n and Supplemental Fig. S8m), meaning that the presence of soft buckled fibers mechanically softened the matrix microenvironment. As noted in Fig. 4m and Supplemental Fig. S8l, the parameter d/D governed the extent of fiber buckling, and these soft buckled fibers modified the stiffness of the matrix in the microenvironment of the dipole (Fig. 4n and Supplemental Fig. S8m). Consequently, the softening of the inclusion-matrix systems could be tuned by changing the value of the system parameter d/D (Fig. 4o and Supplemental Fig. S8n). This observation remains valid as long as the inter-inclusion spacing and the inclusion diameter exceeds the fiber length. Therefore, the microenvironment over which a cell (or an analogous dipole) samples the matrix is independently tuned not by the fiber length, but by the ratio of the inter-inclusion spacing and the inclusion diameter.

The analysis thus far depicted softening in the direction transverse to the center-line of the inclusions, and notably the matrix fibers predominantly buckled in this direction as well (Figs. 4e,h,k and Supplemental Figs. S8d,g,j). Recalling our observations on matrices with many contracting inclusions that dipoles oriented in all directions sensed matrix softening (Fig. 2e), we wanted to investigate how the contracted inclusions altered the mechanics sensed by dipoles in other orientations, in particular in the orientation longitudinal to the center-line of the inclusions, where we would expect the fibers to be predominantly under a state of tension. In this case, one would expect the matrix to stiffen in the region between the inclusions, because tensile force chains exist along this axis (Fig. 4d,g,j). We modeled longitudinal dipoles between the inclusions and repeated the parametric study of Fig. 4o. Contrary to our expectation, results from four independent matrices frequently showed softening (Fig. 4p). Specifically, 18 out of 28 data points (95% confidence interval, [13,23] out of 28 data points) in Fig. 4p exhibited softening with k_s/k_s,ref < 1.

To investigate further, we considered that buckling-induced softening would likely result from negative (contractile) normal strain. Thus, we computed the normal strain along the dipole axis, ε_‖. We first analyzed the case of dipoles oriented transverse to the axis connecting the inclusions, for which the normal strain was negative, with both excess lengths and softening strongly correlated with –ε_‖ (Supplemental Fig. S9a,b). We also compared −ε_‖ against the distance between contracting inclusions, d/D, wherein there was a clear monotonic trend (Supplemental Fig. S9c). Next, we analyzed the surprising case of the dipole aligned along the axis connecting the contracting inclusions, for which the stiffness ratio k_s/k_s,ref was correlated to the strain along the dipole axis ε_‖ (Supplemental Fig. S9d). Importantly, for all cases of softening (k_s/k_s,ref < 1), ε_‖ was negative. A negative value of ε_‖ may seem unexpected but is consistent with the observation that the contracting inclusions are able to move closer together (Supplemental Fig. S5). Finally, we computed ε_‖ for the case of many contracting inclusions (Supplemental Fig. S9e–g), observing that ε_‖ was always negative, consistent with the fact that softening was always observed for the case of multiple contracting inclusions with free external boundaries (Figs. 2–3).

3.4. Experimental Assessment of the Mechanics of PNIPAAm-Embedded Matrices

Next, we designed an experiment to test the model prediction that contractile inclusions lead to softening of matrices having free boundaries. The need for free boundaries mean that experimental testing using a rheometer, as in Fig. 1, was not possible, because gripping the matrices with the rheometer would prevent the global contraction of the matrix. Therefore, we designed an experiment that would measure matrix stiffness in a matrix that was free to contract. To this end, we used an indentation experiment (similar to Ref. [54]). We cast collagen matrices embedded with PNIPAAm microspheres, used glass microspheres as indenters, and imaged them with a confocal microscope to quantify the indentation depths (δ), which were typically in the tens of microns. Before inducing contraction of the PNIPAAm microspheres (i.e., at 22°C), the microscale indentation depth δ was typically ≈ 45 μm, both in the absence and presence of PNIPAAm microspheres (e.g., Fig. 5a,b and Fig. 5c,d). Similarly, at 37°C, the indentation δ for control matrices without the PNIPAAm microspheres was near 45 μm, suggesting that the matrix stiffness was unaffected by the temperature change (Fig. 5e,f), similar to our earlier macroscale observation (Fig. 1c). When heated to 37°C, thereby inducing contraction of the PNIPAAm microspheres, global matrix shrinking occurred with in-plane stretch ratios of ≈ 0.73 and an associated increase in fiber density (Supplemental Fig. S10). The indentation depth δ was larger than before matrix shrinking, typically ≈ 76 μm (e.g., Fig. 5g–l), indicating the occurrence of a mechanically soft microenvironment below the indenters.

Figure 5:

Micro-indentation experiments. (a) A representative cross-section of a control collagen matrix without the PNIPAAm microspheres at room temperature (22°C), when indented by a spherical micro-indenter at its surface. (b) The fitted circle represents the indenter, and its measured diameter (148 μm) was used to compute its weight. The slight inclination of the surface of the matrix (dashed yellow line) with the horizontal (0.66°) was taken into account to calculate the component of the weight perpendicular to the matrix surface. The depth of indentation was the perpendicular indentation depth (δ = 46.2 μm) from the matrix surface. (c,d) A representative cross-section and fitted circle of a collagen matrix with the PNIPAAm microspheres at room temperature (22°C), when these PNIPAAm microspheres did not contract. (e,f) A representative cross-section and fitted circle of a control collagen matrix without the PNIPAAm microspheres at elevated temperature (37°C). (g–l) Representative cross-sections and fitted circles of collagen matrices with the PNIPAAm microspheres at elevated temperature (37°C), when the PNIPAAm microspheres contracted. (m) The values of secant stiffness (k_s) measured by multiple different indenters are plotted. The values of stiffness of the control matrices and PNIPAAm-matrices were plotted at 22°C and 37°C, and they were segregated in four groups. Notably, the matrices with PNIPAAm microspheres at the elevated temperature (37°C) exhibited significant decrease in stiffness in contrast to the other three groups (p = 9.7 × 10⁻⁹, Kruskal-Wallis test). Markers with an identical shape and color indicate independent indenters from one sample. The black lines represent mean values of the respective data set.

The indentation depths δ resulted from the weight of the indenter alone, as there was negligible adhesion between the indenters and the matrix, as verified from the obtuse contact angles between indenters and matrix (Supplemental Fig. S11b). We further noted that in a region below the indenters, the fibrous matrix was always densified (Supplemental Fig. S11c–e). Such densification was expected from the predictions of a recent numerical study [55], which identified that this densification results from nonlinearity in the fibrous matrix, and highlights that the force–displacement relationship for indentation of a fibrous matrix is strongly different than that for a linear elastic matrix. To quantify the stiffness of the matrices, we computed the ratio of indenter weight and indentation depth. The values of stiffness in control and PNIPAAm-embedded matrices at multiple independent locations were quantified. For matrices at 22°C with and without PNIPAAm microspheres, there was no significant difference in stiffness, indicating no effect of the PNIPAAm microspheres themselves. Additionally, control matrices without microspheres that were heated to 37°C had no significant difference in stiffness, indicating no effect of the temperature change on stiffness. Compared to these controls, matrices with contracted PNIPAAm microspheres exhibited a statistically smaller stiffness (k_s), by factor of > 2 (Fig. 5m). Therefore, even though the multiple contracting microspheres increased the local fiber density, the buckling enabled by contraction of these unrestrained matrices outweighed the local increase in density, in turn leading to mechanical softening, confirming our model predictions.

3.5. Cells Sense and Respond to the Mechanics of PNIPAAm-Embedded Matrices

Finally, we questioned whether the reduction in matrix stiffness caused by local contraction of PNIPAAm microspheres could be sensed by cells. To this end, we prepared additional collagen matrices and seeded MDA-MB-231 breast carcinoma cells onto them with the goal of observing cellular functions on both the contracted PNIPAAm-embedded matrices and on the matrices of pure collagen, which served as controls. Before seeding cells, PNIPAAm-embedded matrices exhibited global radial contraction by ≈ 27%, and as the micro-indentation experiments revealed, these contracted matrices were more compliant than control at scales of tens of microns. Matrix stiffness affects the morphology of breast cancer cells [56, 57], with softer matrices generally inducing the cells to become more rounded and less elongated. In this light, we assessed the cellular morphology in control and contracted matrices to determine whether the cells sensed the soft contracted matrices. The cells were more elongated in the stiff control matrices, whereas their morphology was more rounded in the soft contracted matrices (Fig. 6a,b and Supplemental Fig. S12). We quantified the morphology of each cell in the control and contracted matrices by calculating their feret’s diameter (the longest cell axis) and the ratio of the square of the cell perimeter P to the cell area A. In contrast to the cells on the control matrices, cells on the softer contracted matrices had significantly smaller feret’s diameters (Fig. 6c). Also, the cells on the contracted matrices were more circular with significantly smaller values of P²/A (Fig. 6d). As it is unlikely that the change in local fiber density due to microsphere contraction caused these effects on cell morphology (see Supplemental Note 3), these data suggest that the contraction-induced matrix softening was sufficient for the cells to sense it.

Figure 6:

Experiments depicting stiffness sensing and response by MDA-MB-231 cells. (a) Representative elongated morphology of a cell in a control matrix of collagen, with no embedded PNIPAAm microspheres at 37°C. (b) Representative rounded morphology of a cell in a PNIPAAm-embedded collagen matrix, when the embedded PNIPAAm microspheres contracted at 37°C. (c) For many independent cells in control and PNIPAAm-matrices, the length of the longest axis (feret’s diameter) was measured. In the contracted PNIPAAm-matrices, the cells had significantly smaller feret’s diameter (p = 8 × 10⁻²¹, rank sum). (d) For many independent cells in control and PNIPAAm-matrices, the ratio of the square of the cell perimeter (P) to the cell area (A) was computed. In the contracted PNIPAAm-matrices, P²/A was significantly smaller (p = 4 × 10⁻¹⁹, rank sum). (e) A representative image of a cell in a control matrix of collagen, with no embedded PNIPAAm microspheres at 37°C. (f) Cell-induced displacement (|U|) of the matrix shown in panel e. (g) A representative image of a cell in a PNIPAAm-embedded matrix of collagen at 37°C. (h) Cell-induced displacement (|U|) of the matrix shown in panel g. (i) The 80th percentile of the magnitude of displacement |U| inside the dumbell-shaped regions (specified regions in panels e–h) were plotted for cells on control and contracted matrices. PNIPAAm-embedded matrices had significantly less deformation (p = 0.0076, rank sum), meaning that cellular activities (cell generated forces) were drastically reduced due to the softening of the contracted PNIPAAm-matrices. In panels c, d, and i, dots represent independent cells; lines indicate medians.

With this evidence that the cells sensed the softening induced by the contracting PNIPAAm microspheres, we further investigated whether those cells responded by producing different displacement fields within the matrix, as the forces produced by the cells typically increase with an increase in matrix stiffness [7–10]. To this end, we collected time-lapse images of the single cells and the matrix (Fig. 6e and Fig. 6g) and applied image correlation to matrix images separated by 48 min to quantify the cell-induced matrix displacement fields. Considering that the PNIPAAm-embedded matrices were softer than the control matrices, displacements in PNIPAAm-embedded matrices of magnitude equal to or less than those in the stiffer control matrices would indicate smaller cell contractility in PNIPAAm-embedded matrices compared to control. Representative results showed that the cell-induced matrix displacements appeared far smaller in the PNIPAAm-embedded matrices compared to control (Fig. 6e–h). Considering that the cell sampled the fibrous matrix analogously to a contracting dipole, we identified the regions in the matrix most affected by cell forces by using the continuum mechanics solution for the displacement field produced by a dipole to define a dumbell-shaped region within which the displacements were expected to be largest (Fig. 6e–h). The 80th percentile of the magnitude of the matrix displacement within these regions was computed as a measure of the cellular response in the control and PNIPAAm-embedded matrices. Compared to control matrices, the cell-induced matrix displacement was nearly a factor of three smaller in the contracted PNIPAAm-embedded matrices (Fig. 6i). This observation, combined with the fact that the stiffness of PNIPAAm-embedded matrices is smaller than control by a factor > 2 indicates a drastic reduction in cell-generated forces in the PNIPAAm-embedded matrices, which further confirms cell sensing and response to the altered matrix stiffness.

4. Discussion

While the matrix near an individual contractile cell is known to stiffen, the effect of multiple contracting cells on the matrix stiffness at the scale of a cell has not been quantified. Here, using both modeling and experiments of matrices with random fiber organization, we mimicked cell contraction using contracting inclusions and quantified the effects on the stiffness of the matrix, both at the global scale and at the scale of a cell. Upon contraction of the embedded inclusions, matrices with free boundaries consistently softened, which was unexpected given that the contracting inclusions pull on the matrix fibers with tensile forces. We investigated the underlying mechanism behind the softening by studying the deformations undergone by the fibers, noting that the contracting inclusions also induced compressive strains, which led to buckling of the majority of the fibers within the matrix. By mapping the amount of fiber buckling to the extent of softening, we found that a greater extent of fiber buckling resulted in stronger softening. Lastly, we verified that this buckling-induced matrix softening was sufficient for cells to sense and respond with less elongated morphology and dramatically reduced force generation on the soft PNIPAAm-embedded matrices.

Prior work on the stiffness of collagen matrices typically used boundary conditions different than those in our study. Perhaps the most common are the boundary conditions of the commercial rheometer, which applies shear while simultaneously imposing additional constraints on the height and diameter of the matrices. Our rheometer experiments imposed these constraints with the additional condition of contracting inclusions, which produced local tensile forces within the matrix, thereby causing stiffening (Fig. 1). Our observation of stiffening induced by tensile forces is conceptually similar to observations of prior studies in which tensile forces were generated by uniaxial pretension [35, 37, 41] or via cell contraction [11, 12]. It is important to note that our experiments differed fundamentally from recent studies that introduced inclusions in fibrous matrices and observed stiffening under uniaxial global compression [58, 59], because our study used inclusions at a lower concentration that contracted rather than being inert. Our findings are also distinct from the rheometer experiments showing that uniaxial global compression of a fibrous matrix leads to softening in shear [35, 37]. Although our simulations and experiments showed softening as well, they differed from the prior rheometer experiments, because the prior experiments applied only compressive forces to the matrix when observing softening. In our study, the contracting inclusions produced both tension and compression. Although tension would be expected to lead to strain stiffening, compression-induced buckling outweighed effects of stiffening for matrices with free boundaries, as evidenced by the matrix softening in both our model and experiments.

In both our modeling and experiments, the softening was observed when the external boundaries were free, allowing the matrix to shrink as a result of the contracting inclusions. Similar to our simulations and experiments, numerous prior experiments observing contraction of cell-embedded matrices used matrices with free boundaries, in which cell contraction caused global matrix shrinking [2, 20, 21]. Our findings would suggest that in the common experiment quantifying contraction of cell-embedded matrices, the stiffness decreases during contraction. Beyond these experiments, we expect our findings to apply to numerous other scenarios wherein the matrix is able to contract. In considering other applications of our findings, it is important to emphasize that most systems have boundaries that are neither fully free nor fully fixed but rather somewhere in between. Following our findings that the softening results from buckling which in turn results from contractile (negative) strain, our findings suggest softening would occur whenever the embedded inclusions (or cells) cause the matrix to shrink.

Our study elucidated that the buckled fibers played the pivotal role in softening the contracted PNIPAAm-embedded matrices at the scale of a cell. Over the last decade, fiber buckling has been found to have important effects on other mechanical phenomena. For example, buckling induced by the contractile forces of a cell dominantly contribute to long range transmission of cell-induced displacements within the matrix [22, 31, 50, 60, 61]. Interestingly, in long range displacement propagation, effects of fiber buckling supersede those of strain stiffening, which was initially thought to be the key contributor [62] and later found to be insufficient [63]. Fiber buckling also mediates a severe form of nonlinearity in the matrix that induces instabilities at the scale of the fibers leading to strong local densification, akin to a phase transition [5, 28]. Direct evidence for fiber buckling has been obtained from images in matrices under global uniaxial compression [59, 64], global shear [65], and local contraction of a single inclusion [22]. Here, substantial buckling occurred in a system with many contracting inclusions, which is interesting, given that the multiple inclusions supported tensile force chains connecting them, whereas buckling is an instability that occurs under compression. Our results pointed to a new role of fiber buckling, wherein the multiple contracting inclusions buckled the majority of fibers in the matrix. We directly related fiber buckling to softening in the 2D model by quantifying excess lengths as an indicator for buckling and showing that they correlated strongly with the amount of softening. The model predictions were confirmed by experiments, though the extent of softening in the experiments was less than in the 2D model (see Supplemental Note 1 for further discussion). In summary, the effects of buckling of fibers under compression outweighed the effects of stiffening of fibers under tension, causing softening in all directions. In future work, it would be interesting to study the effects of buckling in more complicated systems, such as composites of collagen fibers and hyaluronic acid, wherein fiber buckling is reduced [28].

Prior studies showed that individual contractile cells can locally pull on the matrix fibers, stiffening the matrix at scales of tens of microns. Experiments have measured stiffening near contracting cells in fibrous matrices to be as large as a factor of 10 [13–15]. However, a few recent nanoindentation and microrheometry experiments observed softening of matrices containing multiple contracting cells [16–18, 66], which is an apparent contradiction. The common resolution for this contradiction was to assume that the softening was a consequence of matrix degradation due to cell-secreted metalloproteinases [17, 18]. It is important to identify other potential explanations for softening, however, because clinical trials using metalloproteinase inhibitors in the tumor microenvironment have so far been ineffective [67–69]. Our PNIPAAm-embedded matrices with contracting PNIPAAm microspheres mimicked the forces in these systems and exhibited similar matrix softening at cellular length scales with no chemical degradation present. Together, our experiments and simulations show that fiber buckling induced by the contractile forces of a cell is a potential mechanics-based explanation for the softening.

Our finding of contraction-induced softening raises new questions for investigation in future research. As an example, in wound healing, inflammatory response activates myofibroblast contraction and secretion of new matrix, but overly active myofibroblasts are undesirable as they create stiff fibrotic scars [70]. As myofibroblast activation is correlated to the matrix stiffness [71], it could be desirable for there to be matrix softening to suppress excessive myofibroblast activation, thereby minimizing excessive matrix production and contraction, and reducing the likelihood of scar formation. It would be interesting to determine whether the cell-induced softening found in our study plays a role in achieving homeostatic conditions following wounding. If cell-induced matrix softening has such an effect, a subsequent question is why such a mechanism is ineffective in other situations, such as the tumor microenvironment or progression of fibrotic disease, wherein it is commonly observed that the matrix stiffens rather than softens. These questions, which are enabled by our findings of contraction-induced softening, provide insightful new directions for investigation.

Statement of Significance.

Mechanical interactions between cells and the surrounding matrix strongly influence cellular functions. Cell-induced forces can alter matrix properties, and much prior literature in this area focused on the influence of individual contracting cells. Cells in tissues are rarely solitary; rather, they are interspersed with neighboring cells throughout the matrix. As a result, the mechanics are complicated, leaving it unclear how the multiple contracting cells affect matrix stiffness. Here, we show that multiple contracting inclusions within a fibrous matrix can cause softening that in turn affects cell sensing and response. Our findings provide new directions to determine impacts of cell-induced softening on maintenance of tissue or progression of disease.