Local response and emerging nonlinear elastic length scale in biopolymer matrices

Haiqian Yang, Estelle Berthier, Chenghai Li, Pierre Ronceray, Yu Long Han, Chase Broedersz, Shengqiang Cai, Ming Guo

► To cite this version:

Haiqian Yang, Estelle Berthier, Chenghai Li, Pierre Ronceray, Yu Long Han, et al. Local re-sponse and emerging nonlinear elastic length scale in biopolymer matrices. Proceedings of the National Academy of Sciences of the United States of America, 2023, 120 (23), pp.e2304666120. 10.1073/pnas.2304666120. hal-04211993

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# Local response and emerging nonlinear elastic length scale in biopolymer matrices

Haiqian Yang<sup>a,1</sup>, Estelle Berthier<sup>b,1</sup>, Chenghai Li<sup>f</sup>, Pierre Ronceray<sup>d,1</sup>, Yu Long Han<sup>a</sup>, Chase P. Broedersz<sup>b,e,1</sup>, Shengqiang Cai<sup>f,1</sup>, and Ming Guo<sup>a,1</sup>

Edited by Herbert Levine, Northeastern University, Boston, MA; received March 20, 2023; accepted April 16, 2023

Nonlinear stiffening is a ubiquitous property of major types of biopolymers that make up the extracellular matrices (ECM) including collagen, fibrin, and basement membrane. Within the ECM, many types of cells such as fibroblasts and cancer cells have a spindle-like shape that acts like two equal and opposite force monopoles, which anisotropically stretch their surroundings and locally stiffen the matrix. Here, we first use optical tweezers to study the nonlinear force-displacement response to localized monopole forces. We then propose an effective-probe scaling argument that a local point force application can induce a stiffened region in the matrix, which can be characterized by a nonlinear length scale $R^*$ that increases with the increasing force magnitude; the local nonlinear force-displacement response is a result of the nonlinear growth of this effective probe that linearly deforms an increasing portion of the surrounding matrix. Furthermore, we show that this emerging nonlinear length scale $R^*$ can be observed around living cells and can be perturbed by varying matrix concentration or inhibiting cell contractility.

nonlinear elasticity | cell-matrix interactions | microrheology | biopolymer networks

Nonlinear stiffening is ubiquitous in biopolymers (1, 2), including the ECM polymers such as collagen, fibrin, and basement membrane (3–9). In these ECM materials, cells such as fibroblasts and certain types of cancer cells are known to deform and stiffen the matrix (8, 10–15). The mechanical nonlinearity is a remarkable feature as it allows cells to spontaneously modify their mechanical environments simply by pulling on the matrices. While the mechanical properties of the ECM are known to be important regulators of cell functions (16, 17), the interplay between cells and nonlinear elasticity of the ECM is unclear.

Existing studies on the nonlinear elastic properties of biopolymers primarily focus on the bulk response (3, 5–7, 18–23). One important example is collagen, which is the most abundant constituent protein of the ECM (4, 24). It was found that collagenous tissues can be well characterized by an exponential force-extension relation (21, 22). Later on, it has been proven that the exponential stiffening behavior of collagen gels can be derived from the mechanics of fibrous networks (5). For collagen gels under shear stress $\sigma$ and shear strain $\gamma$, a consequence is that the differential stiffness $K$ ($K = \frac{d\sigma}{d\gamma}$) is directly proportional to stress $\sigma$ in the nonlinear regime for bulk measurements, $K \sim \sigma$ (5).

Cells embedded in the ECM, however, interact with the surrounding matrix locally instead of in bulk. For example, a fibroblast in the collagen matrix extends its filopodia and pulls on the fibers (Fig. 1 A–C). The loading condition from a cell is localized, and it generates a nonuniform deformation field in the matrix, which is distinct from a uniform bulk measurement (25, 26). It remains elusive how much mechanical resistance a cell feels when applying such localized forces: within highly nonlinear elastic matrices, while it is still expected that a cell will face more resistance as it pulls harder, it is not clear whether the resistant force grows exponentially with displacement such that the resistant stiffness $k$ grows linearly with the force $F$ as expected from bulk measurements. In local microrheology by optical and magnetic tweezers, where a particle of diameter $d$ is moved to a displacement $u$ by a force $F$, nominal stress $\sigma = \frac{4F}{\pi d^2}$ and nominal strain $\epsilon = \frac{u}{d}$ are usually calculated (27, 28), which implies that the local measurements are analogous to the bulk measurements (28). While the linear responses can be quite similar (29), our results indicate that local and bulk nonlinear power-law responses are not necessarily consistent. Mimicking this localized loading condition, here, we show experimental results by optical tweezers that the nonlinear response of collagen to a local force monopole is close to $k \sim F^{1/2}$. We explain this power-law relation by a minimal scaling argument of an effective probe: the observed local nonlinear response is similar to the case of using

## Significance

Cells reside in biopolymer matrices that stiffen under mechanical stresses. To understand the mechanics of this nonlinear environment, bulk rheology has been traditionally used; however, cells apply microscopic local forces on the matrix, and it is not clear whether the microscopic local response is the same as that observed in bulk. Here, we report that local monopole forces in a highly nonlinear matrix induce stiffness that grows universally as a square root of the applied force independent of matrix type. This nonlinear stiffening response to localized forces can be explained by a distinct mechanism, the growth of an effective probe. Our results highlight that local forces and bulk forces have different effects in the context of cell-matrix nonlinear mechanics.

Author contributions: H.Y. and M.G. designed research; H.Y. performed research; H.Y., E.B., C.L., P.R., Y.L.H., C.P.B., S.C., and M.G. contributed new reagents/analytic tools; H.Y. analyzed data; and H.Y. and M.G. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Copyright © 2023 the Author(s). Published by PNAS. This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

¹To whom correspondence may be addressed. Email: guom@mit.edu

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2304666120/-/DCSupplemental.

Published May 30, 2023.
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** The force applied by a single cell is highly localized, and local nonlinear stiffening of collagen gels deviates from the scaling law in bulk. (A) Image of a mouse embryonic fibroblast (green, cytoplasm) within 2 mg mL⁻¹ collagen networks (gray) stretching the matrix with filopodia. (B) Image of the same cell with one filopodium released. The dash circle schematically marks the highly deformed region of the network, which is released once the force is removed. (Scale bar, 10µm). (C) Schematics of a cell embedded in 3D fibrous ECM. The filopodium applies a point force to the matrix and creates a highly deformed zone. (D) Confocal reflective image of a particle (2 µm in diameter) embedded in a 4 mg mL⁻¹ collagen gel. A focused 1,064-nm laser is used to trap the particle and apply a point force F. (E and F) Force is a nonlinear function of displacement *u*, which is close to *F* ~ *u*², as measured by both (E) optical tweezers and (F) flat-punch indentation.</figcaption></figure>

a larger effective probe to linearly deform the far-field matrix whose shear modulus is *G*, and the effective probe size *R*\* grows by *F*¹/². Surprisingly, this argument suggests that *k* ~ *F*¹/² is a universal relation for local mechanical measurements independent of bulk stiffening behaviors provided that the stiffening is sufficiently strong. Furthermore, we measure the stiffness and displacement profile around spindle-shaped contractile cancer cells embedded in collagen gel and show that this emerging nonlinear length scale is observable and can be perturbed by changing collagen concentration or inhibiting cell contractility. These results highlight the significant differences between nonlinear responses upon local and bulk force applications, relevant to cell-ECM interactions and thus mechanobiology of multicellular living systems.

## The Local Stiffening *F* ~ *u*² and *k* ~ *F*¹/²

To apply a localized force such as those exerted by filopodia, we use optical tweezers to displace micron-sized particles embedded in collagen gels. A bead is displaced to a distance *u* from its equilibrium position by an optical force *F* (Fig. 1D). The laser trap applies monopole forces through the particle to the gel, and the resulting deformation field is asymmetric, similar to the case of a single protrusion from contractile cells. When we perform this measurement in 4 mg mL⁻¹ collagen gels with 2 µm-in-diameter beads, we find that the material indeed stiffens with increasing force (Fig. 1E); the logarithmic slope changes from 1 to 2 (Fig. 1E), indicating that the relation between force

*F* and displacement *u* changes from a linear function *F* ~ *u* to a quadratic function *F* ~ *u*². Moreover, using flat-punch indentation, we find a similar power-law relation close to *F* ~ *u*² with force on the order of 1 to 100 µN, and the displacement needed to stiffen the matrix is on the order of one-tenth the probe size, which is consistent with our optical-tweezers results (Fig. 1F).

Furthermore, we test how the differential stiffness *k* = *dF/du* scales as a function of applied force *F* at various conditions. We perform the optical-tweezers measurements with 4.5 µm-in-diameter beads in collagen gels of different concentrations (e.g., 1.6, 2.4, and 3.2 mg mL⁻¹), which all show a relation close to *k* ~ *F*¹/² (SI Appendix, Fig. S1C). To compare the power-law relation across different conditions, we normalize the stiffness by the linear stiffness *k*\* and normalize the force by the onset force of nonlinear stiffening *F*\*. The relation between *k* and *F* shows a linear regime followed by a nonlinear regime (Fig. 2). A closer examination reveals that the logarithmic slope is about 1/2, suggesting *k* ~ *F*¹/² (Fig. 2); this is consistent with directly inspecting the force *F* as a function of the displacement *u*, which is close to *F* ~ *u*² (Fig. 1E). Moreover, we perform measurements on fibrin gels. Remarkably, the local nonlinear response of fibrin gels is also close to *k* ~ *F*¹/² (SI Appendix, Fig. S2 and Fig. 2), regardless of the different bulk stiffening response of collagen and fibrin gels (3, 5). These results demonstrate the generality of this weak power-law relation *k* ~ *F*¹/² in local micromechanics measurements in highly nonlinear matrices.
<figure><img src="image_3.png" /><figcaption>Fig. 2. All the experimental and theoretical curves can be collapsed to a single master curve by nondimensionalizing stiffness and force. The master curve shows $k \sim F^{1/2}$ in the nonlinear regime. c (mg mL$^{-1}$) is the polymer concentration, and d (µm) is the diameter of the particle. The exponential free-energy function is used for the finite element simulation. k* is the linear stiffness, and the onset force of nonlinear stiffening k* is estimated by the force at which the differential stiffness is twice the linear stiffness.</figcaption></figure>

## The Emerging Nonlinear Length Scale R*

To understand this power-law nonlinear relation $k \sim F^{1/2}$ ($F \sim u^2$), we follow a dimensional analysis of unidirectionally dragging a rigid particle (radius $R_0$) with resultant force F in a 3D infinite large highly nonlinear matrix, an idealization of which is an “elastic-rigid material” simplification: The material is linear at small stresses but becomes rigid once the stress exceeds a critical value $\sigma^*$ (Fig. 3B). Considering the force balance, the resultant force to perturb the particle is balanced by the integral of traction fields on an arbitrary closed surface containing this particle,

$$F = -\iint \sigma_{ij} n_j e_i^1 dA, \quad [1]$$

where $\sigma_{ij}$ are the stress components, $n_j$ are components of the surface normal vector, and $e_i^1$ are components of the unit vector along the point force. In the matrix, strain and stress fields decay away from the probe. Although F can be comparably large that nonlinearly deforms the vicinity of the probe, because the strain and stress vanish in the far field, the strong nonlinear deformation must be constrained within a finite region above which the material undergoes only moderate deformation. Denoting the length scale of this closed surface as $R^*$, and realizing that the stress scale of nonlinear stiffening is $\sigma^*$, the force balance equation yields

$$F \sim \sigma^* R^{*2}, \quad [2]$$

such that $R^*$ scales as $(F/\sigma^*)^{1/2}$, and highly nonlinear deformation is restricted within a finite region of size $R^*$ in the adjacency of the monopole.*

Because the material within this range $R^*$ is more severely deformed and much stiffer than the outside as a result of nonlinear elastic stiffening, this is analogous to using an effective rigid particle of length scale $R^*$ (larger than the true particle size $R_0$) to probe the linear matrix surrounding this effective particle (Fig. 3A),

$$k \sim GR^*. \quad [3]$$

This is similar to the Stokes-Einstein law (32), which we further verify in fiber network simulations by linearly probing the networks with particles of different sizes (SI Appendix, Fig. S4A). Combine Eqs. 2 and 3, in the nonlinear regime, the stiffness scales as

$$k \sim F^{1/2}. \quad [4]$$

This scaling argument is similar to a boundary layer analysis in fluid mechanics, where a length scale $\delta$ can be derived by balancing viscous force with kinetic force; while the flow inside the boundary layer is highly viscous, the flow outside the boundary layer can be treated as inviscid flow. In addition, it is also similar to the small-scale yielding in linear elastic-plastic fracture, where the material is plastically deformed in the vicinity of the crack with a length scale, but the linear elastic solution can still be applied to the region outside this small-scale yielding region (33). Furthermore, this near-field rigid and far-field linear assumption is also consistent with previous theoretical analysis on spherical dipoles within nonlinear matrices which also indicates an emerging nonlinear lengthscale (34).

To validate this scaling argument, we first use a minimal finite element simulation (Details are provided in the Materials and Methods). We consider a minimal material model of an incompressible Fung-type elastic medium with an elastic free-energy function $W_{exp} = \frac{G}{2b} \exp(b(\bar{I}_1 - 3))$, where G is the linear shear modulus, b is a stiffening parameter, and $\bar{I}_1$ is the first invariant of the distortional Cauchy-Green deformation tensor. As expected, the Fung-type material relation can capture the exact responses in bulk measured by rheometers (SI Appendix, Fig. S1 A and B) (5). Further, we study the local response by performing finite element analysis of an axial symmetric model where a rigid particle of radius $R_0$ is embedded in a cylindrical matrix with radius $R = 250R_0$ and height $H = 2R$. The particle is perturbed in the axial direction from its equilibrium position by a force F to a distance u. Interestingly, we find that this minimal model can recover the power law $k \sim F^{1/2}$ observed in our microrheology experiments of the local response (Fig. 1E and SI Appendix, Fig. S1C).† As we examine the profile of displacement in the direction of the applied force in simulation, the direct adjacency of the particle moves with it almost rigidly; indeed, there is a critical length $R^*$ above which the displacement field starts to rapidly decay (Fig. 3C). To obtain the transition distance from nonlinear to linear deformation (i.e., the effective probe size $R^*$), we choose the distance at which the displacement decays to half that of the rigid particle as an estimation (Fig. 3C); we find that it agrees with Eq. 2 ($R^* \sim F^{1/2}$) at all forces (Fig. 3D). It is worth noting that, a linear system satisfies the superposition principle, in which a half-decay contour of the displacement field should be invariant of the magnitude of loading; the changing of $R^*$ thus indicates a nonlinear deformation.

To further validate this scaling argument, we perform fiber network simulations (8, 31), where a point force F is applied at

<footnote>*While the emerging length scale R* in our work comes from nonlinear stiffening, there potentially exist other emerging length scales. Statistical theory predicts that the buckling happens at a comparable strain to stiffening, but if the buckling were to happen at a much lower strain, an anisotropic length scale might be critical (2). Interestingly, although fiber buckling is not explicitly considered here, the scaling law $R^* \sim F^{1/2}$ also coincides with the range of the buckled fibers in discrete fibrous models (30, 31).</footnote>

<footnote>†In a smaller matrix, the model predicts that this local response eventually becomes a global shear with increasing force; then, the bulk response $k \sim F$ is recovered (SI Appendix, Fig. S3). However, due to limitations in the maximum force, this finite-size effect is not clearly observed in our optical-tweezers experiments.</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** The local stiffening response can be attributed to the growth of an effective probe size $R^*$, almost independent of the constitutive relations. (A) The immediate adjacency of the particle is stiffened over the length scale $R^*$ such that the point force is probing the far-field linear matrix with an effective larger probe of length scale $R^*$. (B) The extreme case is an "elastic-rigid" simplification that the material is linear at small stresses but becomes rigid as the stress grows larger than a threshold $\sigma^*$. $\hat{\sigma}$ and $\hat{c}$ are scales of the magnitude of stress and strain. (C) The axial displacement as a function of the axial distance away from the particle. $R^*$ can be estimated by the transition point of the axial displacement profile near the particle. $R^*$ is estimated as the distance when the axial displacement decays to half of that of the particle. The transition points are marked by red dots. ($G = 1, b = 5$). (D) Effective probe size $R^*$ grows as $F^{1/2}$ regardless of the constitutive relations. (E) Displacement profiles at all different force levels can all be collapsed to the linear solution. (F) Stiffness $k$ grows as $F^{1/2}$ regardless of the constitutive relations ($G = 1$).</figcaption></figure>

the center of the network. A similar power-law relation of the average measured stiffness as a function of the applied force is observed in the network simulations (Fig. 2 and SI Appendix, Fig. S4B) (31). All curves collapse onto a single master curve, demonstrating the robustness of the $k \sim F^{1/2}$ scaling law (Fig. 2). This result also suggests that both the discrete and continuous models capture the underlying stiffening mechanism of monopole forces in highly nonlinear matrices.‡

Interestingly, this scaling argument appears to be independent of how the material is stiffened; the only requirement is that the material stiffens strongly enough such that the material within $R^*$ can be considered as “rigid.” The remaining issue is how strong the stiffening is to be approximated as an “elastic-rigid” model. To investigate this, we further modify the simulation with different types of stiffening: The quadratic energy function is defined as $W_2 = \frac{G}{2}(\bar{I}_1 - 3) + \frac{Gb}{4}(\bar{I}_1 - 3)^2$, and the fourth order energy function is $W_4 = \frac{G}{2}(\bar{I}_1 - 3) + \frac{Gb}{8}(\bar{I}_1 - 3)^4$. Interestingly, except for the quadratic stiffening that is not sufficiently strong and thus slightly deviates from $F^{1/2}$, the local stiffening behavior

$k \sim F^{1/2}$ seems quite robust, regardless of the different types of stiffening in bulk (Fig. 3F).

The solution of $R^*$ may be written as $R^* = C(\frac{F}{G^*})^{1/2}$ where the prefactor $C$ cannot be derived from scaling analysis but should come from solving the boundary value problem either analytically or numerically. The prefactor $C$ should depend on the nonlinear dimensionless parameters. This minimal model suggests that for a material whose nonlinear term grows sufficiently fast, the nonlinear differential stiffness $k$ probed by the particle can be interpreted as coming from the linear response to an effective probe with size $R^*$; this effective probe deforms a larger portion of the matrix with increasing force $F$, that yields $k \sim F^{1/2}$. Therefore, the nonlinear local response is a consequence of both material nonlinearity and this special geometry. Once the materials surrounding the probe become stiff enough compared to the linear far-field matrix, the response becomes dominated by the effective probe (Fig. 3D and F). Nevertheless, the onset of stiffening still indicates material nonlinearity; materials with stronger nonlinearity will stiffen at smaller forces (Fig. 3F, exponential energy with $b = 5$ and $b = 50$). Indeed, the stiffening parameter $b$ determines the stiffening onset nominal strain $\epsilon^* = \frac{2u^*}{d}$; thus, in principle, the onset strain can be used to measure the stiffening factor of the material (SI Appendix, Fig. S5).

## Identify $R^*$ around Living Cells

To explore whether the nonlinear length scale $R^*$ is indeed observable around living cells, we first measure the stiffness profile

<footnote>‡It is worth noting that even though empirically one could relate the spring constant $k$ to shear modulus $G$ by the generalized Stokes relation $G = k/6\pi R_0$, the local linear stiffness does not necessarily match bulk measurements as our optical probe (μm) is approaching the mesh size of the collagen gel (μm). For exact matching of force-displacement data, many other aspects might also be important, such as fiber buckling, heterogeneity, anisotropy, viscoelasticity, and plasticity (6, 30, 31, 35-38). Importantly, it was computationally shown that local forces approaching the mesh size of fibrous matrices can “pluck” the fiber which yields nonlinear stiffening at a much smaller force compared to the bulk response (39). Nevertheless, the focus of this work is the scaling law $k \sim F^{1/2}$ and the emerging nonlinear length scale $R^*$; as demonstrated here, the scaling law can be captured by a minimal continuous model, consistent with both the experiments and the fiber network simulation.</footnote>
around MDA-MB-231 human breast cancer cells in collagen gels at various concentrations (e.g., 1.6, 2.4, 3.2 mg mL⁻¹) by locally pulling preembedded particles (diameter 4.5 µm) with optical tweezers in the spreading direction of the cell (Fig. 4A, red dashed box). Spindle-like cancer cells grab the matrix with their two filopodia, which is similar to two equal and opposite force monopoles and possibly generate two effective probes with the effective size *R*\* that increases the range of deformation fields (Figs. 1 and 4A). At all concentrations, the matrix is stiffened in the near field but decays to the linear stiffness in the far field, which is consistent with our near-field rigid and far-field linear approximation. *R*\* can be approximated by the characteristic length scale at which the stiffness decays to the far-field value. We find that *R*\* increases with decreasing collagen concentration (Fig. 4B). Moreover, we show that this stiffness profile can be perturbed by inhibiting cell contraction with blebbistatin (SI Appendix, Fig. S6), indicating that a significant portion of the stiffening is indeed due to cell contraction.

To further explore whether the nonlinear length scale *R*\* is observable from the displacement profile around living cells as we observed in Fig. 3, we measure the displacement profile around MDA-MB-231 cells in 3D collagen matrices (2 mg mL⁻¹) using traction force microscopy (40–42). To do so, we lyse the cells and record the lysis process with a time-lapse video using confocal microscopy. Fluorescent particles (1 µm in diameter)

are dispersed in the matrix as tracers to visualize cell-induced displacement. After cell spreading, a lysing solution containing cytochalasin D and Triton is used to lyse the cells to fully relax them. The displacement field is calculated by comparing the tracer particle positions at a given time to the end-point lysed configuration (Fig. 4C). Interestingly, similar to Fig. 3C, the half-decay contour of the displacement field gradually shrinks as cell contractility is released during cell lysis (Fig. 4D), and the corresponding nonlinear length scale *R*\* decreases (Fig. 4E). It is worth mentioning again that similar to Fig. 3, the changing half-decay contour indicates the breakdown of the superposition principle, which is the hallmark of linearity; this suggests that the cell indeed nonlinearly deforms the matrix. We normalize the displacement as a function of distance (Fig. 4F) with dimensionless displacement *u*/*u*\* and distance *r*/*R*\*, where *u*\* is the maximum displacement (Fig. 4G). The normalized displacement profiles of different cells have very similar shapes, while the discrepancy possibly suggests a variation among individual cells in the mechanical interaction between the two localized forces applied at the two ends of the spindle-like cell (SI Appendix, Fig. S7).

Living tissues are naturally nonlinear materials, and it is known that cells can trigger the nonlinear stiffening of their matrices and induce long-ranged mechanical fields (8, 43, 44). Unlike on linear elastic substrates, the matrix nonlinearity complicates cell–cell

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Stiffness, displacement profile, and characteristic nonlinear length scales *R*\* around MDA-MB-231 cells in 3D collagen gels. (A) Representative images of MDA-MB-231 cells (green) in collagen networks (gray) of various concentrations, 1.6, 2.4, and 3.2 mg mL⁻¹. (Scale bar, 50 µm). The measurements take place in the cell-spreading direction. The red dashed box schematically shows the probed region. (B) Stiffness profile measured around individual cells in collagen gels of different concentrations. Stiffness is plotted as a function of the distance to the cell boundary. The lines are piece-wise linear fit, which marks the transition from the near-field stiff to the far-field linear matrix. (C) Representative bright-field image overlaid with cell-induced displacement field in the normal and lysed configuration. (Scale bar, 100 µm). (D and E) The half-decay range of the cell-induced displacement shrinks as the cell contractility decreases. (F) Displacement as a function of distance for multiple cells. Each curve represents the displacement profile of a cell comparing the normal state to the lysed state. The black dots (*R*\*, *u*\*) mark the distance at which the displacement decays to half the maximum displacement. (G) Normalized displacement *u*/*u*\* as a function of normalized distance *r*/*R*\*.</figcaption></figure>
interaction and collective cell behaviors within these matrices. The effective probe argument might help to understand cell–cell interaction in highly nonlinear matrices within a simplified linearized picture. For example, the subsequent linearization by the effective probe argument might provide a pathway to extend recent endeavors regarding the cell collective behaviors on the substrate (45, 46) to natural 3D highly nonlinear extracellular matrices.

In conclusion, we find that the local monopole force–displacement response in a highly nonlinear material is closer to $F \sim u^2$ ($k \sim F^{1/2}$) (Figs. 1 *E* and *F* and 2). This weaker power–law stiffening behavior appears to be robust across various types of biopolymers, both in experiments and simulations (Fig. 2). This local effect is distinct from bulk rheology where several different power–law relations can be observed for different types of nonlinear elastic biopolymer matrices; for example, for collagen, $k \sim F$ is observed in bulk rheology which is distinct from $k \sim F^{1/2}$ in microrheology. Furthermore, we show that the local stiffening mechanism is distinct from the bulk: the local response ($k \sim F^{1/2}$) can be explained by introducing an emerging nonlinear length scale $R^*$ that grows with increasing forces (Fig. 3). These results highlight the difference of highly nonlinear ECM gels between responses to local and bulk forces with simple concepts. While the mechanical fields are rather uniform under bulk forces, they decay away under localized forces. We show that the local stiffening behaviors can be explained by considering the mechanical fields as two successive regions, i.e., an effective probe induced by material nonlinear stiffening and a linear matrix that is being deformed by the effective probe (Fig. 3A). The nonlinear response to local forces ($k \sim F^{1/2}$) thus comes from this unique geometry that a growing probe with effective length scale $R^*$ is deforming an increasing portion of the far-field matrix.

Our results, along with previous theoretical analysis on spherical dipoles (30, 34), suggest that cells embedded in 3D strongly nonlinear matrices thus might be abstracted as localized forces with an effective size $R^*$ that grows with the contractility of the cell. Facilitated by the nonlinear material properties, the localized force exerted by a cell can create a stiff effective probe with an effective radius $R^*$ through which a cell can deform farther matrices, and this extended matrix deformation can facilitate mechanical communications between cells. We show that such a nonlinear length scale $R^*$ indeed exists and can be varied by perturbing matrix concentration and cell contractility (Fig. 4). The emerging nonlinear length scale $R^*$ upon local force application may provide insights into understanding cell–matrix interactions and mechanobiology of complex living systems such as tissues and organs.

## Materials and Methods

**Preparation of Collagen Gels.** To make 1 mL 4 mg mL⁻¹ collagen gels, 400 µL 10 mg mL⁻¹ collagen solution (Advance Biomatrix, Cat# 5133) is mixed with 460 µL Dulbecco's Modified Eagle Medium (DMEM) and 100 µL 10× phosphate-buffered saline. Then, 40 µL 0.1 M NaOH is added to the mixture to adjust pH to 7.2 to 7.6. The solution is then polymerized in the cell culture incubator (37 °C, 5% CO₂) for 60 min.

**Preparation of Fibrin Gels.** To make 1 mL 3 mg mL⁻¹ fibrin gels, 500 µL 6 mg mL⁻¹ fibrinogen (F8630-1G, Sigma-Aldrich) solution is mixed with 500 µL 4 U mL⁻¹ thrombin (T4648-1KU, Sigma-Aldrich). The solution is then polymerized in the cell culture incubator (37 °C, 5% CO₂) for 20 min.

**Optical Tweezers Measurements.** Microparticles (2 µm, C37278, Ther-moFisher, and 4.5 µm carboxylate microspheres, Polyscience) are embedded in the gel prior to polymerization. Home-built optical tweezers are used (47). The laser beam is moved at a constant velocity of 1 µm s⁻¹. The relative distance between the laser and the bead is recorded. Raw data are processed as described in ref. 8. The tweezers are calibrated in phosphate-buffered saline solution. The power spectrum density method is used (47). The trap stiffness is 15 pN/µm.

**Flat-Punch Indentation.** A mold is fabricated by bonding a circular polymethyl methacrylate ring (the inner and outer diameters of 7 mm and 12 mm, respectively, and the height of 4.2 mm) to a glass slide. Then, the collagen solution is transferred to the mold, and another glass slide was used to cover the solution. Next, the sample is left in an oven of 37 °C for 60 min to complete the cross-linking. After cross-linking, the top glass slide is removed. The cured collagen gel within the mold is placed on the stage of an inverted microscope (Nikon, Eclipse Ti2-E). An in situ indenter (Bruker, Hysitron BioSoft) with a flat punch (diameter of 500 µm) is used to perform the indentation test. The test profile for controlling the movement of the punch includes five sequential steps: approaching, loading, holding, unloading, and retracting. In the approaching step, the displacement rate is set at 1 µm s⁻¹, and the target value for detecting the sample surface is set at 5 µN. In the following loading step, the displacement rate is set at 0.2 µm s⁻¹ until reaching the maximum displacement of 140 µm. Then, the displacement is held at 140 µm for 0.1 s. Next, in the unloading step, the displacement rate is set at −0.2 µm s⁻¹ until returning to the displacement of 0 µm. Finally, in the retracting step, the force rate is set at −15 µN s⁻¹ for 5 s to move the punch back to its initial position. Right before the indentation test, the tip is manually moved downward toward the sample surface, and the real-time compression force was simultaneously monitored. Due to the limitation of the displacement range, we applied different precompressions to the sample and then followed the above test profile to do the indentation.

**Finite Element Simulations.** For local measurement, an axial symmetric geometry is used, where a rigid spherical domain of radius $R_0$ is embedded in a cylindrical matrix of radius $R = 250R_0$ and height $H = 2R$. The matrix is treated as incompressible, and various forms of hyperelastic constitutive relations are assigned to the matrix. The lateral surface of the matrix is fixed, and the top and bottom surfaces are traction-free. A series of point forces are applied at the center of the rigid spherical domain, and the displacement is recorded. The axial displacement along the axis is also recorded. For the bulk measurement, a rectangular domain with the same material properties is sheared by a shear stress $\sigma$ to a shear strain $\gamma$, and the differential modulus can be defined as $K = d\sigma/d\gamma$. COMSOL Multiphysics is used for the simulation.

**Network Simulations.** Details of the network simulation are in ref. 31. Briefly, a point-force monopole is placed at the center of a 3D disordered network of radius $R = 40$. The 3D network is generated by placing fibers on a face-centered cubic lattice and randomly depleting it such that a bond is present with probability $p = 0.4$. Fibers are discretized with bonds that act as springs with linear stretching modulus $\mu = 100$ and nonlinear longitudinal response $\exp(\mu\delta\epsilon) - 1$, where $\delta\epsilon$ is the bond deformation. The fibers also resist transverse deflection with a bending rigidity $\kappa = 1$. Fibers are connected by freely deforming hinges at their intersection. Results are averaged over 100 random network realizations.

**Cell Culture.** MDA-MB-231 cells are cultured in DMEM supplemented with 10% fatal bovine serum in the cell culture incubator (37 °C, 5% CO₂).

**Stiffness Profile around Living Cells.** We mixed 4.5-µm-diameter beads (Polysciences, 17140-5) with cells in the collagen solution prepared with the same protocol described above. We unidirectionally drag the particle for 200 nm at a speed of 1 µm s⁻¹ to deform local collagen hydrogels, and a force-displacement curve characterizing local matrix micromechanics is obtained; the initial slope of the obtained force-displacement curve is used to describe the elasticity of the local matrix. We drag each particle near the cells in the direction away from the cell to obtain local force-displacement curves and quantify the
local matrix elasticity. For these spindle-shaped cells (MD-MBA-231), we probed the elasticity of the matrix only in the longitudinal direction of the cell because the cell-generated stress is mainly distributed in this direction. The stiffness as a function of distance is fitted by a far-field-constant piece-wise linear function. and the nonlinear length scale $R^*$ is estimated by the transition distance. To reduce the cell contractility, 40 µM blebbistatin was added into the hydrogel 1 h before the measurement.

**Displacement Profile around Living Cells.** First, 1-µm-diameter tracer particles (F13083, FluoSpheres, 1:200) are seeded together with cells in the collagen solution prepared with the same protocol described above. Then, the solution is transferred into a 96-well plate (100 µL per well) and allowed to polymerize for 60 min in the incubator before 200 µL cell culture medium is added to each well. The next day, fresh lysing solution (20 µg mL⁻¹ cytochalasin D, 2% triton) is prepared. Within each well, we identify one isolated spreading cell and add 100 µL lysing solution. Lysis takes place in around 1 h, which is recorded by time-lapse z-stack videos at 20 s per frame using confocal microscopy. We take the maximum projection of the z-stack. Image drift is corrected in ImageJ (48). The displacement fields at each frame compared to the reference (1 h after adding the lysing solution) are computed using PIVLab (49) in MATLAB. To estimate displacement as a function of distance as shown in Fig. 4 E–G, we set a serious of displacement magnitude *u* and calculate the total area *A* with displacement greater than *u*, and then the distance is estimated by $r = \sqrt{A/2\pi}$.

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**ACKNOWLEDGMENTS.** We thank R. Abeyaratne, I. Y. Wong, and S. Wasserman for helpful discussions. We would like to acknowledge the support from the NIH (1R01GM140108), MathWorks, and the Jeptha H. and Emily V. Wade Award at the Massachusetts Institute of Technology. H.Y. acknowledges the MathWorks Mechanical Engineering Fellowship. M.G. acknowledges the Sloan Research Fellowship. This project received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 891217 and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 201269156 - SFB 1032 (Project B12) (E.B. and C.P.B.). P.R. is supported by France 2030, the French National Research Agency (ANR-16-CONV-0001), and the Excellence Initiative of Aix-Marseille University–A*MIDEX.

Author affiliations: aDepartment of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139; bArnold-Sommerfeld-Center for Theoretical Physics and Center for NanoScience, Ludwig-Maximilians-Universität München, München D-80333, Germany; cDepartment of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093; dAix Marseille University, CNRS, CINAM, Turing Center for Living Systems, 13288 Marseille, France; and eDepartment of Physics and Astronomy, Vrije Universiteit Amsterdam, Amsterdam 1081 HV, The Netherlands

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