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# Fiber networks amplify active stress

Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved December 18, 2015 (received for review July 20, 2015)

## Significance

Living organisms generate forces to move, change shape, and maintain their internal functions. These forces are typically produced by molecular motors embedded in networks of fibers. Although these motors are traditionally regarded as the defining elements of biological force generation, here we show that the surrounding network also plays a central role in this process. Indeed, rather than merely propagating forces like a simple elastic medium, fiber networks produce emergent, dramatically amplified stresses and can go so far as reversing small-scale extensile forces into large-scale contraction. Our theory quantitatively accounts for experimental measurements of contraction.

## Abstract

Large-scale force generation is essential for biological functions such as cell motility, embryonic development, and muscle contraction. In these processes, forces generated at the molecular level by motor proteins are transmitted by disordered fiber networks, resulting in large-scale active stresses. Although these fiber networks are well characterized macroscopically, this stress generation by microscopic active units is not well understood. Here we theoretically study force transmission in these networks. We find that collective fiber buckling in the vicinity of a local active unit results in a rectification of stress towards strongly amplified isotropic contraction. This stress amplification is reinforced by the networks’ disordered nature, but saturates for high densities of active units. Our predictions are quantitatively consistent with experiments on reconstituted tissues and actomyosin networks and shed light on the role of the network microstructure in shaping active stresses in cells and tissue.

Living systems constantly convert biochemical energy into forces and motion. In cells, forces are largely generated internally by molecular motors acting on the cytoskeleton, a scaffold of protein fibers (Fig. 1*A*). Forces from multiple motors are propagated along this fiber network, driving numerous processes such as mitosis and cell motility (1) and allowing the cell as a whole to exert stresses on its surroundings. At the larger scale of connective tissue, many such stress-exerting cells act on another type of fiber network known as the extracellular matrix (Fig. 1*B*). This network propagates cellular forces to the scale of the whole tissue, powering processes such as wound healing and morphogenesis. Despite important differences in molecular details and length scales, a common physical principle thus governs stress generation in biological matter: Internal forces from multiple localized “active units”—motors or cells—are propagated by a fiber network to generate large-scale stresses. However, a theoretical framework relating microscopic internal active forces to macroscopic stresses in these networks is lacking. Here we propose such a theory for elastic networks.

This generic stress generation problem is confounded by the interplay of network disorder and nonlinear elasticity. Active units generate forces at the scale of the network mesh size, and force transmission to larger scales thus sensitively depends on local network heterogeneities. In the special case of linear elastic networks, the macroscopic active stress is simply given by the density of active force dipoles, irrespective of network characteristics (2). Importantly, however, this relationship is not applicable to most biological systems, because typical active forces are amply sufficient to probe the nonlinear properties of their constitutive fibers, which stiffen under tension and buckle under compression (3). Indeed, recent experiments on reconstituted biopolymer gels have shown that individual active units induce widespread buckling and stiffening (4, 5), and theory suggests that such fiber nonlinearities can enhance the range of force propagation (6, 7).

Fiber networks also exhibit complex, nonlinear mechanical properties arising at larger scales, owing to collective deformations favored by the networks’ weak connectivity (3, 8). The role of connectivity in elasticity was famously investigated by Maxwell, who noticed that a spring network in dimension *d* becomes mechanically unstable for connectivities

Here we study the theoretical principles underlying stress generation by localized active units embedded in disordered fiber networks (Fig. 1*C*). We find that arbitrary local force distributions generically induce large isotropic, contractile stress fields at the network level, provided that the active forces are large enough to induce buckling in the network. In this case, the stress generated in a biopolymer network dramatically exceeds the stress level that would be produced in a linear elastic medium (2), implying a striking network-induced amplification of active stress. Our findings elucidate the origins and magnitude of stress amplification observed in experiments on reconstituted tissues (4, 18) and actomyosin networks (14, 17). We thus provide a conceptual framework for stress generation in biological fiber networks.

## A Lattice Model for Elastic Fiber Networks

We investigate force transmission, using a lattice-based fiber network model (3). In our model, straight fibers are connected at each lattice vertex by cross-links that do not constrain their relative angles. Each lattice edge represents a “bond” made of two straight segments and can thus stretch, bend, or buckle (Fig. 1*D* and Fig. S1). Segments have stretching rigidity *μ* and a rest length equal to one, implying a stretching energy *θ* between consecutive segments through a bending energy

Network disorder is introduced through bond depletion, i.e., by randomly decimating the lattice so that two neighboring vertices are connected by a bond with probability *p*. This probability controls the network’s connectivity, giving rise to distinct elastic regimes delimited by two thresholds

We model active units as local sets of forces *i* with positions *σ* the trace (i.e., the isotropic component) of the coarse-grained active stress induced in the fiber network by a density *ρ* of such units.

The relationship between this active stress and local forces in homogeneous linear networks is very simple and yields (2)**1** is generically violated in disordered or nonlinear networks, although it holds on average in linear networks with homogeneous disorder**1**, we define the far-field force dipole

## Contractility Robustly Emerges from Large Local Forces

Stress generation by active units integrates mechanical contributions from a range of length scales. We first consider the immediate vicinity of the active unit. Network disorder plays a crucial role at that scale, because forces are transmitted through a random pattern of force lines determined by the specific configuration of depleted bonds (Fig. 2 *A* and *B*, *Insets*). To understand how these patterns affect force transmission, we investigate the probability distribution of the far-field force dipole

We first consider the linear regime **2** and **3**). The fluctuations around this average are strikingly broad compared with this average, shown by plotting the distribution of dipole amplifications (Fig. 2 *A* and *B*). For instance, a significant fraction (*A*, *Inset*). Due to linearity, contractility in response to an extensile dipole is just as likely. Overall, the far-field response in the linear regime is only loosely correlated to the applied force dipole.

The situation is dramatically different in the large force regime (*C* and *D*). Although the detailed shapes of these curves are model dependent, three robust features emerge: First, locally extensile dipoles predominantly undergo negative amplification, implying far-field contractility irrespective of the sign of *D*, *Inset*). Second, the randomization observed in the linear regime is strongly attenuated, and the sign of the amplification is very reproducible (positive for *C* and *D*,

To understand these three effects, we consider contractile and extensile dipoles in a simpler regular network (no bond depletion, Fig. 2 *E* and *F*). Qualitatively, these uniform networks behave similarly to the randomly depleted ones described above: Force dipole conservation holds for *G*). The origin of these behaviors is apparent from the spatial arrangement of the forces in Fig. 2 *E* and *F*. Whereas contractile and extensile active units both induce compressive and tensile stresses in their immediate surroundings, the buckling of the individual bonds prevents the long-range propagation of the former. This results in enhanced tensile stresses in the far field and thus in strongly contractile far-field dipoles. In addition, this nonlinear response renders the far-field stresses uniformly tensile and therefore more isotropic than the active unit forces. We quantify this effect in Fig. 2*G*, *Inset*, using an anisotropy parameter for the far-field stresses *d* is the dimension of space and *Supporting Information*). This anisotropy parameter indeed becomes very small for large local dipoles of either sign as shown in Fig. 2*G*.

Moving to a systematic quantification of force transmission in depleted, bending-dominated networks, we show in Fig. 2 *H* and *I* the same three effects of rectification, amplification, and isotropization, which set in at smaller forces than in regular networks. Overall, these effects are very general and hold in both bending- and stretching-dominated depleted networks, in two and three dimensions, and for active units with complicated force distributions not along lattice directions (Figs. S3 and S4). Thus, beyond the immediate neighborhood of the active force-generating unit, strong isotropic contractile stresses emerge in the system from a generic local force distribution due to the nonlinear force propagation properties of the fiber network.

## A Model for Active Units as Isotropic Pullers

Whereas nonlinear force transmission over large length scales involves large active forces, the model for active units used above can exert only moderate dipoles in soft, weakly connected networks. Indeed, for large enough contractile dipoles the two vertices on which the forces are applied collapse to a point as in Fig. 2*C*, *Inset*, preventing further contraction. In contrast, molecular motors and contractile cells continuously pull fibers in without collapsing. To study the response of fiber networks to large active forces, we thus introduce an active unit model capable of exerting arbitrarily large forces without changing its size. We assume an isotropic force distribution, an approximation valid in the large force regime, where complicated force distributions are locally rectified toward isotropic contraction by the network (Fig. 2 *G* and *I*).

Our model active unit is centered on a vertex *i* and pulls on every vertex *j* within a distance *i* and *j*, and *F*, which we define as the average force per unit area exerted on the surrounding network by the active unit at its outer surface (

## Contractile Forces Are Long Ranged in Bucklable Media

We now study force propagation beyond the immediate vicinity of an active unit, using the above-described isotropic puller. We identify two asymptotic regimes for this propagation. Close to the active unit, forces are large and fiber buckling affects force transmission, whereas beyond a crossover distance

To describe the near-field regime, we note that fiber buckling prevents the network from sustaining compressive stresses above the buckling threshold. Close to the active unit, the network is thus effectively equivalent to a network of floppy ropes, i.e., filaments with tensile strength but no resistance to compression or bending. The active unit pulls on these ropes and thus becomes the center of a radial arrangement of tensed ropes. Force balance on a small portion of a spherical shell centered on the active unit imposes that radial stresses in this rope-like medium decay as*r* is the distance from the active unit and *d* the dimension of space (19). In the far field, stresses are small and buckling does not occur, implying that force transmission crosses over from rope-like to linear elasticity:

To test this two-regime scenario, we simulate force propagation away from a single active unit in both stretching- and bending-dominated networks in two and three dimensions. In all cases, rope-like radial stresses and bond buckling are predominant in the vicinity of the active unit (Fig. 3 *A*–*C*). Monitoring the decay of radial stresses with *r*, we find a crossover from rope-like to linear behavior, consistent with Eqs. **5** and **6** (Fig. 3 *D*, *F*, and *H*).

Visually, the crossover length *A*–*C*, black circles). In stretching-dominated networks, our prediction of Eq. **7** captures the force dependence of this crossover length (Fig. 3*E* and Fig. S5). In contrast, bending-dominated networks display a more complex behavior: Although the system still exhibits a transition from rope-like to linear force transmission, the crossover region is much broader (Fig. 3 *F* and *H*) and rope-like force transmission extends much farther than predicted by Eq. **7**. Instead, we find behavior that is reasonably well described by a power law *G* and *I*). These exponents appear to be insensitive to the exact value of the depletion parameter *p* within the bending-dominated regime (Fig. S5). We speculate that this extended range for nonlinear force transmission is mediated by the strong concentration of tensile stresses along force chains (12, 20) observed in Fig. 3 *B* and *C*. Indeed, such force chains are much more pronounced in bending-dominated than in stretching-dominated networks (Fig. S5). The difference between stretching- and bending-dominated exponents thus suggests that these elastic heterogeneities qualitatively affect force transmission in such soft networks. As a result, contractile forces large enough to induce buckling benefit from an enhanced range of transmission, characterized by the mesoscopic radius of the rope-like region

## Amplification by a Collection of Active Units

Over large length scales, active stresses in biological systems are generated by multiple active units. We thus compute the stress amplification ratio in the presence of a finite density of randomly positioned active units in 2D and 3D for various densities *ρ* and depletion parameters *p* (Fig. 4*A*). In all cases we observe three stress amplification regimes as a function of the unit force *F*: a low-force plateau without amplification, an intermediate regime of increasing amplification, and a saturation of the amplification at a level that depends on *ρ*.

In the low-force regime, linear force transmission prevails (Fig. 4*B*) and the active stress is given by Eq. **1**:

For moderate forces, the fibers in the network buckle in the vicinity of each active unit, up to a distance *C*. To predict the resulting active stress in the system, we model each nonlinear region as an effective active unit of size **5** to describe force propagation within the nonlinear region. As the effective units are themselves embedded in a linear medium, linear force transmission (Eq. **1**) outside of these units implies*E*. Because *F* as previously observed in Fig. 4*A*.

For large forces, the radius of the rope-like regions becomes so large as to exceed the typical distance between adjacent active units *D*). This yields*F*. Strikingly, the stress generated in this large-force regime has a nonlinear dependence on *ρ*, again consistent with Fig. 4*A*. Indeed, the addition or removal of active units leads to large rearrangements of the rope network, resulting in significant local modifications of force transmission.

We summarize the physics of collective stress generation by many active units in a phase diagram (Fig. 4*G*). In each regime, the magnitude of an active unit’s effective force dipole is directly proportional to one of the three length scales **8**–**10**). Whereas we have shown that

## Discussion

In living organisms, microscopic units exert active forces that are transmitted by fibrous networks to generate large-scale stresses. The challenge in analyzing this force-transmission problem stems from the disordered architecture of such fibrous networks and the nonlinearities associated with the strong forces exerted by biological active units. Despite this complexity, we find surprisingly simple and robust behaviors: In response to any distribution of active forces, dramatically amplified contractile stresses emerge in the network on large scales. This remarkable property hinges only on the local asymmetry in elastic response between tensed and compressed fibers and is enhanced by network disorder. Our simple, yet realistic description of individual fibers yields a universal scenario for force transmission: long-ranged, rope-like propagation near a strong active unit and linear transmission in the far field. This generic result should be contrasted with recent studies focused on fibers with special singular force-extension relation (7) and resulting in nonuniversal force transmission regimes.

Our generic phase diagram (Fig. 4*G*) recapitulates our quantitative understanding of stress generation by a collection of active units based on the interplay between three length scales: active unit size *C* and *G*), in reasonable agreement with experiments (17). Finally, we consider a clot comprised of fibrin filaments and contractile platelets as active units (system III). The large forces exerted by platelets allow for long-range nonlinear effects, placing this in vitro system deep in the density-controlled regime (Fig. 4 *D* and *G*). Consequently, we expect stress amplification to be controlled by the distance between active units, irrespective of the large value of the active force

Far from merely transmitting active forces, we show that fiber networks dramatically alter force propagation as contractility emerges from arbitrary spatial distributions of local active forces. This could imply that living organisms do not have to fine-tune the detailed geometry of their active units, because any local force distribution yields essentially the same effects on large length scales. This emergence of contractility sheds a new light on the longstanding debate in cytoskeletal mechanics regarding the emergence of macroscopic contraction in nonmuscle actomyosin despite the absence of an intrinsic contractility of individual myosin motors (20⇓⇓–23). Indeed, although these motors exert equal amounts of local pushing and pulling forces (24, 25), our result suggests that the surrounding network rectifies pushing contributions into uniform contraction. This rectification effect in two and three dimensions contrasts with the behavior of previously studied one-dimensional actomyosin systems, where extensile dipoles are attenuated but not reversed (26). It complements more local effects biasing the effects of a motor toward contractility (25), such as local buckling (24) and polarity-induced treadmilling (27). More broadly, we suggest that this strong propensity for the emergence of contraction in fibrous materials can explain the overwhelming dominance of contractile stresses in active biological materials up to the tissue level. Clearly, this does not mean that it is impossible to generate large-scale expansion in living organisms as required for limb abduction and extension or for lung inflation. Nevertheless, in each of these examples the expansion actually results from the clever harnessing of muscle contraction through lever structures involving the skeleton. Our findings connect widely used phenomenological “active gel” theories (28) to the underlying molecular scale forces, a crucial step in bringing theory and experiments together in the study of active biological matter, and call for further progress in characterizing force transmission in fiber networks. For instance, whereas our results concern the short-time elastic response of the network, it will be interesting to see how they are modified on longer timescales as cross-linker detachment and cytoskeletal remodeling induce flow in the fibrous matrix.

## Methods

### Network Energy Minimization.

Using the network model described in the main text, we investigate the response to localized active forces of the form *i* of the network, as illustrated in Fig. S1. Summing the associated elastic energy with all fiber stretching and bending contributions, our total Hamiltonian reads*i* and *j*, *i* at which the force

### Active Stress Measurements.

The contractility of our active networks is quantified by the value of its macroscopic active stress. To quantify this stress, we use a framework previously developed by us and suited for the analysis of random discrete networks with next-nearest neighbor interactions (2). This framework includes different prescriptions for systems with fixed and periodic boundary conditions.

For fixed boundary conditions, we compute the system’s active stress tensor *i*, and *V* is the volume of the system. Note that

For periodic boundary conditions (used in Fig. 4 of the main text), we use the so-called mean-stress theorem**S3** are straightforward, as both

### Measurements of the Rope-Like Region Radius.

Our definition of the radius **5** and **6** of the main text). We measure

The identification of the crossover length is complicated by the fact that the linear regime is not a pure scaling regime. Indeed, it is affected by the boundary conditions of the system. For an isotropic elastic continuum with spherical symmetry, the generic solution to the linear elastic equations for the radial displacement *d* is the space dimension, and the constants *A* and *B* are set by the boundary conditions. The radial stress thus reads*μ* is the shear modulus and *ν* is the Poisson ratio of the material. In the case of an infinite system, *B*, thus perturbing the scaling regime and complicating the estimation of

In the case of fixed boundary conditions, as in Fig. 3 of the main text, we have *R* is the radius of the system. In addition, the Poisson ratio of elastic fiber networks is easily measured, and we numerically find that it is independent of the precise geometry and connectivity of the network: In *R* and compute the corrected “infinite system” stress as a function of our finite-size “raw” measurement, using the formula

### Experimental Data.

As shown in Table 1 of the main text, our predictions on stress amplification are quantitatively supported by a range of experiments in

Our first example (“system I” in the main text) illustrates the linear regime (Fig. 4 *B* and *G* of the main text) in 3D actomyosin. In ref. 14, a cross-linked actin network with mesh size

Our second example (“system II” in the main text) illustrates force-controlled amplification in 2D actomyosin networks (Fig. 4 *C* and *G* of the main text). Ref. 17 reports active stresses generated by a membrane-supported two-dimensional actomyosin sheet competing with the tension of a bare lipid membrane. This reveals that the active stresses generated by thick filaments (*C* and *G* of the main text). This yields an amplification factor

Our last example (“system III” in the main text) addresses the density-controlled amplification regime (Fig. 4 *D* and *G* of the main text). We consider stress generation in a reconstituted clot composed of a fibrin network rendered contractile by a concentration *ξ*.

## Acknowledgments

We thank Cécile Sykes and Guy Atlan for fruitful discussions. This work was supported by Marie Curie Integration Grant PCIG12-GA-2012-334053, “Investissements d’Avenir” LabEx PALM (ANR-10- LABX-0039-PALM), Agence Nationale de la Recherche Grant ANR-15-CE13-0004-03, and European Research Council Starting Grant 677532 (to M.L.), as well as by the German Excellence Initiative via the program “NanoSystems Initiative Munich” (NIM) and the Deutsche Forschungsgemeinschaft (DFG) via project B12 within the SFB 1032. P.R. is supported by “Initiative Doctorale Interdisciplinaire 2013” from IDEX Paris-Saclay (ANR-11-IDEX-0003-02), and C.P.B. is supported by a Lewis-Sigler fellowship. M.L.'s group belongs to the CNRS consortium CellTiss.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: martin.lenz{at}u-psud.fr or c.broedersz{at}lmu.de.

Author contributions: P.R., C.P.B., and M.L. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1514208113/-/DCSupplemental.

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