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# Stress controls the mechanics of collagen networks

Edited by William Bialek, Princeton University, Princeton, NJ, and approved June 19, 2015 (received for review March 2, 2015)

## Significance

We report nonlinear rheology experiments on collagen type I networks, which demonstrate a surprising concentration independence of the network stiffness in the nonlinear elastic regime. We develop a model that can account for this, as well as the classical observations of an approximate exponential stress–strain relationship in collagenous tissues, for which a microscopic model has been lacking. Our model also demonstrates the importance of normal stresses in controlling the nonlinear mechanics of fiber networks.

## Abstract

Collagen is the main structural and load-bearing element of various connective tissues, where it forms the extracellular matrix that supports cells. It has long been known that collagenous tissues exhibit a highly nonlinear stress–strain relationship, although the origins of this nonlinearity remain unknown. Here, we show that the nonlinear stiffening of reconstituted type I collagen networks is controlled by the applied stress and that the network stiffness becomes surprisingly insensitive to network concentration. We demonstrate how a simple model for networks of elastic fibers can quantitatively account for the mechanics of reconstituted collagen networks. Our model points to the important role of normal stresses in determining the nonlinear shear elastic response, which can explain the approximate exponential relationship between stress and strain reported for collagenous tissues. This further suggests principles for the design of synthetic fiber networks with collagen-like properties, as well as a mechanism for the control of the mechanics of such networks.

Collagen type I is the most abundant protein in mammals where it serves as the primary component of many load-bearing tissues, including skin, ligaments, tendons, and bone. Networks of collagen type I fibers exhibit mechanical properties that are unmatched by manmade materials. A hallmark of collagen and collagenous tissues is a dramatic increase in stiffness when strained. Qualitatively, this property of strain stiffening is shared by many other biopolymers, including intracellular cytoskeletal networks of actin and intermediate filaments (1⇓⇓⇓–5). On closer inspection, however, collagen stands out from the rest: it has been shown that collagenous tissues exhibit a regime in which the stress is approximately exponential in the applied strain (6). The origins of this nonlinearity are still not known (7, 8), and existing models for biopolymer networks cannot account quantitatively for collagen. In particular, it is unknown whether the nonlinear mechanical response of collagen originates at the level of the individual fibers (1, 3, 9, 10) or arises from nonaffine network deformations as suggested by numerical simulations (11⇓⇓⇓⇓⇓–17).

Here, we present both experimental results on reconstituted collagen networks, as well as a model that quantitatively captures the observed nonlinear mechanics. Our model is a minimal one, of random networks of elastic fibers possessing only bending and stretching elasticity. This model can account for our striking experimental observation that the stiffness of collagen becomes independent of protein concentration in the nonlinear elastic regime, over a range of concentrations and applied shear stress. Our model highlights the importance of local network geometry in determining the strain threshold for the onset of nonlinear mechanics, which can account for the concentration independence of this threshold that is observed for collagen (8, 17), in strong contrast to other biopolymer networks. Finally, our model points to the important role of normal stresses in determining the nonlinear shear elastic response, including the approximate exponential relationship between stress and strain reported for collagenous tissues (6).

## Results and Discussion

In contrast to most synthetic polymer materials, biopolymer gels are known to exhibit a strong stiffening response to applied shear stress, in some cases leading to a more than 100-fold increase in the shear modulus, at strains as low as 10% or less, before network failure (1, 3⇓–5). Here, we perform rheology experiments on reconstituted networks of collagen type I, a key component of many tissues. We measure the differential shear modulus *σ* to the strain *γ*. We plot this in Fig. 1*A* as a function of the applied stress. At low stress (and strain), we observe a linear elastic response with *K* above a threshold stress that increases with concentration. Remarkably, for network concentrations ranging from 0.45 to 3.6 mg/mL, the modulus becomes insensitive to concentration in the nonlinear regime, where *K* increases approximately linearly with *σ*: Here, for a given sample preparation (e.g., polymerization temperature), the various *K* vs. *σ* curves overlap, despite the fact that the linear moduli of these samples vary by two orders of magnitude.

Moreover, the approximate linear dependence of *K* on *σ* in our reconstituted networks is consistent with the empirically established exponential dependence of stress on strain in collagenous tissues (6), because *K* on *σ* and the insensitivity of the nonlinear stiffening to network concentration appear to be unique to collagen.

### Physical Picture.

Although surprising at first sight, the features seen in Fig. 1*A* can be understood in simple physical terms for athermal networks of fibers that are soft to bending and where the nonlinear network response is controlled by stress. At low stress, if the elastic energy is dominated by soft bending modes, the linear shear modulus *G* should be proportional to the fiber bending rigidity *κ*. Of course, *G* also depends on the density of collagen, as can be seen in Fig. 1*A*. The concentration can be characterized in geometric terms by *ρ*, the total length of fiber per unit volume. Because *κ* has units of energy *G* has units of energy per volume, we expect that *G*, similar arguments apply to the characteristic stress *σ* becomes large enough to dominate the initial stability due to fiber bending resistance, it is expected that *K* will increase proportional to *σ*, in a way analogous to the linear dependence of magnetization on field in a paramagnetic phase. Combining these observations, one obtains an approximate stiffening given by *G* and *K* that becomes insensitive to concentration. Interestingly, this behavior is neither expected nor observed for F-actin and intermediate filament networks, which are not bend dominated and exhibit a stronger nonlinear stiffening regime, in which

### Model.

To test this simple physical picture, as well as uncover the mechanisms of collagen elasticity in more detail, we study simple/minimal computational models of fiber networks, specifically, 2D and 3D lattice-based networks (26⇓–28) and 2D Mikado networks (11, 16, 29). It is known that the mechanical stability and rigidity of networks depends on their connectivity, which can be characterized by the coordination number *z*, defined by the number of fiber segments meeting at a junction. Prior imaging of collagen networks (30) report an average connectivity *z* from its critical value (22, 25). Thus, we generate our networks within a range of *z*, straddling the experimentally relevant values. Specifically, our 2D and 3D lattice-based networks are created with *G* and the strain threshold for the onset of nonlinear elasticity depend on *z*, although the overall form of the nonlinear regime is unaffected.

In our model, as in our experiments, we impose a volume-preserving simple shear strain *γ* and minimize the total elastic energy *Supporting Information*). The network stiffness *K* is calculated as follows:*V* is the volume of the system. Because *K* depends on the energy per unit volume, and the energy involves an integral along the contour of all fibers in the system, *K* is naturally proportional to the total length of fiber per volume, *ρ*, which is proportional to the protein concentration *c*. Thus, *K* can be expressed as follows (*Supporting Information*):*μ* is the fiber stretching modulus and *σ* can be expressed in a similar fashion as *γ* and

In our simulations, we determine both *K* and *σ* for various values of *κ*. We do this for networks with *d* dimensions. We plot *K* vs. *σ* in Fig. 1*B*. For an elastic rod of diameter *E*, the parameter *ϕ*, because

Consistent with our experiments, our model networks also show an approximately linear relationship between stiffness *K* and shear stress *σ*, as shown in Fig. 1*B* (26). We also study networks under extension, for which our model predicts a linear relationship between the stiffness and extensional stress, as shown in the *Inset* to Fig. 1*B*. Thus, our model can also account for prior experiments on collagenous tissues, which report such a linear relationship (6). Moreover, both experiments and theory show a very surprising result in the stiffening regime, where the *K* vs. *σ* curves for different networks are seen to cluster around a common line, and where networks of varying protein concentrations exhibit the same stiffness at a given level of applied shear stress; i.e., the network stiffness *K* becomes independent of network concentration and appears to be governed only by the applied stress in the nonlinear regime.

For low stress, the linear regime is indicated by a constant stiffness *G* with collagen concentration in the experiments (*Supporting Information*), as well as with prior reports showing an approximate quadratic dependence of *G* on concentration (8, 17). Moreover, to test whether for a given concentration *G* increases with *κ*, we show data with glutaraldehyde (GA) cross-linkers, which increases the bending rigidity of collagen fibers (32) (Fig. 1*A*). Not only are these results consistent with the predicted increase in *G*, but the *K* vs. *σ* curve still collapse onto the corresponding data for non-GA cross-linked networks in the stiffening regime. Thus, our model can account for the features observed in the experiments. For a more direct comparison, we plot theoretical and experimental stiffening curves together in Fig. 1*C*. Moreover, both 2D and 3D results exhibit similar behavior, suggesting that stiffening is independent of dimensionality for a given local network geometry (Fig. 1*B*).

In the nonlinear regime, the observed independence of *κ*. Fig. 2*A* shows that *ρ* and *κ*, throughout the range *z*, as well as the type of the network, i.e., whether lattice-based or Mikado. As we show in *Supporting Information*, in the strongly bending-dominated limit, our model predicts a simple scaling dependence of *L* is the average length of the fibers. In general, *z*. For a given network type, lattice-based or Mikado, this aspect ratio is an entirely equivalent measure of connectivity to *z*: there is a one-to-one relationship between these two quantities, which increase (decrease) together. By construction, our networks have local coordination numbers strictly less than 4, which also represents the physiological upper bound of two fibers crossing at a cross-link. As the aspect ratio *z* decreases toward 3 (a branched structure), the aspect ratio decreases toward unity. Thus, stiffening in our model networks is controlled by geometry, specifically via the aspect ratio *z*.

Collagen is known to form branched network structures (30, 33) (Fig. 1*D*), whose pore size scales as *B*). Although this is consistent with prior experiments on collagen (8, 17), it is in strong contrast to reports for other biopolymer networks (1, 4, 19, 35).

### Role of Local Network Geometry.

We can now understand quantitatively the features in our experiments based on three key assumptions: (*i*) the networks are athermal, (*ii*) the networks are bend dominated, and (*iii*) their geometry at different concentrations is self-similar, i.e., the network structures at different concentrations are scale-invariant in that they are characterized by the same (aspect) ratio *E*). Changing the local geometry, and specifically *A*). Moreover, less branched networks show a lower *B*). This is consistent with simulation results when comparing Mikado with lattice-based networks (Figs. 1*B* and 2*A*). To confirm that this is due to network geometry, and not to the temperature at which the rheology measurements are performed, we polymerize a network at 37 °C and subsequently cool it to 25 °C. We then perform rheology measurements at 25 °C and find that, despite its larger linear modulus, the stiffening regime coincides with networks polymerized at 37 °C, demonstrating that network geometry, indeed, sets the prefactor *Supporting Information*).

To understand the stiffening mechanism, we first examine which of the two modes, stretching or bending, dominates the stiffening regime. Prior work has suggested that stiffening corresponds to a transition from bending- to stretching-dominated behavior (12). Our simulations show that bending is dominant throughout the stiffening regime (Fig. 2*C* and *Supporting Information*). When stretching modes finally become dominant, all *γ* curves converge, as shown in Fig. 2*C*. In most cases, this only occurs after the network stiffness has increased by more than an order of magnitude. Moreover, when stretching dominates, we find a distinct stiffening behavior characterized by *Supporting Information*) (26). Thus, we find three distinct rheological regimes: (*i*) a linear elastic regime, (*ii*) a bend-dominated stiffening regime, and (*iii*) a stretch-dominated stiffening regime. Interestingly, the approximate

The existence of a distinct bend-dominated nonlinear regime and the corresponding concentration-independent nonlinear response in Fig. 1 *A* and *B* depends crucially on the subisostatic nature of the networks, as well as on small values *ϕ* in experiments. The collagen networks we study here are, indeed, all subisostatic with respect to stretching alone (24, 31), because *B* and *Supporting Information*), which includes reported collagen network structures, we consistently see effects of bend-dominated network response in our model, including the concentration-independent nonlinear behavior. Here, *κ* acts as a stabilizing interaction or field for networks in their linear elastic regime, with

### Normal Stresses.

Biopolymer networks, including collagen, have been shown to develop large negative normal stresses (29, 40, 41). This is in contrast to most elastic materials that exhibit positive normal stresses, as first demonstrated by Poynting (42), who showed elongation of wires under torsion. Biopolymer gels have been shown to do the opposite. In experiments, the constraint normal to the sample boundaries leads to the buildup of tensile stress at these boundaries when simple shear is imposed. These normal stresses can stabilize submarginal networks. In Fig. 3*A*, we show that the linear modulus grows in direct proportion to an applied normal stress. We hypothesize that the network stiffness could arise from the normal stresses that develop under shear strain due to the imposed constraint at the boundaries:*B*, we show a direct comparison of *K* and *σ*, where *K* and *A* and *B*. Finally, we confirm our hypothesis by performing further relaxation of the networks when the normal stresses are released. In the *Lower Inset* of Fig. 3*B*, by removing the normal stresses, we observe a significant reduction of the stiffness throughout the stiffening regime. This clearly supports the hypothesis that normal stresses control the stiffening of fiber networks under simple shear. Moreover, upon closer examination of the model predictions, we see a small but systematic evolution of the stiffening exponent *α* with *Upper Left Inset* of Fig. 3*B*. A similar evolution is seen in our experiments as a function of concentration, as shown in the right panel of the *Upper Inset* of Fig. 3*B*. This agreement between experiment and model further justifies our identification of

## Concluding Remarks

The development of normal stresses in these networks is intimately related to the volume-preserving nature of simple shear deformations, both in our rheology experiments and in our simulations. In our model, removal of the normal stress leads to a reduction in the volume of the system and a concomitant reduction in the stiffness. Although collagenous tissues in vivo are subject to more complex deformations, approximate volume conservation is valid in many cases, e.g., due to embedded cells (6). Our results suggest that any volume-preserving deformation should lead to similar behavior in stiffness vs. stress. In particular, in addition to accounting for the approximate exponential stress–strain relationship known empirically for collagen under extension (6), our model also predicts that the nonlinear (Young’s) modulus should become concentration independent for a given extensional stress, in a way similar to the case of simple shear. This can be seen in the *Inset* to Fig. 1*B* and *Supporting Information*.

The concentration independence and collapse of the stress–stiffness curves seen in Fig. 1*A* appears to be unique to collagen networks, at least among biopolymers. Within our model, this property depends on three aspects: (*i*) the athermal and simple elastic response of the constituent fibers, (*ii*) the bend-dominated response of the network in its linear elastic regime, and (*iii*) the linear scaling of stiffness with stress, given by *iii*) is strongly violated: for actin and intermediate filament networks, a stronger stiffening, with *A* and *B*), is that the local deformation of such matrices is expected to become nearly uniform in the nonlinear elastic regime, even in the presence of large local inhomogeneities in network density. This can have a stabilizing effect under excessive mechanical loading. The present work has identified the key properties that can form the basis for design of biomimetic networks with similar nonlinear properties to collagen.

The importance of the nonlinear stiffness of collagen matrices comes in part from the inherent stability that such stiffening can impart to whole tissues: if collagen network elasticity were linear, then such networks would either fail or have to strain by more than 200–300% under the maximum stresses in our experiments. Moreover, an initial soft elastic response of collagen also seems to be important physiologically: high stiffness due to excessive collagen production, e.g., during fibrosis, scar formation, or around tumors, is known to contribute strongly to pathological processes at the cellular level, where it can drive the so-called epithelial-to-mesenchymal transition and affect cell differentiation. Thus, both a soft initial linear response, as well as a strongly nonlinear stiffening regime of collagen matrices are important for individual cells and tissues. Apart from tissues with high content collagen, such as tendon and skin, most soft tissues have collagen content in the range of tenths of percent to a few percent, which corresponds to a range from our densest reconstituted networks up to about a decade higher in concentration (43). In such tissues, we expect nonlinear effects such as we report here to appear at shear stresses of order kilopascal, which is a level of stress easily reached, for instance, by traction forces of fibroblasts (44). Thus, we expect the kind of stiffening reported here to be relevant to many soft tissues. Although collagen networks have been known to exhibit nonlinear mechanics that is qualitatively similar to other biopolymer networks, it has become increasingly clear that the underlying mechanism of collagen stiffening differs from that of other biopolymers (8). Not only does the present work shed light on the origins of collagen matrix mechanics, but it can also form a basis for the design of synthetic networks to mimic collagen and other extracellular matrices for tissue engineering.

## Materials and Methods

### Polymerization of Collagen Networks.

We dilute type I collagen (BD Biosciences) at 4 °C to the desired final concentrations of between 0.45 and 3.6 mg/mL in 1× DMEM (Sigma Aldrich) with 25 mM Hepes added and adjust the pH to 9.5 by addition of 1 M NaOH. We fill the solution into the rheometer geometry preheated to 25 or 37 °C as indicated and allow for at least 2 h of polymerization. To stiffen some samples, we pipette a solution of 0.2% glutaraldehyde (Sigma) in 1× PBS (Lonza) around the rheometer geometry once the networks have polymerized for 45 min and incubate these samples for 3 h before performing experiments.

### Rheometry and Data Analysis.

We perform the experiments on an AR-G2 rheometer (set to strain-controlled mode) or an ARES-G2 strain-controlled rheometer (both TA Instruments) both fitted with a 25-mm poly(methyl methacrylate) disk as top plate and a 35-mm Petri dish as bottom plate and set a gap of 400 μm. We prevent evaporation by sealing the samples with mineral oil, except for experiments on cross-linked collagen; here, we use a custom-built solvent trap, which allows for the addition of the crosslinking solution. We monitor the polymerization of all samples by continuous oscillations with a strain amplitude of 0.005 at a frequency of 1 rad/s. Subsequently, we impose a strain ramp with a rate of 0.01/s and measure the resulting stresses. We fit each stress–strain data set with a cubic spline interpolation and calculate its local derivative, which we then plot vs. stress.

### Generation of Disordered Phantom Networks.

We take a *d*-dimensions, the lattice occupies a volume *p* is the probability of an existing bond. After dilution, fibers that span the system size may still be present and these could lead to unphysical contributions on the macroscopic network stiffness. To avoid this, we make sure that every fiber has at least one diluted bond. All remaining dangling ends are further removed. Finally, nodes are introduced at the midpoint of every lattice bond so that the first bending mode on each bond is represented. The procedure just described effectively reduces both the average connectivity to

### Generation of Mikado Networks.

We generate these networks (29) by random deposition of monodisperse fibers in the form of rods of length *φ* relative to a fixed axis are each drawn from a uniform distribution. The box has periodic boundaries such that, if a rod intersects any side of the box, it crosses over to the opposite side. A cross-link is assigned to the point wherever a given pair of rods intersect. Every time a rod is deposited, the cross-linking density

### Discrete Extensible Wormlike Chain Model.

The internal degrees of freedom in the network is the set of spatial coordinates *μ* and bending rigidity *κ*. When the network is deformed, the nodes undergo a displacement *i* and *j* along a fiber is given by

The bending of a fiber segment involves a triplet of consecutive nodes *j* is estimated as

where

### Rheology Simulation.

We simulate rheology on the networks by imposing an affine simple shear strain *γ*. We fix the fiber stretching modulus *κ* to probe a range of bending rigidities. We steadily increase the strain in *σ* and differential shear modulus *K*:

## Dimensionless Shear Modulus and Bending Rigidity

For a homogeneous elastic rod (47) of radius *a* and Young’s modulus *E*, the stretching modulus *V* and composed of *N* fibers, this can be evaluated as a sum of the elastic energies of all semiflexible fibers *f*:*s* on the fiber with unit tangent *n* segments *n*. Thus, because *ρ* is the network concentration in total fiber length per volume. Differentiating with respect to *γ* yields the shear stress

The concentration *ρ* is also related to the fiber rigidity

## Geometric Dependence of the Critical Strain

The schematic in Fig. S1 shows two fiber strands *L*. Each fiber undergoes a backbone relaxation

We emphasize that Eq. **S1** applies to the asymptotic bend-dominated limit. This we observe in Fig. S2 for

## Acknowledgments

A.J.L., A.S., M.S., and F.C.M. were supported in part by Stichting voor Fundamenteel Onderzoek der Materie/Nederlandse Organisatie voor Wetenschappelijk Onderzoek. S.M. and B.F. were supported by Deutsche Forschungsgemeinschaft. This work was supported in part by the National Science Foundation (DMR-1006546) and the Harvard Materials Research Science and Engineering Center (DMR-0820484).

## Footnotes

↵

^{1}A.J.L., S.M., and A.S. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: fcmack{at}gmail.com.

Author contributions: A.J.L., S.M., A.S., L.M.J., B.F., D.A.W., and F.C.M. designed research; A.J.L., S.M., A.S., M.S., and L.M.J. performed research; A.J.L., S.M., and A.S. analyzed data; and A.J.L., S.M., A.S., M.S., L.M.J., B.F., D.A.W., and F.C.M. 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.1504258112/-/DCSupplemental.

Freely available online through the PNAS open access option.

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