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Knut M.
Heidemann
^{a},
Abhinav
Sharma
^{b},
Florian
Rehfeldt
^{b},
Christoph F.
Schmidt
*^{b} and
Max
Wardetzky
*^{a}
^{a}Institute for Numerical and Applied Mathematics, Georg-August-Universität, Göttingen, Germany. E-mail: wardetzky@math.uni-goettingen.de
^{b}Third Institute of Physics—Biophysics, Georg-August-Universität, Göttingen, Germany. E-mail: cfs@physik3.gwdg.de

Received
12th August 2014
, Accepted 15th October 2014

First published on 15th October 2014

Disordered filamentous networks with compliant crosslinks exhibit a low linear elastic shear modulus at small strains, but stiffen dramatically at high strains. Experiments have shown that the elastic modulus can increase by up to three orders of magnitude while the networks withstand relatively large stresses without rupturing. Here, we perform an analytical and numerical study on model networks in three dimensions. Our model consists of a collection of randomly oriented rigid filaments connected by flexible crosslinks that are modeled as wormlike chains. Due to zero probability of filament intersection in three dimensions, our model networks are by construction prestressed in terms of initial tension in the crosslinks. We demonstrate how the linear elastic modulus can be related to the prestress in these networks. Under the assumption of affine deformations in the limit of infinite crosslink density, we show analytically that the nonlinear elastic regime in 1- and 2-dimensional networks is characterized by power-law scaling of the elastic modulus with the stress. In contrast, 3-dimensional networks show an exponential dependence of the modulus on stress. Independent of dimensionality, if the crosslink density is finite, we show that the only persistent scaling exponent is that of the single wormlike chain. We further show that there is no qualitative change in the stiffening behavior of filamentous networks even if the filaments are bending-compliant. Consequently, unlike suggested in prior work, the model system studied here cannot provide an explanation for the experimentally observed linear scaling of the modulus with the stress in filamentous networks.

Here we analyze 3-dimensional (3D) composite networks theoretically, and we offer physical simulations thereof. Our networks are composed of randomly oriented rigid filaments that are connected by highly flexible crosslinks, each of which is modeled as a wormlike chain (WLC),^{22,23} which has been shown to accurately describe flexible crosslinkers, such as filamin.^{24,25} In our approach we assume that the filaments are much more rigid than the crosslinks, meaning that the network elasticity is dominated by the entropic stretching resistance of the crosslinks.

In our theoretical analysis we adopt the widely employed assumption of affine deformations.^{16,19,26} Under this premise, the network is assumed to deform affinely on the length scale of the filaments, which in turn is assumed to be much longer than the contour length of the crosslinks. Using a single filament description in the limit of a continuous distribution of crosslinks along the filament, we obtain the asymptotic scaling behavior of the elastic modulus with the stress in the nonlinear regime. We show that in 1-dimensional (1D) networks, the elastic modulus scales with the second power of the stress, whereas it scales with the third power in 2-dimensional (2D) networks. Remarkably, there is no power law scaling in 3D—in fact, the elastic modulus of a 3D composite network increases exponentially with the stress. Numerical evaluation of the affine theory at finite crosslink densities—as opposed to a continuous distribution of crosslinks—shows that (i) the only asymptotic scaling is the modulus scaling with an exponent 3/2 with the stress and that (ii) the dependence on dimensionality of the system is limited to an intermediate-stress regime. These findings are in agreement with our extensive physical simulations of 3D composite networks. For all cases, the elastic modulus diverges at a finite strain.

Our theoretical analysis is inspired by the mean-field model proposed by Broedersz et al.^{16,26} In sharp contrast to our theoretical analysis and to the results of our physical simulations, however, these authors predict linear scaling of the elastic modulus with applied stress. In particular, for any finite strain, the elastic modulus remains finite in their model. While this linear scaling of the elastic modulus is in accordance with what has been observed experimentally,^{13,20,21} we here argue that this model does not adequately capture the elastic response of networks with rigid filaments and permanent (i.e., non rupturing or rebinding) crosslinks of finite length.

In ref. 19, the authors ruled out that the experimentally observed approximately linear scaling of the modulus with the stress might be due to enthalpic (linear) stretching compliance of the crosslinks or filaments. Here, we complement their analysis by physical simulations that take into account bending of filaments. Our results empirically show that the inclusion of bending rigidity does not impact the nonlinear stiffening behavior of composite networks either. We therefore conclude that the theoretical explanation for the linear scaling of the modulus with stress in experiments remains a challenging open problem.

By physical simulations, we also study the role of prestress. We show that in contrast to 1D and 2D networks, 3D networks experience an initial tension due to non-intersecting filaments resulting in initially stretched crosslinks, and are therefore prestressed. The modulus in the linear deformation regime is then governed by this prestress; indeed, it is higher than the modulus expected from the affine theory. Our simulations additionally indicate that if the network is allowed to relax initial tension by unbinding and rebinding of crosslinks, the impact of prestress on the elastic modulus in the linear regime becomes insignificant, although the prestress does not relax all the way to zero.

The remainder of the article is organized as follows. First, we present the affine theory of composite networks in Section 2. Under the assumption that deformations of the network are affine on the length scale of the filaments, we derive expressions for the stress and modulus in 1D, 2D, and 3D. We then present our physical simulation model and describe our network generation procedure in Section 3. We expand on the implications of our 3D simulation procedure in Section 4; in particular, we explain the emergence of prestress. We then discuss the results of our simulations in the linear deformation regime in Section 5 and indicate which results from the affine theory are still valid. Finally, we analyze the simulation results in the nonlinear regime in Section 6.

(1) |

(2) |

Imposing affine deformations on the filament level fully determines the deformation field u on the subfilament level. In the following analysis, we consider a single representative filament subject to an extensional strain of the surrounding medium that it is embedded in and crosslinked to.

In the rest frame of the filament, the end-to-end distance of a crosslink at distance x from the center of the filament is given by |u(x,ε)| = |εx| (see Fig. 1(a)). For notational convenience, we consider positive ε only. Under the assumption that the crosslink density is high enough that one can consider the associated distribution as uniformly continuous, the total energy of a filament in 1D is given by

(3) |

Following the described approach for the linear regime of the WLC force–extension relation, i.e., for u ≪ l_{0}, the linear modulus may be extracted as , where E/V is the energy per unit volume V in the network and ε is a small strain.^{27} For a 1D system this yields , with being the linear spring constant of a crosslink and ρ: = NL/V the total length of filaments per unit volume. The same holds for the modulus in 2D and 3D, but with different numerical prefactors: 1/96 and 1/192, respectively.^{16,19,26}

Next we show that one can extract a functional relation between the nonlinear modulus and stress in the nonlinear regime, based on simple asymptotic scaling analysis. It follows from above that there is a strain ε_{d}: = l_{0}/(L/2) at which the outer most crosslink (at x = L/2) reaches maximum extension. For ε → ε_{d} the energy diverges as follows

(4) |

K_{1D} ∼ (σ_{1D})^{2}. | (5) |

This scaling relation between the modulus and stress in 1D has also been derived in previous work.^{19} Next we consider scaling relations in 2D and 3D.

|ε(γ, θ)| = |(γ/2)sin2θ|, | (6) |

Substituting this expression into eqn (4) and averaging over all orientations leads to

(7) |

Differentiating eqn (7) with respect to γ and neglecting the weaker (logarithmically) diverging part of the integrand lead to an expression for the stress, as γ → γ_{d}:

(8) |

(9) |

K_{2D} ∼ (σ_{2D})^{3}. | (10) |

|ε(γ, θ, ϕ)| = |(γ/2)sin2θcosϕ|, | (11) |

(12) |

K_{3D} ∼ e^{cσ3D}, | (13) |

Finite crosslink density.
By considering the limit of infinite crosslink density, we have derived theoretical scaling relations for strain stiffening by integrating along a filament's backbone (see eqn (3)). For any real system, however, the crosslink density is finite and eqn (3) turns into a sum

where {x_{i}} are the crosslink binding sites along the filament. Fig. 2 shows numerical results for the behavior of the corresponding differential modulus K for finite n, obtained by numerical evaluation of eqn (14) and appropriate orientational averaging. Note that the asymptotic scaling behavior of K in the limit of infinite crosslink density influences a finite network's behavior in the intermediate-stress regime (see the inset of Fig. 2); however, near the critical strain, the differential modulus scales as K ∼ σ^{3/2}, i.e., like the response of a single WLC. Furthermore, for 1D and 2D systems the theoretical scaling exponents in the limit of infinite crosslink densities can (in the intermediate regime) indeed be approached by increasing n. In contrast, as shown above, in 3D the theoretically derived scaling of K is exponential in σ. Such an exponential increase is quantified by an (in principle) indefinitely increasing maximal slope with increasing n in the lnK versus lnσ plots; e.g., for n = 60 the maximal slope is 3.49, for n = 3000 it is 5.82. However, for any finite n, eventually there is always universal scaling of K ∼ σ^{3/2}, resulting from the single WLC force–extension relation, independent of the dimensionality of the system. Indeed, for any given n, the integral representation eqn (3) becomes invalid close to γ = γ_{d} due to the divergence of the WLC energy.

(14) |

The numerical results in Fig. 2 have been obtained without the small-strain approximation for the extension of the filaments. However, the redistribution of the filament orientations under shear has not been taken into account in Fig. 2. Calculations including this effect show that it may both decrease and increase the maximum intermediate slope in the lnK versus lnσ plot and shift the peak to larger stress values depending on the maximum strain γ_{d}. In any case, the asymptotic scaling regime remains unchanged.

In the next section we introduce the simulation framework that we use to study 3D networks consisting of many filaments and crosslinks, relaxing the assumption of affine deformations.

Each filament is modeled as perfectly rigid, implying that its configuration can be described by its two endpoints only, which are constraint to stay at distance L. The flexible crosslinks are modeled as a central force acting between the two binding sites. In particular, we use the WLC interpolation formula (eqn (1)) and the corresponding energy (eqn (2)). In all data that are presented, forces are measured in units of (k_{B}T)/l_{p}. There are no additional degrees of freedom introduced through the crosslinks, since their configuration is represented via the endpoints of the filaments, in terms of barycentric coordinates.

In order to generate an initial network configuration we proceed as follows. We generate N randomly distributed filaments by first randomly choosing their centers of mass in our simulation box and by then picking a random orientation for each filament. For crosslinking we apply the following iterative procedure. We randomly choose two points on distinct filaments and insert a crosslink if the corresponding point-to-point distance is shorter than a certain threshold αl_{0}. Here α ∈ [0,1) serves as a parameter to vary the initially allowed crosslink lengths in the system. This procedure is repeated until the desired number of crosslinks is reached; see Fig. 3 for an illustration of the final configuration. Since we perform quasistatic simulations, the system must be at static equilibrium at all times. As practically all crosslinks will be stretched beyond their rest-length after the initial network generation, we minimize the energy (of the crosslinks) before subjecting the simulation box to any deformation.‡ For energy minimization we use the freely available external library IPOPT,^{28} which requires the gradient and the Hessian of the system's energy function. It might happen during the optimization process that individual crosslinks reach extensions u larger than their contour length l_{0}. Acceptance of these solutions is prohibited by setting the energy to infinity (10^{19}) for u ≥ l_{0} in eqn (2); without this modification it would become negative in that regime. The length constraints for the filaments are realized via Lagrange multipliers.

In order to extract elastic properties of the network we perform quasistatic shearing by applying an affine incremental shear strain δγ to the network, with subsequent rescaling of filaments to length L (see Fig. 1). We apply Lees–Edwards shearing periodic boundary conditions.^{29} The magnitude of δγ is determined by calculating the maximum affine shear that leaves all crosslinks below their contour length. Due to the rescaling of filament lengths, a nonaffine deformation component is introduced. This nonaffinity may lead to crosslinks being overstretched after all. In this case, we iteratively halve the shear strain until the length of all crosslinks remains below their contour length. After each shear increment, the energy is minimized. We apply a fixed upper bound of 1% strain on δγ in order to stay reasonably close to the previous solution. This increases numerical efficiency and accelerates convergence because it allows us to use a warm-start procedure that reuses Lagrange multipliers from one minimization as initial guesses for the next one. Moreover, the application of small shear steps reduces the likelihood of discontinuously jumping between local energy minima.

We stop shearing when the achievable increment in shear strain becomes smaller than a chosen threshold due to crosslinks that are very close to their maximum extension. During the entire simulation process, we record network parameters in the equilibrated states—in particular, the energy E as a function of shear strain γ. This allows us to extract the shear stress as well as the differential shear elastic modulus . Derivatives are taken by first interpolating E(γ) with a cubic spline. We define the linear shear elastic modulus as

G_{0} ≔ K|_{γ=0}. | (15) |

In the following section we discuss the implications of our specific simulation model, in particular with respect to the network structure, and contrast it with previous studies that have been carried out mostly in 2D.

In contrast, the initial stretching of crosslinks in our networks results in an initial tension before any deformation. For a quantitative analysis we measure a global variant of this effect by what we call total prestress σ_{0}, which measures the normal stress§ component orthogonal to the shear planes.¶ More precisely, we measure the single sided (e.g., upward) normal component of the force that is acting on a given shear plane, by summing up the normal components of the forces exerted by each crosslink and filament passing through the given shear plane, see Fig. 4(a). The normal stress is then given by dividing by the surface area of the shear plane. Note that σ_{0} does not depend on the choice of a particular shear plane; indeed, if the total stress was changing during vertical movement of a shear plane, then this would immediately contradict force balance in the system.

Fig. 4 (a) Measuring the total prestress σ_{0} by extracting the normal component of the total force acting on a shear plane. We sum up all the forces acting on one side of the plane exerted by (i) the crosslinks passing through (here f_{2} and f_{4}) and (ii) the filaments passing through (here f_{1} + f_{3})—then we project onto the normal vector n. (b) A tensegrity structure (here: Snelson's X^{33}) remains in static equilibrium without the application of boundary conditions. The forces acting on any plane add up to zero, i.e. no plane carries any total prestress although it is under tension locally. |

Intuitively, one might expect negative normal stresses (pulling down on the upper face of the simulation box), since crosslinks are contractile. However, since filaments withstand compression, it is possible to construct systems that exhibit positive normal stress. This suggests the existence of configurations with zero normal stress.|| Indeed, so-called tensegrity structures,^{34} which are in static equilibrium in the absence of boundary conditions satisfy this criterion—while still being able to store arbitrary amounts of energy (see Fig. 4(b)). Empirically, our simulations show that the random networks generated by the procedure described in Section 3 exhibit negative initial normal stresses throughout. Their integrity is provided through the application of periodic boundary conditions. Note, in particular, that our setup enforces conservation of the volume of the simulation box. In general, it would be possible to relax the prestress by letting the volume of the simulation box change. However, we did not follow this approach in the study presented here, in order to ensure that the filament length remains significantly smaller than the size of the simulation box.

In the following, we relate total prestress to the linear elastic response of our networks.

One scenario that clearly demonstrates the dependence of the linear modulus G_{0} (defined in eqn (15)) on the initial tension is shown in Fig. 5 where the admissible maximum initial crosslink length was varied.

For a more quantitative analysis we have designed a method that allows us to change initial tension for a network with a fixed set of simulation parameters. We first randomly generate a network as described above and let it relax into static equilibrium. We then remove a given amount (5%) of the most-stretched crosslinks in the system. Then we reconnect those crosslinks randomly again, and let the network relax. This procedure is repeated N_{rel} times. Thereby, we successively decrease the system's initial tension, and therefore also its total energy, see the inset of Fig. 6. Not only does the total energy decrease, we also observe a change in the distribution of forces (see Fig. 6). As long as one performs the crosslink binding–unbinding procedure over a small enough fraction of crosslinks, the network remains nearly isotropic.

It is apparent from the inset of Fig. 7 that the linear elastic modulus is reduced by increasing the number of relaxation steps, as expected. Fig. 7 also shows the dependence of linear modulus G_{0} on the total prestress σ_{0}, which has been introduced in Section 4. We varied σ_{0}via the above described procedure, and measured G_{0} with the shearing protocol described in Section 3. After a certain number of relaxation steps the empirical value for G_{0} equals the value G^{aff}_{0} expected from affine theory (see Section 2.1). Relaxing initial tension further, we reach moduli even below G^{aff}_{0}. This is possible because the network can rearrange nonaffinely, thereby softening its response. Over a certain range of total prestresses, we observe linear scaling of G_{0} with σ_{0}, a phenomenon, which has been discussed in other contexts before (see for example ref. 35). We explain the linear regime as follows. For small strains the normal component σ_{N} of the stress acting on shear planes is close in magnitude to the total prestress σ_{0}, i.e., σ_{N} ≈ σ_{0}. For small strains given by shear angles ϑ ≈ 0, total forces acting on the shear planes make an angle φ with the direction normal to the shear planes (see Fig. 8). Our simulations show that tanφ ∝ tanϑ and that the constant of proportionality remains unchanged in the linear scaling regime. Therefore, shear satisfies

(16) |

Fig. 8 The initial network carries a total prestress σ_{0}. After a small shear γ = tanϑ has been applied it exhibits shear stress σ_{S} and normal stress σ_{N}, with tanφ = σ_{S}/σ_{N}. |

Furthermore, affine theory predicts linear scaling of the modulus G_{0} with crosslink density n. Fig. 9 shows that this linear scaling is indeed reproduced in our simulations, independent of the prestress. Moreover, by changing the prestress via our relaxation procedure it is possible to reach comparable slopes to what is predicted by the affine theory.

The next section deals with the nonlinear elastic response of the simulated networks, and relates it to the theoretical results that were derived in Section 2.

For ranges that are accessible to our simulations, we obtain the following results. If we fix l_{0}, then we observe linear scaling γ_{c} ∝ 1/L for systems where no relaxation procedure has been applied (see Fig. 10(a)). Relaxed systems, however, sometimes show a less than linear dependence. This effect might be due to anisotropies induced by the relaxation procedure. If we fix L, then the dependence of γ_{c} on l_{0} is slightly less than linear (see Fig. 10(b)).

(17) |

We let δΓ denote the average of the differential nonaffinities over all filaments. Affine approximations imply δΓ = 0. Fig. 12 shows that center of mass deformations are mostly affine for small strains. However, the differential nonaffinity increases starting at a strain around γ_{c} and eventually diverges as γ → γ_{d}. This can be understood, since the networks are strain stiffening, such that small incremental strain can induce a large increase in the forces of individual crosslinks, thereby inducing large local rearrangements during energy minimization.

Fig. 12 Differential nonaffinity δΓ as a function of scaled shear strain γ/γ_{c} for a system with N = 3000, n = 60, L = 0.3, l_{0} = 0.06, and α = 0.05. |

While increasing shear strain, there are force chains^{36–38} developing in the network, which carry most of the tension, and which cannot reduce their strain due to the fact that they span the entire system (see the inset of Fig. 13). We quantify this effect by considering tension profiles along filaments. The tension τ at position x along a filament is given via, where {x_{i}} are the crosslink binding sites and {u_{i}} are their extensions (u_{i} = εx_{i} in affine theory). Fig. 13 shows tension profiles averaged over all filaments for both, theoretical and simulated systems at various strains. In the simulations there is non-zero tension at zero strain due to prestress. With increasing γ, the simulations resemble the profiles expected from affine theory. However, when approaching the maximum strain γ_{d}, the emergence of selective paths (force chains) that carry most of the tension becomes evident. The highly stretched crosslinks dominate the averaged tension profiles and therefore lead to jumps in the tension curves at the respective binding sites along the filament (green solid curve in Fig. 13).

Thus, in isolation, neither bending nor stretching compliance of filaments impacts the nonlinear stiffening regime of composite networks. These findings suggest that the theoretical models at present cannot explain the K ∼ σ scaling observed in experiments.

Based on the affine theory introduced in ref. 19 we derived asymptotic power law scaling exponents for the differential elastic modulus with stress, in the limit of infinite crosslink density. In this case, the scaling exponents depend on the dimensionality of the system. In particular, we showed that 3D systems no longer exhibit a power law. Furthermore, we showed that for finite crosslink densities, the only persistent regime (over several orders of magnitude of stress) is the σ^{3/2} scaling, as it is derived from the single WLC force–extension relation, eqn (1). This is in sharp contrast with the model proposed in ref. 16 and 26, where linear scaling was suggested, independent of the dimensionality of the system. This model implies finite stress at any strain and therefore does not apply to composite networks of rigid filaments with flexible crosslinks of finite length.

We further developed a simulation framework that allows us to measure the elastic response of random filamentous networks with WLC crosslinks. One important property of these 3D networks is that, by construction, they are prestressed due to initial extensions of the crosslinks. In addition to geometrical constraints, active elements such as motors can induce prestress as well.^{39} We showed that the prestress in a network can dominate the linear response and might therefore be a feature that is worthwhile analyzing in experimental systems.

Regarding the nonlinear response, we observed divergence of stress (and differential modulus) at finite strain. Close to this strain we measured power law scaling of the differential modulus with stress, with an exponent 3/2, just as expected for a single WLC. In an intermediate-stress regime we observed local exponents that span the entire range of theoretically derived values for systems of differing dimensionality. The fact that our simulation results do not always resemble the predictions of a 3D affine theory, in this intermediate regime, may be attributed to nonaffine deformations. Extracting the exact set of assumptions—such as uniform crosslink density, isotropy, or zero prestress—that are responsible for these discrepancies is left for future investigation.

Experiments (see, e.g., ref. 13, 20, and 21) have shown that in the nonlinear regime the differential modulus scales approximately linearly with the shear stress. We did not find such a regime in our simulations—neither when working with rigid filaments nor when incorporating finite bending stiffness (or enthalpic stretching as done in ref. 19). Therefore, we argue that none of the currently available theories can adequately explain the linear scaling of the differential modulus observed experimentally. It could possibly be that the WLC model does not accurately describe the elastic response of a single crosslink throughout the whole experimentally accessible regime. We speculate, however, that the linear scaling might be due to thermal fluctuations of the filaments, which have not been considered so far.

(18) |

(19) |

(20) |

(21) |

(22) |

∼ −lnδ, | (23) |

∼ −ln(1 − γ/γ_{d}), | (24) |

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## Footnotes |

† More precisely, it is a free energy, which includes both energetic (bending) and entropic terms for the crosslinks (not for the filaments). |

‡ We neither take into account fluctuations of the filaments nor excluded-volume effects. |

§ Note that our notion of prestress is not to be confused with the constant prestress externally applied in bulk rheology experiments, which is a shear stress in general. |

¶ Although we could in principle define total prestress as the normal component of the stress acting on any plane in our system we prefer to use the shear plane as this simplifies the forthcoming analysis. |

|| Note that individual crosslinks are still under tension; however, the total normal force acting on the shear plane vanishes. |

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