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ξ ≈ cA−1/ 2

Stress σ = Gθ

c

Fig. 8-5. The macroscopic shear stress σ depends on the mean tension in each filament, and on the area density of such filaments passing any plane. There are on average 1/ξ 2 such filaments per unit area. This gives rise to the factor ξ −2 in both Eqs. 8.24 and 8.25. The macroscopic response can also depend strongly on the typical distance c between cross-links, as discussed below.

In the preceding derivation we have assumed a thermal/entropic (Affine and Thermal) response of filaments to longitudinal forces. As we have seen, however, for shorter filament segments (that is, for small enough c), one may expect a mechanical response characteristic of rigid rods that can stretch and compress (with a modulus µ). This would lead to a different expression (Affine, Mechanical) for the shear modulus

GAM	µ	φ,	(8.25)
	ξ 2

which is proportional to concentration. The expectations for the various mechanical regimes is shown in Fig. 8.6 (Head et al., 2003b).

Nonlinear response

In contrast with most polymeric materials (such as gels and rubber), most biological materials, from the cells to whole tissues, stiffen as they are strained even by a few percent. This nonlinear behavior is also quite well established by in vitro studies of a wide range of biopolymers, including networks composed of F-actin, collagen, fibrin, and a variety of intermediate filaments (Janmey et al., 1994; Storm et al., 2005). In particular, these networks have been shown to exhibit approximately ten-fold

Fig. 8-6. A sketch of the expected diagram showing the various elastic regimes in terms of cross-link density and polymer concentration. The solid line represents the rigidity percolation transition where rigidity first develops from a solution at a macroscopic level. The other, dashed lines indicate crossovers (not thermodynamic transitions). NA indicates the non-affine regime, while AT and AM refer to affine thermal (or entropic) and mechanical, respectively.

crosslink concentration

AT

AM

NA

solution

polymer concentration

166 F.C. MacKintosh

Fig. 8-7. The differential modulus K = dσ/dγ describes the increase in the stress σ with strain γ in the nonlinear regime. This was measured for cross-linked actin networks by small-amplitude oscillations at low frequency, corresponding to a nearly purely elastic response, after applying a constant prestress σ0. This was measured for four different concentrations represented by the various symbols. For small prestress σ0, the differential modulus K is nearly constant, corresponding to a linear response for the network. With increasing σ0, the network stiffens, in a way consistent with theoretical predictions (MacKintosh et al., 1995; Gardel et al., 2004), as illustrated by the various theoretical curves. Specifically, it is expected that in the strongly nonlinear regime, the stiffening increases according to the straight line, corresponding to dσ/dγ σ 3/2. Data taken from Gardel et al., 2004.

stiffening under strain. Thus these materials are compliant, while being able to withstand a wide range of shear stresses.

This strain-stiffening behavior can be understood in simple terms by looking at the characteristic force-extension behavior of individual semiflexible filaments, as described above. As can be seen in Fig. 8-4, for small extensions or strains, there is a linear increase in the force. As the strain increases, however, the force is seen to grow more rapidly. In fact, in the absence of any compliance in the arc length of the filament, the force strictly diverges at a finite extension. This suggests that for a network, the macroscopic stress should diverge, while in the presence of high stress, the macroscopic shear strain is bounded and ceases to increase. In other words, after being compliant at low stress, such a material will be seen to stop responding, even under high applied stress.

This can be made more quantitative by calculating the macroscopic shear stress of a strained network, including random orientations of the constituent filaments (MacKintosh et al., 1995; Kroy and Frey, 1996; Gardel et al., 2004; Storm et al., 2005). Specifically, for a given shear strain γ , the tension in a filament segment of length c is calculated, based on the force-extension relation above. This is done within the (affine) approximation of uniform strain, in which the microscopic strain on any such filament segment is determined precisely by the macroscopic strain and the filament’s orientation with respect to the shear. The contribution of such a filament’s tension to the macroscopic stress, in other words, in a horizontal plane in Fig. 8.5, also depends on its orientation in space. Finally, the concentration or number density of such filaments crossing this horizontal plane is a function of the overall polymer concentration, and the filament orientation.

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The full nonlinear shear stress is calculated as a function of γ , the polymer concentration, and c, by adding all such contributions from all (assumed random) orientations of filaments. This can then be compared with macroscopic rheological studies of cross-linked networks, such as done recently by Gardel et al. (2004). These experiments measured the differential modulus, dσ/dγ versus applied stress σ , and found good agreement with the predicted increase in this modulus with increasing stress (Fig. 8-7). In particular, given the quadratic divergence of the single-filament tension shown above (Fixman and Kovac, 1973), this modulus is expected to increase as dσ/dγ σ 3/2, which is consistent with the experiments by Gardel et al. (2004). This provides a strong test of the underlying mechanism of network elasticity.

In addition to good agreement between theory and experiment for densely cross-linked networks, these experiments have also shown evidence of a lack of strain-stiffening behavior of these networks at lower concentrations (of polymer or cross-links), which may provide evidence for a non-affine regime of network response described above.

Discussion

Cytoskeletal filaments play key mechanical roles in the cell, either individually (for example, as paths for motor proteins) or in collective structures such as networks. The latter may involve many associated proteins for cross-linking, bundling, or coupling the cytoskeleton to other cellular structures like membranes. Our knowledge of the cytoskeleton has improved in recent years through the development of new experimental techniques, such as in visualization and micromechanical probes in living cells. At the same time, combined experimental and theoretical progress on in vitro model systems has provided fundamental insights into the possible mechanical mechanisms of cellular response.

In addition to their role in cells, cytoskeletal filaments have also proven remarkable model systems for the study of semiflexible polymers. Their size alone makes it possible to visualize individual filaments directly. They are also unique in the extreme separation of two important lengths, the persistence length p and the size of a single monomer. In the case of F-actin, p is more than a thousand times the size of a single monomer. This makes for not only quantitative but also qualitative differences from most synthetic polymers. We have seen, for instance, that the way in which semiflexible polymers entangle is very different. This makes for a surprising variation of the stiffness of these networks with only changes in the density of cross-links, even at the same concentration.

In spite of the molecular complexity of filamentous proteins as compared with conventional polymers, a quantitative understanding of the properties of single filaments provides a quantitative basis for modeling solutions and networks of filaments. In fact, the macroscopic response of cytoskeletal networks quite directly reflects, for example, the underlying dynamics of an individual semiflexible chain fluctuating in its Brownian environment. This can be seen, for instance in the measured dynamics of microtubules in cells (Caspi et al., 2000).

In developing our current understanding of cytoskeletal networks, a crucial role has been played by in vitro model systems, such as the one in Fig. 8-1. Major challenges,

168F.C. MacKintosh

however, remain for understanding the cytoskeleton of living cells. In the cell, the cytoskeleton is hardly a passive network. Among other differences from the model systems studied to date is the presence of active contractile or force-generating elements such as motors that work in concert with filamentous proteins. Nevertheless, in vitro models may soon permit a systematic and quantitative study of various actinassociated proteins for cross-linking and bundling (Gardel et al., 2004), and even contractile elements such as motors.

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