Nonlinear Elasticity of Composite Networks of Stiff Biopolymers with Flexible Linkers

Abstract

Motivated by recent experiments showing nonlinear elasticity of in vitro networks of the biopolymer actin cross-linked with filamin, we present an effective medium theory of flexibly cross-linked stiff polymer networks. We model such networks by randomly oriented elastic rods connected by flexible connectors to a surrounding elastic continuum, which self-consistently represents the behavior of the rest of the network. This model yields a cross-over from a linear elastic regime to a highly nonlinear elastic regime that stiffens in a way quantitatively consistent with experiment.PACS numbers: 87.16. Ka, 87.15. La, 82.35.Pq The mechanical response of living cells depends largely on their cytoskeleton, a network of stiff protein polymers such as F-actin, along with various associated proteins for cross-linking and force generation. In addition to their importance for cell mechanics, cytoskeletal networks have also demonstrated novel elastic properties, especially in numerous in vitro studies [1,2,3,4,5,6]. The cellular cytoskeleton, however, is an inherently composite structure, consisting of elements with highly varied mechanical properties, and there have been few theoretical or experimental studies of this aspect [8,9,10,11,12]. Recent experiments on F-actin with the physiological crosslinker filamin have demonstrated several striking features while their linear modulus is significantly lower than for rigidly cross-linked actin systems, they can nonetheless withstand remarkably large stresses and can stiffen by a factor of 1000 with applied shear [8,10,13]. This behavior appears to result from the highly flexible nature of filamin, although the basic physics of such a network, in which the elasticity is dominated by cross-linkers, is not understood. Apart from their physiological importance, such networks suggest new principles that may be extended to new synthetic materials with designed crosslinks [9].Here, we develop a theoretical model for composite networks of rigid filaments connected by flexible crosslinkers, in which the macroscopic network elasticity is governed by the cross-links. We examine this model in a limit in which the basic elastic element is a single rigid rod, directly linked by numerous compliant crosslinkers to a surrounding linear elastic medium. We show that such a network stiffens in a manner determined by the mechanics of individual cross-links, which we model both as linear springs with finite extension, and also as wormlike chains. We analyze our model in both a fully 3D network, as well as a simplified 1D representation, which already captures the essential physics of the nonlinear behavior. The finite extension ℓ 0 of the cross-links along with the length of the filaments/rods L implies that there exists a characteristic strain γ c ∼ ℓ 0 /L for the onset of the nonlinear response of the network. Indeed prior in vitro experiments, in which the length of the cross-linkers was varied [9], have reported this linear dependence on ℓ 0 . We extend this model in a fully selfconsistent man...

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