Mechanical Response of Active Gels

We study a model of an active gel of cross-linked semiflexible filaments with additional active linkers such as myosin II clusters. We show that the coupling of the elasticity of the semiflexible filaments to the mechanical properties of the motors leads to contractile behavior of the gel, in qualitative agreement with experimental observations. The motors, however, soften the zero frequency elastic constant of the gel. When the collective motor dynamics is incorporated in the model, a stiffening of the network at high frequencies is obtained. The frequency controlling the crossover between low and high frequency network elasticity is estimated in terms of microscopic properties of motors and filaments, and can be as low as 10Hz. Introduction. – The mechanical properties of cells control many biological functions, including the sensing and generation of forces, cell motility and cell division. The response of the cell to mechanical stimuli is mediated by the cytoskeleton, a network of semiflexible filaments (F-actin, microtubules and intermediate filaments) linked by a variety of passive and active proteins. [1, 2] The cytoskeleton is maintained out of equilibrium by chemical reactions that drive force generation by motor proteins, as well as by filament treadmilling. A variety of recent experiments have measured the remarkable rheological properties of this intrinsically nonequilibrium polymer network. These include bulk and microrheology of in vitro stabilized networks of cytoskeletal filaments with a controlled concentration of various crosslinkers, as well as in vivo whole cell rheology. Cross-linked entangled actin networks are viscoelastic solids, with a time-dependent mechanical response (stress σ) to deformation (strain γ). These networks have both viscous and elastic responses characterized by loss G(ω) ∼ σ/γ̇ and storage moduli G(ω) ∼ σ/γ, respectively. For cross-linked gels, the elastic (storage) modulus dominates the mechanical response and reaches a frequency independent plateau G0 at low frequencies (less than 1Hz). ExperimentallyG0 is found to depend strongly on cross-link density and can vary from 0.1 100 Pa [3]. For frequencies above 1Hz, both the storage and loss moduli show a high frequency behavior G, G ∼ ω characteristic of semiflexible polymer dynamics [4]. Measurements of the mechanical properties of cells yield, however, quite different behaviour [7]. The low frequency (< 10Hz) shear moduli are observed to behave as, G, G” ∼ G∗(ω/ω), with a small exponent α ∼ 0.15− 0.2, G∗ ∼ 10 − 10 Pa and ω ∼ 1 Hz [8–13]. Significantly, the magnitude of G∗ is much higher than the typical plateau moduli of purified in-vitro actin gels. While increasing cross-linker density can significantly enhance the elastic modulus [3], it is surprising that it would have such a dramatic effect on the loss modulus. It was recently suggested that the remarkable stiffening of the low frequency linear response of active gels may be due to the internal stresses generated by the presence of active crosslinkers, such as myosin II minifilaments [14, 15]. Recent quantitative experiments studying the mechanics of in-vitro networks of F-actin, with passive (α−actinin) and active (muscle myosin II) cross-linkers, have shown both stiffening [14] and contractile behaviour [17] of these reconstituted networks. Interestingly the contractile behaviour has been shown to appear only in a narrow concentration range of passive cross-linkers.