Generic phases of cross-linked active gels: Relaxation, Oscillation and Contractility

We study analytically and numerically a generic continuum model of an isotropic active solid with internal stresses generated by non-equilibrium ‘active’ mechano-chemical reactions. Our analysis shows that the gel can be tuned through three classes of dynamical states by increasing motor activity: a constant unstrained state of homogeneous density, a state where the local density exhibits sustained oscillations, and a steady-state which is spontaneously contracted, with a uniform mean density. Introduction. – The mechanical properties of living cells are largely controlled by a variety of filament-motor networks optimized for diverse physiological processes [1]. In the presence of ATP, such networks are capable of generating controlled contractile forces and spontaneous oscillations. These phenomena arise from the presence of groups of motor proteins, such as myosin II, that convert the chemical energy from ATP hydrolysis into mechanical work via a cyclic process of attachment and detachment to associated polar protein filaments, e.g. F-actin [2]. There is great variation in the organization of actin, myosins, and other cross-linking proteins in the actomyosin structures found in cells. Myofibrils in striated muscle cells are examples of highly organized structures [1], composed of repeated subunits of actin and myosin, known as sarcomeres, arranged in series. Each sarcomere consists of actin filament of alternating polarity bound at their pointed end by large clusters of myosins, known as myosin “thick filaments”. The periodic structure of the myofibril allows it to generate forces on large length scales due to the collective dynamics of individual units of microscopic size, giving rise to muscle oscillation and contraction. More difficult is to understand the origin of spontaneous oscillations and contractility in cytoskeletal filament-motor assemblies that lack such a highly organized structure [3]. Oscillations are for instance observed in many organisms during repositioning of the mitotic spindle from the cell center towards the cell pole when unequal cell division occurs [3]. In vitro examples of such phenomena are sustained cilia-like beatings in self-assembled bundles of microtubules and dyneins [4] and spontaneous contractility in isotropic reconstituted actomyosin networks with additional F-actin crosslinking proteins, such as filamin or α-actinin [5, 6]. Theoretical models have shown that spontaneous oscillations can arise from the collective action of groups of molecular motors coupled to a single elastic element [7]. In these models the load dependence of the binding/unbinding kinetics of motor proteins breaks detailed balance and provides the crucial nonlinearity that tunes the system through a Hopf bifurcation to a spontaneously oscillating state. This simple theoretical concept has been adapted to describe the beating of cilia and flagella [8], mitotic spindles during asymmetric cell division [9], and spontaneous waves in muscle sarcomeres [10–13]. In a parallel development, generic continuum theories of active polar ‘gels’ have been constructed by suitable modification of the hydrodynamic equations of equilibrium liquid crystals to incorporate the effect of activity. These continuum models are capable of capturing some of the large-scale consequences of the internal contractile stresses induced by active myosin crosslinkers, including the presence of propagating actin waves in cells adhering to a substrate [14] and the retrograde flow in the lamellipodium of crawling cells [15]. In these studies the acto-myosin network is modeled as a Maxwell viscoelastic p-1 ar X iv :1 10 8. 59 99 v2 [ co nd -m at .s of t] 1 2 O ct 2 01 1 S. Banerjee1 T.B. Liverpool2 M.C. Marchetti1,3 fluid, with short-time elasticity and liquid response at long times [16]. The active viscoelastic liquid cannot, however, support sustained oscillations that require low frequency, long wavelength elastic restoring forces. In this letter we consider a nonlinear version of the generic continuum model of an isotropic active solid discussed recently by two of us [17]. We go beyond the linear stability analysis discussed in [17] and show that the presence of nonlinearities leads to both stable contracted and oscillatory states in different regions of parameters. The acto-myosin network is modeled as an elastic continuum, as appropriate for a cross-linked polymer gel embedded in a permeating viscous fluid, with elastic response at long times and liquid-like dissipation at short times. The active solid model describes the various phases of acto-myosin systems as a function of motor activity, including spontaneous contractility and oscillations. It provides a unified description of both phenomena and a minimal model relevant to many biological systems with motor-filament assemblies that behave as solids at low frequencies. When the active solid is isotropic, as assumed in the present work, the coupling to a non-hydrodynamic mode provided by the binding and unbinding kinetics of motor proteins is essential to generate spontaneous sustained oscillations. The main results are summarized in Fig. 1, where we sketch the steady states of the system as we tune the activity, defined as the difference ∆μ between the chemical potential of ATP and its hydrolysis products, and the compressional modulus B of the passive gel. For a fixed value of B we find a regime where the active gel supports sustained oscillating states and a contracted steady state as ∆μ is increased. For fixed ∆μ spontaneous contractility is only observed below a critical network stiffness. This is consistent with experiments on isotropic acto-myosin networks with additional cross-linking α-actinin where spontaneous contractility was seen only in an intermediate range of α-actinin concentration [5]. Our model does not, however, yield a lower bound on B below which the contracted state does not exist. This may be because, in contrast to the experiments where a minimum concentration of α-actinin is required to provide integrity to the network, our system is always by definition an elastic solid, even at the lowest values of B. The phase diagram resulting from our model also resembles the state space diagram of a muscle sarcomere obtained experimentally [13]. Interestingly, we find that a simplified dynamical system obtained by a one-mode approximation to our continuum theory corresponds to the half-sarcomere model proposed recently by Günther and Kruse [10] for a particular set of parameters. Our analysis shows, however, that the phase behavior described above is generic and can be expected in a wide variety of active elastic systems, as it relies solely on symmetry arguments. It provides a unified description of both spontaneous and oscillatory states and predicts their region of stability as a function of the elastic properties of the network and motor activity. Unstrained Steady State Sustained Oscillations Contracted Steady State 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7