## Abstract

Cells both actively generate and sensitively react to forces through their mechanical framework, the cytoskeleton, which is a nonequilibrium composite material including polymers and motor proteins. We measured the dynamics and mechanical properties of a simple three-component model system consisting of myosin II, actin filaments, and cross-linkers. In this system, stresses arising from motor activity controlled the cytoskeletal network mechanics, increasing stiffness by a factor of nearly 100 and qualitatively changing the viscoelastic response of the network in an adenosine triphosphate–dependent manner. We present a quantitative theoretical model connecting the large-scale properties of this active gel to molecular force generation.

Mechanics directly control many functions of cells, including the generation of forces, motion, and the sensing of external forces (*1*). The cytoskeleton is a network of semiflexible linear protein polymers (actin filaments, microtubules, and intermediate filaments) that is responsible for most of the mechanical functions of cells. It differs from common polymer materials in both the complexity of composition and the fact that the system is not at thermodynamic equilibrium. Chemical nonequilibrium drives mechanoenzymes (motor proteins) that are the force generators in cells. The cytoskeleton is thus an active material that can adapt its mechanics and perform mechanical tasks such as cell locomotion or cell division.

Here, we show how nonequilibrium motor activity controls the mechanical properties of a simple three-component in vitro model cytoskeletal network. The nonequilibrium origin of this active control mechanism can be seen directly in the violation of a fundamental theorem of statistical physics, the fluctuation-dissipation (FD) theorem, which links thermal fluctuations of systems to their response to external forces. The FD theorem is a generalization of Einstein's description of Brownian motion (*2*). Although it is valid only in equilibrium, its possible extension to out-of-equilibrium systems such as granular materials and living cells has been debated (*3*–*5*). Prior studies in cells have suggested violations of the FD theorem (*3*), but this has not been directly observed. We show that an in vitro model system consisting of a cross-linked actin network with embedded force-generating myosin II motors strongly violates the FD theorem and that it does so because of the contractility of the acto-myosin system.

Actin and myosin are key components in muscle contraction and cell motility (*6*, *7*). Myosin motor domains, or heads, bind to actin filaments (F-actin) and generate force via the hydrolysis of adenosine triphosphate (ATP), resulting in motion along the polar actin filaments. At low salt concentrations, myosin II can form multimeric bipolar structures in vitro (Fig. 1A) (*8*). These “minifilaments” can link different actin filaments and move these filaments relative to each other (*9*). In the absence of ATP, these motor complexes statically cross-link F-actin and generate bundles that can be seen in a light microscope (Fig. 1B). In the presence of ATP, minifilaments generate contractile forces that can result in actin aggregation and phase separation (Fig. 1C), a phenomenon known as superprecipitation (*10*). To stabilize the networks and delay the onset of superprecipitation, we used F-actin cross-linked by biotin and neutravidin.

We measured the mechanical properties of these networks by active microrheology (AMR) (*11*–*13*), in which micrometer-sized embedded probe particles are manipulated by a sinusoidally oscillated optical trap, generating a force *F* at frequency ω. The response function α(ω) is obtained from the measured probe particle displacement *u*(ω): (1) For a simple incompressible and homogeneous elastic medium, this response function is related to the shear modulus *G* or stiffness of the medium via a generalization of the Stokes relation (*13*–*17*) α = 1/(6π*Ga*), where *a* is the probe particle radius. For materials with dissipation, the displacement *u* and force *F* are not in phase, which results in a complex response function. In this case, the shear modulus is *G* = *G*′ + *iG*″, where *G*′ is the elastic modulus and *G*″ is the viscous modulus. For cross-linked actin (1 mg/ml) gels, we found a predominantly elastic response in which *G*′ is much larger than *G*″ in the range of frequencies below 100 Hz. The measured moduli are consistent both with experiments on similar actin gels (*18*) and with theoretical predictions for actin networks with an average distance of about 2 to 3 μm between cross-links (*13*, *19*).

To characterize motor-generated activity, we used passive microrheology (PMR), which consists of recording the spontaneous displacement fluctuations of a probe particle without applied forces (*13*–*16*). In an equilibrium system, only thermal forces act on the probe, and the power spectral density (2) of the displacement fluctuations *u*(*t*) is directly related to the mechanical response of the material by the FD theorem, (3) where α″(ω) is the imaginary part of the response function, *k*_{B} is the Boltzmann constant, and *T* is absolute temperature. Because we can independently measure the left side of Eq. 3 with AMR and the right side with PMR, we can search for signatures of motor activity in the form of violations of the FD theorem.

As a control, we first verified the FD theorem as expressed in Eq. 3 for an equilibrium sample by directly comparing α″(ω) measured with AMR and ω*C*(ω)/2*k*_{B}*T* measured with PMR. For cross-linked actin without myosin, the agreement with Eq. 3 is shown in Fig. 2A.

Active processes create additional fluctuations and are expected to make the right side of Eq. 3 larger than the left side, thus violating the FD theorem. Indirect evidence for this has been reported in cells (*3*). We started with experiments at 3.5 mM ATP, where motors are expected to be active. Interestingly, we saw no difference between AMR and PMR results for up to 5 hours (Fig. 2A). At longer times, however, a clear difference developed in the form of strongly enhanced fluctuations at frequencies below 10 Hz (Fig. 2B). The appearance of these nonequilibrium fluctuations after a time lag can be explained by a switching of the myosin minifilaments from a nonprocessive mode, which cannot generate forces between actin filaments, to a processive tension-generating mode. Such a transition is expected because the ratio of attached to unattached time (duty ratio) of myosin increases with decreasing ATP concentration, when motor release induced by ATP binding becomes the rate-limiting step in the chemical cycle (*20*). Consistent with this, the lag time increased with increasing initial ATP concentration. We also performed experiments at low ATP concentrations stabilized with an ATP regenerating system (*13*). At ATP concentrations below ∼60 μM, we observed stable active fluctuations in the gels lasting for several hours.

Along with the nonequilibrium fluctuations, we saw a strong stiffening of the networks due to motor activity. This is apparent in the reduced response or compliance of the network (Fig. 3A). The shear modulus can be calculated using the generalized Stokes formula (*14*–*17*) (Fig. 3B). Here, the network stiffness increases by a factor of almost 100 depending on the ATP concentration. The stiffening of the network is related to the well-known strain stiffening response of actin gels under external stress (*18*, *19*, *21*): The contractile activity of motors results in internal tensile stresses in the actin filaments, which make the network more rigid. Unlike stiffening due to anisotropic shear stress, however, we expect that stresses induced by the motors are isotropic, which should lead to more pronounced stiffening because filaments of all orientations can participate.

The hypothesis that the observed stiffening is due to isotropically tensed filaments can be tested quantitatively by an examination of the frequency dependence of the stiffening. For relaxed actin gels, the stiffness of the networks increases with frequency in the form of a power law as *G* ≈ (–*i*ω)^{3/4} (*15*, *22*, *23*). Given the inverse relationship between response and stiffness, this is consistent with the behavior of our model system either in the absence of motors or at high ATP concentrations (Figs. 2A and 3A). With the onset of nonequilibrium activity, however, the power law changed toward a slope of ½, which is consistent with the prediction for filaments under tension (*24*). The full frequency dependence (green curve in Fig. 3A) is derived in (*13*). This frequency dependence also rules out another possible explanation for the observed stiffening of the network, namely an increased cross-linking of actin by inactive myosin. Cross-links alone will not change the ω^{–3/4} frequency dependence (*22*, *25*). Thus, the increased stiffness seen in Fig. 3B is most likely the result of motor-induced tension in the network strands.

Given the presence of both cross-links and myosin minifilaments in our system, we expect that the myosins generate contractile tension in actin filaments between cross-link points (Fig. 4A). We find rates on the order of a fraction of a micrometer per second in the movement of probe particles (Fig. 4B), consistent with typical rates of myosin motility. This slow buildup of strain implies quasi-static elastic deformations that include network compression. Increased filament density would imply a local reduction of the solvent/buffer, just as squeezing a sponge expels liquid. Given the stochastic binding kinetics of the myosins, the minifilaments eventually let go, which results in a fast relaxation of the network strain (Fig. 4B). The rate at which the network can relax is determined by the dissipation due to the inflow of solvent (like the swelling of a sponge). Time scales for this process are expected to be on the order of η(*r*/ξ)^{2}/*G* (*15*, *17*), where ξ (∼0.3 μm) is the mesh size of the network and *r* is a typical length scale of deformation. If we take the cross-linking distance as *r* (∼3 μm), this gives a relaxation time of ∼0.1 s, which is consistent with the observed relaxation in Fig. 4B and the appearance of motor activity in Fig. 2B.

To further test the hypothesis that the nonequilibrium effects we observe are due to contractile/compressive gel deformations, we also examined the correlated motions of pairs of particles within the network (Fig. 4, C and D). The observed anticorrelations in particular are not expected in equilibrium, but they are consistent with the contractile forces sketched in Fig. 4A.

Thus, actin, myosin, and cross-links are sufficient to capture essential and general features of contractility and mechanical adaptation in cytoskeletal networks. These observations suggest mechanisms by which cells could rapidly modulate their stiffness by flexing their internal “muscles” without changing the density, polymerization, or bundling state of F-actin. Cells can actively adapt their elasticity to the mechanics of the extracellular matrix (*26*) or to an externally applied force (*1*), and motors could be the cause for that. From a materials perspective, this in vitro model system exhibits an active state of matter that adjusts its own mechanical stiffness via internal forces. This work can be a starting point for exploring both model systems and cells in quantitative detail, with the aim of uncovering the physical principles underlying the active regulation of the complex mechanical functions of cells.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/315/5810/370/DC1

Materials and Methods

SOM Text

Movies S1 to S4

References