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.

💡 Research Summary

In this paper the authors develop a generic continuum description of an isotropic active gel—a cross‑linked polymer network permeated by a viscous fluid and driven by motor proteins that consume ATP. Building on earlier linear theories of active polar gels, they incorporate essential nonlinearities: a fourth‑order elastic free energy in the strain, strain‑dependent motor unbinding kinetics, and nonlinear active stresses that depend on both the local density and bound motor concentration. The basic fields are the one‑dimensional displacement u(x,t) and the dimensionless bound‑motor concentration φ(x,t). The elastic stress follows from fₑ=½Bs²+αs³/3+βs⁴/4 (with s=∂ₓu), while the active pressure is taken linear in the ATP consumption rate Δµ and expanded in powers of density and motor fluctuations with coefficients ζ_i that encode contractility, excluded‑volume, and higher‑order effects. Motor unbinding is assumed to increase exponentially with load, k(s)=k₀e^{γs}, which is expanded to second order for analytical tractability.

The governing equations are a force balance with friction Γ∂ₜu=∂ₓσₑ−∂ₓpₐ and a convection‑reaction equation for φ that includes advection by the gel velocity and a strain‑dependent unbinding term. After nondimensionalisation the key control parameters become the effective compressional modulus B, the activity Δµ, the strain‑nonlinearity coefficients α,β, the motor‑feedback parameters γ and ζ_i, and the friction‑scaled unbinding rate k₀.

Linear stability analysis reveals three homogeneous steady states: (i) an unstrained state (s=0, φ=0), (ii) two strained states s_{±} (solutions of a cubic equation). For φ=0 the eigenvalue is simply z=−Bq², so stability reduces to the sign of B: positive B yields a linearly stable unstrained state, while negative B drives a mechanical instability toward a contracted configuration. When motor fluctuations are retained, the dispersion relation becomes z_{u,φ}(q)=−b(q)²±½q√