Long-range order in two-dimensional systems with fluctuating active stresses†

Abstract

In two-dimensional tissues, such as developing germ layers, pair-wise forces (or active stresses) arise from the contractile activity of the cytoskeleton, with dissipation provided by the three-dimensional surroundings. We show analytically how these pair-wise stochastic forces, unlike the particle-wise independent fluctuating forces usually considered in active matter systems, produce conserved centre-of-mass dynamics and so are able to damp large-wavelength displacement fluctuations in elastic systems. A consequence of this is the stabilisation of long-range translational order in two dimensions, in clear violation of the celebrated Mermin–Wagner theorem, and the emergence of hyperuniformity with a structure factor S(q) ∼ q² in the q → 0 limit. We then introduce two numerical cell tissue models which feature these pair-wise active forces. First a vertex model, in which the cell tissue is represented by a tiling of polygons where the edges represent cell junctions and with activity provided by stochastic junctional contractions. Second an active disk model, derived from active Brownian particles, but with pairs of equal and opposite stochastic forces between particles. We study the melting transition of these models and find a first-order phase transition between an ordered and a disordered phase in the disk model with active stresses. We confirm our analytical prediction of long-range order in both numerical models and show that hyperuniformity survives in the disordered phase, thus constituting a hidden order in our model tissue. Owing to the generality of this mechanism, we expect our results to be testable in living organisms, and to also apply to artificial systems with the same symmetry.