Generic Phase Diagram of Active Polar Films

Abstract

We study theoretically the phase diagram of compressible active polar gels such as the actin network of eukaryotic cells. Using generalized hydrodynamics equations, we perform a linear stability analysis of the uniform states in the case of an infinite bidimensional active gel to obtain the dynamic phase diagram of active polar films. We predict in particular modulated flowing phases, and a macroscopic phase separation at high activity. This qualitatively accounts for experimental observations of various active systems, such as acto-myosin gels, microtubules and kinesins in vitro solutions, or swimming bacterial colonies.PACS numbers: 87.10.+e, 83.80.Lz, 61.25.Hq Active materials are a challenging class of systems driven out of equilibrium by an internal or an external energy source. Examples of active systems are selfpropelled particle assemblies in bacterial colonies [1,2], or the membrane and the cytoskeleton of eukaryotic cells [3]. The cell cytoskeleton is a network of long filaments made by protein assembly, interacting with other proteins [4] which can, among other things, crosslink or cap the filaments. Motor proteins, myosins, kinesins or dyneins use the chemical energy of Adenosinetriphosphate (ATP) hydrolysis to "walk" along the filaments, and exert stresses that deform the network [5], leading to an active behavior. The active character of the cytoskeleton plays a major role in most cell functions such as intracellular transport, motility and cell division.The cell cytoskeleton has a rich and complex dynamical behavior [5,6,7,8,9]. Self-organized patterns, such as asters, vortices, and rotating spirals, microscopic and macroscopic phase separations ("superprecipitation" [10] ) have been observed as a function of motor and ATP concentrations in a thin film [5]. This two-dimensional geometry gives for example a good description of the thin lamellipodium of a cell spreading or moving on a substrate. Some of these structures have been recently explained theoretically [11,12], but a full phase diagram of active polar films is still missing.The passive visco-elastic properties of the cytoskeleton are similar to that of a physical gel made of the cross-linked semi-flexible filaments. Recently, Kruse et al. [12] have proposed a generalized hydrodynamic theory to describe macroscopically the active character of incompressible polar gels, based on conservation laws and symmetry considerations. In this letter, we use the generic model of Ref.[12] to study the stability of compressible active polar films. We perform a linear stability analysis of the uniform states in the case of an infinite two-dimensional geometry, and obtain the dynamic phase diagram. Our results qualitatively account for the experimental observations on various active systems, such as acto-myosin gels, microtubules and kinesin solutions in vitro or swimming bacterial colonies. We choose the example of infinite acto-myosin films that we consider as two-dimensional, and non interacting with the environment. Such an example could be realiz...

Hydrodynamics of isotropic and liquid crystalline active polymer solutions

“…For a more precise comparison we refer to Ref. [38], where the equations are written in the simpler form appropriate to an active viscous solution, with no viscoelastic effects. In Ref.…”

“…[21,22] and [38] explicitly incorporates the dynamics of the solvent, which is assumed quiescent in our work (see, however, Ref. [41]), but consider systems near equilibrium by only keeping terms of first order in the chemical potential which controls the rate of ATP consumption.…”

“…It is useful to compare our work to others in some detail. Kruse et al [21,22] and Voituriez et al [38] have developed a continuum phenomenological descriptions of the polarized state of active polymer solutions where the hydrodynamic equations are written down on the basis of general symmetry considerations. Our work, in contrast, derives such equations from a systematic coarse-graining of a more microscopic kinetic equation.…”

“…It is therefore a far from equilibrium contribution to active filament dynamics. Kruse et al [21,22] and Voituriez et al [38] have considered the hydrodynamics of an active polymer solution including explicitly the flow of the solvent. In their formulation activity enters via a chemical potential proportional to ATP concentration.…”

“…In our approach this corresponds to the densitym of active motors. The polarization equation considered in [38] contains a term like our ∼ β∂ j ρ in Eq. (5.22) and is obtained there by allowing a coupling between density and splay deformations in the free energy of the system.…”