Key model ingredients

(a) The local state is defined by areal density ρ and by orientational order quantified by the nematic parameter S and by the nematic direction n. (b) Isotropic active tension λ when the network is isotropic (S = 0), and (c) anisotropic tension when S ≠ 0, controlled by κ. Active tension is positive (contractile) in all directions whenever |κ| < 1, but its deviatoric part is contractile when κ > 0 and extensile when κ < 0. Orientational order is driven by (d) active forces conjugate to nematic order and characterized by parameter λ⊙ and by (e) passive flow-induced alignment in the presence of deviatoric rate-of-deformation with coupling parameter β.

Active patterns coupling nematic order and density driven by self-reinforcing flows.

(a) Illustration of the dimensionless parameters characterizing active tension anisotropy (κ) and pattern architecture, quantified by the relative orientation of nematic order and high-density structures . (b) Order parameter of pattern architecture ω as a function of active tension anisotropy κ obtained from nonlinear simulations, showing transition from states with nematic direction parallel to high-density structures (ω < 0, fibrillar patterns) for κ < 0 to states with nematic direction perpendicular to high-density structures (ω > 0, banded patterns with perpendicular nematic organization) for κ > 0. Because |κ| < 1, the active tension is always positive in all directions. (c) Map of density, nematic order S, nematic direction (red segments) and flow field (green arrows) for quasi-steady fibrillar (I) and banded (III) patterns, and for a transition pattern of high density droplets with high nematic order (II) corresponding to nearly isotropic active tension. (d) Illustration of the out-of-equilibrium quasi-steady states, maintained by self-reinforcing flows, diffusion and turnover.

Pattern formation for a range of values of anisotropic active parameter κ in the limit λ⊙ → 0.

(a) Pattern formation for material parameters used in Fig. 2 except for λ⊙ = 0 while leaving c0 unchanged. (b) Here, in addition to λ⊙ = 0, we set β2 = 2ηηrot to the largest value allowed by the entropy production inequality. (c) Same parameters as in (b), except for an increase in friction as detailed in Table I.

Control of nematic bundle pattern orientation, connectivity and dynamics.

(a) Effect of orientational bias. (I) A uniform isotropic gel self-organizes into a labyrinth pattern with defects. (II) A small background anisotropic strain-rate efficiently orients nematic bundles. (III) A slight initial network alignment (S0 = 0.05) orients bundles, which later loose stability, bend, and generate/anneal defects. See Movie 4. We recall that the nematic order parameter in the quiescent and uniform equilibrium state is S0 = 0 if c0 ≥ 0 and otherwise, with c0 = 2a − ρ0λ⊙. (b) Depending on active tension anisotropy, nematic bundles are contractile and straighten (I, κ = −0.2), leading to quasi-steady networks, or extensile and wrinkle (II, κ = −0.8), leading to bundle breaking and recombination, and persistently dynamic networks (III). See Movie 5. The contractility or extensibility of the nematic bundles can be appreciated in the maps of the difference between the stress along the nematic direction (σ||) and the stress perpendicular to it (σ⊥), normalized by the constant σcrit = λcritρ0. (c) Promoting mechanical interaction between bundles. (I) Dynamical pattern obtained by reducing friction, and thereby increasing and . Time sequence in the bottom indicates a typical reconfiguration event during which weak bundles (dashed) become strong ones (solid) and vice-versa. (II) Nearly static pattern obtained increasing , (III) which becomes highly dynamic by further increasing . Time sequence in the bottom indicates illustrate the collapse (black), expansion (purple) and splitting (green) of cells in the network. See Movie 6. We indicate by * dynamical patterns exhibiting spatiotemporal chaos.

Assessment of activity parameters κ and λ⊙ through discrete network simulations.

(a) Illustration of the computational domain of the discrete network as a uniform representative volume element of the gel. (b) Sketch of model ingredients and setup to compute tension along and perpendicular to the nematic direction. (c) Typical time-signal for parallel and perpendicular tensions following addition of crosslinkers and motors (translucent lines) along with time average (solid lines) for isotropic and anisotropic networks. Tension is normalized by mean tension computed from time-averages and time by actin turnover time. (d) Mean tension as a function of network density for several nematic parameters S0, where both quantities are normalized by their values for the lowest density. With this normalization, Eq. (11) predicts a linear dependence with slope λ = 1 (dashed line). Error bars span two standard deviations. (e) Deviatoric tension as a function of nematic order for different densities. The dashed line is a linear regression to simulation data. (f) Dynamics of nematic order in a periodic network following addition of crosslinkers and motors for three initial values of nematic order. (g) Rate of change of nematic order normalized by turnover rate as a function of initial nematic order at zero and finite temperature.

Lengthscale of the pattern.

Illustration of the lengthscale of the pattern close to the onset of the instability (left) and deeper into de nonlinear regime (right). The linear stability estimate of the lengthscale of the pattern is ℓpattern = 2π/vcrit, where Vcrit is given by Eq. (10) in terms of material parameters. The side length of the square periodic computational domain is 8ℓpattern. Close to the onset (left), the pattern along the black arrow, with exactly 8 repeats, shows that the lengthscale in the nonlinear simulations closely follows the theoretical estimate. At activities 30% above the threshold (right), we sampled the typical separation between dense structures (green arrows) to approximate from simulations, finding that (mean and standard deviation). We robustly found across the parameter space that the pattern lengthscale from nonlinear simulations 30% above the threshold was a little smaller than the theoretical estimate ‘pattern, see density maps in Figs. 2 and 4.

Tension distribution along (σ||) and perpendicular to (σ⊥) the dense nematic bundles.

The left column shows the density distribution, the middle column the total tension along (solid) and perpendicular (dashed) to nematic bundles, and the right column the different contributions to the total tension dominated by the active (black) and the viscous (blue) components. In all cases, we consider for convenience fully nonlinear simulations in 1D to easily define the orthogonal directions relative to the self-organized pattern. (a) to (c) show patterns obtained for negative tension anisotropy κ of increasing magnitude and correspond to Fig. 3b in the main text, whereas (d) shows a chaotic pattern resulting from a large hydrodynamic length and corresponds to Fig. 3c(I) in the main text. The model parameters used in the plots are the same as in Figs. 2 and 3 and are described in Tables I and II

Discrete network simulations.

(I) Microstructural modeling approach using cytosim. (a) The nematic ordering of the network, S, measured with respect to a director , is controlled by a system-wide restraining energy. (b) Actin filaments are represented by connected points kept at a fixed distance and have finite bending rigidity κA. Crosslinkers are Hookean springs with stiffness ℓX and resting length KX. Myosin motors are Hookean springs with stiffness ℓM and resting length ℓM; their ends can walk on filaments at a speed υM, which is affected by force application. (II) Simulation protocol to quantify active tension anisotropy. (a) Initial fiber seeding without orientational bias. (b) Athermal imposition of desired order parameter S0 using a restraining potential. (c) Introduction of boundary anchors. (d) Deactivation of orientational restraining potential and addition of crosslinkers and myosin motors, which drive the system out-of-equilibrium. Reactions at anchors allow us to track tensions at the boundary, (e,f). (III) Protocol to quantify the orientational activity parameter ρλ⊙. (a) Initial fiber seeding without orientational bias in a periodic box. (b) Athermal imposition of desired order parameter S0 by restraining potential. (c,d) Activation of temperature, deactivation of orientational restraining potential, and addition of crosslinkers and myosin motors. The dynamics of orientational order of the system driven out-of-equilibrium are then tracked during a time period (c,d). (e) Estimation of the model parameter ηrot, which characterizes the athermal dynamics of network reorientation in the presence of a restraining potential of stiffness KS; note that a single value of ηrot fits the dynamics for all considered S0. (f) The rate of change of nematic order, Ṡ, is estimated from a 10 s time interval.

Model parameters used in figures.

Model parameters used in movies that do not directly reproduce figures.

Global parameters adopted in this study and in previous microstructural models that used cytosim.

Actin filament parameters adopted in this study and in previous microstructural models that used cytosim.

Myosin motor parameters adopted in this study and in previous microstructural models that used cytosim.