Actomyosin pulsation and flows in an active elastomer with turnover and network remodeling

Tissue remodeling requires cell shape changes associated with pulsation and flow of the actomyosin cytoskeleton. Here we describe the hydrodynamics of actomyosin as a confined active elastomer with turnover of its components. Our treatment is adapted to describe the diversity of contractile dynamical regimes observed in vivo. When myosin-induced contractile stresses are low, the deformations of the active elastomer are affine and exhibit spontaneous oscillations, propagating waves, contractile collapse and spatiotemporal chaos. We study the nucleation, growth and coalescence of actomyosin-dense regions that, beyond a threshold, spontaneously move as a spatially localized traveling front. Large myosin-induced contractile stresses lead to nonaffine deformations due to enhanced actin and crosslinker turnover. This results in a transient actin network that is constantly remodeling and naturally accommodates intranetwork flows of the actomyosin-dense regions. We verify many predictions of our study in Drosophila embryonic epithelial cells undergoing neighbor exchange during germband extension.

The hydrodynamic variables describing an active elastomer embedded in a fluid solvent are -(i) density of filamentous mesh, ρ; (ii) displacement field of filamentous mesh, u; (iii) density of bound myosin minifilaments, ρ b , and (iv) fluid (solvent) velocity v. The linearized elastic strain is defined as ij = 1/2(∂ i u j + ∂ j u i ). Our treatment closely follows [1,2].
The hydrodynamic equations of the active elastomer mesh in bulk are given by, where v is the fluid velocity and the rest of the parameters have been defined in the main text. The friction Γ between the mesh and cytosol can in principle depend on the mesh density. The dynamics of both actin filaments and myosin minifilaments show a turnover over a scale of minutes, which we refer to as S a and S m , respectively. The constitutive relations for the elastic (σ e ) and dissipative (σ d ) stresses are, Here λ, ν are the Lamé coefficients of the elastic mesh and η b , η s are the bulk and shear viscosities of the mesh, respectively. The form of the active stress σ a is described in the main text. The fluid hydrodynamics is given by, where ρ f is the density of the fluid and η s f and η b f are the shear and bulk viscosities of the fluid. Finally, the pressure P is eliminated by demanding total incompressibility, where φ denotes the volume fraction of the actomyosin mesh. For a gel, we expect φ 1, yielding, These are the complete set of equations, which we display here for completeness. Since the cytosol is a low Reynolds number fluid, Eq. (6) reduces to a force balance condition, where the right hand side of the equation equals zero. In the main manuscript, we study the overdamped or Rouse limit where we ignore the hydrodynamics of the fluid v; this is suggested by our experiments that show the actin mesh moves with respect to the fluid, and does not carry (advect) the fluid and other soluble molecules along with it, except for those which are bound to the mesh [3,4].
Since we are interested in the over-damped dynamics of the mesh, we ignore inertia and drop theü term in Eq. (1). This leads to Eqs. (1), (2) and (3) in the main text.
These equations can be rewritten in dimensionless form with time (t) and space (x) in units of k −1 b and l = η Γ , respectively, leading to the redefinitions, and the following equations in dimensionless form Upon linearizing about the homogeneous, unstrained fixed point (u 0 , ρ b0 , ρ 0 ), we obtain, where we have used the fact that the fluctuation in ρ is slaved to the compression, Solving (11) for the two eigenvalues, λ + and λ − , we obain the general solution, where, From the dispersion relations, we obtain 4 phases (a) stable (Im[λ ± ] = 0), (b) damped oscillations, (c) unstable oscillations, and (d) contractile instability (Im[λ ± ] = 0), corresponding to the dispersion curves shown in Fig. S2. The phase boundary in Fig. 1b of the main text is obtained by specifying a particular value of q (we have taken q = 2), or by restricting q to lie between [q min , q max ], where q min = 2π/L and q max = 2π/l (= 2π, in rescaled units).
The phase boundary between the stable and damped traveling wave phases can be obtained from the solution of Im [λ ± ] = 0, The stable phase crosses over to the unstable phase, when (obtained from Re [λ ± ] = 0), where q c , is the fastest growing mode, The phase boundary between the unstable oscillatory and contractile instability phases can be obtained from the solution of Im [λ ± ] = 0, Precisely at the stable-unstable phase boundary, since Re [λ ± ] = 0, the solutions correspond to left and right traveling waves for ρ b (and u), of the form where the wavelength q c and frequency w c of the wave are given by,

Supplementary Note 3. Strain dependent unbinding
For the turnover of bound myosin filament density, we allow for a possible strain-induced unbinding of the Hill-form, k u = k u0 e α∇·u . The sign of the coefficient α can be taken to be either positive or negative (Fig. S3a) : α > 0 implies a local extension (compression) of the mesh will increase (decrease) the myosin unbinding, while α < 0 implies a local compression (extension) of the mesh will increase (decrease) the myosin unbinding. The choice α = 0 implies that the myosin unbinding rate is a constant, independent of mesh deformation. We thus cover all possibilities.
Changing the sign of α only affects the placement of the phase boundaries, but not the qualitative aspects of the phases, see Fig. S3b,c. We find that the oscillatory phase exists, as long as the eigenvalue Im[λ ± ] < 0, from which get the maximal (positive) value α max (B, ζ 1 , . . .), beyond which there are no oscillations. Thus to get oscillations for a given set of (other) parameters, we have to set −∞ < α < α max . We have taken α > 0 (but smaller than α max ) in all numerical results presented in the main text.   . (a,b) We collect the 2d intensity map (I(x, y, t)) of myosin (green dots) at different times, e.g., t 1 , t 2 (with t 2 > t 1 ), within a thin rectangular strip (green rectangle) chosen so that it is not contaminated by signals at the cell boundary. After background subtraction, we integrate the intensity I(x, y, t) along y between the limits y min and y max , along each thin rectangular strip (blue). This gives us the 1-dimensional profile I(x, t) vs. x, as shown schematically here. (c,d)(i) We carry out the above protocol for the labeled myosin-dense images obtained in two different experiments. (ii) After background subtracting the intensity maps, we plot I(x, y, t) in the x − y plane. (iii) The 1D projection I(x, t) is plotted versus x.  Fig. 4 of [3]). (a)-(c) shows that advection is crucial to obtain oscillations of bound myosin. (a) Space-time plot (kymograph) of the bound myosin density (colour bar) against a foreground of arrows indicating the local velocity vectoru (whose magnitude is given by the size of the arrows). This velocity vector describes the advection of myosin by the actin mesh, and shows that local convergence of velocity is associated with increased myosin density (and vice versa). This recapitulates the advection-myosin density profiles shown in Fig. 4a of [3]. (b) This panel shows spatial profiles extracted from the above figure at three specific time instants. There is a definite correlation between convergent velocity vectors (black arrows) and increased myosin density (blue graph) between t = 0.5 and t = 1, and divergent velocity vectors (black arrows) and reduced myosin density (blue graph) between t = 1 and t = 1.5. Green arrows show overall convergence (divergence) of advection. Our results here are consistent with Fig. 4a-c of [3]. (c) Time variation of spatially averaged myosin densityρ b (black line) and advection speedv (obtained from magnitude ofu, red line) during an oscillation cycle. The graph ofv is shifted to the left with respect to the myosin density graph by an amount 0.2, indicating that the advection is a cause for local enhancement of myosin density. Our results are entirely consistent with Fig. 4m of [3]. (d) The amplitude of the oscillation decreases when we reduce actin mesh density ρ 0 , here we compare ρ 0 = 1 with ρ 0 = 0.7. Compare this to the actin perturbation experiments, Fig. 4f of [3]. Parameter used here are, B = 8 , −ζ 1 ∆µ = 5.2, k = 0.2 , D = 0.25, α = 3, c = 0.1.