Adhesion-mediated heterogeneous actin organization governs apoptotic cell extrusion

Apoptotic extrusion is crucial in maintaining epithelial homeostasis. Current literature supports that epithelia respond to extrusion by forming a supracellular actomyosin purse-string in the neighbors. However, whether other actin structures could contribute to extrusion and how forces generated by these structures can be integrated are unknown. Here, we found that during extrusion, a heterogeneous actin network composed of lamellipodia protrusions and discontinuous actomyosin cables, was reorganized in the neighboring cells. The early presence of basal lamellipodia protrusion participated in both basal sealing of the extrusion site and orienting the actomyosin purse-string. The co-existence of these two mechanisms is determined by the interplay between the cell-cell and cell-substrate adhesions. A theoretical model integrates these cellular mechanosensitive components to explain why a dual-mode mechanism, which combines lamellipodia protrusion and purse-string contractility, leads to more efficient extrusion than a single-mode mechanism. In this work, we provide mechanistic insight into extrusion, an essential epithelial homeostasis process.

(a) Confocal time-lapse of cells extruding from monolayer without laser induction (naturallyoccurred extrusion). Scale bar = 10μm. T = 0 is defined as the time point right before caspase was first turned on induction.
(a') Statistics of extrusion cases with neighboring cells used lamellipodia protrusion (by detecting increased PBD signals) over a total number of observed extrusion events. Source data are provided as a Source Data file.    (a) Average caspase signal as a function of extrusion time. The shaded area represented s.e.m. N = 10 extrusions. Note that the caspase signal became saturated at t ~ 40 min, which corresponded to the drop of extrusion area function in for Figure 1D -WT to 10% of the original area. Source data are provided as a Source Data file.
(a') Time cross-correlation analysis between two time series: caspase-3 signal and the changes of the extruding area at the apical plane. The correlation coefficient was at -0.9 at time lag = 0 min indicated that these two time series are significantly anti-correlative to each other. As such, caspase-3 can be used as the surrogate marker for the extruding stage. n = 6 extrusion event from m = 1 experiment. Source data are provided as a Source Data file.  (a) Schematic of force decomposition. Traction force measured was decomposed into radial and orthoradial components (with respect to the extrusion center).
(b) Representative traction force distribution during extrusion. The cosine of angles at which the traction forces formed with respect to the center of extrusion (white + sign) was colorcoded. +1 indicates forces pointing towards the center of the extruding cell (inwards), and -1 indicates forces pointing away from the center of the extruding cell (outwards). Scale bar = 10 μm. How angle θ was defined is shown in the schematic in (a). Note that the force distribution was anisotropic at the start of extrusion. When the purse-string became more visible and uniform, the force distribution is gradually inwards.   Table 1). The color scale bars represent the predicted probability of successful extrusion by numerical simulation.
(d) Case of a constant and mild amplitude of apico-basal radial fibers D A = μ as a function of the purse-string contractility.
(e) Case of a modulated amplitude of apico-basal radial fibers D A = μ.γ 0 /γ fluctuations as a function of the purse-string contractility, where γ 0 is the purse string contractility in the wildtype case.
(f) Case of a modulated amplitude of apico-basal radial fibers D A = 30μ.γ 0 /γ , showing that an increased level of fluctuations could lead to a significant increase in the extrusion probability -even in the no-extrusion phase space region of a purely deterministic framework based on averaged quantities.

Supplementary Note -Theoretical Appendix
A. P. Le et al.
Our objective is to understand the variability in the frequency of extrusion over both adherent and non-adherent patches. Based an hydrodynamic assumptions for the cellular material surrounding the extruding cell, we derive a stochastic equation for the extruding cell area reduction rate from which we obtain the probability distribution of the extrusion times. Based on this result, we construct a phase diagram for the extrusion success rate that can be readily compared to the experimental results (see main Figure 5). Compared to the WT case, we interpret the effect of chemical perturbations in terms of a decrease (αcad-KD,L344P) or increase (∆-Mod) in the strength of the apical purse-string cable.
Here, we closely follow the general arguments presented in [1] and [2] which were applied to wound healing experiments. However, in the wound healing experiments analysed in [1], the initial hole radii are in the R wound 0 ≈ 50−100 µm range which is significantly larger than the patch size used here in the context of cell extrusion (R 0 ≈ 5 − 30 µm). Noting such discrepancy is particularly important since, in [1], R wound 0 is found to be significantly larger than two characteristic length scales: (1) R wound 0 R τ = τ /σ p ≈ 10 µm whereby R τ is the length scale over which the pursestring mechanism provides a negligible contribution to the overall healing dynamics, and (2) R wound 0 R e whereby R e is a length scale over which dissipation through friction with the underlying substrate dominates over internal viscous dissipation within the cell-monolayer (though cell-type dependent, a typical estimate is R η ≈ 40 µm [3]).
In the current context of cell extrusion, we expect the initial cell radius R 0 to be comparable to both R τ and R η . Here we present a general framework that encompasses the contribution from two possible closing modes (purse string contractility and lamellipodia protrusions) and show that tensile yet viscous apico-basal cables can generate forces that prevent full extrusion.

I. STOCHASTIC DYNAMICS
Following [1], we propose to model the cellular material surrounding the extruding cell as being a bulk twodimensional fluid that is: 1. incompressible, i.e. the in-plane flows within the monolayer are such that ∂ α v α = 0. In [1], this is justified as a consequence of the absence of cell division and cell death in the vincinity of the extruding cell at the time scale of the experiments.

satisfying a force balance equation that reads
where ξ is the friction coefficient of the monolayer on the substrate; P is the tissue pressure, acting as a Lagrange multiplier imposing incompressibility; η is a homogenized bulk viscosity (i.e. including the dissipative contributions from both the bulk of cells and from their interfaces); the shear viscous stress is expressed in terms of the deviatoric partṽ αβ of the symmetric shear rate v αβ = 1 2 (∂ α v β + ∂ β v α ). One notices that incompressibility implies that v αα = 0, leads to the identity on the deviatoric strain rateṽ αβ = v αβ (see also [1]).

Incompressibility: kinetic equation in the presence of a radially symmetric flow profile
We consider that the epithelium is irradiated by UV light over a disk of radius R 0 at t 0 = 0 whose center defines the origin O. We assume that the circular shape is preserved during the closure process. We denote by R(t) the radius of the extruding cell at the time t post laser irradiation.
Assuming rotational invariance of the flow, we express the velocity field as v = v r (r, t) e r , where the non-vanishing radial component depends only on the distance r relative to the center O of the initial wound. Using the incompressibility constraint ∂ α v α = 0, we obtain v r (r, t) = A(t)/r, where A(t) is to be determined from the kinematic boundary condition at the margin: since v r (r = R(t), t) =Ṙ(t), we obtain that: Such 1/r velocity profiles is consistent with the experimental data obtained through PIV.
Supplementary Figure 14. Sketch of the adherent versus non-adherent geometries. (a) As seen from a top-view perspective, the area of the extruding cell radius decreases with time (b) In a side view perspective, one main difference between the adherent and non-adherent substrates situations lie in the existence of apico-basal radial fibers, which we assume to be contractile with a mean active stress µ.

Viscous stress equation
Within the bulk of the tissue, using rotational invariance (P = P (r, t)) and Eq. (3) for the velocity field, the force balance (1) becomes ∂ r P = −ξRṘ/r . Following [1], one expects the pressure to be given by P = P c − ξRṘ ln r/R max , where R max is a constant, corresponding to a long-range cut-off.
At the interface between the tissue and the extruding cell, we define the stress boundary condition as where P c is the initial mean pressure within the monolayer before ablation of the cell; σ p accounts for the lamellipodia protrusive stresses exerted by the cells around the extruding cell; τ is an effective tension that describes purse-string forces (see next subsection for a discussion of the adherent case); σ µ (R) corresponds to contractile stresses exerted by apico-basal actomyosin cables, whose sign is the opposite in the adherent situation compared to the non-adherent one (see the next section for a further discussion); Θ corresponds to correlated stress fluctuations (see further discussion). Following [1], we find that a dynamical equation for the wound radius R(t) follows from the stress boundary condition at the margin, Eq. (4), and with the above expression for P we find that the closing radius rate reads: where R max is a hydrodynamic length found to be in the 100µm range [1]; we expect R max ∝ η/ξ whereη is a bulk viscous modulus.
3. Adherent versus non-adherent cases: two models for σp(R) • In the non-adherent case, we observe (1) the absence of lamellipodia protrusions, hence we set σ p = 0; (2) the existence of a large purse-string contractile ring, hence we set τ > 0 and (3) actin cables connecting the apical contractile purse-string to the basal adhesion complexes, which we refer to as apico-basal/radial cables (see main text Fig. 5.). The geometry is sketched in Supplementary Fig. 14. These radial actin cables are expected to contribute to the overall stress through a negative stress σ µ < 0 providing a resistive contribution to extrusion. Based on a three-dimensional perspective, we consider that σ µ = µ sin(θ), where θ = arctan [(R(t) − R 0 )/h], with h the height of the monolayer and µ a parameter quantifying the tension along the radial fiber cable; equivalently, such stress reads:  Table I. For large enough purse-string tension τ , the extrusion always complete, corresponding to Rc > R0.
so that σ µ < 0 as soon as cell extrusion begins (R(t) < R 0 ). In the absence of fluctuations, we find that there is a critical radius R c above which the extrusion process cannot be completed. In the regime where R 0 > R c , the time required for cell extrusion diverges; elastic forces due to the radial actin cables prevent hole closing. In contrast, for R 0 < R c , the extrusion happens in a finite time. Equation (6) is similar to the one presented in [1] based on an elastic material assumption; however our model is consistent with a model of tensile yet viscous apico-basal cables.
The precise value of R c can be precisely estimated numerically (see Supplementary Fig. 15). In the limit of a strong apico-basal stress fibers tension µ τ /R 0 , we find that: such that R 0 > R c , meaning that in such limit, extrusion is never completed, in agreement with expectations.
• In the adherent case, we model the effect of the stresses exerted by the lamellipodia protrusions by considering that a constant positive contribution to the stress σ p > 0, as proposed in [1]. Since, here in the cell extrusion context, no multi-cellular purse-string is observed, we expect that the overall apical edge surface tension to be negligible compared to other contributions; hence we set τ = 0 in the adherent case. However, we consider the additional closing effect caused by the local contractile rings observed within the cells surrounding the the extruding cell; we implement an additional contribution to the stress σ p in the form of Eq. (6); in stark contrast to the non-adherent case, the resulting stress has a positive sign since the cable orientation is then favoring the extrusion process (i.e. see the main text Fig. 2f' corresponding to θ > π/2 in Supplementary Fig. 15). Since both the protrusion and apico-basal cables contribute to the extrusion process in the adherent case, we find that the extrusion process always occurs in a finite time for any radius R 0 of the cell to be extruded, in that in the absence of a purse-string contractility (τ = 0).

Predicting traction force intensity at mechanical equilibrium
We now focus on the non-adherent case. As defined in the Method section, we estimate the overall radial force at the edge of the non-adherent patch as: where T r (r, θ) is the radial projection of the measured traction field (expressed in Pa). In our theoretical model, we consider a similar quantity defined in the limit a vanishing small patch δ → 0. In this limit, the only non-vanishing contribution to forces come from the apico-basal cables localized at the edge boundary, which, based on the model defined Eq. (6), leads to the following expression for the total radial force: We point that σ (non−adherent) µ is to be expressed in the units of a two-dimensional stress, i.e. Pa.m 1 . Experimentally, we are interested in the value of the traction force at a final steady-state configuration. Neglecting at first fluctuation terms, such steady state corresponds to the assumption of a local mechanical equilibrium between apico-basal radial forces and the orthoradial purse-string contraction. Whenever the purse-string value is large, we have R(t) ∝ R 0 h = 3µm, hence that: Assuming that F theory ≈ F exp , we can estimate for the apico-basal contractility µ and the purse-string line tension τ . The extracted value for τ and µ can be compared to an estimated line tension defined as F = 2πR 0 γ and estimated in [4] as γ = 15 mJm −2 = 15.10 −3 Nm −1 = 15 nNµm −1 .

II. EXTRUSION SUCCESS RATE PROBABILITY
Based on the stochastic equation 5, we use the framework of [2] to obtain the extrusion success rate probability. We start by rewriting Eq. (5) as an equation on the dynamics ofṘ: where: • the deterministic force reads • the drag reads • the noise term Θ is modelled as a Gaussian distributed noise with zero mean, unit variance and a characteristic correlation time, e.g. Θ(t)Θ(t ) = exp(|t − t |/τ A )/τ A ; (iv) the amplitude of the noise can be expressed in terms of the diffusion coefficient: where D, D τ and D µ are constants quantifying the intensity of fluctuations in the force production arising from pressure, purse-string and cable force fluctuations, respectively. We expect stress fluctuations to be correlated over a timescale τ A ≈ 100 − 1000 s that is a relatively short compared to the overall extrusion timescale. Considering the limit τ A → 0 requires some care; the corresponding Langevin equation is said to be multiplicative, e.g. here whereby the noise amplitude D T depends on the value of R t , which can lead to some ambiguity on the interpretation of the noise.
Following the framework of [5], we show that Eq. (11) is equivalent to the following Langevin equation: where χ is a centered Gaussian white noise ( χ(t)χ(t ) = δ(t − t ); Eq. (15) is now to be interpreted with the Ito convention in which fluctuations provide no deterministic contribution (i.e. such that 2D T (R)χ = 0). Equation (15) is equivalent to a differential equation on the probability distribution function p(R, t|R 0 , 0) between a radius R 0 at the initial time 0 and the radius R at a given time t: The latter backward Fokker-Planck equation (16) is complemented by two boundary conditions: we consider that R = R 0 is a reflecting boundary (after a sufficiently long time post irradiation, the extruding cell does not spread over its initial size) and that R = 0 is an absorbing state (corresponding to a complete cell extrusion). We then solve Eq. (16) numerically to obtain the phase diagram presented in the main text; we have used the parameter values provided in Supplementary Table 1. For simplicity, we set the value of the cable diffusion coefficient to zero (D µ = 0).

Description
Value (