Supplementary Material

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1 1 2 3 Topological defects in confined populations of spindle-shaped cells by G. Duclos et al. Supplementary Material Supplementary Note 1: Characteristic time associated with the activity of the system... 2 Supplementary Note 2: Modeling the confined cell layer as a nematic drop... 3 References of Supplementary Material Legends of Supplementary Figures and Video Supplementary Figures NATURE PHYSICS 1

2 Supplementary Note 1: Hydrodynamic screening length and characteristic time associated with the activity of the system The hydrodynamic screening length λ is defined as : 18 (1) where η is the effective 3d monolayer viscosity, h is the height of the cells and γ is the cell-substrate friction. From the literature, we get η ~ 10 5 Pa s (ref 1,2 ), ~ N m -3 s (ref 3 5 ); we estimate h ~ 5 µm. Therefore, λ ~ 10 µm. The hydrodynamic screening length is of the order of a cell size Since, λ is of the order of a cell size, the characteristic time τ results from a balance between activity and cell-substrate friction and we compare this value with the experimentally measured time of annihilation between two defects of opposite charges. We define the characteristic time τ 28 (2) where is the cell-substrate friction, a cell size, and is the active stress in 2D. Using the relationship between the activity and the Frank constant K 6, we get: where L is the distance between defects. (3) To estimate K, we reason that confluent cells must physically deform when the nematic order is distorted. Therefore, K is a function of the Young modulus E. The cell size been the only relevant length scale at this level, we take by dimensional arguments, where E is the Young modulus of the cells (E ~ 1 kpa) 7, and is the surface occupied by a cell. We ignore here the other contributions to the Frank constant. Therefore, (4) We estimate ~ 20µm and, to compare with annihilation dynamics reported in Fig. S4B, we set L = 150 µm. Therefore, τ ~ 3 h, in good agreement with the experimental annihilation time of typically 4h (Fig. S4B) NATURE PHYSICS

3 SUPPLEMENTARY INFORMATION 3 45 Supplementary Note 2: Modeling the confined cell layer as a nematic disk The passive nematic disk We treat the cell layer as a 2-dimensional liquid crystal. In the one-constant approximation, the energy of the system is then given by ( ) (Equation1) with Frank-constant K and nematic director n = cos(ω) e x + sin(ω) e y and ω [0,π[. For sake of simplicity, we set K=1 in the following. A variational calculation shows that the orientation field that minimizes the total energy of the nematic disk satisfies 2 ω = 0. The nematic equations are defined on a circular patch of radius R. We use polar coordinates (r,φ). At the boundaries of the circle, we force the orientation of the director field to be tangential to the surface, as we observe in the experimental situation. Thus, we have the boundary conditions ω(r) = φ + π/2. In the following, we first define defects as point sources in the vector potential associated with the director field. Then, we solve the constitutive equations in terms of Fourier-modes in the angular variable φ and a power-series in the radial variable r and use the boundary conditions and a continuity condition to determine the unknown Fourier-coefficients. A particular choice of the integration region then allows us to determine the energy F of the system as a function of a small cutoff length ε. From this, we determine the optimum defect position as a function of ε. We will see that this position converges towards a finite distance r 0 * from the patch center and a tilt angle θ*=π in the limit ε 0. Eventually, we use the force on the defect in the limit ε 0 to determine the distribution of defect positions in a thermal bath. Defects in nematic liquids The defects are introduced as point sources in the potential that can be associated with the director field. We define the vector potential A for which ω = Α. Since the nematic director lies in the x-y plane, we write A= Ψ e z with a scalar function Ψ. A +1/2-defect at r 0 can now be defined as a point source in Ψ. Here, we put two defects of topological order +1/2 at position r 1 and r 2, hence (Equation 2) with δ(.) being the Delta-distribution. One easily convinces oneself that, employing the Stokes theorem and using 2 Α= - Α, ) NATURE PHYSICS 3

4 = (Equation 3) on any closed curve C encompassing a single defect. In the following, we place the two defects at the same distance from the center of the circular patch. Their positions are then parametrized by a radial component, r 0, and an angle, θ. For sake of simplicity: = o θ 2 in θ 2 and = o θ 2 in θ 2. Solution on the circle Expanding the angular variable in Fourier-modes, we note. (Equation 4) We immediately see that a n = 0 for all n and Ψ0 = 0 (for r < r 0 ) since Ψ must not diverge at r = 0. Note that Ψ/ r has a finite jump at r = r 0 and expanding the right hand side of 2 in a Fourier-series, (Equation 5) we find. (Equation 6) Imposing continuity of Ψn at r 0, it follows Ψ0 = -1 (for r r 0 ), = For θ, we then have. ( ) and (Equation 7) NATURE PHYSICS

5 SUPPLEMENTARY INFORMATION and we can apply the boundary condition θ(r) = φ + π/2 to determine b n and b. Together, we note 121 (Equation 8) Summing the Fourier-series, we obtain a closed form for θ, (Equation 9) The energy of the system At the defect positions, ω diverges such that the system's energy given by Eq. (1) diverges as well. One can circumvent this problem by introducing a finite area around the defect that is excluded from the integration; see Fig. 1 for an illustration. The value of F then crucially depends on the typical cutoff length ε of that region. In particular, F ~ -log(ε) as ε 0. Since we are not interested in the value of F itself but rather the position of its minimum, we still attempt to derive an expression for F as a function of the defect's distance from the center, r 0, the inclination angle θ, and the cutoff length ε. The integration can be significantly simplified when carried out in polar coordinates and breaking the integration over r into two parts, ranging from 0 to r 0 -ε and r 0 +ε to R, respectively, see Fig. A. We find (Equation 10) NATURE PHYSICS 5

6 Figure A: Area of analytical integration For all fixed values of ε, F ε (r 0,θ) has a single minimum with respect to variations of r 0, θ, and ε. Successively reducing the value of ε, the area of integration approximates the full circle and in the limit ε 0, the position of the minimum converges to a well-defined point at and (Equation 11) (Equation 12) Both values are in excellent agreement with the experimentally determined values (Fig. 4D- F). Since the diverging terms in Eq.(10) appear in a logarithm, we can separate them as offsets that don't contribute to the forces exerted on the defects. Neglecting these terms when taking the limit ε 0, the energy of the system can be written as (Equation 13) A surface plot of the ensuing energy landscape is shown in Fig. S8. Finite size of the defects For all values ε>0, the energy of the nematic liquid as a function of r 0 remains finite and * possesses a single minimum. The position of the minimum changes with ε but converges to r 0 = 5-1/4 R as ε 0, see Fig. B, C. We find experimentally that this position of the minimum of the energy coincides well with the most likely radial position of the defects. 6 NATURE PHYSICS

7 SUPPLEMENTARY INFORMATION Figure B: Energy of the nematic drop as a function of the defect position, ε = 0.05 (blue, circle) 0.02 (green, squares) 0.01 (red, diamonds). Symbols are results of the numeric integration, solid lines show equation 10. The dashed line goes through the minima under variation of ε Figure C: Minimum position under variation of defect size ε Taking the defect size (ε) of a cell size (~10µm), which is compatible with the width of the order parameter at a defect (Figure D), we get: ε(r=400µm) = < ε < ε(r=250µm) = And therefore (Fig. C) 0.68 < r * 0 < 0.69 For all practical purposes of comparison with the experimental results, these values do not differ significantly from the ε = 0 value. NATURE PHYSICS 7

8 Figure D: Left: Map of the order parameter near a +1/2 defect core (average map over 6 (+1/2) defects from 2 independent experiments). Right: profile of the order parameter along and perpendicular to the comet tail. Defects are slightly anisotropic but their width is of the order of 10 µm Fluctuations of the defect position In the experimental system, the final position of the defects fluctuates around a finite value. The distribution's maximum, however, coincides well with the predicted minimum of the energy at (r 0 *,θ*). Here, we interpret the position of the defect as a stochastic variable subject to fluctuations by the stochastic movement of fibroblasts as well as a drift force from the potential F 0 (r 0,θ). Without caring too much about the microscopic details, we assume the movement to be equivalent to that of a diffusive particle in a potential and describe the probability for the defects to be at position x at time t, p(x,t), by the Fokker-Planck equation (Equation 14) For stationary situations without any probability fluxes, the probability density is given by a Boltzmann law. (Equation 15) When fitting the experimental values, we observe that the agreement is best for very small values of ε (ε 10-6 R). We therefore consider the results of the case ε = 0 in the following. In this description, Τ is a fitting parameter that contains information about the defect mobility and the amplitude of fluctuations. It is measured in units of energy, which are here given as multiples of Frank constant K. In thermal systems, Τ would be proportional to a temperature, 8 NATURE PHYSICS

9 SUPPLEMENTARY INFORMATION hence the naming convention. We fit the solution of p(r) to the experimentally found distributions of defect positions to estimate the degree of noise in the system, see Fig.4D-G in the main text. We find qualitative and quantitative agreement between experiment and theory for T=0.10 (R = 250μm), T=0.21 (R = 300μm), T=0.20 (R = 350μm), and T=0.15 (R = 400μm) in the absence of Blebbistatin, see Fig. 4G in the main text. The temperature Τ is roughly constant as a function of system size with an average of Τ = 0.16, much smaller than 1. When Blebbistatin is added at concentrations c = 1µM and c = 3µM, the estimated values of T are T=0.16 and T=0.17, respectively NATURE PHYSICS 9

10 References of Supplementary Material 1. Marmottant, P. et al. The role of fluctuations and stress on the effective viscosity of cell aggregates. Proc. Natl. Acad. Sci. 106, (2009). 2. Guevorkian, K., Colbert, M.-J., Durth, M., Dufour, S. & Brochard-Wyart, F. Aspiration of Biological Viscoelastic Drops. Phys. Rev. Lett. 104, 1 4 (2010). 3. Cochet-Escartin, O., Ranft, J., Silberzan, P. & Marcq, P. Border forces and friction control epithelial closure dynamics. Biophys. J. 106, (2014). 4. Mayer, M. et al. Anisotropies in cortical tension reveal the physical basis of polarizing cortical flows. Nature 467, (2010). 5. Hannezo, E., Prost, J. & Joanny, J.-F. Growth, homeostatic regulation and stem cell dynamics in tissues. J. R. Soc. Interface 11, (2014). 6. Ramaswamy, S. The Mechanics and Statistics of Active Matter. Annu. Rev. Condens. Matter Phys. 1, (2010). 7. Solon, J., Levental, I., Sengupta, K., Georges, P. C. & Janmey, P. a. Fibroblast adaptation and stiffness matching to soft elastic substrates. Biophys. J. 93, (2007) NATURE PHYSICS

11 SUPPLEMENTARY INFORMATION Legends of Supplementary Figures and Video Supplementary Figure 1 : The system is at quasi-equilibrium. By using low Calcium conditions ([Ca 2+ ] = 50 µm), proliferation of RPE1 cells was impaired, causing the order parameter to remain constant (black points) whereas it increased in normal conditions (red points). Error bars are SDs. N= Supplementary Figure 2: Decrease of the number of defects with time. The color codes for the local order parameter. The orientation field is represented by LIC (Line Integral Convolution). Low order parameter spots are associated with the presence of defects. As times goes, defects of opposite charges pairwise annihilate. NIH-3T3 cells Supplementary Figure 3: Directed motion of +1/2 defects. These defects migrate along lines of low order but where the order parameter changes continuously (lines of kinks). NIH3T3 cells Supplementary Figure 4: defects of opposite charges pairwise annihilate. A/ Details of the annihilation. B/ The separation between two defects in the process of annihilation varies like t 1/2. t a is the time at which the two defects annihilate (N=8 annihilation events, the green line is the average and the colored area is the SD. The red line is a fit by (t a -t) 1/2 in the 4 hours preceding annihilation). NIH-3T3 cells Supplementary Figure 5: Splay and bend distortion energies. A/ Time evolution of the orientation field, and the local splay and bend distortion energies. NIH-3T3 cells. The positions of the defects are highlighted with the yellow circles in the LIC images (scale bar: 500 µm). B/ Time evolution of the splay distortion energy in the cellular nematic layer. The energy has been integrated over the field of view. The energy decreases until reaching a plateau 40 hours after confluence, once the defects density is stabilized. (N=130, colored area: SD) Supplementary Figure 6: After 60h post-confluence in a circular domain, the majority of the domains contains two facing +1/2 defects. NIH-3T3 cells, N=145 domains, R 0 =350µm Supplementary Figure 7: The position of the defects in the disk patterns is independent of the activity of the system (via the Blebbistatin concentration) and even of the cell type. Error bars=sd, N= Supplementary Figure 8: The orientation field observed in a representative experiment (A) and the one computed from the nematic disk model (B) are very similar. LIC representations. ((A): R=350µm). 277 NATURE PHYSICS 11

12 Supplementary Figure 9: Energy profile in a disk of radius R 0 =1, computed from the nematic disk model. The first defect is placed at the position [r ; 0]. The minimum of this energy landscape on the left hand side of the domain corresponds to [r~0.668 ; θ=π] Supplementary Video 1: Evolution of a Fibroblasts monolayer confined in a disk-shape domain (R 0 =400µm). Left: phase contrast; Right: orientation field. With time, defects annihilate until only +1/2 disclinations remain NATURE PHYSICS

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