suppeentary inratin€¦ · conditions and a continuity condition to determine the unknown...

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“Topological defects in confined populations of spindle-shaped cells” by G. Duclos et al. 1

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SupplementaryMaterial3

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Supplementary Note 1: Characteristic time associated with the activity of the system ........................ 2 8

Supplementary Note 2: Modeling the confined cell layer as a nematic drop ........................................ 3 9

References of Supplementary Material ................................................................................................ 10 10

Legends of Supplementary Figures and Video ...................................................................................... 11 11

Supplementary Figures ......................................................................................................... ................. 13 12

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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS3876

NATURE PHYSICS | www.nature.com/naturephysics 1

© 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

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Supplementary Note 1: Hydrodynamic screening length and characteristic time associated 15 with the activity of the system 16

The hydrodynamic screening length λ is defined as : 17

� � �� (1) 18

where η is the effective 3d monolayer viscosity, h is the height of the cells and γ is the cell-substrate 19 friction. 20

From the literature, we get η ~ 105 Pa∙s (ref 1,2), � ~ 1010 N∙m-3∙s (ref 3–5); we estimate h ~ 5 µm. 21 Therefore, λ ~ 10 µm. The hydrodynamic screening length is of the order of a cell size. 22

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Since, λ is of the order of a cell size, the characteristic time τ results from a balance between activity 24 and cell-substrate friction and we compare this value with the experimentally measured time of 25 annihilation between two defects of opposite charges. 26

We define the characteristic time τ 27

� � �� (2) 28

where � is the cell-substrate friction, �� a cell size, and � is the active stress in 2D. Using the 29 relationship between the activity and the Frank constant K 6, we get: 30

� � �� (3) 31

where L is the distance between defects. 32

To estimate K, we reason that confluent cells must physically deform when the nematic order is 33 distorted. Therefore, K is a function of the Young modulus E. The cell size been the only relevant 34 length scale at this level, we take by dimensional arguments � � � � �� , where E is the Young 35 modulus of the cells (E ~ 1 kPa) 7, and �� is the surface occupied by a cell. We ignore here the 36 other contributions to the Frank constant. 37

Therefore, 38

� � ��

(4) 39

We estimate �� ~ 20µm and, to compare with annihilation dynamics reported in Fig. S4B, we set L = 40 150 µm. 41

Therefore, τ~ 3 h, in good agreement with the experimental annihilation time of typically 4h (Fig. 42 S4B). 43

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Supplementary Note 2: Modeling the confined cell layer as a nematic disk 45

The passive nematic disk 46 We treat the cell layer as a 2-dimensional liquid crystal. In the one-constant approximation, 47 the energy of the system is then given by 48

49

� � �2 �� (��)�(Equation1)

50 with Frank-constant K and nematic director n = cos(ω) ex + sin(ω) ey and ω [0,π[. For sake 51 of simplicity, we set K=1 in the following. A variational calculation shows that the orientation 52 field that minimizes the total energy of the nematic disk satisfies ∇2ω = 0. 53

54 The nematic equations are defined on a circular patch of radius R. We use polar coordinates 55 (r,φ). At the boundaries of the circle, we force the orientation of the director field to be 56 tangential to the surface, as we observe in the experimental situation. Thus, we have the 57 boundary conditions ω(R) = φ + π/2. 58

59 In the following, we first define defects as point sources in the vector potential associated 60 with the director field. Then, we solve the constitutive equations in terms of Fourier-modes in 61 the angular variable φ and a power-series in the radial variable r and use the boundary 62 conditions and a continuity condition to determine the unknown Fourier-coefficients. A 63 particular choice of the integration region then allows us to determine the energy F of the 64 system as a function of a small cutoff length ε. From this, we determine the optimum defect 65 position as a function of ε. We will see that this position converges towards a finite distance 66 r0* from the patch center and a tilt angle θ*=π in the limit ε → 0. Eventually, we use the force 67 on the defect in the limit ε → 0 to determine the distribution of defect positions in a thermal 68 bath. 69

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Defects in nematic liquids 71 The defects are introduced as point sources in the potential that can be associated with the 72 director field. We define the vector potential A for which ∇ω = ∇× Α. Since the nematic 73 director lies in the x-y plane, we write A= Ψ ez with a scalar function Ψ. A +1/2-defect at r074 can now be defined as a point source in Ψ. Here, we put two defects of topological order +1/2 75 at position r1 and r2, hence 76

77 (Equation 2) 78

79

with δ(.) being the Delta-distribution. One easily convinces oneself that, employing the Stokes 80

theorem and using ∇2Α= -∇×∇×Α,81 82

)83

2 NATURE PHYSICS | www.nature.com/naturephysics

SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS3876


2

Supplementary Note 1: Hydrodynamic screening length and characteristic time associated 15 with the activity of the system 16

The hydrodynamic screening length λ is defined as : 17

� � �� (1) 18

where η is the effective 3d monolayer viscosity, h is the height of the cells and γ is the cell-substrate 19 friction. 20

From the literature, we get η ~ 105 Pa∙s (ref 1,2), � ~ 1010 N∙m-3∙s (ref 3–5); we estimate h ~ 5 µm. 21 Therefore, λ ~ 10 µm. The hydrodynamic screening length is of the order of a cell size. 22

23

Since, λ is of the order of a cell size, the characteristic time τ results from a balance between activity 24 and cell-substrate friction and we compare this value with the experimentally measured time of 25 annihilation between two defects of opposite charges. 26

We define the characteristic time τ 27

� � �� (2) 28

where � is the cell-substrate friction, �� a cell size, and � is the active stress in 2D. Using the 29 relationship between the activity and the Frank constant K 6, we get: 30

� � �� (3) 31

where L is the distance between defects. 32

To estimate K, we reason that confluent cells must physically deform when the nematic order is 33 distorted. Therefore, K is a function of the Young modulus E. The cell size been the only relevant 34 length scale at this level, we take by dimensional arguments � � � � �� , where E is the Young 35 modulus of the cells (E ~ 1 kPa) 7, and �� is the surface occupied by a cell. We ignore here the 36 other contributions to the Frank constant. 37

Therefore, 38

� � ��

(4) 39

We estimate �� ~ 20µm and, to compare with annihilation dynamics reported in Fig. S4B, we set L = 40 150 µm. 41

Therefore, τ~ 3 h, in good agreement with the experimental annihilation time of typically 4h (Fig. 42 S4B). 43

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Supplementary Note 2: Modeling the confined cell layer as a nematic disk 45

The passive nematic disk 46 We treat the cell layer as a 2-dimensional liquid crystal. In the one-constant approximation, 47 the energy of the system is then given by 48

49

� � �2 �� (��)�(Equation1)

50 with Frank-constant K and nematic director n = cos(ω) ex + sin(ω) ey and ω [0,π[. For sake 51 of simplicity, we set K=1 in the following. A variational calculation shows that the orientation 52 field that minimizes the total energy of the nematic disk satisfies ∇2ω = 0. 53

54 The nematic equations are defined on a circular patch of radius R. We use polar coordinates 55 (r,φ). At the boundaries of the circle, we force the orientation of the director field to be 56 tangential to the surface, as we observe in the experimental situation. Thus, we have the 57 boundary conditions ω(R) = φ + π/2. 58

59 In the following, we first define defects as point sources in the vector potential associated 60 with the director field. Then, we solve the constitutive equations in terms of Fourier-modes in 61 the angular variable φ and a power-series in the radial variable r and use the boundary 62 conditions and a continuity condition to determine the unknown Fourier-coefficients. A 63 particular choice of the integration region then allows us to determine the energy F of the 64 system as a function of a small cutoff length ε. From this, we determine the optimum defect 65 position as a function of ε. We will see that this position converges towards a finite distance 66 r0* from the patch center and a tilt angle θ*=π in the limit ε → 0. Eventually, we use the force 67 on the defect in the limit ε → 0 to determine the distribution of defect positions in a thermal 68 bath. 69

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Defects in nematic liquids 71 The defects are introduced as point sources in the potential that can be associated with the 72 director field. We define the vector potential A for which ∇ω = ∇× Α. Since the nematic 73 director lies in the x-y plane, we write A= Ψ ez with a scalar function Ψ. A +1/2-defect at r074 can now be defined as a point source in Ψ. Here, we put two defects of topological order +1/2 75 at position r1 and r2, hence 76

77 (Equation 2) 78

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with δ(.) being the Delta-distribution. One easily convinces oneself that, employing the Stokes 80

theorem and using ∇2Α= -∇×∇×Α,81 82

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� �� = �(Equation3)�

on any closed curve C encompassing a single defect. In the following, we place the two 84 defects at the same distance from the center of the circular patch. Their positions are then 85 parametrized by a radial component, r0, and an angle, θ. For sake of simplicity: 86

87 �� = �� o��θ 2� � � �� in�θ 2� �� and �� = �� o��θ 2� � � �� in��θ 2� ��.88

89 90

Solution on the circle 91 Expanding the angular variable in Fourier-modes, 92

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we note 96 97

. (Equation 4) 98

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We immediately see that an = 0 for all n and Ψ0 = 0 (for r < r0) since Ψ must not diverge at r = 100

0. Note that ∂Ψ/∂r has a finite jump at r = r0 and expanding the right hand side of 2 in a 101

Fourier-series, 102 103

(Equation 5) 104

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we find 106 107

. (Equation 6) 108

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Imposing continuity of Ψn at r0, it follows Ψ0 = -1 (for r ≥ r0), �� =��(�� )��

and 110

.111

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For θ, we then have 113 114

(Equation 7) 115

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and we can apply the boundary condition θ(R) = φ + π/2 to determine bn and b�� . 117

118 Together, we note 119

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(Equation 8) 121

122 Summing the Fourier-series, we obtain a closed form for θ,123

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(Equation 9) 126 127 128

The energy of the system 129 At the defect positions, ∇ω diverges such that the system's energy given by Eq. (1) diverges 130 as well. One can circumvent this problem by introducing a finite area around the defect that is 131 excluded from the integration; see Fig. 1 for an illustration. The value of F then crucially 132 depends on the typical cutoff length ε of that region. In particular, F ~ -log(ε) as ε → 0. Since 133 we are not interested in the value of F itself but rather the position of its minimum, we still 134 attempt to derive an expression for F as a function of the defect's distance from the center, r0,135 the inclination angle θ, and the cutoff length ε. The integration can be significantly simplified 136 when carried out in polar coordinates and breaking the integration over r into two parts, 137 ranging from 0 to r0-ε and r0+ε to R, respectively, see Fig. A. We find 138

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(Equation 10) 141

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Figure A: Area of analytical integration144

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For all fixed values of ε, Fε(r0,θ) has a single minimum with respect to variations of r0, θ, and 146 ε. Successively reducing the value of ε, the area of integration approximates the full circle and 147 in the limit ε → 0, the position of the minimum converges to a well-defined point at 148

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(Equation 11) 150

and 151

(Equation 12) 152

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Both values are in excellent agreement with the experimentally determined values (Fig. 4D-154 F).155

Since the diverging terms in Eq.(10) appear in a logarithm, we can separate them as offsets 156 that don't contribute to the forces exerted on the defects. Neglecting these terms when taking 157 the limit ε → 0, the energy of the system can be written as 158

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(Equation 13) 161

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A surface plot of the ensuing energy landscape is shown in Fig. S8.163 164

Finite size of the defects 165 For all values ε>0, the energy of the nematic liquid as a function of r0 remains finite and 166 possesses a single minimum. The position of the minimum changes with ε but converges to r0

*167 = 5-1/4 R as ε → 0, see Fig. B, C. We find experimentally that this position of the minimum of 168 the energy coincides well with the most likely radial position of the defects. 169

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Figure B: Energy of the nematic drop as a function of the defect position, ε = 0.05 (blue, circle) 0.02 171 (green, squares) 0.01 (red, diamonds). Symbols are results of the numeric integration, solid lines show 172

equation 10. The dashed line goes through the minima under variation of ε. 173

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Figure C: Minimum position under variation of defect size ε 176

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Taking the defect size (ε) of a cell size (~10µm), which is compatible with the width of the 178 order parameter at a defect (Figure D), we get: 179

ε(R=400µm) = 2.5∙10-2 < ε < ε(R=250µm) = 4∙10-2 180

And therefore (Fig. C) 181

0.68 < r0* < 0.69 182

For all practical purposes of comparison with the experimental results, these values do not 183 differ significantly from the ε = 0 value. 184




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Figure D: Left: Map of the order parameter near a +1/2 defect core (average map over 6 (+1/2) 186 defects from 2 independent experiments). Right: profile of the order parameter along and 187

perpendicular to the comet tail. Defects are slightly anisotropic but their width is of the order of 10 188 µm. 189

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Fluctuations of the defect position 191 In the experimental system, the final position of the defects fluctuates around a finite value. 192 The distribution's maximum, however, coincides well with the predicted minimum of the 193 energy at (r0*,θ*). Here, we interpret the position of the defect as a stochastic variable subject 194 to fluctuations by the stochastic movement of fibroblasts as well as a drift force from the 195 potential F0(r0,θ). Without caring too much about the microscopic details, we assume the 196 movement to be equivalent to that of a diffusive particle in a potential and describe the 197 probability for the defects to be at position x at time t, p(x,t), by the Fokker-Planck equation 198

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(Equation 14) 200

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For stationary situations without any probability fluxes, the probability density is given by a 202 Boltzmann law 203

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. (Equation 15) 205

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When fitting the experimental values, we observe that the agreement is best for very small 207

values of ε (ε ≅ 10-6 R). We therefore consider the results of the case ε = 0 in the following. In 208

this description, Τ is a fitting parameter that contains information about the defect mobility 209

and the amplitude of fluctuations. It is measured in units of energy, which are here given as 210

multiples of Frank constant K. In thermal systems, Τ would be proportional to a temperature, 211

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hence the naming convention. 212

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We fit the solution of p(r) to the experimentally found distributions of defect positions to 214 estimate the degree of noise in the system, see Fig.4D-G in the main text. We find qualitative 215 and quantitative agreement between experiment and theory for T=0.10 (R = 250μm), T=0.21 216 (R = 300μm), T=0.20 (R = 350μm), and T=0.15 (R = 400μm) in the absence of Blebbistatin, 217 see Fig. 4G in the main text. The temperature Τ is roughly constant as a function of system 218 size with an average of Τ∗ = 0.16, much smaller than 1. When Blebbistatin is added at 219 concentrations c = 1µM and c = 3µM, the estimated values of T are T=0.16 and T=0.17, 220 respectively. 221

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References of Supplementary Material 225

1. Marmottant, P. et al. The role of fluctuations and stress on the effective viscosity of cell 226 aggregates. Proc. Natl. Acad. Sci. 106, 17271–17275 (2009). 227

2. Guevorkian, K., Colbert, M.-J., Durth, M., Dufour, S. & Brochard-Wyart, F. Aspiration of 228 Biological Viscoelastic Drops. Phys. Rev. Lett. 104, 1–4 (2010). 229

3. Cochet-Escartin, O., Ranft, J., Silberzan, P. & Marcq, P. Border forces and friction control 230 epithelial closure dynamics. Biophys. J. 106, 65–73 (2014). 231

4. Mayer, M. et al. Anisotropies in cortical tension reveal the physical basis of polarizing cortical 232 flows. Nature 467, 617–21 (2010). 233

5. Hannezo, E., Prost, J. & Joanny, J.-F. Growth, homeostatic regulation and stem cell dynamics 234 in tissues. J. R. Soc. Interface 11, 20130895–20130895 (2014). 235

6. Ramaswamy, S. The Mechanics and Statistics of Active Matter. Annu. Rev. Condens. Matter 236 Phys. 1, 323–345 (2010). 237

7. Solon, J., Levental, I., Sengupta, K., Georges, P. C. & Janmey, P. a. Fibroblast adaptation and 238 stiffness matching to soft elastic substrates. Biophys. J. 93, 4453–61 (2007). 239

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Legends of Supplementary Figures and Video 242

Supplementary Figure 1 : The system is at quasi-equilibrium. By using low Calcium conditions ([Ca2+] 243 = 50 µM), proliferation of RPE1 cells was impaired, causing the order parameter to remain constant 244 (black points) whereas it increased in normal conditions (red points). Error bars are SDs. N=25. 245

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Supplementary Figure 2: Decrease of the number of defects with time. The color codes for the local 247 order parameter. The orientation field is represented by LIC (Line Integral Convolution). Low order 248 parameter spots are associated with the presence of defects. As times goes, defects of opposite 249 charges pairwise annihilate. NIH-3T3 cells. 250

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Supplementary Figure 3: Directed motion of +1/2 defects. These defects migrate along lines of low 252 order but where the order parameter changes continuously (lines of kinks). NIH3T3 cells. 253

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Supplementary Figure 4: defects of opposite charges pairwise annihilate. A/ Details of the 255 annihilation. B/ The separation between two defects in the process of annihilation varies like t1/2. ta is 256 the time at which the two defects annihilate (N=8 annihilation events, the green line is the average 257 and the colored area is the SD. The red line is a fit by (ta-t)1/2 in the 4 hours preceding annihilation). 258 NIH-3T3 cells. 259

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Supplementary Figure 5: Splay and bend distortion energies. A/ Time evolution of the orientation 261 field, and the local splay and bend distortion energies. NIH-3T3 cells. The positions of the defects are 262 highlighted with the yellow circles in the LIC images (scale bar: 500 µm). B/ Time evolution of the 263 splay distortion energy in the cellular nematic layer. The energy has been integrated over the field of 264 view. The energy decreases until reaching a plateau 40 hours after confluence, once the defects’ 265 density is stabilized. (N=130, colored area: SD). 266

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Supplementary Figure 6: After 60h post-confluence in a circular domain, the majority of the domains 268 contains two facing +1/2 defects. NIH-3T3 cells, N=145 domains, R0=350µm. 269

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Supplementary Figure 7: The position of the defects in the disk patterns is independent of the 271 activity of the system (via the Blebbistatin concentration) and even of the cell type. Error bars=SD, 272 N=25. 273

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Supplementary Figure 8: The orientation field observed in a representative experiment (A) and the 275 one computed from the nematic disk model (B) are very similar. LIC representations. ((A): R=350µm). 276

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Supplementary Figure 9: Energy profile in a disk of radius R0=1, computed from the nematic disk 278 model. The first defect is placed at the position [r ; 0]. The minimum of this energy landscape on the 279 left hand side of the domain corresponds to [r~0.668 ; θ=π]. 280

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Supplementary Video 1: Evolution of a Fibroblasts monolayer confined in a disk-shape domain 282 (R0=400µm). Left: phase contrast; Right: orientation field. With time, defects annihilate until only 283 +1/2 disclinations remain. 284

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Supplementary Figures 286

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