Tissue Homeostasis: a Tensile State

March 2015 EPL, 109 (2015) 58005 www.epljournal.org doi: 10.1209/0295-5075/109/58005

Tissue homeostasis: A tensile state

N. Podewitz1,M.Delarue2 and J. Elgeti1 1 Theoretical Soft Matter and Biophysics, Institute of Complex Systems, Forschungszentrum Juelich 52425 Juelich, Germany 2 Physics Department, Stanley Hall, University of Berkeley - Berkeley, USA

received 10 December 2014; accepted in ﬁnal form 23 February 2015 published online 13 March 2015

PACS 87.19.R- –Mechanicalandelectricalpropertiesoftissuesandorgans PACS 87.17.Aa –Modeling,computersimulationofcellprocesses PACS 87.17.Ee –Growthanddivision

Abstract –Mechanicsplayasignificantroleduringtissuedevelopment.Oneofthekeychar- acteristics that underlies this mechanical role is the homeostatic pressure, which is the pressure stalling growth. In this work, we explore the possibility of a negative bulk homeostatic pressure by means of a mesoscale simulation approach and experimental data of several cell lines. We show how different cell properties change the bulk homeostatic pressure, which could explain the benefit of some observed morphological changes during cancer progression. Furthermore, we study the dependence of growth on pressure and estimate the bulk homeostatic pressure of five cell lines. Four out of five result in a bulk homeostatic pressure in the order of minus one or two kPa.

Introduction. – While growth of eukaryotic cells is one of the walls acts as a movable piston connected to a mainly determined by signaling and nutrition, the notion spring. Cells are placed inside the chamber and the tis- that mechanics also play a role has been advancing con- sue grows, compressing the spring. Eventually, the force tinuously over the years. Ingber and colleagues noted exerted by the piston on the cells is large enough to slow already thirty years ago that tensile stresses can regu- down division, yielding a steady state, where apoptosis late tissue architecture [1]. Today, mechanotransduction balances cell division. and mechanobiology are important fields of research [2,3]. It is a challenging task to measure this homeostatic pres- From a physics point of view, a cell has to increase its sure experimentally. In some attempts, researchers em- volume in order to accommodate for new cell material. bedded cells in an elastic shell that can be deformed by In terms of thermodynamics, pressure is the conjugated the tissue, but the tissue never reached a steady state and variable of volume. Therefore, it seems natural to investi- therefore a well-defined pressure [7]. Another approach gate the pressure cells can develop as well as its feedback exerted a compressive stress directly onto multicellular onto growth. Mechanical feedback on growth has been spheroids [5,8,9], which resulted in a slow down in growth. implemented in many different ways [4,5]. One intuitive Adetailedanalysisshowedthatspheroidgrowthis approach is to expand the growth rate in powers of the highly dominated by surface effects in the sense that cell pressure around the zero growth rate pressure —the home- division occurs mainly at thefirst2–3celllayersofthe ostatic pressure [6]. In this theory, the homeostatic state free surface. Although some physiological gradients of nu- of a tissue is characterized by the balance of cell division trients or growth factors exist [10], this surface growth and cell death (apoptosis) rate, which reflects a dynamic effect is also seen in simulations [9], where we can de- steady state with respect to the number of cells. The tis- couple the mechanics from the biochemistry. A possible sue maintains a well-defined pressure, called homeostatic intuitive explanation by simple mechanics would be as fol- pressure PH ,andtheoverallgrowthratek around the lows. A growing cell increases its volume, creating a strain homeostatic state depends on the difference between PH dipole. The energy required to insert such a dipole at the and the externally imposed pressure P i: surface of the tissue is smaller than in the bulk as in the former case part of the necessary strain field is cut away. k = κ(P P i). (1) H − Therefore, division is favored at free surfaces. The concept is best understood with a simple gedankenex- The net growth in the bulk of the aforementioned tissue periment :atissueisconfinedinasmallchamber,where experiments was determined to be negative. This leads to

58005-p1 N. Podewitz et al. adynamicsteadystate,wheretheapoptoticcoreisbal- t =0d t =3d t = 15 d cell age [d] 0 anced by an influx of cells from the surface [11]. Could we thus deduce the homeostatic pressure to be negative? 0.2 Does a negative homeostatic pressure even make sense? On the one hand, many working in the field of tissue me- 0.4 chanics consider cells to be in a tensile state and, therefore, could indeed expect the homeostatic pressure to be nega- 0.6 tive. On the other hand, from a physical point of view, a F [a.u.] 0.8 material under tension seems to contradict expansion and | z| only yield unstable states. 0 3 15 t [d] We used the same mesoscopic simulation model as in [8,9,12] to perform a gedankenexperiment of tissue Fig. 1: (Color online) Visualization of the concept of a nega- growth in a box with a piston (see fig. 1 and Supplemen- tive homeostatic pressure. Sticky walls are modeled by parti- tary Movie S1 neghp.avi), demonstrating the concept cles similar to normal cells, except they are fixed with a strong and stability of a negative homeostatic pressure. Initially, harmonic potential (purple particles). The piston at the top is, cells are seeded at the bottom of the box, adhere to the furthermore, subjected to a weak harmonic potential depicted wall and start to grow upwards due to the free surface. by the spring. Snapshots at different times t are shown for The piston, acting as a sticky surface, however, cuts away the same simulation with (left) and without (right) piston (see this free surface and leaves only the on-average apoptotic Supplementary Movie S1 neghp.avi). Cells are color coded ac- bulk. The resulting decrease in cell number and the ad- cording to their age (red corresponds to recently divided cells, hesion forces between cells pull down the piston. Stress while blue corresponds to cells that did not divide for a certain in the bulk becomes tensile, and the growth rate increases amount of time). Note that most cell divisions occur at free surfaces. The sticky piston cuts away this free surface and is until a different steady state (under tension) is reached. pulled down due to the negative bulk homeostatic pressure. Methods. – In order to study the behavior of growing tissues, we use the same mesoscopic simulation model is fixed, whereas the second derivative of the potential as in [8,9,12]. Briefly, each cell consists of two point par- around this minimum is varied, leading to softer/stiffer ticles that repel each other with a growth force F g = ij tissues. The compressibility K is defined as 1/f ∗. 2 0 B/(rij + r0) rîj ,whereB is the growth force strength, In order to measure the bulk dependences on pressure, rij the distance between the particles i and j, r0 some we implement a constant pressure ensemble as detailed constant and rîj the unit displacement between i and j. in [13]. The described method imposes a defined pressure Upon exceeding a certain critical distance, the cell divides. on a system with full periodic boundaries. This pressure is Apoptosis is introduced as a constant probability for each maintained through a periodic rescaling of the box volume cell to disappear. These two are the only active com- by a factor ponents in the simulations. The volume exclusion force ∆t i v 5 5 χ =1 βT (P P ), (2) Fij = f0 Rpp/rij 1 rîj ensures impenetrability of the − tP − cells with the cell-cell− potential coefficient f .Allinterac- 0 where β is the isothermal compressibility, ∆t the sim- tions are! short ranged" with a cut-offradius R ,beyond T pp ulation time step, t arelaxationtimeconstantandP i which all interactions are set to zero. Cell adhesion is p the pressure we impose on the system (in simulations: represented by a simple constant force F a = f rˆ and ij 1 ij β /t =1d 1kPa 1 with a time step of ∆t =10 3 d). tuned by the adhesion strength f . − T P − − − 1 Furthermore, the center of mass of all cells is rescaled by To account for dissipation and random fluctuations, we √3 χ.Thereareingeneralnoconstraintsontheimposed use a dissipative particle dynamics (DPD)-type thermo- r R pressure, which makes it possible to simulate systems un- stat. It consists of a random force Fij = σω (rij )ξij rîj der tension (P i < 0) as well. Following [14], the current with σ,thestrengthofthisforce,ξij = ξji asymmetric pressure P inside the full periodic simulation box is cal- Gaussian random variable with zero mean and unit vari- R culated through the evaluation of the virial stress ance and a weight function ω (rij )andadissipativeforce F d = γωD(r )(v rˆ )rˆ with γ the strength of this ij ij ij ij ij 1 force, v− = v v the· relative velocity between i and i i ij ij ij j i σαβ = mivαvβ + rα fβ , (3) − D −V ⎡ ⎤ j and a weight function ω (r ). Unless explicitly men- i i,j ij % % tioned, we use the same standard parameter set and nota- ⎣ ⎦ i tion as in [8,9], i.e. all quantities denoted with a ∗ are in where i is the sum over all particles, vα the α-th com- terms of the standard parameter set (e.g. B∗ = B/Bref). ponent of the velocity of particle i, i,j the sum over all ( ij i j To model different compressibilities, the volume exclu- pair interactions that act on i, rα = rα rα the α-th ( − ij sion force prefactor f0∗ and the adhesion force prefactor f1∗ component of the distance vector between i and j and fβ are varied simultaneously, while f0∗/f1∗ is kept constant. the β-th component of the force acting on particle i.We Thus, the minimum of the potential between particles define the mean pressure as P = 1/3Trσ .Without − αβ

58005-p2 Tissue homeostasis: A tensile state rescaling, the system reaches its homeostatic state, where a) the measured pressure equals the homeostatic pressure. * b H B =0.8 * Experimentally, pressure is applied to spheroids by B =1.0 20 * B =1.2 * adding a biocompatible polymer, Dextran, to the cul- B =1.4 ture medium. Dextran does not penetrate the spheroid, and rather exerts an osmotic stress onto the whole struc- 0 ture. This method, however, can only apply compressive stresses onto the surface of a growing spheroid. The eﬀect

of Dextran can be mimicked in the simulations by a purely Bulk homeostatic pressure P -20 repulsive van der Waals gas, which then applies the stress 0.8 0.9 1 1.1 1.2 1.3 on the system [8,9,11]. This method, however, requires a * Adhesion strength f1 complete tissue spheroid placed in a simulation box with b) hard walls. The pressure applied on the spheroid due to the additional gas particles is then calculated from the mo- b H 20 mentum transfer onto the walls. A direct comparison of the results of the constant pressure ensemble and the van der Waals gas shows a perfect agreement. However, the 0 constant pressure ensemble is computationally much more eﬃcient (due to its smaller system size) and additionally * B =0.8 * it can also impose tensile stresses onto the system. There- B =1.0 * B =1.2

Bulk homeostatic pressure P -20 * fore, all bulk-growth-rate–related results shown in this pa- B =1.2 per were obtained with this method. 0.5 1 1.5 2 The experimental data consists of 5 distinct cell lines of Compressibility K immortalized cells. The CT26 and HT29 cell lines both come from colon carcinoma (HT29 originate from a human Fig. 2: (Color online) Bulk homoestatic pressure dependence b tumor and have a more epithelial phenotype, while CT26 on model parameters. (a) Bulk homeostatic pressure PH in originate from a murine tumor and have a mesenchymal simulation units as a function of adhesion strength f1∗ and growth force B∗.Solidlinesrepresentlinearfits.(b)Bulk phenotype). The AB6 cell line comes from the immor- b homeostatic pressure PH as a function of cell compressibility K. talization of murine spontaneous sarcoma and also has a Solid lines represent linear fits. All data were measured in full mesenchymal phenotype. The FHI cell line, coming from periodic boundaries. Error bars represent standard deviations. Schwannoma, and the BC52 cell line, coming from human breast cancer, both have an epithelial phenotype. R of such a tissue spheroid reads Results. – With the gedankenexperiment described 3 3 2 above, we have seen how a negative homeostatic pressure ∂tR = kbR +3λδksR , (4) could still form a macroscopic tissue and develop a tensile state (see fig. 1). However, to quantify the stress rep- where kb is the growth rate in the bulk and δks > 0isthe sonse of the tissue this setup might still contain boundary growth rate incrementinasmallregionλ at the surface. A 3 effects. We thus utilize the constant pressure ensem- stable steady state (∂tR =0)canonlyariseifkb < 0and ble to better understand the response of growing tissues δk > k .Anunstablesteadystateexistsfork > 0and s − b b to mechanical stresses and describe experimental data in δks < kb.Pluggingeq.(1)intoeq.(4)leadstoasteady more detail. Starting from the standard parameter set state radius− R 1/P b ,whichcanberewrittentoR ss ∝ H ss ∝ as defined in [8,9], we investigated the bulk homeostatic 1/(f1∗ fc∗)usingourearliersimulationresults,diverging pressure dependence on different model parameters. We at the− same critical adhesion strength. This dependence is focused on the growth force strength B∗,theadhesion perfectly reproduced in the simulations as shown in fig. 3. strength f1∗ and the cell stiffness K,sincetheseparame- As a next step, we analyzed the bulk growth rate kb for ters have a straightforward analogon in real tissues and can awiderangeofimposedpressuresP i.Althoughexperi- in principle be measured [15–23]. The results are shown mentally not yet accessible, we simulated not only tissues b i in fig. 2 and display a linear relationship between PH and under compression (P > 0) but also tissues under tension i f1∗ and B∗,respectively.Thecriticaladhesionstrength (P < 0). Our studies reveal a distinct nonlinearity of the i fc∗(B∗)characterizesthetransition from a negative to a bulk growth rate kb with the imposed pressure P . b b positive PH .Finally,wefindPH to increase with the com- In fig. 4 the bulk growth rate is displayed as a func- pressibility K (see fig. 2(b)). This increase is somewhat tion of the imposed pressure P i for different parameter nonlinear and the slope changes with B∗. sets. The general behavior separates into three distinct b AnegativePH naturally leads to a stable steady state regimes. Equation (1) holds in the domain close to the for a freely growing tissue. Assuming a similar two-rate homeostatic pressure (with slope κ), which extends a bit growth model as in [8], the time evolution of the radius further towards cell compression. Under stronger tension,

58005-p3 N. Podewitz et al.

* B =0.8 Table 1: Bulk growth rate kb of several different cell lines with * B =1.0 ss 20 * and without pressure. The pressure PDex exerted by the added B =1.2 * B =1.4 Dextran is 10 kPa for HT29 and CT26 and 5 kPa otherwise. Furthermore, the bulk homeostatic pressure estimation via linear interpolation is shown. Growth rates are taken from [8,24]. 10 1 1 b kb(0) (d− ) kb(PDex)(d− ) PH (kPa) Steady state radius R AB6 0.02(1) 0.04(2) 5(7) 0 BC52 −0.07(1) −0.15(1) −4(1) 0.8 1 1.2 1.4 1.6 − − − * Adhesion strength f1 FHI 0.59(5) 1.08(37) 6(5) HT29 −0.0020(1) −0.13(12) −0.16(15) − − − Fig. 3: (Color online) Steady state radius R of a freely CT26 0.24(1) 0.49(10) 14(4) ss − − − growing spheroid as a function of adhesion strength f1∗ for different growth forces B∗.Thesolidlinesrepresentfitsof Rss = α/(f ∗ f ∗). All data were measured in full periodic 1 − c a) boundaries. Error bars represent standard deviations. 100 2 χ a) 60 tension compression

] * * 10 -3 B =1.0, f1=1.0 B*=0.8, f*=0.8 [10 40 1 b * * B =1.2, f1=1.2 * * B =1.4, f1=1.1

20 Sum of squared residuals 1 -8 -6 -4 -2 0 Bulk homeostatic pressure Pb [kPa] Bulk growth rate k 0 H b) b -30 -20 -10 0 10 20 Bulk homeostatic pressure PH [kPa] -0.1 -1 -10 Imposed pressure Pi

100 2 χ

b) * 1.4 60 tension compression

] * * 1.2 -3 B =1.0, f1=1.0 B*=0.8, f*=0.8 10 [10 40 1 b * * B =1.2, f1=1.2 1 * * κt B =1.4, f1=1.1 linear fits 20 0.8 Growth force strength B

1 Sum of squared residuals κ 0.9 1 1.1 1.2 1.3

Bulk growth rate k * 0 Adhesion strength f1

-10 0 10 20 Fig. 5: (Color online) Sum of squared residuals. (a) The sum of Reduced pressure Pi-Pb H squared residuals χ2 as a function of the bulk homeostatic pres- b sure PH .Thereexistsaclearminimumaround 1to 2kPa. 2 − − Fig. 4: (Color online) Growth rate under compression and ten- (b) The sum of squared residuals χ as a function of growth sion. (a) Bulk growth rate kb as a function of the imposed force B∗ and adhesion strength f ∗ (heat map directly below i 1 pressure P for several parameter sets. (b) Same as in (a) but black crosses with scale on the right) and the bulk homeostatic shifted by the bulk homeostatic pressure P b ,whichcollapses b H pressure PH as a function of the same parameters (heat map all curves onto one. Note the two linear regimes. above black crosses with scale above plot). The black crosses mark the exact values for B∗ and f1∗ used in the according simulations. Note that a low χ2 (blue) always coincides with a b another linear regime with slope κt is observed. The PH of 1to 2kPa(red/orange). 3 − − slopes κ are very similar (κ ( 0.60 0.07) 10− ) ≈ − ± · for all tested parameter sets, while κt seems to change 3 (κt ( 8.0 1.9) 10− ), however, not statistically sig- Former experiments measured the growth of cellular ag- nificant.≈ − The± third region· reveals an asymptotic behavior, gregates under pressure [8,24]. From the growth curves where cell division is mostly suppressed due to the imposed and the surface growth model of eq. (4), the bulk growth pressure and therefore the growth rate kb approaches the rate and the surface rate increment have been extracted fixed apoptosis rate. All curves fall on top of each other, for 5 different cell lines under various intensities of pres- when we shift them by their bulk homeostatic pressure. sure (see table 1). All studied cell lines have a negative

58005-p4 Tissue homeostasis: A tensile state

a) t =0d t = 12 d t =13d t =15d t =24d 0 [1/d] b -0.5 AB6 CT26 BC52 FHI HT29 Bulk growth rate k -1

tension compression Fig. 7: (Color online) Virtual 3d laser cut experiment. Set-up 0 5 10 and colors are the same as in ﬁg. 1. The tissue grows, ﬁlls the Pressure P [kPa] b) compartment and pulls down the piston, until a steady state 0 under tension is reached. Upon laser exposure (visualized by AB6 the red box), cells are taken out of the simulation and the

(P=0)| CT26 b BC52 piston relaxes back to its equilibrium position. The free surface /|k b FHI increases the growth rate of the tissue until the wound is closed -1 HT29 and the same steady state under tension emerges again. linear fit simulation In ﬁg. 5(a) χ2 is plotted as a function of the resulting bulk b homeostatic pressure PH ,withaclearminimumaround 2 -2 1to 2kPa. Althoughweﬁnda f ∗ with a low χ for − − 1

tension compression each B∗, the resulting homeostatic pressure always lies Relative bulk growth rate k around 1to 2kPa(seefig.5(b)). 0 5 10 − − Pressure P [kPa] Surprisingly, a rescaling of the fitted simulations according to the bulk growth rate at zero pressure nicely repro- Fig. 6: (Color online) Experimental data and simulation fits. duces the data of three of the remaining four cell lines (see (a) Experimentally determined absolute bulk growth rate kb as fig. 6(a)). Therefore, rescaling the experimental data with afunctionofthepressureP for different cell lines (closed sym- the zero pressure growth rate collapses all curves onto one bols). The open symbols show the same simulation data that (except HT29). This is shown in fig. 6(b) and suggests a was fitted to CT26, rescaled according to kb(0). The dashed lines represent a linear extrapolation of each cell line. Note certain universality to the shape. that all cell lines have a negative growth rate at zero pressure. Conclusions. – In summary we showed that a nega- (b) Same as in (a) but rescaled with the growth rate at zero tive homeostatic pressure is possible and stable. Indeed, pressure kb(0). The dashed line represents a linear fit of the such a tensile state is suggested by experimental data and data of CT26 and the open symbols show simulation data with fitting our simulations estimates the homeostatic pressure P b 1.3kPa. TherelativebulkgrowthrateofHT29under H ≃− pressure is offscale ( 65). Error bars obtained from jackknife of four out of five analyzed cell lines to be of the order of − estimation. minus one to two kPa. The homeostatic pressure grows with compressibility, growth force and the reduction of bulk growth rate at zero external pressure, indicative of a adhesion. It is known that as cancer evolves, tumor cells negative homeostatic pressure. have a lower expression of the cell-cell adhesive protein To estimate its magnitude, we extrapolate the bulk E-cadherin [25]. Here, we show that a reduction of cell-cell growth rate linearly to zero, using eq. (1). This yields adhesion could increase the homeostatic pressure, which anegativebulkhomeostaticpressureroughlyaround would favor cancer progression [6]. Furthermore, the in- 5kPa. The only exception, for the HT29 cell line, re- crease of homeostatic pressure with cell compressibility sults− in a much smaller value of 0.16 kPa (see table 1). hints for an explanation as to why cancer cells across many However, simulations show that the− linear extrapolation is different origins are consistently found to be softer [17–23]. not valid for all applied pressures, and the data of CT26 The negative homeostatic pressure offers a novel and further underlines this nonlinear trend. simple explanation of how tensile homeostasis is main- In order to match the simulations to the experimental tained. Indeed, many epithelia are found to be under tension in vivo [2,3,26–28]. Laser ablation experiments data of CT26, we varied the parameters growth force B∗ typically show recoil velocities after the cut, clearly prov- and adhesion strength f1∗ around their standard values from [8] and measured the growth rate for several different ing the epithelia to be under tension. Intuitively one could imposed pressures. The pressure rescaling factor Pˆ was think that a tensile homeostasis is unstable: a simple cut then determined for each parameter set by minimizing the relaxes stress, leading to an elevated pressure and thus sum of squared residuals more apoptosis. However, as we have shown, the surface growth effect stabilizes the tensile state. In the simula- 2 χ2 = kCT26(P ) ksim(P /Pˆ) . (5) tions we can perform a 3d virtual laser cut. While a cut b i − b i i leads to a certain recoil, cells grow faster at the new free % ) *

58005-p5 N. Podewitz et al. surface, closing the wound (see fig. 7 and Supplementary [11] Delarue M., Montel F., Caen O., Elgeti J., Movie S2 gcg.avi). Siaugue J.-M., Vignjevic D., Prost J., Joanny J.-F. One question that remains unanswered is: what could and Cappello G., Phys. Rev. Lett., 110 (2013) 138103. be the evolutionary advantage of a tensile homeostatic [12] Basan M., Prost J., Joanny J.-F. and Elgeti J., state? While we can only speculate, a few ideas suggest Phys. Biol., 8 (2011) 026014. [13] Allen M. P., Computer Simulation of Liquids themselves. The combinationofsurfacegrowthandbulk (Clarendon Press, Oxford University Press, Oxford, death naturally lead to finite-sized compartments. The England; New York) 1987. mechanism described here could be a simple method of [14] Tsai D. H., J. Chem. Phys., 70 (1979) 1375. size control. From a mechanics point of view, a tensile [15] Minc N., Boudaoud A. and Chang F., Curr. Biol., 19 tissue connected to a stiffskeleton seems more capable of (2009) 1096. sustaining its shape and integrity under constantly chang- [16] Gonzalez-Rodriguez D., Bonnemay L., Elgeti J., ing external forces. Dufour S., Cuvelier D. and Brochard-Wyart F., Soft Matter, 9 (2013) 2282. [17] Runge J., Reichert T., Fritsch A., Kas¨ J., REFERENCES Bertolini J. and Remmerbach T., Oral Dis., 20 (2014) e120. [1] Ingber D. E., Madri J. A. and Jamieson J. D., Proc. [18] Jonas O., Mierke C. T. and Kas¨ J. A., Soft Matter, 7 Natl. Acad. Sci. U.S.A., 78 (1981) 3901. (2011) 11488. [2] Wozniak M. A. and Chen C. S., Nat. Rev. Mol. Cell [19] Fritsch A., Hockel¨ M., Kiessling T., Nnetu K. D., Biol., 10 (2009) 34. Wetzel F., Zink M. and Kas¨ J. A., Nat. Phys., 6 (2010) [3] Butcher D. T., Alliston T. and Weaver V. M., Nat. 730. Rev. Cancer, 9 (2009) 108. [20] Remmerbach T. W., Wottawah F., Dietrich J., [4] Drasdo D. and Hoehme S., New J. Phys., 14 (2012) Lincoln B., Wittekind C. and Guck J., Cancer Res., 055025. 69 (2009) 1728. [5] Taloni A., Alemi A. A., Ciusani E., Sethna J. P., [21] Cross S. E., Jin Y.-S., Rao J. and Gimzewski J. K., Zapperi S. and La Porta C. A. M., PLoS ONE, 9 Nat. Nanotechnol., 2 (2007) 780. (2014) e94229. [22] Guck J., Schinkinger S., Lincoln B., Wottawah F., [6] Basan M., Risler T., Joanny J.-F., Sastre-Garau Ebert S., Romeyke M., Lenz D., Erickson H. M., X. and Prost J., HFSP J., 3 (2009) 265. Ananthakrishnan R., Mitchell D., Kas¨ J., Ulvick [7] Alessandri K., Sarangi B. R., Gurchenkov V. V., S. and Bilby C., Biophys. J., 88 (2005) 3689. Sinha B., Kießling T. R., Fetler L., Rico F., [23] Lekka M., Laidler P., Gil D., Lekki J., Stachura Scheuring S., Lamaze C., Simon A., Geraldo S., Z. and Hrynkiewicz A. Z., Eur. Biophys. J., 28 (1999) Vignjevi D., Domejean´ H., Rolland L., Funfak A., 312. Bibette J., Bremond N. and Nassoy P., Proc. Natl. [24] Delarue M., Montel F., Vignjevic D., Prost J., Acad. Sci. U.S.A., 110 (2013) 14843. Joanny J.-F. and Cappello G., Biophys. J., 106 (2014) [8] Montel F., Delarue M., Elgeti J., Malaquin L., 42a. Basan M., Risler T., Cabane B., Vignjevic D., [25] Weinberg R., The Biology of Cancer (Garland Science, Prost J., Cappello G. and Joanny J.-F., Phys. Rev. New York) 2007. Lett., 107 (2011) 188102. [26] Trepat X., Wasserman M. R., Angelini T. E., [9] Montel F., Delarue M., Elgeti J., Vignjevic D., Millet E., Weitz D. A., Butler J. P. and Fredberg Cappello G. and Prost J., New J. Phys., 14 (2012) J. J., Nat. Phys., 5 (2009) 426. 055008. [27] Heisenberg C.-P. and Bella¨ıche Y., Cell, 153 (2013) [10] Hirschhaeuser F., Menne H., Dittfeld C., West 948. J., Mueller-Klieser W. and Kunz-Schughart L. A., [28] Martin A. C. and Goldstein B., Development, 141 J. Biotechnol., 148 (2010) 3. (2014) 1987.

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