A tug-of-war between stretching and bending in living cell sheets
Pierre Recho, Jonathan Fouchard, Tom Wyatt, N. Khalilgharibi, Guillaume Charras, Alexandre Kabla
AA tug-of-war between stretching and bending in living cell sheets
P. Recho , , J. Fouchard , T. Wyatt , , N. Khalilgharibi , , G. Charras , , , A. Kabla . LIPhy, CNRS–UMR 5588, Universit´e Grenoble Alpes, F-38000 Grenoble, France Department of Engineering, Cambridge University, Cambridge, UK London Centre for Nanotechnology, University College London, London, UK Centre for Computation, Mathematics and Physics in the Life Sciencesand Experimental Biology, University College London, London, UK Institute for the Physics of Living Systems, University College London, London, UK Department of Cell and Developmental Biology, University College London, London, UK ∗ (Dated: June 11, 2020)The balance between stretching and bending deformations characterizes shape transitions of thinelastic sheets. While stretching dominates the mechanical response in tension, bending dominatesin compression after an abrupt buckling transition. Recently, experimental results in suspendedliving epithelial monolayers have shown that, due to the asymmetry in surface stresses generated bymolecular motors across the thickness e of the epithelium, the free edges of such tissues spontaneouslycurl out-of-plane, stretching the sheet in-plane as a result. This suggests that a competition betweenbending and stretching sets the morphology of the tissue margin. In this study, we use the frameworkof non-euclidean plates to incorporate active pre-strain and spontaneous curvature to the theory ofthin elastic shells. We show that, when the spontaneous curvature of the sheet scales like 1 /e ,stretching and bending energies have the same scaling in the limit of a vanishingly small thicknessand therefore both compete, in a way that is continuously altered by an external tension, to definethe three-dimensional shape of the tissue. I. INTRODUCTION
Active surfaces are ubiquitous in biology, ranging fromsub-cellular organelles to the complex multi-layered wallscompartmentalising organs. An important feature ofsuch surfaces compared to classical visco-elastic mate-rials is that their mechanical properties depend of thecontrolled cellular metabolic processes that continuouslyinject energy and maintain mechanical tension in the sys-tem [1]. Such activity is responsible for the appearanceof a cleavage cytokinetic furrow driving the division of asingle cell [2, 3], or multicellular topological transitionsduring development such as mesoderm invagination in
Drosophila [4, 5] and inversion of the
Volvox embryo [6]both involving hundreds of cells. The detailed patternsresulting from these mechanical interactions often involveinstabilities where elastic stretching and bending defor-mations play an important role, such as in the ruptureand subsequent curling of single red blood cells [7, 8] orthe formation of villi in the gut [9] and gyri and sulci inthe brain [10–12] which shape entire organs.We focus in this paper on epithelial cell monolayers.These tissues are composed of a single layer of cells lat-erally attached to one another via specialised adhesionproteins, as illustrated on Fig. 1. The inner surface of thecells is covered by a thin cortex composed of a dynamicpolymer meshwork that can contract thanks to molecularmotors which act as active cross-linkers [13]. Epithelialtissues line the surface of organs and vessels, physiologi-cally defining compartments and regulating transport ac-cros them. As such, epithelial monolayers are polarized, ∗ [email protected] with anatomical differences between the two sides on themonolayer. This includes molecular motors which oftenexhibit an asymmetric distribution along the sheet thick-ness axis and are preferentially located to one of the sides[14, 15]. In continuum mechanics theories, this unevendistribution of motors gives rise to both active in-planetensions and out-of-plane torques [16–23]. e FIG. 1. Sketch of a polar cohesive cell monolayer. Darkellipses are the cell nuclei. Black lines show cell-cell junctions.The thin grey line indicates the basal side which developed incontact with the substrate while the apical side was free. Thesubstrate is then removed in our mechanical experiments. Theapico-basal polarity entails a mechanical polarity stemmingfrom the inhomogeneous distribution of active stress alongthe thickness (thick red arrows). This results in the presenceof active tension along mid-plane (black arrows) as well asactive torques out-of plane (rotating arrows).
Our experimental system consists of a suspended cellmonolayer devoid of its substrate and clamped betweentwo cantilevers, one fixed and one mobile, whose spac-ing can be adjusted (Fig. 2). The protocol is detailed in[24, 25]. This experimental condition allows us to specif-ically probe the mechanical properties of the epithelial a r X i v : . [ phy s i c s . b i o - ph ] J un sheets in the absence of confounding effects stemmingfrom the substrate. Since the stiffness of the mobile can-tilever is known, the total traction force on the cantilevercan be measured and the active non-linear rheology ofsuch suspended monolayer can be probed across varioustimescales [26, 27]. We shall focus here on an intermedi-ate timescale ranging roughly between 30 s and 10 minwhere the macroscopic monolayer stress-strain curve iswell captured by an elastic model with an active pre-stress [28]. However, examining the shape of the freeedge of the suspended layer also revealed that the mar-gin locally curls out-of-plane with a high spontaneouscurvature of the order of the inverse of tissue thickness[29] (see also Fig. 2 (b) ).We therefore adopt the framework of Non-Euclideanelastic Plates (NEP) [18, 30] which generalizes the the-ory of F¨oppl-von K´arm´an [31] to account for the presenceof both in-plane pre-strain as well as spontaneous curva-ture. The peculiarity of our analysis is that, followingexperimental observations, we assume that the sponta-neous curvature scales with the inverse of the layer thick-ness leading to a direct competition between stretchingand bending energies to set the shape of the free edgeof the monolayer. We then study how this competitionis controlled by the active pre-strain and spontaneouscurvature using a simple one parameter ansatz which wequalitatively compare to experimental results. Note thatwe do not aim at quantitatively capturing the experimen-tal results which we rather use to motivate our theoreticalstudy.The paper is organized as follows. In section II and III,we present the elastic film framework that we use tomodel the passive behavior of the monolayer and exem-plify in section IV that an external compression is neededto observe a buckling behavior characterized by a tran-sition from a stretching to a bending dominated regime.We then complete our model in section V by account-ing for tissue activity. Next, in section VI, we assume asimple deformation ansatz that reduces the mechanicalproblem to a single parameter characterizing the shapeof the free edge. We use this parameter in section VII toshow that unlike in the passive case, there is a continu-ous competition between stretching and bending energieseven in the absence of external loading. We study in sec-tion VIII how applying an external tension changes thebalance between stretching and bending, therefore mod-ifying the film free edge. We finally discuss our results insection IX. II. LARGE DEFORMATIONS KINEMATICS OFTHE MONOLAYER
We denote w the width of the monolayer at its con-tact with the cantilever and L the separation betweenthe two cantilevers. A displacement L − L can be uni-axially imposed from the initial separation L . The bulkof the layer at a given time t is denoted Ω, the free lat- FIG. 2. (a) Picture of the cellular layer suspended betweentwo cantilevers. The dimensions of the layer are L and w =1 . R and the deflection in the center by d . Thecontact lines with the cantilevers are Γ ± s and the free edges areΓ l . (b) Profile of the cellular layer in the ( e , e ) plane where d is measured. Cell-cell junctions are marked in green viaECadherin-GFP. Medium is marked in red. Note the radiusof curvature of the curl is on the same order of magnitudeas the layer thickness [29]. Profile is taken at the positionindicated by the dashed red line on the diagram above. Scalebar: 30 µm . eral surface (i.e. the tissue margin) Γ l , the fixed left sidesurface Γ − s and the right side Γ + s . See Fig. 2 (a). Thefree margin of the layer bridges the gap between the twocantilevers. The deflection at the middle of the bridge isdenoted d . See Fig. 2 (a).The applied displacement may become large (of the or-der of the size of the initial layer) so we do not impose therestriction of small displacements in the theory presentedbelow. In the lab frame ( e , e , e ), the displacement ofthe monolayer is measured from a flat rectangular ref-erence configuration Ω = (0 , L ) × (0 , w ) where all thereference boundaries of the domain Γ l, , Γ ± s, are straightlines. This physically corresponds to the configurationof the cell monolayer in the absence of any internal ac-tive stress or external loading. The position of a materialpoint in the current configuration can thus be written f ( x ) = x + u ( x ) + v ( x ) e , where the reference configuration coordinate x ∈ Ω : x ∈ (0 , L ) , x ∈ (0 , w ) and u = u e + u e is thein-plane ( e , e ) displacement while v is the out-of-planedisplacement. The deformation F is the gradient of f : F = ∂ x u ∂ x u ∂ x u ∂ x u ∂ x v ∂ x v . (1) F is not a square matrix because we consider a twodimensional object embedded in a three dimensionalspace. From the deformation, we compute the membraneCauchy-Green tensor C = F T F and E = C − I I denotes the identity).The unit normal to the monolayer reads n = F e ∧ F e √ det C , where ∧ denote the vector product. The local curvaturetensor at each point of the monolayer is then defined by K = n. ∇ f , that is in index notation [32] K i,j = (cid:88) k =1 .. n k ∂ x i x j f k . (3)Having defined the strain and curvature of the film,we now use these two variables to specify its mechanicalbehavior. III. PASSIVE RHEOLOGY AND BOUNDARYCONDITIONS
We describe the passive response of the cell monolayeras purely elastic. Due to the thin film approximation,the stored elastic energy U can be decomposed into mem-branal and bending terms [33]: U [ u, v ] = (cid:90) Ω [ u s ( E ) + u b ( K )] dx dx . For simplicity, we consider only physically linear elas-ticity (meaning that the energy functionals neglect theterms that are higher than quadratic in the strain andcurvature), isotropic and 3D-incompressible (because thevolume of each cell remains constant in the regime tested[24, 28]). The stretching elastic energy U therefore re-duces to the classical Saint-Venant Kirchhoff expression u s ( E ) = Y e (cid:2) (tr E ) + tr( E T E ) (cid:3) , where Y is the 3D Young’s elastic modulus and e thethickness of the layer in the reference configuration (smallcompared to L and h ). Similarly, the bending energytakes the form, u b ( K ) = Y e (cid:2) (tr K ) + tr( K T K ) (cid:3) . Note that retaining only quadratic contributions in theenergies (i.e. assuming a linear material) is compati-ble with considering large deformations (i.e. geometricalnon-linearities).Using the internal energy, we can define the Piola-Kirchhoff stress N = ∂ E u s and the torque M = ∂ K u b .On the reference boundary Γ l, , the traction stress as well as the torque vanish. Note the absence of a work term inthe above expression of the potential energy U becausethere is no surface where a non-zero traction stress is im-posed. Locally, it is always the displacement which isimposed on Γ ± s, , i.e. ( L − L ) e on Γ + s, and clampedconditions on Γ − s, . The local traction stress on Γ + s, , eT = F N F T .e , which is opposite to the one on Γ − s, cannot be imposed with this device. Instead, we imposea certain displacement such that a target global tractionforce T m = | Γ + s, | − (cid:82) Γ + s, T.e is applied [28]. IV. THE STRETCHING TO BENDINGTRANSITION IN BUCKLING
Before moving to our main results, we re-derive in thissection some classical results about the stretching andbending behavior of a plate in plane strain (i.e. equiva-lent to a one dimensional beam in the e direction) forour specific theoretical setting.To do so, we consider the case where u = 0 and u ( x )and v ( x ) do not depend on x . Then, setting 1+ ∂ x u =Λ cos( φ ) and ∂ x v = Λ sin( φ ) the strain and the curvaturebecome scalar quantities: E = Λ −
12 and K = − Λ ∂ x φ. (4)The variable Λ therefore represents the stretch along the x direction and φ represents the angle of the plate withits tangent. Using Λ and φ we can re-express the bendingand stretching energies: u s = 2 Y e (cid:18) Λ − (cid:19) and u b = Y e
18 (Λ ∂ x φ ) such that the total elastic energy reads, U [Λ , φ ] = (cid:90) L (cid:32) Y e (cid:18) Λ − (cid:19) + Y e
18 (Λ ∂ x φ ) (cid:33) dx . In our problem, the plate is clamped at x = 0 and x = L : v (0) = v ( L ) = 0, u (0) = 0 and u ( L ) = L − L with a slope that we assume null ( ∂ x v (0) = ∂ x v ( L ) =0). In the new variables Λ and φ , these boundary condi-tions become the integrals constraints (cid:90) L (Λ cos( φ ) − − (cid:15) m ) dx = 0 and (cid:90) L Λ sin( φ ) dx = 0 , where (cid:15) m = ( L − L ) /L and the boundary conditions φ (0) = φ ( L ) = 0 . The solution of this problem is therefore obtained byminimization of the Lagrangian L [Λ , φ ] = U [Λ , φ ] − P (cid:90) L (Λ cos( φ ) − − (cid:15) m ) dx − Q (cid:90) L Λ sin( φ ) dx , (5)where the Lagrange multipliers P and Q represent theforces at the boundary in the e and e directions. Thefirst variation of L provides the two coupled equationsdetermining the equilibrium shape: Y e Λ( ∂ x φ ) + Y e Λ(Λ −
1) = P cos( φ ) + Q sin( φ ) Y e ∂ x (cid:0) Λ ∂ x φ (cid:1) = Q Λ cos( φ ) − P Λ sin( φ ) . (6)When (cid:15) m > φ s = 0, Λ s = L (1+ (cid:15) m ), Q s = 0 and P s = 2 Y e s (Λ s − . The total bending energy U b = (cid:90) L u b ( x ) dx therefore vanishes and the total stretching energy U s = (cid:90) L u s ( x ) dx , scales with e leading to a stretching energy dominatedregime where the bending term is irrelevant.The case (cid:15) m < (cid:15) m is lower than the deformationcorresponding to the critical loading threshold (cid:15) mc = (cid:115) − π e L − , the trivial solution φ s , Λ s , Q s , P s stops to be stable andbifurcates through a second order phase transition toa non-homogeneous solution which can be expanded inpower series close to the bifurcation point using theLyapunov-Schmidt reduction technique [34, 35]. Follow-ing this approach, the normal form up to second orderreads: φ c ( x ) (cid:39) φ s + νφ ( x ) + ν φ ( x )Λ c (cid:39) Λ s + ν Λ ( x ) + ν Λ ( x ) Q c (cid:39) Q s + νQ + ν Q P c (cid:39) P s + νP + ν P . (7)As it is classical for a second order phase transition (i.e. asuper-critical pitchfork bifurcation), the small parameterin the expansion is given by ν = (cid:115) (cid:15) m − (cid:15) mc (cid:15) m and in our specific problem, φ ( x ) = φ ( x ) = √ (cid:18) πx L (cid:19) , Λ = P = Q = Q = 0whileΛ ( x ) = π e ( (cid:15) mc + 1) (cid:16) (cid:16) π e L − (cid:17) cos (cid:16) πx L (cid:17) + π e L − (cid:17) L (cid:16) π e L − π e L + 3 (cid:17) , (8) P = π e Y (cid:16) − π e L (cid:17) ( (cid:15) mc + 1)9 L (cid:16) π e L − (cid:17) and (cid:15) m = (cid:16) π e L + π e L − (cid:17) ( (cid:15) mc + 1)4 (cid:16) π e L − π e L + 3 (cid:17) . Using the above expressions we obtain the scaling of thestretching and bending energies for the buckled solutionclosed to the critical threshold: U s (cid:39) π e Y (cid:0) δ (cid:15) + 12 δ (cid:15) + 4 (cid:1) e L and U b (cid:39) π e Y δ (cid:15) (cid:0) δ (cid:15) + 2 √ √ δ (cid:15) + 1 (cid:1) eL where δ (cid:15) = (cid:15) mc − (cid:15) c > e/L of the power series.This shows that as soon as δ (cid:15) > e/L (cid:28) e direction, as imposed with our experimental device, ini-tiates the formation of wrinkles in the e direction [33].Indeed, volume conservation implies a certain level ofcompression which activates the bending energy in thatdirection. Similar to the classical buckling case presentedabove, these wrinkles happen through a bifurcation indi-cating a transition -driven by the external loading- froma regime dominated by the stretching energy to a regimewhere minimization of the bending energy becomes morefavorable.In the following sections, we will show that, due to ac-tive effects, this situation changes as the film exhibits atug-of-war between the stretching and the bending ener-gies even when the film is put under an external tension.This is because both energies scale in the same way withrespect to the small parameter e/L to determine thefilm shape. V. INCORPORATION OF THE ACTIVERHEOLOGY
The biological activity in the monolayer here refersto a contractile acto-myosin polymer network generat-ing mechanical tension in the plane of the monolayer.This active tension, combined with the elastic modulusof the monolayer, leads to the emergence of an effectivepre-strain that can be controlled by modulating the acto-myosin dynamics [28]. In addition to the in-plane com-ponent of the tension, an asymmetry of myosin activityacross the thickness of the monolayer leads to an activetorque, which in turn manifests itself as a spontaneouscurvature of the monolayer [14, 15].Building on the idea of a stretching and bending de-composition, we therefore speculate that the total poten-tial energy reads, U [ u, v ] = (cid:90) Ω [ u s ( E − E a ) + u b ( K − K a )] dx dx , (9)where we suppose that the minimum of the internal en-ergy (i.e. the ground state) is shifted by active effects[17, 18, 30, 36]. In particular, we do not consider herethe fact that activity may modify the functional form ofthe energies u s and u b themselves. This expression ofthe elastic energy has been justified under the classicalKirchhoff-Love assumptions in the limit of small thick-ness of a bulk elastic material with embedded pre-strain[18]. However for a spontaneous curvature of the orderof 1 /e created between the apical and basal side of thecell monolayer, one of the Kirchhoff-Love assumptions(the plane-stress assumption) is no longer verified andwe therefore use this form of the elastic energy as an ef-fective way to capture the competition between stretch-ing and bending that we experimentally observed, ratherthan the one originating from a generic thin film limit.We also assume that the active contributions areisotropic in the monolayer plane: E a = (cid:18) (cid:15) a (cid:15) a (cid:19) and K a = (cid:18) R − a R − a (cid:19) , where (cid:15) a < R a is a spontaneous radius of curvature.The total potential energy U then needs to be mini-mized in the proper kinematically admissible field of dis-placement ( u, v ) (displacements satisfying the imposeddisplacements boundary conditions) to solve the prob-lem. To gain some analytical insight, we follow below amore simple single parameter analysis that captures themononolayer shape. VI. PARAMETRIC MODEL OF THE TISSUEMARGIN CURLING
In experiments, we noticed the presence of a strongcurling at the tissue margin with more pronounced curl-ing in the center of the margin (of the order of 1 /e ) [29].Our hypothesis is that such curling localized at the tissuemargin creates the deflection d by relaxing some bendingenergy. The deflection however remains finite since thisoperation costs stretching energy as it leads to stretchingin the monolayer tangential to the tissue margin. Thedeflection is thus a compromise between stretching andbending of the cell monolayer.To make this reasoning quantitative but keep analyti-cal computations tractable, we postulate that the defor-mation field is an isotropic planar stretch of the rectan-gular configuration Ω = (0 , L ) × (0 , w ) correspondingto a relaxed state in the absence of external stretch andactivity into a rectangular configuration with the actualsize (0 , L ) × (0 , w ). This configuration is then combinedwith a curling normal to the free margins of the mono-layer with the constant radius of curvature R a . A morerefined ansatz would take into account some expected[37] self-similar curling at the margin. The shape of thefree interface is assumed to be an arc circle of radius R .See Fig. 2 and 3. Given the symmetry of the problem,we only consider the lower half of the monolayer in thefollowing analysis.The initial isotropic stretching is related to the defor-mation ansatz in the ( e , e , e ) frame: f stretch1 = x LL f stretch2 = x f stretch3 = 0 . Next, the lower edge curling is captured in the Frenetframe (˜ e , ˜ e , e ) attached to the free margin (SeeFig. 3 (c)) by f curl1 = R a sin (cid:16) ˜ x R a (cid:17) f curl2 = 0 f curl3 = R a (cid:16) − cos (cid:16) ˜ x R a (cid:17)(cid:17) . The curling is normal to the free margin (direction ˜ e )and encompasses the material points denoted as Ω onFig. 3. Thus ˜ x ∈ [ − r m , r m is the length ofmaterial curled at a given point of the interface. A morerefined ansatz involving a non constant curvature of thefree margin would modify the expression of f curl . Pointsoutside of the domain Ω are unaffected by the curling.The final deformation is then the composition of the twodeformations specified above: f = f curl ◦ f stretch .Based on f , we need to evaluate the total energy U inthe reference configuration. To this end, we separatelydefine the deformation into the two domains Ω (wherethere is no curling) and Ω (where there is curling) inthe current configuration and we map them back into Ω and Ω in the reference configuration. We parametrize FIG. 3. Ansatz of the deformation of a flat monolayer from its natural configuration (a) in the absence of active terms(membranal pre-strain and spontaneous curvature) to its curled configuration in presence of activity (c). We decompose thedisplacement into two steps, f stretch and f curl . f stretch brings the monolayer to its actual in-plane dimensions (b) and f curl curlsthe margins to reach the current configuration (c). The hatched regions therefore indicate the material points correspondingto the curled margins in the current configuration. The last panel (d) shows the 3D projections of the ansatz, focusing on thebottom free edge only. ( x , x ) ∈ Ω by using a mapping j transforming [0 , to Ω j : ( λ , λ ) (cid:55)→ (cid:16) x = L λ , x = w λ (cid:104) s ( λ ) − w (cid:105)(cid:17) , where variables λ and λ vary in the unit interval( λ , λ ) ∈ [0 , . The expression of the local deflection s (see Fig. 3 (c)) is given by, s ( λ ) = L (cid:112) − (2 λ − ξ (cid:16) − (cid:113) − ξ − (2 λ − ξ (cid:17) ξ . In the above formula, ξ = L R (10)is a convenient non-dimensional quantity ranging be-tween 0 and 1 that parametrizes the deflection at thecenter of the layer: d = L (cid:16) − (cid:112) − ξ (cid:17) ξ − R a sin L (cid:16) − (cid:112) − ξ (cid:17) R a ξ . (11)Next, we parametrize the domain ( x , x ) ∈ Ω usingpolar coordinates mapping, j : ( r, θ ) (cid:55)→ (cid:18) x = L (cid:18) sin( θ ) (cid:18) rL + 1 ξ (cid:19) + 1 (cid:19) ,x = cos( θ )( L + 2 rξ ) − L (cid:112) − ξ ξ (cid:33) . The angle θ thus varies in the range [ − θ m , θ m ] where θ m = arcsin( ξ ) ∈ [0 , π/
2] and the radius r variesin the range [ − r m ( θ ) ,
0] where r m ( θ ) = L (1 − (cid:112) − ξ sec( θ )) / (2 ξ ). When ξ = 0 (i.e. θ m = 0), themargin is flat and uncurled while the deflection is maxi-mal when ξ = 1 (i.e. θ m = π/ e , e , e ) frame. Namely, deformation inΩ is: f = Lλ f = λ (cid:0) s ( λ ) − w (cid:1) + w f = 0 . (12)and in Ω it reads, f = L sin( θ )2 ξ + L + R a sin( θ ) sin (cid:16) rR a (cid:17) f = cos( θ ) (cid:18) L ξ − L √ − ξ sec( θ )2 ξ (cid:19) + R a cos( θ ) sin (cid:16) rR a (cid:17) f = R a (cid:16) − cos (cid:16) rR a (cid:17)(cid:17) . (13)With L , w and R a given, the deformation f is fully char-acterized by ξ which controls the amount of curling atthe margin as we illustrate on Fig. 4. With our ansatzof f , we can now compute the total elastic energy of thelayer¯ U = U (cid:90) Ω [ u s ( E − E a ) + u b ( K − K a )] dx dx + (cid:90) Ω [ u s ( E − E a ) + u b ( K − K a )] dx dx . (14) FIG. 4. Different views of the margin curling when theparameter ξ increases with L , w and R a fixed. For this we use the deformation fields (12) and (13) tocompute the strain tensor and the curvature tensor ac-cording to formulas (1)-(2)-(3). The integrals and partialderivatives entering in all formulas have to be computedusing the mappings j and j respectively in Ω andΩ . All these computations, though giving potentiallylengthy expressions can be carried out explicitly exceptfor the final integration of local stretching and bendingenergies.However, this last step can also be made explicit usingthe fact that e is very small compared to all other lengths(there are two orders of magnitude between e (cid:39) µ mand L (cid:39) w (cid:39) R a is of theform R a = ek a , where k a is a non-dimensional parameter quantifying themagnitude of the spontaneous curvature [38]. The impli-cation of this last assumption is that, despite the small-ness of the thickness e , the stretching and bending ener-gies have terms contributing at first order in e and canlocally balance each other even in the limit where e isvanishingly small. Indeed, both (12) and (13) can gener- ically be written as f ( x ) = f ( x ) + ef (cid:16) x, xe (cid:17) such that the bending term in (9) contributes to U through f even at first order in e . The computationof ¯ U at first order in e essentially involves asymptotic ex-pansions in the small parameters e/L and e/w as well assome averaging over the variable x/e which varies veryquickly compared to x (similar to the technique employedin vibrational mechanics [39] to extract the slow part ofthe motion).Note that the average tension on the cantilevers onlyinvolves the elementary deformation field in Ω and thustakes a simple form T = eY λ L (cid:0) λ L − (cid:15) a + 1 (cid:1) (cid:0) λ L − (cid:15) a − (cid:1) , where the stretch variable λ L = L/L , can be experimentally adjusted by moving the can-tilevers. For small applied strains the tension reads, T = T a + eY a ( λ L − T a = 4 eY ( (cid:15) a − (cid:15) a / Y a = 2 Y (cid:0) (cid:15) a − (cid:15) a + 1 (cid:1) /
3. It isknown that the effective stiffness of a single cell [40] ora tissue [28, 41] can change when some of its molecularmotors are inhibited or activated.The value of (cid:15) a was measured in [28] to be (cid:15) a (cid:39) − . Y a (cid:39)
650 Pa.
VII. INITIAL DEFLECTION
Before any stretching is applied, we consider that λ L = 1 such that the only source of tension T = T a > U − U eY L = 1144 ξ (1 − ξ ) (cid:2)(cid:0) − k a (cid:0) − ξ (cid:1) + 36 (cid:0) − ξ (cid:1) (cid:15) a + 3 (cid:1) × (cid:16) sin − ( ξ ) − ξ (cid:112) − ξ (cid:17) + 12 (cid:0) ξ − (cid:1) (3 (cid:15) a + 1) M ( ξ ) + 6 ξ (cid:112) − ξ (cid:105) where the special function M can be expanded in powerseries, M ( ξ ) = ∞ (cid:88) k =1 ( − k (cid:18) − ( ξ ) cos(2 k sin − ( ξ )) k − sin(2 k sin − ( ξ )) k (cid:19) . (15) FIG. 5. Dependence of the energy U on the deflection vari-able ξ for a given value of (cid:15) a = − .
4. The inset shows thedecay of the energy with ξ for experimentally relevant valuesof k a ∼ and U = ¯ U ( ξ = 0) is a constant independent of ξ . Weshow on Fig. 5 the typical behavior of the energy forseveral values of k a . For small values of ξ , it decreases as¯ U − U ∼ ξ → − eY L k a ξ because curling more material reduces the bending en-ergy while the stretching energy is negligible. This isa bending dominated regime. However, in the oppositelimit where ξ approaches 1 the energy diverges when as¯ U − U ∼ ξ → eY L π − ξ )and is dominated by the stretching contribution. Thebalance between these two regimes determines the equi-librium shape of the free margin of the monolayer.To find the balancing point between bending andstretching, we therefore minimize ¯ U with respect to ξ .It is important to note that Y drops out from the min-imization and therefore does not influence ξ eq , the equi-librium value that minimizes the energy. We show onFig. 6 the dependence of ξ eq on the spontaneous curva-ture parameter k a . As we expect, ξ eq increases with k a asan increase of spontaneous curvature favors curling. Thedependence of ξ eq on k a can be analytically captured intwo asymptotic cases. When k a (cid:28) ξ eq ∼ (cid:114) k a √− (cid:15) a , (16)which degenerates as a square root dependence when (cid:15) a = 0: ξ eq ∼ (7 / / (cid:112) k a /
2. Interestingly, this limitstill accurately captures the value of ξ eq up to moderatevalues of k a (cid:46) k a (cid:28) ξ eq by:24 π (1 − ξ eq ) ∼ √ (cid:0) k a − (cid:15) a − (cid:1)(cid:112) − ξ eq + 288 π (2 (cid:15) a + 1)1 − ξ eq . (17) FIG. 6. The full green line shows the equilibrium value of ξ as a function of k a obtained by numerical minimization ofthe potential energy ¯ U . The dashed lines correspond to theasymptotic formulas (16) (in the k a (cid:28) k a (cid:29) (cid:15) a = − .
4. The inset showshow the deflection associated to the value of ξ eq varies as afunction of the motor activity which proportionally affectsboth the pre-strain and the spontaneous curvature (see (18)).Parameters ¯ (cid:15) a = − . k a = 5. Note that when k a >
1, the spontaneous radius of curva-ture is larger than the monolayer thickness which is ad-missible since it is induced by a mismatch of apical andbasal tension [38] and not related to the actual curvatureof a single cell. However, the applicability of our ansatzmay be questioned in this case where mechanical contactsbetween the folds of the curled region may play an im-portant role. The account of such non-penetration con-straints would require a complex numerical treatment.The equilibrium value of ξ eq can be easily translatedinto a measured deflection through equation (11) which,at zeroth order in e/L reads, dL = 1 − (cid:112) − ξ ξ and increases from 0 when ξ = 0 to 1 / ξ = 1.It is not directly obvious how an inhibition or promo-tion of molecular motors activity will affect the deflectionbecause motors control both (cid:15) a and k a which can haveantagonistic roles on ξ eq (see (16)). However, a simpleassumption is that (cid:15) a scales with the average of the ac-tivity of the motor on the apical and basal side of themonolayer while k a scales with the difference betweenthe activities on both faces of the monolayer. In this re-spect, it is reasonable to assume that both k a and (cid:15) a areaffected in the same proportion if the motor activity ismodified genetically or with drugs. We formally expressthis proportionality as k a = ¯ k a A and (cid:15) a = ¯ (cid:15) a A, (18)where A is a non-dimensional measure of the motor ac-tivity and we show on the inset of Fig. 6 the dependence FIG. 7. Shape of the free edge of monolayers (Γ l on Fig.2) ina control untreated situation (blue line, N=5) and with twodrug treatments (Y27, red line, N=6) and (Latrunculin-B,purple line, N=6) that impair the active tensions in the cellscortex. The thick lines represent the mean behavior whilethe light lines are directly extracted from experiments. Thedashed black line corresponds to a straight bridge between thetwo cantilevers. Misalignment of the monolayers boundariesappears as a result of the uneven spreading of the monolayerson each of the two plates. of d eq /L on A . We observe that the effect of spontaneouscurvature surpasses that of in-plane pre-stress to increasethe deflection of the margin when activity increases. Inagreement with this trend, we show on Fig. 7 the equi-librium shape of the monolayer free edge in response totwo pharmacological treatments that reduce the activityof the cell monolayer either by partially inhibiting themolecular motors (Y27 curve) or by partially depolymer-izing the polymers that serve as scaffolds for molecularmotor contractility (Lat B curve). VIII. DEFLECTION TO ELONGATIONRELATION
From the initial configuration, we can experimentallyapply a finite stretch to the mobile cantilever and observethat the deflection decreases, see Fig. 8, while we wouldexpect an increase of the necking for a passive elasticsheet. To rationalize this observation, we can computeagain the elastic energy which takes a more complex formin this case (see expression in Appendix. A)As for the initial case, this form has a single mini-mum in ξ corresponding to the equilibrium deflection ofthe free tissue margin. Fig. 9 shows how the deflectiondepends on the applied stretch for small, moderate andlarge values of k a .For a small value of k a (cid:28)
1, we can compute the de-flection for small strains d eq L ∼ (cid:113) k a √− (cid:15) a − (cid:113) ( λ L − k a − (cid:15) a ) / (19) FIG. 8. Shape of the free edge of monolayers (Γ l on Fig.2)for different applied tensions. The blue line corresponds tothe initial state where λ L = 0 and T = T a . The black linecorresponds to a smaller tension T = T a . The cyan line corre-sponds to a larger tension T = T a . The thick lines representthe mean behavior while the light lines are directly extractedfrom experiments (N=6). The dashed black line correspondsto a straight bridge between the two cantilevers. Misalign-ment of the monolayers boundaries appears as a result of theuneven spreading of the monolayers on each of the two plates.FIG. 9. Deflection of the layer margin as a function of thestretch. The dashed lines are related to the asymptotic formu-las (19) (when k a (cid:28)
1, red) and (20) (when k a (cid:29)
1, black).We show in inset the value of the deflection normalized bythe initial length L instead of the actual length. Parameter (cid:15) a = − . and in the k a (cid:29) d eq L ∼ − √ π k / a − √ π ( λ L − k a ) / . (20)While the value of the deflection itself is larger for ahigher motor activity A , we expect the slope of the de-flection under stretch to decrease with the motor activity,because such slope scales as A − / for A (cid:28) A − / for A (cid:29) d eq /L , d eq /L does not have tobe smaller than 1 /
2. In the large spontaneous curvature0regime, this ratio starts to increase for small strains. Thisis because ξ eq remains close to 1 since the energy neededto uncurl the margin is very large while the length in-creases. Ultimately, as the stretch becomes large, it canagain balance the bending energy and uncurls the mar-gin leading to a decrease of d eq /L . This behavior is notpresent in the k a (cid:28) d eq /L immediatelystarts to decrease by uncurling the margin in response toeven small stretches. Note that the large tension limitis not necessarily well captured by our ansatz since itdoes not account for stress concentration phenomena in-volved in necking and/or mechanical damage of the cellmonolayer under loading. IX. DISCUSSION
We begun by studying the case of the buckling of anelastic film suspended between two cantilevers and sub-jected to in-plane strain to illustrate the fact that there isa transition from a regime dominated by the stretchingenergy when the film is put under tension to a regimedominated by the bending energy when the film is com-pressed beyond a critical threshold. While the two ener-gies do compete to set the value of this buckling thresh-old, only one of the two is important in each regime todetermine at least qualitatively the object shape. An-other signature in this passive case is that buckling doesnot happen continuously as the compression is graduallyincreased but suddenly through a bifurcation at the crit-ical loading threshold.Next, to model the activity of the cellular monolayer,following the framework of NEP, we have augmentedthe passive film model by introducing a spontaneous in-plane contractility and out-of-plane curvature that orig-inate from the presence of molecular motors unevenlydistributed along the film thickness. As a result, thespontaneous curvature scales with the inverse of the filmthickness leading, even in the absence of an external load-ing, to a competition between the stretching and bendingenergies to set the shape of the free edge of the film.More precisely, by assuming that the shape of the freemargin is an arc of a circle, the elastic energy dependson only a single free parameter that quantifies the centraldeflection of the film. We then show that the minimum ofthe energy corresponding to the mechanical equilibriumof the film exhibits a deflection that balances stretchingand bending. We obtain the expression of this deflectionas a function of the active parameters quantifying thecontractility and spontaneous curvature and conclude, inagreement with experiments, that increasing the molec-ular motor activity leads to a larger deflection.Interestingly, increasing the external stretch applied tothe monolayer continuously modifies the balancing pointbetween stretching and bending in a non-trivial manner.If the spontaneous curvature is not too high, the pre-vailing effect is to uncurl the tissue margin leading toa decrease of the deflection as observed in experiments. However, in the limit of a high spontaneous curvature,we predict that the deflection will first increase as for apassive material because uncurling the layer requires alot of energy until the stretching is enough to uncurl themargin and the deflection decreases again.Overall, our results suggest that unlike in the case ofpassive slender elastic objects where the transition frombending to stretching happens through a sharp transi-tion when the loading is changed, the presence of a spon-taneous curvature scaling with the inverse of the filmthickness leads to a competition between stretching andbending that is continuously affected by an external load-ing. Such competition may be crucial to understand somethree-dimensional mechanical events that happen duringmorphogenesis such as the formation of folds and invagi-nations for instance during gastrulation; or the fractureof an epithelium which happens during the
Drosophila leg disc eversion [29].One interesting follow-up of this work would be to solvethe full mechanical problem with the new assumption ofa small spontaneous curvature scaling like the inverse ofthe film thickness formulated above instead of using anansatz for the deformation. By doing so, one would beable to find the real equilibrium shape of the tissue mar-gin (i.e. not approximating it by an arc of circle) whichwill be characteristic of the competition between stretch-ing and bending. Other non-linear effects could also beinvestigated in this way such as necking under tension orwrinkling [42] . A more fundamental perspective that issuggested by our results is to rigorously develop a theoryfor elastic plates with a spontaneous curvature that scaleswith the inverse of the thickness instead of postulatingthe NEP type energy (9) used in this work. This maygeneralize the framework developed [43] that assumes afinite spontaneous curvature.Another important generalization of this work wouldbe to account for cell-cell rearrangements that are knownto happen over a long timescale during many develop-mental processes [44], such as convergence and extension[45]. This would require to specify in a self-consistentway the time evolution of the target metric controlled by E a and K a as is done for instance in the framework ofmorpho-elasticity [46]. However, in the experiments pre-sented here, cell-cell rearrangements have been shown tobe negligible over hour long time-scales [47]. ACKNOWLEDGMENTS
P.R. acknowledges support from a CNRS-Momentumgrant. J.F. and P.R. were funded by BBSRC grant(BB/M003280 and BB/M002578) to G.C. and A.K. J.F,T.W., N.K. and G.C.were supported by a consolidatorgrant from the European Research Council to G.C. ( Mol-CellTissMech, agreement 647186). T.W. and N.K. werefunded by the UCL Graduate School and the EPSRCfunded doctoral training programme CoMPLEX. N.K.was also in receipt of a UCL Overseas Research Scholar-1ship.
Appendix A: Expression of the elastic energy when λ L (cid:54) = 0 ¯ U − U eY L = δ L ξ ( ξ − (cid:16) ξ (cid:112) − ξ (cid:0) k a (cid:0)(cid:0) ξ − ξ + 30 (cid:1) δ L + (cid:0) − ξ + 92 ξ − (cid:1) δ L + 4 ξ − ξ + 22 (cid:1) + δ L (cid:0) (cid:0) ξ − (cid:1) (cid:15) a − ξ + 248 ξ − (cid:1) − ξ (cid:15) a + 576 (cid:15) a + (cid:0) ξ − ξ + 45 (cid:1) δ L + 4 ξ − ξ + 225 (cid:1) − sin − ( ξ ) (cid:16) k a (cid:0) (cid:0) ξ − ξ + 5 (cid:1) δ L + (cid:0) − ξ + 74 ξ − (cid:1) δ L + 20 ξ − ξ + 11 (cid:1) + 3 (cid:16) − δ L (cid:16) (cid:0) ξ − (cid:1) (cid:15) a + 32 ξ − ξ +41) + 96 (cid:0) ξ − ξ + 2 (cid:1) (cid:15) a + (cid:0) ξ − ξ + 15 (cid:1) δ L + 40 ξ − ξ + 75 (cid:1)(cid:1) + 12 (cid:0) ξ − (cid:1) (cid:0) δ L + 1 (cid:1) (cid:0) − (cid:15) a + δ L − (cid:1) M ( ξ ) (cid:1) . (A1) [1] T. Lecuit and P.-F. Lenne, Nature reviews Molecular cellbiology , 633 (2007).[2] H. Turlier, B. Audoly, J. Prost, and J.-F. Joanny, Bio-physical journal , 114 (2014).[3] A.-C. Reymann, F. Staniscia, A. Erzberger, G. Salbreux,and S. W. Grill, Elife , e17807 (2016).[4] A. C. Martin, M. Gelbart, R. Fernandez-Gonzalez,M. Kaschube, and E. F. Wieschaus, The Journal of cellbiology , 735 (2010).[5] G. W. Brodland, V. Conte, P. G. Cranston, J. Veld-huis, S. Narasimhan, M. S. Hutson, A. Jacinto, F. Ul-rich, B. Baum, and M. Miodownik, Proceedings of theNational Academy of Sciences , 22111 (2010).[6] S. H¨ohn, A. R. Honerkamp-Smith, P. A. Haas, P. K.Trong, and R. E. Goldstein, Physical review letters ,178101 (2015).[7] D. Kabaso, R. Shlomovitz, T. Auth, V. L. Lew, andN. S. Gov, Biophysical journal , 808 (2010).[8] A. Callan-Jones, O. E. A. Arriagada, G. Massiera,V. Lorman, and M. Abkarian, Biophysical journal ,2475 (2012).[9] A. E. Shyer, T. Tallinen, N. L. Nerurkar, Z. Wei, E. S. Gil,D. L. Kaplan, C. J. Tabin, and L. Mahadevan, Science , 212 (2013).[10] T. Tallinen, J. Y. Chung, F. Rousseau, N. Girard,J. Lef`evre, and L. Mahadevan, Nature Physics , 588(2016).[11] M. Holland, S. Budday, A. Goriely, and E. Kuhl, Phys-ical review letters , 228002 (2018).[12] E. Karzbrun, A. Kshirsagar, S. R. Cohen, J. H. Hanna,and O. Reiner, Nature physics , 515 (2018).[13] G. Salbreux, G. Charras, and E. Paluch, Trends in cellbiology , 536 (2012).[14] D. St Johnston and B. Sanson, Current opinion in cellbiology , 540 (2011).[15] A. Asnacios and O. Hamant, Trends in cell biology ,584 (2012).[16] H. Liang and L. Mahadevan, Proceedings of the NationalAcademy of Sciences , 22049 (2009).[17] J. Dervaux, P. Ciarletta, and M. B. Amar, Journal ofthe Mechanics and Physics of Solids , 458 (2009).[18] E. Efrati, E. Sharon, and R. Kupferman, Journal of the Mechanics and Physics of Solids , 762 (2009).[19] H. Berthoumieux, J.-L. Maˆıtre, C.-P. Heisenberg, E. K.Paluch, F. J¨ulicher, and G. Salbreux, New Journal ofPhysics , 065005 (2014).[20] N. Murisic, V. Hakim, I. G. Kevrekidis, S. Y. Shvarts-man, and B. Audoly, Biophysical journal , 154(2015).[21] M. Krajnc and P. Ziherl, Physical Review E , 052713(2015).[22] G. Salbreux and F. J¨ulicher, Physical Review E ,032404 (2017).[23] P. A. Haas and R. E. Goldstein, Physical Review E ,022411 (2019).[24] A. R. Harris, L. Peter, J. Bellis, B. Baum, A. J. Kabla,and G. T. Charras, Proceedings of the National Academyof Sciences , 16449 (2012).[25] A. R. Harris, J. Bellis, N. Khalilgharibi, T. Wyatt,B. Baum, A. J. Kabla, and G. T. Charras, Nature pro-tocols , 2516 (2013).[26] N. Khalilgharibi, J. Fouchard, N. Asadipour, R. Bar-rientos, M. Duda, A. Bonfanti, A. Yonis, A. Harris,P. Mosaffa, Y. Fujita, et al. , Nature Physics , 1 (2019).[27] A. Bonfanti, J. Fouchard, N. Khalilgharibi, G. Charras,and A. Kabla, BioRxiv , 543330 (2019).[28] T. P. Wyatt, J. Fouchard, A. Lisica, N. Khalilgharibi,B. Baum, P. Recho, A. J. Kabla, and G. T. Charras,Nature Materials (2019).[29] J. Fouchard, T. P. Wyatt, A. Proag, A. Lisica, N. Khalil-gharibi, P. Recho, M. Suzanne, A. Kabla, and G. Char-ras, Proceedings of the National Academy of Sciences , 9377 (2020).[30] M. Pezzulla, N. Stoop, X. Jiang, and D. P. Holmes, Pro-ceedings of the Royal Society A: Mathematical, Physicaland Engineering Sciences , 20170087 (2017).[31] B. Audoly and Y. Pomeau, Elasticity and Geometry:From Hair Curls to the Non-linear Response of Shells (Oxford University Press, 2010).[32] P. G. Ciarlet, Journal of Elasticity , 1 (2005).[33] Q. Li and T. J. Healey, Journal of the Mechanics andPhysics of Solids , 260 (2016).[34] J. C. Amazigo, B. Budiansky, and G. F. Carrier, Inter-national Journal of Solids and Structures , 1341 (1970). [35] W. Koiter, Current trends in the theory of buckling (Springer, 1976).[36] Y. Klein, E. Efrati, and E. Sharon, Science , 1116(2007).[37] A. Callan-Jones, P.-T. Brun, and B. Audoly, Physicalreview letters , 174302 (2012).[38] E. Hannezo, J. Prost, and J.-F. Joanny, Proceedings ofthe National Academy of Sciences , 27 (2014).[39] I. I. Blekhman,
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