The mechanical properties of a cell-based numerical model of epithelium

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The mechanical properties of a cell-based numerical model of epithelium

MERZOUKI, Fatma Aziza, MALASPINAS, Orestis Pileas, CHOPARD, Bastien

Abstract

In this work we use a computational cell-based model to study the influence of the mechanical properties of cells on the mechanics of epithelial tissues. We analyze the effect of the model parameters on the elasticity and the mechanical response of tissues subjected to stress loading application. We compare our numerical results with experimental measurements of epithelial cell monolayer mechanics. Unlike previous studies, we have been able to estimate in physical units the parameter values that match the experimental results. A key observation is that the model parameters must vary with the tissue strain. In particular, it was found that, while the perimeter contractility and the area elasticity of cells remain constant at lower strains (20%). However, above a threshold of 50% extension, the cells stop counteracting the tissue strain and reduce both their perimeter contractility and area elasticity.

MERZOUKI, Fatma Aziza, MALASPINAS, Orestis Pileas, CHOPARD, Bastien. The mechanical properties of a cell-based numerical model of epithelium. Soft Matter , 2016, vol. 12, no. 21, p.

4745-4754

DOI : 10.1039/C6SM00106H

Available at:

http://archive-ouverte.unige.ch/unige:88088

Disclaimer: layout of this document may differ from the published version.

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Cite this:Soft Matter,2016, 12, 4745

The mechanical properties of a cell-based numerical model of epithelium

Aziza Merzouki,* Orestis Malaspinas and Bastien Chopard

In this work we use a computational cell-based model to study the influence of the mechanical properties of cells on the mechanics of epithelial tissues. We analyze the eﬀect of the model parameters on the elasticity and the mechanical response of tissues subjected to stress loading application. We compare our numerical results with experimental measurements of epithelial cell monolayer mechanics.

Unlike previous studies, we have been able to estimate in physical units the parameter values that match the experimental results. A key observation is that the model parameters must vary with the tissue strain.

In particular, it was found that, while the perimeter contractility and the area elasticity of cells remain constant at lower strains (o20%), they must increase to respond to larger strains (420%). However, above a threshold of 50% extension, the cells stop counteracting the tissue strain and reduce both their perimeter contractility and area elasticity.

1 Introduction

Epithelium is one of the four basic types of animal tissues, along with the connective, muscle and nervous tissues. They line cavities, glands and surfaces throughout the body. The skin and the intestinal system are major examples thereof.

An important work is done to understand the eﬀect of cell biophysics on the emergence of higher scale properties, such as tissue’s mechanics, shape and size. An increasing number of experiments are performed to characterize the mechanics of single cells and cell monolayers,^1–4and to understand the way the mechanical behavior of tissues results from cellular and subcellular properties.^5,6

In addition to experimental research, computational models of epithelium are used to investigate the contribution of cell mechanics in many biological processes, such as cell sorting,^7–10wound healing,^11–13embryo development,¹⁴and organ morphogenesis.^15,16 Various cell-based modeling approaches are commonly used for these purposes, including Cellular Potts Models, Boundary Cell Models and Vertex Models. They generally depend on three independent energy terms, related to internal cell pressure, actomyosin cortex contractility and intercellular adhesion.¹⁷They differ in their cell representation and dynamics. The Cellular Potts Model¹⁸ is lattice-based, which allows the representation of arbitrary cell shapes, but may be computationally expensive, especially when cells exert forces with non-local effects, such as apical constriction. The Boundary Cell Model¹⁴represents cells

by finely resolved polygons, where each cell has its own edges.

Although it allows neighbor cells to adhere and detach from each other, it needs a permanent detection and management of undesired overlapping of neighbor cells, which increases its computational cost. The Vertex Model,¹⁹which is the approach we use in this paper, represents cells by polygons sharing common edges with neighbor cells, and allows us to effectively model densely packed epithelium tissues with low complexity and reasonable implementation effort.

An important question is to calibrate the parameters of such models. For instance, Farhadifaret al.²⁰estimated the dimensionless parameters of a vertex cell-based model based on the cell packing geometries of Drosophila wing epithelium and its reaction to laser ablation experiments.

In this work, we modify the model proposed by Farhadifar et al.²⁰ to study the mechanical properties of simulated cell monolayers. By comparing our numerical results with experimental measurements obtained from cultured cell monolayers,^1,5 we determine how the mechanical properties of cells evolve with increasing tissue strain. This paper is organized as follows.

Section 2 introduces our cell-based model of epithelial cell monolayers. Results and discussion are presented in Section 3.

In Section 3.1, we compute the Young’s modulus and Poisson’s ratio for a wide range of dimensionless cell contractility and intercellular adhesion parameters. In Section 3.2, we compare numerical simulations with experimental results. In Section 3.3, we explain the algorithm used to calibrate the model parameters from experimental data. In Section 3.4, we show that the experimental results can be better reproduced by adapting the value ofG andKas the tissue strain increases. Finally, we conclude in Section 4 by summarizing the results and presenting the future work.

Department of Computer Science, University of Geneva, Battelle, Building A, Carouge, 7 route de Drize, 1227 Carouge, Switzerland. E-mail: [email protected];

Fax:+41 (0)22 3790250; Tel:+41 22 3790174 Received 14th January 2016,

Accepted 19th April 2016 DOI: 10.1039/c6sm00106h

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2 Numerical model

Our model is based on the 2D model proposed by Farhadifar et al.,²⁰ to investigate the way the mechanical properties of cells in conjunction with cell proliferation determine the development of cell packing geometries.

In this model, a cell is represented by a closed polygon defined by successive vertices interconnected by edges. Two neighbor cells share a common edge and a monolayer epithelial tissue is represented by a network of vertices (see Fig. 1). Its topology is determined by minimizing the energy functionH given by eqn (1).

H¼ X

all cellsa

1

2K_aA_aA⁰_a2

þ X

all cellsa

1

2G_aL_a²þ X

all edgeseij

L_i;jL_i;_j

(1) whereAais the area of a cella,Lais its perimeter andL_i,_jis the length of the edge e_ij connecting verticesv_i and v_j. The first term of H represents the cell area elasticity, where Ka is the corresponding coeﬃcient andA⁰ais the preferred area of the cell a. With Ka being positive, this first term is minimized when the area of the cellAatends to its preferred areaA⁰a. The second term of the energy represents the cell perimeter contractility, produced by the cell cytoskeleton, such as the actomyosin contractile ring running along the cell apical perimeter,^20,21 andGais the corresponding coeﬃcient. WithGabeing positive, the minimization of this second term reduces the perimeter of the cellLaand makes the cell more round-shaped. Finally, the third term describes the line tensionsL_i,jalong edgese_ij, which represent forces along cell/cell junctions resulting from adhesion molecules, mainly Cadherin, connecting neighbor cells. In contrast to the cell perimeter contractilityG_a, which is specific to a cella, the line tensionsL_i,_jmay vary from edge to edge. The value of L_i,_j can be positive or negative. With a negative value ofL_i,_j, the edge is likely to be longer to minimize H, which represents a junction between two cells that have a tendency to adhere to each other. On the other hand, whenL_i,j is positive, the edge is likely to be shorter to minimizeH, which represents a junction between two cells that tend to detach from each other (as used in ref. 20). In summary, reducing here the line tension along an edge separating two cells represents an increase of their adhesion and the expansion of their interface contact. While this last term is motivated by the idea that intercellular junction lengths depend on the potential energy of cell/cell adhesion,^20,21recent studies found that adhesion has

little direct eﬀect on contact expansion. However, it seems to control it indirectly by reducing the actomyosin contractility along the cell/cell contact.^22–24 We emphasize that in this model, since two neighbor cells share a common edge, they cannot detach from each other and rupture the tissue even if the line tension is positive.

To determine the position of vertices that minimizes H, Farhadifar et al.²⁰ used the Conjugate Gradient Method and studied the equilibrium state of the tissue only. In addition, the model proposes a simple way to simulate the growth and division of cells to implement cell proliferation. Because this paper focuses on studying the elasticity of simulated cell monolayers at short timescales, where no cell division or topology rearrange- ment takes place, in agreement with the experiments performed by Harriset al.,⁵we will not consider cell division in our simulations and therefore this part of the model is not described any further.

2.1 Time dynamics and boundaries

Our model implementation diﬀers from that of Farhadifar et al.²⁰Instead of minimizing the energy functionHusing the Conjugate Gradient Method, we use Newtonian dynamics in the same spirit as used in the study of Tamuloniset al.¹⁴This dynamics allows us to follow the physical time evolution (in seconds) of the tissue.

A velocity-dependent friction is added to the force equation to dissipate energy and prevent the system to oscillate forever.

The forceFiacting on each vertexviat a positionri=hx_i,yii, is derived from the energy functionH(see eqn (1)),

F_i¼ dH dr_i¼ 1

2 X

cellacontainsvi

K_aA_aA⁰_a

y_a_iþ1yai1;x_a_i1x_a_iþ1

X

cellacontainsvi

G_aL_a x_ixaiþ1

L_i;a_iþ1 þxixai1

L_a_i1_;i ;y_iy_a_iþ1

L_i;a_iþ1 þyiyai1

L_a_i1_;i

X

edgeeij

L_i;j x_ixj

L_i;j ;y_iy_j L_i;j

;

(2) wherehxa_i1,ya_i1iandhxa_i+1,ya_i+1iare the positions of the previous and next vertices,va_i1andva_i+1, of vertexviin cella, when the vertices ofaare ordered counterclockwise (see Fig. 2).

Newton mechanics are used to determine the acceleration d²r_i

dt² of the vertexv_i,

d²r_i dt² ¼F_i

m_i; (3)

wheremiis the mass of the vertexvi.

Fig. 1 Polygonal representation of cells with shared edges between

neighbor cells. Fig. 2 Counterclockwise ordering of cell vertices.

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To determine the positionr_iof a vertexv_iat the time step t+ dt based on its accelerationd²ri

dt², the diﬀerential equation (eqn (3)) is solved using a damped Verlet integration method with a time stepdt,

riðtþdtÞ ¼ ð2ZdtÞ riðtÞ ð1ZdtÞ riðtdtÞ þdt²d²ri

dt² (4) where Z is the damping parameter controlling the viscosity of the vertex movement during the simulation. In this model there are two additional parameters, the mass mof a vertex and the frictionZ. Their values do not aﬀect the steady state but are important for the time dependent situations discussed below.

A second diﬀerence with respect to the approach of Farhadifar et al.²⁰involves the boundary conditions. Instead of a periodic tissue, we choose to implement open boundary conditions, as illustrated in Fig. 3. The tissue has borders that separate it from its environment (black line). A cell which is located at the border of the tissue is referred to as a boundary cell (red cells) and has at least one boundary edge (black edges). In contrast to a bulk edge (green edges), which is always shared by two neighbor cells, a boundary edge belongs to one cell only. It represents the interface between the cell and the tissue’s environment. The two vertices linked by a boundary edge are referred to as boundary vertices (black points). It is through these boundary vertices that external mechanical constraints are easily applied on the simulated tissue. In addition to the external force, a boundary vertex is also subject to internal constraints generated by the tissue itself (see eqn (2)).

2.2 Model normalized parameters

In eqn (1), the energy function depends on the model parameters Ka, A⁰a, G_a and L_ij.Ka and A⁰a are related to the area elasticity of the cells.G_arepresents their perimeter contractility.

L_ijis the line tension along cell/cell junctions.

As described by Farhadifaret al.,²⁰in a tissue where all the cells are identical in terms of mechanical propertiesKa = K, A⁰a = A⁰, G_a = G, and cell/cell interaction L_ij = L, the model parameters can be reduced to two free and dimensionless variablesLandG. These parameters represent the normalized perimeter contractility of cells and the normalized line tension along cell/cell junctions, respectively. UsingK(A⁰)²and ffiffiffiffiffiffi

A⁰ p

as

units of energy and lengths, and dividing both sides of eqn (1) byK(A⁰)², we get the dimensionless energy function,

H¼1 2

X

cella

A_a1 ð Þ²þ1

2GX

cella

L_a

ð Þ²þL X

edgeei;j

L_i;j (5)

with, H¼ H

KðA⁰Þ², A_a¼Aa

A⁰, G¼ G

KA⁰, L_a¼ L ffiffiffiffiffiffi A⁰ p , L¼ L

KðA⁰Þ³²

andLi;j¼ L_i;j ffiffiffiffiffiffi A⁰ p

In the normalized unit system defined above, the dimensionless force is given byF¼ F

KðA⁰Þ³² :

3 Results and discussion

Our vertex model represents the mechanical properties of single cells and their interactions with neighboring cells. The resulting elasticity at the tissue scale cannot be easily deduced and its dependence upon the model parameters is non-trivial.

In particular, we observe that our vertex model produces a non- linear elasticity behavior for large deformations.

The main goal of our study is to investigate the eﬀect ofK, A⁰,GandLon the mechanical properties of simulated tissues.

For this purpose, we analyze the response of the model to stretching simulations and we compare our results to experimental data.

In Section 3.1, we compute the sensitivity of the Young’s modulus and the Poisson’s ratio toLandG. We compare the result with experimental studies that examine how subcellular perturbations changing the mechanical properties of cells aﬀect the elasticity of a cell monolayer.

In Section 3.2, we analyze the time-dependent deformation of numerically simulated cell monolayers subject to a given stress loading, as well as the equilibrium stress–strain relation.

We compare our numerical results to existing experiments on cultured cell monolayers.

Finally, in Sections 3.3 and 3.4, we propose values forK,A⁰, GandL, in physical units, that make the model best match the experimental cell monolayer mechanics. These values diﬀer from those proposed by Farhadifaret al.²⁰

3.1 Influence of model parameters on the elastic properties of simulated cell monolayers

The parameters of our model influence the elastic properties of the simulated tissues, namely their Young’s modulus Eand Poisson’s ration. To analyze this eﬀect, we measureEandnfor cell monolayers generated with diﬀerent values of normalized cell perimeter contractility G and normalized line tensionL along cell/cell junctions.

Eandnare obtained by stretching a simulated tissue along the X-axis with a given constant stress s (see Fig. 4). Each stretching simulation starts with the creation of a square tissue made up of 4141 regular hexagonal cells with an initial cell area equal to their preferred areaA⁰. First, the created tissue, which is free from external constraint, undergoes an initial Fig. 3 Sheet of cells with open boundary conditions.

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relaxation toward its steady state minimizing its energy. The area of cells at this stage is denoted by Asteady. When Gand L are null, Asteady is equal to A⁰. The length and the width of the tissue at this stage are denoted by Xsteady and Ysteady. Then, the tissue is stretched with a given constant stress s.

When the deformation of the loaded tissue stabilizes, the resulting elongationdXof the tissue along the stretching axis and its shrinkagedYalong the opposite direction are measured (see Fig. 5).

We consider that the steady state is reached and that the deformation of the loaded tissue stabilizes when the normalized standard deviation of the tissue size along the stretching axis stdðXÞ

avgðXÞ over the last 10³

dt iterations is smaller than a threshold value 10⁴.

The stresss(t) at an instanttis the external forceFext(t) per unit area (see eqn (6)). The external force is distributed along the width of the tissueY(t) and its thicknessZ(t). Because we use a 2D-model, the thickness of the cells is considered approximately constantZ(t) =Zand estimated by the diameter of the cell at its steady unconstrained state. The estimation of the applied stress using an approximate constant cell monolayer thickness was used to compute the experimental results⁵ that are compared to our simulations throughout this article.

The Young’s modulusEis the ratio of the linear stresssto the linear strain dX

X_steady (see eqn (7)). It represents the stiﬀness of

the cell monolayer. The Poisson’s ratio n is the ratio of the lateral strain dY

Y_steady to the axial strain dX

X_steady. It measures the thinning of the cell monolayer in the lateral direction with respect to its axial extension. For big deformations, the Poisson’s ratio is computed by eqn (8).

sðtÞ ¼ F_extðtÞ

YðtÞZðtÞ (6)

E¼ s dX X_steady

; (7)

n¼

log Y_steadyþdY Y_steady

log X_steadyþdX X_steady

; (8)

wheredX=XstressXsteadyanddY=YstressYsteady.

The stretching simulations are performed on tissues where all cells are given the same preferred areaA⁰= 9010¹²m², area elasticity coeﬃcient K = 10⁹ N m³, and perimeter contractility G. In ref. 3, the area compressibility modulus KA=KA₀is found to be on the order of 10¹N m¹. Therefore, we estimate our model parameterKcorresponding to the area elasticity of the cell on the order of 10⁹ N m³. All the bulk edges have a line tensionL. The boundary edges are assigned with a line tension L

2. This is a way to keep a similar global tension along all edges. Because a boundary edge belongs to one cell only, it is subjected to one cell cortical tension, compared to the bulk edges, which are shared by two cells, and are therefore subject to a double cortical tension.

Subjecting the boundary edges to a higher tension than bulk edges is possible by assigning them a higher line tensionL than bulk edges. This does not change the qualitative impact of the model parameters on the elastic properties of the simulated tissues. The results stated in what follows remain true. Quantitatively, it is observed that increasing the line tension along boundary edges, while keeping cell contractility and line tension along bulk edges constant, increases the tissue stiﬀness. However, the importance of the tissue stiﬀness increase becomes negligible as the tissue size is bigger.

The value of the applied stress iss= 700 Pa. It corresponds to a dimensionless stresss¼ s

K ðA⁰Þ¹⁼²¼0:0738. This value is considered by Harriset al.⁵as a low stress intensity and yields an extension of 6% with the parameter values (L= 0.12 and G= 0.04) proposed by Farhadifaret al.²⁰

Our results on the influence of the model parameters on the Young’s modulus and Poisson’s ratio are displayed in Fig. 6a and b, respectively. In these figures, each blue dot is a pair (L,G) of normalized parameter values characterizing a cell monolayer whose elastic properties were measured. The contour lines, obtained by interpolating the values computed at each blue dots, are also shown.

Fig. 4 Relaxed tissue and stretching forces applied through boundary vertices.

Fig. 5 Resulting tissue after stretching along theX-axis.

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In both Fig. 6a and b, four regions are visible. No simulations were possible in Regions 1 and 4 due to a non-physical behavior (e.g.the cell size minimizing the energy is 0).²⁵

In Region 2, the increase of the perimeter contractility or the line tension induces an increase in the Young’s modulus.

WhenLandGincrease, the tissue gets stiﬀer. On the right of Region 2, we find a thin region, referred to as Region 3. There, increasing perimeter contractility of cells or the line tension along edges decreases the Young’s modulus.

Fig. 6b shows a large region, called Region 2⁰, where increasingLor G decreases the Poisson’s ratio. In this case less thinning is measured laterally with respect to the linear strain. A thin region of parameter, Region 3⁰, appears between Region 2⁰and Region 4. There, we observe negative values of Poisson’s ratio. They are due to an extension along both stretching and lateral directions, instead of extending along the stretching direction and shrinking along the lateral direction, as it is the case in Region 2⁰. This corresponds to cases where the cell contractility would be so important (and unbalanced by cell/cell adhesion) that stretching the tissue along one direction (X) leads to its extension along the transversal direction (Y) to minimize the energy equation and maintain the cell shape as round as possible.

Laboratory experiments were conducted by Harriset al.⁵on cultured cell monolayers. Actin cytoskeleton was depolymerized and myosin II activity was inhibited using Latranculin B and

Y27632, respectively. It was observed that these two actions decreased the stiﬀness of the tissue. This is consistent with the result of Fig. 6a, in which decreasing cell perimeter contractility Greduces the Young’s modulus of the simulated cell monolayer.

To study the eﬀect of intercellular adhesion on the cultured monolayer mechanics, Harris et al.⁵ disrupted the cell/cell adhesion using a treatment with EDTA, a divalent cation chelator that blocks cadherin-mediated adhesion. They observed that it reduced the stiﬀness of the tissue. This is not consistent with the numerical result of Fig. 6a, in which decreasing the intercellular adhesion (increasing L) increases the Young’s modulus. One explanation of this inconsistency is that the treatment with EDTA did not only disrupt the intercellular adhesion, but it also altered the actin cytoskeleton.²⁶We should therefore compare this experiment with a simultaneous increase of Land decrease ofG. As shown in Fig. 6a, the variation rate of L andG can make the numerical simulation compatible with the experiment. To confirm this hypothesis, additional experiments acting solely on the intercellular adhesion should be performed.

Our results seem to contradict those obtained using a very similar model by Xu et al.²⁷ They found that reducing the contractility of actin-myosin rings and enhancing the cell adhesion increase the stiﬀness of the monolayers. One explanation of this discrepancy is that they used diﬀerent boundary conditions.

In addition to stretching the simulated tissues to study their mechanical response, the upper and lower boundaries (in the transverse direction to stretching) were constrained such that the tissue does not deform in this direction. This additional constraint biases the results obtained when analyzing the eﬀect of the cellular mechanical properties on the tissue mechanical response. In particular this is not consistent with the experiments of Harris et al.,⁵ where indeed little or no tissue deformation was observed along the transversal direction of stretching, but where no experimental constraint was applied on the tissue to force it. Our results show that the tissues generated using this model generally shrink along the Ydirection when they are stretched along the X direction, in contrast to what was observed experimentally. This needs to be highlighted, so that the diﬀerences between the experimental observations and the simulations generated using this model are clear, and so that it questions the validity of the 2D numerical model in this context.

A second point is that Xuet al.studied the system for a range of parameters, which according to our findings (see below) does not correspond to the experimental behavior.

3.2 Numerical and experimental cell monolayer mechanics during stress loading application

In this section, we analyze the mechanical behavior of numerically simulated cell monolayers subjected to diﬀerent stress loading intensities. We compare these numerical results to the mechanical properties of cultured cell monolayers that were experimentally measured.⁵

Fig. 7 represents the time-dependent deformation of an experimentally cultured cell monolayer subjected to stress loadings of 700 Pa and 3000 Pa (see grey and black curves).

These experimental results are compared to the time-dependent Fig. 6 Mechanical properties of a simulated tissue as a function of the

normalized cell perimeter contractilityGand normalized line tensionL, for a stresss= 0.0738. (a) Normalized Young’s modulus of the simulated cell monolayer. (b) Poisson’s ratio of the simulated cell monolayer.

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deformation of a simulated cell monolayer subjected to the same stress loading intensities (see green and red curves). At low-stress loading (700 Pa), it was shown that the strain of cultured cell monolayers increases rapidly in response to stress application before reaching a plateau that subsists over 200 s (see the grey curve of Fig. 7). In contrast, when a high stress (3000 Pa) is applied, no plateau is reached and strain increases continually with time (see the black curve of Fig. 7).

The simulated cell monolayer shown in Fig. 7 is made up of cells withG= 0.16 andL=1.10. The area elasticity coeﬃcient and the preferred area are set to K= 2.67 10⁹ N m³ and A⁰= 7010¹²m². These parameters are estimated using the iterative algorithm presented in Section 3.3, such that a stress loading of 700 Pa yields an extension of 14% (see the green curve) and thus be consistent with the experimental data (see the grey curve).

The figure shows that in the simulations the strain reaches a plateau for both low and high stress-loading (green and red curves), at about the same timescale O(10 s). This result indicates that the model is consistent with experiments for low- stress loading but that it is not behaving realistically under high stress application. The inability of the model to reproduce the creep-response of tissues subjected to high-stress loading is discussed later in the conclusion.

Moreover, it was found (see the red dotted curve in Fig. 8) that the stress-extension curve of cultured cell monolayers displays three diﬀerent regimes:⁵ (i) a first region of low stiﬀness (E5 kPa) for an extension between 0 andE20%; (ii) a second region between approximately 25% and 50% of extension, where the slope of the stress–strain curve is about 20 kPa, almost four times bigger than in the first region, and (iii) a mechanical failure for extensions 470%, following a plateau of the curve.

In addition to these experimental data, Fig. 8 also displays the stress-extension curves for four numerically simulated cell monolayers. They have been produced with the four pairs of normalized parameters (G = 0.005, L = 0.04), (G = 0.03, L=0.14), (G= 0.19,L=1.30) and (G= 0.35,G=2.47).

For all four of them, the parametersKandA⁰were set to 2.3 10⁹N m³and 7010¹²m², respectively. The figure shows

that the relation between the stress loading and the extension of the simulated monolayer is non-linear and depends on the normalized parameters (G, L). While the four pairs of normalized parameters were chosen to match the experimental stress–strain curve for less than 20% extension, we observe that the four curves are different at higher strains. We notice that for strains larger than 0.2, the slope of the curve increases as GandLget larger.

This numerical observation is consistent with the results presented in the study of Harriset al.,^4,5where the stiﬀness of a monolayer was computed from the slope of the stress–strain curve between approximately 25% and 50% extension, and where it was observed that the depolymerization of the actin cytoskeleton, the inhibition of the myosin contractility as well as the disruption of the intercellular adhesion lead to a decrease in the stiffness of the tissue.

In summary, we found that our model is able to match the experiments at low-stress loading. It also reproduces the experimental observations^4,5that the stiﬀness of the tissue at higher strains (from 0.25 to 0.50 strain) increases with the increase of the cell contractility and the cell/cell adhesion. But, in its current state, the numerical model does not reproduce well the slope of the stress–strain curve for strains comprised between 25% and 50% extension. Therefore, in Section 3.4, we will assume that the values of the model parameters can change with the strain to better match the experimental stress–strain measurements of Harris et al.⁵ Before that, we propose in Section 3.3 a systematic method to fit the parameters of the model from the experimental data.

3.3 Estimation of values of the model parameters

From the above results, we found that no constant model parameters (K,A⁰,L, and G) match the experimental stress–

strain curve of cultured cell monolayers from lower strains (o20%) to higher strains (420%) (see Fig. 8). Therefore we investigate the possibility that the model parameters change with stress loading and strain. For this reason we developed an iterative process giving the value of the parameters reproducing a given strain–stress pair, without the need for an exhaustive exploration of the parameter space.

Fig. 7 Strainversustime of a simulated cell monolayer subject to diﬀerent stress loadings, and comparison with experimental data.⁵The parameters of the simulation areK= 2.6710⁹N m³,A⁰= 7010¹²m², (G= 0.16, L=1.10),dt= 0.005 s,Z= 10 s¹, andm= 10⁵kg.

Fig. 8 Stress–strain curves for different cell monolayers simulated using our model and compared to the experimental average stress–strain curve of cultured cell monolayers.⁵

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We suppose here that the line tensionLremains constant despite the tissue extension. It was estimated experimentally⁵ that the average force required to separate two cells within the cultured cell monolayer is 1.7 10⁶N. We deduce that it corresponds to a line tension along edges separating two neighbor cells L = 1.7 10⁶ N. Moreover, from the estimation of the cell diameteraE10⁵m, we deduce that the steady area of an unconstrained cell A_steady is the area of a regular hexagon with a circumradiusr= 510⁶m, and thus Asteady= 6510¹²m².

In addition, we also assume that the ratio A⁰

A_steady does not change with strain.A⁰is estimated as the cell area when the actomyosin cortex is inhibited, and Asteady when it is active.

Indeed, it was observed¹that disrupting the actomyosin cortex increases the cell volume by 7 4% and abolishes the rounding-pressure. We consider that with a normal activity of the cytoskeleton, the area of a cell corresponds toAsteady, while the inhibition of the actomyosin activity makes the cell area increase and reach the cell preferred areaA⁰. We can use the quantitative measure of the cell volume increase resulting from actomyosin cortex disruption to estimate the ratio between the preferred area of a cellA₀ and its unconstrained steady area Asteady. Hence, we may estimate that A⁰

A_steady¼1:07andA⁰= 70 10¹²m², if we approximate the volumeVof the cell byV=Ah, where the height of the cellhremains approximately constant.

Our parameter estimator takes as input the experimentally determined value of L, A⁰, as well as A⁰

Asteady

expected

and e¼ dX

X_steady

expected

, the expected tissue strain when a given stresssis applied.

Then we use the following iterative process. We start with i= 0 and a random value of the area elasticityKi, for instance Ki= 1 N m³. Then, we compute the normalized line tensionL_i as a function of our estimated values of L and A0, and the current value ofKi;L_i¼ L_i

K_iðA⁰Þ³⁼². Given this value ofL_i, we look for the corresponding normalized contractilityG_i, such as the ratio between the preferred area of the cells and their steady area A⁰

AsteadyðL_i;G_iÞ¼ A⁰ A_steady

expected

. For this purpose, we can refer to Fig. 9, which represents the contour lines of the normalized steady area of cells as a function of the normalized parametersLandG.

We then determine the Young’s modulusEi(see (7)) corresponding to the pair of parameters (L_i,G_i) and an applied stress s. While dX

Xsteady

L_i;G_i

ð Þ is diﬀerent from the wanted value dX

Xsteady

expected

, we correct the value ofKby computingK_i+1as

Kiþ1¼E

E_iK_i (9)

withE¼s

, dX Xsteady

expected

, the expected Young’s modulus.

Then we iterate the process withi=i+ 1.

In summary, we loop on (a) computing the normalized line tension L_i with the new value of K_i, (b) finding the corresponding normalized contractilityG_i producing the right ratio between the preferred area of the cells and their steady area

A⁰

A_steady, and (c) correcting the value of Ki until we obtain the experimental strain value dX

X_steady

expected

corresponding to the applied stress loadings.

For a line tension L = 1.7 10⁶ N, a steady area A_steady= 65 10¹² m², a ratio between the preferred area of cells and their steady area A⁰

A_steady¼1:07 and a strain of 14%

when a stress-loading of 700 Pa is applied (which is equivalent to a Young’s modulusE= 5000 Pa), we estimateKto be equal to 2.6710⁹N m³,L=1.10 andG= 0.16.

In summary, for a Young’s Modulus E = 5000 Pa corresponding to a 14% strain and a stress-loading ofs= 700 Pa, we have obtained

A⁰= 7010¹²m² L=1.710⁶N K= 2.6710⁹N m³ G= 3.0110²N m¹ Asteady= 6510¹²m² (10)

We can compare our results with experimental findings,² where the contractility stress is reported to be 3.410³N m². However, this value corresponds to 3D cells. To roughly estimate the contractility tension for 2D cells, we can use the cell diameteraE10⁵m, which is a natural length scale of the system. By dividing our resultG = 3.01 10² N m¹ by a, we obtain an equivalent cortical contractility stress of 3.01 10³N m², which is compatible with ref. 2.

Similarly, the area compressibility modulusK_A=KA₀is found³ to be on the order of 10¹N m¹. To obtain an estimation ofK, we divideKAbya²and we obtainKE10⁹N m³, in agreement with our value ofKin eqn (10).

Fig. 9 Ratio between the steady area of cells and their preferred area as a function of the normalized line tension Lat cell interfaces and the normalized contractilityGof cells.

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3.4 Evolution of cell mechanical properties as a response to an increasing strain

In this section, we use the iterative process described above to find how the parameters of our model should evolve as a function of the tissue straine, in order to match the mechanical properties observed experimentally. It was observed that the tissue becomes stiﬀer and the slope of the stress–strain curve becomesE2010³Pa, when the tissue is subjected to a high strain (e425%).⁵For a set of experimental strain–stress pairs taken over a large range of strains, from 5% to 75% (see Fig. 10), we estimate the corresponding model parameters.

Experimental observations showed that, unlike the inter- mediate filaments, whose aspect changed during increasing monolayer extension, the intercellular adhesion of cells does not seem to be aﬀected by cell monolayer extension. Thus, we consider that the parameter related to cell contractility may vary as a response to stretching, while we make the hypothesis that the line tension along cell/cell interfaces remains constant L=1.710⁶N, as well as the ratio between the preferred area of the cells and their unloaded steady area A⁰

A_steady¼1:07.

In this case, for each pair of applied stress and resulting strain, two parameters of the model remain to be estimated, namelyKandG.

We find a first region ranging from 0% to 15% strain, where KandGare constant;K= 2.25610⁹N m³andG= 2.98 10²N m¹. In a second region, from 15% to 50% strain,Kand Gincrease with the strain. In the last region, when the strain is higher than 50%, K andG decrease with the increase of the strain (see Fig. 11a and b). The evolution of the parametersK andGas a function of the strain can be fitted with polynomial functions of degree 3, represented by the red curves in Fig. 11a and b. Withedenoting the strain, we have

K(e) = (3.7150e³+ 2.8795e²0.1307e+ 0.1865)10¹⁰ (11) and

G(e) =0.0244e³+ 0.0190e²0.0009e+ 0.0295 (12)

The chosen polynomial function order is the smallest one able to reproduce the two inflection points at strains 0.15 and 0.5.

We notice thatKandGevolve identically over all the strain range. When G increases (or decreases), K increases (or decreases) as well. More precisely, increasing Gincreases the stiﬀness of the tissue and the ratio A⁰

A_steady. In order to fulfill the hypothesis that A⁰

A_steady remains constant during the stress loading process, K should increase to maintain the size of the cell.

In other words, for a given cell/cell adhesionL, the strain corresponding to a given stress depends only on the cell

Fig. 10 Experimental values of applied stress and resulting strain.⁵

Fig. 11 Change of the cell mechanical properties with an increasing tissue strain. (a) Evolution of the parameterGas a function of the tissue strain.

(b) Evolution of the parameterKas a function of the tissue strain.

Fig. 12 Stress–strain curves. Black dots: experimental results. Red curve:

model with constant parameters (K= 2.25610⁹N m³,A⁰= 7010¹²m², L=1.710⁶N andG= 0.0298 N m¹). Green line: model with strain- dependent values forKandG, andA⁰= 7010¹²m²andL=1.710⁶N.

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perimeter contractilityG. The smaller the strain, the larger the cell perimeter contractility. The cell area elasticity modulusKis involved in maintaining the ratio between the preferred areas of cells and their unconstrained steady areas.

With the proposed dependency of K and G on e, the experimentally measured stress–strain curve of cultured cell monolayers can be successfully reproduced by our simulations (see Fig. 12).

4 Conclusions

In this work, we studied the mechanical properties of a vertex cell-based model for epithelium and compared the numerical simulations with experiments in which mechanical constraints are applied to a cell monolayer.

We analyzed the eﬀect of cell contractility and intercellular adhesion on the elasticity of the tissue. Stretching simulations showed that the increase of cell contractility and the decrease of cell/cell adhesion lead to an increase of the stiﬀness of the tissue and a reduction of its Poisson’s ratio.

We also investigated the ability of the model to reproduce the stress–strain relation observedin vitro. Both real epithelium and simulated tissues exhibit a non-linear elastic behavior. The stiffness is more pronounced at larger strains especially when the cell perimeter contractility and the intercellular adhesion are high.

Experiments indicate that the slope of the stress–strain curve increases by a factor four when the strain gets larger than 25%. These low and high strain regimes can be reproduced numerically provided one introduces an adequate dependency of the cell perimeter contractility and area elasticity upon the strain. This dependency was determined by an iterative algorithm computing the model parameters that match a given stress–

strain pair.

The numerical results suggest that the cells adapt their perimeter contractility and area elasticity to the strain. We found that while the cell area elasticity and perimeter contractility are constant under 15% extension, the first reaction of cells to address an increasing strain is to increase their perimeter contractility. In order to maintain a constant cell size, an increase of the area elasticity also occurs. But over a given threshold of 50%

extension, cells stop counteracting the stretching and decrease their perimeter contractility and area elasticity, dissipating the energy excess induced by the stress loading, probably through cytoskeleton remodeling. This regime change at higher strains (over 50% extension) may pinpoint a permanent and irreversible change in the mechanical properties of cells. Indeed Harriset al.⁵ described in their Supplementary Material that the stiﬀness of tissues decreases after being subjected to cyclic deformations.

However, even though the variation of the model parameters with the tissue strain allowed us to reproduce the experimental stress–strain curve, we could not reproduce the creep response at high stress (3000 Pa) application (see Section 3.2 and Fig. 7).

It may be that the application of high stress-loading leading to a large extension of the tissue over a short period of time causes

irreversible alteration of the cell cytoskeleton, reducing therefore the tissue stiffness and yielding larger extensions. The simulation of such phenomena should be the subject of future investigations.

Finally, we made the hypothesis in this study that the intercellular adhesion is constant when the strain increases, and that the change of the perimeter contractility is balanced by a change of the area elasticity. But, other possibilities can be considered, for instance, keeping the area elasticityKconstant and balancing the increase of the perimeter contractility G by an increase of the intercellular adhesion L. The other scenarios will be investigated in a future work.

An important future task to consider as well is the development of a 3D model which would correct the weaknesses of the 2D model investigated in this paper. It would allow us to simulate the cell monolayer height thinning during stretching, and to reduce the shrinkage along theYdirection that was visible in the 2D model stretching simulations, but was negligible during experiments. Indeed, the extension of the tissue along the stretching axis Xwould be balanced to keep the cell volume constant by a thinning of the cell monolayer height instead of its shrinkage along theYdirection.

Acknowledgements

We thank Michel Milinkovitch, Aurelien Roux, Marcos Gonzalez- Gaitan and Andreas Wagner for discussions and comments. This work was supported by SystemsX.ch initiative (project EpiPhysX).

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