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The effect of substrate stiffness on tensile force transduction in the epithelial monolayers

In recent years, the importance of mechanical signaling and the cellular mechanical microenvironment in affecting cellular behavior has been widely accepted. Cells in epithelial monolayers are mechanically connected to each other and the underlying extracellular matrix (ECM), forming a highly connected mechanical system subjected to various mechanical cues from their environment, such as the ECM stiffness. Changes in the ECM stiffness have been linked to many pathologies, including tumor formation. However, our understanding of how ECM stiffness and its heterogeneities affect the transduction of mechanical forces in epithelial monolayers is lacking. To investigate this, we used a combination of experimental and computational methods. The experiments were conducted using epithelial cells cultured on an elastic substrate and applying a mechanical stimulus by moving a single cell by micromanipulation. To replicate our experiments computationally and quantify the forces transduced in the epithelium, we developed a new model that described the mechanics of both the cells and the substrate. Our model further enabled the simulations with local stiffness heterogeneities. We found the substrate stiffness to distinctly affect the force transduction as well as the cellular movement and deformation following an external force. Also, we found that local changes in the stiffness can alter the cells’ response to external forces over long distances. Our results suggest that this long-range signaling of the substrate stiffness depends on the cells’ ability to resist deformation. Furthermore, we found that the cell’s elasticity in the apico-basal direction provides a level of detachment between the apical cell-cell junctions and the basal focal adhesions. Our simulation results show potential for increased ECM stiffness, e.g. due to a tumor, to modulate mechanical signaling between cells also outside the stiff region. Furthermore, the developed model provides a good platform for future studies on the interactions between epithelial monolayers and elastic substrates. Author summary Cells can communicate using mechanical forces, which is especially important in epithelial tissues where the cells are highly connected. Also, the stiffness of the material under the cells, called the extracellular matrix, is known to affect cell behavior, and an increase in this stiffness is related to many diseases, including cancers. However, it remains unclear how the stiffness affects intercellular mechanical signaling. We studied this effect using epithelial cells cultured on synthetic deformable substrates and developed a computational model to quantify the results better. In our experiments and simulations, we moved one cell to observe how the substrate stiffness impacts the deformation of the neighboring cells and thus the force transduction between the cells. Our model also enabled us to study the effect of local stiffness changes on the force transduction. Our results showed that substrate stiffness has an apparent impact on the force transduction within the epithelial tissues. Furthermore, we found that the cells can communicate information on the local stiffness changes over long distances. Therefore, our results indicate that the cellular mechanical signaling could be affected by changes in the substrate stiffness which may have a role in the progression of diseases such as cancer.


Introduction
Our understanding of the importance of mechanical forces and microenvironment in 2 cellular processes and signaling alongside biochemistry has drastically improved during 3 the last decades [1][2][3]. Biomechanics has a vital role in embryogenesis, stem cell 4 differentiation, tissue homeostasis, and migration [4][5][6][7][8][9][10]. In addition, abnormal changes 5 in the biomechanics of the cells and the extracellular matrix (ECM) are linked to many 6 pathological processes, including tumor-and carcinogenesis [11][12][13][14][15]. While the research 7 on the effect of physical cues and the role of the mechanical environment on cell 8 functions and signaling is active, the understanding of this mechanical system is not 9 complete. In epithelial tissues, the cellular mechanical microenvironment is formed by 10 the neighboring cells and the ECM on the basal side of the cells. In some instances, the 11 apical side of the cells is subjected to shearing forces from fluid flow [16]. Thus, 12 epithelial cells are exposed to various physical cues rising from their environment. The 13 high interconnectivity of the epithelial tissues enables the cells to distribute exogeneous 14 mechanical energy and transmit endogenous forces to their environment [17,18]. Here, 15 we investigated how the propagation of tensile forces in the epithelial monolayer is 16 affected by the stiffness of the substrate under the cells. 17 Cells can use their actomyosin cytoskeleton to generate contractile forces that enable 18 cells to change their shape and move [18,19]. Since the actomyosin cytoskeletons of 19 neighboring epithelial cells are connected via adherens junctions, these contractile forces 20 can be transmitted between cells over distances [20]. Cells have various responses to 21 these exogenous forces, behaving elastically over a short time scale by deforming and 22 viscously over sustained stress by dissipating the stress via various processes [21][22][23][24]. 23 Endo-and exogenous forces can also alter the structural state of different proteins, 24 which may lead to the conversion of forces to biochemical signals via a process known as 25 mechanotransduction [3,25]. Furthermore, mechanical forces have been indicated to 26 have an even more direct effect on the cell behavior since they are transmitted directly 27 to the cell nucleus along the actin stress fibers, where they have been shown to be able 28 to modulate gene expression [26][27][28][29]. Therefore, understanding how forces are 29 transmitted between cells is essential to understand the mechanical system formed by 30 the epithelial tissues. 31 Epithelial cells are connected to the ECM on their basal side via focal adhesions, 32 which connect the actomyosin cytoskeleton to the basal lamina. The focal adhesions 33 contain mechanosensitive proteins that enable the cells to sense the external forces and 34 ECM stiffness [30][31][32], which has been shown to affect many cell functions during 35 development, homeostasis, and diseases. For example, ECM stiffness has been heavily 36 linked to cell differentiation [6,33] and the metastatic potential of tumors [34][35][36]. It is 37 well established that tumor stroma, the adjacent tissue surrounding the tumor, is often 38 considerably stiffer than native tissue [37,38]. In tissues, this leads to stiffness gradients 39 and interfaces between the stroma and the surrounding healthy tissue, which can 40 influence cellular mechanosignaling, especially during cancer invasion [14,39]. However, 41 we do not fully understand how the ECM stiffness or stiffness gradients affect the immersed-boundary method [45][46][47] provide a more nuanced description of the cells and 48 their mechanical properties but differ on how they describe the cells and solve the 49 cellular movement. Only a few cell-based models have included a description of a 50 deformable substrate under the cells [42,43]. 51 This study aimed to describe how strain and forces propagate in an elastic 52 mechanical system formed by the epithelial monolayer and a deformable substrate with 53 different stiffnesses. The work was conducted experimentally using Madin-Darby canine 54 kidney (MDCK) II cell model on polyacrylamide (PAA) hydrogel substrates and 55 computationally using a cell-based model we developed. To investigate the propagation 56 of tensile forces between the neighboring cells and the cells and the ECM, we, 57 experimentally and in the computational model, moved a single cell, causing a local 58 stretching in the epithelium. Furthermore, we used our computational model in 59 combination with data from the literature to study the mechanics related to subcellular 60 changes in cell shapes. 61 63 varying stiffness 64 To experimentally study the effect of environmental stiffness on the propagation of 65 forces and deformation in epithelial tissue, we used an in vitro model of MDCK II cells 66 expressing tight junction marker mEmerald-Occludin cultured on PAA hydrogel 67 substrates with embedded fluorescent beads. We used collagen-I-coated PAA substrates 68 with four stiffnesses (Young's moduli): 1.1, 4.5, 11, and 35 kPa. We manipulated a 69 single cell with a sharp pipette attached to a piezo-driven micromanipulator as a 70 mechanical stimulus. The pipette was brought into contact with the cell, and the 71 micromanipulator was used to move the pipette 30 µm parallel to the surface in 1 72 second (speed 30 µm/s) while simultaneously imaging both the mEmerald-Occludin and 73 the fluorescent beads. 74 The micromanipulation led to large deformation of the epithelium and displacement 75 of the cells and the PAA substrate. We visualized the movement by comparing the 76 images of the epithelium and the substrate before and following the micromanipulation 77 (Fig. 1). It is clear from the movement of the cell boundaries ( Fig. 1A) that the PAA 78 stiffness profoundly affects the distance that the mechanical strain spreads around the 79 manipulated cell. For example, the movement of the cell boundaries along the axis of 80 the pipette movement (Fig. 1B, dashed lines in Fig. 1A) showed that for the 1.1-kPa 81 substrate, the cell boundary at the edge of the imaged field (approximately 80 µm from 82 the initial pipette position) moves 3.3 µm. This was in stark contrast with the 11-kPa 83 and the 35-kPa substrates, for which the discernible cell boundary movement happened 84 only at the distance of approximately 50 µm along the axis of its movement (Fig. 1B). 85 Similar to the cells, the displacement of the substrate was naturally affected by their 86 stiffness. There was deformation in the whole imaged field for both the 1.1-and 4.5-kPa 87 substrates (Fig. 1C) in the direction parallel to the pipette movement. However, the 88 displacement in the perpendicular direction was more limited with the 4.5-kPa substrate 89 than with 1.1 kPa. The deformation was even smaller with the stiffer (11 and 35 kPa) 90 substrates (Fig. 1C). , and 35 kPa following the movement of the micromanipulated pipette for 30 µm in 1 s (pipette movement shown by the white arrow). The boundaries of cells were indicated using mEmerald-Occludin and shown in green before the micromanipulation and in magenta following the 30 µm pipette movement. The cell displacement on the right side of the pipette (white arrow) is partly due to the pipette affecting part of the image. (B) Line plot along the dashed lines for cell boundaries in A in arbitrary units (AU) before (green) and following the micromanipulation (magenta) for each gel stiffness to better show the magnitude of the cell movement along the pipette movement axis. The data was smoothed using 10 pixel moving average. The pipette movement is indicated by the black arrows. (C) The movement of the fluorescent beads embedded in the PAA hydrogel substrates with the corresponding stiffness underlying the epithelia shown in A. The pipette movement of 30 µm is shown by the white arrow and the bead locations before and following the micromanipulation in green and magenta, respectively. The pipette shadow was affecting the results on the right side of the white arrow. Scale bar, 20 µm.

Micromanipulation of epithelial monolayers on substrates with
To quantify the cell displacements, we segmented the cell imaging data before and 92 following the micromanipulation to obtain the cell outlines. Using the outlines, we 93 defined a geometrical cell center, which we then used to measure the displacement of 94 the individual cell during the micromanipulation. This provided us with a spatial map 95 of the cell center movements in relation to their distance and direction from the initial 96 position of the pipette (p 0 ). We then interpolated the cell center movement data over 97 the whole imaging area to obtain a continuous distribution for each measurement and 98 calculated the average distribution for each substrate stiffness ( Fig. 2A and B). In order 99 to do the same for the substrate data, we used particle image velocimetry (PIV) 100 analysis to find the displacement of the substrate beads between images taken before 101 and following the micromanipulation. Similar to the cell data, this was averaged and 102 plotted in relation to p 0 ( Fig. 2C and D). 103 Interestingly, the 30-µm pipette movement translated to a cell center movement of a 104 similar range independent of the substrate stiffness with values of 15.4 ± 3.2, 15.8 ± 2.3, 105 14.5 ± 2.6, and 14.8 ± 3.1 µm (mean ± SD), from the softest to the stiffest substrate. 106 This difference between the pipette movement and the cell displacement can be 107 explained by the deformation and stretching of the manipulated cell. The pipette 108 movement caused substantial deformation to the adjacent cells in the direction of the 109 movement, and thus the displacement of these cells was difficult to quantify. Therefore, 110 we mainly concentrated our analysis on the area where the pipette pulled and stretched 111 the cells and present the results mainly as a function of the location on the negative 112 y-axis from p 0 . Parallel to the pipette movement (Fig. 2B, along the red, dashed line in 113 2A), the cell centers move 5 µm or more within a distance of 47, 34, 26, and 20 µm from 114 p 0 respectively for 1.1, 4.5, 11, and 35-kPa substrates. Interestingly, the three stiffest 115 substrates have the same amount of cell center displacement perpendicular to the 116 pipette movement. In contrast, with the softest substrate, the displacement of at least 5 117 µm extends to approximately 1.5 times farther away than the rest (Fig. 2B).

118
The amount of displacement of the substrate was considerably smaller than that of 119 the cells (Fig. 2D). The maximum displacements, located near p 0 , were, from the softest 120 to the stiffest, 8.9 ± 0.8, 7.6 ± 2.5, 2.7 ± 1.0, and 2.6 ± 1.1 µm. Therefore, the relative 121 magnitude of the maximum substrate displacement compared to that of the cells 122 corresponding to the stiffnesses from 1.1 to 35 kPa were 0.58, 0.48, 0.19, and 0.17. This 123 difference in the maximum displacements partly originated from the facts that the to the initial pipette position (p 0 ) for the stiffnesses 1.1 (n = 11), 4.5 (n = 7), 11 (n = 11), and 35 kPa (n = 7). The field is limited to the left of micromanipulation axis since the movement was symmetric on either side of the axis. The area of the shown displacement field varies between the stiffnesses since p 0 in relation to the imaging area varied between measurements. (B) The cell center displacement along the y-axis (red dashed line in A) away from the direction of the pipette movement (left) and along x-axis (red dotted line in A) perpendicular to the pipette movement direction (right) for each stiffness. (C) Average displacement of PAA hydrogel substrates based on the particle image velocimetry (PIV) analysis as function of location in relation to p 0 for the stiffnesses 1.1 (n = 11), 4.5 (n = 7), 11 (n = 11), and 35 kPa (n = 7). The field is limited to the left of micromanipulation axis since the movement was symmetric on either side of the axis and the pipette causes artefacts in the PIV data on the right side of the pipette. The area of the shown displacement field varies between the stiffnesses since the pipette position in relation to the imaging area varied between measurements. Note that maximum displacement is different than with the cells. (D) The PAA substrate displacement along the red dashed line in C away from the direction of the pipette movement (left) and along the red dotted line in C perpendicular to the pipette movement direction (right) for each stiffness. The shaded region represents the SD for each stiffness. (E) Distance of pipette movement before cells detached from the substrate estimated from the live imaging data for the different stiffnesses 1.1 (n = 11), 4.5 (n = 7), 11 (n = 11), and 35 kPa (n = 7). The indicated cases with the distance to detachment of 30 µm did not detach from the substrate during the experiment.  there was more variance in the detachment distance for the 11-kPa substrate compared 135 to the others. The detachment with the two stiffer substrates explained the minuscule 136 difference between the substrate displacements in these measurements.  al. [48]. We described the deformable substrate under the cells as a triangular grid of 143 points whose movement was solved similar to that of the cell vertices. For a detailed 144 explanation of the model, the fitting, and the simulations, see the description in S2 Text. 145 We used the experimental results from our in vitro cell model and data from the 146 literature to fit the model parameters. The computational model was first used to grow 147 virtual epithelia, followed by the simulation of single-cell mechanical manipulation.
During the simulated manipulation, we restricted the remodeling of the cell properties 149 to describe the purely elastic properties of the experimental time scale. 150 We fitted the model parameters by comparing the cell center and substrate 151 displacements between the in vitro experiments and the computational model. Due to 152 the similarity of the experimental 11 and 35-kPa results in cellular displacements (Fig. 153 2B and D) and detachment distances (Fig. 2E), we decided to omit the 35-kPa substrate 154 from our simulations. We assumed that the elastic properties of the epithelium are 155 similar between the substrates, and therefore we used the same cell parameter values for 156 each substrate stiffness in the fitting process. However, the only exception was the focal 157 adhesion strength parameter, which we assumed to depend on the substrate stiffness 158 and was thus altered accordingly. Similar to the experiments, in the simulation data 159 analysis, we focused on the area of the epithelium under tension (i.e., y < 0). The most 160 drastic movement of the cell boundaries during the micromanipulation was visible for 161 the softest substrate (Fig. 3A). We measured the average cell and substrate   were small even though we ran each simulation using a different epithelial system.

190
The model was then used to compare the propagation of different forces depending 191 on the substrate stiffness. To describe the forces following the micromanipulation as a 192 function of position in relation to the p 0 , we averaged the forces over the vertices of 193 each cell and assigned them to the original cell center positions. Next, we interpolated 194 the averaged force magnitudes between the cell centers to obtain continuous spatial 195 distributions and then averaged over multiple simulations. 196 We concentrated on the forces that arise from the interactions between the cells and 197 the substrate (focal adhesion forces, Fig. 3F

215
The results indicate that the stiffness of the substrate had an apparent effect on the 216 propagation of cell movement following an external tensile force. However, while the 217 forces between cells and cell deformation in the apical plane, as indicated by the 218 increased cortical force, were higher near the manipulated cell on stiffer substrates, 219 there were only minor differences in the forces at longer distances. On the other hand, 220 the focal adhesion forces on the softest substrate remained lower over the simulated 221 distance.

222
Propagation of cell-cell and cell-substrate forces over substrate 223 stiffness gradients 224 Next, we investigated how stiffness gradients -describing those between stiff tumorous 225 tissues and healthy soft tissue -affect the transduction of tensile forces between the 226 cells. Our computational model enabled the generation of epithelial monolayers 227 attached to substrates with stiffness gradients with different slopes. We first 228 concentrated on studying strain propagation from a soft substrate to an area of a stiff 229 substrate. To do this, we simulated the pipette micromanipulations with substrates that 230 included one of the following three different types of stiffness gradients between 1.1 and 231 11 kPa: a stiffness interface (change in stiffness in 2 µm, Fig. 4H), a sharp or a shallow 232 gradient (changes in stiffness in 10 µm or 50 µm, respectively, Fig. B in S1 Appendix). 233 The pipette was moved only 20 µm in these simulations to minimize the cell detachment 234 from the substrate. We also simulated the 20-µm micromanipulations with the uniform 235 September 3, 2021 7/25 stiffnesses of 1.1 and 11 kPa for comparison ( Fig. A and B in S1 Appendix). We 236 analyzed the results by calculating how the cell and substrate displacements and the 237 focal adhesion, cortical, and junction forces changed compared to the situation with a 238 uniform 1.1-kPa substrate under the cells. Therefore, we calculated how much the 239 displacements and forces changed by comparing results from simulations with a stiffness 240 gradient against those with a uniform stiffness. This absolute difference was defined by 241 subtracting the latter from the first at each distance from p 0 . We also calculated the 242 relative difference between the two cases for the forces by dividing the results with the 243 stiffness changes by those with the uniform stiffness at each distance. An example 244 visualizes these steps in Fig. 4A. We used the relative differences for the forces to Compared to the 1.1-kPa uniform substrate, the rapid increase in stiffness at the 249 various distances from p 0 led to a reduced cell displacement with the most prominent 250 effect near the stiffness interface (Fig. 4B). While larger when the interface was closer 251 to p 0 , the reduction still occurred even when the interface was up to 80 µm away.

252
However, with the interface farther away, the difference in displacement remained 253 smaller than 1 µm. The difference in the substrate displacement was slightly higher but 254 otherwise similar to that of the cells (Fig. 4C). Interestingly, the substrate was 255 displaced more at p 0 when the stiffness interface was farther away.

256
The decreased cell and substrate displacement was accompanied by increased forces 257 (Fig. 4D-G). The cortical and junction forces were generally increased between p 0 and 258 the interface and some distance beyond the interface when closer to p 0 . However, only 259 minor changes were visible near the interface at 80 µm away. On the other hand, the The observed behavior was similar with the shallow and sharp stiffness gradients 274 compared to the interface gradients in equal distances from p 0 (Fig. B in S1 Appendix). 275 These gradients also produced similar relative peaks in the focal adhesion forces; 276 however, the wider the stiffness gradient was, the more spread out and lower the peak 277 was.

278
Next, we wanted to investigate how the force transduction is altered when the increased well before the interface itself, also for those farther away from p 0 . The 286 general behavior of the difference in the substrate displacement was similar to that of 287 the cells but with slightly higher peak values (Fig. 5B). The substrate displacement was 288 also increased at p 0 , unlike with the cells. interface gradient location (Fig. 5C-F). The cortical and junction forces were decreased 292 close to p 0 with the interface at 20 µm (Fig. 5C). On the other hand, focal adhesion 293 forces were increased near p 0 but showed generally reduced values at longer distances. 294 Furthermore, following the interfaces at 20 and 40 µm, there were inverse peaks in these 295 forces compared to the uniform substrate ( Fig. 5C and D), which were not as clearly  activation. Therefore, we implemented the optogenetic activation into our model based 310 on the experimental and theoretical work by Staddon et al. [24]. We obtained the model 311 parameters either directly from Staddon et al. or by fitting as described in S2 Text. We 312 did not consider the strain-based remodeling of the cortical tension since we wanted to 313 concentrate on the effect of the substrate stiffness on the local movement of cell 314 boundaries, and the tension remodeling primarily affects the reduced junction length 315 following the optogenetic activation [24]. We also allowed the remodeling of the cell 316 structures in these simulations due to the long experiment duration compared to the 317 micromanipulation. We increased the contractility of cell vertices forming the junctions 318 between two cells to observe how the length of this junction reduced following the 319 activation (Fig. 6A). We ran the simulations on substrates with uniform stiffnesses of During the simulated 20-min activation, the increased cortical contractility reduced 322 the junction length the most with the softest 1.1-kPa substrate with the final relative 323 length of around 0.63 ± 0.08 (mean ± SD) (Fig. 6B). The relative junction length was 324 reduced to similar values for the two stiffer substrates with the values of 0.66 ± 0.07 325 and 0.67 ± 0.07 for 4.5 and 11 kPa, respectively (Fig. 6B). However, the initial length 326 reduction was faster with the 4.5-kPa substrate than the 11-kPa, with a similar slope to 327 the 1.1-kPa substrate.  We first studied how a substrate with a uniform stiffness affected the propagation of 362 cell displacements and strain, and therefore forces, following an exogenous 30-µm 363 movement of a single cell within one second. Logically, both the cell and the substrate 364 displacement spanned over longer distances with soft substrates. The higher substrate 365 stiffnesses, on the other hand, greatly reduced both displacements. The high cell 366 displacement perpendicular to the pipette movement with the soft 1.1-kPa substrate can 367 be explained by the reported stiffnesses of the MDCK monolayers, that are between 1 368 and 5 kPa, when measured with atomic force microscopy [53][54][55]. This means that the 369 stiffness of the epithelial monolayer in this system has a similar or slightly higher 370 stiffness than the softest substrate and, therefore, can more readily displace the 371 substrate than the monolayers on the stiffer substrates. We observed only minor 372 differences between the cell displacements on the stiffer substrates, suggesting that the 373 propagation of forces began to saturate. Therefore, having a substrate stiffer than 35 374 kPa would most likely not have a further effect on the cell displacements.

375
The difference between the maximum cell and substrate displacements can be 376 explained by the different displacements of the apical and basal surfaces of the cell.

377
Both our imaging of the mEmerald-Occludin-expressing MDCK cells and our 378 computational model describe the cells by their apical surface. The displacement of the 379 basal substrate-binding surface is more related to that of the substrate. This suggests 380 that the cell shape in the apico-basal axis is heavily distorted, especially near the 381 micromanipulated cell. Furthermore, the variabilities in the displacements of both the 382 cells and the substrate predicted by our computational model were considerably smaller 383 than those seen in the experimental results. Therefore, the variability in the epithelial 384 morphology -i.e., the cell sizes and shapes -was not enough to explain the 385 experimental displacement variability, and our simulation results thus only reflect an 386 average epithelium. In reality, the mechanical properties are more varied. 387 We used our computational model to study the cortical, cell-cell, and cell-substrate 388 forces during the micromanipulations. It is noteworthy to mention that the focal 389 adhesion forces describe both the tension in the focal adhesions and the apico-basal axis 390 of the cell. We found that all of these forces were increased by the increase in substrate 391 stiffness. This was expected because the cells were subjected to the same external strain 392 on the stiffer, less deformable substrates as those on the soft substrates. Therefore, the 393 smaller substrate deformation meant that the cells were subjected to a larger portion of 394 this strain, which led to larger cell deformation in the apical plane and higher cell strain 395 in the apico-basal cells axis. These changes corresponded to the increases in the cortical 396 and focal adhesion forces. Importantly, for the apical cell deformation to occur, the next 397 cell opposite to the incoming strain must resist movement or deformation. If the next 398 cell can be readily moved, less of the mechanical energy goes to cell deformation since it 399 is easier to transmit onward. Therefore, the cortical and junction forces depend on each 400 other since higher resistance against deformation leads to higher junction forces as less 401 of the mechanical energy is absorbed by the cell cortex. However, the differences in the 402 cortical and junction forces between the stiffnesses disappeared beyond the distance of 403 50 µm, indicating that the bulk of the mechanical energy is absorbed closer to p 0 on the 404 stiff substrates.

405
The focal adhesion forces remained higher for the cells on stiffer substrates due to 406 the more considerable difference between the cell and substrate displacement and the 407 higher focal adhesion strength. Similar results were found by Goodwin et al. [56] in the 408 developing Drosophila embryos, as they showed that the amount, and thus the strength, 409 of basal cell-ECM adhesions was inversely correlated with the displacement of the apical 410 surface. They hypothesized that the increased apical displacement was the result of 411 more efficient apical force transmission. On the other hand, our results indicated higher 412 forces transmitted between cells with stronger focal adhesions and smaller apical 413 displacements in an elastic system with an exogenous mechanical stimulus. 414 We also studied how local changes in the substrate stiffness affect the propagation of 415 forces in the epithelium. Stiffness interfaces have been shown to affect the integrity of 416 endothelial monolayers and impact their behavior over a distance of more than a 417 hundred micrometers [57]. Similarly, we found that the cell displacement was affected 418 near p 0 even if the stiffness interface was at a distance of 80 µm. Interestingly, when the 419 forces propagate from soft to stiff substrate, the more distant interfaces led to a higher 420 substrate displacement near p 0 . The origin of this effect is unclear. The cortical and 421 junction forces on the soft region between the stiffness interfaces and p 0 were increased, 422 indicating that they experience more strain than on a substrate with a uniform stiffness. 423 This is because the cells on the stiff region were more difficult to displace due to their 424 stronger binding to the substrate and the stiffer substrate itself. A similar effect 425 occurred for the force propagation from stiff to soft since the cells on the soft region 426 were easier to move, meaning that more of the strain could be transmitted to them.

427
Thus, this led to decreased cortical and junction forces in the stiff region.

428
The predicted peaks in the focal adhesion forces near the interface gradients with the 429 increases in stiffnesses were a combination of two factors. First, the cells on the stiff 430 substrate near the interface were subjected to a larger apical displacement from their 431 close neighbors from the soft side. Second, the focal adhesion strength of these cells was 432 higher due to the stiffer substrate under them. This combination thus led to the high 433 relative increase in the focal adhesion forces. The situation was similar when the forces 434 were transmitted from the stiff to the soft substrate. In this case, the cells on the soft 435 side of the interface sensed smaller focal adhesion forces compared to the stiff uniform 436 substrate due to the lower focal adhesion strength. However, since the cell displacement 437 was already diminished at more distant interfaces, this effect was not as visible.

438
Furthermore, we observed only minor differences in the cell displacement or forces in 439 relation to the slope of the increase or decrease in stiffness. Therefore, whether the 440 change in stiffness occurs within 2 or 50 µm, the main factors that affected the cells were the change in stiffness and its distance. 442 We also studied how the substrate stiffness affects the small local changes in cell 443 shape by implementing an optogenetic control of the myosin activation into our 444 computational model. The results suggested that the substrate stiffness has only a 445 minor effect on the small changes in the cell-cell junction elastic behavior. This can be 446 explained by the cellular apico-basal connections, as small morphological changes in the 447 apical side were not greatly restricted by the substrate-binding in the basal side of the 448 cells. The observed difference in the relative substrate and cell displacement between 449 the optogenetic and micromanipulation simulations show that the displacement of the 450 apical side of the cells has to be extensive enough to visibly deform the substrate due to 451 the compliance provided by the cells' apico-basal axis.

452
The factors affecting the displacement and deformation of cells following some 453 external force can be summarized as follows (Fig. 7). First, the stiffness of the substrate 454 to which the cell is attached -together with the strength of this attachment -   [32,58,59], but it seems to be important also in the long-range mechanical 471 signaling within an epithelial monolayer. This same effect is seen with fibrous substrates 472 since separate cells have been shown to be able to communicate via the substrate over 473 long distances [60][61][62]. compared to the experimental data, it seems that the elastic springs' ability to describe 500 the cell mechanics might be limited to cases with smaller strains. Furthermore, the 501 description of the focal adhesion forces is challenging since they included both the focal 502 adhesions themselves as well as the stiffness of the cell in the apico-basal axis.

503
Separating these two components into their own forces could better describe the 504 mechanics during the micromanipulation.

505
In summary, results from our in vitro cell model and computational simulations 506 suggest that the mechanical properties of the substrate have a significant effect on the 507 distance over which forces are transmitted in the epithelium. Furthermore, we found 508 that the cells can communicate information of the substrate stiffness over long distances 509 based on their ability to resist deformations. This indicates that, for example, the 510 increased ECM stiffness in tumors can affect the mechanical signaling also outside the 511 tumor itself. However, further studies are needed to better understand the role of each 512 component in this phenomenon. The computational cell-based model presented here 513 forms a valuable platform for futures studies on epithelial mechanics. In the future, the 514 model would benefit from adding the tension remodeling described by Staddon et al. [24] 515 and the inclusion of the cell nuclei. The latter would also allow the study of the forces 516 felt by the nucleus and thus their possible role in regulating gene expression [29,49,64]. 517 Furthermore, since the more fibrous nature of the natural ECM has been shown to 518 transmit forces over longer distances [60][61][62], it would be interesting to study the ability 519 of a fibrous substrate to propagate strain in the epithelial monolayer. washed twice with PBS, and cell seeding was conducted immediately.

583
MDCK II cells stably expressing mEmerald-Occludin were maintained in 75 cm 2 cell 584 culture flasks. The protein-coated gels were placed on a sterile 6-well plates with PBS 585 and sterilized in the laminar under UV light for 15 min. The cells were trypsinated and 586 suspended into 10 ml of cell culture medium, and 100 µl of the cell suspension was then 587 pipetted on each gel, and 2 ml of medium was added to the well. Cells were cultured for 588 7 d prior to the micromanipulation experiments.

589
Imaging and micromanipulation 590 We imaged the epithelial mechanics during micromanipulation using Nikon FN1 upright 591 microscope (Nikon Europe BV, Amsterdam, Netherlands) with CFI Apo 40x/0. The experimental imaging data before and after the micromanipulation pipette 615 movement was initially segmented using the Trainable Weka Segmentation [67] plugin of 616 ImageJ Fiji [68]. We randomly selected six images from the imaging data set to train 617 the classifier to segment the cells based on the mEmerald-Occludin data to obtain the 618 cell boundaries. Next, the probability maps were converted to binary masks using Find 619 maxima and then skeletonized. We manually fixed any errors in the skeleton images 620 based on comparison with the original images. Finally, BioVoxxel Toolbox's Extended 621 Particle Analyzer [69] was used to analyze the final segmented binary images. We 622 tracked the movement of the cell centers between the segmented images before and after 623 the pipette movements using a custom, semi-automated MATLAB script (R2020b, The 624 MathWorks Inc., Natick, Massachusetts). The movement data was then used to 625 interpolate the cell movement in relation to the original pipette position to obtain a cell 626 September 3, 2021 16/25 movement map. Finally, we averaged the movement maps over the data from each gel 627 stiffness. 628 We analyzed the gel deformation based on the fluorescent microbead data using 629 Fiji's particle image velocimetry (PIV) analysis plugin between the images before and 630 after the micromanipulation. Similar to the cell data, the gel deformation maps were 631 centered on the original pipette position and averaged over the same stiffnesses.

632
Computational modeling 633 A detailed description of the model, the fitting, and the simulations are available in S2 634 Text. In our model, the epithelium was described as a two-dimensional monolayer, with 635 each cell represented by a closed polygon (Fig. 8A). The model was based mainly on 636 the boundary-based model by Tamulonis et al. [48] but borrowed features from the 637 vertex models [40][41][42]. Cell structures and processes were incorporated into the model 638 as forces affecting the polygon vertices. These include cortical actomyosin, cell-cell 639 junction dynamics, intracellular pressure, cell division, focal adhesions, and membrane 640 elasticity. Some of these forces are depicted in Fig. 8B. Furthermore, the cortical

646
The top surface of the underlying substrate was represented by a two-dimensional 647 triangular grid of points (Fig. ??). As with the cells, the substrate mechanics were 648 represented by forces acting on the grid points. The forces were related to the internal 649 mechanics of the substrate as well as to the focal adhesions.
where η is the dampening coefficient (kg s -1 ), r i is the position of the cell vertex i 656 (m), s m is the position of the substrate point m (m), t is time (s), and F i,tot is the total 657 force acting on cell vertex i (N) and F m,tot that on the substrate point m (N). The total 658 force for each cell vertex i was calculated as the sum of these component forces: where ⃗ F i,cort is the cortical actomyosin force, ⃗ F i,junc the cell-cell junction force, 660 ⃗ F i,area the area force that describes the internal pressure, ⃗ F i,div the division force, ⃗ F i,f a 661 the focal adhesion force, ⃗ F i,mem the membrane force, ⃗ F i,cont the contact force, and 662 ⃗ F i,edge is the edge force. The last two forces had an auxiliary role: the contact force 663 described contact between cells and prevented cell overlap, and the edge force described 664 the continuity of the epithelium outside the simulated area.

665
The substrate mechanics were divided into three forces: a central force between 666 neighboring points, a repulsive force between a point and the connection between two of 667 its neighbors, and a restorative force that sought to move a point to its original location. 668 The second force was included to prevent the collapse of the substrate during large 669 deformations [71], and the third to describe the fact that the substrate was attached to 670 rigid glass at its bottom surface in our experiments.
where ⃗ F m,cent is the central force between closest neighboring points, ⃗ F m,rep is the 674 repulsive force to prevent material collapse, ⃗ F m,rest is a restorative force, and ⃗ F m,f a is 675 the force from the focal adhesions.

676
The model was used to simulate epithelial growth and the tissue response to two 677 different mechanical stimuli: 1) pointlike micromanipulation in a short time scale and 2) 678 a local increase in actomyosin tension by optogenetics over a longer time scale. 679 We used the model to grow epithelia from a single cell (Fig. 8C) to produce 680 epithelium of sufficient size without the substrate. The randomness in the tissue was 681 produced by normally distributed times between divisions and cell area distribution 682 based on our in vitro MDCK cell data. The size of the grown epithelium was chosen 683 based on the assumed effect of each mechanical stimulus to minimize the impact of the 684 tissue edges. Following the growth, the epithelia were given time to relax without 685 division to remove any stresses. Next, the grown epithelia were placed on the substrate, 686 and the focal adhesions were defined between the two.

687
Corresponding to our micromanipulation experiments, we moved a single cell by an 688 external force with a known speed over a distance. Since we wanted to describe the 689 elastic behavior, we prohibited any changes in the number of cell vertices and cell-cell 690 junctions in these simulations, justified by the short time scale of these measurements. 691 The values of the model parameters governing the cell mechanics were fitted using our 692 in vitro micromanipulation data with the uniform 1.1, 4.5, and 11-kPa substrates by 693 iteratively changing the parameter values and comparing the cell center and substrate 694 displacements between the experimental data and simulations results. The fitted model 695 was then used to study the force propagation on the uniform substrates and those with 696 stiffness interfaces and gradients. The interfaces and the gradients were defined along 697 the direction of the virtual pipette movement and characterized by the gradient slope 698 and distance from the initial pipette position.

699
In the optogenetic activation simulations, the contractility of the cortex in a section 700 between two randomly chosen cells was increased to describe the experimental myosin 701 activation. This was done by increasing the value of cortical tension constants for the 702 cortical forces within the activation region. The parameters for these simulations were 703 obtained from Staddon et al. [24] and by fitting our model to their data.

704
The model was solved using either 2nd or 4th order Runge-Kutta methods with 705 variable time steps. During the growth simulations when the substrate was excluded, 706 2nd order Runge-Kutta was used since it was sufficiently accurate. These simulations 707 also omitted the focal adhesion and the cell edge forces. During the simulations that 708 included the substrate, 4th order Runge-Kutta was used to evolve the system.

709
The model is implemented in MATLAB, where we also created a graphical user