Apical constriction requires patterned apical surface remodeling to synchronize cellular deformation

  1. Satoshi Yamashita
  2. Shuji Ishihara
  3. François Graner

  • Curated by eLife

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    eLife Assessment

    The results from this study, which investigates the mechanisms necessary for initiating tissue invagination using a cellular Potts modelling approach, suggest that apical constriction is not sufficient to drive the process by itself. The study highlights how choices inherent to modelling - such as permitting straight or curved cell edges - may affect the outcome of simulations and, consequently, their biophysical interpretation. Despite incomplete evidence supporting their major claims due to a rather coarse-grained exploration of the model, this work is useful for biophysicists investigating complex tissue deformation through computational frameworks.

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Abstract

Apical constriction is a basic mechanism for epithelial morphogenesis, making columnar cells into wedge shape and bending a flat cell sheet. It has long been thought that an apically localized myosin generates a contractile force and drives the cell deformation. However, when we tested the increased apical surface contractility in a cellular Potts model simulation, the constriction increased pressure inside the cell and pushed its lateral surface outward, making the cells adopt a drop shape instead of the expected wedge shape. To keep the lateral surface straight, we considered an alternative model in which the cell shape was determined by cell membrane elasticity and endocytosis, and the increased pressure is balanced among the cells. The cellular Potts model simulation succeeded in reproducing the apical constriction, and it also suggested that a too strong apical surface tension might prevent the tissue invagination.

Article activity feed

  1. Version published to 10.7554/elife.93496.4 on eLife
  2. Version published to 10.7554/elife.93496 on eLife
  3. Version published to 10.7554/elife.93496.3 on eLife
  4. eLife

    eLife Assessment

    The results from this study, which investigates the mechanisms necessary for initiating tissue invagination using a cellular Potts modelling approach, suggest that apical constriction is not sufficient to drive the process by itself. The study highlights how choices inherent to modelling - such as permitting straight or curved cell edges - may affect the outcome of simulations and, consequently, their biophysical interpretation. Despite incomplete evidence supporting their major claims due to a rather coarse-grained exploration of the model, this work is useful for biophysicists investigating complex tissue deformation through computational frameworks.

  5. eLife

    Joint Public Review:

    Satoshi Yamashita et al., investigate the physical mechanisms driving tissue bending using the cellular Potts Model, starting from a planar cellular monolayer. They argue that apical length-independent tension control alone cannot explain bending phenomena in the cellular Potts Model, contrasting with previous works, particularly Vertex Models. They conclude that an apical elastic term, with zero rest value (due to endocytosis/exocytosis), is necessary to achieve apical constriction and that tissue bending can be enhanced by adding a supracellular myosin cable. Additionally, a very high apical elastic constant promotes planar tissue configurations, opposing bending.

    Strengths:

    - The finding of the required mechanisms for tissue bending in the cellular Potts Model provides a natural alternative for studying bending processes in situations with highly curved cells.

    - Despite viewing cellular delamination as an undesired outcome in this particular manuscript, the model's capability to naturally allow T1 events might prove useful for studying cell mechanics during out-of-plane extrusion.

    [Editors' note: The previous reviews have not been updated, as the changes to the manuscript were restricted to refining the text. The authors addressed all of the minor points raised by the reviewers. Some of the major points such as the lack of a summary quantification still stand. The previous reviews are here: https://doi.org/10.7554/eLife.93496.2.sa1]

  6. eLife

    Author response:

    The following is the authors’ response to the previous reviews.

    Public Reviews:

    Reviewer #1 (Public Review):

    Summary:

    Satoshi Yamashita et al., investigate the physical mechanisms driving tissue bending using the cellular Potts Model, starting from a planar cellular monolayer. They argue that apical length-independent tension control alone cannot explain bending phenomena in the cellular Potts Model, contrasting with previous works, particularly Vertex Models. They conclude that an apical elastic term, with zero rest value (due to endocytosis/exocytosis), is necessary to achieve apical constriction, and that tissue bending can be enhanced by adding a supracellular myosin cable. Additionally, a very high apical elastic constant promotes planar tissue configurations, opposing bending.

    Strengths:

    - The finding of the required mechanisms for tissue bending in the cellular Potts Model provides a natural alternative for studying bending processes in situations with highly curved cells.

    - Despite viewing cellular delamination as an undesired outcome in this particular manuscript, the model's capability to naturally allow T1 events might prove useful for studying cell mechanics during out-of-plane extrusion.

    We thank the reviewer for the careful comments and suggestions.

    Weaknesses:

    - The authors claim that the cellular Potts Model (CPM) is unable to achieve the results of the vertex model (VM) simulations due to naturally non-straight cellular junctions in the CPM versus the VM. The lack of a substantial comparison undermines this assertion. None of the references mentioned in the manuscript are from a work using vertex model with straight cellular junctions, simulating apical constriction purely by a enhancing a length-independent apical tension. Sherrard et al and Pérez-González et al. use 2D and 3D Vertex Models, respectively, with a "contractility" force driving apical constriction. However, their models allow cell curvature. Both references suggest that the cell side flexibility of the CPM shouldn't be the main issue of the "contractility model" for apical constriction.

    We appreciate the comment.

    For the reports by Sherrard et al and Pérez-Gonález et al, lack of the cell rearrangement (T1 transition) might have caused the difference. Other than these, Muñoz et al. (doi:10.1016/j.jbiomech.2006.05.006), Polyakov et al. (doi:10.1016/j.bpj.2014.07.013), Inoue et al.

    (doi:10.1007/s10237-016-0794-1), Sui et al.

    (doi:10.1038/s41467-018-06497-3), and Guo et al. (doi:10.7554/eLife.69082) used simulation models with the straight lateral surface.

    We updated an explanation about the difference between the vertex model and the cellular Potts model in the discussion.

    P12L318 “An edge in the vertex model can be bent by interpolating vertices or can be represented with an arc of circle (Brakke, 1992). Even in cases where vertex models were extended to allow bent lateral surfaces, the model still limited cell rearrangement and neighbor changes (Pérez-González et al., 2021), limiting the cell delamination. Thus the difference in simulation results between the models could be due to whether the cell rearrangement was included or not. However, it is not clear how the absence of the cell rearrangement affected cell behaviors in the simulation, and it shall be studied in future. In contrast to the vertex model, the cellular Potts model included the curved cell surface and the cell rearrangement innately, it elucidated the importance of those factors.”

    - The myosin cable is assumed to encircle the invaginated cells. Therefore, it is not clear why the force acts over the entire system (even when decreasing towards the center), and not locally in the contour of the group of cells under constriction. The specific form of the associated potential is missing. It is unclear how dependent the results of the manuscript are on these not-well-motivated and model-specific rules for the myosin cable.

    A circle radius decreases when the circle perimeter shrinks, and this was simulated with the myosin cable moving toward the midline in the cross section.

    We added an explanation in the introduction and the results.

    P2L74 “In the same way with the contracting circumferential myosin belt in a cell decreasing the cell apical surface, the circular supracellular myosin cable contraction decreases the perimeter, the radius of the circle, and an area inside the circle.”

    P6L197 “In the cross section, the shrinkage of the circular supracellular myosin cable was simulated with a move of adherens junction under the myosin cable toward the midline.”

    - The authors are using different names than the conventional ones for the energy terms. Their current attempt to clarify what is usually done in other works might lead to further confusion.

    The reviewer is correct. However we named the energy terms differently because the conventional naming would be misleading in our simulation model.

    We added an explanation in the results.

    P4L140 “Note that the naming for the energy terms differs from preceding studies. For example, Farhadifar et al. (2007) named a surface energy term expressed by a proportional function "line tensions" and a term expressed by a quadratic function "contractility of the cell perimeter". In this study, however, calling the quadratic term "contractility" would be misleading since it prevents the contraction when < _0. Therefore we renamed the terms accordingly.”

    Reviewer #2 (Public Review):

    Summary:

    In their work, the Authors study local mechanics in an invaginating epithelial tissue. The work, which is mostly computational, relies on the Cellular Potts model. The main result shows that an increased apical "contractility" is not sufficient to properly drive apical constriction and subsequent tissue invagination. The Authors propose an alternative model, where they consider an alternative driver, namely the "apical surface elasticity".

    Strengths:

    It is surprising that despite the fact that apical constriction and tissue invagination are probably most studied processes in tissue morphogenesis, the underlying physical mechanisms are still not entirely understood. This work supports this notion by showing that simply increasing apical tension is perhaps not sufficient to locally constrict and invaginate a tissue.

    We thank the reviewer for the careful comments.

    Weaknesses:

    Although the Authors have improved and clarified certain aspects of their results as suggested by the Reviewers, the presentation still mostly relies on showing simulation snapshots. Snapshots can be useful, but when there are too many, the results are hard to read. The manuscript would benefit from more quantitative plots like phase diagrams etc.

    We agree with the comment.

    However, we could not make the qualitative measurement for the phase diagram since 1) the measurement must be applicable to all simulation results, and 2) measured values must match with the interpretation of the results. To do so, the measurement must distinguish a bent tissue, delaminated cells, a tissue with curved basal surface and flat apical surface, and a tissue with closed invagination. Such measurement is hardly designed.

    Recommendations for the authors:

    Reviewing Editor (Recommendations For The Authors):

    I see that the authors have worked on improving their paper in the revision. However, I agree with both reviewer #1 and reviewer #2 that the presentation and discussion of findings could be clearer.

    Concrete recommendations for improvement:

    (1) I find the observation by reviewer #1 on cell rearrangement very illuminating: It is indeed another key difference between the Cellular Potts Model that the authors use compared to typical Vertex Models, and could very well explain the different model outcomes. The authors could expand on the discussion of this point.

    We updated an explanation about the difference between the vertex model and the cellular Potts model in the discussion.

    P12L318 “An edge in the vertex model can be bent by interpolating vertices or can be represented with an arc of circle (Brakke, 1992). Even in cases where vertex models were extended to allow bent lateral surfaces, the model still limited cell rearrangement and neighbor changes (Pérez-González et al., 2021), limiting the cell delamination. Thus the difference in simulation results between the models could be due to whether the cell rearrangement was included or not. However, it is not clear how the absence of the cell rearrangement affected cell behaviors in the simulation, and it shall be studied in future. In contrast to the vertex model, the cellular Potts model included the curved cell surface and the cell rearrangement innately, it elucidated the importance of those factors.”

    (2) In lines 161-164, the authors write "Some preceding studies assumed that the apical myosin generated the contractile force (Sherrard et al, 2010: Conte et al., 2012; Perez-Mockus et al., 2017; Perez-Gonzalez et al., 2021), while others assumed the elastic force (Polyakov et al., 2014; Inoue et al. 2016; Nematbakhsh et al., 2020)."

    Similarly, in lines 316-319 the authors write "In the preceding studies, the apically localized myosin was assumed to generate either the contractile force (Sherrard et al, 2010: Conte et al., 2012; Perez-Mockus et al., 2017; Perez-Gonzalez et al., 2021), or the elastic force (Polyakov et al., 2014; Inoue et al. 2016; Nematbakhsh et al., 2020)."

    The phrasing here is poor, as it suggests that the latter three studies (Polyakov et al., 2014; Inoue et al. 2016; Nematbakhsh et al., 2020) do not use the assumption that apical myosin generated contractile forces. This is wrong. All three of these studies do in fact assume apical surface contractility mediated by myosin. In addition, they also include other factors such as elastic restoring forces from the cell membrane (but not mediated by myosin as far as I understand).

    These statements should be corrected.

    We named the energy term expressed with the proportional function “contractility” and the energy term expressed with the quadratic function “elasticity”. Here we did not define what biological molecules correspond with the contractility or the elasticity.

    For the three studies, the effect of myosin was expressed by the quadratic function, and Polyakov et al. (2014) named it “springlike elastic properties”, Inoue et al. (2016) named it “Apical circumference elasticity”, and Nematbakhsh et al. (2020) named it “Actomyosin contractility”. To explain that the for generated by myosin was expressed with the quadratic function in these studies, we wrote that they “assumed the elastic force”.

    We assumed the myosin activity to be approximated with the proportional function in later parts and proposed that the membrane might be expressed with the quadratic function and responsible for the apical constriction based on other studies.

    To clarify this, we added it to the results.

    P4L175 “Some preceding studies assumed that the apical myosin generated the contractile force (Sherrard et al., 2010; Conte et al., 2012; Perez-Mockus et al., 2017; Pérez-González et al., 2021), while the others assumed the myosin to generate the elastic force (Polyakov et al., 2014; Inoue et al., 2016; Nematbakhsh et al., 2020).”

    (3) Lines 294-296: The phrasing suggests that the "alternative driving mechanism" consists of apical surface elasticity remodelling alone. This is not true, it's an additional mechanism, not an alternative. The authors' model works by the combined action of increased apical surface contractility and apical surface elasticity remodelling (and the effect can be strengthened by including a supracellular actomyosin cable).

    We agree with the comment that the surface remodeling is not solely driving the apical constriction but with myosin activity. However, if we wrote it as an additional mechanism, it might look like that both the myosin activity alone and the surface remodeling alone could drive the apical constriction, and they would drive it better when combined together. So we replaced “mechanism” with “model”.

    P12L311 “In this study, we demonstrated that the increased apical surface contractility could not drive the apical constriction, and proposed the alternative driving model with the apical surface elasticity remodeling.”

    (4) In general, the part of the results section encompassing equations 1-5 should more explicitly state which equations were used in all simulations (Eqs1+5), and which ones were used only for certain conditions (Eqs2+3+4).

    We added it as follows.

    P4L153 “While the terms Equation 1 and Equation 5 were included in all simulations since they were fundamental and designed in the original cellular Potts model (Graner and Glazier, 1992), the other terms Equation 2-Equation 4 were optional and employed only for certain conditions.”

    (5) Lines 150-152: Please state which parameters were examined. I assume Equation 4 was also left out of this initial simulation, as it is the potential energy of the actomyosin cable that was only included in some simulations.

    We added it as follows.

    P4L163 “The term Equation 4 was not included either. For a cell, its compression was determined by a balance between the pressure and the surface tension, i.e., the heigher surface tension would compress the cell more. The bulk modulus 𝜆 was set 1, the lateral cell-cell junction contractility 𝐽_𝑙 was varied for different cell compressions, and the apical and basal surface contractilities 𝐽_𝑎 and 𝐽_𝑏 were varied proportional to 𝐽_𝑙.”

    (6) Lines 118-122: The sentence is very long and hard to parse. I suggest the following rephrasing:

    “In this study, we assumed that the cell surface tension consisted of contractility and elasticity. We modelled the contractility as constant to decrease the surface, but not dependent on surface width or strain. We modelled the elasticity as proportional to the surface strain, working to return the surface to its original width."

    We updated the explanation as follows.

    P3L121 “In this study, we assumed that the cell surface tension consisted of contractility and elasticity. We modeled the contractility as a constant force to decrease the surface, but not dependent on surface width or strain. We modeled the elasticity as a force proportional to the surface strain, working to return the surface to its original width.”

    (7) Lines 270-274: Another long sentence that is difficult to understand.

    Suggested rephrasing:

    "Note that the supracellular myosin cable alone could not reproduce the apical constriction (Figure 2c), and cell surface elasticity in isolation caused the tissue to stay almost flat. However, combining both the supracellular myosin cable and the cell surface elasticity was sufficient to bend the tissue when a high enough pulling force acted on the adherens junctions."

    We updated the sentence as follows.

    P9L287 “Note that the supracellular myosin cable alone could not reproduce the apical constriction (Figure 2c), and that with some parameters the modified cell surface elasticity kept the tissue almost flat (Figure 4). However, combining both the supracellular myosin cable and the cell surface elasticity made a sharp bending when the pulling force acting on the adherens junction was sufficiently high.”

    (8) Lines 434-435: Unclear what is meant with sentence starting with "Rest of sites"

    We update the sentence as follows.

    P17L456 “At the initial configuration and during the simulation, sites adjacent to medium and not marked as apical are marked as basal.”

    (9) Fixing typos and other minor grammar and wording changes would improve readability. Following is a list in order of appearance in the text with suggestions for improvement.

    We greatly appreciate the careful editing, and corrected the manuscript accordingly.

    Line 14: "a" is not needed in the phrase "increased a pressure"

    Line 15: "cell into not the wedge shape" --"cell not into the wedge shape" In fact it might be better to flip the sentence around to say, e.g. "making the cells adopt a drop shape instead of the expected wedge shape".

    Line 24: "cells decrease its apical surface" --"cells decrease their apical surface"

    Line 25: instead of "turn into wedge shape", a more natural-sounding expression could be "adopt a wedge shape"

    Line 28: "which crosslink and contract" --because the subject is the singular "motor protein", the verb tense needs to be changed to "crosslinks and contracts"

    Line 29: I suggest to use the definite article "the" before "actin filament network" as this is expected to be a known concept to the reader.

    Line 31: "adherens junction and tight junction" --use the plural, because there are many per cell: "adherens junctions and tight junctions"

    Line 42: "In vertebrate" --"In vertebrates"

    Line 46: "Since the interruption to" --"Since the interruption of"

    Line 56: "the surface tension of the invaginated cells were" --since the subject is "the surface tension", the verb "were" needs to be changed to "was" Line 63: "extra cellular matrix" --generally written as "extracellular matrix" without the first space

    Line 66: "many epithelial tissues" --"in many epithelial tissues"

    Line 70: "This supracellular cables" --"These supracellular cables"

    Line 72: "encircling salivary gland" --either "encircling the salivary gland" or "encircling salivary glands"

    Lines 76-77: "investigated a cell physical property required" --"investigated what cell physical properties were required"

    Line 78: "was another framework" --"is another framework" (it is a generally and currently valid true statement, so use the present tense)

    Line 79: "simulated an effect of the apically localized myosin" --for clarity, I suggest rephrasing as "simulated the effect of increased apical contractility mediated by apically localized myosin"

    Similarly, in Line 80: "did not reproduce the apical constriction" --"did not reproduce tissue invagination by apical constriction", as technically the cells in the model do reduce their apical area, but fail to invaginate as a tissue.

    Line 82: "we found that a force" --"we found that the force"

    Line 101: "apico-basaly" --"apico-basally"

    Lines 107-108: "in order to save a computational cost" --"in order to save on computational cost"

    Line 114: "Therefore an area of the cell" --"Therefore the interior area of the cell"

    Line 139: "formed along adherens junction" --"formed along adherens junctions"

    Line 166: "we ignored an effect" --"we ignored the effect"

    Line 167: "and discussed it later" --"and discuss it later"

    Lines 167-168: "an experiment with a cell cultured on a micro pattern showed that the myosin activity was well corresponded by the contractility" --"an experiment with cells cultured on a micro pattern showed that the myosin activity corresponded well to the contractility"

    Line 172: "success of failure" --"success or failure"

    Figure 1 caption: "none-polar" --"non-polarized"; "reg" --"red"

    Line 179: "To prevented the surface" --"To prevent the surface"

    Line 180: "It kept the cells surface" --"It kept the cells' surface" (apostrophe missing)

    Line 181: "cells were delaminated and resulted in similar shapes" --"cells were delaminated and adopted similar shapes"

    Line 190: "To investigate what made the difference" --"To investigate the origin of the difference"

    Line 203: For clarity, I would suggest to add more specific wording. "the pressure, and a difference in the pressure between the cells resulted in" --"the internal pressure due to cell volume conservation, and a difference in the pressure between the contracting and non-contracting cells resulted in"

    Line 206: "by analyzing the energy with respect to a cell shape" --"by analyzing the energy with respect to cell shape"

    Line 220: "indicating that cell could shrink" --"indicating that a cell could shrink"

    Line 224: For clarity, I would suggest more specific wording "lateral surface, while it seems not natural for the epithelial cells" --"lateral surface imposed on the vertex model, a restriction that seems not natural for epithelial cells"

    Line 244: "succeeded in invaginating" --"succeeding in invaginating"

    Line 247: "were checked whether the cells" --"were checked to assess whether the cells"

    Line 250: "cells became the wedge shape" --"cells adopted the wedge shape"

    Line 286: "there were no obvious change in a distribution pattern" --"there was no obvious change in the distribution pattern"

    Lines 296-297: "When the cells were assigned the high apical surface contractility, the cells were rounded" --"When the cells were assigned a high apical surface contractility, the cells became rounded"

    Line 298: "This simulation results" --"These simulation results"

    Lines 301-302: I suggest to increase clarity by somewhat rephrasing. "Even when the vertex model allowed the curved lateral surface, the model did not assume the cells to be rearranged and change neighbors" --"Even in cases where vertex models were extended to allow curved lateral surfaces, the model still limited cell rearrangement and neighbor changes"

    Line 326: "high surface tension tried to keep" --"high surface tension will keep"

    Line 334: "In many tissue" --"In many tissues"

    Line 345: "turned back to its original shape" --"turned back to their original shape" (subject is the plural "cells")

    Lines 348-349: "resembles the result of simulation" --"resembles the result of simulations"

    Line 352: "how the myosin" --"how do the myosin"

    Line 356: "it bears the surface tension when extended and its magnitude" What does the last "its" refer to? The surface tension?

    Line 365: "the endocytosis decrease" --"the endocytosis decreases"

    Line 371: "activatoin" --"activation"

    Line 374 "the cells undergoes" --"the cells undergo"

    Line 378: "entier" --"entire"

    Line 389: "individual tissue accomplish" --"individual tissues accomplish"

    Line 423: "is determined" --"are determined" (subject is the plural "labels")

    Line 430: "phyisical" --"physical"

    Table 6 caption: "cell-ECN" --cell-ECM

    Line 557: "do not confused" --"should not be confused"

    Reviewer #1 (Recommendations For The Authors):

    - The phrase "In addition, the encircling supracellular myosin cable largely promoted the invagination by the apical constriction, suggesting that too high apical surface tension may keep the epithelium apical surface flat." is not clear to me. It sounds contradictory.

    This finding was unexpected and surprising for us too. However, it is actually not contradictory since stronger surface tension will make the surface flatter in general. Figure 4 shows the flat apical surface with the wedge shape cells for the too strong apical surface tension. On the other hand, the supracellular myosin cable promoted the cell shape changes without raising the surface tension, and thus it could make a sharp bending (Figure 5).

    We updated the explanation for the effect of the supracellular myosin cable as follows.

    P2L74 “In the same way as the contracting circumferential myosin belt in a cell decreasing the cell apical surface, the circular supracellular myosin cable contraction decreases the perimeter, the radius of the circle, and an area inside the circle.”

    P6L197 “In the cross section, the shrinkage of the circular supracellular myosin cable was simulated with a move of adherens junction under the myosin cable toward the midline.”

    - Even when the authors now avoid to say "in contrast to vertex model simulations" in pg.4, in the next section there is still the intention to compare VM to CPM. Idem in the Discussion section. The conclusion in that section is that the difference between the results arising with VM (achieving the constriction) and the CPM (not achieving the constriction, and leading to cell delamination) are due to the straight lateral surfaces. However, Sherrard et at could achieve the constriction with an enhanced apical surface contractility using a 2D VM that allows curvatures. Therefore, I don't think the main difference is given by the deformability of the lateral surfaces. Instead, it might be due to the facility of the CPM to drive cellular rearrangements, coupled to specific modeling rules such as the permanent lost of the "apical side" once a delamination occurs and the boundary conditions. A clear example is the observation of loss of cell-cell adherence when all the tensions are set the same. Instead, in a VM cells conserve their lateral neighbors in the uniform tension regime (Sherrard et at). Is it noteworthy that the two mentioned works using vertex models to achieve apical constriction (Sherrard et at. (2D) and Pérez-González (3D) et al.) seem to neglect T1 transitions. I specifically think the added discussion on the impact of the T1 events (fundamental for cell delamination) is quite poor. A more detailed description would help justify the differences between model outcomes.

    We updated an explanation about the difference between the vertex model and the cellular Potts model in the discussion.

    P12L318 “ An edge in the vertex model can be bent by interpolating vertices or can be represented with an arc of circle (Brakke, 1992). Even in cases where vertex models were extended to allow bent lateral surfaces, the model still limited cell rearrangement and neighbor changes (Pérez-González et al., 2021), limiting the cell delamination. Thus the difference in simulation results between the models could be due to whether the cell rearrangement was included or not. However, it is not clear how the absence of the cell rearrangement affected cell behaviors in the simulation, and it shall be studied in future. In contrast to the vertex model, the cellular Potts model included the curved cell surface and the cell rearrangement innately, it elucidated the importance of those factors.”

    - Fig6c: cell boundary colors are quite difficult to see.

    The images were drawn by custom scripts, and those scripts do not implement a method to draw wide lines.

    - Title Table 1: "epitherila".

    We corrected the typo.

    Reviewer #2 (Recommendations For The Authors):

    The Authors have addressed most of my initial comments. In my opinion, the results could be better represented. Overall, the manuscript contains too many snapshots that are hard to read. I am sure the Authors could come up with a parameter that would tell the overall shape of the tissue and distinguish between a proper invagination and delamination. Then they could plot this parameter in a phase diagram using color plots to show how varying values of model parameters affects the shape. Presentation aside, I believe the manuscript will be a valuable piece of work that will be very useful for the community of computational tissue mechanics.

    We agree with the comment.

    However, we could not make a suitable qualitative measurement method. For the phase diagrams, the measurement must be applicable to simulation results, otherwise each figure introduce a new measurement and a color representation would just redraw the snapshots but no comparison between the figures. So the different measurements would make the figures more difficult to read.

    The single measurement must distinguish the cell delamination by the increased surface contractility from the invagination by the modified surface elasticity and the supracellular contractile ring, even though the center cells were covered by the surrounding cells and lost contact with apical side extracellular medium in both cases.

    With the center of mass, the delaminated cells would return large values because they were moved basally. With the tissue basal surface curvature, it would not measure if the tissue apical surface was also curved or kept flat. If the phase diagram and interpretation of the simulation results do not match with each other, it would be misleading.

    A measurement meeting all these conditions was hardly designed.

  7. Version published to 10.1101/2023.12.19.572310v3 on bioRxiv
  8. Version published to 10.7554/elife.93496.2 on eLife
  9. eLife

    eLife assessment

    The work investigates mechanisms necessary and sufficient for initiating tissue bending in the Cellular Potts Model. The authors emphasize how differences in implicit model assumptions, such as different constraints on cell shape change and cell rearrangement, may explain different outcomes in Cellular Potts Model and Vertex Model simulations. Despite incomplete evidence supporting the major claims due to a rather coarse-grained exploration of the model, the findings are valuable for the biophysics and computational biology communities, and cautions toward greater care in interpretation of model results.

  10. eLife

    Reviewer #1 (Public Review):

    Summary:

    Satoshi Yamashita et al., investigate the physical mechanisms driving tissue bending using the cellular Potts Model, starting from a planar cellular monolayer. They argue that apical length-independent tension control alone cannot explain bending phenomena in the cellular Potts Model, contrasting with previous works, particularly Vertex Models. They conclude that an apical elastic term, with zero rest value (due to endocytosis/exocytosis), is necessary to achieve apical constriction, and that tissue bending can be enhanced by adding a supracellular myosin cable. Additionally, a very high apical elastic constant promotes planar tissue configurations, opposing bending.

    Strengths:

    - The finding of the required mechanisms for tissue bending in the cellular Potts Model provides a natural alternative for studying bending processes in situations with highly curved cells.
    - Despite viewing cellular delamination as an undesired outcome in this particular manuscript, the model's capability to naturally allow T1 events might prove useful for studying cell mechanics during out-of-plane extrusion.

    Weaknesses:

    - The authors claim that the cellular Potts Model (CPM) is unable to achieve the results of the vertex model (VM) simulations due to naturally non-straight cellular junctions in the CPM versus the VM. The lack of a substantial comparison undermines this assertion. None of the references mentioned in the manuscript are from a work using vertex model with straight cellular junctions, simulating apical constriction purely by a enhancing a length-independent apical tension. Sherrard et al and Pérez-González et al. use 2D and 3D Vertex Models, respectively, with a "contractility" force driving apical constriction. However, their models allow cell curvature. Both references suggest that the cell side flexibility of the CPM shouldn't be the main issue of the "contractility model" for apical constriction.
    - The myosin cable is assumed to encircle the invaginated cells. Therefore, it is not clear why the force acts over the entire system (even when decreasing towards the center), and not locally in the contour of the group of cells under constriction. The specific form of the associated potential is missing. It is unclear how dependent the results of the manuscript are on these not-well-motivated and model-specific rules for the myosin cable.
    - The authors are using different names than the conventional ones for the energy terms. Their current attempt to clarify what is usually done in other works might lead to further confusion.

  11. eLife

    Reviewer #2 (Public Review):

    Summary:

    In their work, the Authors study local mechanics in an invaginating epithelial tissue. The work, which is mostly computational, relies on the Cellular Potts model. The main result shows that an increased apical "contractility" is not sufficient to properly drive apical constriction and subsequent tissue invagination. The Authors propose an alternative model, where they consider an alternative driver, namely the "apical surface elasticity".

    Strengths:

    It is surprising that despite the fact that apical constriction and tissue invagination are probably most studied processes in tissue morphogenesis, the underlying physical mechanisms are still not entirely understood. This work supports this notion by showing that simply increasing apical tension is perhaps not sufficient to locally constrict and invaginate a tissue.

    Weaknesses:

    Although the Authors have improved and clarified certain aspects of their results as suggested by the Reviewers, the presentation still mostly relies on showing simulation snapshots. Snapshots can be useful, but when there are too many, the results are hard to read. The manuscript would benefit from more quantitative plots like phase diagrams etc.

  12. eLife

    Author response:

    The following is the authors’ response to the original reviews.

    Public Reviews:

    Reviewer #1 (Public Review):

    Summary:

    Satoshi Yamashita et al., investigate the physical mechanisms driving tissue bending using the cellular Potts Model, starting from a planar cellular monolayer. They argue that apical length-independent tension control alone cannot explain bending phenomena in the cellular Potts Model, contrasting with the vertex model. However, the evidence supporting this claim is incomplete. They conclude that an apical elastic term, with zero rest value (due to endocytosis/exocytosis), is necessary in constricting cells and that tissue bending can be enhanced by adding a supracellular myosin cable. Notably, a very high apical elastic constant promotes planar tissue configurations, opposing bending.

    Strengths:

    - The finding of the required mechanisms for tissue bending in the cellular Potts Model provides a more natural alternative for studying bending processes in situations with highly curved cells.

    - Despite viewing cellular delamination as an undesired outcome in this particular manuscript, the model's capability to naturally allow T1 events might prove useful for studying cell mechanics during out-of-plane extrusion.

    We thank the reviewer for the careful comments and insightful suggestions.

    Weaknesses:

    - The authors claim that the cellular Potts Model is unable to obtain the vertex model simulation results, but the lack of a substantial comparison undermines this assertion. No references are provided with vertex model simulations, employing similar setups and rules, and explaining tissue bending solely through an increase in a length-independent apical tension.

    Studies cited in a previous paragraph included the simulations employing the increased length-independent apical tension. For the sake of clarity, we added the citation to them as below.

    P4L174: “In contrast to the simulations in the preceding studies (Sherrard et al., 2010; Conte et al., 2012; Perez-Mockus et al., 2017; Pérez-González et al., 2021), our simulations could not reproduce the apical constriction”.

    We did not copy the parameters of the vertex models in the preceding studies because we also found that the apical, lateral, and basal surface tensions must be balanced otherwise the epithelial cell could not maintain the integrity (Figure 1—figure supplement 1), while the ratio was outside of the suitable range in the preceding studies.

    - The apparent disparity between the two models is attributed to straight versus curved cellular junctions, with cells with a curved lateral junction achieving lower minimum energies at steady-state. However, a critical discussion on the impact of T1 events, allowing cellular delamination, is absent. Note that some of the cited vertex model works do not allow T1 events while allowing curvature.

    We appreciate the comment and added it to the discussion as suggested.

    P12L301: “Even when the vertex model allowed the curved lateral surface, the model did not assume the cells to be rearranged and change neighbors, limiting the cell delamination (Pérez-González et al., 2021).”

    P12L311: “Note that the vertex model could also be extended to incorporate the curved edges and rearrangement of the cells by specifically programming them, and would reproduce the cell delamination. That is, we could find the importance of the balanced pressure because the cellular Potts model intrinscally included a high degree of freedom for the cell shape, the cell rearrangement, and the fluctuation.”

    - The suggested mechanism for inducing tissue bending in the cellular Potts Model, involving an apical elastic term, has been utilized in earlier studies, including a cited vertex model paper (Polyakov 2014). Consequently, the physical concept behind this implementation is not novel and warrants discussion.

    The reviewer is correct but Polyakov et al. assumed “that the cytoskeletal components lining the inside membrane surfaces of the cells provide these surfaces with springlike elastic properties” without justification. We assumed that the myosin activity generated not the elasticity but the contractility based on Labouesse et al. (2015), and expected that the surface elasticity corresponded with the membrane elasticity. Also, in the physical concept, we clarified how the contractility and the elasticity differently deformed the cells and tissue, and demonstrated why the elasticity was important for the apical constriction. We added it to the discussion as below.

    P12L316: “In the preceding studies, the apically localized myosin was assumed to generate either the contractile force (Sherrard et al., 2010; Conte et al., 2012; Perez-Mockus et al., 2017; Pérez-Vonzález et al., 2021) or the elastic force (Polyakov et al., 2014; Inoue et al., 2016; Nematbakhsh et al., 2020). However, the limited cell shape in the vertex model made them similar in terms of the energy change during the apical constriction, i.e., the effective force to decrease the apical surface. In this study, we showed that the contractile force and the elastic force differently deformed the cells and tissue, and demonstrated why and how the elasticity was important for the apical constriction.”

    - The absence of information on parameter values, initial condition creation, and boundary conditions in the manuscript hinders reproducibility. Additionally, the explanation for the chosen values and their unit conversion is lacking.

    We agree with the comment.

    For the initial configuration, we added an explanation to Tissue deformation by increased apical contractility with cellular Potts model section in the Results as below.

    P4L170: “A simulation started from a flat monolayer of cells beneath the apical ECM, and was continued until resulting deformation of cells and tissue could be evaluated for success of failure of reproducing the apical constriction.”

    For the parameter values we added a section “Parameters for the simulations” in the Methods.

    For the parameters unit conversion, we did not measure the surface tension and cell pressure in an actual tissue and thus could not compare the parameters to the actual forces. Instead, we varied the parameters and demonstrated that the apical constriction was reproduced with the wide range of the parameter values. We added it to the discussion as below.

    P12L310: “It succeeded with a wide range of parameter values, indicating a robustness of the model.”

    Reviewer #2 (Public Review):

    Summary:

    In their work, the authors study local mechanics in an invaginating epithelial tissue. The mostly computational work relies on the Cellular Potts model. The main result shows that an increased apical "contractility" is not sufficient to properly drive apical constriction and subsequent tissue invagination. The authors propose an alternative model, where they consider an alternative driver, namely the "apical surface elasticity".

    Strengths:

    It is surprising that despite the fact that apical constriction and tissue invagination are probably most studied processes in tissue morphogenesis, the underlying physical mechanisms are still not entirely understood. This work supports this notion by showing that simply increasing apical tension is perhaps not sufficient to locally constrict and invaginate a tissue.

    We thank the reviewer for recognizing the importance and novelty of our work.

    Weaknesses:

    The findings and claims in the manuscript are only partially supported. With the computational methodology for studying tissue mechanics being so well developed in the field, the authors could probably have done a more thorough job of supporting the main findings of their work.

    We thank the reviewer for the careful assessment and suggestions. However our simulation was computationally expensive, modeling the epithelium in an analytically calculable expression requires a lot of work, and it is beyond the scope of the present study.

    Recommendations for the authors:

    Reviewer #1 (Recommendations For The Authors):

    (1) Reference line 648: Correct the author's name (Pérez-González).

    We thank the reviewer and corrected the reference.

    (2) "Pale" colors are challenging to discern.

    We updated the figures.

    (3) Figure 1j: What does the yellow color in the cellular junction represent?

    We used the apical lateral site colored yellow in Fig. 1e-f’ to simulate the effect of the adherens junction. We updated the figure legend.

    (4) Figure 2c - left: Why is there a red apical junction?

    Our simulation model marked the apical junction in the initial configuration and updated the marking based on connectedness to surrounding other site marked as apical in the same cell. But when a cell was once delaminated and lost its apical junction, any surface site not adjacent to other epithelial cells were marked as basal junction because they were not adjacent to the apical junction.

    We added it to Cellular Potts model with partial surface elasticity section in the Methods as below.

    P17L430: “To simulate the differential phyisical properties of the apical, lateral, and basal surfaces, the subcellular locations are marked automatically, and the marking is updated during the simulation. In each cell, sites adjacent to different cells but not to the medium are marked as lateral.

    At the initial configuration, sites adjacent to the apical ECM are marked as apical, and during the simulation, sites adjacent to medium and other apical sites in the same cell are marked as apical.

    Rest of sites which are adjacent to medium but not marked as apical are marked as basal.

    Therefore, once a cell is delaminated and loses its apical surface, afterwards all sites in the cell adjacent to the medium are marked as basal even if it is adjacent to the apical ECM or the outer body fluid.”

    (5) Figure 4a: The snapshots are not in a steady state but in the middle of deformation. Is the time the same for all snapshots? The motivation to change P_0a is related to endocytosis. However, this could be achieved by decreasing P_0a to a non-zero value. Here, in the more drastic limit, the depth (a measure of bending) is very slight, approximately half of a cell size. What physically limits further invagination? Is it the number of cells or the range of parameters under study?

    The time length was the same for simulations in each figure, and we add it to Parameters for the simulations section in Method as below.

    P18L466: “In each figure, snapshots of the simulations show deformation by the same time length unless specified.”

    For P_0a, the reviewer is correct and the iterated ratcheting may decrease P_0a step by step instead of making it 0 immediately. Still, with P_a0 >0, the energy function and its derivative are both increasing with respect to the apical width as long as P_a > P_a0, and thus the apical shrinkage would be synchronized, even though the deformation would be smaller. We also run simulations by decreasing P_0a to 0.6 times the initial P_a, and observed smaller deformation as expected. On the other hand, the non-zero P_0a made the invagination deeper when it was combined with the effect of surrounding supracellular myosin cable, maybe due to a resistance of the apical surface against compression. One of the novel and important finding in this study is the synergetic effect of the elasticity-based apical constriction and the surrounding supracellular myosin cable. To demonstrate that the deep invagination was not due to the apical surface resistance against the compression, we showed the simulations with P_a0 = 0.

    For the conditions for further invagination, it may include the number of cells, a ratio between the cell height and width (Figure 5—figure supplement 1), interaction with ECM (Figure 5—figure supplement 2), etc. For the parameter, there might be an upper limit (Figure 4). We did not test the number of cells because of its computational cost. Among the conditions we tested, we found the planar compression by surrounding supracellular myosin the most influential rather than the mechanical property of apically constricting cells themselves.

    How each condition and parameter contributes to the invagination shall be studied in future. We added it to the conclusion as below.

    P15L395: “The depth, curvature, and speed of the invagination might be influenced by the cell shape, configuration, and parameters, and how each condition contributes to the invagination shall be studied in future.”

    (6) Figure 6b: What does the cell-surface color represent? If the idea was to represent junction tension, it would be clearer to color the junctions only.

    The junction tension may vary differently in different situations. For example, T1 transition is accompanied by enriched myosin along a shrinking cell-cell junction, and the junction bears higher tension, but other junctions of the same cell do not and thus the cell does not decrease its apical surface. In chick embryo neural tube closure, the junction tension is also polarized, and the cells shrink the apical surface along medial-lateral axis, driving the apical constriction (Nishimura et al., 2012, doi:10.1016/j.cell.2012.04.021). In the case of Drosophila embryo tracheal invagination, the cells shrank their apical surface isotropically (Figure 6a). If the junction tension was responsible for the shrinkage, all junctions of the cell must bear higher tension. Based on this assumption, the junction tension was averaged in each cell to check if the tracheal cells bore the higher average tension than surrounding cells.

    We also plotted stress tensor and calculated nematic order to check if there was radial or encircling tension alignment in the tracheal pit, but there was not.

    (7) Figure 6c: What does the junction color represent here?

    The junction color represent the relative junctional tension. We updated the figure legend.

    (8) Figure 6d-e: It is challenging to understand which error bar corresponds to each dataset.

    We updated the figure.

    (9) What is the definition of relative pressure?

    The geometrical tension inference method assumes that the tissue is in mechanical equilibrium and a sum of the junctional tensions and cell pressures pulling/pushing a vertex (tricellular junction) is 0. Therefore the calculated tensions and pressures are proportional to each other but not absolute values. We added it to the 3D Bayesian tension inference section of Methods as below.

    P24L567: “Since Equation 13 and Equation 14 only evaluate the balance among the forces, it cannot estimate an absolute value but a relative value of the tension and pressure.”

    (10) In the main text, it is mentioned that a large Es (apical elastic constant) leads to flat surfaces, avoiding bending, but the abstract says "strong apical surface tension," which, according to the rest of the text, would seem to be J_apical. Clarification is needed.

    The surface tension includes both of the surface contractility and the surface elasticity.

    We added it to Extended cellular Potts model to simulate epithelial deformations section in the Results as below.

    P3L122: “Note that in some studies the tension and the contractility are considered as equivalent, but they are distinguished in this study.”

    and

    P4L151: “The energy H included only the terms of the contact energy (Equation 1) and the area constraint (Equation 5), but the surface elasticity (Equation 2) nor (Equation 3) was not included, and thus the surface tension was determined by the contact energy.”

    Reviewer #2 (Recommendations For The Authors):

    (1) The model used is rather specific and it is rather confusing whether the issue is in the methodology or fundamental biophysics of apical constriction. For instance, one of the main narratives of the manuscript is that the Cellular Potts model better predicts apical constriction and tissue invagination than the vertex model. As I understand it, and as the authors state in p7 (line 210), "the difference between the vertex model and the cellular Potts model results was due to the straight lateral surface...". I assume that if apical constriction and tissue invagination were modelled with a vertex model with curved edges, while also allowing for cell rearrangements out of the tissue plane (some sort of epithelium-to-mesenchyme transition), the vertex model would yield exactly the same results as in the authors' cellular Potts model. If my understanding is correct, the authors should change the narrative of their manuscript and focus more on the comparison of a model with flat vs. curved edges, with "contractility" vs. "surface elasticity", with patterned apical contractility vs. non-patterned contractility (see my comment in point 2 below)... and not on comparison between CPM and VM.

    We appreciate the comments. The reviewers is correct that the vertex model can include the curved edges and the cell rearrangement, and it would reproduce the result of our cellular Potts model simulations. For the cellular Potts model, there was no need to specifically design how much the cell surface could be curved in a large arc, zigzag, or other shape, and that enabled us to find the conditions of delamination and bending.

    We added it to the discussion as below.

    P12L311: “Note that the vertex model could also be extended to incorporate the curved edges and rearrangement of the cells by specifically programming them, and would reproduce the cell delamination. That is, we could find the importance of the balanced pressure because the cellular Pott’s model intrinscally included a high degree of freedom for the cell shape, the cell rearrangement, and the fluctuation.”

    (2) About physics... and I think this is a really important point: one of the observations in the model was that in the "contractilty" model, only "edge cells" shrank its apical surface, while inner cells remained quadrilateral. Related to this, the authors say that one of the requirements for proper apical constriction is a mechanism that "simulataneously shrinks the apical surface among cells in a cluster". What would happen if the authors assumed patterned contractility, meaning that cells in the center of the cluster would be most apically-contractile, while those further away from the center, would not be contractile? Features like this were investigated in studies of ventral-furrow invagination [see, for instance, Spahn and Reuater PLOS ONE (2013) and Rauzi et al. Nat Commun (2015)-Fig. S13d].

    We thank the reviewer for the critical comment, and ran simulations with the patterned apical contractility. The apical contractility following a gradient of parabola shape succeeded in the simultaneous apical shrinkage. However, it was weak against fluctuations and the cells were delaminated by chance.

    We added it to Apical constriction by modified apical elasticity section in the result as below.

    P9L252: “We also tested another model for the simultaneous apical shrinkage, a gradient contractility model (Spahn and Reuter, 2013; Rauzi et al., 2015). If the inner cells bear higher apical surface contractility than the edge cells, that inner cells may shrink their apical surface. To synchronize the apical shrinkage, the apical contractility must follow a parabola shape gradient. Even though the gradient contractility enabled the cells to shrink the apical surface simultaneously, often some of the cells shrank faster than neighbors and were delaminated by chance (Figure 4—figure Supplement 1).”

    (3) The quality of the figures should be improved. Especially, Figure 3 and the related explanation in lines 183-192. This explanation is way too complicated and it is not clear what Figure 3c shows. For instance: if the arrows are indeed showing contractile forces (as written in the caption) then they are not illustrated correctly, but should be tangential to the cell membrane.

    We updated the figure.

    (4) The figures mostly show steady-state cross-sections from simulations. I miss a more dedicated study with model parameters being varied through wider ranges and some phase diagrams being shown etc. Also, some results could probably be supported by analytic calculations. For instance, the condition for stability (discussed in p4 lines 145-151), cells' preferred aspect ratio, cells' preferred "wedgeness" i.e., local curvature etc... I am sure some of these, if not all, could be calculated analytically and then these analytic results could help to interpret the phase diagrams.

    For the simulation results shown in the figures, we were not sure if the simulations results were in a steady state or not. We added it to Tissue deformation by increased apical contractility simulated with cellular Potts model section in the Results as below.

    P4L170: “A simulation started from a flat monolayer of cells beneath the apical ECM, and was continued until resulting deformation of cells and tissue could be evaluated for success of failure of reproducing the apical constriction.”

    For the ranges of parameters, we ran the simulation in wider range and showed results from sub-range. We added it to Parameters for the simulations section in Methods as below.

    P18L464: “The parameters were varied in a range, and the figures showed simulations with parameter values within a sub-range so that the results showed both success and failure in a development of interest.”

    For the analytical calculations, the Figure 3f shows a kind of phase diagram for shapes of a single cell. To clarify this, we rephrased “map of cell shapes” to “Phase diagram of cell shapes” in the figure legend, and added an explanation to the Results section as below.

    P6L207: “For the analysis of the cell shape in motion, we plotted a phase diagram for shapes of a single cell (Figure 3f).”

    For the analytical evaluation of the cellular Potts model simulations, there was a study doing similar but it concerned a cell of isotropic shape in a steady state (Magno et al., 2015, doi:10.1186/s13628-015-0022-x). Also, our simulation framework is computationally expensive and we could not vary the parameters in fine resolution. Therefore we could not include it in this study.

    (5) I am not sure about the terminology "contractility" vs. "elasticity". In Farhadifar et al. (2007) "contractility" is described by a squared apical-perimeter energy term, while in this work, the authors describe it by a surface-energy-like term.

    In general, elasticity is the ability of a material to resist against deformation and to return to its original shape/size. In Farhadifar et al. (2007), the cell apical area was assigned the area elasticity in this meaning. For the contractility, it is the ability to decrease the size/length, and thus it could be either expressed in linear or quadratic dependent on the modeling. In this study, we assumed cell-cell/cell-ECM adhesion and myosin activity to generate the surface contractility, and thus employed the linear expression. In Farhadifar et al. (2007) it was described as a line tension.

    We used the terms surface ‘elasticity’ and ‘contractility’ as distinctive elements composing the surface ‘tension’. We added it Extended cellular Potts model to simulate epithelial deformations section in the Results as below.

    P3L122: “Note that in some studies the tension and the contractility are considered as equivalent, but they are distinguished in this study.”

    (6) It is not entirely clear what are apical, basal, lateral, and cell "perimeters". This is a 2D model, so I assume all P-s are in fact interface lengths. In either case, this needs to be explained more clearly.

    We updated the explanation in Extended cellular Potts model to simulate epithelial deformations section in the Results as below.

    P3L111: “The cell's perimeter was partitioned automatically based on adjacency with other cells, and it was marked as apical, lateral, basal. Also, apico-lateral sites were marked as a location for the adherens junction. This cell representation also cast the vertical section of the cell. Therefore an area of the cell corresponded with a body of the cell, and a perimeter of the cell corresponded with the cell surface. Likewise the apical, lateral, and basal parts of the perimeter corresponded with the apical surface, cell-cell interface, and the basal surface of the cell respectively.”

    (7) The term H_{mc} is not clear at all. Why is this term called potential energy? What is U(i)? What is the exact biophysical interpretation of this term in 2D vs 3D?

    In 3D, the supracellular myosin cable is formed encircling the cells deformed by the apical constriction. Shrinking of the supracellular myosin cable makes the circle small, and it moves the cable toward the center of the circle. To simulate this motion of the supracellular myosin cable in the 2D cross section, we assigned the force exerted on the adherens junction of the boundary cells pulling toward the center, and because the force is relative to the position of the adherens junction and the center, it was expressed by the potential energy in the simulation.

    We updated Extended cellular Potts model to simulate epithelial deformation section in Results and Cellular Potts model with potential energy section in Methods as below.

    P4L140: “The potential energy was defined by a scalar field which made a horizontal gradient decreasing toward the center,”

    and

    P17L449: “In 3D, tension on a circular actomyosin cable would shrink the circle, and the shrinkage would pull the cable toward the center of the circle. In 2D cross section, the cable is pulled horizontally toward the middle line.”

    (8) Highten->increased

    We updated the text.

    (9) "It seems natural to consider that the myosin generates a force proportional to its density but not to the surface width nor the strain". This sentence should be supported by a reference. Also, if the force is proportional to myosin density, then it must depend on surface width, since density, I assume, is the number of motors per area.

    For the myosin density and generated force, in all preceding studies cited in this manuscript and others in the extent of our knowledge, the myosin and actin filaments density visualized by staining or labeling had been assumed relevant to the generated contractility without references. Therefore it might be well established and shared assumption.

    For the independence from the surface width and strain, the review comment is correct, but the results would be the same. If we presumed that the number of motors on the apical surface was constant in a cell during the apical constriction, then the density would increase when the apical surface was contracted, and thus it would make the apical contractility more unbalanced and promote the delamination. We added it to the results and discussion as below.

    P4L166: “For the sake of simplicity, we ignored an effect of the constriction on the apical myosin density, and discussed it later.”

    P14L328: “In our model, for the sake of simplicity, we ignored an effect of the constriction on the apical myosin density. If we presumed that the apical myosin would be condensed by the shrinkage of the apical surface, it would increase the apical tension in the shrinking cell and is expected to promote the cell delamination further. Therefore it would not change the results.”

    Reviewing Editor (Recommendations For The Authors):

    Please note also the following excerpts from discussions amongst the reviewers and the Reviewing Editor:

    Regarding Reviewer #2's Point 2:

    I believe the authors have assumed patterned contractility in their simulations, and this is shown by the "pale blue" cell color (see also lines 162-163). However, as Reviewer #2 points out in their point 2), the pale colors are very hard to see and therefore easy to miss.

    We updated figure coloring and also add the gradient pattern of contractility.

    Regarding Reviewer #2's point 5:

    It is indeed unconventional to call the "J" terms contractility, they are usually called contact energy or adhesive energy.

    In this study, we included both of the contact energy of cell-cell/cell-ECM adhesion and actomyosin activity in the surface contractility, and used the “J” term as it was conventional in the cellular Potts model.

    On the other hand, due to the parameters chosen for J_apical and J_basal in the pale blue cells, the apical membrane area will tend to shrink and the basal membrane will tend to enlarge. Because the lateral membrane energy J_lateral is constant among all cells (I think?), this will effectively drive cells to apically contract in the center.

    That expectation was an initial motivation of our study, but we found that the differential J alone could not drive the cells to apically contract in the center.

    I agree that extra clarification by the authors would be very helpful here.

    Reviewer #2:

    Regarding the patterned contractility: indeed, I missed this point (the pale blue region is really poorly visible).

    Nevertheless, it seems that contractility in the authors' model changes in a step-like fashion.

    [...] There may be important differences between furrowing under step-like patterning profile versus smooth "bell-like" patterning (see Supplementary Figure 13 in Rauzi et al. Nat Commun 2015). In particular, in the case of a step-like patterning, [there are] constrictions of side cells (similar to what the authors in this manuscript report), whereas in the bell-like patterning, [...] such side constrictions [do not occur].

    As replied to the reviewer #2 comment (2), we added the simulations with gradient-pattern contractility.

  13. Version published to 10.1101/2023.12.19.572310v2 on bioRxiv
  14. Version published to 10.7554/elife.93496.1 on eLife
  15. eLife

    Author Response

    Public Reviews:

    Reviewer #1 (Public Review):

    Summary:

    Satoshi Yamashita et al., investigate the physical mechanisms driving tissue bending using the cellular Potts Model, starting from a planar cellular monolayer. They argue that apical length-independent tension control alone cannot explain bending phenomena in the cellular Potts Model, contrasting with the vertex model. However, the evidence supporting this claim is incomplete. They conclude that an apical elastic term, with zero rest value (due to endocytosis/exocytosis), is necessary in constricting cells and that tissue bending can be enhanced by adding a supracellular myosin cable. Notably, a very high apical elastic constant promotes planar tissue configurations, opposing bending.

    Strengths:

    • The finding of the required mechanisms for tissue bending in the cellular Potts Model provides a more natural alternative for studying bending processes in situations with highly curved cells.
    • Despite viewing cellular delamination as an undesired outcome in this particular manuscript, the model's capability to naturally allow T1 events might prove useful for studying cell mechanics during out-of-plane extrusion.

    We thank the reviewer for the careful comments and insightful suggestions.

    Weaknesses:

    • The authors claim that the cellular Potts Model is unable to obtain the vertex model simulation results, but the lack of a substantial comparison undermines this assertion. No references are provided with vertex model simulations, employing similar setups and rules, and explaining tissue bending solely through an increase in a length-independent apical tension.

    We did not copy the parameters of the vertex models in the preceding studies because we also found that the apical, lateral, and basal surface tensions must be balanced otherwise the epithelial cell could not maintain the integrity (Supplementary Figure 1), while the ratio was outside of the suitable range in the preceding studies.

    • The apparent disparity between the two models is attributed to straight versus curved cellular junctions, with cells with a curved lateral junction achieving lower minimum energies at steady-state. However, a critical discussion on the impact of T1 events, allowing cellular delamination, is absent. Note that some of the cited vertex model works do not allow T1 events while allowing curvature.

    We appreciate the comment, and will add it to the discussion.

    • The suggested mechanism for inducing tissue bending in the cellular Potts Model, involving an apical elastic term, has been utilized in earlier studies, including a cited vertex model paper (Polyakov 2014). Consequently, the physical concept behind this implementation is not novel and warrants discussion.

    The reviewer is correct but Polyakov et al. assumed “that the cytoskeletal components lining the inside membrane surfaces of the cells provide these surfaces with springlike elastic properties” without justification. We assumed that the myosin activity generated not the elasticity but the contractility based on Labouesse et al. (2015), and expected that the surface elasticity corresponded with the membrane elasticity. Also, in the physical concept, we clarified how the contractility and the elasticity differently deformed the cells and tissue, and demonstrated why the elasticity was important for the apical constriction. We will add it to the discussion.

    • The absence of information on parameter values, initial condition creation, and boundary conditions in the manuscript hinders reproducibility. Additionally, the explanation for the chosen values and their unit conversion is lacking.

    We agree with the comment, and will add them to the methods.

    Reviewer #2 (Public Review):

    Summary:

    In their work, the authors study local mechanics in an invaginating epithelial tissue. The mostly computational work relies on the Cellular Potts model. The main result shows that an increased apical "contractility" is not sufficient to properly drive apical constriction and subsequent tissue invagination. The authors propose an alternative model, where they consider an alternative driver, namely the "apical surface elasticity".

    Strengths:

    It is surprising that despite the fact that apical constriction and tissue invagination are probably most studied processes in tissue morphogenesis, the underlying physical mechanisms are still not entirely understood. This work supports this notion by showing that simply increasing apical tension is perhaps not sufficient to locally constrict and invaginate a tissue.

    We thank the reviewer for recognizing the importance and novelty of our work.

    Weaknesses:

    The findings and claims in the manuscript are only partially supported. With the computational methodology for studying tissue mechanics being so well developed in the field, the authors could probably have done a more thorough job of supporting the main findings of their work.

    We thank the reviewer for the careful assessment and suggestions. However our simulation was computationally expensive, modeling the epithelium in an analytically calculable expression requires a lot of work, and it is beyond the scope of the present study.

  16. eLife

    eLife assessment

    The results from this study, which investigates the mechanisms necessary for initiating tissue invagination using a cellular Potts modelling approach, suggests that apical constriction is not sufficient to drive the process by itself. The study highlights how choices inherent to modelling - such as permitting straight or curved cell edges - may affect the outcome of simulations and, consequently, their biophysical interpretation. Despite incomplete evidence supporting their major claims due to a rather coarse-grained exploration of the model, this work is useful for biophysicists investigating complex tissue deformation through computational frameworks.

  17. eLife

    Reviewer #1 (Public Review):

    Summary:

    Satoshi Yamashita et al., investigate the physical mechanisms driving tissue bending using the cellular Potts Model, starting from a planar cellular monolayer. They argue that apical length-independent tension control alone cannot explain bending phenomena in the cellular Potts Model, contrasting with the vertex model. However, the evidence supporting this claim is incomplete. They conclude that an apical elastic term, with zero rest value (due to endocytosis/exocytosis), is necessary in constricting cells and that tissue bending can be enhanced by adding a supracellular myosin cable. Notably, a very high apical elastic constant promotes planar tissue configurations, opposing bending.

    Strengths:

    - The finding of the required mechanisms for tissue bending in the cellular Potts Model provides a more natural alternative for studying bending processes in situations with highly curved cells.

    - Despite viewing cellular delamination as an undesired outcome in this particular manuscript, the model's capability to naturally allow T1 events might prove useful for studying cell mechanics during out-of-plane extrusion.

    Weaknesses:

    - The authors claim that the cellular Potts Model is unable to obtain the vertex model simulation results, but the lack of a substantial comparison undermines this assertion. No references are provided with vertex model simulations, employing similar setups and rules, and explaining tissue bending solely through an increase in a length-independent apical tension.

    - The apparent disparity between the two models is attributed to straight versus curved cellular junctions, with cells with a curved lateral junction achieving lower minimum energies at steady-state. However, a critical discussion on the impact of T1 events, allowing cellular delamination, is absent. Note that some of the cited vertex model works do not allow T1 events while allowing curvature.

    - The suggested mechanism for inducing tissue bending in the cellular Potts Model, involving an apical elastic term, has been utilized in earlier studies, including a cited vertex model paper (Polyakov 2014). Consequently, the physical concept behind this implementation is not novel and warrants discussion.

    - The absence of information on parameter values, initial condition creation, and boundary conditions in the manuscript hinders reproducibility. Additionally, the explanation for the chosen values and their unit conversion is lacking.

  18. eLife

    Reviewer #2 (Public Review):

    Summary:

    In their work, the authors study local mechanics in an invaginating epithelial tissue. The mostly computational work relies on the Cellular Potts model. The main result shows that an increased apical "contractility" is not sufficient to properly drive apical constriction and subsequent tissue invagination. The authors propose an alternative model, where they consider an alternative driver, namely the "apical surface elasticity".

    Strengths:

    It is surprising that despite the fact that apical constriction and tissue invagination are probably most studied processes in tissue morphogenesis, the underlying physical mechanisms are still not entirely understood. This work supports this notion by showing that simply increasing apical tension is perhaps not sufficient to locally constrict and invaginate a tissue.

    Weaknesses:
    The findings and claims in the manuscript are only partially supported. With the computational methodology for studying tissue mechanics being so well developed in the field, the authors could probably have done a more thorough job of supporting the main findings of their work.

  19. Version published to 10.1101/2023.12.19.572310v1 on bioRxiv