# Mathematical Modeling of Tissue Folding and Asymmetric Tissue Flow during Epithelial Morphogenesis

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## Abstract

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## 1. Introduction

## 2. Cell-Autonomous Mechanisms that Induce Tissue Folding: Apical Constriction and Basal–Lateral Modulation

## 3. Unidirectional Tissue Flow Induced by a Combination of Chirality and Junctional Remodeling

#### 3.1. Vertex Model in an Apical Plane

**r**(t), and the segment labeled by <ij> indicates the bond connecting the i-th and j-th vertices. The dynamics of the vertices is given by

_{i}_{i}is the coefficient of linear frictional resistance on the vertex i. In this Section, instead of Equation (1), we apply

#### 3.2. Junctional Remodeling and Tissue Morphogenesis

_{0}, the bond is rotated by 90 degrees, and at the same time, the bonds are rearranged as shown in Figure 2d. Various morphogenetic processes have been reproduced using a cell vertex model with junctional remodeling [45]. For example, combining anisotropy of line tension strength with junctional remodeling, the vertex model in two dimensions can recapitulate the convergent-extension process [46].

#### 3.3. Theory and Numerical Simulation of Genitalia Rotation

_{kl}and δ

_{kl}are the frequency given by random variables and the initial phase, respectively, and ${\gamma}_{C}$ is a positive constant. Here, there is a second important assumption that this strengthening of line tension in the tilted junctions is a cell-autonomous activity. Mathematically, this assumption can be realized by the following manipulation on the junctional tension: we first take the derivative of the potential function $U\left(\left\{{\mathbf{r}}_{i}\right\},\left\{{\gamma}_{ij}\right\}\right)$ with respect to ${\mathbf{r}}_{i}$ for a given ${\gamma}_{ij}$, and then incorporate the direction dependence of tension ${\gamma}_{ij}={\widehat{\gamma}}_{ij}$ given by Equation (4). While such chiral line tension is still speculative, recently increasing evidence has been found for possible chirality in force generation of the actin cytoskeleton, e.g., based on actin’s chirality combined with formin [48] and myosin 1D [49].

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Umetsu, D.; Kuranaga, E. Planar polarized contractile actomyosin networks in dynamic tissue morphogenesis. Curr. Opin. Genet. Dev.
**2017**, 45, 90–96. [Google Scholar] [CrossRef] [PubMed] - Khan, Z.; Wang, Y.-C.; Wieschaus, E.F.; Kaschube, M. Quantitative 4D analyses of epithelial folding during Drosophila gastrulation. Development
**2014**, 141, 2895–2900. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sun, Z.; Amourda, C.; Shagirov, M.; Hara, Y.; Saunders, T.E.; Toyama, Y. Basolateral protrusion and apical contraction cooperatively drive Drosophila germ-band extension. Nat. Cell Biol.
**2017**, 19, 375–383. [Google Scholar] [CrossRef] [PubMed] - Hutson, M.S. Forces for Morphogenesis Investigated with Laser Microsurgery and Quantitative Modeling. Science
**2003**, 300, 145–149. [Google Scholar] [CrossRef] [PubMed] - Toyama, Y.; Peralta, X.G.; Wells, A.R.; Kiehart, D.P.; Edwards, G.S. Apoptotic Force and Tissue Dynamics During Drosophila Embryogenesis. Science
**2008**, 321, 1683–1686. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hara, Y.; Shagirov, M.; Toyama, Y. Cell Boundary Elongation by Non-autonomous Contractility in Cell Oscillation. Curr. Biol.
**2016**, 26, 2388–2396. [Google Scholar] [CrossRef] [PubMed] - Polyakov, O.; He, B.; Swan, M.; Shaevitz, J.W.; Kaschube, M.; Wieschaus, E. Passive Mechanical Forces Control Cell-Shape Change during Drosophila Ventral Furrow Formation. Biophys. J.
**2014**, 107, 998–1010. [Google Scholar] [CrossRef] [Green Version] - Bertet, C.; Sulak, L.; Lecuit, T. Myosin-dependent junction remodelling controls planar cell intercalation and axis elongation. Nature
**2004**, 429, 667–671. [Google Scholar] [CrossRef] - Nishimura, T.; Honda, H.; Takeichi, M. Planar Cell Polarity Links Axes of Spatial Dynamics in Neural-Tube Closure. Cell
**2012**, 149, 1084–1097. [Google Scholar] [CrossRef] [Green Version] - Sato, K.; Hiraiwa, T.; Maekawa, E.; Isomura, A.; Shibata, T.; Kuranaga, E. Left–right asymmetric cell intercalation drives directional collective cell movement in epithelial morphogenesis. Nat. Commun.
**2015**, 6, 10074. [Google Scholar] [CrossRef] [Green Version] - Taniguchi, K.; Maeda, R.; Ando, T.; Okumura, T.; Nakazawa, N.; Hatori, R.; Nakamura, M.; Hozumi, S.; Fujiwara, H.; Matsuno, K. Chirality in Planar Cell Shape Contributes to Left-Right Asymmetric Epithelial Morphogenesis. Science
**2011**, 333, 339–341. [Google Scholar] [CrossRef] [PubMed] - Perez-Mockus, G.; Mazouni, K.; Roca, V.; Corradi, G.; Conte, V.; Schweisguth, F. Spatial regulation of contractility by Neuralized and Bearded during furrow invagination in Drosophila. Nat. Commun.
**2017**, 8, 1594. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Martin, A.C.; Goldstein, B. Apical constriction: Themes and variations on a cellular mechanism driving morphogenesis. Development
**2014**, 141, 1987–1998. [Google Scholar] [CrossRef] [PubMed] - Heisenberg, C.-P.; Bellaïche, Y. Forces in Tissue Morphogenesis and Patterning. Cell
**2013**, 153, 948–962. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Heller, E.; Fuchs, E. Tissue patterning and cellular mechanics. J. Cell Biol.
**2015**, 211, 219–231. [Google Scholar] [CrossRef] [Green Version] - Sánchez-Corrales, Y.E.; Röper, K. Alignment of cytoskeletal structures across cell boundaries generates tissue cohesion during organ formation. Curr. Opin. Cell Biol.
**2018**, 55, 104–110. [Google Scholar] [CrossRef] - Hara, Y. Contraction and elongation: Mechanics underlying cell boundary deformations in epithelial tissue. Dev. Growth Differ.
**2017**, 59, 340–350. [Google Scholar] [CrossRef] - Lee, J.-Y. Uncorking gastrulation: The morphogenetic movement of bottle cells. Wiley Interdiscip. Rev. Dev. Biol.
**2012**, 1, 286–293. [Google Scholar] [CrossRef] - Popov, I.K.; Ray, H.J.; Skoglund, P.; Keller, R.; Chang, C. The RhoGEF protein Plekhg5 regulates apical constriction of bottle cells during gastrulation. Development
**2018**, 145, dev.168922. [Google Scholar] [CrossRef] - Fletcher, A.G.; Cooper, F.; Baker, R.E. Mechanocellular models of epithelial morphogenesis. Philos. Trans. R. Soc. Lond. B. Biol. Sci.
**2017**, 372, 20150519. [Google Scholar] [CrossRef] [Green Version] - Ninomiya, H.; Elinson, R.P.; Winklbauer, R. Antero-posterior tissue polarity links mesoderm convergent extension to axial patterning. Nature
**2004**, 430, 364–367. [Google Scholar] [CrossRef] [PubMed] - Williams, M.L.; Solnica-Krezel, L. Regulation of gastrulation movements by emergent cell and tissue interactions. Curr. Opin. Cell Biol.
**2017**, 48, 33–39. [Google Scholar] [CrossRef] [PubMed] - Kuranaga, E.; Matsunuma, T.; Kanuka, H.; Takemoto, K.; Koto, A.; Kimura, K.-I.; Miura, M. Apoptosis controls the speed of looping morphogenesis in Drosophila male terminalia. Development
**2011**, 138, 1493–1499. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hatori, R.; Ando, T.; Sasamura, T.; Nakazawa, N.; Nakamura, M.; Taniguchi, K.; Hozumi, S.; Kikuta, J.; Ishii, M.; Matsuno, K. Left–right asymmetry is formed in individual cells by intrinsic cell chirality. Mech. Dev.
**2014**, 133, 146–162. [Google Scholar] [CrossRef] [PubMed] - Inaki, M.; Hatori, R.; Nakazawa, N.; Okumura, T.; Ishibashi, T.; Kikuta, J.; Ishii, M.; Matsuno, K.; Honda, H. Chiral cell sliding drives left-right asymmetric organ twisting. Elife
**2018**, 7, e32506. [Google Scholar] [CrossRef] [PubMed] - Spéder, P.; Ádám, G.; Noselli, S. Type ID unconventional myosin controls left–right asymmetry in Drosophila. Nature
**2006**, 440, 803–807. [Google Scholar] [CrossRef] - Hiraiwa, T.; Kuranaga, E.; Shibata, T. Wave Propagation of Junctional Remodeling in Collective Cell Movement of Epithelial Tissue: Numerical Simulation Study. Front. Cell Dev. Biol.
**2017**, 5, 66. [Google Scholar] [CrossRef] - Sato, K. Direction-dependent contraction forces on cell boundaries induce collective migration of epithelial cells within their sheet. Dev. Growth Differ.
**2017**, 59, 317–328. [Google Scholar] [CrossRef] - Wen, F.-L.; Wang, Y.-C.; Shibata, T. Epithelial Folding Driven by Apical or Basal-Lateral Modulation: Geometric Features, Mechanical Inference, and Boundary Effects. Biophys. J.
**2017**, 112, 2683–2695. [Google Scholar] [CrossRef] - Osterfield, M.; Du, X.; Schüpbach, T.; Wieschaus, E.; Shvartsman, S.Y. Three-Dimensional Epithelial Morphogenesis in the Developing Drosophila Egg. Dev. Cell
**2013**, 24, 400–410. [Google Scholar] [CrossRef] [Green Version] - Murisic, N.; Hakim, V.; Kevrekidis, I.G.; Shvartsman, S.Y.; Audoly, B. From discrete to continuum models of three-dimensional deformations in epithelial sheets. Biophys. J.
**2015**, 109, 154–163. [Google Scholar] [CrossRef] [PubMed] - Ogura, Y.; Wen, F.-L.; Sami, M.M.; Shibata, T.; Hayashi, S. A Switch-like Activation Relay of EGFR-ERK Signaling Regulates a Wave of Cellular Contractility for Epithelial Invagination. Dev. Cell
**2018**, 46, 162–172.e5. [Google Scholar] [CrossRef] [PubMed] - Hočevar Brezavšček, A.; Rauzi, M.; Leptin, M.; Ziherl, P. A Model of Epithelial Invagination Driven by Collective Mechanics of Identical Cells. Biophys. J.
**2012**, 103, 1069–1077. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Štorgel, N.; Krajnc, M.; Mrak, P.; Štrus, J.; Ziherl, P. Quantitative Morphology of Epithelial Folds. Biophys. J.
**2016**, 110, 269–277. [Google Scholar] [CrossRef] [PubMed] - Misra, M.; Audoly, B.; Kevrekidis, I.G.; Shvartsman, S.Y. Shape Transformations of Epithelial Shells. Biophys. J.
**2016**, 110, 1670–1678. [Google Scholar] [CrossRef] [PubMed] - Hannezo, E.; Prost, J.; Joanny, J.-F. Theory of epithelial sheet morphology in three dimensions. Proc. Natl. Acad. Sci. USA
**2014**, 111, 27–32. [Google Scholar] [CrossRef] - Krueger, D.; Tardivo, P.; Nguyen, C.; De Renzis, S. Downregulation of basal myosin-II is required for cell shape changes and tissue invagination. EMBO J.
**2018**, 37, e100170. [Google Scholar] [CrossRef] [PubMed] - Okuda, S.; Takata, N.; Hasegawa, Y.; Kawada, M.; Inoue, Y.; Adachi, T.; Sasai, Y.; Eiraku, M. Strain-triggered mechanical feedback in self-organizing optic-cup morphogenesis. Sci. Adv.
**2018**, 4, eaau1354. [Google Scholar] [CrossRef] [PubMed] - Nagai, T.; Honda, H. A dynamic cell model for the formation of epithelial tissues. Philos. Mag. B
**2001**, 81, 699–719. [Google Scholar] [CrossRef] - Li, B.; Sun, S.X. Coherent Motions in Confluent Cell Monolayer Sheets. Biophys. J.
**2014**, 107, 1532–1541. [Google Scholar] [CrossRef] [Green Version] - Bi, D.; Yang, X.; Marchetti, M.C.; Manning, M.L. Motility-driven glass and jamming transitions in biological tissues. Phys. Rev. X
**2016**, 6, 021011. [Google Scholar] [CrossRef] [PubMed] - Coburn, L.; Lopez, H.; Caldwell, B.J.; Moussa, E.; Yap, C.; Priya, R.; Noppe, A.; Roberts, A.P.; Lobaskin, V.; Yap, A.S.; et al. Contact inhibition of locomotion and mechanical cross-talk between cell–cell and cell–substrate adhesion determine the pattern of junctional tension in epithelial cell aggregates. Mol. Biol. Cell
**2016**, 27, 3436–3448. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lecuit, T.; Lenne, P.-F.; Munro, E. Force Generation, Transmission, and Integration during Cell and Tissue Morphogenesis. Annu. Rev. Cell Dev. Biol.
**2011**, 27, 157–184. [Google Scholar] [CrossRef] - Jha, A.; van Zanten, T.S.; Philippe, J.-M.; Mayor, S.; Lecuit, T. Quantitative Control of GPCR Organization and Signaling by Endocytosis in Epithelial Morphogenesis. Curr. Biol.
**2018**, 28, 1570–1584.e6. [Google Scholar] [CrossRef] - Fletcher, A.G.; Osterfield, M.; Baker, R.E.; Shvartsman, S.Y. Vertex Models of Epithelial Morphogenesis. Biophys. J.
**2014**, 106, 2291–2304. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Collinet, C.; Rauzi, M.; Lenne, P.-F.; Lecuit, T. Local and tissue-scale forces drive oriented junction growth during tissue extension. Nat. Cell Biol.
**2015**, 17, 1247–1258. [Google Scholar] [CrossRef] - Sato, K.; Hiraiwa, T.; Shibata, T. Cell Chirality Induces Collective Cell Migration in Epithelial Sheets. Phys. Rev. Lett.
**2015**, 115, 188102. [Google Scholar] [CrossRef] [PubMed] - Tee, Y.H.; Shemesh, T.; Thiagarajan, V.; Hariadi, R.F.; Anderson, K.L.; Page, C.; Volkmann, N.; Hanein, D.; Sivaramakrishnan, S.; Kozlov, M.M.; et al. Cellular chirality arising from the self-organization of the actin cytoskeleton. Nat. Cell Biol.
**2015**, 17, 445–457. [Google Scholar] [CrossRef] - Lebreton, G.; Géminard, C.; Lapraz, F.; Pyrpassopoulos, S.; Cerezo, D.; Spéder, P.; Ostap, E.M.; Noselli, S. Molecular to organismal chirality is induced by the conserved myosin 1D. Science
**2018**, 362, 949–952. [Google Scholar] [CrossRef]

**Figure 1.**Modulation at different cell surfaces induces tissue-scale fold formation. The five cells at the center of a cell sheet (13 cells) are modulated at the apical (

**a**) or basal–lateral (

**b**) surface. Depending on the apical or basal–lateral modulations, characteristic folds with a shallow V-shape or deep U-shape are formed. Modified from [29].

**Figure 2.**

**Cellular vertex model in an apical plane.**(

**a**) Top view of an epithelial tissue at the apical plane. Cell–cell junctions are white (E-cad). (

**b**) In the vertex model in two dimensions, each cell is regarded as a polygon [33]. (

**c**) The process in which two cells (B and D) collide with each other. (

**d**) In the cell vertex model, this process is represented by the T1 transition, a type of junctional remodeling. Reprinted and modified from [27].

**Figure 3.**Numerical simulation of Drosophila genitalia rotation. (

**a**) Schematic diagram of the mathematical model. Particularly strong tension is applied only at the junctions tilted clockwise, as indicated by the red lines. (

**b**) Tissue rotation in our simulation. Each polygon indicates each cell [33]. (

**c**) Rotation angle around the tissue center of each cell measured from the AP axis going through the cell at the initial time, t = 20. Red, blue, and green lines correspond to the color of individual cells in b. (

**d**) Mechanical wave propagation. Junctional remodeling (colored dots) propagates clockwise, at a much faster speed than the migration speed of each cell. In (

**d**), the simulation is carried out in the absence of tension oscillation (f

_{kl}= 0) for the sake of visibility. Arrow heads indicate the junctions undergoing remodeling. Reprinted and modified from [27].

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Hiraiwa, T.; Wen, F.-L.; Shibata, T.; Kuranaga, E.
Mathematical Modeling of Tissue Folding and Asymmetric Tissue Flow during Epithelial Morphogenesis. *Symmetry* **2019**, *11*, 113.
https://doi.org/10.3390/sym11010113

**AMA Style**

Hiraiwa T, Wen F-L, Shibata T, Kuranaga E.
Mathematical Modeling of Tissue Folding and Asymmetric Tissue Flow during Epithelial Morphogenesis. *Symmetry*. 2019; 11(1):113.
https://doi.org/10.3390/sym11010113

**Chicago/Turabian Style**

Hiraiwa, Tetsuya, Fu-Lai Wen, Tatsuo Shibata, and Erina Kuranaga.
2019. "Mathematical Modeling of Tissue Folding and Asymmetric Tissue Flow during Epithelial Morphogenesis" *Symmetry* 11, no. 1: 113.
https://doi.org/10.3390/sym11010113