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Signaling gradients in surface dynamics as basis for planarian regeneration

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Abstract

Based on experimental data, we introduce and analyze a system of reaction-diffusion equations for the regeneration of planarian flatworms. We model dynamics of head and tail cells expressing positional control genes that translate into localized signals which in turn guide stem cell differentiation. Tissue orientation and positional information are encoded in a long range wnt-related signaling gradient. Our system correctly reproduces typical cut and graft experiments, and improves on previous models by preserving polarity in regeneration over orders of magnitude in body size during growth phases. Key to polarity preservation in our model flatworm is the sensitivity of cell differentiation to gradients of wnt-related signals relative to the tissue surface. This process is particularly relevant in small tissue layers close to cuts during their healing, and modeled in a robust fashion through dynamic boundary conditions.

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  • 28 March 2022

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References

  • Achermann J, Sugiyama T (1985) Genetic analysis of developmental mechanisms in hydra: X. Morphogenetic potentials of a regeneration-deficient strain (reg-16). Dev Biol 107(1):13–27

    Google Scholar 

  • Adell T, Cebrià F, Saló E (2010) Gradients in planarian regeneration and homeostasis. Cold Spring Harb Perspect Biol 2(1):a000505

    Google Scholar 

  • Almuedo-Castillo M, Bläßle A, Mörsdorf D, Marcon L, Soh GH, Rogers KW, Schier AF, Müller P (2018) Scale-invariant patterning by size-dependent inhibition of nodal signalling. Nat Cell Biol 20(9):1032–1042

    Google Scholar 

  • Almuedo-Castillo M, Sureda-Gómez M, Adell T (2012) WNT signaling in planarians: new answers to old questions. Int J Dev Biol 56(1—-2—-3):53–65

    Google Scholar 

  • Alt W (1980) Biased random walk models for chemotaxis and related diffusion approximations. J Math Biol 9(2):147–177

    MathSciNet  MATH  Google Scholar 

  • Baguñà J (1976) Mitosis in the intact and regenerating planarian Dugesia mediterranea n. sp. II. Mitotic studies during regeneration, and a possible mechanism of blastema formation. J Exp Zool 195(1):65–79

    Google Scholar 

  • Baguñà J, Romero R (1981) Quantitative analysis of cell types during growth, degrowth and regeneration in the planarians Dugesia mediterranea and Dugesia tigrina. Hydrobiologia 84(1):181–194

    Google Scholar 

  • Baguñà J, Romero R, Saló E, Collet J, Auladell C, Ribas M, Riutort M, García-Fernàndez J, Burgaya F, Bueno D (1990) Growth, degrowth and regeneration as developmental phenomena in adult freshwater planarians. In: Experimental embryology in aquatic plants and animals. Springer, Boston, MA, pp 129–162

  • Baguñà J, Carranza S, Pala M, Ribera C, Giribet G, Arnedo MA, Ribas M, Riutort M (1999) From morphology and karyology to molecules. New methods for taxonomical identification of asexual populations of freshwater planarians. A tribute to professor Mario Benazzi. Ital J Zool 66(3):207–214

    Google Scholar 

  • Bardeen CR, Baetjer F (1904) The inhibitive action of the roentgen rays on regeneration in planarians. J Exp Zool Part A Ecol Genet Physiol 1(1):191–195

    Google Scholar 

  • Bode HR (2003) Head regeneration in hydra. Dev Dyn 226(2):225–236

    Google Scholar 

  • Bowen I, Ryder T, Thompson J (1974) The fine structure of the planarian Polycelis tenuis Iijima. Protoplasma 79(1–2):1–17

    Google Scholar 

  • Cramer von Laue C (2004) Untersuchungen zur dualen Funktion von beta-Catenin im Wnt-Signalweg und der Cadherin-vermittelten Zelladhäsion bei Hydra. Ph.D. Thesis, Darmstadt. http://tuprints.ulb.tu-darmstadt.de/421/

  • Cusseddu D, Edelstein-Keshet L, Mackenzie J, Portet S, Madzvamuse A (2018) A coupled bulk-surface model for cell polarisation. J Theor Biol 481:119–135

    MathSciNet  MATH  Google Scholar 

  • De Vries E, Baguñà J, Ball I (1984) Chromosomal polymorphism in planarians (Turbellaria, Tricladida) and the plate tectonics of the western Mediterranean. Genetica 62(3):187–191

    Google Scholar 

  • Elliott CM, Ranner T, Venkataraman C (2017) Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics. SIAM J Math Anal 49(1):360–397

    MathSciNet  MATH  Google Scholar 

  • Fife PC (2000) Models for phase separation and their mathematics. Electron J Differ Equ (26), 48

  • Gagliardi M, Hernandez A, McGough IJ, Vincent J-P (2014) Inhibitors of endocytosis prevent WNT/Wingless signalling by reducing the level of basal \(\beta \)-catenin/Armadillo. J Cell Sci 127(22):4918–4926

    Google Scholar 

  • Gee L, Hartig J, Law L, Wittlieb J, Khalturin K, Bosch TC, Bode HR (2010) \(\beta \)-catenin plays a central role in setting up the head organizer in hydra. Dev Biol 340(1):116–124

    Google Scholar 

  • Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30–39

    MATH  Google Scholar 

  • Gurley KA, Elliott SA, Simakov O, Schmidt HA, Holstein TW, Alvarado AS (2010) Expression of secreted WNT pathway components reveals unexpected complexity of the planarian amputation response. Dev Biol 347(1):24–39

    Google Scholar 

  • Härting S, Marciniak-Czochra A (2014) Spike patterns in a reaction-diffusion ode model with Turing instability. Math Meth Appl Sci 37(9):1377–1391

    MathSciNet  MATH  Google Scholar 

  • Härting S, Marciniak-Czochra A, Takagi I (2017) Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Disc Cont Dyn Sys A 38:757–800

    MathSciNet  MATH  Google Scholar 

  • Hobmayer B, Rentzsch F, Kuhn K, Happel CM, von Laue CC, Snyder P, Rothbächer U, Holstein TW (2000) WNT signalling molecules act in axis formation in the diploblastic metazoan Hydra. Nature 407(6801):186–189

    Google Scholar 

  • Iijima M, Huang YE, Devreotes P (2002) Temporal and spatial regulation of chemotaxis. Dev Cell 3:469–478

    Google Scholar 

  • Kakugawa S, Langton PF, Zebisch M, Howell SA, Chang T-H, Liu Y, Feizi T, Bineva G, O’Reilly N, Snijders AP et al (2015) Notum deacylates WNT proteins to suppress signalling activity. Nature 519(7542):187–192

    Google Scholar 

  • Lengfeld T, Watanabe H, Simakov O, Lindgens D, Gee L, Law L, Schmidt HA, Özbek S, Bode H, Holstein TW (2009) Multiple WNTs are involved in Hydra organizer formation and regeneration. Dev Biol 330(1):186–199

    Google Scholar 

  • Levine H, Rappel W-J (2005) Membrane-bound Turing patterns. Phys Rev E 72(6):061912

    MathSciNet  Google Scholar 

  • Li Y, Marciniak-Czochra A, Takagi I, Wu B et al (2017) Bifurcation analysis of a diffusion-ode model with Turing instability and hysteresis. Hiroshima Math J 47(2):217–247

    MathSciNet  MATH  Google Scholar 

  • Li Y, Marciniak-Czochra A, Takagi I, Wu B (2019) Steady states of FitzHugh-Nagumo system with non-diffusive activator and diffusive inhibitor. Tohoku Math J 71(2):243–279. https://doi.org/10.2748/tmj/1561082598

    Article  MathSciNet  MATH  Google Scholar 

  • Lobo D, Beane WS, Levin M (2012) Modeling planarian regeneration: a primer for reverse-engineering the worm. PLoS Comput Biol 8(4):e1002481

    Google Scholar 

  • MacWilliams HK (1983) Hydra transplantation phenomena and the mechanism of hydra head regeneration. II: Properties of the head activation. Dev Biol 96(1):239–257

    Google Scholar 

  • Marciniak-Czochra A (2003) Receptor-based models with diffusion-driven instability for pattern formation in hydra. J Biol Syst 11(03):293–324

    MATH  Google Scholar 

  • Marciniak-Czochra A (2006) Receptor-based models with hysteresis for pattern formation in hydra. Math Biosci 199(1):97–119

    MathSciNet  MATH  Google Scholar 

  • Marciniak-Czochra A, Härting S, Karch G, Suzuki K (2018) Dynamical spike solutions in a nonlocal model of pattern formation. Nonlinearity 31(5):1757–1781

    MathSciNet  MATH  Google Scholar 

  • Marciniak-Czochra A, Karch G, Suzuki K (2017) Instability of Turing patterns in reaction-diffusion-ode systems. J Math Biol 74(3):583–618

    MathSciNet  MATH  Google Scholar 

  • Marciniak-Czochra A, Karch G, Suzuki K, Zienkiewicz J et al (2016) Diffusion-driven blowup of nonnegative solutions to reaction-diffusion-ode systems. Differ Integr Equ 29(7/8):715–730

    MathSciNet  MATH  Google Scholar 

  • Marciniak-Czochra A, Nakayama M, Takagi I et al (2015) Pattern formation in a diffusion-ode model with hysteresis. Differ Integr Equ 28(7/8):655–694

    MathSciNet  MATH  Google Scholar 

  • Meinhardt H (2012) Turing’s theory of morphogenesis of 1952 and the subsequent discovery of the crucial role of local self-enhancement and long-range inhibition. Interface Focus 2(4):407–416

    Google Scholar 

  • Morgan TH (1898) Experimental studies of the regeneration of Planaria maculata. Dev Genes Evol 7(2):364–397

    Google Scholar 

  • Morgan TH (1904) The control of heteromorphosis in Planaria maculata. Archiv für Entwicklungsmechanik der Organismen 17(4):683–695

    Google Scholar 

  • Morgan TH (1905) Polarity considered as a phenomenon of gradation of materials. J Exp Zool 2(4):495–506

    Google Scholar 

  • Mori Y, Jilkine A, Edelstein-Keshet L (2008) Wave-pinning and cell polarity from a bistable reaction-diffusion system. Biophys J 94(9):3684–3697

    Google Scholar 

  • Murray JD (2002) Mathematical biology I: an introduction. Interdisciplinary applied mathematics. Springer, New York

    Google Scholar 

  • Murray JD (2003) Mathematical biology II: spatial models and biomedical applications. Interdisciplinary Applied mathematics. Springer, New York, p 2003

    Google Scholar 

  • Newmark PA, Alvarado AS (2002) Not your father’s planarian: a classic model enters the era of functional genomics. Nat Rev Genet 3(3):210–219

    Google Scholar 

  • Noda K (1971) Reconstitution of dissociated cells of hydra. Zool Mag 80:99–101

    Google Scholar 

  • Nüsslein-Volhard C, Wieschaus E (1980) Mutations affecting segment number and polarity in Drosophila. Nature 287(5785):795–801

    Google Scholar 

  • Othmer HG, Pate E (1980) Scale-invariance in reaction-diffusion models of spatial pattern formation. Proc Natl Acad Sci 77(7):4180–4184

    Google Scholar 

  • Owlarn S, Bartscherer K (2016) Go ahead, grow a head! a planarian’s guide to anterior regeneration. Regeneration 3(3):139–155

    Google Scholar 

  • Petersen CP, Reddien PW (2008) SMED-\(\beta \)catenin-1 is required for anteroposterior blastema polarity in planarian regeneration. Science 319(5861):327–330

    Google Scholar 

  • Petersen CP, Reddien PW (2009) A wound-induced WNT expression program controls planarian regeneration polarity. Proc Natl Acad Sci 106(40):17061–17066

    Google Scholar 

  • Petersen CP, Reddien PW (2011) Polarized notum activation at wounds inhibits WNT function to promote planarian head regeneration. Science 332(6031):852–855

    Google Scholar 

  • Philipp I, Aufschnaiter R, Özbek S, Pontasch S, Jenewein M, Watanabe H, Rentzsch F, Holstein TW, Hobmayer B (2009) WNT/\(\beta \)-catenin and noncanonical WNT signaling interact in tissue evagination in the simple eumetazoan Hydra. Proc Natl Acad Sci 106(11):4290–4295

    Google Scholar 

  • Plickert G, Jacoby V, Frank U, Müller WA, Mokady O (2006) WNT signaling in hydroid development: formation of the primary body axis in embryogenesis and its subsequent patterning. Dev Biol 298(2):368–378

    Google Scholar 

  • Randolph H (1892) The regeneration of the tail in Lumbriculus. J Morphol 7(3):317–344

    Google Scholar 

  • Randolph H (1897) Observations and experiments on regeneration in planarians. Dev Genes Evol 5(2):352–372

    Google Scholar 

  • Rätz A, Röger M (2012) Turing instabilities in a mathematical model for signaling networks. J Math Biol 65(6–7):1215–1244

    MathSciNet  MATH  Google Scholar 

  • Rätz A, Röger M (2014) Symmetry breaking in a bulk-surface reaction-diffusion model for signalling networks. Nonlinearity 27(8):1805–1827

    MathSciNet  MATH  Google Scholar 

  • Reddien PW (2011) Constitutive gene expression and the specification of tissue identity in adult planarian biology. Trends Genet 27(7):277–285

    Google Scholar 

  • Reddien PW, Alvarado AS (2004) Fundamentals of planarian regeneration. Annu Rev Cell Dev Biol 20:725–757

    Google Scholar 

  • Reuter H, März M, Vogg MC, Eccles D, Grífol-Boldú L, Wehner D, Owlarn S, Adell T, Weidinger G, Bartscherer K (2015) \(\beta \)-catenin-dependent control of positional information along the AP body axis in planarians involves a teashirt family member. Cell Rep 10(2):253–265

    Google Scholar 

  • Rink JC (2018) Stem cells, patterning and regeneration in planarians: self-organization at the organismal scale. Springer, New York, pp 57–172

    Google Scholar 

  • Saló E, Baguñà J (1984) Regeneration and pattern formation in planarians. Development 83(1):63–80

    Google Scholar 

  • Saló E, Baguñà J (1985) Cell movement in intact and regenerating planarians. Quantitation using chromosomal, nuclear and cytoplasmic markers. Development 89(1):57–70

    Google Scholar 

  • Santos FV (1929) Studies on transplantation in planaria. Biol Bullet 57(3):188–197

    Google Scholar 

  • Scimone ML, Lapan SW, Reddien PW (2014) A forkhead transcription factor is wound-induced at the planarian midline and required for anterior pole regeneration. PLoS Genet 10(1):e1003999

    Google Scholar 

  • Sherratt J, Maini P, Jäger W, Müller W (1995) A receptor based model for pattern formation in hydra. Forma 10(2):77–95

    Google Scholar 

  • Shimizu H, Sawada Y, Sugiyama T (1993) Minimum tissue size required for hydra regeneration. Dev Biol 155(2):287–296

    Google Scholar 

  • Shostak S (1972) Inhibitory gradients of head and foot regeneration in Hydra viridis. Dev Biol 28(4):620–635

    MathSciNet  Google Scholar 

  • Smales LR, Blankespoor HD (1978) The epidermis and sensory organs of Dugesia tigrina (Turbellaria: Tricladida). Cell Tissue Res 193(1):35–40

    Google Scholar 

  • Stückemann T, Cleland JP, Werner S, Vu HT-K, Bayersdorf R, Liu S-Y, Friedrich B, Jülicher F, Rink JC (2017) Antagonistic self-organizing patterning systems control maintenance and regeneration of the anteroposterior axis in Planarians. Dev Cell 40(3):248–263

    Google Scholar 

  • Sureda-Gómez M, Martín-Durán JM, Adell T (2016) Localization of planarian \(\beta \)-CATENIN-1 reveals multiple roles during anterior-posterior regeneration and organogenesis. Development 143(22):4149–4160

    Google Scholar 

  • Tasaki J, Shibata N, Nishimura O, Itomi K, Tabata Y, Son F, Suzuki N, Araki R, Abe M, Agata K et al (2011) ERK signaling controls blastema cell differentiation during planarian regeneration. Development 138(12):2417–2427

    Google Scholar 

  • Tenbrock C (2017) Mathematical models for regeneration on the example of planarians. University of Münster (WWU), Germany, PhD-Thesis

  • Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond Ser B 237(641):37–72

    MathSciNet  MATH  Google Scholar 

  • Turing AM (1978) Webcollection: Turing unpublished manuscripts and drafts. http://www.turingarchive.org/browse.php/C

  • Umulis DM (2009) Analysis of dynamic morphogen scale invariance. J R Soc Interface 6(41):1179–1191

    Google Scholar 

  • Umulis DM, Othmer HG (2013) Mechanisms of scaling in pattern formation. Development 140(24):4830–4843

    Google Scholar 

  • Webster G (1966) Studies on pattern regulation in hydra. Development 16(1):123–141

    Google Scholar 

  • Wenemoser D, Lapan SW, Wilkinson AW, Bell GW, Reddien PW (2012) A molecular wound response program associated with regeneration initiation in planarians. Genes Dev 26(9):988–1002

    Google Scholar 

  • Wenemoser D, Reddien PW (2010) Planarian regeneration involves distinct stem cell responses to wounds and tissue absence. Dev Biol 344(2):979–991

    Google Scholar 

  • Werner S, Stückemann T, Beirán Amigo M, Rink JC, Jülicher F, Friedrich BM (2015) Scaling and regeneration of self-organized patterns. Phys Rev Lett 114:138101

    Google Scholar 

  • Wilby OK, Webster G (1970) Experimental studies on axial polarity in hydra. Development 24(3):595–613

    Google Scholar 

  • Witchley JN, Mayer M, Wagner DE, Owen JH, Reddien PW (2013) Muscle cells provide instructions for planarian regeneration. Cell Rep 4(4):633–641

    Google Scholar 

  • Wolpert L (1969) Positional information and the spatial pattern of cellular differentiation. J Theor Biol 25(1):1–47

    Google Scholar 

  • Wolpert L (1994) Positional information and pattern formation in development. Dev Genet 15(6):485–490

    Google Scholar 

  • Wurtzel O, Cote LE, Poirier A, Satija R, Regev A, Reddien PW (2015) A generic and cell-type-specific wound response precedes regeneration in planarians. Dev Cell 35(5):632–645

    Google Scholar 

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Acknowledgements

CT was supported by a fellowship of the Graduate School of the Cells-in-Motion Cluster of Excellence EXC 1003–CiM, University of Münster (WWU). ASc was partially supported through NSF grants DMS–1311740 and DMS–1907391, a Research Award from the Alexander-von-Humboldt Foundation, and a WWU Fellowship. ASc and ASt were supported by the DFG (German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics–Geometry–Structure. CT and ASt gratefully acknowledge numerous valuable discussions about the biology of planarians with Kerstin Bartscherer.

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Scheel, A., Stevens, A. & Tenbrock, C. Signaling gradients in surface dynamics as basis for planarian regeneration. J. Math. Biol. 83, 6 (2021). https://doi.org/10.1007/s00285-021-01627-w

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