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Simulation of Morphogen and Tissue Dynamics

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Morphogen Gradients

Part of the book series: Methods in Molecular Biology ((MIMB,volume 1863))

Abstract

Morphogenesis, the process by which an adult organism emerges from a single cell, has fascinated humans for a long time. Modeling this process can provide novel insights into development and the principles that orchestrate the developmental processes. This chapter focuses on the mathematical description and numerical simulation of developmental processes. In particular, we discuss the mathematical representation of morphogen and tissue dynamics on static and growing domains, as well as the corresponding tissue mechanics. In addition, we give an overview of numerical methods that are routinely used to solve the resulting systems of partial differential equations. These include the finite element method and the Lattice Boltzmann method for the discretization as well as the arbitrary Lagrangian-Eulerian method and the Diffuse-Domain method to numerically treat deforming domains.

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References

  1. Iber D (2011) Inferring Biological Mechanisms by Data-Based Mathematical Modelling: Compartment-Specific Gene Activation during Sporulation in Bacillus subtilis as a Test Case. Adv Bioinformatics 2011:1–12

    Article  CAS  Google Scholar 

  2. Iber D, Karimaddini Z, Ãœnal E (2015) Image-based modelling of organogenesis. Brief Bioinform.

    Google Scholar 

  3. Gómez HF, Georgieva L, Michos O, Iber D (2017) Image-based in silico models of organogenesis. In: Systems Biology, vol 6.

    Chapter  Google Scholar 

  4. Mogilner A, Odde D (2011) Modeling cellular processes in 3d. Trends Cell Biol 21(12):692–700

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  5. Sbalzarini IF (2013) Modeling and simulation of biological systems from image data: Prospects & Overviews. BioEssays 35(5):482–490

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. Alberts B, Johnson A, Lewis J, Morgan D, Raff M, Roberts K, Walter P (2014) Molecular biology of the cell, 6th edn. Taylor & Francis, London

    Google Scholar 

  7. van den Hurk R, Zhao J (2005) Formation of mammalian oocytes and their growth, differentiation and maturation within ovarian follicles. Theriogenology 63(6):1717–1751

    Article  CAS  PubMed  Google Scholar 

  8. Worley MI, Setiawan L, Hariharan IK (2013) Tie-dye: a combinatorial marking system to visualize and genetically manipulate clones during development in drosophila melanogaster. Development 140(15):3275–3284

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Ricklefs RE (2010) Embryo growth rates in birds and mammals. Funct Ecol 24(3):588–596

    Article  Google Scholar 

  10. Liang X, Michael M, Gomez GA (2016) Measurement of mechanical tension at cell-cell junctions using two-photon laser ablation. Bio Protoc 6(24):e2068

    Google Scholar 

  11. Eden E, Geva-Zatorsky N, Issaeva I, Cohen A, Dekel E, Danon T, Cohen L, Mayo A, Alon U (2011) Proteome half-life dynamics in living human cells. Science 331(6018):764–768

    Article  CAS  PubMed  Google Scholar 

  12. Müller P, Rogers KW, Shuizi RY, Brand M, Schier AF (2013) Morphogen transport. Development 140(8):1621–1638

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  13. Forgacs G, Foty RA, Shafrir Y, Steinberg MS (1998) Viscoelastic properties of living embryonic tissues: a quantitative study. Biophys J 74(5):2227–2234

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  Google Scholar 

  15. Dierick HA, Bejsovec A (1998) Functional analysis of wingless reveals a link between intercellular ligand transport and dorsal-cell-specific signaling. Development 125(23):4729–4738

    Google Scholar 

  16. Ramírez-Weber FA, Kornberg TB (1999) Cytonemes: cellular processes that project to the principal signaling center in drosophila imaginal discs. Cell 97(5):599–607

    Article  PubMed  Google Scholar 

  17. Rodman J, Mercer R, Stahl P (1990) Endocytosis and transcytosis. Curr Opin Cell Biol 2(4):664–672

    Article  CAS  PubMed  Google Scholar 

  18. Entchev EV, Schwabedissen A, González-Gaitán M (2000) Gradient formation of the TGF-β homolog dpp. Cell 103(6):981–992

    Article  CAS  PubMed  Google Scholar 

  19. Lander AD, Nie Q, Wan FY (2002) Do morphogen gradients arise by diffusion? Dev Cell 2(6):785–796

    Article  CAS  PubMed  Google Scholar 

  20. Schwank G, Dalessi S, Yang SF, Yagi R, de Lachapelle AM, Affolter M, Bergmann S, Basler K (2011) Formation of the long range dpp morphogen gradient. PLoS Biol 9(7):1–13

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Kornberg TB, Roy S (2014) Cytonemes as specialized signaling filopodia. Development 141(4):729–736

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  22. Bischoff M, Gradilla AC, Seijo I, Andrés G, Rodríguez-Navas C, González-Méndez L, Guerrero I (2013) Cytonemes are required for the establishment of a normal hedgehog morphogen gradient in drosophila epithelia. Nat Cell Biol 15(11):1269–1281

    Article  CAS  PubMed  Google Scholar 

  23. Sanders TA, Llagostera E, Barna M (2013) Specialized filopodia direct long-range transport of shh during vertebrate tissue patterning. Nature 497(7451):628–632

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  25. Gregor T, Bialek W, de Ruyter van Steveninck RR, Tank DW, Wieschaus EF (2005) Diffusion and scaling during early embryonic pattern formation. Proc Natl Acad Sci USA 102(51):18403–18407

    Article  CAS  Google Scholar 

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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  27. Umulis DM (2009) Analysis of dynamic morphogen scale invariance. J R Soc Interface

    Google Scholar 

  28. Wartlick O, Mumcu P, Kicheva A, Bittig T, Seum C, Jülicher F, Gonzalez-Gaitan M (2011) Dynamics of dpp signaling and proliferation control. Science 331(6021):1154–1159

    Article  CAS  PubMed  Google Scholar 

  29. Fried P, Iber D (2014) Dynamic scaling of morphogen gradients on growing domains. Nat Commun 5:5077

    Google Scholar 

  30. Fried P, Iber D (2015) Read-out of dynamic morphogen gradients on growing domains. PloS ONE 10(11):e0143226

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  31. Gillespie DT (2007) Stochastic simulation of chemical kinetics. Annu Rev Phys Chem 58:35–55

    Article  CAS  PubMed  Google Scholar 

  32. Berg HC (1993) Random walks in biology. Princeton University Press, Princeton

    Google Scholar 

  33. Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361

    Article  CAS  Google Scholar 

  34. Kondo S, Asai R (1995) A reaction-diffusion wave on the skin of the marine angelfish pomacanthus. Nature 376(6543):765–768

    Article  CAS  PubMed  Google Scholar 

  35. Henderson J, Carter D (2002) Mechanical induction in limb morphogenesis: the role of growth-generated strains and pressures

    Google Scholar 

  36. Iber D, Tanaka S, Fried P, Germann P, Menshykau D (2014) Simulating tissue morphogenesis and signaling. In: Nelson CM (ed) Tissue morphogenesis: methods and protocols. Springer, New York, pp 323–338

    Google Scholar 

  37. Iber D, Menshykau D (2013) The control of branching morphogenesis. Open Biol 3(9):130088

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  38. Menshykau D, Iber D (2013) Kidney branching morphogenesis under the control of a ligan-receptor-based Turing mechanism. Phys Biol 10(4):046003

    Article  PubMed  CAS  Google Scholar 

  39. Dillon R, Gadgil C, Othmer HG (2003) Short-and long-range effects of sonic hedgehog in limb development. Proc Natl Acad Sci 100(18):10152–10157

    Article  CAS  Google Scholar 

  40. Bittig T, Wartlick O, Kicheva A, González-Gaitán M, Jülicher F (2008) Dynamics of anisotropic tissue growth. New J Phys 10(6):063001

    Article  Google Scholar 

  41. Tanaka S, Iber D (2013) Inter-dependent tissue growth and Turing patterning in a model for long bone development. Phys Biol 10(5):056009

    Article  PubMed  CAS  Google Scholar 

  42. Fried P, Sánchez-Aragón M, Aguilar-Hidalgo D, Lehtinen B, Casares F, Iber D (2016) A model of the spatio-temporal dynamics of drosophila eye disc development. PLoS Comput Biol 12(9)

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  43. Boehm B, Westerberg H, Lesnicar-Pucko G, Raja S, Rautschka M, Cotterell J, Swoger J, Sharpe J (2010) The role of spatially controlled cell proliferation in limb bud morphogenesis. PLoS Biol 8(7):e1000420

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  44. Forgacs G (1998) Surface tension and viscoelastic properties of embryonic tissues depend on the cytoskeleton. Biol Bull 194:328–329

    Article  CAS  PubMed  Google Scholar 

  45. Foty RA, Forgacs G, Pfleger CM, Steinberg MS (1994) Liquid properties of embryonic tissues: measurement of interfacial tensions. Phys Rev Lett 72(14):2298–2301

    Article  CAS  PubMed  Google Scholar 

  46. Marmottant P, Mgharbel A, Käfer J, Audren B, Rieu JP, Vial JC, van der Sanden B, Mareé AFM, Graner F, Delanoë-Ayari H (2001) The role of fluctuations and stress on the effective viscosity of cell aggregates. Proc Natl Acad Sci USA 106(41):17271–17275

    Article  CAS  Google Scholar 

  47. Dillon RH, Othmer HG (1999) A mathematical model for outgrowth and spatial patterning of the vertebrate limb bud. J Theor Biol 197:295–330

    Article  CAS  PubMed  Google Scholar 

  48. Fung YC (1993) Biomechanics: mechanical properties of living tissues. Springer, New York

    Google Scholar 

  49. Taber LA (2004) Nonlinear theory of elasticity: applications in biomechanics. World Scientific, Singapore

    Google Scholar 

  50. Ciarlet PG (1988) Mathematical elasticity volume 1: three-dimensional elasticity. Elsevier, Amsterdam

    Chapter  Google Scholar 

  51. Holzapfel G (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, Chichester

    Google Scholar 

  52. Peters MD, Iber D (2017) Simulating organogenesis in COMSOL: tissue mechanics. arXiv:1710.00553v2

    Google Scholar 

  53. Meyer-Hermann ME, Maini PK, Iber D (2006) An analysis of b cell selection mechanisms in germinal centers. Math Med Biol 23(3):255–277

    Article  PubMed  Google Scholar 

  54. Merks RMH, Koolwijk P (2009) Modeling morphogenesis in silico and in vitro : towards quantitative, predictive, cell-based modeling. Math Model Nat Phenom 4(4):149–171

    Article  Google Scholar 

  55. Tanaka S (2015) Simulation frameworks for morphogenetic problems. Computation 3(2):197–221

    Article  Google Scholar 

  56. Van Liedekerke P, Palm MM, Jagiella N, Drasdo D (2015) Simulating tissue mechanics with agent-based models: concepts, perspectives and some novel results. Comput Part Mech 2(4):401–444

    Google Scholar 

  57. Ising E (1925) Beitrag zur Theorie des Ferromagnetismus. Z Phys 31(1):253–258

    Article  CAS  Google Scholar 

  58. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092

    Google Scholar 

  59. Graner F, Glazier JA (1992) Simulation of biological cell sorting using a two-dimensional extended Potts model. Phys Rev Lett 69(13):2013–2016

    Article  CAS  PubMed  Google Scholar 

  60. Hirashima T, Iwasa Y, Morishita Y (2009) Dynamic modeling of branching morphogenesis of ureteric bud in early kidney development. J Theor Biol 259(1):58–66

    Article  PubMed  Google Scholar 

  61. Hester SD, Belmonte JM, Gens JS, Clendenon SG, Glazier JA (2011) A multi-cell, multi-scale model of vertebrate segmentation and somite formation. PLoS Comput Biol 7(10):e1002155

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  62. Poplawski NJ, Swat M, Gens JS, Glazier JA (2007) Adhesion between cells, diffusion of growth factors, and elasticity of the AER produce the paddle shape of the chick limb. Physica A 373:521–532

    Article  PubMed  Google Scholar 

  63. Swat MH, Thomas GL, Belmonte JM, Shirinifard A, Hmeljak D, Glazier JA (2012) Multi-scale modeling of tissues using CompuCell3D. Methods Cell Biol 110:325–366

    Google Scholar 

  64. Hoehme S, Drasdo D (2010) A cell-based simulation software for multi-cellular systems. Bioinformatics 26(20):2641–2642

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  65. Honda H, Eguchi G (1980) How much does the cell boundary contract in a monolayered cell sheet? J Theor Biol 84(3):575–588

    Article  CAS  PubMed  Google Scholar 

  66. Weliky M, Oster G (1990) The mechanical basis of cell rearrangement. I. Epithelial morphogenesis during Fundulus epiboly. Development 109(2):373–386

    Google Scholar 

  67. Nagai T, Honda H (2001) A dynamic cell model for the formation of epithelial tissues. Philos Mag Part B 81(7):699–719

    Google Scholar 

  68. Fletcher AG, Osborne JM, Maini PK, Gavaghan DJ (2013) Implementing vertex dynamics models of cell populations in biology within a consistent computational framework. Prog Biophys Mol Biol 113(2):299–326

    Article  CAS  PubMed  Google Scholar 

  69. Fletcher AG, Osterfield M, Baker RE, Shvartsman SY (2014) Vertex models of epithelial morphogenesis. Biophys J 106(11):2291–2304

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  70. Pitt-Francis J, Pathmanathan P, Bernabeu MO, Bordas R, Cooper J, Fletcher AG, Mirams GR, Murray P, Osborne JM, Walter A, Chapman, S.J., Garny A, van Leeuwen IMM., Maini PK, Rodríguez B, Waters SL, Whiteley JP, Byrne HM, Gavaghan DJ (2009) Chaste: a test-driven approach to software development for biological modelling. Comput Phys Commun 180(12):2452–2471

    Article  CAS  Google Scholar 

  71. Mirams GR, Arthurs CJ, Bernabeu MO, Bordas R, Cooper J, Corrias, A., Davit Y, Dunn SJ, Fletcher AG, Harvey DG, Marsh ME, Osborne JM, Pathmanathan P, Pitt-Francis J, Southern J, Zemzemi N, Gavaghan DJ (2013) Chaste: an open source C++ library for computational physiology and biology. PLoS Comput Biol 9(3):e1002970

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  72. Rejniak KA (2007) An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development. J Theor Biol 247(1):186–204

    Article  CAS  PubMed  Google Scholar 

  73. Tanaka S, Sichau D, Iber D (2015) LBIBCell: a cell-based simulation environment for morphogenetic problems. Bioinformatics 31(14):2340–2347

    Article  CAS  PubMed  Google Scholar 

  74. Eymard R, Gallouët T, Herbin R (2000) Finite volume methods. In: Solution of equation in \( {\mathrm{\mathbb{R}}}^n \) (Part 3), techniques of scientific computing (Part 3). Volume 7 of handbook of numerical analysis. Elsevier, New York, pp 713–1018

    Google Scholar 

  75. Braess D (2007) Finite elements: theory, fast solvers, and applications in solid mechanics, 3rd edn. Cambridge University Press, Cambridge

    Google Scholar 

  76. Courant R (1943) Variational methods for the solution of problems of equilibrium and vibrations. Bull Am Math Soc 49(1):1–23

    Article  Google Scholar 

  77. Brenner S, Scott LR (2008) The mathematical theory of finite element methods. Texts in applied mathematics. Springer, New York

    Book  Google Scholar 

  78. Szabo BA, Babuška I (1991) Finite element analysis. Wiley, Chichester

    Google Scholar 

  79. Zienkiewicz O, Taylor R, Zhu JZ (2013) The finite element method: its basis and fundamentals, 7th edn. Butterworth-Heinemann, Oxford

    Chapter  Google Scholar 

  80. Gander MJ, Wanner G (2012) From Euler, Ritz, and Galerkin to modern computing. SIAM Rev 54(4):627–666

    Article  Google Scholar 

  81. Gelfand IM, Fomin SV (1963) Calculus of variations. Prentice-Hall, Upper Saddle River

    Google Scholar 

  82. Donea J, Huerta A, Ponthot J, Rodríguez-Ferran A (2004) Arbitrary Lagrangian–Eulerian methods. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics. Wiley, Hoboken

    Google Scholar 

  83. Hirt CW, Amsden AA, Cook JL (1974) An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J Comput Phys 14(3):227–253

    Article  Google Scholar 

  84. MacDonald G, Mackenzie J, Nolan M, Insall R (2016) A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: application to a model of cell migration and chemotaxis. J Comput Phys 309:207–226

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  85. Karimaddini Z, Unal E, Menshykau D, Iber D (2014) Simulating organogenesis in COMSOL: image-based modeling. arXiv:1610.09189v1

    Google Scholar 

  86. Menshykau D, Iber D (2012) Simulation organogenesis in COMSOL: deforming and interacting domains. arXiv:1210.0810

    Google Scholar 

  87. Kockelkoren J, Levine H, Rappel WJ (2003) Computational approach for modeling intra- and extracellular dynamics. Phys Rev E Stat Nonlin Soft Matter Phys 68(3–2)

    Google Scholar 

  88. Li X, Lowengrub J, Rätz A, Voigt A (2009) Solving PDEs in complex geometries: a diffuse domain approach. Commun Math Sci 7(1):81–107

    Google Scholar 

  89. Lervåg KY, Lowengrub J (2014) Analysis of the diffuse-domain method for solving PDEs in complex geometries. arXiv:1407.7480v3

    Google Scholar 

  90. Lowengrub J, Allard J, Aland S (2016) Numerical simulation of endocytosis: viscous flow driven by membranes with non-uniformly distributed curvature-inducing molecules. J Comput Phys 309:112–128

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  91. Wittwer LD, Croce R, Aland S, Iber D (2016) Simulating organogenesis in COMSOL: phase-field based simulations of embryonic lung branching morphogenesis. arXiv:1610.09189v1

    Google Scholar 

  92. Aland S (2012) Modelling of two-phase flow with surface active particles. PhD thesis, Technische Universität Dresden

    Google Scholar 

  93. Eck C, Garcke H, Knabner P (2011) Mathematische Modellierung. Springer, Berlin

    Book  Google Scholar 

  94. Folch R, Casademunt J, Hernández-Machado A, Ramírez-Piscina L (1999) Phase-field model for Hele-Shaw flows with arbitrary viscosity contrast. I. Theoretical approach. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 60(2-B):1724–1733

    Article  CAS  Google Scholar 

  95. Aland S, Landsberg C, Müller R, Stenger F, Bobeth M, Langheinrich AC, Voigt A (2014) Adaptive diffuse domain approach for calculating mechanically induced deformation of trabecular bone. Comput Methods Biomech Biomed Engin 17(1):31–38

    Article  PubMed  Google Scholar 

  96. Teigen KE, Li X, Lowengrub J, Wang F, Voigt A (2009) A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Commun Math Sci 4(7):1009–1037

    Google Scholar 

  97. Aland S, Lowengrub J, Voigt A (2011) A continuum model of colloid-stabilized interfaces. Phys Fluids 23:062103

    Article  CAS  Google Scholar 

  98. Teigen KE, Song P, Lowengrub J, Voigt A (2011) A diffuse-interface method for two-phase flows with soluble surfactants. J Comput Phys 230(2):375–393

    Google Scholar 

  99. Wittwer LD, Peters M, Aland S, Iber D (2017) Simulating organogenesis in COMSOL: comparison of methods for simulating branching morphogenesis. arXiv:1710.02876v1

    Google Scholar 

  100. Frisch U, Hasslacher B, Pomeau Y (1986) Lattice-gas automata for the Navier-Stokes equation. Phys Rev Lett 56(14):1505–1508

    Article  CAS  PubMed  Google Scholar 

  101. Frouzakis CE (2011) Lattice Boltzmann methods for reactive and other flows. In: Echekki T, Mastorakos E (eds) Turbulent combustion modeling, fluid mechanics and its applications. Springer Science+Business Media, Berlin

    Chapter  Google Scholar 

  102. McNamara GR, Zanetti G (1988) Use of the Boltzmann equation to simulate lattice-gas automata. Phys Rev Lett 61(20):2332–2335

    Article  CAS  PubMed  Google Scholar 

  103. Wolf-Gladrow DA (2000) Lattice-gas cellular automata and lattice boltzmann models - an introduction. Springer, Berlin

    Google Scholar 

  104. Chen S, Doolen GD (1998) Lattice Boltzmann method for fluid flows. Annu Rev Fluid Mech 30:329–364

    Article  Google Scholar 

  105. He X, Luo L (1997) A priori derivation of the lattice Boltzmann equation. Phys Rev E 55(6)

    Article  CAS  Google Scholar 

  106. Peskin C (2002) The immersed boundary method. Acta Numer 11:479–517

    Article  Google Scholar 

  107. Feng ZG, Michaelides EE (2004) The immersed boundary-lattice Boltzmann method for solving fluid–particles interaction problems. J Comput Phys 195(2):602–628

    Article  Google Scholar 

  108. Zhang J, Johnson PC, Popel AS (2007) An immersed boundary lattice Boltzmann approach to simulate deformable liquid capsules and its application to microscopic blood flows. Phys Biol 4(4):285–295

    Article  CAS  PubMed  Google Scholar 

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Authors and Affiliations

  1. Department of Biosystems Science and Engineering, ETH Zurich, Basel, Switzerland

    Michael D. Multerer, Lucas D. Wittwer, Anna Stopka, Diana Barac, Christine Lang & Dagmar Iber

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  1. Michael D. Multerer
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  2. Lucas D. Wittwer
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  3. Anna Stopka
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  4. Diana Barac
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  5. Christine Lang
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  6. Dagmar Iber
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Correspondence to Dagmar Iber .

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Editors and Affiliations

  1. Integrated Microscopy Core, Baylor College of Medicine, Houston, TX, USA

    Julien Dubrulle

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Multerer, M.D., Wittwer, L.D., Stopka, A., Barac, D., Lang, C., Iber, D. (2018). Simulation of Morphogen and Tissue Dynamics. In: Dubrulle, J. (eds) Morphogen Gradients. Methods in Molecular Biology, vol 1863. Humana Press, New York, NY. https://doi.org/10.1007/978-1-4939-8772-6_13

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  • DOI: https://doi.org/10.1007/978-1-4939-8772-6_13

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