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Discrete Mesh Approach in Morphogenesis Modelling: the Example of Gastrulation

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Abstract

Morphogenesis is a general concept in biology including all the processes which generate tissue shapes and cellular organizations in a living organism. Many hybrid formalizations (i.e., with both discrete and continuous parts) have been proposed for modelling morphogenesis in embryonic or adult animals, like gastrulation. We propose first to study the ventral furrow invagination as the initial step of gastrulation, early stage of embryogenesis. We focus on the study of the connection between the apical constriction of the ventral cells and the initiation of the invagination. For that, we have created a 3D biomechanical model of the embryo of the Drosophila melanogaster based on the finite element method. Each cell is modelled by an elastic hexahedron contour and is firmly attached to its neighbouring cells. A uniform initial distribution of elastic and contractile forces is applied to cells along the model. Numerical simulations show that invagination starts at ventral curved extremities of the embryo and then propagates to the ventral medial layer. Then, this observation already made in some experiments can be attributed uniquely to the specific shape of the embryo and we provide mechanical evidence to support it. Results of the simulations of the “pill-shaped” geometry of the Drosophila melanogaster embryo are compared with those of a spherical geometry corresponding to the Xenopus lævis embryo. Eventually, we propose to study the influence of cell proliferation on the end of the process of invagination represented by the closure of the ventral furrow.

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Acknowledgments

We thank VHP NoE (EC) and MEC grant from CONICYT (Chile) for financially aiding our research.

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Authors and Affiliations

  1. AGIM, Faculty of Medicine of Grenoble, University J. Fourier, 38700, La Tronche, France

    J. Demongeot

  2. TIMC-IMAG, Faculty of Medicine of Grenoble, University J. Fourier, 38700, La Tronche, France

    A. Lontos & E. Promayon

Authors
  1. J. Demongeot
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  2. A. Lontos
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  3. E. Promayon
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Corresponding author

Correspondence to J. Demongeot.

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Annex: P. Tracqui’s Obituary

Annex: P. Tracqui’s Obituary

P. Tracqui died in 2015 leaving the French community of Theoretical Biology very sad and orphan. Philip left a very important work covering many fields of modelling in biomedicine: osteogenesis, wound healing, cell movement, tissue morphogenesis, carcinogenesis, with many seminal articles. He published a total of 40 articles in international journals of high level cited as reference works [cf. for example (Tracqui et al. 1995; Tracqui and Mendjeli 1999; Tracqui 2006; Tracqui et al. 2005; Tranqui and Tracqui 2000; Vailhe et al. 1997; Woodward et al. 1996)], usually with prominent co-authors. His students, his friends, his colleagues in Paris and Grenoble, in the Dynacell team of the laboratory TIMC-IMAG he founded and he directed with great science and humanity, will infinitely regret him.

figure a

P. Tracqui in 2014

Eventually, the authors of the present paper, friends of Philippe, recall some words in French pronounced by Philippe’s spouse Valérie at the funerals (with her permission):

1.1 Philippe, toi le CHERCHEUR qui a marché ta Vie sur de multiples chemins…

Tu es né à Modane en 1956, dans une famille venant de Bessans, au fond de la vallée de la Maurienne. Tu étais un Être aux multiples talents, doué d’une grande intelligence et d’une formidable mémoire, ton exigence était à la hauteur de ton « commandeur » intérieur.

1.2 Tes passions?

La photographie

Très tôt, ton regard fin a eu envie de photographier les instants d’éternité, les couleurs, les paysages de nature, les ambiances, les visages et tu as ainsi capté des milliers d’images, pour qu’elles restent pour toujours gravées en toi. Et pourquoi choisir un appareil photo plutôt qu’un autre? Tu préférais les posséder tous, avec tous les objectifs qui allaient avec, histoire de garder tous les possibles…

La musique et les sons

Tu avais une oreille hors du commun, capable de repérer la moindre dissonance et de faire des réglages infinis pour améliorer la qualité des sons. Et ce fut une vraie joie quand à la fin de ta vie, tu révélais ta voix, si chaleureuse et juste, aussi bien sur tes poèmes que pour faire des sons. Tu jouais du piano, de la guitare, un peu de basse, des percussions… et possédais un labo d’enregistrement digne d’un professionnel…

Le sport

Pour maintenir ton corps actif et surtout ne pas vieillir, ta hantise ! Marcher vite, pédaler dur, courir longtemps… l’exigence est toujours là!

La recherche scientifique

Chercheur, tu étais très apprécié de tes collaborateurs. Capable à la fois d’une vision large et d’une détermination à aller dans les détails, tu allais chercher derrière les phénomènes de surface leurs processus et leurs causes. Tous les sujets t’intéressaient et tu menais tout de front: l’ordre et le chaos, la cicatrisation et… les mécanismes du cancer. Très reconnu dans le monde scientifique, tu as reçu en 1998 un prix de l’Académie des Sciences.

En tout, tu préférais la diversité, peut être par peur de manquer quelque chose?

Homme aux multiples talents, grand poète, musicien, peintre, photographe et sportif, tu marchais sur toutes les routes à la fois, au risque de t’épuiser et de ne pas te trouver. Ce cancer qui a touché tes os, c’est-à-dire la structure de qui tu croyais être (et que tu as tant étudiée au début de ta carrière de chercheur), fut terrible physiquement, mais ce fut une métamorphose pour ton éveil de conscience: tu as découvert le pouvoir de tes pensées sur le corps et le monde des illusions. Tu te réjouissais de témoigner de ton expérience au monde scientifique, mais, tu n’en as pas eu le temps, car la guérison ne touche pas forcément le corps physique, quand elle se dévoile à l’Esprit…

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Demongeot, J., Lontos, A. & Promayon, E. Discrete Mesh Approach in Morphogenesis Modelling: the Example of Gastrulation. Acta Biotheor 64, 427–446 (2016). https://doi.org/10.1007/s10441-016-9301-4

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