Collective mechanics of embryogenesis: Formation of ventral furrow in Drosophila

📝 Abstract

We propose a 2D mechanical model of the ventral furrow formation in Drosophila that is based on undifferentiated epithelial cells of identical properties whose energy resides in their membrane. Depending on the relative tensions of the apical, basal, and lateral sides, the minimal-energy states of the embryo cross-section includes circular, elliptical, biconcave, and buckled furrow shapes. We discuss the possible shape transformation consistent with reported experimental observations, arguing that generic collective mechanics may play an important role in the embryonic development in Drosophila.

💡 Analysis

We propose a 2D mechanical model of the ventral furrow formation in Drosophila that is based on undifferentiated epithelial cells of identical properties whose energy resides in their membrane. Depending on the relative tensions of the apical, basal, and lateral sides, the minimal-energy states of the embryo cross-section includes circular, elliptical, biconcave, and buckled furrow shapes. We discuss the possible shape transformation consistent with reported experimental observations, arguing that generic collective mechanics may play an important role in the embryonic development in Drosophila.

📄 Content

arXiv:1108.4795v1 [q-bio.CB] 24 Aug 2011 Collective mechanics of embryogenesis: Formation of ventral furrow in Drosophila A. Hoˇcevar1,2 and P. Ziherl1,3 1Joˇzef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia 2Department of Physics and Astronomy, University of Pennsylvania, 209 S. 33rd St., Philadelphia PA 19104-6396, USA and 3Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia (Dated: November 5, 2018) We propose a 2D mechanical model of the ventral furrow formation in Drosophila that is based on undifferentiated epithelial cells of identical properties whose energy resides in their membrane. Depending on the relative tensions of the apical, basal, and lateral sides, the minimal-energy states of the embryo cross-section includes circular, elliptical, biconcave, and buckled furrow shapes. We discuss the possible shape transformation consistent with reported experimental observations, ar- guing that generic collective mechanics may play an important role in the embryonic development in Drosophila. PACS numbers: 87.17.Pq, 87.19.lx, 87.19.rd The single most important step in the embryonic de- velopment of all multicellular animals is gastrulation [1]. It establishes the future body plan of the organism and transforms a topologically spherical embryo into a topo- logical torus. In the beginning of this crucial process, a part of the convex shell-like single layer of cells called the blastula folds in, and the invaginated part of the embryo develops into the digestive system [1]. The ge- ometry of invagination differs quite considerably among the animals as does the shape of the embryo itself [1–4]. One of the best-studied types of gastrulation is the for- mation of the ventral furrow in the fruit fly (Drosophila melanogaster) [4–6]. At the beginning of this process, the Drosophila embryo is an elongated ellipsoidal one- cell-thick epithelium of about 6000 cells coating the yolk. During the first stage of the furrow formation, a length- wise invagination occurs as the cells in a stripe on ventral side deform and buckle inwards [4] (Fig. 1a,b). This pri- mary invagination globally alters the shape of the embryo and is one of the vital factors for determining the future head-to-tail body axis [1]. The role of mechanics in the early embryonic develop- ment is universally appreciated [7], and a range of models have been proposed to explain the formation of ventral furrow. To the best of our knowledge, all of them rec- ognize that the invaginating cells — the mesoderm — are different from the rest of the embryo [6, 8–11]. By forcing the mesoderm cells into a keystone shape with a small outer (apical) face and a large inner (basal) face, the ventral section of the embryo can be driven rather naturally into an invaginated shape. This can be done by superposing active and passive deformations; the former are generated by an inherent preferred keystone shape of cells and the latter result from the elasticity of the ep- ithelium [6, 8–10]. Alternatively, invagination can be re- produced by hydrodynamic motion of embryo immersed in a surrounding fluid and consisting of cells character- ized by surface energy [11]. In this theory, formation of FIG. 1: Drosophila ventral furrow (micrograph reproduced with permission from Ref. [4]) is the lengthwise invagination of the embryo on ventral side (panel a). The cross-sections of the central part (reproduced with permission from Ref. [2]) show the inward buckling of the initially circular epithelial wall (panel b). In our 2D model (panel c), the cross-section consists of a ring of N quadrilateral cells of area Ac enclosing the yolk of area Ay. All apical, basal, and lateral edges are characterized by line tensions Γa, Γb, and Γl, respectively. ventral furrow is triggered by an increase of the apical membrane tension of the ventral cells. These sophisti- cated models have been elaborated both in the simplified 2D variants describing the predominantly cylindrical cen- tral section of the embryo [8, 9, 11] as well as in the full 3D geometry [6, 10], and they also included the effect of the vitelline membrane which covers the embryo [9–11]. In this paper, we challenge the importance of cell dif- ferentiation as the main postulate of the existing theo- ries of ventral furrow formation in Drosophila. Instead we explore the primary invagination as a collective phe- nomenon in a system of cells of identical mechanical prop- erties. In our 2D model of the embryo cross-section that 2 is based on surface energy alone, each type of cell side is characterized by a specific line tension: The apical, the basal, and the lateral line tensions are all different from one another but each of them is the same in all cells. We compute the phase diagram of the model epithelium sub- ject to constraints associated with the incompressibility of the yolk and each cell to find that it does include the invaginated shapes even though the model is devoid of any

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