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A Cell-Based Computational Model of Early Embryogenesis Coupling Mechanical Behaviour and Gene Regulation

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A Cell-Based Computational Model of Early Embryogenesis Coupling Mechanical Behaviour and Gene Regulation

Julien Delile et al. Nat Commun.

Abstract

The study of multicellular development is grounded in two complementary domains: cell biomechanics, which examines how physical forces shape the embryo, and genetic regulation and molecular signalling, which concern how cells determine their states and behaviours. Integrating both sides into a unified framework is crucial to fully understand the self-organized dynamics of morphogenesis. Here we introduce MecaGen, an integrative modelling platform enabling the hypothesis-driven simulation of these dual processes via the coupling between mechanical and chemical variables. Our approach relies upon a minimal 'cell behaviour ontology' comprising mesenchymal and epithelial cells and their associated behaviours. MecaGen enables the specification and control of complex collective movements in 3D space through a biologically relevant gene regulatory network and parameter space exploration. Three case studies investigating pattern formation, epithelial differentiation and tissue tectonics in zebrafish early embryogenesis, the latter with quantitative comparison to live imaging data, demonstrate the validity and usefulness of our framework.

Figures

Figure 1
Figure 1. Schematic overview of the MecaGen model coupling the cell's biomechanical properties to its biochemical activity.
Mechanical parameters are specified by the gene expression dynamics and molecular state. Conversely, spatial rearrangements among cells impact protein synthesis via signalling and mechanical stress. (a) Meca: cell shapes are idealized as ellipsoids (pale grey) represented by a centre (black dot) and two radii (Supplementary Fig. 1). Edges connecting centres materialize cell neighbourhoods, derived from metric and topological criteria. Neighbouring cells exert ‘passive' and ‘active' forces on one another (Supplementary Note 2, equation (3)). Passive relaxation forces (solid arrows), in particular attraction-repulsion formula image, maintain volume integrity via adhesion and cortical tension coefficients (equation (14) and Supplementary Figs 3–5). Attractive forces (orange) point towards the neighbours, while repulsive forces (blue) point away from them. Active behavioural forces (dashed arrows), exerted at the level of protrusions or apical constriction and involved in cell intercalation, comprise pairs of ‘intrinsic' components formula image (green) and ‘extrinsic' components formula image (red; equation (21) and Fig. 5e). During protrusion, intrinsic forces result from the cell's cortical cytoskeleton maintaining its shape, while extrinsic forces result from the traction on neighbouring cells through protrusive activity (here, to the left). (b) Gen: the biochemical model relies on a gene regulatory network (GRN), associated with concentration variables of intracellular proteins and extracellular ligands, driven by chemical kinetics (synthesis, secretion, binding) and reaction-diffusion equations (30)–(47). (c) MecaGen: both sides are coupled via a cell behaviour ontology of three cell ‘archetypes': epithelial, mesenchymal and idle (Fig. 2) corresponding to mutual relationships between Meca and Gen variables (equations (48)–(58)).
Figure 2
Figure 2. Cell behaviour ontology (CBO) foundations of the MecaGen coupling between mechanical and molecular states.
Cells can transition between three ‘archetypes': mesenchymal (M), epithelial (E) and the default, idle (I), controlled by the GRN topology and dynamics. Protrusion, most apparent in M cells, is a treadmilling activity based on adhesion (similar to tracked vehicles), which is regulated to avoid sliding at the contact surface area, and can induce an intercalation motion between neighbour cells (Supplementary Figs 7 and 8). It rests on a polarization axis formula image, generally created by an asymmetrical distribution of external ligands and internal substances (Fig. 5c). Like mesenchymal cells, epithelial cells are able to intercalate themselves between other E cells, except that they remain in the tangential plane of the epithelium. This is due to a property of ‘planarity conservation' supported by another type of passive relaxation forces formula image (Supplementary Note 2, equation (17)) and an additional apicobasal polarization axis formula image (Fig. 4c). They can also exhibit active apical constriction (similar to purse strings), but this behaviour is not implemented in the current version of the model. Differentiation into E requires signalling by other surrounding epithelial cells to create and maintain apicobasal polarity (equation (9) and Supplementary Fig. 2), otherwise an isolated cell reverts to I. Finally, both M and E cells can be polarized by multiple ‘potential' mechanisms throughout development: chemotaxis along concentration gradients, propagation of alignment via cell contacts, polar induction from nearby protruding cells, and randomized orientation by blebbing (equations (25), (52)–(54)).
Figure 3
Figure 3. Example of genetic regulation and molecular signalling.
This example based on Drosophila shows the possibility of (neural) cell fate specification and pattern formation in a spatially explicit simulation of tissue (Supplementary Movie 1). Cells divide sporadically, increasing their number from 880 to 1,125 (Supplementary Note 2, equations (24)–(29)). (a) The GRN contains a signal-mediated toggle switch using a single Target gene, upregulated (resp. downregulated) by a Tcf+ (resp. Tcf−) protein complex (equations (30)–(32)). This complex results from the reaction (black circle) of Tcf with an intracellular cofactor, β-catenin (resp. Groucho, or Gro; equations (33)–(35)). The internal release of β-catenin (resp. XIAP) is triggered by ligand-receptor binding (chevrons) and transduction (equation (40)). In turn, XIAP induces the ubiquitination of Gro and its degradation (equation (36)). (b) Partial snapshot at t=95 min (showing about 120 cells). Production of Wnt protein was turned on at t=20 min in one region (asterisks; equation (38)), provoking the secretion of Wnt ligand (equation (39)) and its extracellular diffusion (pink gradient; equations (43)–(46)). At t>50 min, the expression of Target shoots up in cells that have received enough Wnt ligand (blue dots). (ce) Temporal evolution of protein concentrations in different regions. Initially, Gro, Tcf, Tcf− and receptors are ubiquitously present. (c) Cells bathing in high levels of Wnt ligand express Target. (d) Cells bordering the source region receive less Wnt ligand, thus Target remains silent. (e) Cells far from the Wnt sources display no activity. See parameters in Supplementary Table 2.
Figure 4
Figure 4. Example of boundary formation and epithelialization.
See Supplementary Movie 2. (a) The GRN implements a Delta-Notch mechanism of mutual induction of the type found in the Drosophila wing. Right half: Intracellular protein X stimulates Anterior, a self-activating transcription factor (left cells in b,c). Anterior promotes the synthesis of a transmembrane ligand Delta (circles), which binds to a Notch receptor on the nearest cells. This produces Epi-inducer by transduction (Supplementary Note 3, equation (41)) and, in the absence of Anterior, leads to the synthesis of Epi. The consequence is that only cells in contact with, but outside of the X domain express Epi (green cells) and adopt a polarized epithelial type oriented along the gradient of Delta. Left half: Symmetrically, the Epi cells synthesize another ligand, Delto (asterisks), which leads to the epithelialization of Anterior cells in contact with them, via Epi2 (orange cells). (b) Zoom on a 2D slice of tissue. After applying a pulse of X on the left, two adjacent rows of cells have differentiated into Epi2 and Epi at the boundary. (c) View of the 1,683-cell cubic domain at a late stage (neighbourhood edges partially visible). Cells do not divide here. Inset: Apicobasal polarization axes (grey) and partial planarity conservation forces (blue) on three Epi2 cells (Supplementary Fig. 6). (d,e) Temporal evolution of protein concentrations in and around one boundary cell. The irregular profiles of certain curves are caused by spatial rearrangements and consequent fluctuations in transduction signals and messengers, as E cells elongate and align in a planar way. See parameters in Supplementary Table 3.
Figure 5
Figure 5. Example of collective behaviour during the zebrafish epiboly.
See Supplementary Movie 3. (a) 2D section from live imaging 3D data (‘oblong' stage, 3.7 hpf), highlighting the EVL, the external and internal YSL, and the interface between the blastoderm and the yolk cell (credit: BioEmergences). Scale bar 150 μm. (b) Simplified GRN controlling the bipolar protrusion of mesenchymal deep cells via protein Ubi, oriented by a gradient of extracellular ligand Lig. (c) Their polarization axes formula image are oriented by chemotaxis (Supplementary Note 3, equations (43)–(46)) along a radial gradient of ligand (purple) released from the EVL (not shown). (d) Sagittal section of the whole simulated embryo (4 hpf), containing 1,595 deep cells (purple and grey polyhedra) and showing the EVL cell centres (purple dots), yolk particles (yellow dots) and yolk membrane (peripheral yellow edges). EVL and yolk take part only in passive relaxation forces. (e) In the bipolar domain of cell i (green cones) containing three neighbours, protrusive forces comprise ‘intrinsic' (dashed green arrows) and ‘extrinsic' components (dashed red; equation (21) and Supplementary Fig. 9), resulting in formula image (solid green arrow).
Figure 6
Figure 6. Parameter exploration of the zebrafish epiboly study.
See Fig. 5. (a) Macroscopic measurements of the epibolic deformation (pasted on a Nomarski 2D picture at the level of the sagittal plane by Karlstrom and Kane, with permission). Landmarks: vegetal pole (VP), animal pole (APe), yolk animal pole (APy). Distances: embryo height (EH), yolk height (YH), margin height (MH) and margin diameter (MD). Scale bar 200 μm. (b) Temporal evolution of the last three measurements normalized by EH in the live embryo (dashed lines) and the best simulated embryo (solid lines). (c) Snapshots at four intermediate stages (time arrows in b). As deep cells divide (mitosis equations (24)–(29) based on empirical data2932), their number increases to 3,095. See parameters in Supplementary Table 4. (d) Fitness landscapes as a function of the protrusive force intensity φ and a noise factor λran controlling the regularity of the polarization axes' orientation. The global fitness function F is the average of the yolk height fitness FYH, the margin height fitness FMH and the margin diameter fitness FMD. A lower fitness value (green) means a better similarity with the live embryo. The blue cross highlights the parameter values used in b, which are: formula image ≈ (0.025, 28.5E-04).

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