Computational simulations using a versatile 3D vertex model of optic-cup formation
This protocol is extracted from research article:
Strain-triggered mechanical feedback in self-organizing optic-cup morphogenesis
Sci Adv, Nov 21, 2018; DOI: 10.1126/sciadv.aau1354

The 3D multicellular dynamics were expressed using the versatile 3D vertex model (fig. S2). The main symbols in this model are listed in table S1. In this model, each cell shape is represented by a single polyhedron, and each cell-cell boundary is represented by a polygonal face (fig. S2, A and B) (13). Because the polygonal faces are shared by neighboring polyhedrons, the entire tissue structure is represented by a single network composed of vertices and edges (fig. S2C).

Topological dynamics of 3D multicellular dynamics such as cell rearrangement, division, and apoptosis were expressed by several types of network reconnections. Cell rearrangement is expressed by reconnecting local network patterns using the reversible network reconnection model (fig. S2D). Previously, we have mathematically proved a part of physical validities of this model; i.e., this model can express continuous cell rearrangements with geometric and energetic reversibility (18). Cell proliferation was expressed using a cell proliferation model that expresses cell proliferation by cell growth (increase in cell volume) and cell division (increase in cell number; fig. S2E) (46). Cell apoptosis was expressed using a cell apoptosis model that expresses cell apoptosis by cell shrinkage (decrease in cell volume) and cell disappearance (decrease in cell number; fig. S2F) (47). Previously, we have also mathematically proved a part of topological validities of this model; i.e., this model can express the entire pattern of multicellular topological dynamics in 3D space (23). This model has been applied to several 3D multicellular dynamics (4851) and can potentially be applied to various physiologies in morphogenesis, homeostasis, and disease.

The 3D multicellular dynamics were expressed by the vertex movements according to the total energy, represented by U. As the deformation process of the optic-cup formation takes approximately 2 days, we regarded it as the quasi-static process from a thermodynamic point of view, where viscous forces are relaxed in an instance. Under this process, the energetic force satisfies the force balance at each time point as U = 0. Here, we defined the total energy as U = U({ri}, {ζjα}, {ξjkβ}), where {…} indicates a set of components, ri is the position vector of the ith vertex, ζjα is the αth physical parameter of the jth cell, and ξjkβ is the βth physical parameter of the jth and kth cells. Under U = 0, the vertex locations {ri} at each time point are obtained by seeking the local minimum of U, and thereby the time displacements of the vertex locations {ri} are obtained as the changes in {ri} against those in physical parameters {ζjα} and {ξjkβ}. The local minimum of U is solved by the Euler method of the overdamped equation (52).

We introduced the jth cell volume vj, height hj, apical perimeter length pj, basal surface area sbj, total surface area stj, and the apical edge between jth and kth cells lajk, as a function of {ri}. The total energy function U is expressed as followsEmbedded Image(3)where H is the Heaviside step function with a value of 1 when the variable is positive and 0 otherwise. In Eq. 3, the first term denotes the elastic energy of individual cell volumes, where Kvj and veqj are the jth cell volume elasticity and equilibrium volume, respectively. The second term is the elastic energy of individual cell heights (fig. S2G), where Khj and heqj are the jth cell height elasticity and equilibrium height, respectively. The third term is the elastic energy of individual cell apical perimeters (fig. S2H), where Kaj and peqj are the jth cell apical perimeter elasticity and equilibrium apical perimeter length, respectively. The fourth term is the boundary length energy between cells on the apical surface (fig. S2I), where λgjk is the apical edge length energy between the jth and kth cells. The fifth term is the elastic energy of individual cell basal surfaces (fig. S1J), where Kbj and peqj are the jth cell basal surface elasticity and equilibrium basal surface area, respectively. The last term is the total surface energy of individual cells (fig. S2K), where κsj is the jth cell surface elasticity. Hence, in Eq. 3, {ζjα} is described as a set of parameters Kvj, Khj, Kaj, Kbj, κsj, veqj, heqj, peqj, and sbeqj, and {ξjkβ} is a set of λgjk.

The optic cup is formed even under the condition where the root of the OV is cut (11), and the shape of the root of the OV is not important for morphogenesis. Hence, the initial condition was set to be a monolayer spherical OV. The initial OV is composed of 1600 cells, which was estimated from the tissue size and cell density measured in experiments. In the initial OV, the dorsal-ventral and distal-proximal axes are defined to be orthogonal in the 3D orthogonal coordinates. Moreover, the optic cup is formed even under conditions where a hole is made in the epithelial sheet or the surrounding tissues are removed (11); therefore, the inner pressure and extrinsic forces are unnecessary. Hence, the boundary condition was set to be free with fixing the center of all vertex positions on the coordinates.

By varying all the unknown parameters, we obtained a standard set of physical parameter values (table S4) that recapitulates the optic-cup formation (Fig. 1, I to L). We also varied all of the unknown parameters around the standard values and obtained many phenotypes (table S5 and fig. S4). The resulting optic cups were screened by two evaluation parameters: the NR curvature and the NR-RPE boundary curvature, represented by cNR and cB, respectively (closed regions in fig. S4). By varying cNR and cB, we determined several dominant physical parameters for the proper optic-cup formation, from which Figs. 2B and 3 (B to E) were extracted.

To test whether the strain-triggered mechanical feedback is inherited in the optic-cup formation, we modeled this cell response and implemented it to the above model for the in silico screening by replacing the part of lateral constriction (Fig. 4Q). Because the lateral constriction oscillates in the optic-cup formation, we expressed that cells transiently constrict depending on the basal surface strain and relax for a while. In biological experiments, the basal calcium transients in NR are much lower than those in OV and RPE, since the gap junction is poor in NR (53); hence, we assumed that this strain-triggered constriction occurs only in the OV and RPE regions. By varying the relevant physical parameters, we obtained the optic-cup morphogenesis that contains the mechanical feedback (Fig. 4, R to T). Details of the model are described in Supplementary Text.

Numerical calculations were performed using customized C++ software on a computer comprising 2.9-GHz Intel Xeon dual processors and 64-GB random-access memory and RIKEN Super Combined Cluster. The results were visualized with ParaView (54).

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