Adapting a Plant Tissue Model to Animal Development: Introducing Cell Sliding into VirtualLeaf

Cell-based, mathematical modeling of collective cell behavior has become a prominent tool in developmental biology. Cell-based models represent individual cells as single particles or as sets of interconnected particles and predict the collective cell behavior that follows from a set of interaction rules. In particular, vertex-based models are a popular tool for studying the mechanics of confluent, epithelial cell layers. They represent the junctions between three (or sometimes more) cells in confluent tissues as point particles, connected using structural elements that represent the cell boundaries. A disadvantage of these models is that cell–cell interfaces are represented as straight lines. This is a suitable simplification for epithelial tissues, where the interfaces are typically under tension, but this simplification may not be appropriate for mesenchymal tissues or tissues that are under compression, such that the cell–cell boundaries can buckle. In this paper, we introduce a variant of VMs in which this and two other limitations of VMs have been resolved. The new model can also be seen as on off-the-lattice generalization of the Cellular Potts Model. It is an extension of the open-source package VirtualLeaf, which was initially developed to simulate plant tissue morphogenesis where cells do not move relative to one another. The present extension of VirtualLeaf introduces a new rule for cell–cell shear or sliding, from which cell rearrangement (T1) and cell extrusion (T2) transitions emerge naturally, allowing the application of VirtualLeaf to problems of animal development. We show that the updated VirtualLeaf yields different results than the traditional vertex-based models for differential adhesion-driven cell sorting and for the neighborhood topology of soft cellular networks. Electronic supplementary material The online version of this article (10.1007/s11538-019-00599-9) contains supplementary material, which is available to authorized users.


Video S2
Differential-adhesion driven cell rearrangement in VirtualLeaf. Engulfment as in Figure  Differential-adhesion driven cell rearrangement in VirtualLeaf. Incomplete cell sorting with only T1 transitions as in Figure 2D. θ T1 = 0.25; other parameters as in Video S3. Simulation length: 500,0000 Monte Carlo Steps (MCS)

Video S5
Effect of interface specific cortical tension. Simulation with cell-typespecific cortical tension applied only at cell medium interfaces as in Figure 3, top-left panel. P T (red) = 20 at cell-medium interfaces and P T (green) = 20 at cell-medium interfaces. All other parameters have default values (see Supporting Text S1). This figure shows the tissues after a simulation of 500,000 MCS.

Video S6
Effect of interface specific cortical tension. Simulation with cell-typespecific cortical tension applied only at cell medium interfaces as in Figure 3, bottom-right panel. P T (red) = 40 at cell-medium interfaces and P T (green) = 40 at cell-medium interfaces. All other parameters have default values (see Supporting Text S1). This figure shows the tissues after a simulation of 500,000 MCS.

Video S7
Effect of interface specific cortical tension. Simulation with cell-typespecific cortical tension applied only at cell medium interfaces as in Figure 3, bottom-right panel. P T (red) = 40 at cell-medium interfaces and P T (green) = 20 at cell-medium interfaces. All other parameters have default values (see Supporting Text S1). This figure shows the tissues after a simulation of 500,000 MCS.

Video S8
Simulation of epithelial cell packing Case I with T1 transitions and straight walls; λ cortical = 10, J (e → L, e → R) = 500. MCS 0 to 40000 with stride 500; cell colors indicate number of neighbors as in Figure 4C-D.

Video S9
Simulation of epithelial cell packing Case II with T1 transitions and straight walls; λ cortical = 26, J (e → L, e → R) = 0. MCS 0 to 40000 with stride 500; cell colors indicate number of neighbors as in Figure 4C-D.

Video S10
Simulation of epithelial cell packing Case III with T1 transitions and straight walls; λ cortical = 26, J (e → L, e → R) = −3560. MCS 0 to 40000 with stride 500; cell colors indicate number of neighbors as in Figure 4C-D.

Video S11
Simulation of epithelial cell packing Case I with sliding and flexible walls; λ cortical = 10, J (e → L, e → R) = 500. MCS 0 to 40000 with stride 500; cell colors indicate number of neighbors as in Figure 4C-D.

Video S12
Simulation of epithelial cell packing Case II with sliding and flexible walls; λ cortical = 26, J (e → L, e → R) = 0. MCS 0 to 40000 with stride 500; cell colors indicate number of neighbors as in Figure 4C-D.

Video S13
Simulation of epithelial cell packing Case III with sliding and flexible walls; λ cortical = 26, J (e → L, e → R) = −3560. MCS 0 to 40000 with stride 500; cell colors indicate number of neighbors as in Figure 4C-D.

Video S14
Effect of cell resolution on cell sorting kinetics. Left, control simulation of cell mixing (cf. Figure 2A) with default values of l min = 6 and l max = 8; Right, refined simulation of cell mixing with reduced values of l min = 3 and l max = 4 such that twice the number of edges and nodes is used for each cell. Bottom panel shows the summed length of red-green cell-cell interfaces relative to the total length of all cell-cell interfaces in the configuration, 1 e∈E e {e∈E|e is red-green interface} e , as a function of time. The moving dot indicates the present time.