Voronoi diagram description of the maternal surface of the placenta: preliminary report.

AIM: Voronoi diagram is defined as a diagram of a collection of geometric points that defines a partition of space into cells, each of which consists of the points closer to one particular point than to any other. The distinctive feature of a placentone is the fetomaternal circulatory unit which is composed of one villous tree with a corresponding, centrifugally perfused portion of the intervillous space. Based on this placental architecture, in this study we generated Voronoi diagram from the photographic images of the maternal surface of the placenta and compared them with the shapes of the actual placentones. MATERIAL AND METHODS: We simulated the placentones of 12 placentas based on Voronoi diagram using a computer program, and compared them with the photographic images of the actual maternal surface of the placentas. The point was carefully selected and adjusted so that the generated Voronoi diagram could represent the actual maternal surface of the placenta. RESULTS: Voronoi diagram simulated satisfactorily the borders of placentones in 10 placentas. However, we could not succeed in the simulation in two cases. CONCLUSION: It seems that development and formation of placentones are not only theoretically and physiologically explainable by Voronoi diagram, but also actually achieved by this mechanism. We believe that mathematical Voronoi diagram holds a promise for evaluating physiological and pathological development of the placenta.

Two different mechanisms of planar cell intercalation leading to tissue elongation.

During development, certain cells intercalate with each other towards tissue-elongation, exemplified in sea-urchin gut-elongation, amphibian gastrulation, and Drosophila germ-band extension. Their mechanism is not universal among intercalation events. To clarify the minimal cellular properties required for cell-intercalation, we computer-simulated the process using three-dimensional geometrical cell-models. We identified two different mechanisms: (1) cell-junction-remodeling by cell-junction contraction along a specific direction, as observed in Drosophila germ-band extension, and (2) cell-shuffling by orientated cell-extension of bipolar cells, as observed in amphibian gastrulation. The cell-junction-remodeling was characterized by well-defined accumulation of contractile molecules along a specific direction of cell-junctions. Length contraction of approximately one cell-junction per cell is enough for the entire tissue-elongation. The cell-shuffling was characterised by rhythmic cell-extension and orientated movement of cytoskeleton within the elongated cells. Furthermore, tissue-elongation along a polarized axis was limited to a 2.5-fold increase in the cell-junction-remodeling, while no limit was defined for the cell-shuffling.

Computer simulation of emerging asymmetry in the mouse blastocyst.

The mechanism of embryonic polarity establishment in mammals has long been controversial. Whereas some claim prepatterning in the egg, we recently presented evidence that mouse embryonic polarity is not established until blastocyst and proposed the mechanical constraint model. Here we apply computer simulation to clarify the minimal cellular properties required for this morphology. The simulation is based on three assumptions: (1) behavior of cell aggregates is simulated by a 3D vertex dynamics model; (2) all cells have equivalent mechanical properties; (3) an inner cavity with equivalent surface properties is gradually enlarged. However, an initial attempt reveals a requirement for an additional assumption: (4) the surface of the cavity is firmer than intercellular surfaces, suggesting the presence of a basement membrane lining the blastocyst cavity, which is indeed confirmed by published data. The simulation thus successfully produces a structure recapitulating the mouse blastocyst. The axis of the blastocyst, however, remains variable, leading us to an additional assumption: (5) the aggregate is enclosed by a capsule, equivalent to the zona pellucida in vivo. Whereas a spherical capsule does not stabilize the blastocyst axis, an ellipsoidal capsule eventually orients the axis in accordance with its longest diameter. These predictions are experimentally verified by time-lapse recordings of mouse embryos. During simulation, equivalent cells form two distinct populations composed of smaller inner cells and larger outer cells. These results reveal a unique feature of early mammalian development: an asymmetry may emerge autonomously in an equivalent population with no need for a priori intrinsic differences.

A three-dimensional vertex dynamics cell model of space-filling polyhedra simulating cell behavior in a cell aggregate.

We developed a three-dimensional (3D) cell model of a multicellular aggregate consisting of several polyhedral cells to investigate the deformation and rearrangement of cells under the influence of external forces. The polyhedral cells fill the space in the aggregate without gaps or overlaps, consist of contracting interfaces and maintain their volumes. The interfaces and volumes were expressed by 3D vertex coordinates. Vertex movements obey equations of motion that rearrange the cells to minimize total free energy, and undergo an elementary process that exchanges vertex pair connections when vertices approach each other. The total free energy includes the interface energy of cells and the compression or expansion energy of cells. Computer simulations provided the following results: An aggregate of cells becomes spherical to minimize individual cell surface areas; Polygonal interfaces of cells remain flat; Cells within the 3D cell aggregate can move and rearrange despite the absence of free space. We examined cell rearrangement to elucidate the viscoelastic properties of the aggregate, e.g. when an external force flattens a cell aggregate (e.g. under centrifugation) its component cells quickly flatten. Under a continuous external force, the cells slowly rearrange to recover their original shape although the cell aggregate remains flat. The deformation and rearrangement of individual cells is a two-step process with a time lag. Our results showed that morphological and viscoelastic properties of the cell aggregate with long relaxation time are based on component cells where minimization of interfacial energy of cells provides a motive force for cell movement.