Computational approaches for simulating luminogenesis.

Lumens, liquid-filled cavities surrounded by polarized tissue cells, are elementary units involved in the morphogenesis of organs. Theoretical modeling and computations, which can integrate various factors involved in biophysics of morphogenesis of cell assembly and lumens, may play significant roles to elucidate the mechanisms in formation of such complex tissue with lumens. However, up to present, it has not been documented well what computational approaches or frameworks can be applied for this purpose and how we can choose the appropriate approach for each problem. In this review, we report some typical lumen morphologies and basic mechanisms for the development of lumens, focusing on three keywords - mechanics, hydraulics and geometry - while outlining pros and cons of the current main computational strategies. We also describe brief guidance of readouts, i.e., what we should measure in experiments to make the comparison with the model's assumptions and predictions.

Left-handed cardiac looping by cell chirality is mediated by position-specific convergent extensions.

In the embryonic heart development of mammals and birds, a straight initial heart tube undergoes left-handed helical looping, which is a remarkable and puzzling event. We are interested in the mechanism of this chiral helical looping. Recently, observations were reported that myocardial cells in the embryonic chick heart show intrinsic chirality of rotation. The chirality of myocardial cells, via anisotropic polarization of Golgi inside the cells, leads to a left-right (LR) asymmetry of cell shape. On cell boundaries of LR asymmetric cells, phosphorylated myosin and N-cadherin are enriched. Such LR asymmetric cellular circumstances lead to a large-scale three-dimensional chiral structure, the left-handed helical loop. However, the physical mechanism of this looping is unclear. Computer simulations were performed using a cell-based three-dimensional mathematical model assuming an anterior-rightward-biased contractile force of the cell boundaries on the ventral surface of the heart (orientation of a clock hand pointing to 10 to 11 o'clock). An initially straight heart tube was successfully remodeled to the left-handed helical tube via frequent convergent extension (CE) of collective cells, which corresponds to the previously reported observations of chick heart development. Although we assumed that the biased boundary contractile force was uniform all over the ventral side, orientations of the CEs became position specific on the anterior, posterior, right, and left regions on the ventral tube. Such position-specific CEs produced the left-handed helical loop. In addition, our results suggest the loop formation process consists of two distinct phases of preparation and explicit looping. Intrinsic cell properties of chirality in this investigation were discussed relating to extrinsic factors investigated by other researches. Finally, because CE is generally exerted in the axial developmental process across different animal species, we discussed the contribution of CE to the chiral heart structure across species of chick, mouse, Xenopus, and zebrafish.

The Chiral Looping of the Embryonic Heart Is Formed by the Combination of Three Axial Asymmetries.

In mammals and birds, embryonic development of the heart involves conversion of a straight tubular structure into a three-dimensional helical loop, which is a chiral structure. We investigated theoretically the mechanism of helical loop formation of the mouse embryonic heart, especially focusing on determination of left-/right-handedness of the helical loop. In geometrical terms, chirality is the result of the combination of three axial asymmetries in three-dimensional space. We hypothesized the following correspondences between axial asymmetries and morphogenesis (bending and displacement): the dorsal-ventral asymmetry by ventral bending of a straight tube of the initial heart and the left-right and anterior-posterior asymmetries, the left-right asymmetry by rightward displacement of the heart tube, which is confined to the anterior region of the tube. Morphogenesis of chiral looping of the embryonic heart is a large-scaled event of the multicellular system in which substantial physical force operates dynamically. Using computer simulations with a cell-based physico-mechanical model and experiments with mouse embryos, we confirmed the hypothesis. We conclude that rightward displacement of the tube determines the left-handed screw of the loop. The process of helix loop formation consists of three steps: 1) the left-right biasing system involving Nodal-related signals that leads to left-right asymmetry in the embryonic body; 2) the rightward displacement of the tube; and finally 3) the left-handed helical looping. Step 1 is already established. Step 3 is elucidated by our study, which highlights the need for step 2 to be clarified; namely, we explore how the left-right asymmetry in the embryonic body leads to the rightward displacement of the heart tube.

Self-organization at the first stage of honeycomb construction: Analysis of an attachment-excavation model.

Honeybees construct nests that consist of regularly arrayed hexagonal cylinders. In the first stage of honeycomb construction, they build a linear sequence of tetrapod structures that form the basis of the comb. However, considering their physiological limitations, it is unknown how honeybees produce that initial pattern. Herein, in an attempt to understand the mechanisms of honeycomb construction, we propose an agent-based model, the attachment-excavation model, in which worker honeybees are classified into attachers who secrete and attach wax, and excavators who excise the attached wax. The model assumes that workers instinctively refrain from digging through the thin parts of a wax cluster. We then conduct two-dimensional (2D) simulations that show how a tripod pattern can be seen as a projection of tetrapods onto a plane. The simulation results show that the tripod pattern emerges due to competition between the attachers and excavators. As time advances, the isotropic wax growth causes the tripods to connect planarly. Because the homogeneously broadened structures do not match that of a natural comb, we employ anisotropic wax growth to obtain a linear sequence of constructed tripods, thus suggesting that anisotropy is a significant contributor to the first stage of honeycomb construction. From our simulation results, we conclude that honeybees utilize self-organization to achieve complexity during the first stage of honeycomb construction. It is anticipated that the results of our study will provide insights into how complexity can be achieved within a hierarchy.

Chiral cell sliding drives left-right asymmetric organ twisting.

Polarized epithelial morphogenesis is an essential process in animal development. While this process is mostly attributed to directional cell intercalation, it can also be induced by other mechanisms. Using live-imaging analysis and a three-dimensional vertex model, we identified 'cell sliding,' a novel mechanism driving epithelial morphogenesis, in which cells directionally change their position relative to their subjacent (posterior) neighbors by sliding in one direction. In Drosophila embryonic hindgut, an initial left-right (LR) asymmetry of the cell shape (cell chirality in three dimensions), which occurs intrinsically before tissue deformation, is converted through LR asymmetric cell sliding into a directional axial twisting of the epithelial tube. In a Drosophila inversion mutant showing inverted cell chirality and hindgut rotation, cell sliding occurs in the opposite direction to that in wild-type. Unlike directional cell intercalation, cell sliding does not require junctional remodeling. Cell sliding may also be involved in other cases of LR-polarized epithelial morphogenesis.

Simulation of Cell Patterning Triggered by Cell Death and Differential Adhesion in Drosophila Wing.

The Drosophila wing exhibits a well-ordered cell pattern, especially along the posterior margin, where hair cells are arranged in a zigzag pattern in the lateral view. Based on an experimental result observed during metamorphosis of Drosophila, we considered that a pattern of initial cells autonomously develops to the zigzag pattern through cell differentiation, intercellular communication, and cell death (apoptosis) and performed computer simulations of a cell-based model of vertex dynamics for tissues. The model describes the epithelial tissue as a monolayer cell sheet of polyhedral cells. Their vertices move according to equations of motion, minimizing the sum total of the interfacial and elastic energies of cells. The interfacial energy densities between cells are introduced consistently with an ideal zigzag cell pattern, extracted from the experimental result. The apoptosis of cells is modeled by gradually reducing their equilibrium volume to zero and by assuming that the hair cells prohibit neighboring cells from undergoing apoptosis. Based on experimental observations, we also assumed wing elongation along the proximal-distal axis. Starting with an initial cell pattern similar to the micrograph experimentally obtained just before apoptosis, we carried out the simulations according to the model mentioned above and successfully reproduced the ideal zigzag cell pattern. This elucidates a physical mechanism of patterning triggered by cell apoptosis theoretically and exemplifies, to our knowledge, a new framework to study apoptosis-induced patterning. We conclude that the zigzag cell pattern is formed by an autonomous communicative process among the participant cells.

The world of epithelial sheets.

An epithelium is a layer of closely connected cells covering the body or lining a body cavity. In this review, several fundamental questions are addressed regarding the epithelium. (i) While an epithelium functions as barrier against the external environment, how is barrier function maintained during its construction? (ii) What determines the apical and basal sides of epithelial layer? (iii) Is there any relationship between the apical side of the epithelium and the apical membrane of an epithelial cell? (iv) Why are hepatocytes (liver cells) called epithelial, even though they differ completely from column-like shape of typical epithelial cells? Keeping these questions in mind, multiple shapes of epithelia were considered, extracting a few of their elemental processes, and constructing a virtual world of epithelia by combining them. Epithelial cells were also classified into several types based on the number of apical domains of each cell. In addition, an intracellular organelle was introduced within epithelial cells, the vacuolar apical compartment (VAC), which is produced within epithelial cells surrounded by external cell matrix (ECM). The VAC interacts with areas of cell-cell contact of the cell surface membrane and is converted to apical membrane. The properties of VACs enable us to answer the initial questions posed above. Finally, the genetic and molecular mechanisms of epithelial morphogenesis are discussed.

Synergistic action of nectins and cadherins generates the mosaic cellular pattern of the olfactory epithelium.

In the olfactory epithelium (OE), olfactory cells (OCs) and supporting cells (SCs), which express different cadherins, are arranged in a characteristic mosaic pattern in which OCs are enclosed by SCs. However, the mechanism underlying this cellular patterning is unclear. Here, we show that the cellular pattern of the OE is established by cellular rearrangements during development. In the OE, OCs express nectin-2 and N-cadherin, and SCs express nectin-2, nectin-3, E-cadherin, and N-cadherin. Heterophilic trans-interaction between nectin-2 on OCs and nectin-3 on SCs preferentially recruits cadherin via α-catenin to heterotypic junctions, and the differential distributions of cadherins between junctions promote cellular intercalations, resulting in the formation of the mosaic pattern. These observations are confirmed by model cell systems, and various cellular patterns are generated by the combinatorial expression of nectins and cadherins. Collectively, the synergistic action of nectins and cadherins generates mosaic pattern, which cannot be achieved by a single mechanism.

Cell models lead to understanding of multi-cellular morphogenesis consisting of successive self-construction of cells.

Morphogenesis of multi-cellular organisms occurs through cell behaviours within a cell aggregate. Cell behaviours have been described using cell models involving equations of motion for cells. Cells in cell models construct shapes of the cell aggregate by themselves. Here, a history of cell models, the cell centre model and the vertex cell model, which we have constructed, are described. Furthermore, the application of these cell models is explained in detail. These cell models have been applied to transformation of cell aggregates to become spherical, formation of mammalian blastocysts and cell intercalation in elongating tissues. These are all elemental processes of morphogenesis and take place in succession during the whole developmental process. A chain of successive elemental processes leads to morphogenesis. Finally, we highlight that cell models are indispensable to understand the process whereby genes direct biological shapes.

Planar cell polarity links axes of spatial dynamics in neural-tube closure.

Neural-tube closure is a critical step of embryogenesis, and its failure causes serious birth defects. Coordination of two morphogenetic processes--convergent extension and neural-plate apical constriction--ensures the complete closure of the neural tube. We now provide evidence that planar cell polarity (PCP) signaling directly links these two processes. In the bending neural plates, we find that a PCP-regulating cadherin, Celsr1, is concentrated in adherens junctions (AJs) oriented toward the mediolateral axes of the plates. At these AJs, Celsr1 cooperates with Dishevelled, DAAM1, and the PDZ-RhoGEF to upregulate Rho kinase, causing their actomyosin-dependent contraction in a planar-polarized manner. This planar-polarized contraction promotes simultaneous apical constriction and midline convergence of neuroepithelial cells. Together our findings demonstrate that PCP signals confer anisotropic contractility on the AJs, producing cellular forces that promote the polarized bending of the neural plate.

Computer simulation of wound closure in epithelial tissues: cell-basal-lamina adhesion.

The mechanism of wound closure in epithelial tissues, i.e., cell monolayer sheets, is investigated through computer simulations. A wound means an area in which some cells have been removed from the normal tissue. The vertex dynamics cell model [T. Nagai and H. Honda, Philos. Mag. B 81, 699 (2001)], which describes morphogenesis of epithelial tissues using the concepts of statistical physics, is modified and applied to the closure of small wounds without mitosis. It is shown that cell-basal-lamina adhesion governs the wound closure competing with cell-cell adhesion and cell elasticity. The simulation results reproduce the actual wound closure process qualitatively and partly quantitatively. The closing proceeds with the translation of the edges of wound polygons toward the wound center and the intermittent reduction in the number of polygon edges. Over time, the process leads to an exponential decrease in the wound area. A shape factor is introduced to describe the wound shape quantitatively and is used to examine the time variation thereof. A method for determining model parameters by comparison with the experiments is given.

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.

Competitive interactions between retinal ganglion axons for tectal targets explain plasticity of retinotectal projection in the servomechanism model of retinotectal mapping.

The mechanism of topographic mapping of retinal ganglion cells to the midbrain was previously elucidated by the servomechanism model, which is based on the fact that cells expressing Eph-receptors respond specifically to surface expressing membrane-bound ephrin-ligands at a critical level. The retina has increased nasal-to-temporal gradient of Eph receptor-density, and the optic tectum/superior colliculus has increased rostral-to-caudal gradient of membrane-bound ephrin-ligand. An axon from the retina has an identification tag of a certain level of Eph-receptor density depending on its retinal position, and adheres to the site on the tectum/superior colliculus expressing ephrin-ligands at a critical ligand-density level. The servomechanism model rigidly defines positions of axon terminals on the midbrain. However, optic nerve regeneration experiments combined with halved retina or tectum show a plastic or flexible mapping (expansion, compression and transposition of tectal projections). To reconcile the discrepancy between the rigid model and the plastic behavior, competition between retinal axon terminals for a target site was introduced to the servomechanism. The servomechanism/competition model succeeded in computer simulations of the plastic mapping of retinal axons on the tectum. Recent experiments of upregulated ligand-density on the tectum during nerve regeneration and the role of axonal competition are discussed.

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.

Competition between retinal ganglion axons for targets under the servomechanism model explains abnormal retinocollicular projection of Eph receptor-overexpressing or ephrin-lacking mice.

Topographic mapping of retinal ganglion axons to the midbrain is computed by the servomechanism model, which is based on the experimental result of cell attachment. Cells expressing a certain level of Eph proteins (receptors for ephrin ligands) optimally attach to a surface that expresses a specific level of ephrin ligand density. The retina has an increasing nasal-to-temporal gradient of Eph receptor density, and the optic tectum/superior colliculus has an increasing rostral-to-caudal gradient of membrane-bound ephrin ligand. An axon from the retina has an identification tag of a certain level of Eph receptor density depending on its retinal position and adheres to the site on the tectum/superior colliculus expressing ephrin ligands at a critical ligand density level. Quantitatively, a retinal axon has a receptor density (R) that is determined by its retinal position, and the axon terminal is induced to adhere to the tectal site of ligand density (L = S/R), where S is a constant. Consequently, the servomechanism model defines positions of axon terminals on the midbrain. Abnormal topographic maps are reported in a knock-in experiment with elevated density of Eph receptors and a knock-out experiment lacking ephrin ligands using gene-targeting technology. By adding competition between axon terminals for target sites to the servomechanism model, the abnormal maps became easy to understand. Furthermore, the servomechanism-competition model allowed conjecture of the gradient shapes of receptor and ligand densities and estimation of the capacity of the midbrain surface to accept retinal axon terminals.

Formation and maintenance of distinctive cell patterns by coexpression of membrane-bound ligands and their receptors.

We show that graded or checkerboard-like cell patterns, and segmental domains along a body axis, can be generated by cell behaviors involving differences in intercellular repulsion. A membrane-bound signal transduction system mediating contact-dependent cell interactions includes membrane-bound ligands (ephrins) and their receptors with tyrosine-kinase activity (Eph proteins). These molecules mediate both repulsive and attractive interactions under bilateral threshold control, i.e., cells expressing the receptors adhere to a surface bearing a critical density of ligand reciprocal to the density of receptor but are repelled by a surface with other densities of ligand (Honda [1998] J. Theor. Biol. 192:235-246). We extend this model. General membrane-bound ligands (not always ephrins) and their receptors are presumably coexpressed in a single cell under bilateral threshold control. Computer simulations of cell pattern formation showed that when coexpression of the ligand and receptor is reciprocal, the cells self-organize into a pattern of segmental domains or a graded cell arrangement along the body axis. The latter process interprets positional information in terms of protein molecules. When coexpression of the two species of molecules is not always reciprocal, the cells generate various patterns including checkerboard and kagome (star) patterns. The case of separate expression of ligands and receptors in different cells is also examined. The mechanism of differences in cell repulsion is compared with the differential cell adhesion hypothesis, which has been used to explain cell sorting.