Tiling mechanisms of the Drosophila compound eye through geometrical tessellation.

Tiling patterns are observed in many biological structures. The compound eye is an interesting example of tiling and is often constructed by hexagonal arrays of ommatidia, the optical unit of the compound eye. Hexagonal tiling may be common due to mechanical restrictions such as structural robustness, minimal boundary length, and space-filling efficiency. However, some insects exhibit tetragonal facets.1-4 Some aquatic crustaceans, such as shrimp and lobsters, have evolved with tetragonal facets.5-8 Mantis shrimp is an insightful example as its compound eye has a tetragonal midband region sandwiched between hexagonal hemispheres.9,10 This casts doubt on the naive explanation that hexagonal tiles recur in nature because of their mechanical stability. Similarly, tetragonal tiling patterns are also observed in some Drosophila small-eye mutants, whereas the wild-type eyes are hexagonal, suggesting that the ommatidial tiling is not simply explained by such mechanical restrictions. If so, how are the hexagonal and tetragonal patterns controlled during development? Here, we demonstrate that geometrical tessellation determines the ommatidial tiling patterns. In small-eye mutants, the hexagonal pattern is transformed into a tetragonal pattern as the relative positions of neighboring ommatidia are stretched along the dorsal-ventral axis. We propose that the regular distribution of ommatidia and their uniform growth collectively play an essential role in the establishment of tetragonal and hexagonal tiling patterns in compound eyes.

Oscillations and bifurcation structure of reaction-diffusion model for cell polarity formation.

We investigate the oscillatory dynamics and bifurcation structure of a reaction-diffusion system with bistable nonlinearity and mass conservation, which was proposed by (Otsuji et al., PLoS Comp Biol 3:e108, 2007). The system is a useful model for understanding cell polarity formation. We show that this model exhibits four different spatiotemporal patterns including two types of oscillatory patterns, which can be regarded as cell polarity oscillations with the reversal and non-reversal of polarity, respectively. The trigger causing these patterns is a diffusion-driven (Turing-like) instability. Moreover, we investigate the effects of extracellular signals on the cell polarity oscillations.

Intracellular trafficking of Notch orchestrates temporal dynamics of Notch activity in the fly brain.

While Delta non-autonomously activates Notch in neighboring cells, it autonomously inactivates Notch through cis-inhibition, the molecular mechanism and biological roles of which remain elusive. The wave of differentiation in the Drosophila brain, the 'proneural wave', is an excellent model for studying Notch signaling in vivo. Here, we show that strong nonlinearity in cis-inhibition reproduces the second peak of Notch activity behind the proneural wave in silico. Based on this, we demonstrate that Delta expression induces a quick degradation of Notch in late endosomes and the formation of the twin peaks of Notch activity in vivo. Indeed, the amount of Notch is upregulated and the twin peaks are fused forming a single peak when the function of Delta or late endosomes is compromised. Additionally, we show that the second Notch peak behind the wavefront controls neurogenesis. Thus, intracellular trafficking of Notch orchestrates the temporal dynamics of Notch activity and the temporal patterning of neurogenesis.

Effective nonlocal kernels on reaction-diffusion networks.

A new method to derive an essential integral kernel from any given reaction-diffusion network is proposed. Any network describing metabolites or signals with arbitrary many factors can be reduced to a single or a simpler system of integro-differential equations called "effective equation" including the reduced integral kernel (called "effective kernel") in the convolution type. As one typical example, the Mexican hat shaped kernel is theoretically derived from two component activator-inhibitor systems. It is also shown that a three component system with quite different appearance from activator-inhibitor systems is reduced to an effective equation with the Mexican hat shaped kernel. It means that the two different systems have essentially the same effective equations and that they exhibit essentially the same spatial and temporal patterns. Thus, we can identify two different systems with the understanding in unified concept through the reduced effective kernels. Other two applications of this method are also given: Applications to pigment patterns on skins (two factors network with long range interaction) and waves of differentiation (called proneural waves) in visual systems on brains (four factors network with long range interaction). In the applications, we observe the reproduction of the same spatial and temporal patterns as those appearing in pre-existing models through the numerical simulations of the effective equations.

Noise-induced scaling in skull suture interdigitation.

Sutures, the thin, soft tissue between skull bones, serve as the major craniofacial growth centers during postnatal development. In a newborn skull, the sutures are straight; however, as the skull develops, the sutures wind dynamically to form an interdigitation pattern. Moreover, the final winding pattern had been shown to have fractal characteristics. Although various molecules involved in suture development have been identified, the mechanism underlying the pattern formation remains unknown. In a previous study, we reproduced the formation of the interdigitation pattern in a mathematical model combining an interface equation and a convolution kernel. However, the generated pattern had a specific characteristic length, and the model was unable to produce a fractal structure with the model. In the present study, we focused on the anterior part of the sagittal suture and formulated a new mathematical model with time-space-dependent noise that was able to generate the fractal structure. We reduced our previous model to represent the linear dynamics of the centerline of the suture tissue and included a time-space-dependent noise term. We showed theoretically that the final pattern from the model follows a scaling law due to the scaling of the dispersion relation in the full model, which we confirmed numerically. Furthermore, we observed experimentally that stochastic fluctuation of the osteogenic signal exists in the developing skull, and found that actual suture patterns followed a scaling law similar to that of the theoretical prediction.

A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices.

In this paper, we introduce a continuation method for the spatially discretized models, while conserving the size and shape of the cells and lattices. This proposed method is realized using the shift operators and nonlocal operators of convolution types. Through this method and using the shift operator, the nonlinear spatially discretized model on the uniform and nonuniform lattices can be systematically converted into a spatially continuous model; this renders both models point-wisely equivalent. Moreover, by the convolution with suitable kernels, we mollify the shift operator and approximate the spatially discretized models using the nonlocal evolution equations, rendering suitable for the application in both experimental and mathematical analyses. We also demonstrate that this approximation is supported by the singular limit analysis, and that the information of the lattice and cells is expressed in the shift and nonlocal operators. The continuous models designed using our method can successfully replicate the patterns corresponding to those of the original spatially discretized models obtained from the numerical simulations. Furthermore, from the observations of the isotropy of the Delta-Notch signaling system in a developing real fly brain, we propose a radially symmetric kernel for averaging the cell shape using our continuation method. We also apply our method for cell division and proliferation to spatially discretized models of the differentiation wave and describe the discrete models on the sphere surface. Finally, we demonstrate an application of our method in the linear stability analysis of the planar cell polarity model.

Reduced model of a reaction-diffusion system for the collective motion of camphor boats.

The unidirectional motion of a camphor boat along an annular water channel is observable. When camphor boats are placed in a water channel, both homogeneous and inhomogeneous states occur as collective motions, depending on the number of boats. The inhomogeneous state is a type of congestion, that is, the velocities of the boats change with temporal oscillation, and the shock wave appears along the line of travel of the boats. The unidirectional motion of a single camphor boat and the homogeneous state can be represented by traveling wave solutions in a mathematical model. Because the experimental results described here are thought of as a type of bifurcation phenomenon, the destabilization of traveling wave solutions may indicate the emergence of congestion. We previously attempted to study a linearized eigenvalue problem associated with a traveling wave solution. However, the problem is too difficult to analyze rigorously, even for just two camphor boats. Therefore we developed a center manifold theory and derived a reduced model in our previous work. In the present paper, we study the reduced model and show that the original model and our reduced model qualitatively exhibit the same properties by applying numerical techniques. Moreover, we demonstrate that the numerical results obtained in our models for camphor boats are quite similar to those in a car-following model, the OV model, but there are some different features between our reduced model and a typical OV model.

JAK/STAT guarantees robust neural stem cell differentiation by shutting off biological noise.

Organismal development is precisely regulated by a sequence of gene functions even in the presence of biological noise. However, it is difficult to evaluate the effect of noise in vivo, and the mechanisms by which noise is filtered during development are largely unknown. To identify the noise-canceling mechanism, we used the fly visual system, in which the timing of differentiation of neural stem cells is spatio-temporally ordered. Our mathematical model predicts that JAK/STAT signaling contributes to noise canceling to guarantee the robust progression of the differentiation wave in silico. We further demonstrate that the suppression of JAK/STAT signaling causes stochastic and ectopic neural stem cell differentiation in vivo, suggesting an evolutionarily conserved function of JAK/STAT to regulate the robustness of stem cell differentiation.

Mathematical modeling for meshwork formation of endothelial cells in fibrin gels.

Vasculogenesis is the earliest process in development for spontaneous formation of a primitive capillary network from endothelial progenitor cells. When human umbilical vein endothelial cells (HUVECs) are cultured on Matrigel, they spontaneously form a network structure which is widely used as an in vitro model of vasculogenesis. Previous studies indicated that chemotaxis or gel deformation was involved in spontaneous pattern formation. In our study, we analyzed the mechanism of vascular pattern formation using a different system, meshwork formation by HUVECs embedded in fibrin gels. Unlike the others, this experimental system resulted in a perfusable endothelial network in vitro. We quantitatively observed the dynamics of endothelial cell protrusion and developed a mathematical model for one-dimensional dynamics. We then extended the one-dimensional model to two-dimensions. The model showed that random searching by endothelial cells was sufficient to generate the observed network structure in fibrin gels.

'Initial state' coordinations reproduce the instant flexibility for human walking.

An important feature of human locomotor control is the instant adaptability to unpredictable changes of conditions surrounding the locomotion. Humans, for example, can seamlessly adapt their walking gait following a sudden ankle impairment (e.g., as a result of an injury). In this paper, we propose a theoretical study of the mechanisms underlying flexible locomotor control. We hypothesize that flexibility is achieved by modulating the posture at the beginning of the stance phase-the initial state. Using a walking model, we validate our hypothesis through computer simulations.

Emergence of adaptability to time delay in bipedal locomotion.

Based on neurophysiological evidence, theoretical studies have shown that locomotion is generated by mutual entrainment between the oscillatory activities of central pattern generators (CPGs) and body motion. However, it has also been shown that the time delay in the sensorimotor loop can destabilize mutual entrainment and result in the failure to walk. In this study, a new mechanism called flexible-phase locking is proposed to overcome the time delay. It is realized by employing the Bonhoeffer-Van der Pol formalism - well known as a physiologically faithful neuronal model - for neurons in the CPG. The formalism states that neurons modulate their phase according to the delay so that mutual entrainment is stabilized. Flexible-phase locking derives from the phase dynamics related to an asymptotically stable limit cycle of the neuron. The effectiveness of the mechanism is verified by computer simulations of a bipedal locomotion model.