八杉 徹雄

Notch signalling is a well-conserved signalling pathway that regulates cell fate through cell-cell communication. A typical feature of Notch signalling is 'lateral inhibition', whereby two neighbouring cells of equivalent state of differentiation acquire different cell fates. Recently, mathematical and computational approaches have addressed the Notch dynamics in Drosophila neural development. Typical examples of lateral inhibition are observed in the specification of neural stem cells in the embryo and sensory organ precursors in the thorax. In eye disc development, Notch signalling cooperates with other signalling pathways to define the evenly spaced positioning of the photoreceptor cells. The interplay between Notch and epidermal growth factor receptor signalling regulates the timing of neural stem cell differentiation in the optic lobe. In this review, we summarize the theoretical studies that have been conducted to elucidate the Notch dynamics in these systems and discuss the advantages of combining mathematical models with biological experiments.

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.

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.

Immunity is considered to be involved in the prevention of cancer. Although both humoral and cellular immune reactions may participate, underlying mechanisms have yet to be clarified. The present study was conducted to clarify this issue using a Drosophila model, in which neoplastic transformation was induced through the simultaneous inhibition of cell-cycle checkpoints and apoptosis. We first determined the location of hemocytes, blood cells of Drosophila playing a role of immune cells, in neoplasia-induced and normal larvae, but there was no significant difference between the two groups. When gene expression pattern in larval hemocytes was determined, the expression of immunity-related genes including those necessary for phagocytosis was reduced in the neoplasia model. We then asked the involvement of phagocytosis in the prevention of neoplasia examining animals where the expression of engulfment receptors instead of apoptosis was retarded. We found that the inhibition of engulfment receptor expression augmented the occurrence of neoplasia induced by a defect in cell-cycle checkpoints. This suggested a role for phagocytosis in the prevention of neoplastic transformation in Drosophila.

The brain is organized morphologically and functionally into a columnar structure. According to the radial unit hypothesis, neurons from the same lineage form a radial unit that contributes to column formation. However, the molecular mechanisms that link neuronal lineage and column formation remain elusive. Here, we show that neurons from the same lineage project to different columns under control of Down syndrome cell adhesion molecule (Dscam) in the fly brain. Dscam1 is temporally expressed in newly born neuroblasts and is inherited by their daughter neurons. The transient transcription of Dscam1 in neuroblasts enables the expression of the same Dscam1 splice isoform within cells of the same lineage, causing lineage-dependent repulsion. In the absence of Dscam1 function, neurons from the same lineage project to the same column. When the splice diversity of Dscam1 is reduced, column formation is significantly compromised. Thus, Dscam1 controls column formation through lineage-dependent repulsion.

Notch-mediated lateral inhibition regulates binary cell fate choice, resulting in salt-and-pepper pattern formation during various biological processes. In many cases, Notch signaling acts together with other signaling systems. However, it is not clear what happens when Notch signaling is combined with other signaling systems. Mathematical modeling and the use of a simple biological model system will be essential to address this uncertainty. A wave of differentiation in the Drosophila visual center, the "proneural wave," accompanies the activity of the Notch and EGF signaling pathways. Although all of the Notch signaling components required for lateral inhibition are involved in the proneural wave, no salt-and-pepper pattern is found during the progression of the proneural wave. Instead, Notch is activated along the wave front and regulates proneural wave progression. How does Notch signaling control wave propagation without forming a salt-and-pepper pattern? A mathematical model of the proneural wave, based on biological evidence, has demonstrated that Notch-mediated lateral inhibition is implemented within the proneural wave and that the diffusible action of EGF cancels salt-and-pepper pattern formation. The results from numerical simulation have been confirmed by genetic experiments in vivo and suggest that the combination of Notch-mediated lateral inhibition and EGF-mediated reaction diffusion enables a novel function of Notch signaling that regulates propagation of the proneural wave. Similar mechanisms may play important roles in diverse biological processes found in animal development and cancer pathogenesis.