Correction to: Theoretical models for branch formation in plants.

The article Theoretical models for branch formation in plants.

Theoretical models for branch formation in plants.

Various branch architectures are observed in living organisms including plants. Branch formation has traditionally been an area of interest in the field of developmental biology, and theoretical approaches are now commonly used to understand the complex mechanisms involved. In this review article, we provide an overview of theoretical approaches including mathematical models and computer simulations for studying plant branch formation. These approaches cover a wide range of topics. In particular, we focus on the importance of positional information in branch formation, which has been especially revealed by theoretical research in plants including computations of developmental processes.

Asymmetries in leaf branch are associated with differential speeds along growth axes: A theoretical prediction.

BACKGROUND: Morphogenesis, when accompanied by continuous growth, requires stable positional information to create a balanced shape in an organism. Evenly spaced branches are examples of such morphogenesis. Previously, we created a model that showed when a one-dimensional (1D) ring (a boundary of a 2D field) was periodically deformed based on a stable, doubled iterative pattern during expansion; a nested, regularly spaced, symmetrically branched structure was generated. The characteristic divaricating pattern is common in the leaves of many plant species; however, the divarication symmetry was often broken. To evaluate this type of asymmetry, we investigated several species with dissected or compound leaves. RESULTS: Sometimes these leaves showed asymmetries in the number of lobes or segments positioned on either side of the secondary axes. The direction of the asymmetry, i.e., which side of a secondary axis has more axes, appeared to be species-specific. CONCLUSIONS: When different growth speeds along axes of a divaricating leaf were introduced into our previous model, robust and directed asymmetries were reproduced. The differences in growth speed could be predicted from the distributions of leaf segments in actual leaves. Developmental Dynamics 246:981-991, 2017. © 2017 Wiley Periodicals, Inc.

Interactions between zebrafish pigment cells responsible for the generation of Turing patterns.

The reaction-diffusion system is one of the most studied nonlinear mechanisms that generate spatially periodic structures autonomous. On the basis of many mathematical studies using computer simulations, it is assumed that animal skin patterns are the most typical examples of the Turing pattern (stationary periodic pattern produced by the reaction-diffusion system). However, the mechanism underlying pattern formation remains unknown because the molecular or cellular basis of the phenomenon has yet to be identified. In this study, we identified the interaction network between the pigment cells of zebrafish, and showed that this interaction network possesses the properties necessary to form the Turing pattern. When the pigment cells in a restricted region were killed with laser treatment, new pigment cells developed to regenerate the striped pattern. We also found that the development and survival of the cells were influenced by the positioning of the surrounding cells. When melanophores and xanthophores were located at adjacent positions, these cells excluded one another. However, melanophores required a mass of xanthophores distributed in a more distant region for both differentiation and survival. Interestingly, the local effect of these cells is opposite to that of their effects long range. This relationship satisfies the necessary conditions required for stable pattern formation in the reaction-diffusion model. Simulation calculations for the deduced network generated wild-type pigment patterns as well as other mutant patterns. Our findings here allow further investigation of Turing pattern formation within the context of cell biology.

Developmental analyses of divarications in leaves of an aquatic fern Microsorum pteropus and its varieties.

Plant leaves occur in diverse shapes. Divarication patterns that develop during early growths are one of key factors that determine leaf shapes. We utilized leaves of Microsorum pteropus, a semi-aquatic fern, and closely related varieties to analyze a variation in the divarication patterns. The leaves exhibited three major types of divarication: no lobes, bifurcation, and trifurcation (i.e., monopodial branching). Our investigation of their developmental processes, using time-lapse imaging, revealed localized growths and dissections of blades near each leaf apex. Restricted cell divisions responsible for the apical growths were confirmed using a pulse-chase strategy for EdU labeling assays.

Perfusable Vascular Network with a Tissue Model in a Microfluidic Device.

A spheroid (a multicellular aggregate) is regarded as a good model of living tissues in the human body. Despite the significant advancement in the spheroid cultures, a perfusable vascular network in the spheroids remains a critical challenge for long-term culture required to maintain and develop their functions, such as protein expressions and morphogenesis. The protocol presents a novel method to integrate a perfusable vascular network within the spheroid in a microfluidic device. To induce a perfusable vascular network in the spheroid, angiogenic sprouts connected to microchannels were guided to the spheroid by utilizing angiogenic factors from human lung fibroblasts cultured in the spheroid. The angiogenic sprouts reached the spheroid, merged with the endothelial cells co-cultured in the spheroid, and formed a continuous vascular network. The vascular network could perfuse the interior of the spheroid without any leakage. The constructed vascular network may be further used as a route for supply of nutrients and removal of waste products, mimicking blood circulation in vivo. The method provides a new platform in spheroid culture toward better recapitulation of living tissues.

Integrating perfusable vascular networks with a three-dimensional tissue in a microfluidic device.

Creating vascular networks in tissues is crucial for tissue engineering. Although recent studies have demonstrated the formation of vessel-like structures in a tissue model, long-term culture is still challenging due to the lack of active perfusion in vascular networks. Here, we present a method to create a three-dimensional cellular spheroid with a perfusable vascular network in a microfluidic device. By the definition of the cellular interaction between human lung fibroblasts (hLFs) in a spheroid and human umbilical vein endothelial cells (HUVECs) in microchannels, angiogenic sprouts were induced from microchannels toward the spheroid; the sprouts reached the vessel-like structures in a spheroid to form a continuous lumen. We demonstrated that the vascular network could administer biological substances to the interior of the spheroid. As cell density in the spheroid is similar to that of a tissue, the perfusable vasculature model opens up new possibilities for a long-term tissue culture in vitro.

A developmental model for branching morphogenesis of lake cress compound leaf.

Lake cress, Rorippa aquatica (Brassicaceae), is a semi-aquatic plant that exhibits a variety of leaf shapes, from simple leaves to highly branched compound leaves, depending on the environment. Leaf shape can vary within a single plant, suggesting that the variation can be explained by a simple model. In order to simulate the branched structure in the compound leaves of R. aquatica, we implemented reaction-diffusion (RD) patterning onto a theoretical framework that had been developed for serration distribution in the leaves of Arabidopsis thaliana, with the modification of the one-dimensional reaction-diffusion domain being deformed with the spatial periodicity of the RD pattern while expanding. This simple method using an iterative pattern could create regular and nested branching patterns. Subsequently, we verified the plausibility of our theoretical model by comparing it with the experimentally observed branching patterns. The results suggested that our model successfully predicted both the qualitative and quantitative aspects of the timing and positioning of branching in growing R. aquatica leaves.

High-frequency affine mechanics and nonaffine relaxation in a model cytoskeleton.

The cytoskeleton is a network of crosslinked, semiflexible filaments, and it has been suggested that it has properties of a glassy state. Here we employ optical-trap-based microrheology to apply forces to a model cytoskeleton and measure the high-bandwidth response at an anterior point. Simulating the highly nonlinear and anisotropic stress-strain propagation assuming affinity, we found that theoretical predictions for the quasistatic response of semiflexible polymers are only realized at high frequencies inaccessible to conventional rheometers. We give a theoretical basis for determining the frequency when both affinity and quasistaticity are valid, and we discuss with experimental evidence that the relaxations at lower frequencies can be characterized by the experimentally obtained nonaffinity parameter.