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.

Diffusion-driven destabilization of spatially homogeneous limit cycles in reaction-diffusion systems.

We study the diffusion-driven destabilization of a spatially homogeneous limit cycle with large amplitude in a reaction-diffusion system on an interval of finite size under the periodic boundary condition. Numerical bifurcation analysis and simulations show that the spatially homogeneous limit cycle becomes unstable and changes to a stable spatially nonhomogeneous limit cycle for appropriate diffusion coefficients. This is analogous to the diffusion-driven destabilization (Turing instability) of a spatially homogeneous equilibrium. Our approach is based on a reaction-diffusion system with mass conservation and its perturbed system considered as an infinite dimensional slow-fast system (relaxation oscillator).

Perturbations and dynamics of reaction-diffusion systems with mass conservation.

In some reaction-diffusion systems where the total mass of their components is conserved, solutions with initial values near a homogeneous equilibrium converge to a simple localized pattern (spike) after exhibiting Turing-like patterns near the equilibrium for appropriate diffusion coefficients. In this study, we investigate the perturbed reaction-diffusion systems of such conserved systems. We show that a reaction-diffusion model with a globally stable homogeneous equilibrium can exhibit large amplitude Turing-like patterns in the transient dynamics. Moreover, we propose a three-component model, which exhibits an alternating repetition of spatially (almost) homogeneous oscillations and large amplitude Turing-like patterns.

Turing instabilities in prey-predator systems with dormancy of predators.

In this paper, we study the stationary and oscillatory Turing instabilities of a homogeneous equilibrium in prey-predator reaction-diffusion systems with dormant phase of predators. We propose a simple criterion which is useful in classifying these Turing instabilities. Moreover, numerical simulations reveal transient spatio-temporal complex patterns which are a mixture of spatially periodic steady states and traveling/standing waves. In this mixture, the steady part is the stable Turing pattern bifurcated primarily from the homogeneous equilibrium, while wave parts are unstable oscillatory solutions bifurcated secondarily from the same homogeneous equilibrium. Although our criterion does not exclude the occurrence of oscillatory Turing instability, we have not yet found stable traveling/standing waves due to oscillatory Turing instability in our simulations. These results suggest that dormancy of predators is not a generator but an enhancer of spatio-temporal Turing patterns in prey-predator reaction-diffusion systems.

Mathematical modelling and experiments for the proliferation and differentiation of Drosophila intestinal stem cells II.

The Drosophila posterior midgut epithelium mainly consists of intestinal stem cells (ISCs); semi-differentiated cells, i.e. enteroblasts (EBs); and two types of fully differentiated cells, i.e. enteroendocrine cells (EEs) and enterocytes (ECs), which are controlled by signalling pathways. In [M. Kuwamura, K. Maeda, and T. Adachi-Yamada, Mathematical modeling and experiments for the proliferation and differentiation of Drosophila intestinal stem cells I, J. Biol. Dyn. 4 (2009), pp. 248-257], on the basis of the functions of the Wnt and Notch signalling pathways, we studied the regulatory mechanism for the proliferation and differentiation of ISCs under the assumption that the Wnt proteins are supplied from outside the cellular system of ISCs. In this paper, we experimentally show that the Wnt proteins are specifically expressed in ISCs, EBs, and EEs, and theoretically show that the cellular system of ISCs can be self-maintained under the assumption that the Wnt proteins are produced in the cellular system of ISCs. These results provide a useful basis for determining whether an environmental niche is required for maintaining the cellular system of tissue stem cells.

Mathematical modelling and experiments for the proliferation and differentiation of Drosophila intestinal stem cells I.

We study the proliferation and differentiation of stem cells in the Drosophila posterior midgut epithelium, which mainly consists of intestinal stem cells (ISCs); semi-differentiated cells, i.e. enteroblasts (EBs); and two types of fully differentiated cells, i.e. enteroendocrine cells (EEs) and enterocytes (ECs). The cellular system of ISCs is controlled by Wnt and Notch signalling pathways. In this article, we experimentally show that EBs are not capable of efficiently differentiating into ECs in the absence of Wnt signalling. On the basis of the experimental results and known facts, we propose a scheme and a simple ordinary differential equation (ODE) model for the proliferation and differentiation of ISCs. This is a first step towards understanding the universal mechanism for the maintenance of the cellular system of tissue stem cells controlled by signalling pathways.

A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment.

In this paper, a mathematical model of a prey-predator system is proposed to resolve the paradox of enrichment in ecosystems. The model is based on the natural strategy that a predator takes, i.e, it produces resting eggs in harsh environment. Our result gives a criterion for a functional response, which ensures that entering dormancy stabilizes the population dynamics. It is also shown that the hatching of resting eggs can stabilize the population dynamics when the switching between non-resting and resting eggs is sharp. Furthermore, the bifurcation structure of our model suggests the simultaneous existence of a stable equilibrium and a large amplitude cycle in natural enriched environments.

Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators.

It is shown that the dormancy of predators induces mixed-mode oscillations and chaos in the population dynamics of a prey-predator system under certain conditions. The mixed-mode oscillations and chaos are shown to bifurcate from a coexisting equilibrium by means of the theory of fast-slow systems. These results may help to find experimental conditions under which one can demonstrate chaotic population dynamics in a simple phytoplankton-zooplankton (-resting eggs) community in a microcosm with a short duration.