The Clock and Wavefront Self-Organizing model recreates the dynamics of mouse somitogenesis in vivo and in vitro.

During mouse development, presomitic mesoderm cells synchronize Wnt and Notch oscillations, creating sequential phase waves that pattern somites. Traditional somitogenesis models attribute phase waves to a global modulation of the oscillation frequency. However, increasing evidence suggests that they could arise in a self-organizing manner. Here, we introduce the Sevilletor, a novel reaction-diffusion system that serves as a framework to compare different somitogenesis patterning hypotheses. Using this framework, we propose the Clock and Wavefront Self-Organizing model that considers an excitable self-organizing region where phase waves form independent of global frequency gradients. The model recapitulates the change in relative phase of Wnt and Notch observed during mouse somitogenesis and provides a theoretical basis for understanding the excitability of mouse presomitic mesoderm cells in vitro.

Phase waves and trigger waves: emergent properties of oscillating and excitable networks in the gut.

The gut is enmeshed by a number of cellular networks, but there is only a limited understanding of how these networks generate the complex patterns of activity that drive gut contractile functions. Here we review two fundamental types of cell behaviour, excitable and oscillating, and the patterns that networks of such cells generate, trigger waves and phase waves, respectively. We use both the language of biophysics and the theory of nonlinear dynamics to define these behaviours and understand how they generate patterns. Based on this we look for evidence of trigger and phase waves in the gut, including some of our recent work on the small intestine.

Competitive aggregation dynamics using phase wave signals.

Coupled equations of the phase equation and the equation of cell concentration n are proposed for competitive aggregation dynamics of slime mold in two dimensions. Phase waves are used as tactic signals of aggregation in this model. Several aggregation clusters are formed initially, and target patterns appear around the localized aggregation clusters. Owing to the competition among target patterns, the number of the localized aggregation clusters decreases, and finally one dominant localized pattern survives. If the phase equation is replaced with the complex Ginzburg-Landau equation, several spiral patterns appear, and n is localized near the center of the spiral patterns. After the competition among spiral patterns, one dominant spiral survives.

Phase Gradients and Anisotropy of the Suprachiasmatic Network: Discovery of Phaseoids.

Biological neural networks operate at several levels of granularity, from the individual neuron to local neural circuits to networks of thousands of cells. The daily oscillation of the brain's master clock in the suprachiasmatic nucleus (SCN) rests on a yet to be identified network of connectivity among its ∼20,000 neurons. The SCN provides an accessible model to explore neural organization at several levels of organization. To relate cellular to local and global network behaviors, we explore network topology by examining SCN slices in three orientations using immunochemistry, light and confocal microscopy, real-time imaging, and mathematical modeling. Importantly, the results reveal small local groupings of neurons that form intermediate structures, here termed "phaseoids," which can be identified through stable local phase differences of varying magnitude among neighboring cells. These local differences in phase are distinct from the global phase relationship, namely that between individual cells and the mean oscillation of the overall SCN. The magnitude of the phaseoids' local phase differences is associated with a global phase gradient observed in the SCN's rostral-caudal extent. Modeling results show that a gradient in connectivity strength can explain the observed gradient of phaseoid strength, an extremely parsimonious explanation for the heterogeneous oscillatory structure of the SCN.

Constraints on somite formation in developing embryos.

Segment formation in vertebrate embryos is a stunning example of biological self-organization. Here, we present an idealized framework, in which we treat the presomitic mesoderm (PSM) as a one-dimensional line of oscillators. We use the framework to derive constraints that connect the size of somites, and the timing of their formation, to the growth of the PSM and the gradient of the somitogenesis clock period across the PSM. Our analysis recapitulates the observations made recently in ex vivo cultures of mouse PSM cells, and makes predictions for how perturbations, such as increased Wnt levels, would alter somite widths. Finally, our analysis makes testable predictions for the shape of the phase profile and somite widths at different stages of PSM growth. In particular, we show that the phase profile is robustly concave when the PSM length is steady and slightly convex in an important special case when it is decreasing exponentially. In both cases, the phase profile scales with the PSM length; in the latter case, it scales dynamically. This has important consequences for the velocity of the waves that traverse the PSM and trigger somite formation, as well as the effect of errors in phase measurement on somite widths.