A non-linear analysis of Turing pattern formation

Stationary periodic patterns in a model of growing population of motile bacteria

Biological pattern formation is one of the most intriguing phenomena in nature. Simple examples include travelling waves and stationary periodic patterns, which arise in various biological processes, including morphogenesis and population dynamics. The emergence of such patterns in populations of motile microorganisms, such as Dictyostelium discoideum and E. coli, has been demonstrated in numerous experimental studies. The conditions required for different types of pattern formation are commonly explored in mathematical studies of dynamical systems incorporating diffusion and advection terms. In this work, we introduce a prototype model for a growing population of motile bacteria that move in response to a chemical signal. We conduct a linear analysis of this model to determine the conditions for the formation of stationary periodic patterns, followed by a nonlinear (Fourier) analysis to characterise their properties, such as amplitude and wavelength. Our analytical findings are further validated through numerical simulations.

Self-sustained planar intercalations due to mechanosignaling feedbacks lead to robust axis extension during morphogenesis

Tissue elongation is a necessary process in metazoans to implement their body plans that is not fully understood. Here we propose a mechanism based on the interplay between cellular mechanics and primordia patterning that results in self-sustained planar intercalations. Thus, we show that a location-dependent modulation of the mechanical properties of cells leads to robust axis extension. To illustrate the plausibility of this mechanism, we test it against different patterning models by means of computer simulations of tissues where we implemented mechano-signaling feedbacks. Our results suggest that robust elongation relies on a trade-off between cellular and tissue strains that is orchestrated through the cleavage orientation. In the particular context of axis extension in Turing-patterned tissues, we report that different directional cell activities cooperate synergetically to achieve elongation. Altogether, our findings help to understand how the axis extension phenomenon emerges from the dynamics of individual cells.

From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ

Pattern formation from homogeneity is well studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is non-trivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction-diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions. We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction-diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing's original thesis to a far wider and more realistic class of systems.