Effects of multistability, absorbing boundaries and growth on Turing pattern formation

Abstract

Turing patterns are a fundamental concept in developmental biology, describing how homogeneous tissues develop into self-organized spatial patterns. However, the classical Turing mechanism, which relies on linear stability analysis, often fails to capture the complexities of real biological systems, such as multistability, non-linearities, growth, and boundary conditions. Here, we explore the impact of these factors on Turing pattern formation, contrasting linear stability analysis with numerical simulations based on a simple reaction-diffusion model, motivated by synthetic gene-regulatory pathways. We demonstrate how non-linearities introduce multistability, leading to unexpected pattern outcomes not predicted by the traditional Turing theory. The study also examines how growth and realistic boundary conditions influence pattern robustness, revealing that different growth regimes and boundary conditions can disrupt or stabilize pattern formation. Our findings are critical for understanding pattern formation in both natural and synthetic biological systems, providing insights into engineering robust patterns for applications in synthetic biology. Author summaryDuring development, tissues self-organize to go from a single cell to a structured organism. In this process, simple chemical reactions lead to the emergence of the intricate designs we see in nature, like the stripes on a zebra or the labyrinths on a brain cortex. Although multiple theories have been proposed to model this phenomenon, one of the most simple and popular ones was introduced in the 1950s by the mathematician Alan Turing. However, his theory oversimplifies the biological conditions and ignores properties such as non-linearities, boundary effects, or growth in the tissue. In this work, we used a combination of mathematical models and computer simulations to investigate how these real-world factors influence pattern formation. Our findings show that when we account for these realistic effects, the patterns that emerge can be very different from what Turings theory would predict. Thus, this work may help us better understand the laws behind pattern formation and could have practical applications in tissue engineering for medical or environmental applications.