Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

2002 ◽  
Vol 64 (4) ◽  
pp. 747-769 ◽  
Author(s):  
E Crampin
Keyword(s):  
Pattern Formation ◽  
Reaction Diffusion ◽  
Diffusion Models ◽  
Domain Growth

scholarly journals A contraction-reaction-diffusion model for circular pattern formation in embryogenesis

2021 ◽  
Author(s):  
Tiankai Zhao ◽  
Yubing Sun ◽  
Xin Li ◽  
Mehdi Baghaee ◽  
Yuenan Wang ◽  
...  
Keyword(s):  
Pattern Formation ◽  
Cell Fate ◽  
Diffusion Model ◽  
Early Stage ◽  
Reaction Diffusion ◽  
Diffusion Models ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Model ◽  
Stress Dependent

Reaction-diffusion models have been widely used to elucidate pattern formation in developmental biology. More recently, they have also been applied in modeling cell fate patterning that mimic early-stage human development events utilizing geometrically confined pluripotent stem cells. However, the traditional reaction-diffusion equations could not satisfactorily explain the concentric ring distributions of various cell types, as they do not yield circular patterns even for circular domains. In previous mathematical models that yield ring patterns, certain conditions that lack biophysical understandings had been considered in the reaction-diffusion models. Here we hypothesize that the circular patterns are the results of the coupling of the mechanobiological factors with the traditional reaction-diffusion model. We propose two types of coupling scenarios: tissue tension-dependent diffusion flux and traction stress-dependent activation of signaling molecules. By coupling reaction-diffusion equations with the elasticity equations, we demonstrate computationally that the contraction-reaction-diffusion model can naturally yield the circular patterns.

scholarly journals Revealing new dynamical patterns in a reaction–diffusion model with cyclic competition via a novel computational framework

2018 ◽  
Vol 474 (2213) ◽  
pp. 20170608 ◽  
Author(s):  
A. Cangiani ◽  
E. H. Georgoulis ◽  
A. Yu. Morozov ◽  
O. J. Sutton
Keyword(s):  
Pattern Formation ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Diffusion Models ◽  
Adaptive Finite Element Method ◽  
Clear Understanding ◽  
Adaptive Finite Element ◽  
Reaction Diffusion Model ◽  
High Regularity ◽  
Dynamical Patterns

Understanding how patterns and travelling waves form in chemical and biological reaction–diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction–diffusion models, such as Lotka–Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction–diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.

scholarly journals Pattern formation in spatially heterogeneous Turing reaction–diffusion models

2003 ◽  
Vol 181 (1-2) ◽  
pp. 80-101 ◽  
Author(s):  
Karen Page ◽  
Philip K. Maini ◽  
Nicholas A.M. Monk
Keyword(s):  
Pattern Formation ◽  
Reaction Diffusion ◽  
Diffusion Models

scholarly journals Influence of survival, promotion, and growth on pattern formation in zebrafish skin

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Christopher Konow ◽  
Ziyao Li ◽  
Samantha Shepherd ◽  
Domenico Bullara ◽  
Irving R. Epstein
Keyword(s):  
Pattern Formation ◽  
Organizational Behavior ◽  
Long Range ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Survival Model ◽  
Domain Growth ◽  
Pigment Cells ◽  
Diffusion Scheme ◽  
Experimental Findings

AbstractThe coloring of zebrafish skin is often used as a model system to study biological pattern formation. However, the small number and lack of movement of chromatophores defies traditional Turing-type pattern generating mechanisms. Recent models invoke discrete short-range competition and long-range promotion between different pigment cells as an alternative to a reaction-diffusion scheme. In this work, we propose a lattice-based “Survival model,” which is inspired by recent experimental findings on the nature of long-range chromatophore interactions. The Survival model produces stationary patterns with diffuse stripes and undergoes a Turing instability. We also examine the effect that domain growth, ubiquitous in biological systems, has on the patterns in both the Survival model and an earlier “Promotion” model. In both cases, domain growth alone is capable of orienting Turing patterns above a threshold wavelength and can reorient the stripes in ablated cells, though the wavelength for which the patterns orient is much larger for the Survival model. While the Survival model is a simplified representation of the multifaceted interactions between pigment cells, it reveals complex organizational behavior and may help to guide future studies.

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