Turing Patterning in Stratified Domains
- PDF / 1,813,606 Bytes
- 37 Pages / 439.37 x 666.142 pts Page_size
- 102 Downloads / 200 Views
Turing Patterning in Stratified Domains Andrew L. Krause1 · Václav Klika2 · Jacob Halatek3 · Paul K. Grant3 Thomas E. Woolley4 · Neil Dalchau3 · Eamonn A. Gaffney1
·
Received: 11 June 2020 / Accepted: 18 September 2020 © The Author(s) 2020
Abstract Reaction–diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal–mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction–diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction–diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of patternforming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role
B
Andrew L. Krause [email protected]
1
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
2
Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00 Prague, Czech Republic
3
Microsoft Research, 21 Station Rd, Cambridge CB1 2FB, UK
4
Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK 0123456789().: V,-vol
123
136
Page 2 of 37
A. L. Krause et al.
of stratified media in modulating pattern-forming processes in developmental biology and beyond. Keywords Turing instabilities · Stratified media · Pattern formation · Synthetic biology
1 Introduction Since Turing’s initial insights into reaction–diffusion-driven morphogenesis (Turing 1952), a substantial research effort has elucidated various mathematical and biophysical aspects of such symmetry-breaking instabilities leading from homogeneity to patterned s
Data Loading...
