scholarly journals Turing Patterning in Stratified Domains

2020 ◽  
Vol 82 (10) ◽  
Author(s):  
Andrew L. Krause ◽  
Václav Klika ◽  
Jacob Halatek ◽  
Paul K. Grant ◽  
Thomas E. Woolley ◽  
...  
Keyword(s):  
Diffusion Processes ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Cell Polarization ◽  
The Other ◽  
Modeling Framework ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Coupling Parameters ◽  
Pattern Forming

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 pattern-forming 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 of stratified media in modulating pattern-forming processes in developmental biology and beyond.

CHEMICAL PATTERNS IN SIMPLE FLOW SYSTEMS

2003 ◽  
Vol 06 (01) ◽  
pp. 155-162 ◽  
Author(s):  
ANNETTE TAYLOR
Keyword(s):  
Chemical Reaction ◽  
Reaction Diffusion ◽  
Stationary Concentration ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Differential Flow ◽  
Chemical Waves ◽  
Pattern Forming ◽  
Simple Flow ◽  
Concentration Patterns

The addition of flow to chemical reaction-diffusion systems provides robust pattern-forming mechanisms which are expected to occur in a wide variety of natural and artificial systems. Experiments demonstrating some of these mechanisms are presented here, including the differential-flow-induced chemical instability (DIFICI), which gives rise to traveling chemical waves, and flow-distributed oscillations (FDO), which produce stationary concentration patterns.

scholarly journals Multi-spike solutions of a hybrid reaction–transport model

2021 ◽  
Vol 477 (2247) ◽  
pp. 20200829
Author(s):  
P. C. Bressloff
Keyword(s):  
Transport Model ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Strongly Nonlinear ◽  
Diffusion Systems ◽  
Spike Solution ◽  
Classical Pattern ◽  
Final Steady State ◽  
Reaction Transport ◽  
Pattern Forming

Simulations of classical pattern-forming reaction–diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady state consisting of a spatially repeating pattern of localized spikes. In activator–inhibitor systems such as the two-component Gierer–Meinhardt (GM) model, one can consider the singular limit D a  ≪  D h , where D a and D h are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse multi-spike solutions in a hybrid reaction–transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate α between right-moving and left-moving velocity states. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in Caenorhabditis elegans . We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit α  → ∞, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for α  → ∞ with a graphical construction for finite α .

scholarly journals The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guido Schneider ◽  
Matthias Winter
Keyword(s):  
Numerical Simulation ◽  
Analytical Approach ◽  
Reaction Diffusion ◽  
Short Wave ◽  
Reaction Diffusion Systems ◽  
Long Wave ◽  
Diffusion Systems ◽  
Schnakenberg Model ◽  
Pattern Forming ◽  
Amplitude System

<p style='text-indent:20px;'>We consider reaction-diffusion systems for which the trivial solution simultaneously becomes unstable via a short-wave Turing and a long-wave Hopf instability. The Brusseletor, Gierer-Meinhardt system and Schnakenberg model are prototype biological pattern forming systems which show this kind of behavior for certain parameter regimes. In this paper we prove the validity of the amplitude system associated to this kind of instability. Our analytical approach is based on the use of mode filters and normal form transformations. The amplitude system allows us an efficient numerical simulation of the original multiple scaling problems close to the instability.</p>

scholarly journals Pattern formation from spatially heterogeneous reaction–diffusion systems

2021 ◽  
Vol 379 (2213) ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Heterogeneous Systems ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Instability Criterion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatially Homogeneous ◽  
Turing Systems

First proposed by Turing in 1952, the eponymous Turing instability and Turing pattern remain key tools for the modern study of diffusion-driven pattern formation. In spatially homogeneous Turing systems, one or a few linear Turing modes dominate, resulting in organized patterns (peaks in one dimension; spots, stripes, labyrinths in two dimensions) which repeats in space. For a variety of reasons, there has been increasing interest in understanding irregular patterns, with spatial heterogeneity in the underlying reaction–diffusion system identified as one route to obtaining irregular patterns. We study pattern formation from reaction–diffusion systems which involve spatial heterogeneity, by way of both analytical and numerical techniques. We first extend the classical Turing instability analysis to track the evolution of linear Turing modes and the nascent pattern, resulting in a more general instability criterion which can be applied to spatially heterogeneous systems. We also calculate nonlinear mode coefficients, employing these to understand how each spatial mode influences the long-time evolution of a pattern. Unlike for the standard spatially homogeneous Turing systems, spatially heterogeneous systems may involve many Turing modes of different wavelengths interacting simultaneously, with resulting patterns exhibiting a high degree of variation over space. We provide a number of examples of spatial heterogeneity in reaction–diffusion systems, both mathematical (space-varying diffusion parameters and reaction kinetics, mixed boundary conditions, space-varying base states) and physical (curved anisotropic domains, apical growth of space domains, chemicalsimmersed within a flow or a thermal gradient), providing a qualitative understanding of how spatial heterogeneity can be used to modify classical Turing patterns. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

scholarly journals pSpatiocyte: a high-performance simulator for intracellular reaction-diffusion systems

2019 ◽  
Author(s):  
Satya N. V. Arjunan ◽  
Atsushi Miyauchi ◽  
Kazunari Iwamoto ◽  
Koichi Takahashi
Keyword(s):  
High Performance ◽  
Large Scale ◽  
Diffusion Processes ◽  
Direct Method ◽  
Network Motif ◽  
Computational Cost ◽  
Reaction Diffusion ◽  
Cellular Systems ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

ABSTRACTBackgroundStudies using quantitative experimental methods have shown that intracellular spatial distribution of molecules plays a central role in many cellular systems. Spatially resolved computer simulations can integrate quantitative data from these experiments to construct physically accurate models of the systems. Although computationally expensive, microscopic resolution reaction-diffusion simulators, such as Spatiocyte can directly capture intracellular effects comprising diffusion-limited reactions and volume exclusion from crowded molecules by explicitly representing individual diffusing molecules in space. To alleviate the steep computational cost typically associated with the simulation of large or crowded intracellular compartments, we present a parallelized Spatiocyte method called pSpatiocyte.ResultsThe new high-performance method employs unique parallelization schemes on hexagonal close-packed (HCP) lattice to efficiently exploit the resources of common workstations and large distributed memory parallel computers. We introduce a coordinate system for fast accesses to HCP lattice voxels, a parallelized event scheduler, a parallelized Gillespie’s direct-method for unimolecular reactions, and a parallelized event for diffusion and bimolecular reaction processes. We verified the correctness of pSpatiocyte reaction and diffusion processes by comparison to theory. To evaluate the performance of pSpatiocyte, we performed a series of parallelized diffusion runs on the RIKEN K computer. In the case of fine lattice discretization with low voxel occupancy, pSpatiocyte exhibited 74% parallel efficiency and achieved a speedup of 7686 times with 663552 cores compared to the runtime with 64 cores. In the weak scaling performance, pSpatiocyte obtained efficiencies of at least 60% with up to 663552 cores. When executing the Michaelis-Menten benchmark model on an eight-core workstation, pSpatiocyte required 45- and 55-fold shorter runtimes than Smoldyn and the parallel version of ReaDDy, respectively. As a high-performance application example, we study the dual phosphorylation-dephosphorylation cycle of the MAPK system, a typical reaction network motif in cell signaling pathways.ConclusionspSpatiocyte demonstrates good accuracies, fast runtimes and a significant performance advantage over well-known microscopic particle simulators for large-scale simulations of intracellular reaction-diffusion systems. The source code of pSpatiocyte is available at https://spatiocyte.org.

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

2020 ◽  
Vol 17 (162) ◽  
pp. 20190621 ◽  
Author(s):  
Andrew L. Krause ◽  
Václav Klika ◽  
Thomas E. Woolley ◽  
Eamonn A. Gaffney
Keyword(s):  
Symmetry Breaking ◽  
Heterogeneous Media ◽  
Heterogeneous Reaction ◽  
Reaction Diffusion ◽  
Spatial Patterning ◽  
Instability Mechanism ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Turing Instabilities ◽  
Pattern Forming

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.

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