Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics

Experimental studies of protein-pattern formation have stimulated new interest in the dynamics of reaction-diffusion systems. However, a comprehensive theoretical understanding of the dynamics of such highly nonlinear, spatially extended systems is still missing. Here, we show how a description in p...

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Autores principales:	Fridtjof Brauns, Jacob Halatek, Erwin Frey
Formato:	article
Lenguaje:	EN
Publicado:	American Physical Society 2020
Materias:	Physics QC1-999
Acceso en línea:	https://doaj.org/article/a601dd2cf24d4842938bd95b44dc428e
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spelling	oai:doaj.org-article:a601dd2cf24d4842938bd95b44dc428e2021-12-02T12:57:26ZPhase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics10.1103/PhysRevX.10.0410362160-3308https://doaj.org/article/a601dd2cf24d4842938bd95b44dc428e2020-11-01T00:00:00Zhttp://doi.org/10.1103/PhysRevX.10.041036http://doi.org/10.1103/PhysRevX.10.041036https://doaj.org/toc/2160-3308Experimental studies of protein-pattern formation have stimulated new interest in the dynamics of reaction-diffusion systems. However, a comprehensive theoretical understanding of the dynamics of such highly nonlinear, spatially extended systems is still missing. Here, we show how a description in phase space, which has proven invaluable in shaping our intuition about the dynamics of nonlinear ordinary differential equations, can be generalized to mass-conserving reaction-diffusion (MCRD) systems. We present a comprehensive analysis of two-component MCRD systems, which serve as paradigmatic minimal systems that encapsulate the core principles and concepts of the local equilibria theory introduced in the paper. The key insight underlying this theory is that shifting local (reactive) equilibria—controlled by the local total density—give rise to concentration gradients that drive diffusive redistribution of total density. We show how this dynamic interplay can be embedded in the phase plane of the reaction kinetics in terms of simple geometric objects: the reactive nullcline (line of reactive equilibria) and the diffusive flux-balance subspace. On this phase-space level, physical insight can be gained from geometric criteria and graphical constructions. The effects of nonlinearities on the global dynamics are simply encoded in the curved shape of the reactive nullcline. In particular, we show that the pattern-forming “Turing instability” in MCRD systems is a mass-redistribution instability and that the features and bifurcations of patterns can be characterized based on regional dispersion relations, associated to distinct spatial regions (plateaus and interfaces) of the patterns. In an extensive outlook section, we detail concrete approaches to generalize local equilibria theory in several directions, including systems with more than two components, weakly broken mass conservation, and active matter systems.Fridtjof BraunsJacob HalatekErwin FreyAmerican Physical SocietyarticlePhysicsQC1-999ENPhysical Review X, Vol 10, Iss 4, p 041036 (2020)
institution	DOAJ
collection	DOAJ
language	EN
topic	Physics QC1-999
spellingShingle	Physics QC1-999 Fridtjof Brauns Jacob Halatek Erwin Frey Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics
description	Experimental studies of protein-pattern formation have stimulated new interest in the dynamics of reaction-diffusion systems. However, a comprehensive theoretical understanding of the dynamics of such highly nonlinear, spatially extended systems is still missing. Here, we show how a description in phase space, which has proven invaluable in shaping our intuition about the dynamics of nonlinear ordinary differential equations, can be generalized to mass-conserving reaction-diffusion (MCRD) systems. We present a comprehensive analysis of two-component MCRD systems, which serve as paradigmatic minimal systems that encapsulate the core principles and concepts of the local equilibria theory introduced in the paper. The key insight underlying this theory is that shifting local (reactive) equilibria—controlled by the local total density—give rise to concentration gradients that drive diffusive redistribution of total density. We show how this dynamic interplay can be embedded in the phase plane of the reaction kinetics in terms of simple geometric objects: the reactive nullcline (line of reactive equilibria) and the diffusive flux-balance subspace. On this phase-space level, physical insight can be gained from geometric criteria and graphical constructions. The effects of nonlinearities on the global dynamics are simply encoded in the curved shape of the reactive nullcline. In particular, we show that the pattern-forming “Turing instability” in MCRD systems is a mass-redistribution instability and that the features and bifurcations of patterns can be characterized based on regional dispersion relations, associated to distinct spatial regions (plateaus and interfaces) of the patterns. In an extensive outlook section, we detail concrete approaches to generalize local equilibria theory in several directions, including systems with more than two components, weakly broken mass conservation, and active matter systems.
format	article
author	Fridtjof Brauns Jacob Halatek Erwin Frey
author_facet	Fridtjof Brauns Jacob Halatek Erwin Frey
author_sort	Fridtjof Brauns
title	Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics
title_short	Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics
title_full	Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics
title_fullStr	Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics
title_full_unstemmed	Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics
title_sort	phase-space geometry of mass-conserving reaction-diffusion dynamics
publisher	American Physical Society
publishDate	2020
url	https://doaj.org/article/a601dd2cf24d4842938bd95b44dc428e
work_keys_str_mv	AT fridtjofbrauns phasespacegeometryofmassconservingreactiondiffusiondynamics AT jacobhalatek phasespacegeometryofmassconservingreactiondiffusiondynamics AT erwinfrey phasespacegeometryofmassconservingreactiondiffusiondynamics
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Phase-Space Geometry of Mass-Conserving Reaction-Diffusion Dynamics

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