scholarly journals On the Interplay of Geometrical Shapes and the Analysis of a Dispersal Model for Pattern Formations

2019 ◽  
pp. 1-10
Author(s):  
Zakir Hossine ◽  
Md. Kamrujjaman
Keyword(s):  
Reaction Mechanism ◽  
Finite Difference Method ◽  
Reaction Diffusion ◽  
Diffusion Kinetics ◽  
Dispersal Model ◽  
Reaction Diffusion Model ◽  
Chemical Reaction Mechanism ◽  
Geometrical Shapes ◽  
Pattern Formations ◽  
Selection Of

A reaction-diffusion model is put forward which is capable of generating chemical maps whose concentration contours are similar to the patterns seen on the flanks of zebras, leopards and other mammals. Initially, this type of reaction diffusion kinetics model was introduced by Turing and later Murray applied it to animal coat patterns. Among many chemical reaction mechanism, we consider Schnackenberg reaction mechanism in details and show that the geometry and scale of the domain, the relevant part of the integument, during the time of laying down plays a crucial role in the structural patterns which result. Patterns which exhibit a limited randomness are obtained for a selection of geometries. Finally the system was solved numerically using finite difference method.

scholarly journals Correspondences between parameters in a reaction-diffusion model and connexin functions during zebrafish stripe formation

2021 ◽  
Author(s):  
Akiko Nakamasu
Keyword(s):  
Molecular Mechanisms ◽  
Reaction Diffusion ◽  
The Body ◽  
Pigment Cells ◽  
Turing Pattern ◽  
Reaction Diffusion Model ◽  
Spatial Periodicity ◽  
Pattern Formations ◽  
Mathematical Analyses ◽  
Spotted Pattern

Abstract Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifested as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models have demonstrated that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish revealed apparent interactions among pigment cells, which allowed for a three-components RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system had been remained unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formations. Here, molecular-level evidence derived from connexin analyses was linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells were generalized as “gates,” and the effects of respective gates were deduced. The model used partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model included nonlinear reaction terms, enabled the consideration of the behavior of components.

scholarly journals Mode selection mechanism in traveling and standing waves revealed by Min wave reconstituted in artificial cells

2022 ◽  
Author(s):  
Sakura Takada ◽  
Natsuhiko Yoshinaga ◽  
Nobuhide Doi ◽  
Kei Fujiwara
Keyword(s):  
Mode Selection ◽  
Standing Waves ◽  
Reaction Diffusion ◽  
Wave Mode ◽  
Spatiotemporal Pattern ◽  
Selection Mechanism ◽  
Artificial Cells ◽  
Pattern Formations ◽  
Selection Of ◽  
Wave Modes

Reaction-diffusion coupling (RDc) generates spatiotemporal patterns, including two dynamic wave modes: traveling and standing waves. Although mode selection plays a significant role in the spatiotemporal organization of living cell molecules, the mechanism for selecting each wave mode remains elusive. Here, we investigated a wave mode selection mechanism using Min waves reconstituted in artificial cells, emerged by the RDc of MinD and MinE. Our experiments and theoretical analysis revealed that the balance of membrane binding and dissociation from the membrane of MinD determines the mode selection of the Min wave. We successfully demonstrated that the transition of the wave modes can be regulated by controlling this balance and found hysteresis characteristics in the wave mode transition. These findings highlight a novel role of the balance between activators and inhibitors as a determinant of the mode selection of waves by RDc and depict a novel mechanism in intracellular spatiotemporal pattern formations.

scholarly journals Geometric cues stabilise long-axis polarisation of PAR protein patterns in C. elegans

2018 ◽  
Author(s):  
Raphaela Geßele ◽  
Jacob Halatek ◽  
Laeschkir Würthner ◽  
Erwin Frey
Keyword(s):  
Membrane Surface ◽  
Reaction Diffusion ◽  
Protein Patterns ◽  
C Elegans ◽  
Reaction Diffusion Model ◽  
Local Ratio ◽  
Diffusive Fluxes ◽  
Anterior Posterior ◽  
Selection Of ◽  
Anterior Posterior Axis

AbstractIn the Caenorhabditis elegans zygote, PAR protein patterns, driven by mutual anatagonism, determine the anterior-posterior axis and facilitate the redistribution of proteins for the first cell division. Yet, the factors that determine the selection of the polarity axis remain unclear. We present a reaction-diffusion model in realistic cell geometry, based on biomolecular reactions and accounting for the coupling between membrane and cytosolic dynamics. We find that the kinetics of the phosphorylation-dephosphorylation cycle of PARs and the diffusive protein fluxes from the cytosol towards the membrane are crucial for the robust selection of the anterior-posterior axis for polarisation. The local ratio of membrane surface to cytosolic volume is the main geometric cue that initiates pattern formation, while the choice of the long-axis for polarisation is largely determined by the length of the aPAR-pPAR interface, and mediated by processes that minimise the diffusive fluxes of PAR proteins between cytosol and membrane.

scholarly journals Correspondences between parameters in a reaction-diffusion model and connexin functions during zebrafish stripe formation

2021 ◽  
Author(s):  
Akiko Nakamasu
Keyword(s):  
Molecular Mechanisms ◽  
Reaction Diffusion ◽  
The Body ◽  
Pigment Cells ◽  
Turing Pattern ◽  
Reaction Diffusion Model ◽  
Spatial Periodicity ◽  
Pattern Formations ◽  
Mathematical Analyses ◽  
Spotted Pattern

Abstract Different diffusivities among interacting substances actualize the potential instability of a system. When these elicited instabilities manifested as forms of spatial periodicity, they are called Turing patterns. Simulations using general reaction-diffusion (RD) models have demonstrated that pigment patterns on the body trunk of growing fish follow a Turing pattern. Laser ablation experiments performed on zebrafish revealed apparent interactions among pigment cells, which allowed for a three-components RD model to be derived. However, the underlying molecular mechanisms responsible for Turing pattern formation in this system had been remained unknown. A zebrafish mutant with a spotted pattern was found to have a defect in Connexin41.8 (Cx41.8) which, together with Cx39.4, exists in pigment cells and controls pattern formations. Here, molecular-level evidence derived from connexin analyses was linked to the interactions among pigment cells described in previous RD modeling. Channels on pigment cells were generalized as “gates,” and the effects of respective gates were deduced. The model used partial differential equations (PDEs) to enable numerical and mathematical analyses of characteristics observed in the experiments. Furthermore, the improved PDE model included nonlinear reaction terms, enabled the consideration of the behavior of components.

scholarly journals A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations

2020 ◽  
Vol 19 ◽  
pp. 103462 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Imtiaz Ahmad ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu
Keyword(s):  
Diffusion Model ◽  
Reaction Diffusion ◽  
Reaction Diffusion Model ◽  
Model Equations
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