Pattern Formation and Oscillations in Reaction–Diffusion Model with p53-Mdm2 Feedback Loop

2019 ◽  
Vol 29 (14) ◽  
pp. 1930040 ◽  
Author(s):  
Qianqian Zheng ◽  
Jianwei Shen ◽  
Zhijie Wang
Keyword(s):  
Pattern Formation ◽  
Feedback Loop ◽  
Reaction Diffusion ◽  
Vital Role ◽  
Reaction Diffusion Equation ◽  
Biological Mechanism ◽  
Reaction Diffusion Model ◽  
The Impact ◽  
Theoretical Results ◽  
Diffusive System

P53 plays a vital role in DNA repair, and several mathematical models of the p53-Mdm2 feedback loop were used to explain the biological mechanism. In this paper, a p53-Mdm2 model described by a delay reaction–diffusion equation is studied both analytically and numerically. This research aims to provide an understanding of the impact of delay and sustained pressure on the p53-Mdm2 dynamics and tries to explain some biological mechanism. It is found that the type of pattern formation is affected by Hopf bifurcation. Also, the amplitude equation in delay diffusive system is derived and it is shown that sustained stress plays an essential role in the function of p53. Finally, simulation is used to verify the theoretical results.

scholarly journals Pattern formation on a growing oblate spheroid. an application to adult sea urchin development

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Deborah Lacitignola ◽  
Massimo Frittelli ◽  
Valerio Cusimano ◽  
Andrea De Gaetano
Keyword(s):  
Sea Urchin ◽  
Adult Stage ◽  
Reaction Diffusion ◽  
Oblate Spheroid ◽  
Gene Control ◽  
Control Networks ◽  
Model Driven ◽  
Reaction Diffusion Model ◽  
The Impact ◽  
Theoretical Results

<p style='text-indent:20px;'>In this study, the formation of the adult sea urchin shape is rationalized within the Turing's theory paradigm. The emergence of protrusions from the expanding underlying surface is described through a reaction-diffusion model with Gray-Scott kinetics on a growing oblate spheroid. The case of slow exponential isotropic growth is considered. The model is first studied in terms of the spatially homogenous equilibria and of the bifurcations involved. Turing diffusion-driven instability is shown to occur and the impact of the slow exponential growth on the resulting Turing regions adequately discussed. Numerical investigations validate the theoretical results showing that the combination between an inhibitor and an activator can result in a distribution of spot concentrations that underlies the development of ambulacral tentacles in the sea urchin's adult stage. Our findings pave the way for a model-driven experimentation that could improve the current biological understanding of the gene control networks involved in patterning.</p>

Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system

2020 ◽  
Vol 21 (7-8) ◽  
pp. 749-759
Author(s):  
Rachida Mezhoud ◽  
Khaled Saoudi ◽  
Abderrahmane Zaraï ◽  
Salem Abdelmalek
Keyword(s):  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Linearization Method ◽  
Direct Lyapunov Method ◽  
Fractional Systems ◽  
Reaction Diffusion Model ◽  
Theoretical Results

AbstractFractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the time–fractional version of the Degn–Harrison reaction–diffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.

scholarly journals The influence of density in population dynamics with strong and weak Allee effect

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kamrun Nahar Keya ◽  
Md. Kamrujjaman ◽  
Md. Shafiqul Islam
Keyword(s):  
Population Dynamics ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Logistic Growth ◽  
Allee Effects ◽  
Reaction Diffusion Model ◽  
Positive Steady States ◽  
The Stability ◽  
The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

scholarly journals An in vitro–in silico interface platform for spatiotemporal analysis of pattern formation in collective epithelial cells

2016 ◽  
Vol 8 (8) ◽  
pp. 861-868 ◽  
Author(s):  
M. Hagiwara
Keyword(s):  
Cell Culture ◽  
Pattern Formation ◽  
Epithelial Cells ◽  
In Silico ◽  
Reaction Diffusion ◽  
Spatiotemporal Analysis ◽  
Bronchial Epithelial ◽  
Reaction Diffusion Model ◽  
2D Pattern

The mechanisms of 2D pattern formation in bronchial epithelial cells were dynamically analyzed by controlled cell culture and a reaction-diffusion model.

scholarly journals HIERARCHICAL MODELS FOR SPATIAL PATTERN FORMATION IN BIOLOGY

1995 ◽  
Vol 03 (04) ◽  
pp. 987-997 ◽  
Author(s):  
P. K. MAINI
Keyword(s):  
Pattern Formation ◽  
Spatial Heterogeneity ◽  
Spatial Pattern ◽  
Diffusion Model ◽  
Hierarchical Models ◽  
Reaction Diffusion ◽  
Skeletal Patterning ◽  
Reaction Diffusion Model ◽  
Spatial Pattern Formation ◽  
Set Up

We review some recent work investigating a hierarchy of patterning processes in which a reaction-diffusion model forms the top level. In one such hierarchy, it is assumed that the boundary is differentiated, and it is shown that this can greatly enhance the robustness of the patterns subsequently formed by the reaction-diffusion model. In the second, a spatial heterogeneity in background environment is first set-up by a simple gradient model. The resulting patterns produced by the reaction-diffusion system may be isolated to specific parts of the domain. The application of such hierarchical models to skeletal patterning in the tetrapod limb is considered.

Receptor-Based Models with Diffusion-Driven Instability for Pattern Formation in Hydra

2003 ◽  
Vol 11 (03) ◽  
pp. 293-324 ◽  
Author(s):  
Anna Marciniak-Czochra
Keyword(s):  
Pattern Formation ◽  
Feedback Loop ◽  
De Novo ◽  
Initial Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Variable Model ◽  
Dissociated Cells ◽  
Cutting Experiments ◽  
Positional Value

The aim of this paper is to show under which conditions a receptor-based model can produce and regulate patterns. Such model is applied to the pattern formation and regulation in a fresh water polyp, hydra. The model is based on the idea that both head and foot formation could be controlled by receptor-ligand binding. Positional value is determined by the density of bound receptors. The model is defined in the form of reaction-diffusion equations coupled with ordinary differential equations. The objective is to check what minimal processes are sufficient to produce patterns in the framework of a diffusion-driven (Turing-type) instability. Three-variable (describing the dynamics of ligands, free and bound receptors) and four-variable models (including also an enzyme cleaving the ligand) are analyzed and compared. The minimal three-variable model takes into consideration the density of free receptors, bound receptors and ligands. In such model patterns can evolve only if self-enhancement of free receptors, i.e., a positive feedback loop between the production of new free receptors and their present density, is assumed. The final pattern strongly depends on initial conditions. In the four-variable model a diffusion-driven instability occurs without the assumption that free receptors stimulate their own synthesis. It is shown that gradient in the density of bound receptors occurs if there is also a second diffusible substance, an enzyme, which degrades ligands. Numerical simulations are done to illustrate the analysis. The four-variable model is able to capture some results from cutting experiments and reflects de novo pattern formation from dissociated cells.

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