scholarly journals A new mechanism for spatial pattern formation via lateral and protrusion-mediated lateral signalling

2016 ◽  
Vol 13 (124) ◽  
pp. 20160484 ◽  
Author(s):  
Zena Hadjivasiliou ◽  
Ginger L. Hunter ◽  
Buzz Baum
Keyword(s):  
Reaction Diffusion ◽  
Cell Contact ◽  
Receptor Ligand ◽  
Reaction Diffusion Systems ◽  
Single Receptor ◽  
Self Organized ◽  
Tissue Organization ◽  
Diffusion Systems ◽  
Different Types ◽  
Spatial Pattern Formation

Tissue organization and patterning are critical during development when genetically identical cells take on different fates. Lateral signalling plays an important role in this process by helping to generate self-organized spatial patterns in an otherwise uniform collection of cells. Recent data suggest that lateral signalling can be mediated both by junctional contacts between neighbouring cells and via cellular protrusions that allow non-neighbouring cells to interact with one another at a distance. However, it remains unclear precisely how signalling mediated by these distinct types of cell–cell contact can physically contribute to the generation of complex patterns without the assistance of diffusible morphogens or pre-patterns. To explore this question, in this work we develop a model of lateral signalling based on a single receptor/ligand pair as exemplified by Notch and Delta. We show that allowing the signalling kinetics to differ at junctional versus protrusion-mediated contacts, an assumption inspired by recent data which show that the cleavage of Notch in several systems requires both Delta binding and the application of mechanical force, permits individual cells to act to promote both lateral activation and lateral inhibition. Strikingly, under this model, in which Delta can sequester Notch, a variety of patterns resembling those typical of reaction–diffusion systems is observed, together with more unusual patterns that arise when we consider changes in signalling kinetics, and in the length and distribution of protrusions. Importantly, these patterns are self-organizing—so that local interactions drive tissue-scale patterning. Together, these data show that protrusions can, in principle, generate different types of patterns in addition to contributing to long-range signalling and to pattern refinement.

scholarly journals Collision patterns on mollusc shells

1997 ◽  
Vol 1 (1) ◽  
pp. 57-76 ◽  
Author(s):  
P. J. Plath ◽  
J. K. Plath ◽  
J. Schwietering
Keyword(s):  
Reaction Diffusion ◽  
Growth Front ◽  
Cell System ◽  
One Dimensional ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Different Types ◽  
Classical Waves ◽  
Cellular Behaviour ◽  
Mollusc Shells

On mollusc shells one can find famous patterns. Some of them show a great resemblance to the soliton patterns in one-dimensional systems. Other look like Sierpinsky triangles or exhibit very irregular patterns. Meinhardt has shown that those patterns can be well described by reaction–diffusion systems [1]. However, such a description neglects the discrete character of the cell system at the growth front of the mollusc shell.We have therefore developed a one-dimensional cellular vector automaton model which takes into account the cellular behaviour of the system [2]. The state of the mathematical cell is defined by a vector with two components. We looked for the most simple transformation rules in order to develop quite different types of waves: classical waves, chemical waves and different types of solitons. Our attention was focussed on the properties of the system created through the collision of two waves.

scholarly journals Control of traveling localized spots

2021 ◽  
Author(s):  
Steffen Martens ◽  
Christopher Ryll ◽  
Jakob Löber ◽  
Fredi Tröltzsch ◽  
Harald Engel
Keyword(s):  
Reaction Diffusion ◽  
Open Loop ◽  
Open Loop Control ◽  
Loop Control ◽  
Reaction Diffusion Systems ◽  
Self Organized ◽  
Nonlinear Reaction ◽  
Diffusion Systems ◽  
Goldstone Modes ◽  
Analytical Expressions

Traveling localized spots represent an important class of self-organized two- dimensional patterns in reaction-diffusion systems. We study open-loop control intended to guide a stable spot along a desired trajectory with desired velocity. Simultaneously, the spot's concentration profile does not change under control. For a given protocol of motion, we first express the control signal analytically in terms of the Goldstone modes and the propagation velocity of the uncontrolled spot. Thus, detailed information about the underlying nonlinear reaction kinetics is unnecessary. Then, we confirm the optimality of this solution by demonstrating numerically its equivalence to the solution of a regularized, optimal control problem. To solve the latter, the analytical expressions for the control are excellent initial guesses speeding-up substantially the otherwise time-consuming calculations.

Different Types of Instabilities and Complex Dynamics in Reaction-Diffusion Systems With Fractional Derivatives

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Vasyl Gafiychuk ◽  
Bohdan Datsko
Keyword(s):  
Complex Dynamics ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Different Types ◽  
Linearized System ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Activator Inhibitor ◽  
Fractional Reaction

In this article we analyze conditions for different types of instabilities and complex dynamics that occur in nonlinear two-component fractional reaction-diffusion systems. It is shown that the stability of steady state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. The results of the linear stability analysis are confirmed by computer simulations of the FitzHugh-Nahumo-like model. On the basis of this model, it is demonstrated that the conditions of instability and the pattern formation dynamics in fractional activator- inhibitor systems are different from the standard ones. As a result, a richer and a more complicated spatiotemporal dynamics takes place in fractional reaction-diffusion systems. A common picture of nonlinear solutions in time-fractional reaction-diffusion systems and illustrative examples are presented. The results obtained in the article for homogeneous perturbation have also been of interest for dynamical systems described by fractional ordinary differential equations.

Wawelet transform application for classification of solutions of reaction-diffusion systems

2017 ◽  
Vol 2017 (45) ◽  
pp. 50-55
Author(s):  
Z.I. Vasjunyk ◽  
◽  
Y.I. Maksymiv ◽  
V.V. Meleshko ◽  
◽  
...  
Keyword(s):  
Wavelet Transforms ◽  
Type System ◽  
Reaction Diffusion ◽  
Oscillatory Solutions ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Classification Of Solutions ◽  
Different Types ◽  
Wavelet Transformations

Girer–Meynxardt type system of reaction-diffusion with classical derivatives and Bryusselyator system with fractional time derivatives are investigated. On the basis of computer simulations it is shown that qualitatively different types of oscillatory solutions may arise due to instability in these systems. Wavelet transformations are applied to analyze and classify the solutions of such systems, A comparative estimation of wavelet transforms of oscillatory and chaotic solutions is given and it is shown that such method of classification of solutions is effective.

Self-organization at the frictional interface for green tribology

2010 ◽  
Vol 368 (1929) ◽  
pp. 4755-4774 ◽  
Author(s):  
Michael Nosonovsky
Keyword(s):  
Reaction Diffusion ◽  
Self Organization ◽  
Irreversible Processes ◽  
Protective Film ◽  
Self Healing ◽  
Structure Property ◽  
Reaction Diffusion Systems ◽  
Self Organized ◽  
Diffusion Systems ◽  
Green Tribology

Despite the fact that self-organization during friction has received relatively little attention from tribologists so far, it has the potential for the creation of self-healing and self-lubricating materials, which are important for green or environment-friendly tribology. The principles of the thermodynamics of irreversible processes and of the nonlinear theory of dynamical systems are used to investigate the formation of spatial and temporal structures during friction. The transition to the self-organized state with low friction and wear occurs through destabilization of steady-state (stationary) sliding. The criterion for destabilization is formulated and several examples are discussed: the formation of a protective film, microtopography evolution and slip waves. The pattern formation may involve self-organized criticality and reaction–diffusion systems. A special self-healing mechanism may be embedded into the material by coupling the corresponding required forces. The analysis provides the structure–property relationship, which can be applied for the design optimization of composite self-lubricating and self-healing materials for various ecologically friendly applications and green tribology.

Different Types of Instabilities and Complex Dynamics in Fractional Reaction-Diffusion Systems

2009 ◽  
Author(s):  
Vasyl Gafiychuk ◽  
Bohdan Datsko
Keyword(s):  
Fractional Derivatives ◽  
Complex Dynamics ◽  
Reaction Diffusion ◽  
Main Attention ◽  
Van Der Pol ◽  
Reaction Diffusion Systems ◽  
Brusselator Model ◽  
Diffusion Systems ◽  
Different Types ◽  
Pattern Formations

In this article we analyze a influence of possible instabilities on pattern formations in the reaction-diffusion systems with fractional derivatives. The results of qualitative analysis are confirmed by numerical simulations. The main attention is paid to two models: a fractional order reaction diffusion system with Bonhoeffer-van der Pol kinetics and to Brusselator model.

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