scholarly journals Emergence of form in embryogenesis

2018 ◽  
Vol 15 (148) ◽  
pp. 20180454
Author(s):  
Murat Erkurt
Keyword(s):  
Mirror Symmetry ◽  
Rotational Symmetry ◽  
Spatial Dynamics ◽  
Reaction Diffusion ◽  
Symmetry Operator ◽  
Three Dimensions ◽  
Dirac Delta ◽  
Integrated Simulation ◽  
Delta Type ◽  
Finite State

The development of form in an embryo is the result of a series of topological and informational symmetry breakings. We introduce the vector–reaction–diffusion–drift (VRDD) system where the limit cycle of spatial dynamics is morphogen concentrations with Dirac delta-type distributions. This is fundamentally different from the Turing reaction–diffusion system, as VRDD generates system-wide broken symmetry. We developed ‘fundamental forms’ from spherical blastula with a single organizing axis (rotational symmetry), double axis (mirror symmetry) and triple axis (no symmetry operator in three dimensions). We then introduced dynamics for cell differentiation, where genetic regulatory states are modelled as a finite-state machine (FSM). The state switching of an FSM is based on local morphogen concentrations as epigenetic information that changes dynamically. We grow complicated forms hierarchically in spatial subdomains using the FSM model coupled with the VRDD system. Using our integrated simulation model with four layers (topological, physical, chemical and regulatory), we generated life-like forms such as hydra. Genotype–phenotype mapping was investigated with continuous and jump mutations. Our study can have applications in morphogenetic engineering, soft robotics and biomimetic design.

scholarly journals General up to next-nearest-neighbour elasticity of triangular lattices in three dimensions

2006 ◽  
Vol 462 (2073) ◽  
pp. 2695-2713 ◽  
Author(s):  
Cyril Dubus ◽  
Ken Sekimoto ◽  
Jean-Baptiste Fournier
Keyword(s):  
Blood Cell ◽  
Red Blood Cell ◽  
Triangular Lattice ◽  
Soft Matter ◽  
Rotational Symmetry ◽  
Three Dimensions ◽  
Nearest Neighbour ◽  
Two Dimensional ◽  
Biological Physics ◽  
Crystalline System

We establish the most general form of the discrete elasticity of a two-dimensional triangular lattice embedded in three dimensions, taking into account up to next-nearest-neighbour interactions. Besides crystalline system, this is relevant to biological physics (e.g. red blood cell cytoskeleton) and soft matter (e.g. percolating gels, etc.). In order to correctly impose the rotational invariance of the bulk terms, it turns out to be necessary to take into account explicitly the elasticity associated with the vertices located at the edges of the lattice. We find that some terms that were suspected in the literature to violate rotational symmetry are, in fact, admissible.

Radiolaria: validating the Turing theory

2017 ◽  
Author(s):  
Bernard Richards
Keyword(s):  
Diffusion Mechanism ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Black And White ◽  
Alan Turing ◽  
School Sports ◽  
The University ◽  
University Of Manchester

In his 1952 paper ‘The chemical basis of morphogenesis’ Turing postulated his now famous Morphogenesis Equation. He claimed that his theory would explain why plants and animals took the shapes they did. When I joined him, Turing suggested that I might solve his equation in three dimensions, a new problem. After many manipulations using rather sophisticated mathematics and one of the first factory-produced computers in the UK, I derived a series of solutions to Turing’s equation. I showed that these solutions explained the shapes of specimens of the marine creatures known as Radiolaria, and that they corresponded very closely to the actual spiny shapes of real radiolarians. My work provided further evidence for Turing’s theory of morphogenesis, and in particular for his belief that the external shapes exhibited by Radiolaria can be explained by his reaction–diffusion mechanism. While working in the Computing Machine Laboratory at the University of Manchester in the early 1950s, Alan Turing reignited the interests he had had in both botany and biology from his early youth. During his school-days he was more interested in the structure of the flowers on the school sports field than in the games played there (see Fig. 1.3). It is known that during the Second World War he discussed the problem of phyllotaxis (the arrangement of leaves and florets in plants), and then at Manchester he had some conversations with Claude Wardlaw, the Professor of Botany in the University. Turing was keen to take forward the work that D’Arcy Thompson had published in On Growth and Form in 1917. In his now-famous paper of 1952 Turing solved his own ‘Equation of Morphogenesis’ in two dimensions, and demonstrated a solution that could explain the ‘dappling’—the black-and-white patterns—on cows. The next step was for me to solve Turing’s equation in three dimensions. The two-dimensional case concerns only surface features of organisms, such as dappling, spots, and stripes, whereas the three-dimensional version concerns the overall shape of an organism. In 1953 I joined Turing as a research student in the University of Manchester, and he set me the task of solving his equation in three dimensions. A remarkable journey of collaboration began. Turing chatted to me in a very friendly fashion.

A high resolution multi-turn TOF mass analyzer

2019 ◽  
Vol 34 (36) ◽  
pp. 1942005 ◽  
Author(s):  
Vyacheslav Shchepunov ◽  
Michael Rignall ◽  
Roger Giles ◽  
Ryo Fujita ◽  
Hiroaki Waki ◽  
...  
Keyword(s):  
High Resolution ◽  
Mirror Symmetry ◽  
Rotational Symmetry ◽  
Optical Design ◽  
Resolving Power ◽  
Mass Analyzer ◽  
Rotationally Symmetric ◽  
Mass Resolving Power ◽  
Operational Modes ◽  
Drift Direction

An ion optical design of a high resolution multi-turn time-of-flight mass analyzer (MT-TOF MA) is presented. The analyzer has rotationally symmetric main electrodes with additional mirror symmetry about a mid-plane orthogonal to the axis of symmetry. Rotational symmetry allows a higher density of turns in the azimuthal (drift) direction compared to MT-TOF MAs that are linearly extended in the drift direction. Mirror symmetry about a mid-plane helps to achieve a high spatial isochronicity of the ions’ motion. The analyzer comprises a pair of polar-toroidal sectors S1 and S3, a pair of polar (trans-axial) lenses, and a pair of conical lenses for longitudinal and lateral focusing. A toroidal sector S2 located at the mid-plane of the analyzer has a set of embedded drift focusing segments providing focusing and spatial isochronicity in the drift direction. The ions’ drift in the azimuthal direction can be reversed by using dedicated reversing deflectors. This gives the possibility of several operational modes with different numbers of turns and passes in the drift direction. According to numerical simulations, the mass resolving power of the analyzer ranges from [Formula: see text]40 k (fwhm) at small (typically below ten) numbers of turns to [Formula: see text]450 k (fwhm) at 96 turns.

scholarly journals Efficient Sampling in Event-Driven Algorithms for Reaction-Diffusion Processes

2013 ◽  
Vol 13 (4) ◽  
pp. 958-984 ◽  
Author(s):  
Mohammad Hossein Bani-Hashemian ◽  
Stefan Hellander ◽  
Per Lötstedt
Keyword(s):  
Diffusion Processes ◽  
Reaction Diffusion ◽  
Three Dimensions ◽  
Cumulative Distribution ◽  
Brownian Diffusion ◽  
Time Interval ◽  
Efficient Sampling ◽  
Event Driven ◽  
Analytical Formulas ◽  
Chemically Reacting

AbstractIn event-driven algorithms for simulation of diffusing, colliding, and reacting particles, new positions and events are sampled from the cumulative distribution function (CDF) of a probability distribution. The distribution is sampled frequently and it is important for the efficiency of the algorithm that the sampling is fast. The CDF is known analytically or computed numerically. Analytical formulas are sometimes rather complicated making them difficult to evaluate. The CDF may be stored in a table for interpolation or computed directly when it is needed. Different alternatives are compared for chemically reacting molecules moving by Brownian diffusion in two and three dimensions. The best strategy depends on the dimension of the problem, the length of the time interval, the density of the particles, and the number of different reactions.

Turing Instability and Hopf Bifurcation in a Modified Leslie–Gower Predator–Prey Model with Cross-Diffusion

2018 ◽  
Vol 28 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Walid Abid ◽  
R. Yafia ◽  
M. A. Aziz-Alaoui ◽  
Ahmed Aghriche
Keyword(s):  
Spatial Dynamics ◽  
Real Life ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model

This paper is concerned with some mathematical analysis and numerical aspects of a reaction–diffusion system with cross-diffusion. This system models a modified version of Leslie–Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator–prey model to detect the spatial dynamics in the real life.

scholarly journals Self-organized criticality and pattern emergence through the lens of tropical geometry

2018 ◽  
Vol 115 (35) ◽  
pp. E8135-E8142 ◽  
Author(s):  
N. Kalinin ◽  
A. Guzmán-Sáenz ◽  
Y. Prieto ◽  
M. Shkolnikov ◽  
V. Kalinina ◽  
...  
Keyword(s):  
Mirror Symmetry ◽  
Tropical Geometry ◽  
Scaling Limit ◽  
Reaction Diffusion ◽  
Auction Theory ◽  
Discrete Models ◽  
Self Organized ◽  
Reaction Diffusion Model ◽  
Pure Mathematics ◽  
Self Organized Criticality

Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy model (cf. Turing reaction-diffusion model), requiring further investigation.

SPIRAL WAVES IN MEDIA WITH COMPLEX-EXCITABLE DYNAMICS

1999 ◽  
Vol 09 (11) ◽  
pp. 2243-2247 ◽  
Author(s):  
ANDREI GORYACHEV ◽  
RAYMOND KAPRAL
Keyword(s):  
Phase Space ◽  
Temporal Dynamics ◽  
Rotational Symmetry ◽  
Reaction Diffusion ◽  
Structural Features ◽  
Spiral Waves ◽  
Space Trajectories ◽  
Reaction Diffusion Systems ◽  
Oscillatory Dynamics ◽  
Diffusion Systems

The structure of spiral waves is investigated in super-excitable reaction–diffusion systems where the local dynamics exhibits multilooped phase-space trajectories. It is shown that such systems support stable spiral waves with broken rotational symmetry and complex temporal dynamics. The main structural features of such waves, synchronization defect lines, are demonstrated to be similar to those of spiral waves in systems with complex-oscillatory dynamics.

scholarly journals Simulation of reaction–diffusion processes in three dimensions using CUDA

2011 ◽  
Vol 108 (1) ◽  
pp. 76-85 ◽  
Author(s):  
Ferenc Molnár ◽  
Ferenc Izsák ◽  
Róbert Mészáros ◽  
István Lagzi
Keyword(s):  
Diffusion Processes ◽  
Reaction Diffusion ◽  
Three Dimensions

RELAXATION BEHAVIOR AND PATTERN FORMATION IN REACTION-DIFFUSION SYSTEMS

1994 ◽  
Vol 04 (05) ◽  
pp. 1183-1191 ◽  
Author(s):  
PATRICK HANUSSE ◽  
VICENTE PEREZ-MUÑUZURI ◽  
MONCHO GOMEZ-GESTEIRA
Keyword(s):  
Rotational Symmetry ◽  
Relaxation Oscillation ◽  
Reaction Diffusion ◽  
Relaxation Behavior ◽  
Reaction Diffusion Systems ◽  
Phase Dynamics ◽  
Diffusion Systems ◽  
Universal Definition ◽  
Definition Of ◽  
Mathematical Definitions

The notions of relaxation oscillation and hard excitation have been extensively used and early recognized as important qualitative features of many nonlinear systems. Nevertheless, there seems to exist so far no clear mathematical definitions of these notions. We consider the description of relaxation behavior in oscillating or excitable systems resulting from symmetry breaking of the rotational symmetry of the velocity vector field of the Hopf normal form. From symmetry considerations we detect the first terms responsible for the relaxation character of the phase dynamics in such systems and show that they provide a good general, if not universal, definition of the relaxation properties. We analyze their consequence in the modeling of spatiotemporal patterns such as spiral waves.

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