Turing's model and branching tip growth: relation of time and spatial scales in morphogenesis, with application to Micrasterias

1987 ◽  
Vol 65 (7) ◽  
pp. 1308-1319 ◽  
Author(s):  
Thurston C. Lacalli ◽  
Lionel G. Harrison
Keyword(s):  
Diffusion Theory ◽  
Spatial Scales ◽  
Diffusion Mechanism ◽  
Effective Diffusivity ◽  
Reaction Diffusion ◽  
Time And Space ◽  
Branch Formation ◽  
Simplifying Assumptions ◽  
Growth Activity ◽  
Probable Site

Morphogenesis following cell division in Micrasterias rotata is by outgrowth and repeated branching of a series of semicell lobes. Though successive branching events are qualitatively similar, they display changes in time and space scales, and these can be quantitated with the aid of autoradiographic patterns of labelled wall precursors that appear late in morphogenesis but which seem to represent its history. This enables us to consider branching as the conversion of a single centre of growth activity into two and to attempt to locate these centres precisely, in terms of both position and time of establishment. Temporal and spatial scales both decrease, by 75%, through a sequence of five branching events, in linear functional relationship to each other. This correlation points toward kinetic control of morphogenesis, i.e., the involvement of something like a reaction–diffusion mechanism. We analyse this possibility in terms of available reaction–diffusion theory to show how, after various simplifying assumptions, and if the time and space scales of branch formation are known, an effective diffusivity, [Formula: see text], for the patterning mechanism can be estimated. For M. rotata we obtain orders of magnitude: [Formula: see text], with an upper limit on the diffusivity of the faster diffusing of the two morphogenetic substances in the mechanism of ca. 1 × 10−7 cm2/s. These values implicate the cell membrane as the most probable site of pattern formation.

scholarly journals Travelling colourful patterns in self-organized cellulose-based liquid crystalline structures

2021 ◽  
Vol 2 (1) ◽  
Author(s):  
Pedro E. S. Silva ◽  
Ricardo Chagas ◽  
Susete N. Fernandes ◽  
Pawel Pieranski ◽  
Robin L. B. Selinger ◽  
...  
Keyword(s):  
Simple Model ◽  
Temporal Variation ◽  
Liquid Crystalline ◽  
Diffusion Mechanism ◽  
Reaction Diffusion ◽  
Self Organization ◽  
Spatial And Temporal Variation ◽  
Living Matter ◽  
Equilibrium Conditions ◽  
Self Organized

AbstractCellulose-based systems are useful for many applications. However, the issue of self-organization under non-equilibrium conditions, which is ubiquitous in living matter, has scarcely been addressed in cellulose-based materials. Here, we show that quasi-2D preparations of a lyotropic cellulose-based cholesteric mesophase display travelling colourful patterns, which are generated by a chemical reaction-diffusion mechanism being simultaneous with the evaporation of solvents at the boundaries. These patterns involve spatial and temporal variation in the amplitude and sign of the helix´s pitch. We propose a simple model, based on a reaction-diffusion mechanism, which simulates the observed spatiotemporal colour behaviour.

Reaction-Diffusion Modeling of Photopolymerization During Femtosecond Projection Two-Photon Lithography

2021 ◽  
Author(s):  
Rushil Pingali ◽  
Sourabh K. Saha
Keyword(s):  
Diffusion Mechanism ◽  
Chemical Properties ◽  
Reaction Diffusion ◽  
Element Model ◽  
Polymerization Process ◽  
Layer By Layer ◽  
Length Scales ◽  
Two Photon ◽  
Reaction Diffusion Model ◽  
Direct Laser

Abstract Two-photon lithography (TPL) is a polymerization-based direct laser writing process that is capable of fabricating arbitrarily complex three-dimensional (3D) structures with submicron features. Traditional TPL techniques have limited scalability due to the slow point-by-point serial writing scheme. The femtosecond projection TPL (FP-TPL) technique increases printing rate by a thousand times by enabling layer-by-layer parallelization. However, parallelization alters the time and the length scales of the underlying polymerization process. It is therefore challenging to apply the models of serial TPL to accurately predict process outcome during FP-TPL. To solve this problem, we have generated a finite element model of the polymerization process on the time and length scales relevant to FP-TPL. The model is based on the reaction-diffusion mechanism that underlies polymerization. We have applied this model to predict the geometry of nanowires printed under a variety of conditions and compared these predictions against empirical data. Our model accurately predicts the nanowire widths. However, accuracy of aspect ratio prediction is hindered by uncertain values of the chemical properties of the photopolymer. Nevertheless, our results demonstrate that the reaction-diffusion model can accurately capture the effect of controllable parameters on FP-TPL process outcome and can therefore be used for process control and optimization.

scholarly journals Reaction Diffusion and Chemotaxis for Decentralized Gathering on FPGAs

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
Bernard Girau ◽  
César Torres-Huitzil ◽  
Nikolaos Vlassopoulos ◽  
José Hugo Barrón-Zambrano
Keyword(s):  
Large Scale ◽  
Complex Dynamics ◽  
Diffusion Mechanism ◽  
Reaction Diffusion ◽  
Fpga Implementation ◽  
Cellular Slime Mold ◽  
Reaction Diffusion Model ◽  
Cellular Slime ◽  
Computational Resources ◽  
Embedded Implementation

We consider here the feasibility of gathering multiple computational resources by means of decentralized and simple local rules. We study such decentralized gathering by means of a stochastic model inspired from biology: the aggregation of theDictyostelium discoideumcellular slime mold. The environment transmits information according to a reaction-diffusion mechanism and the agents move by following excitation fronts. Despite its simplicity this model exhibits interesting properties of self-organization and robustness to obstacles. We first describe the FPGA implementation of the environment alone, to perform large scale and rapid simulations of the complex dynamics of this reaction-diffusion model. Then we describe the FPGA implementation of the environment together with the agents, to study the major challenges that must be solved when designing a fast embedded implementation of the decentralized gathering model. We analyze the results according to the different goals of these hardware implementations.

scholarly journals ASPECTS OF THE DIFFUSION THEORY OF INNOVATIONS AND LOGISTISATION OF AGRICULTURAL INDUSTRY OF UKRAINE

2018 ◽  
pp. 49-56
Author(s):  
Yevhen Nikishyn
Keyword(s):  
Diffusion Of Innovations ◽  
Diffusion Theory ◽  
Reaction Diffusion ◽  
Industrial Complex ◽  
Competitive Advantages ◽  
Agricultural Industry ◽  
Regulatory Factors ◽  
Diffusion Structure ◽  
Cascade Effect ◽  
Separate System

The article is devoted to theoretical aspects of diffusion of innovations, as the conditions of logistics of the agro industrial complex of Ukraine. The concept of innovation-economic niche as a separate system with the potential of making innovations, the development of which creates competitive advantages, is formulated. New types of diffusion are classified on the basis of decision-making mechanisms by innovators. The diffusion models are considered, the descriptions of specific features of the behaviour of the dissemination of innovations in the reaction-diffusion structure are studied and made taking into account the system-regulatory factors. The principle of informational conditionality of economic phenomena as the basis of distribution of diffusion is formulated. The existence of a cascade effect in the diffusion of basic innovations has been determined; the necessity of the accompanying innovations has been substantiated. The causal relationship between the influence of system-regulatory factors on diffusion, the emergence of a cascade effect, the formation of clusters of innovations and the general influence on the Kondratiev cycles have been investigated.

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.

scholarly journals Computational modeling of hydrogel cross‐linking based on reaction‐diffusion theory

PAMM ◽  
2019 ◽  
Vol 19 (1) ◽  
Author(s):  
Aidin Hajikhani ◽  
Michele Marino ◽  
Peter Wriggers
Keyword(s):  
Computational Modeling ◽  
Diffusion Theory ◽  
Reaction Diffusion ◽  
Cross Linking

Pyrolysis behaviors dominated by the reaction–diffusion mechanism in the fluorine-free metal–organic decomposition process

2020 ◽  
Vol 8 (48) ◽  
pp. 17417-17428
Author(s):  
Jiangtao Shi ◽  
Yue Zhao ◽  
Yue Wu ◽  
Jingyuan Chu ◽  
Xiao Tang ◽  
...  
Keyword(s):  
Diffusion Model ◽  
Diffusion Mechanism ◽  
Reaction Diffusion ◽  
Decomposition Process ◽  
One Dimensional ◽  
Reaction Diffusion Model ◽  
Metal Organic ◽  
Free Metal ◽  
Organic Decomposition

In this work, pyrolysis behaviors dominated by the reaction–diffusion mechanism were investigated. And one-dimensional reaction–diffusion model is proposed.

scholarly journals Renewal Reward Perspective on Linear Switching Diffusion Systems in Models of Intracellular Transport

2020 ◽  
Vol 82 (10) ◽  
Author(s):  
Maria-Veronica Ciocanel ◽  
John Fricks ◽  
Peter R. Kramer ◽  
Scott A. McKinley
Keyword(s):  
Transport Properties ◽  
Intracellular Transport ◽  
Motor Proteins ◽  
Molecular Motor ◽  
Effective Diffusivity ◽  
Reaction Diffusion ◽  
Model Parameters ◽  
Switching Diffusion ◽  
Diffusion Dynamics ◽  
Effective Transport

Abstract In many biological systems, the movement of individual agents is characterized having multiple qualitatively distinct behaviors that arise from a variety of biophysical states. For example, in cells the movement of vesicles, organelles, and other intracellular cargo is affected by their binding to and unbinding from cytoskeletal filaments such as microtubules through molecular motor proteins. A typical goal of theoretical or numerical analysis of models of such systems is to investigate effective transport properties and their dependence on model parameters. While the effective velocity of particles undergoing switching diffusion dynamics is often easily characterized in terms of the long-time fraction of time that particles spend in each state, the calculation of the effective diffusivity is more complicated because it cannot be expressed simply in terms of a statistical average of the particle transport state at one moment of time. However, it is common that these systems are regenerative, in the sense that they can be decomposed into independent cycles marked by returns to a base state. Using decompositions of this kind, we calculate effective transport properties by computing the moments of the dynamics within each cycle and then applying renewal reward theory. This method provides a useful alternative large-time analysis to direct homogenization for linear advection–reaction–diffusion partial differential equation models. Moreover, it applies to a general class of semi-Markov processes and certain stochastic differential equations that arise in models of intracellular transport. Applications of the proposed renewal reward framework are illustrated for several case studies such as mRNA transport in developing oocytes and processive cargo movement by teams of molecular motor proteins.

A model for the longitudinal patterns shaped by water on erodible rocks

2020 ◽  
Author(s):  
Matteo Bernard Bertagni ◽  
Carlo Camporeale
Keyword(s):  
Mathematical Model ◽  
Pattern Formation ◽  
Laminar Flow ◽  
Spatial Scales ◽  
Concentration Field ◽  
Water Film ◽  
Time And Space ◽  
Longitudinal Patterns ◽  
Flow Intensity ◽  
Alpine Environments

<p>The interactions between water and rocks create an extensive variety of marvelous patterns, which span on several classes of time and space scales. In this work, we provide a mathematical model for the formation of longitudinal erosive patterns commonly found in karst and alpine environments. The model couples the hydrodynamics of a laminar flow of water (Orr-Somerfield equation) to the concentration field of the eroded-rock chemistry. Results show that an instability of the plane rock wetted by the water film leads to a longitudinal channelization responsible for the pattern formation. The spatial scales predicted by the model span over different orders of magnitude depending on the flow intensity and this may explain why similar patterns of different sizes are observed in nature (millimetric microrills, centimetric rillenkarren, decametric solution runnels).</p>

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