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Related Concept Videos

Metallic Solids02:37

Metallic Solids

Metallic solids such as crystals of copper, aluminum, and iron are formed by metal atoms. The structure of metallic crystals is often described as a uniform distribution of atomic nuclei within a “sea” of delocalized electrons. The atoms within such a metallic solid are held together by a unique force known as metallic bonding that gives rise to many useful and varied bulk properties.
All metallic solids exhibit high thermal and electrical conductivity, metallic luster, and malleability. Many...

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Continuum limits of pattern formation in hexagonal-cell monolayers.

R D O'Dea1, J R King

  • 1Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. reuben.odea@ntu.ac.uk

Journal of Mathematical Biology
|May 21, 2011
PubMed
Summary
This summary is machine-generated.

This study develops continuum models for intercellular signalling in hexagonal cells, revealing how cell shape influences differentiation patterns. While powerful, these models have limitations in capturing all dynamics of discrete systems.

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Area of Science:

  • Mathematical Biology
  • Developmental Biology
  • Cellular Dynamics

Background:

  • Intercellular signalling, particularly juxtacrine signalling, is crucial for cell fate determination and spatial pattern generation in developing epithelia.
  • Negative feedback mechanisms in signalling and cell shape are known to influence pattern formation, but analysis has been limited to discrete models.
  • Previous work established continuum models for square cells, but complex hexagonal cell geometries remained challenging.

Purpose of the Study:

  • To extend continuum modelling approaches to hexagonal cell arrangements for analysing fine-grained spatial patterns in cell differentiation.
  • To develop models capturing pattern generation from homogeneous states and transitions between patterning modes in hexagonal cells.
  • To construct a generalized continuum representation of patterning behaviour under simultaneous feedback parameter variations.

Main Methods:

  • Derivation of continuum models for hexagonal cell geometries based on juxtacrine signalling principles.
  • Analysis of pattern emergence from homogeneous states and mode transitions.
  • Construction of a general continuum model incorporating simultaneous feedback parameter variations.
  • Comparison of continuum model predictions with discrete system dynamics, including travelling waves and mode competition.

Main Results:

  • Successfully derived continuum models for fine-grained patterning in hexagonal cells.
  • Developed a generalized model capturing the bifurcation structure of patterning.
  • Demonstrated that continuum models can represent pattern generation and transitions in hexagonal cells.
  • Identified limitations of continuum models in fully capturing discrete system dynamics like travelling waves and mode competition.

Conclusions:

  • Continuum models provide a robust framework for analysing juxtacrine signalling and pattern formation in hexagonal cell arrangements.
  • Cell shape significantly impacts the type and emergence of differentiation patterns.
  • While continuum models offer insights, they possess inherent deficiencies in representing certain complex dynamics present in discrete systems.