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Related Concept Videos

Positive and Negative Feedback Loops01:18

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Animal organs and organ systems constantly adjust to internal and external changes through a process called homeostasis ("steady state"). Examples of these changes include regulation of the level of glucose or calcium in the blood or internal responses to external temperatures. Homeostasis requires  maintaining an internal dynamic equilibrium:
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Using Caenorhabditis elegans as a Model System to Study Protein Homeostasis in a Multicellular Organism
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Homeostasis, singularities, and networks.

Martin Golubitsky1, Ian Stewart2

  • 1Mathematical Biosciences Institute, Ohio State University, 364 Jennings Hall, Columbus, OH, 43210, USA. mg@mbi.osu.edu.

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|June 4, 2016
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Summary
This summary is machine-generated.

This study mathematically defines homeostasis using singularity theory and explores its invariance in biological networks. Findings aid in understanding biological system stability and network dynamics.

Keywords:
Catastrophe theoryCoupled cell systemsHomeostasisNetworksSingularity theory

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Area of Science:

  • Mathematical Biology
  • Systems Biology
  • Dynamical Systems Theory

Background:

  • Homeostasis is crucial for biological and chemical systems, maintaining stable output despite input variations.
  • Understanding homeostasis stability under perturbations is key to biological system function.
  • Previous models often lack a rigorous mathematical framework for analyzing homeostasis under coordinate changes.

Purpose of the Study:

  • To reformulate homeostasis using singularity theory for a precise mathematical definition.
  • To investigate the invariance of homeostasis under coordinate transformations in network dynamics.
  • To apply these theoretical frameworks to biological examples like thermoregulation.

Main Methods:

  • Reformulation of homeostasis using singularity theory and unfolding theory.
  • Analysis of 'chair' singularities and their universal unfolding.
  • Application of implicit differentiation to identify critical points in mathematical models.
  • Combinatorial characterization of network topology for homeostasis invariance.

Main Results:

  • A precise mathematical definition of homeostasis is established via singularity theory.
  • The 'chair' singularity is analyzed, providing insights into approximate homeostasis regions.
  • A method for identifying homeostasis-critical points using implicit differentiation is presented.
  • Conditions for homeostasis invariance under structure-preserving coordinate changes in networks are characterized.

Conclusions:

  • Singularity theory offers a powerful framework for studying homeostasis mathematically.
  • The study provides tools to analyze stability and robustness in biological systems.
  • Network topology plays a critical role in determining the invariance of homeostasis.