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Turing Instabilities are Not Enough to Ensure Pattern Formation.

Andrew L Krause1, Eamonn A Gaffney2, Thomas Jun Jewell2

  • 1Department of Mathematical Sciences, Durham University, Upper Mountjoy Campus, Stockton Road, Durham, DH1 3LE, UK. andrew.krause@durham.ac.uk.

Bulletin of Mathematical Biology
|January 22, 2024
PubMed
Summary
This summary is machine-generated.

Turing instabilities can initiate pattern formation but are insufficient for sustained patterns in systems with multiple stable states. Further analysis is needed for self-organization mechanisms in complex biological systems.

Keywords:
MultistabilityPattern formationTuring instabilities

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Area of Science:

  • Theoretical biology
  • Mathematical modeling
  • Chemical kinetics

Background:

  • Symmetry-breaking instabilities drive pattern diversity in nature, crucial for processes like cellular signaling and development.
  • Turing's reaction-diffusion theory and linear stability analysis are standard tools for understanding self-organization.
  • Existing models often assume Turing instabilities are sufficient for pattern formation.

Purpose of the Study:

  • To investigate whether Turing instability conditions are sufficient for the emergence of stable patterns.
  • To explore the role of multistability and nonlinearity in pattern formation.
  • To question the predictive power of linear stability analysis in complex biological systems.

Main Methods:

  • Analysis of canonical transport models with mild multistable nonlinearities.
  • Application of linear stability analysis to identify Turing instability conditions.
  • Examination of pattern dynamics in systems with multiple stable homogeneous equilibria.

Main Results:

  • Selected models satisfy Turing instability conditions but exhibit only transient patterns.
  • Turing-like instability alone is insufficient to guarantee a stable patterned state.
  • Linear theory's predictive failures for pattern formation are robust in multistable systems.

Conclusions:

  • The presence of multiple stable equilibria can lead to transient patterns despite satisfying Turing instability criteria.
  • Rethinking the analysis of self-organization mechanisms is necessary for systems with high multistability and nonlinearity.
  • This challenges the direct application of linear stability analysis in fields like gene regulatory networks and ecosystems.