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Patterning of Microorganisms and Microparticles through Sequential Capillarity-assisted Assembly
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Delayed Pattern Formation in Two-Dimensional Domains.

Nirmali Das1, István Balázs2,3,4, Bornali Das5

  • 1Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, H-6720, Hungary. nimi.szegeduni22@gmail.com.

Bulletin of Mathematical Biology
|June 11, 2026
PubMed
Summary
This summary is machine-generated.

Gene expression time delay linearly impacts pattern formation time in reaction-diffusion systems. Domain size shows a non-monotonic effect, with initial conditions influencing pattern emergence.

Keywords:
Gene expressionLI modelSchnakenberg modelTime delayTuring pattern

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Area of Science:

  • Systems Biology
  • Mathematical Biology
  • Computational Biology

Background:

  • Spatio-temporal pattern formation is crucial in biological systems.
  • Reaction-diffusion models, like the Schnakenberg model, are used to study these patterns.
  • Gene expression time delays and domain size are key factors influencing pattern dynamics.

Purpose of the Study:

  • To investigate the interplay between gene expression time delay and domain size in spatio-temporal pattern formation.
  • To analyze pattern emergence in a modified Schnakenberg model (LI model) across different dimensions.
  • To characterize the relationship between parameters and pattern formation dynamics.

Main Methods:

  • Utilized a modified Schnakenberg model (LI model) for reaction-diffusion simulations.
  • Extended analysis from one-dimensional to two-dimensional domains.
  • Investigated diverse initial conditions, including steady-state perturbations and unstable manifold functions.
  • Computed a two-parameter chart mapping patterns against time delay and domain size.

Main Results:

  • A linear relationship was found between gene expression time delay and pattern formation time in 1D and 2D domains.
  • Domain size exhibited a non-monotonic correlation with the time required for pattern emergence.
  • Initial conditions significantly influenced the observed pattern formation dynamics.

Conclusions:

  • Gene expression time delay is a critical determinant of pattern formation speed.
  • Domain size has a complex, non-monotonic influence on pattern emergence.
  • The LI model provides insights into how delays and domain geometry affect biological pattern formation.