Mathematical Modeling of Plant Root Hair Initiation: Dynamics of Localized Patches

Victor F Brena Medina, A. R. Champneys, C. Grierson, M. J. Ward

Research output: Contribution to journalArticle (Academic Journal)peer-review

17 Citations (Scopus)

Abstract

A mathematical analysis is undertaken of a Schnakenberg reaction-diffusion system in one dimension with a spatial gradient governing the active reaction. This system has previously been proposed as a model of the initiation of hairs from the root epidermis Arabidopsis, a key cellular-level morphogenesis problem. This process involves the dynamics of the small G-proteins, Rhos of plants, which bind to form a single localized patch on the cell membrane, prompting cell wall softening and subsequent hair growth. A numerical bifurcation analysis is presented as two key parameters, involving the cell length and the overall concentration of the auxin catalyst, are varied. The results show hysteretic transitions from a boundary patch to a single interior patch, and to multiple patches whose locations are carefully controlled by the auxin gradient. The results are confirmed by an asymptotic analysis using semistrong interaction theory, leading to closed form expressions for the patch locations and intensities. A close agreement between the numerical bifurcation results and the asymptotic theory is found for biologically realistic parameter values. Insight into the initiation of transition mechanisms is obtained through a linearized stability analysis based on a nonlocal eigenvalue problem. The results provide further explanation of the recent agreement found between the model and biological data for both wild-type and mutant hair cells.

Original languageEnglish
Pages (from-to)210-248
Number of pages39
JournalSIAM Journal on Applied Dynamical Systems
Volume13
Issue number1
DOIs
Publication statusPublished - 2014

Keywords

  • nonhomogeneous reaction-diffusion system
  • localized structures
  • nonlocal eigenvalue problem
  • numerical bifurcation analysis
  • REACTION-DIFFUSION SYSTEMS
  • GIERER-MEINHARDT MODEL
  • GRAY-SCOTT MODEL
  • PATTERN-FORMATION
  • ARABIDOPSIS-THALIANA
  • SCHNAKENBERG MODEL
  • AUXIN TRANSPORT
  • TURING PATTERNS
  • SPIKE SOLUTIONS
  • CELL POLARITY

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