An auxin-driven polarized transport model for phyllotaxis

Abstract

Recent studies show that plant organ positioning may be mediated by localized concentrations of the plant hormone auxin. Auxin patterning in the shoot apical meristem is in turn brought about by the subcellular polar distribution of the putative auxin efflux mediator, PIN1. However, the question of what signals determine PIN1 polarization and how this gives rise to regular patterns of auxin concentration remains unknown. Here we address these questions by using mathematical modeling combined with confocal imaging. We propose a model that is based on the assumption that auxin influences the polarization of its own efflux within the meristem epidermis. We show that such a model is sufficient to create regular spatial patterns of auxin concentration on systems with static and dynamic cellular connectivities, the latter governed by a mechanical model. We also optimize parameter values for the PIN1 dynamics by using a detailed auxin transport model, for which parameter values are taken from experimental estimates, together with a template consisting of cell and wall compartments as well as PIN1 concentrations quantitatively extracted from confocal data. The model shows how polarized transport can drive the formation of regular patterns.

Fig. 1.

Template extraction from a confocal image. Shown is a horizontal optical section through the epidermal layer of cells at the shoot apex. (A) Original confocal data. The red signal shows a membrane marker, and green shows a PIN1::GFP construct. P1 and P2 marks visible primordia, I1 and I2 show the locations of the next primordia (modified from ref. 20). (Scale bar, 30 μm.) (B) Walls extracted by watershed algorithm (cellular compartments inside) visualized on the membrane marker image. (C) PIN1 intensities in extracted cell compartments. (D) PIN1 intensities in cell/membrane compartments.

Fig. 2.

Auxin equilibrium concentrations for simulations on the template using constant extracted PIN1 concentrations (Fig. 1D). (A) Wall pH equal to 5, which is the experimentally estimated value. (B) Wall pH equal to 4.5. (C) Wall pH equal to 5.5.

Fig. 3.

Simulations of the simplistic model, for which the initial auxin concentrations are [0.999:1.001] and periodic boundary conditions are used. The plots show equilibrium auxin concentrations for simulations with different values of D/TP.

Fig. 4.

Example of an optimized PIN1 cycling model. The optimized parameter values are K = 0.4, n = 3.0, and k₂/k₁ = 0.4. (A) PIN1 concentration resulting from the optimized model plotted on the template (compare with Fig. 1D). (B) Quantitative comparison of PIN1 between the template and the model, where the resulting optimized model values are plotted vs. the extracted values for individual compartments.

Fig. 5.

Simulation of the phyllotaxis model on a half-sphere cylinder surface including cellular growth and proliferation. The main images show a top view (with the insets showing a side view), and time is increasing from top to bottom. The two simulations have different values for the size of the defined central zone. (A) Peaks formed in a spiral pattern. (B) Peaks formed in a whorled (decussate) pattern.

Fig. 6.

Simulation of the phyllotaxis model on a two-dimensional plane. Shown is the PIN1 polarization in cells as a new peak is formed. PIN1 polarization (P_ij) is shown as bars, with lengths corresponding to a measure of its value, and the color coding shows the auxin concentration. Cells reversing polarity are marked with an asterisk.