Model of polar auxin transport coupled to mechanical forces retrieves robust morphogenesis along the $\mathit{Arabidopsis}$ root

Stem cells are identical in many scales, they share the same molecular composition, DNA, genes, and genetic networks, yet they should acquire different properties to form a functional tissue. Therefore, they must interact and get some external information from their environment, either spatial (dynamical fields) or temporal (lineage). In this paper we test to what extent coupled chemical and physical fields can underlie the cell's positional information during development. We choose the root apical meristem of Arabidopsis thaliana to model the emergence of cellular patterns. We built a model to study the dynamics and interactions between the cell divisions, the local auxin concentration, and physical elastic fields. Our model recovers important aspects of the self-organized and resilient behavior of the observed cellular patterns in the Arabidopsis root, in particular, the reverse fountain pattern observed in the auxin transport, the PIN-FORMED (protein family of auxin transporters) polarization pattern and the accumulation of auxin near the region of maximum curvature in a bent root. Our model may be extended to predict altered cellular patterns that are expected under various applied auxin treatments or modified physical growth conditions.

Figure 1

Confocal microscopy image of the Arabidopsis thaliana root tip. The elongation zone (EZ), the transition domain (TD), the cell proliferation domain (CPD) with actively proliferating cells, and the stem-cell niche (SCN) with the quiescent cells [QC, in green (light gray)] and surrounding stem cells are indicated. The root apical meristem (RAM) contains the TD, CPD, and SCN domains.

Figure 2

Flow diagram of the platform built for numerical calculations of the root growth. The parameters of the model are in red (with $*)$ and the initial conditions in blue (with $•)$.

Figure 3

Numerical calculation of the root growth. We show snapshots of the auxin concentration (top) and the potential (bottom) for an early time (left), a time when many proliferation events have taken place (right), and a time between the two (middle). The QC is indicated with dark asterisks and the epidermal cells are indicated with black circles.

Figure 4

Numerical results for the positional distributions of various quantities. We show the auxins (orange continuous line), the potential (red dots), and the number of cell divisions (stair lines) along the apical-basal axis of the root. The scale of auxins and the potential has been amplified to help the eye.

Figure 5

Snapshots of auxin distributions. The times correspond to the ones in Fig. 3 and represent the root growth dynamics when the position QC is not fixed. The epidermal cells are indicated with black circles. The simulation was made for $K_e=120$ and $\beta =15$ in both cases. (See video V1 in the Supplemental Material [38].)

Figure 6

Snapshots of PIN polarization direction and Auxin Flux. The calculation corresponds to the one shown in Fig. 5 (QC not fixed). (a) The blue (dark gray) segments represent basal polarization, and the orange (light gray) ones represent apical polarization. Black lines without decoration represent unpolarized walls. In the case where two segments of different color meet, diffusion through the wall is increased. (b) The blue (dark gray) arrows represent auxin flowing upwards, and downward flux are orange (light gray) ones. The size of the arrows is fixed to help the eye, but the real amount of auxins being transported is very small in most cells. Observe the reverse fountain pattern in which the auxin in the outer cells flow upwards while in the center they flow mostly downwards, particularly on the CPD. Notice also the formation of vortices in the region of maximum auxin concentration near the SCN. (See videos V2 and V3 in the Supplemental Material [38]).

Figure 7

Comparison between experimental data and numerical simulations for the length of the cells along the apicobasal axis of the root. We show the cell length obtained with three values of the elastic constant of the epidermal cells, using $\beta =15$, for two cases, first when the QC is fixed (top) and the other one when the QC is not fixed (bottom). The continuous red lines correspond to the experimental data reported in Ref. [34]. In the insets we show the effect of altering the cell-cycle duration (inner graphs), keeping the elastic constant $K_e=120$. The images on the right correspond to the final state of the root with $\beta =15$ and $K_e=120$ [blue (dark gray) circles].

Figure 8

Distribution of auxins for a U-shape root bent away from QC. We show three cases, when the bend is near the tip of the root (left), when it is far from the tip (right), and an intermediate case. The figures show the distribution of the relaxed quantities before cell proliferation takes place at the same time of the dynamics. Observe that auxins mainly concentrate near the curved region in all configurations. Epidermal cells are indicated with black circles. The color scale is the same as in former figures.

Figure 9

Histogram of the number of cell divisions as a function of the distance from the tip of the root (a) in the straight root shown in Fig. 3, (b) in the bent root shown on the right of Fig. 8. In both figures the potential profile is shown as red dots. Observe the different horizontal scale.