Auxin-driven patterning with unidirectional fluxes

Highlight The key modes of auxin-driven patterning— the formation of convergence points and canals—depend on whether auxin efflux and influx feed back on polar auxin transport synergistically or antagonistically.


S1 Parameters used in simulations
S9 S10 S11 S12 S13 S14      Figure S1: Simulation of canalization using Mitchison's model implemented with tally molecules. The model is defined by the Petri net in Fig. 4a. The cells in the top row are sources of auxin. The centre cell produces auxin at a relatively higher rate. The cell in the centre of the bottom row is an auxin sink.  Differential equations for our models are derived directly from their Petri net representation. We provide the equations for the model shown in Fig. 6a as an example.

S2 Supplementary figures
The change in the concentration of intracellular auxin, [A], is where T out is the efflux rate coefficient, and are extracellular auxin concentrations on the left and right sides of the cell. Finally, σ a is the auxin production rate, and µ a is the degradation rate.
We focus on the left membrane segment, as the equations for the right segment are analogous. The change in auxin concentration in the extracellular compartment on the left side of the cell, The change in PIN concentration in the left membrane is where σ px is the rate of PIN exocytosis dependent on tally molecule X L , [PIN] is the concentration of PIN in the cell interior, σ p is the constitutive rate of exocytosis, and µ p is the rate of endocytosis. We assume that the total concentration of PIN in the cell, [PIN total ], is constant, so that Finally, changes in the concentrations of tally molecules are described by the equations where µ x is the degradation rate of X, and ν xy is the rate with which X and Y annihilate each other.

S3.2 Equations describing steady state of the network from Figure 6c
In Section 'Dual polarization' (main text) we noted that that the steady-state concentration of tally molecules in the network from where J I→L denotes auxin efflux from the cell interior I to the extracellular space L, and J L→I denotes auxin influx from L to I. If ν xy = 0, these equations have solutions: In this case, the tally molecules measure directly the efflux and influx of auxin, which reduces the network from Fig. 6c to that from Fig. 6b. To find concentration [X L ] in the general case ν xy = 0, we : After substituting this value into Equation S7, we obtain where φ I→L = J I→L − J L→I . This quadratic equation has positive solution which has positive solution Unfortunately, Equations S13 and S15 are difficult to interpret due to their relatively complex form.
For this reason, we have not analyzed the network from Fig. 6c directly, but exploited its similarities to the networks from Fig. 6a and 6b instead.

S3.3 Model used to generate Figure 15
In Section 'Discussion' (main text) we observed that the key modes of auxin-driven patterning can be distinguished by contrasting roles of the auxin influx, which may act either antagonistically of synergistically with the efflux. These possibilities are illustrated in Fig. 15. To create it, we developed a simplified model of cell polarization, in which PIN allocation to the membranes is controlled directly by unidirectional fluxes. Here we present the equations defining this model (due to the simplification, it cannot be specified by a Petri net).
We consider a single cell sandwiched between two extracellular compartments, top (T ) and bottom (B). PIN concentration in the membrane facing either compartment is controlled by a linear combination of auxin efflux and influx. In the case of the top membrane (an analogical formula applies to the bottom membrane), this combination is Auxin efflux J I→E T and influx J E T →I thus act synergistically if α > 0 and antagonistically if α < 0.
Given Q I→E T , PIN allocation to the top membrane changes according to the equation where [PIN] is the concentration of PIN not allocated to either membrane: The model is completed by the equations describing unidirectional fluxes. The efflux from the cell to the top extracellular compartment is calculated as where T out is an efflux rate constant and [A] is auxin concentration in the cell. Changes in this concentration are calculated as the sum of net fluxes between the cell and the extracellular compartments, as well as the production (with rate σ a ) and turnover (with rate µ a ) of auxin: The influx from the top extracellular compartment is calculated as where T in is an influx rate constant, [A T ] is auxin concentration in the extracellular compartment, and [AUX] is the AUX/LAX concentration in the cell (assumed to be constant).
The values of model parameters, initial conditions and boundary conditions used to create Fig. 15    and below each cell). These propagating polarizations converge at the farthest location from the first cell and produce an auxin maximum there (Fig. S7d). Compared to the initial state, concentration differences are now reversed: the cell with the highest initial concentration has become the cell with the lowest concentration. The pattern in Fig. S7e represents a stable steady-state configuration of the system.

S3.4 Analysis of pattern emergence in a file of cells with antagonistic or synergistic polarization mechanisms
The described process is robust, in the sense that a similar pattern, with a single convergence point at the location opposing the first cell, is produced in shorter or longer files of cells (results not shown). If the initial conditions are changed so that the first cell has a lower auxin concentration than others, the convergence point emerges in the first cell (Fig. S8). In either case, the antagonistic action of auxin efflux and influx results in a coordinated polarization of cells, with PIN oriented towards a single convergence point. This point is located away from the initial auxin maximum, or at the initial auxin minimum.  Figure S9 and Video S16 illustrate pattern formation with the auxin efflux and influx acting synergistically. The simulations of antagonistic and synergistic polarization differ only by the value of parameter α, which is now positive (α = 0.135). The initial state is the same, with the first cell having higher auxin concentration than other cells (Fig. S9a). As in the previous simulation, the initial fluxes are directed away from the first cell and propagate to the neighbouring cells. A gradient of auxin concentrations and total fluxes results, decreasing away from the first cell (Fig. S9b). Synergistic polarization implies that PIN will be localized preferentially to the membranes with a higher total flux. As the gradient of fluxes coincides with the gradient of auxin concentration, PIN becomes polarized towards the neighbouring cell with a higher auxin concentration (Fig. S9c). This polarization is opposite to the net auxin flux (note the opposite slope of red and blue lines in Fig. S9c). Eventually, the direction of net flux reverses to coincide with polarization ( Fig. S9d; note that the slopes of red and blue lines now concur). As a result, a convergence point forms in the first cell, i.e., at the location of the initial auxin maximum (Fig. S9e).

S3.4.2 Synergistic polarization
If the first cell has a lower initial auxin concentration than other cells, the convergence point emerges in the cell that is furthest away from the first cell (Fig. S10). Independently of the initial conditions, in longer cell files additional, approximately equally spaced convergence points may emerge, as the depletion of auxin in the proximity of one convergence points can lead to the formation of the next convergence point at some distance away. The number of convergence points increases as the length of the file increases (Fig. S11)  Video S11: Canalization with exocytosis controlled by influx and efflux carriers that act via a mediating molecule. The model is defined by the Petri net in Fig. 11a. Selected frames from this simulation are shown in Fig. 12a.
Video S12: Convergence point formation with exocytosis controlled by influx and efflux carriers that act via a mediating molecule. The model is defined by the Petri net in Fig. 11a. Selected frames from this simulation are shown in Fig. 12b.
Video S13: Canalization with endocytosis controlled by influx and efflux carriers that act via a mediating molecule. The model is defined by the Petri net in Fig. 11b. Selected frames from this simulation are shown in Fig. 13a.
Video S14: Convergence point formation with endocytosis controlled by influx and efflux carriers that act via a mediating molecule. The model is defined by the Petri net in Fig. 11b. Selected frames from this simulation are shown in Fig. 13b.
Video S15: Dynamics of pattern formation in a file of cells, with PIN polarized by auxin efflux and influx acting antagonistically. The model is defined in Section S3.3. Selected frames from this simulation are shown in Fig. S7.
Video S16: Dynamics of pattern formation in a file of cells, with PIN polarized by auxin efflux and influx acting synergistically. The model is defined in Section S3.3. Selected frames from this simulation are shown in Fig. S9.