A recovery principle provides insight into auxin pattern control in the Arabidopsis root

Regulated auxin patterning provides a key mechanism for controlling root growth and development. We have developed a data-driven mechanistic model using realistic root geometry and formulated a principle to theoretically investigate quantitative auxin pattern recovery following auxin transport perturbation. This principle reveals that auxin patterning is potentially controlled by multiple combinations of interlinked levels and localisation of influx and efflux carriers. We demonstrate that (1) when efflux carriers maintain polarity but change levels, maintaining the same auxin pattern requires non-uniform and polar distribution of influx carriers; (2) the emergence of the same auxin pattern, from different levels of influx carriers with the same nonpolar localisation, requires simultaneous modulation of efflux carrier level and polarity; and (3) multiple patterns of influx and efflux carriers for maintaining an auxin pattern do not have spatially proportional correlation. This reveals that auxin pattern formation requires coordination between influx and efflux carriers. We further show that the model makes various predictions that can be experimentally validated.


Figure S
the QC, w centre; C (B) Conc both PIN the root t S1. Wildtype a which is set to COR: cortex; S centration colo N1 and PIN2 at tip.   (b) show the same two regions of the root, and they demonstrate that, although auxin patterning is recovered for both cases, this ratio of ratios is generally not unity. This result implies that PIN and AUX1/LAX patterns that maintain the same auxin pattern do not exhibit spatially proportional correlation. Comparing figures A and B shows that this ratio of ratios is different for 100% loss of PIN3,4,7 from that for 100% gain of PIN3,4,7. This result implies that 100% loss of PIN3,4,7 and 100% gain of PIN3,4,7 require different relationships between PIN and AUX1/LAX to maintain the same auxin patterning.  (B) show the same two regions of the root, and they demonstrate that, although auxin patterning is recovered for both cases, this ratio of ratios is generally not unity. This result implies that PIN and AUX1/LAX patterns that maintain the same auxin pattern do not exhibit spatially proportional correlation. Comparing figures A and B shows that this ratio of ratios is different for 50% loss of PIN3,4,7 from that for 50% gain of PIN3,4,7. This result implies that 50% loss of PIN3,4,7 and 50% gain of PIN3,4,7 require different relationships between PIN and AUX1/LAX to maintain the same auxin patterning.
Figure S12. Modelling predictions on the combined PIN1 and PIN2 concentration patterns for 100% loss of PIN3 or PIN4 or PIN7 and for the combined 100% loss of PIN3, PIN4 and PIN7. This figure shows percentage difference of PIN1,2 concentration from wildtype. From left to right, PIN3, PIN4, PIN7 or total PIN3,4,7 concentration is set to be zero, respectively.  A data-driven mechanistic model for studying the control of auxin patterning in Arabidopsis root development The data-driven mechanistic model developed in this work integrates actual cell geometries, the level and polar or nonpolar localisation of auxin influx and efflux carriers, with a variety of experimental data about hormonal crosstalk, as described below.

Root structure with actual cell geometries, polar localisation of efflux carriers and nonpolar localisation of influx carriers
The digitised root structure was created using an image ( Figure 1A) downloaded from www.simuplant.org (Band et al., 2014) which was generated from stacks of confocal images of roots stained with propidium iodide to define root geometry and cell organization. Software (SurfaceProject) was developed (Band et al., 2014) to process the confocal image data to produce a 2D root structure ( Figure 1B). . LRC -lateral root cap; S1 to S5 -columella; CE cortical endodermis; QC quiescent centre.
The initial downloaded SimuPlant image ( Figure 1A) was scanned with ImageJ and the grid point data saved as an Excel file. The scanned image contained multiple imperfections and discontinuities in the

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cell wall structures. To correct these, the file was processed by a series of error-checking MATLAB programs. The digital image was first searched for discontinuities in the cell walls which were corrected by interpolation from the end of a wall along points of weak ImageJ signal until another wall point was encountered. The resulting image was searched again for abnormal groups of cell wall grid points (GPs) and discontinuities, which were identified and removed. These processes were repeated several times since it was possible for the correction of one type of error to create another. The resulting file was formatted to highlight the cell walls to allow a visual check of the digitised root for remaining abnormalities or discontinuities. Finally all cytosolic GPs were set to 0 and cell wall GPs to 1, with GPs outside the root set to 9 and then the image was visually checked against the 2D map from Band et al., 2014 ( Figure 1B). The next step was to create an individual cell wall for each cell. This was achieved by duplicating all cell wall GPs and then replacing the double wall at the exterior of the root with a single wall. The resulting root image was again searched for larger blocks of cell wall GPs which were each individually checked and corrected if necessary. The final image was visually checked for any wall discontinuities or abnormalities and a few manual adjustments made. In the basic root map, some cell wall GPs were adjacent to cytosolic points, having a nearest neighbour (NN) cytosolic point to the N, S, E or W. Other wall GPs did not have a NN cytosolic point, especially at multiple cell junctions, and could therefore be regarded as forming an extracellular space. The above process resulted in a basic digital root map which required further properties to be assigned to each GP for efficient use in the model. For example it would be necessary to define polarised efflux and non-polar influx carrier properties to the plasma membrane in selected cells. To allow automation of these steps, all cytosolic points were assigned cell numbers to allow automated identification of specific cells (Table 1) and apical, basal, and inner and outer lateral cell faces were defined for each cell. The final root map matrix contains 374,900 grid points defining 501 cells. Once the digital root map had been defined, with an individual cell wall structure for each cell and unique cell numbers (Figure 2A), it was necessary to assign the levels and localisation of the polar PIN efflux carriers and the non-polar AUX1/LAX influx carriers to the plasma membrane (included in the cell wall properties). PIN1 and 2 carrier levels are regulated in exactly the same way by the crosstalk network which also regulates the rate that the cytosolic PIN proteins are placed and removed from the plasma membrane, as previously described . Since the model does not differentiate between PIN1 and PIN2 (apart from polarity localisation), for the purposes of this section PIN1 and 2 will be jointly referred to as PIN12. Depending on the location in the root, the cell wall GPs are assigned different properties to define the rate of transfer of the PIN12 proteins from the cytosolic GPs to the neighbouring plasma membrane . The PIN12 transfer rates (low, medium or high) were based on experimental images from various publications. It was assumed that PIN12 are differentially expressed everywhere in the root tip , therefore as a default, all PIN12 transfer rates to the plasma membrane were initialised at a low level and then transfer rates at selected cell faces were reset to medium or high based on experimental images. Figure  2B summarises the transfer rate settings for PIN12 which define PIN12 polarity (1 = Blue/Low, 2 = Green/Medium, 3 = Red/High).
PIN12 transfer rates in the lateral root cap cells were set at medium (Green) on the apical cell faces from approximately the 200 level from the distal end of the root ( Figure 2B) up to approximately the 325 level after which they were set to high (Red) on the apical cell faces (Laskowski et al., 2008). In the epidermal cells, the transfer rates were left at low (Blue) on all cell faces for the first 3 cells apical to the QC. For the next 3 cells they were set at medium on the apical faces and low on the lateral and basal faces, and thereafter set at high on the apical face and low on the lateral and basal faces (Laskowski et al., 2008;Muller et al., 1998). For the cortical cell files for the first 3 cells apical to the QC the transfer rates were set to low at all faces. For the next 3 cells the rates were set to medium on the basal face and low on the inner and outer lateral and apical faces. The rates were then set to high on the basal face, medium on the outer lateral and low on the inner lateral and apical faces through the meristematic zone (MZ) to approximately the 650 level. As the cells move out of the MZ there is a basal to apical shift and the transfer rates were then set at high on the apical cell face, medium on the outer lateral face and low on the basal and inner lateral faces (Kleine-Vehn et al., 2008;Muller et al., 1998). The outer lateral settings in the cortical cells were based on Figure 1D from Kleine-Vehn et al., 2008. In the vascular cylinder and pericycle, the transfer rates were set to high at the basal face of all cells and high on the inner lateral of the pericycle cells (Friml et al., 2002) while the rates at the remaining vascular and pericycle cell faces were set at low. The transfer rates were set at medium on the basal face and at low on the inner and outer lateral and apical faces of the endodermis cells (Friml et al., 2002).
A B Figure   The PIN3, PIN4 and PIN7 efflux carrier concentration levels and polar localisation are not regulated by the network but have prescribed concentrations at selected cell faces based on experimental imaging from the literature (Blilou et al., 2005) with concentrations levels adjusted to produce WT auxin patterning (Figure 3). The model is set up with 4 possible concentration levels (however only 3 are used) for each efflux carrier for which concentrations are assigned by the user at model run time so that concentrations can be easily adjusted when searching for WT auxin patterning. PIN3 has nonpolar localisation at a high level in the columella S1 and S2. It is localised at a high level at the basal face and at a medium level at the inner and outer lateral and apical faces of the vascular cells in the elongation zone (EZ). In the pericycle cells in the EZ, it has a high level of localisation at the inner lateral and basal faces and a medium level at the outer lateral and apical faces (Figure 3). It has nonpolar localisation at a low level in all other cells. PIN4 has non-polar localisation at a medium level in the QC and initials and their immediate neighbours. In the other cells in the distal meristematic zone (MZ) it is localised at a high level at the basal face and at a medium level at the apical and inner and outer lateral faces. In all other cells, PIN4 has non-polar localisation at a low level at all cell faces (Figure 3). PIN7 is localised at a medium or high level at all cell faces in the pericycle and vascular cells in the MZ and EZ and in certain columella cells. In the pericycle cells it is localised at a high level on the inner lateral and basal faces and at a medium level at the apical and outer lateral faces. In the vascular cells it is localised at a high level at the inner and outer lateral and basal faces and at a medium level on the apical face. It is localised in a non-polar distribution at a high level at all faces of the columella S1 and S2 cells. At all other cells PIN7 has non-polar low level localisation (Figure 3).
The non-polar localisation of the auxin influx carriers AUX1, LAX2 and LAX3 (Figure 4) is again based on experimental imaging (Band et al., 2014) with concentrations adjusted to achieve WT auxin patterning. The model allows 15 possible concentration levels for AUX1, 8 possible levels for LAX2 and 4 levels for LAX3 (however only 3 were used for each carrier to define WT), and specific concentrations are assigned to each level by the user at model run time. AUX1 is localised at a medium level in the lateral root cap, at a medium level in the EZ and apical MZ of the epidermis, and at a medium level in the cortical cells in the EZ. It is localised at a medium level in the cortical and epidermal cells just apical to the QC and at a high level in the columella S1-S4. In all other cells AUX1 is localised at a low level. LAX2 is localised at a high level in the vascular and pericycle cells in the mid to distal region of the MZ, and in the QC and columella initials. There is zero LAX2 concentration in all other cells. LAX3 is localised at a high level in the columella S2 and has a zero concentration in all other cells.

Hormonal crosstalk between PIN1,2 and three hormones (auxin, ethylene and cytokinin)
Crosstalk between auxin, ethylene, cytokinin and PIN1 and PIN2 was previously described using a network (Liu et al., 2010(Liu et al., , 2013Moore et al., 2015), which was constructed by iteratively combining modelling and experimental measurements. In previous research , we considered that AUX1 activity is positively regulated by the downstream ethylene signalling based on experimental observation ( Figure 7B in Ruzicka et al., 2007). Model results for AUX1 patterning (Fig. S12 in Moore et al., 2015) are in part similar to experimental imaging (Fig. S8 in Band et al. 2014) with AUX1 levels increasing proximally in the epidermis, and higher AUX1 levels in the outer cell layers compared to the central cell cylinder. Experimentally, it has been shown that, within the epidermis, AUX1 is present mainly in the elongation zone cells (Band et al. 2014). However, the model does not exhibit the elevated experimental AUX1 levels in the columella and near the QC or the proximally declining AUX1 levels in the central cylinder. We concluded that the differences between modelling and experimental results may indicate that, in addition to ethylene, other effectors may also regulate AUX1 activity . Therefore, we consider that the crosstalk between AUX1 and three hormones (auxin, ethylene and cytokinin) cannot be fully established so that AUX1 patterning can be fully predicted using the model. Thus, in this research, AUX1 localisation is prescribed using experimental data.
Although the regulatory relationships between auxin, ethylene, cytokinin and polar PIN1 and PIN2 proteins were previously established by iteratively combining experimental measurements with modelling analysis (Liu et al., 2010(Liu et al., , 2013Moore et al., 2015;Rowe et al., 2016), it is currently not possible to construct the crosstalk network between the three hormones and other auxin carriers, due to insufficient data and crosstalk complexity. For example, experimental and modelling analysis has suggested that spatial expression patterning of the influx carrier LAX3 is affected by expression of the efflux carrier PIN3 (Peret et al., 2013). However, other modelling and experimental analysis suggests that the induction of PIN3 is not required to explain the switch-like expression of LAX3 (Mellor et al., 2015). The feedback of GH3, which is an important component in the auxin-degradation pathway, may also have a role in LAX3 expression (Mellor et al., 2016). Furthermore, a recent study has shown that the regulation of PIN3 and PIN7 expression by auxin and cytokinin in root development follows different mechanisms (Lavenus et al., 2016;Wang et al., 2015). These examples show the complexity of crosstalk between auxin, its carriers and other hormones.
In addition, extensive examination of published experimental data reveals that it is currently not possible to construct a network between the three hormones and other carriers, due to insufficient data and the complexity of crosstalk between hormones and auxin carriers. Therefore PIN3,4,7, AUX1 and LAX2 and 3 localisation is prescribed using experimental data.
Therefore, in the model developed in this work, the following crosstalk network ( Figure 5) controls metabolism of two efflux carriers (PIN1and PIN2) and the hormones auxin, ethylene and cytokinin in the cytosolic spaces. The network allows quantitative description of PIN1 and PIN2 regulation by the three hormones and enables study of the relationship between auxin, PIN1 and PIN2 patterning.

A model that integrates actual cell geometries, the level and polar or nonpolar localisation of auxin influx and efflux carriers, with a variety of experimental data about hormonal crosstalk
The current research develops a model integrates actual cell geometries, the level and polar or nonpolar localisation of auxin influx and efflux carriers, with a variety of experimental data about hormonal crosstalk. We note that our previous research  did not include LAX2,3 and PIN3,4,7, and it used a rectangular root structure that does not describe realistic cellular geometries. Moreover, the rectangular root structure used in our previous research  does not have a root cap, and it does not include extracellular space. The model developed in the current research integrates a root structure with actual cell geometries, the level and polar or nonpolar localisation of all auxin influx (AUX1, LAX2,3) and efflux carriers (PIN1,2,3,4,7), with a variety of experimental data about hormonal crosstalk. Since the ABCB family of auxin carriers can reversibly redirect auxin flux, the role of ABCB transporters has been incorporated into PIN and AUX1/LAX activity to simplify modelling analysis. Therefore, the current research has integrated all known important auxin transporters for cell to cell communication with a wide range of experimental data about the crosstalk between PIN1,2 and three hormones (auxin, ethylene and cytokinin) (Liu et al., 2010(Liu et al., , 2013Moore et al., 2015 and references therein).