Brassinosteroid signaling delimits root gravitropism via sorting of the Arabidopsis PIN2 auxin transporter

Arabidopsis PIN2 protein directs transport of the phytohormone auxin from the root tip into the root elongation zone. Variation in hormone transport, which depends on a delicate interplay between PIN2 sorting to and from polar plasma membrane domains, determines root growth. By employing a constitutively degraded version of PIN2, we identify brassinolides as antagonists of PIN2 endocytosis. This response does not require de novo protein synthesis, but involves early events in canonical brassinolide signaling. Brassinolide-controlled adjustments in PIN2 sorting and intracellular distribution governs formation of a lateral PIN2 gradient in gravistimulated roots, coinciding with adjustments in auxin signaling and directional root growth. Strikingly, simulations indicate that PIN2 gradient formation is no prerequisite for root bending but rather dampens asymmetric auxin flow and signaling. Crosstalk between brassinolide signaling and endocytic PIN2 sorting, thus, appears essential for determining the rate of gravity-induced root curvature via attenuation of differential cell elongation.


Supplementary Note 1 -A model of IAA transport in the root apex
To assess effects of PIN2 asymmetry in the root apex on intracellular auxin concentration and its temporal dynamics, we simulated auxin transport in the root tip. For this purpose, we transformed our recently developed 3D model 1 into 2D (Supplementary Fig.11a). The modeled root tip comprises a multicellular structure with isolated intracellular compartments and an extracellular compartment (the cell wall), within which auxin can diffuse freely, following the framework used in previous models of auxin transport 2,3,4 . Auxin transport between adjacent compartments is governed by its concentration gradient and membrane permeability.
There are three distinct paths for auxin to permeate the cell membrane: diffusion, influx carriers and efflux carriers. A major governing factor for auxin transport is the distribution of membrane permeabilities for auxin, which depend on localization of the carrier proteins (AUX1, PINs and others) 2, 3, 4 . Our model also contains a media compartment around the root that makes it possible to apply boundary conditions simulating experiments, in which roots are placed on top of the agar block and thus have contact with agar gel, where auxin can diffuse (see Methods).

Supplementary Note 2 -Model Geometry
Presenting the root tip as an array of rectangular cells has computational advantages and provided useful quantitative results in the past 2, 4, 5 . We draw a 2D rectangular geometry of a root tip, containing cells of the root apex: meristematic zone, elongation zone and the beginning of the differentiation zone ( Supplementary Fig.11a). This schematic root apex is ~ 1200 μm in length and 160 μm in width. The outmost domain represents the surrounding growth medium (agar block). Widths of the cell files in our model correspond to those in the elongation zone, taken from sectioned confocal images 3 . The heights of the cells are also taken from confocal images. Lateral root cap cells are drawn as an additional cell file on the outer side of the meristematic zone. Cell wall thickness is set to 200 nm.
The geometry presented in Supplementary Fig.11a serves as a main model. In order to test dependence of the results of our simulations on root geometry, we transformed the main geometry, and also tested a geometry, which is based on the image of a root cross section ( Supplementary Fig.11cd, Supplementary Fig.12). These model variants are described below in Supplementary Note 2.2.
as: 2789 = 0.35 ± 0.07 µm/s. We assume ;28),= = 2789 3 . Permeability for PINs has been previously estimated as ~ 0.3 µm/s 2 . One strategy to estimate PIN efflux permeabilities in epidermis cell files is based on polar auxin transport (PAT) velocities 8 . Measured auxin velocities in the Arabidopsis root are in a range of 0.3 to 3 µm/s [1]. In our model, we set the efflux permeability of PIN2 value, such that the shootward auxin flux in the epidermis equals 1 / . Such velocity is produced by DEF) = 0.5 ± 0.2 µm/s. This is close to the value estimated by Rutschow et al. 7 . We assume that all PIN-containing cells have the same DEF , disregarding different PIN types and their overlapping expression in certain cell files. Thus, G1H9 = G1H=/G1HI = G1H) = 0.5 µm/s. This assumption is rough, however, as we aim to calculate [IAA]cell mainly in the epidermis, we choose 2789 and G1H) as the critical values for this model. (ii) Auxin synthesis and degradation are neglected in the model, since no clearly defined values are available for the analyzed cell files. Nevertheless, Kramer and Ackelsberg estimated that synthesis contributes less than 1% of the auxin amount present in fast transporting tissues such as roots 11 . Auxin degradation is a slow process compared to transport 3,11 . As we have an infinite pool of auxin (coming from the shoot), we considered degradation as negligible.
For the complete list of the parameters used in the model, see Supplementary Table 1.

Supplementary Note 4 -Governing equations: auxin diffusion within compartments and membrane auxin fluxes
Auxin concentration is calculated by solving a system of Partial Differential Equations (PDEs) that describes how the auxin concentration IAA( , ) within each compartment evolves due to bulk auxin diffusion (Fick's law) and the mass conservation law: Normal diffusive auxin flux at the boundaries of compartments equals auxin flux across membrane between compartments: (=molecules passed through membrane per unit time per unit area), which is governed by membrane permeabilities: which depend on acidity of the wall j'"" , cytoplasmic i""" , the logarithmic dissociation constant of auxin and the membrane potential / . W is the Faraday constant, is the gas constant and is the temperature. Derivation of • , • is provided in detail in 2 and 3 . For j'"" = 5.3, i""" = 7.2, / = −120 , typical for the root, these constants have the following values: 9 = 0.24, 9 Fig. 8b,e).
Case 2 Gravistimulation: When simulating gravistimulated roots, the initial conditions are the same as in case of no stimulation, except for columella cells containing G1H=/G1HI (see Fig. 8c for a scheme). Permeability on the membrane of the lower side of these cells is set higher: We assumed that PIN2 effects on auxin membrane permeability are proportional to PIN2 abundance and, accordingly, the ratio of permeabilities is proportional to the PIN2:Venus signal ratio, as presented in Fig Thus, in this variant of the model we assume that both permeabilities, on the upper and on the lower side depend on : for example, when PIN2 becomes asymmetric with < 1, G1H) JKKLM NOPL decreases and G1H) TU,LM NOPL increases. This is an unverified assumption, because experimental data in our present study provides only values of the PIN2 signal ratio ( ).
However, previous studies 12 presented some evidence that indeed PIN2 abundance is affected on both sides of the root apex. Supplementary Eq. 11 and 12 are applied to all PIN2containing membranes in groups at the "upper side" and "lower side", as depicted in Fig. 8a.  (Fig. 8gh, Supplementary Fig. 12c, Supplementary Fig.13d, Supplementary   Fig.14e, Supplementary Fig.15ab, Supplementary Fig.16ad, Supplementary Fig.17). For the calculation of the temporal dynamics of IAA after the onset of PIN2 asymmetry, the steady state solution was used as an initial IAA concentration (Fig. 8e,f). Time-dependent solver was then used to calculate IAA concentration at each time step, which were determined by the solver.

Supplementary Note 10 -Effect of the root geometry
We tested several root geometries to assess the sensitivity of the accumulation ratio   Second, in all the models we have observed a slight change in the slope 122 §¨©ª« 122 ¬--ª« ( ) in case of gravistimulation ( Supplementary Fig.11g, Supplementary Fig.14e). This result shows that the initial 122 §¨©ª« 122 ¬--ª« caused by PIN3/7 asymmetry depends not only on the degree of PIN3/7 asymmetry (and corresponding gravistimulation strength), but also on cell shapes. This is due to the fact that relative lengths of the sides of the cell defines the proportion of influx and efflux carriers and thus influences differential auxin fluxes. 2) Realistic image-based geometry ( Supplementary Fig.12a).

Resulting graphs for
To evaluate the influence of the squarish cell geometry on auxin distribution, we constructed a realistic root geometry using a published 2D root image, modified from the Simuplant model 122 ¬--ª« . This is due to very short geometry and influence of the sink boundary conditions close to the EZ cells.
3) eBL-treated root geometry ( Supplementary Fig.12b) e-BL-treated roots have more elongated cell shapes in meristem, which consists of fewer cells, and overall thickness of the root is smaller. We simulated these morphological changes in our squarish geometry and found only a slight enhancement of the initial

4) Geometry with wide lateral root cap cells (for better visualization).
LRC cells width is defined in the models (main, elongated and eBL-treated geometries) based on thickness of LRC cells in real roots (width~ 5um). We tested how auxin concentrations in EZ will depend on 2x times increase in width of LRC, and found that it leads only to slight enhancement of the initial These examples demonstrate that the distance to the sink and source boundary influences absolute auxin concentrations, but does not influence the auxin ratio functions are presented in Supplementary Fig.13d.

Supplementary Note 11 -Effect of boundary conditions
We further have tested the sensitivity of auxin concentration ratios to changes in external boundary conditions in the main model (depicted on Supplementary Fig.11e).
1) No-flux condition around the root surface ("close BC") leads to higher auxin concentrations, and columella-induced gravistimulation has a stronger effect on 122 §¨©ª« 122 ¬--ª« , compared to the "Main BC", meaning that less input by gravistimulation is required to achieve the same 122 §¨©ª« 122 ¬--ª« as under default boundary conditions. 2) Accordingly, "far BC" lead to lower auxin accumulation (higher auxin loss) and columella-induced gravistimulation has a weaker effect on 3) The distance from the root tip to the "source" and "sink" boundaries ('shootward end') also influences absolute concentrations of auxin, but has no effect on auxin ratios induced by Auxin concentration in the "no stimulus" and "gravistimulation" scenarios, appear to be similar to the main model (Fig. 8c,d vs. Supplementary Fig.14c); (except for the LRC cells, in which highest auxin concentration appears to be not in the last LRC cell in a row). Accordingly, dependence of the auxin ratio on PIN2 asymmetry 122 §¨©ª« 122 ¬--ª« ( ) coincide completely with the one of the main model ( Supplementary Fig.14e).
Next, we completely eliminated PIN2 in these cortical cells. This results in auxin accumulation in these cells ( Supplementary Fig.14d). However, calculations of OrO¡OsT ≤ 10 ( Supplementary Fig.15a,b). In this case, required increase in G1H) OrO¡OsT is within the experimentally observed range (~1.5-fold at most). For extreme values of G1H) OrO¡OsT , the effect of PIN2 asymmetry diminishes (see next section).

Supplementary Note 14 -Effect of values on the slope of the
We tested the sensitivity of the accumulation ratio OrO¡OsT permeability values, whilst permeabilities of other transporters remained unchanged. Initial conditions for the 'No gravistimulation' case were used. Results are presented in Supplementary Fig.16a. The slope of the curve OrO¡OsT (Fig. 8h).
Next, we investigated the dependence of the auxin ratio on the overall AUX1 permeability.  Supplementary Fig.17a), and 2) there is a minimal gravistimulation strength (= 60% / 40%) needed to achieve the observed auxin ratio in eBL-treated roots (  Fig.17a, red line).
Importantly, PIN2 asymmetry leads to diminished 122 §¨©ª« 122 ¬--ª« in case of gravistimulation compared to no asymmetry (compare graphs for = 1 and = 0.5) for any value of 2789 , thus AUX1 abundance does not influence this main conclusion of our model (Supplementary Fig.17b).
Next, Supplementary Fig.17c    OrO¡OsT and ß 2789 (bar #10). Alternitevely, simultaneous threefold increase in ß G1H) OrO¡OsT and ß 2789 is required for that, if PIN2 asymmetry is unchanged, when compared to the control (bar #11), which is much higher increase, than measured experimentally in Supplementary Fig.   15c. 2) The coefficient of this proportionality depends on the initial IAA ratio; therefore, effects are higher in gravistimulated roots:
3) The effect of PIN2 asymmetry on  Fig.16a) and AUX1 permeability is in the order of 0.1 -3 um/s ( Supplementary Fig.17b). We conclude that the control roots might display this maximum theoretical effect of PIN2 asymmetry on 122 §¨©ª« 122 ¬--ª« in vivo, because permeability values, reported in literature (see Supplementary Table 1)  6) The time frame required for [IAA]cell establishing steady-state is ~1000 sec (Fig. 8ef) All models pH in cytoplasm pHcell 5.3 [3] All models pH in the cell wall pHwall 7.2 [3] All models IAA dissociation constant pK 4.8 [3] All models IAA diffusion in the cell i 600 µm ) /s [2] All models IAA diffusion in the media k 60 µm ) /s [2] All models IAA diffusion in the wall j 32 µm ) /s [6] All models Plasma membrane potential Vm -120 mV [3] All models Cell wall thickness 0.2 µm rounded from [3] All models