© 2022. Published by The Company of Biologists Ltd | Development (2022) 149, dev200510. doi:10.1242/dev.200510 RESEARCH ARTICLE BRAVO self-confined expression through WOX5 in the Arabidopsis root stem-cell niche Josep Mercadal1,2, Isabel Betegó n-Putze3, Nadja Bosch3, Ana I. Caño-Delgado3 and Marta Ibañes1,2,* ABSTRACT Xie and Spradling, 1998; Zhang and Kalderon, 2001). Overproduction of these signals can drive an increase in the In animals and plants, stem-cell niches are local microenvironments number of stem cells, resulting in enlarged niches and often leading that are tightly regulated to preserve their unique identity while to malfunctioning of the surrounding tissue or even whole organs communicating with adjacent cells that will give rise to specialized cell (Spradling et al., 2001). Knowing the origin and function of these types. In the primary root of Arabidopsis thaliana, two transcription signals is therefore essential to understand the role of stem cells in factors, BRAVO and WOX5, among others, are expressed in the the processes underlying organism development and sustenance. stem-cell niche. Intriguingly, BRAVO, a repressor of quiescent center In the model plant Arabidopsis thaliana, highly mobile hormones, divisions, confines its own gene expression to the stem-cell niche, as such as auxin (Aida et al., 2004; Ding and Friml, 2010; Müller and evidenced in a bravo mutant background. Here, we propose through Sheen, 2008), as well as short-range transcription factors like mathematical modeling that BRAVO confines its own expression WUSCHEL and WUSCHEL-RELATED HOMEOBOX 5 (WOX5) domain to the stem-cell niche by attenuating a WOX5-dependent (Sarkar et al., 2007; Yadav et al., 2011) are involved in specifying diffusible activator of BRAVO. This negative feedback drives WOX5 stem-cell niches. In the root apical meristem of Arabidopsis, the activity to be spatially restricted as well. The results show that WOX5 SCN lies at the tip of the root, a location known to be established diffusion and sequestration by binding to BRAVO are sufficient to by positional information conferred by the overlapping of the drive the experimentally observed confined BRAVO expression at the SCARECROW and SHORTROOT transcription factors, together stem-cell niche. We propose that the attenuation of a diffusible with the activation of PLETHORA genes by the hormone auxin, the activator can be a general mechanism acting at other stem-cell niches levels of which peak at the position where the SCN is established to spatially confine genetic activity to a small region while maintaining (Kidner et al., 2000; Shimotohno et al., 2018; van den Berg et al., signaling within them and with the surrounding cells. 1995). This positional signaling allows for the necessary plasticity KEY WORDS: BRAVO, WOX5, Root, Stem-cell niche, Diffusion, to establish a new niche when it has been destroyed or damaged, by Sequestration a continual supply and renewal of stem cells at the same location (Sena et al., 2009). INTRODUCTION The SCN is composed of a small group of rarely dividing Both in animals and plants, stem cells are maintained in tightly pluripotent cells called the quiescent center (QC) and of the regulated microenvironments called stem-cell niches (SCNs), in immediately surrounding stem cells, i.e. the vascular initials (VIs), which they remain in an undifferentiated state and provide a columella stem cells (CSCs) and cortex-endodermis initials (CEIs) continuous flux of precursors of more specialized cells that sustain (Fig. 1A) (Dolan et al., 1993). Direct cell-cell contact from the QC to growth and replace old or damaged tissues (Sablowski, 2007). its surrounding stem cells is important for stem cell identity (Liu et al., SCNs usually consist of a few stem cells that are maintained by 2017b; van den Berg et al., 1997) and can involve the transport of short-range signals produced by localized sources or ‘organizing short-range signals from the QC. The homeodomain transcription centers’, groups of cells that sustain neighboring cells in a stem-cell factor WOX5 is specifically expressed at the QC (Sarkar et al., 2007) state (Scadden, 2006; Spradling et al., 2001). As stem-cell daughters and can move towards adjacent cells (Berckmans et al., 2020; Clark are placed outside the reach of these signals, they begin to et al., 2020; Pi et al., 2015). WOX5 itself has been proposed to act as differentiate, giving rise to more specialized cell types (Sablowski, the long-sought short-range signal to repress columella stem cell 2004). In animals, diffusible ligands are common signals that differentiation (Pi et al., 2015), albeit recent results challenge this preserve stem cells (Chen and McKearin, 2003; Han et al., 2008; view (Berckmans et al., 2020). Although short-range signaling is thought to ensure that stem-cell numbers are restrained and the SCN 1 Departament de Fı́sica de la Matèria Condensada, Facultat de Fı́sica, Universitat does not become displaced from the growing root tip, it is yet unclear de Barcelona, 08028 Barcelona, Spain. 2Universitat de Barcelona Institute of how this is achieved (Matosevich and Efroni, 2021; Rahni et al., Complex Systems (UBICS), Universitat de Barcelona, 08028 Barcelona, Spain. 3 Department of Molecular Genetics, Centre for Research in Agricultural Genomics 2016; Scheres, 2007). (CRAG), CSIC-IRTA-UAB-UB, Campus UAB (Cerdanyola del Vallès), 08193 The R2R3-MYB transcription factor BRASSINOSTEROIDS DEVELOPMENT Barcelona, Spain. AT VASCULAR AND ORGANIZING CENTER (BRAVO) has *Author for correspondence (mibanes@ub.edu) recently been linked to the maintenance of QC homeostasis (Betegón-Putze et al., 2021; Vilarrasa-Blasi et al., 2014). BRAVO M.I., 0000-0002-7913-7936 is expressed at the QC and VI (Vilarrasa-Blasi et al., 2014) and This is an Open Access article distributed under the terms of the Creative Commons Attribution colocalizes with WOX5 in the QC (Betegón-Putze et al., 2021). License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, Both BRAVO and WOX5 have been shown to individually promote distribution and reproduction in any medium provided that the original work is properly attributed. quiescence, as mutant roots with disrupted BRAVO or WOX5 Handling Editor: Ykä Helariutta exhibit increased QC divisions, supporting their role as essential Received 8 January 2022; Accepted 23 June 2022 factors for QC homeostasis (Forzani et al., 2014; Vilarrasa-Blasi 1 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 Fig. 1. Spatial confinement of BRAVO promoter expression. (A) Cartoon of the root apical meristem depicting its organization in cell-types: quiescent center, QC; vascular initials, VI; vascular cells, V; cortex/endodermis initials, CEI; cortex, CX; endodermis, END; epidermis, EPI; columella stem cells, CSC; columella cells, CC; lateral root cap, LRC; pericycle, PER. (B-D) Confocal images of propidium iodide-stained (magenta) 6-day-old roots. GFP-tagged protein expression is shown in yellow. Activity of the BRAVO promoter in WT (B), and in bravo-2 (C) and wox5-1 (D) loss-of-function backgrounds is shown. Scale bar: 50 μm. The promoter activity of BRAVO expands its domain in the bravo-2 mutant but not in the wox5-1 mutant. Data and images in B-D are taken from experiments conducted for Betegó n-Putze et al. (2021). (E) Total GFP fluorescence intensity along the vertical axis for the images shown in B (WT, solid line in E) and C (bravo-2 mutant, dashed line in E), indicated as the sum of the fluorescence intensity measured in the horizontal direction. The images were analyzed using a custom Python script. (F) Data from Betegó n-Putze et al. (2021) on fold-changes of spatially integrated pBRAVO-GFP expression over the whole SCN in bravo-2, wox5-1 and bravo-2 wox5-1 mutants relative to that in WT (indicated as mean±s.d.). (G) Transcript abundance (read counts) of BRAVO and WOX5 in QC cells from RNA-sequencing data in Betegó n-Putze et al. (2021) and Clark et al. (2019) [Gene Expression Omnibus (GEO) accession numbers GSE173945 and GSE98204, respectively]. et al., 2014). We have recently shown that BRAVO and WOX5 are common targets (Betegón-Putze et al., 2021) to propose an co-dependent, promoting expression of each other and physically additional mechanism for self-confinement. To assess whether interacting, presumably through their binding into a protein these different scenarios can drive a spatial confinement that complex (e.g. as a heterodimer) (Betegón-Putze et al., 2021). is compatible with the reported experimental data, we first studied These results suggested that BRAVO promotes WOX5 expression them through minimal models in one spatial dimension, separately indirectly through WOX5, via the formation of the complex and the and in combination. These simplified schemes enabled thorough subsequent disruption of a WOX5 negative feedback on itself. In parameter space exploration. Finally, to overcome the limitations set contrast, since the expression of BRAVO increases upon WOX5 by the simplified geometry, we translated these scenarios to a realistic overexpression, WOX5 activates (directly or through targets) two-dimensional geometry. Our results support that WOX5 diffusion BRAVO expression (Betegón-Putze et al., 2021). (Berckmans et al., 2020; Clark et al., 2020; Pi et al., 2015) is sufficient Our previous study revealed the effective regulations between to drive the spatial self-confinement of BRAVO expression. BRAVO BRAVO and WOX5, but did not address the mechanisms driving self-confinement also reduces the spatial domain of WOX5 activity, spatial changes in their expression. The expression of the BRAVO overall providing a natural way for stem-cell-specific factors to locally promoter, restricted to the QC and VI in wild-type (WT) roots, regulate SCN maintenance. Altogether, our results shed light on the increases and expands towards the vasculature, cortex and regulatory principles balancing the confinement of transcription endodermis in the bravo-2 loss-of-function mutant (Betegón- factors to a microenvironment and their communication with the Putze et al., 2021) (Fig. 1B,C). This expansion suggests that surrounding cells. BRAVO actively confines in space its own expression domain. Although our previous study considered the increase in total RESULTS (spatially integrated) BRAVO promoter expression in the bravo-2 BRAVO can confine its own expression domain by DEVELOPMENT mutant through negative feedback (Betegón-Putze et al., 2021), it immobilizing WOX5 did not address how the expression of BRAVO is regulated in space The expansion of BRAVO promoter expression towards the to generate self-confinement. Here, we address this problem through vasculature in the bravo-2 loss-of-function mutant compared with mathematical and computational modeling. the WT (Betegón-Putze et al., 2021) (Fig. 1B-E) suggests that, in To this end, we used the finding that WOX5 can activate BRAVO the WT, BRAVO confines its own expression to the SCN. Because expression and can bind to BRAVO protein (Betegón-Putze et al., there is no expansion in the loss-of-function wox5-1 mutant 2021) to evaluate how these interactions can be regulated in space to (Fig. 1D) nor in the bravo-2 wox5-1 mutant (Betegón-Putze drive spatial self-confinement of BRAVO expression. We also used et al., 2021), we considered that the mechanism for BRAVO the finding that BRAVO and WOX5 can act antagonistically on self-confinement requires WOX5. To decipher how this self- 2 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 Fig. 2. ‘Immobilization by sequestration’ model. (A) Cartoon displaying the interactions in the model. WOX5 diffuses (wavy lines) and activates (arrow) BRAVO. In turn, BRAVO immobilizes WOX5 by sequestering it into an immobile and inactive complex (depicted by ;). The main parameters characterizing the regulations are shown. (B) Cartoon of the model in the one- dimensional spatial framework. Proteins are denoted by circles and promoters by rectangles. (C) Stationary profiles of BRAVO and WOX5 promoter activities ( pB and pW) obtained with this model in the WT (continuous lines) and in the bravo mutant (dashed lines). The QC region (dark gray) has a size of LQC and is where pW is active. For simplicity, the VI region (light gray) is defined with this same size. The pB value in the WT at the end of the VI region is denoted by   VI . The quantity Δx measures pB expansion in the bravo mutant j ¼ LQC . pBWT Dx If BRAVO confines its own expression in the WT, then Δx>0. (D,E) Stationary protein profiles of BRAVO and WOX5 (not bound to each other) in WT (continuous lines) and in the bravo mutant (dashed lines) corresponding to the simulations in C. (F,G) Parameter domains (yellow) compatible with experimental observations for (F) BRAVO synthesis rate αL and binding strength λ and (G) WOX5 diffusion coefficient DW and binding strength λ parameter spaces. Solid, dashed and dotted lines denote the observables ξ, RB and RBW, respectively, taking the maximum or minimum value that we considered to be compatible with observations, as defined in the main text. Red stars in F and G mark the values used for the simulations in C-E. In C-G, activation of BRAVO by WOX5 is linear. (H) Cartoon of the model with WOX5 self-repression added. (I,J) Analogous to panels F and G, parameter domains (yellow) compatible with experimental observations are shown for the model in H: (I) BRAVO synthesis saturation rate α and binding strength λ, and (J) WOX5 diffusion coefficient DW and binding strength λ parameter spaces, with RW (dot-dashed line) included. Productions are nonlinear functions with saturation [α (saturated rate) is used instead of αL]. The values of observables for each parameter space are shown in Figs S1 and S2. Parameter values are given in Table S1. a.u., arbitrary units. (Berckmans et al., 2020; Clark et al., 2020), no evidence of intercellular BRAVO transport has been reported. Due to its larger size (BRAVO has a molecular mass of ∼36 kDa compared with ∼20 kDa for WOX5; The Arabidopsis Information Resource, TAIR, loci AT5G17800 and AT3G11260, respectively), BRAVO proteins are expected to be less mobile than WOX5 proteins, if mobile at all. Moreover, the BRAVO-WOX5 complex, owing to its even larger size, is not expected to move very much from cell to cell. In this respect, although the bounds of the size exclusion limit (SEL) of the plasmodesmata vary, estimates place the SEL lower bound to be 27 kDa and upper bound to be <54 kDa between the QC and the cortex and between the QC and columella stem cells (Rim et al., 2011), yet these values may change with environmental conditions and developmental stages. Taken together, these observations allow us to propose the following mechanism for BRAVO to confine its own expression (Fig. 2A,B). WOX5 proteins, produced in QC cells and mobile towards the VI cells, can activate BRAVO expression in the VI. In turn, BRAVO proteins sequester WOX5 into an immobile (i.e. from cell to cell) and inactive complex, thus disrupting WOX5 movement confinement can be attained, we first considered that BRAVO and impeding the activation of BRAVO expression in the VI. Hence, transcription is ultimately activated by WOX5 (either directly or the activation of the BRAVO promoter by WOX5 is spatially confined through WOX5 targets) (Betegón-Putze et al., 2021) and that by BRAVO proteins, a restriction which can be released when WOX5 proteins are able to move from the QC to the VI (Berckmans BRAVO can no longer immobilize WOX5, e.g. in the bravo mutant. DEVELOPMENT et al., 2020; Clark et al., 2020). Thus, WOX5, by moving to the VI To evaluate this mechanism, we constructed a minimal cells and shootwards, is expected to induce BRAVO expression in mathematical model, hereafter named the ‘immobilization by these cells. Additionally, we considered that BRAVO and WOX5 sequestration’ model (see Materials and Methods; Fig. 2A,B) in can physically interact at the protein level, presumably by binding which the main assumptions are: (1) WOX5 induces BRAVO together, as suggested by co-immunoprecipitation and fluorescence expression (Betegón-Putze et al., 2021); (2) the WOX5 protein resonance energy transfer by fluorescence lifetime imaging (FRET- diffuses (Berckmans et al., 2020; Clark et al., 2020), whereas FLIM) analysis (Betegón-Putze et al., 2021). BRAVO does not; (3) the BRAVO and WOX5 proteins reversibly Although the mobility of WOX5 ( possibly through the bind together forming a complex that does not induce BRAVO plasmodesmata) has been experimentally tested in planta expression (Betegón-Putze et al., 2021); (4) the BRAVO-WOX5 3 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 complex is immobile, i.e. it cannot diffuse; and (5) all molecules, As sequestration is a necessary factor for this mechanism to work, the complex included, are degraded. one could argue that the presence of other molecules also binding to We studied the model in a one-dimensional space in which WOX5 BRAVO and/or WOX5 might impair or even destroy the is produced only at a localized region (dark grey shaded area in confinement. Indeed, it is known that WOX5 and BRAVO Fig. 2B,C) that we identify as the QC, and can diffuse in both proteins can bind to additional molecules, such as TOPLESS directions, namely towards regions that could be identified as the (TPL) and BES1 (Betegón-Putze et al., 2021; Vilarrasa-Blasi et al., CSCs and columella cells (negative values of the position in Fig. 2B, 2014). These are relatively large molecules (∼39 kDa and C), or towards the vasculature (positive values of the position in ∼124 kDa, respectively; The Arabidopsis Information Resource, Fig. 2B,C, where the light gray shaded region is identified as VI TAIR) and it would be possible for them to immobilize WOX5. We cells). Conversely, BRAVO was set to be activated by WOX5 only in modeled this situation assuming the most limiting case, i.e. that regions from the QC and towards the vasculature, but not towards the these molecules (S) only act as competitors for the binding of columella, thus mimicking the asymmetric activity of the BRAVO BRAVO and WOX5 (see supplementary Materials and Methods; promoter in the Arabidopsis primary root. Table S2). The results show that BRAVO self-confinement takes In Fig. 2C-E, we show the stationary activity profiles of BRAVO place if S is much less produced than BRAVO (for the same and WOX5 promoters ( pB and pW, respectively) and the stationary degradation rate, Fig. S3). If instead S production is similar to that of BRAVO and WOX5 protein concentrations (B and W, respectively), BRAVO, then the BRAVO promoter expression domain is similarly obtained by numerically solving the model equations with linear confined in the WT and in the bravo mutant. In this mutant, S activation of BRAVO by WOX5 (see Materials and Methods). overcomes the absence of BRAVO by binding to WOX5 and The concentration of the protein complex (which is proportional to immobilizing it (Fig. S3). As TPL and BES1 transcript abundance the product BW) is shown in Fig. S1. The stationary profiles at the QC is not very high [ranging 0.57-3.08 times that of BRAVO obtained when modeling the bravo mutant condition (see Materials and 0.13-0.39 times that of WOX5, based on read counts (Clark and Methods) are also depicted. These results support that the et al., 2019; Betegón-Putze et al., 2021)], the results suggest that the ‘immobilization by sequestration’ mechanism can drive self- ‘immobilization by sequestration’ mechanism may take place. confinement and constitute a potential way for BRAVO to To further assess the plausibility of the ‘immobilization by spatially confine its own expression in the WT root. sequestration’ mechanism in the context of BRAVO and WOX5, we To address whether this mechanism drives BRAVO expression included the fact that BRAVO promotes WOX5 expression, as the and self-confinement consistent with experimental observations, expression of GFP under the WOX5 promoter decreases in the we defined three different observables (see Materials and Methods). bravo-2 mutant (Betegón-Putze et al., 2021; Vilarrasa-Blasi et al., The first observable, ξ, focuses on the spatial expansion of 2014). Specifically, we previously proposed that BRAVO promotes BRAVO promoter expression in the bravo mutant compared with WOX5 expression through a mechanism that involves a WOX5 that in the WT (Fig. 2C). We considered that expansions within negative feedback (modeled as self-repression) and the binding of 0.5 and five times the QC size were consistent with experimental the WOX5 and BRAVO proteins (Betegón-Putze et al., 2021). observations (Fig. 1E). The second observable, RB, evaluates the WOX5 self-repression is consistent with increased WOX5 promoter increase of BRAVO expression levels in the bravo mutant (Fig. 2C). expression in the wox5-1 mutant (Betegón-Putze et al., 2021). The model results are considered compatible with experimental Therefore, as the ‘immobilization by sequestration’ model already data when BRAVO promoter activity at the QC increases up to included the binding between BRAVO and WOX5, we only added twofold in the bravo mutant (Fig. 1F). The third observable, RBW, WOX5 self-repression to the model (Fig. 2H, Materials and compares BRAVO and WOX5 expressions in WT, as BRAVO Methods) and defined a fourth observable, RW, which measures transcripts are less abundant (Fig. 1G) (Betegón-Putze et al., 2021; the ratio between WOX5 promoter activity in the bravo mutant and Clark et al., 2019). We considered that the simulation results also were that in the WT. We considered that simulation results within the compatible with observations from real roots if BRAVO promoter range of 0.5 and 0.9 are compatible with experimental data expression at the end of the VI over WOX5 promoter expression at the (Betegón-Putze et al., 2021). The results show that the BRAVO QC, both in the WT, was within the range of 0.05 and 0.5. expression profile remains similar when WOX5 self-repression is Exploration of different parameter spaces shows large domains in included (Fig. S2). RW sets additional constraints to the lower bound which the model is compatible with experimental data, for linear of BRAVO production rate (Fig. 2I) and limits the diffusion (Fig. 2F,G; Fig. S1A) and nonlinear regulatory functions (Fig. S2A, coefficient of WOX5 (Fig. 2J). This last restriction is not expected to B). In this mechanism of self-confinement, the binding between be very limiting as the experimental data suggest a small amount of WOX5 and BRAVO (mediated by the binding strength λ) as well as WOX5 diffusion (Berckmans et al., 2020; Clark et al., 2020). WOX5 diffusion are necessary (Fig. 2F). However, too much Altogether, the ‘immobilization by sequestration’ mechanism sequestration of WOX5 by BRAVO (large λ and high BRAVO constitutes a plausible candidate for explaining the self- synthesis rate αL) can lead to an overly exaggerated confinement, with confined BRAVO expression to both the QC and VI cells in the BRAVO being highly expressed at the QC but not in VI cells (Fig. 2G; WT. Notably, the WT stationary profile of the WOX5 protein Fig. S1C), a situation which would disagree with the experimentally (W, not bound to BRAVO) obtained for this model is not symmetric, DEVELOPMENT observed WT expression (Fig. 1B). The results also show that if but decays differently above and below the QC (Fig. 2E). This WOX5 is set to have a large diffusion coefficient, BRAVO expression behavior is caused by the presence and absence, respectively, in the WT expands but gets dimmer (Fig. S1E). This suggests that of BRAVO proteins in the two distinct spatial regions. Above WOX5 diffusion might be rather small, in agreement with observed the QC, the gradient of WOX5 proteins is steeper than below the protein distributions of WOX5 within the root SCN, which primarily QC, where BRAVO cannot be activated. Hence, the presence reach cells adjacent to the QC (Berckmans et al., 2020; Clark et al., of BRAVO makes the WOX5 gradient more abrupt, restricting 2020). Overall, our results indicate that the ‘immobilization by the spatial domain in which WOX5 proteins are concentrated, sequestration’ mechanism is a plausible candidate model to explain and consequently confining the BRAVO expression domain. the observed self-confinement of BRAVO expression. Therefore, this mechanism not only drives self-confined 4 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 BRAVO expression but also results in a confined action of WOX5 (1) WOX5 activates Z, which activates BRAVO; (2) Z and WOX5 proteins. diffuse, whereas the BRAVO protein does not; (3) BRAVO and WOX5 proteins reversibly bind together forming a complex that A mobile intermediary can facilitate BRAVO cannot induce Z; (4) the BRAVO-WOX5 complex is immobile, self-confinement i.e. it cannot diffuse; (5) all molecules are degraded; and, The self-confinement of BRAVO expression in the ‘immobilization (6) additionally, WOX5 self-represses as in Fig. 2H. by sequestration’ model requires WOX5 to diffuse. Experiments The results show that Z drives a wider BRAVO expression, as suggest that although it is mobile, WOX5 does not move very large expected, which is still self-confined (Fig. 3C,D). The mobility of Z distances (Berckmans et al., 2020; Clark et al., 2020). Hence, its low enables BRAVO self-confinement at lower diffusion coefficients of mobility may not be sufficient to explain the self-confinement of WOX5 (compare Fig. 3E with Fig. 2J, Fig. 3J) as BRAVO, when BRAVO expression observed in real roots. As activation of BRAVO binding to WOX5, confines its expression by also attenuating the by WOX5 could occur through intermediary molecules, we asked production of Z (Fig. S4). whether realistic BRAVO self-confinement can arise for smaller WOX5 diffusion coefficients if WOX5 activates BRAVO through a BRAVO can confine its own expression domain by repressing mobile intermediary molecule, Z. To evaluate this scenario, we its mobile activator included such an intermediary molecule in the ‘immobilization by The previous model showed that a transcription factor can confine sequestration’ model and studied its implications by numerically its own expression by reducing the production of its mobile activator simulating the WT and bravo mutant in one dimension, as Z. Based on this, we envisaged an additional scenario that drives a previously done. In this model (Fig. 3A,B) the assumptions are reduction of the mobile activator but is not based on the analogous to those mentioned earlier but include Z as follows: sequestration of BRAVO and WOX5. In this case, BRAVO represses the production of its activator Z, which is mobile and is activated by WOX5 (Fig. 4A,B). Reported transcriptomics of QC cells have revealed that several of the genes of which the mRNA levels are de-regulated in wox5-1 and bravo-2 mutants show opposite regulations (Betegón-Putze et al., 2021), thus opening the possibility of these genes to express or regulate the intermediate factor Z. We called this the ‘repression’ model (see Materials and Methods) and its assumptions are: (1) WOX5 activates Z, which activates BRAVO; (2) Z and WOX5 diffuse, whereas BRAVO does not; (3) BRAVO represses the production of Z (but the BRAVO- WOX5 complex does not); and (4) all molecules degrade. Note that all these assumptions except (3) were already present in the previous model. Simulations indicate that the ‘repression’ mechanism is able to induce self-confinement of BRAVO expression without requiring WOX5 to be mobile (Fig. 4C-G). The action of WOX5 mediated by its target Z, but not the WOX5 protein, becomes spatially restricted in the WT, whereas it expands in the bravo mutant (Fig. 4F,E). Strong repression enhances the self-confinement, but can result in unrealistic BRAVO expression profiles in the WT, limited only to the QC and not reaching the VI cells (Fig. S5). In addition, strong repression involves not only spatial confinement but also a dramatic reduction of BRAVO expression at the QC in the WT compared with the bravo mutant (Fig. S5), a situation which is not observed experimentally (Betegón-Putze et al., 2021) (Fig. 1B,C). This sets strong limiting conditions on the repression strength and on the rate of Z production when assuming linear regulatory functions (Fig. S5), but not as much for nonlinear functions, as expected Fig. 3. ‘Immobilization by sequestration’ model with intermediary. (Fig. 4H; Fig. S6). (A) Cartoon of the model, same as Fig. 2H except that WOX5 activates a Finally, we considered that BRAVO promotes WOX5 expression diffusible intermediary Z, which in turn activates BRAVO. (B) Cartoon of the and assumed that this occurs through binding with WOX5 and model in the one-dimensional spatial framework. WOX5 self-repression is not WOX5 self-repression. We termed this the ‘mixed’ model, as it represented for clarity. (C) Stationary activity profiles pB and pW (inset) in WT (continuous lines) and in the bravo mutant (dashed lines) for equal WOX5 and involves both the ‘repression’ and the ‘immobilization by DEVELOPMENT Z diffusion coefficients (DW=DZ=2). (D) Stationary pB profiles in the WT sequestration’ models (Fig. 5A,B). Despite the sequestering of (continuous) and in bravo mutant (dashed) for non-mobile Z (violet) and for BRAVO by (large amounts of ) WOX5 reduces the BRAVO- stronger mobile Z ( pink). (E,F) Parameter domains (yellow) compatible with mediated repression, the presence of both the repression and experimental observations for (E) WOX5 diffusion coefficient DW and binding immobilization mechanisms enhances self-confinement (Fig. 5C). strength λ, and (F) WOX5 diffusion coefficient DW and Z diffusion coefficient In addition, BRAVO self-confinement that is compatible with DZ parameter spaces. Conditions for RW are satisfied in all the depicted spaces (indicated by the absence of a dot-dashed line). Fig. S4 depicts the values of experimental observations happens in larger parameter domains, i.e. the observables. Red stars mark the parameter values used in C. when repression is not sufficient, sequestration takes its place Production functions are nonlinear. Parameter values are given in Table S1. (Fig. 5D-F). Note that sequestration is required to account for a.u., arbitrary units. decreased WOX5 expression in the bravo mutant but not for BRAVO 5 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 Fig. 5. Mixed model. (A) Cartoon of the interactions, which involve the ‘immobilization by sequestration’ and the ‘repression’ models. (B) Cartoon of the model in the one-dimensional spatial framework. (C) Stationary pB and pW (inset) profiles in WT (continuous lines) and in the bravo mutant (dashed lines) for several repression strengths. All cases have the same profile in the bravo mutant. (D-F) Parameter domains (yellow) compatible with experimental observations in (D) WOX5 and Z diffusion coefficients, DW and DZ, respectively, (E) Z saturated activation rate β and repression strength c, and (F) BRAVO production saturation rate α and binding strength λ parameter spaces. Red stars mark the parameter values used in C for c=0.01. Fig. S7 Fig. 4. ‘Repression’ model. (A) Cartoon of the interactions. WOX5 diffuses depicts the values of the observables. Nonlinear production functions used. and activates a diffusible factor Z, which activates BRAVO. BRAVO, in turn, Parameter values are given in Table S1. a.u., arbitrary units. represses (blunt arrow) Z. (B) Cartoon of the model in the one-dimensional spatial framework. WOX5 movement is not depicted. (C) Stationary activity profiles of pB and pW in WT (continuous lines) and in the bravo mutant (dashed Arabidopsis root SCN. However, it remains unclear whether lines) obtained with the ‘repression’ model assuming negligible diffusion of WOX5 (DW=0) for DZ=4. (D-F) Stationary profiles of BRAVO, WOX5 and Z the degree of diffusion of WOX5 is sufficiently large to generate proteins as well as promoter activity pZ in WT (continuous lines) and in the realistic expansions. To address this issue, we modeled the mixed bravo mutant (dashed lines) corresponding to the simulations in B. model on realistic two-dimensional root layouts, for which the (G,H) Parameter domains (yellow) compatible with experimental observations cellular geometry of the roots, as well as the presence of cell walls, for (G) WOX5 and Z diffusion coefficients, DW and DZ, respectively, and for was explicitly incorporated. We also modeled the dynamics of GFP (H) Z saturated activation rate β and repression strength c. C-G use linear produced by the BRAVO and the WOX5 promoters ( pBRAVO:GFP activation functions. H uses nonlinear functions and DW=0. Figs S5 and S6 and pWOX5:GFP, respectively; see Materials and Methods). depict the values of the observables for G and H, respectively. Parameter values are given in Table S1. a.u., arbitrary units. The main features included in this new framework can be enumerated as follows (for further details, see Materials and Methods and supplementary Materials and Methods). (1) The spatial discretization of the realistic root layout was made at the pixel self-confinement. Similar results are found when basal production scale. Hence, the size of the cytoplasm and cell walls is determined of BRAVO, i.e. independent of WOX5, is set at the VI region by the number and localization of their corresponding pixels DEVELOPMENT (Fig. S7) to mimic the situation in which these cells still show (Fig. S8). (2) The dynamics of the molecular components in the BRAVO promoter expression in the wox5-1 mutant (Betegón-Putze interior of the cells is distinct from the dynamics in the cell walls. et al., 2021) (Fig. 1D). Specifically, in all the pixels belonging to the interior of the cell, molecules can be produced, regulated or degraded, can sequester The ‘immobilization by sequestration’ mechanism other molecules and are able to diffuse. Inside the cell walls, is sufficient and the ‘repression’ mechanism enhances however, molecules are only able to diffuse. (3) The diffusion self-confinement in Arabidopsis coefficient in cell walls is set to be smaller than in the interior of the Our previous results suggest that the mixed model is a conceivable cells. With this assumption, the physical boundaries between cells candidate to explain the self-confined expression of BRAVO in the are naturally incorporated. These are to be expected for any 6 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 molecule diffusing freely inside the cytoplasm but moving only action on QC divisions (Betegón-Putze et al., 2021). Here, we show occasionally from cell to cell, as it may happen, for example, in that these interactions, together with the diffusion of WOX5, are plasmodesmata-mediated transport. For simplicity (and lack of sufficient to drive the spatially self-confined expression of BRAVO, as evidence), we assume that BRAVO and the BRAVO-WOX5 revealed by the ‘immobilization by sequestration’ mechanism. complex can only move inside cells, being unable to diffuse Furthermore, the opposite regulation of common targets by WOX5 through cell walls. Conversely, WOX5 and Z proteins can move and BRAVO, as inferred by transcriptional profiling (Betegón-Putze inside the cytoplasm with diffusion coefficients Dcyt W and DZ , cyt et al., 2021), proposes a complementary scenario in which this wall respectively, and between cells with diffusion coefficients DW and confinement is induced by negative feedback between BRAVO and a Dwall Z , respectively. Because GFP proteins are set to diffuse from cell factor activated by WOX5 (the ‘repression’ mechanism). According to cell, they can reach cells in which there is no promoter activity. to transcriptomics analyses at the QC, only the AT3G41762 gene has However, the degree of GFP diffusion is set sufficiently low so that a significantly decreased expression in the wox5-1 mutant and this effect is small, such that GFP expression under the WOX5 increased expression in the bravo-2 mutant, making it a potential promoter is mostly localized at the QC (Betegón-Putze et al., 2021; candidate to act as Z if it induces BRAVO. Notably, this gene is Vilarrasa-Blasi et al., 2014). (4) Cells are classified by cell type. We strongly expressed in the root SCN of WT Arabidopsis (Fig. S15) defined nine different cell types: quiescent center (QC), vascular (Clark et al., 2019), but its function awaits to be characterized. The initials (VI), vascular cells (V), cortex-endodermis initials (CEI), remaining genes that are antagonistically and significantly regulated pericycle (PER), cortex (CX), endodermis (END), epidermis (EPI), by BRAVO and WOX5 show increased expression in the wox5-1 columella stem cells (CSC), columella cells (CC) and lateral root cap mutant and decreased expression in the bravo-2 mutant (Betegón- cells (LRC) (Fig. 1A). This classification enables setting different Putze et al., 2021). Hence, they are repressed by WOX5 and activated dynamics for the proteins in distinct cell types, as well as different by BRAVO. If any of these genes inhibits a particular factor, this transport properties (through different diffusion coefficients). factor might be acting as Z (Fig. S15). Such a potential case is However, we set the diffusion coefficient of each molecular species exemplified by peroxidase proteins, which are repressed by WOX5 to be the same in all cell types. In contrast, different cell types differ in but activated by BRAVO, and which remove the mobile hydrogen their protein production dynamics. WOX5 only activates BRAVO in peroxide. Based on their expression patterns in the SCN (Fig. S15) the QC, VI, vascular cells, cortex and endodermis, but not in the (Clark et al., 2019), they may play a role in confining BRAVO remaining cell types (Fig. S9). This assumption is based on the expression through hydrogen peroxide, pointing to the relevance of experimental observation that BRAVO is expressed only in inner preventing oxidative stress in the stem-cell microenvironment. tissues, i.e. from the SCN upwards. Basal production of BRAVO – Whether any of these candidates regulates BRAVO and can drive independent of WOX5 – is set in a few of these cells (Fig. S10). For the ‘repression’ mechanism remains to be elucidated. simplicity, WOX5 is set to be only produced at the QC [albeit low The ‘immobilization by sequestration’ and ‘repression’ amounts of promoter expression are also present in the vascular mechanisms share the idea that the emergence of self-confinement initials (Berckmans et al., 2020)]. For each species, the degradation involves negative feedback of BRAVO on itself with a mobile rate was assumed to be the same in all cell types. activator (WOX5 or Z) and an immobile inhibitor (BRAVO). As We chose parameter values for WOX5 dynamics so that WOX5 such, they mechanistically account, at least partially, for the effective proteins, in the absence of any regulatory factor, remain mostly negative self-regulation of BRAVO expression in the whole SCN that localized at the QC, VI and CSCs (Fig. S11). This is consistent with was previously proposed (Betegón-Putze et al., 2021). Each of these reported data of GFP bound to WOX5 proteins (Berckmans et al., two mechanisms can be understood as a different regulatory way for 2020; Clark et al., 2020). The other parameter values were chosen to this negative feedback to be accomplished: by immobilizing the drive pWOX5:GFP and pBRAVO:GFP expression patterns activator or by reducing the production of the activator. Therefore, compatible with those observed in real WT roots (Table S3). The they can be encompassed within a general framework of self- mixed model can explain the behavior of pBRAVO:GFP in the WT confinement, i.e. an ‘attenuation of a mobile activator’ model. Both and the expansion of its domain in the bravo mutant (compare mechanisms drive the self-confinement of both the repressor and the Fig. 6A with Fig. 1E). BRAVO self-confinement additionally activator. Yet, the mechanisms are not equivalent, each of them involves the confinement of WOX5 proteins and of the diffusible having distinct characteristics, as our analysis revealed. The most target Z (Fig. S12). Although different layouts (Fig. S10) are used to noticeable feature is perhaps the fact that the ‘immobilization by model WT and bravo mutant roots (Fig. S10), this does not sequestration’ mechanism involves a change in the gradient profile of introduce significant differences (Fig. S13). the activator (Fig. 2E; supplementary Materials and Methods), The small diffusion of WOX5 is sufficient to drive a realistic whereas the ‘repression’ mechanism does not. confinement of pBRAVO:GFP through the ‘immobilization by A different mechanism to drive self-confined expression has been sequestration’ mechanism without the need of Z mobility (and proposed for WUSCHEL in the shoot apical meristem of thereby, of Z) (Fig. 6B). Yet, the presence of a mobile Z cannot be Arabidopsis. According to this mechanism, the WUSCHEL discarded. For such a mobile Z, sequestration and repression can protein confines its own expression by activating its repressor, provide an attenuation of Z that contributes to confine BRAVO CLAVATA3, which is highly mobile (Yadav et al., 2011; Zhou DEVELOPMENT expression (Fig. 6A,C,D; Fig. S14). Taken together, the results in et al., 2018). Both this mechanism and the ‘attenuation of a mobile the realistic root layout support the idea that the immobilization of activator’ mechanism studied here share the idea that negative WOX5 by BRAVO is sufficient for the self-confined expression of feedback is responsible for the self-confinement. However, in the BRAVO in the root SCN. ‘attenuation of a mobile activator’ mechanism, the strongly mobile component is the activator and not the repressor. This mechanism is DISCUSSION thus a distinct mechanism for self-confinement and, because of its We have previously shown that BRAVO and WOX5 regulate the minimal assumptions, we expect it, or a variant thereof, to take place expression of each other, and that their binding into a protein complex in very distinct developmental contexts. For instance, Hedgehog can be relevant for these regulations and for BRAVO and WOX5 signaling in the Drosophila wing confines its own expression by 7 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 Fig. 6. pBRAVO:GFP simulated in realistic root layouts. Simulation results for the stationary activity of pBRAVO:GFP in WT (left root) and in bravo mutant (right root) for different regulatory interactions (each depicted as a cartoon on the right). The middle panel depicts pBRAVO:GFP (integrated transversally) along the vertical axis, from simulation images on the left. (A) Mixed model. In the WT, pBRAVO:GFP is confined to the SCN, and it is expanded in the bravo mutant. Fig. S12 depicts the stationary profiles of all other molecules. (B) ‘Immobilization by sequestration’ model with non-diffusible Z. Self-confined expression of pBRAVO:GFP is also obtained. (C) Mixed model without repression (i.e. ‘immobilization by sequestration’ model with diffusible Z). (D) Mixed model without BRAVO-WOX5 binding. C and D show similar results to A. Hence, sequestration can compensate for the absence of repression and vice versa. Simulations in the bravo mutant lead to the same pBRAVO:GFP for A,C and D. In all panels, cell walls are superimposed (in black with transparency) over the colormap so that they can be easily visualized. Parameter values are given in Table S3. For other parameter values, repression can be more relevant (Fig. S14). a.u., arbitrary units. activating one of its repressors, the nuclear zinc finger master sequestering of the morphogen by receptors might lead to an effective of thickveins (mtv, also known as sbb) (Bejarano et al., 2007). non-linear degradation of the ligand, resulting in concentration Thus, this Hedgehog self-confinement may be framed within the profiles that are robust to changes in the rate of production at the ‘attenuation of a mobile activator’ mechanism, in which Hedgehog source (Eldar et al., 2003). We could make the correlation between DEVELOPMENT acts as the mobile activator (Z in the repressor model) and master of such models and ours by identifying WOX5 as the morphogen and thickveins as the immobile repressor. BRAVO as the receptor. In agreement with what has been described The mechanism of ‘immobilization by sequestration’ can be for morphogen gradients induced by non-linear degradation (Eldar related to mechanisms that generate robust morphogen gradient et al., 2003), we find that the gradient of WOX5 decays more abruptly profiles (Eldar et al., 2003; Shilo and Barkai, 2016). Morphogens are in the presence of BRAVO than without, thus suggesting that the ligand molecules that are produced at localized sources but can specific regulations between BRAVO and WOX5 may be tuned to diffuse, generating an activity gradient that can then be interpreted by achieve robust activity profiles. Yet, we have not seen a robust different genes, leading to their activation or repression in a response to changes in WOX5 expression levels for parameter values concentration-dependent manner. It has been proposed that the compatible with experimental observations (Fig. S16) and further 8 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 analysis is required to address whether this model enhances BRAVO which the transport of the mobile proteins is modeled through diffusion. In robustness to WOX5. the models in which an intermediate factor (Z) is present, its dynamics are Modeling of root tissues has been most commonly done in terms of also considered. We only focus on the stationary solutions of the models, idealized realistic simplified geometries in which cells and cell walls assuming these account for the experimentally reported expressions. In all the models, the rate of synthesis of each protein is assumed to be are each subdivided in squares or rectangles with molecules diffusing proportional to its corresponding promoter activity and the quasi-stationary between them (Grieneisen et al., 2007; Tian et al., 2014), or by approximation for mRNAs (i.e. mRNAs dynamics are assumed to be very considering real cellular layouts with diffusing molecules between fast compared with the dynamics of proteins) is done (see supplementary but not within cells (Jönsson et al., 2005; Voß et al., 2014; Yadav Materials and Methods). For simplicity, proteins are assumed to degrade et al., 2011). By using pixels as the basic unit for discretization, we are linearly. Mass action law is set for the rate at which two proteins form a able to model the shapes of cells in a realistic manner and consider complex (i.e. ∝ BW for the BRAVO-WOX5 complex). We only consider both the interactions within and between cells. A similar pixel-based pairwise interactions, omitting higher order reactions. Complexes are taken approach has been used to model hormonal crosstalk in the to be unable to transcriptionally regulate any of the proteins considered (for Arabidopsis root (Liu et al., 2017a). Mathematically, our framework simplicity, we name them inactive complexes, albeit they could regulate can be characterized as a reaction-diffusion model in heterogeneous other factors not modeled herein). The supplementary Materials and Methods contain the derivation of the equations of the minimal models from media, in which the spatial inhomogeneities appear due to the the full set of equations that include mRNAs and complexes. The stationary presence of cell walls, which involve different diffusion coefficients. solutions are the same for both (full and minimal) models. The realistic root layout used for the simulations can be extended to The following subsections describe the one-dimensional minimal models include internal structures within the cells (such as the nucleus) as in WT scenarios, the modeling of bravo mutants, the parameter space well as structures in the cell walls (e.g. specific communication exploration with the definition of the observables and their comparison with channels). Therefore, it has the potential to implement and evaluate experimental data, the construction of the realistic root layout, the equations much more complex scenarios in a manageable way. used for the simulations in the realistic root layout and the numerical details The similarities between stem-cell niche organization in animals of all models simulated. and plants may represent the outcome of convergent evolution (Dinneny and Benfey, 2008). Multicellularity – a necessary condition ‘Immobilization by sequestration’ mathematical model for stem-cell niches to emerge – is thought to have evolved In this case, WOX5 activates BRAVO, and both proteins are able to form independently in both kingdoms (Grosberg and Strathmann, 2007; an inactive complex, which is degraded. In the WT, the dynamics of B and W are: Knoll, 2011), implying that the presence of stem-cell niches in widely disparate organisms may be a direct consequence of developmental @Bðx; tÞ constraints and not of historical contingencies. Similar mechanisms ¼ PB ðW ; xÞ  lBðx; tÞW ðx; tÞ  dB Bðx; tÞ; ð1Þ @t of niche regulation are therefore to be expected, not through common genes or molecules, but through more general regulatory principles. @W ðx; tÞ @ 2 W ðx; tÞ The fact that stem-cell niches consist of narrow regions of a few cells ¼ PW ðW ; xÞ  lBðx; tÞW ðx; tÞ  dW W ðx; tÞ þ DW ; @t @x2 clustered together, in opposition to large numbers of cells distributed ð2Þ over the whole organism, possibly emerged as a way to ensure a proper balance between centralized renewal and genome integrity, by where x denotes the spatial position in one dimension and t denotes time. minimizing deleterious mutations that may be able to spread across Here B and W indicate BRAVO and WOX5 protein concentrations, whole cell lineages (Sena et al., 2009; Yoshiyamaa et al., 2009). respectively, when they are not bound to each other, and the complex they Indeed, the smaller the population of stem cells and the lower their form is not explicitly modeled as a variable (see supplementary Materials and Methods). PB(W, x) is the production of BRAVO through WOX5, which division rate, the less likely for deleterious mutations to accumulate in is only allowed from the QC to VI direction. For the simplest scenario of differentiated tissues (Cannataro et al., 2017). The mechanism linear regulatory functions we use PB(W,x)=αLWΘ(x+LQC/2), where the proposed in this paper establishes a balance between the parameter αLmeasures the production rate of BRAVO per unit concentration communication mediated by the activator (WOX5) with confining of WOX5 (in units of inverse time) and the Heaviside function Θ(x+LQC/2) its action, ensuring communication remains local. denotes that production can only occur at x≥− LQC/2 (i.e. from the QC In the Arabidopsis SCN, these signals allow cells to communicate towards the VI direction). For nonlinear regulations with saturation, we use between them and with other cell types, at the same time as they Wn PB ðW ; xÞ ¼ a n Qðx þ LQC =2Þ, where α is the saturated rate of create boundaries within which local information can be W þ kBn transmitted. Indeed, QC cells have been shown to influence production (in units of concentration over time), kB is the concentration of neighboring cell types such as the CSCs, in which WOX5 can WOX5 at which the production is half the saturated value and exponent n play a crucial role as a signaling agent (Pi et al., 2015). We propose sets the cooperativity. WOX5 is only produced at the QC. In the simplest scenario, WOX5 production is constant (independent of WOX5 itself ) such that towards the vasculature, BRAVO can be a signaling molecule, that PW(W, x)=γQC(x), where γQC(x) denotes that production is only at the which, by actively restraining its own expression from reaching cells QC. We choose γQC(x) to be a Heaviside function: far away from its source, ensures that the small microenvironment of ( the SCN remains confined within the root. The molecular processes 0 if jxj . LQC =2 gQC ðxÞ ¼ ; ð3Þ DEVELOPMENT underlying this spatial restriction and their implications for proper g if jxj  LQC =2 stem-cell renewal are just beginning to be uncovered. Mechanisms like the ones proposed here involve very general principles that where LQCis the total length of the QC region. This implies that WOX5 contribute to the understanding of stem cell populations not only in production only occurs in the region delimited by − LQC/2≤x≤LQC/2, with plants, but in multicellular organisms on the whole. constant production rate γ. When WOX5 negative feedback is included, we kn use PW ðW ; xÞ ¼ k n þWWðx;tÞn gQC ðxÞ, kW being the WOX5 concentration at W MATERIALS AND METHODS which WOX5 production at the QC is γ/2. Degradation of BRAVO and The models formulated set the rate of change of protein concentrations of WOX5 is controlled by the parameters dB and dW, respectively. Complex BRAVO (B) and WOX5 (W ), by using partial differential equations in formation between BRAVO and WOX5 is mediated by the parameter λ, 9 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 which sets the rate at which the two factors interact per concentration unit of pB(x)=PB(Zs(x), x), pZ(x)=PZ(Ws(x)) and pW(x)=γQC(x), where again the each of them and involves binding, unbinding and degradation rates (see subscript s indicates that the concentration profiles are those corresponding supplementary Materials and Methods). Finally, WOX5 is able to diffuse to the stationary state. from the QC with rate DW. As explained in detail in the supplementary Materials and Methods, in order to have confinement through the Mixed model ‘immobilization by sequestration’ mechanism, it is essential for the This model combines the immobilization by sequestration with intermediary formation of the BRAVO-WOX5 complex to be either irreversible, or Z and the repression models. It reads: reversible but being subject to degradation. The ‘immobilization by sequestration’ model constitutes a spatially dependent version of the @Bðx; tÞ ¼ PB ðZ; xÞ  lBðx; tÞW ðx; tÞ  dB Bðx; tÞ; ð10Þ ‘complex formation model’ proposed in Betegón-Putze et al. (2021), which @t explained the regulations between BRAVO and WOX5 in the whole @Zðx; tÞ @ 2 Zðx; tÞ Arabidopsis stem-cell niche and did not contain any transport or spatial ¼ PZ ðW ; BÞ  dZ Zðx; tÞ þ DZ ; ð11Þ @t @x2 framework. @W ðx; tÞ The promoter activities of BRAVO and WOX5, which are computed at ¼ PW ðW ; xÞ  lBðx; tÞW ðx; tÞ  dW W ðx; tÞ the stationary state (i.e. when all time derivatives are zero), are defined as @t pB(x)=PB(Ws(x), x) and pW(x)=PW(Ws(x), x), respectively, where Ws(x) @ 2 W ðx; tÞ þ DW ; ð12Þ denotes the spatial profile of WOX5 in the stationary state, as indicated by @x2 the subscript s. We also refer to these activities as promoter expressions. n with the production  terms PB ðZ; xÞ ¼ a Z nZþk n Qðx þ LQC =2Þ, Another version of this model where an additional sequestrator (S) affects n n kW B the dynamics of BRAVO and WOX5 is described in the supplementary PZ ðW ; BÞ ¼ b WWn þk n 1þcB and PW ðW ; xÞ ¼ k n þW n gQC ðxÞ. When basal 1 Z W Materials and Methods (results in Fig. S3). activation of BRAVO at the VI regions is included (Fig. S7C), we use n PB ðZ; xÞ ¼ a Z nZþk n Qðx þ LQC =2Þ þ a0 ðxÞ, as done for the immobilization B ‘Immobilization by sequestration with intermediary Z’ model. The stationary promoter activities (expressions) of BRAVO, WOX5 mathematical model and Z are defined as pB(x)=PB(Zs(x), x), pW(x)=PW(Ws(x), x) and The activation of BRAVO by WOX5 is set through the intermediate pZ(x)=PZ(Ws(x), Bs(x)), respectively, where the subscript s denotes the factor Z: protein profile in the stationary state. @Bðx; tÞ ¼ PB ðZ; xÞ  lBðx; tÞW ðx; tÞ  dB Bðx; tÞ; ð4Þ Modeling bravo mutants @t The same type of approach as in Betegón-Putze et al. (2021) is used to @Zðx; tÞ @ 2 Zðx; tÞ simulate the bravo mutant. Specifically, to model this mutant, the very same ¼ PZ ðW Þ  dZ Zðx; tÞ þ DZ ; ð5Þ dynamical equations and the same parameter values are used as those to @t @x2 model the WT (as described in the models above), except for BRAVO, @W ðx; tÞ which is set as B(x, t)=0 for all x and t. This leads to stationary values of ¼ PW ðW ; xÞ  lBðx; tÞW ðx; tÞ  dW W ðx; tÞ @t WS(x) and ZS(x) that are different than in the WT. Although no dynamical @ 2 W ðx; tÞ equation for B(x, t) is set, there is a promoter activity of BRAVO in the þ DW ; ð6Þ stationary state, pB(x), which is as defined for the WT but with the stationary @x2 n profiles of the mutant. We exemplify this with the ‘immobilization by with non-linear regulatory functions PB ðZ; xÞ ¼ a Z nZþk n Qðx þ LQC =2Þ, sequestration’ model defined by Eqns 1 and 2. Setting B(x, t)=0 into Eqns 1 n B n kW PZ ðW Þ ¼ b W W n þk n and PW ðW ; xÞ ¼ k n þW n gQC ðxÞ, where Z and W stand for and 2, we get: Z W Z(x,t) and W(x,t), respectively. The parameters β, dz and Dz describe the production, degradation and diffusion of Z, respectively. The stationary @W bravo ðx; tÞ ¼ PW ðW bravo ; xÞ  dW W bravo ðx; tÞ promoter activities (expressions) of BRAVO, WOX5 and Z are defined as @t pB(x)=PB(Zs(x), x), pW(x)=PW(Ws(x), x) and pZ(x)=PZ(Ws(x)), respectively, @ 2 W bravo ðx; tÞ where the subscript s denotes the protein profile in the stationary state. þ DW ; ð13Þ @x2 Fig. S4B evaluates the effect of introducing a basal production of BRAVO at n where the new superscript bravo indicates that the model corresponds to the the VI (i.e. PB ðZ; xÞ ¼ a Z nZþk n Qðx þ LQC =2Þ þ a0 ðxÞ, where the term α0(x) B bravo mutant and PW(W bravo, x) is the same function as the one used for the corresponds to a basal production of value α0 only at the VI region). WT, but using the variable W bravo instead of W. The stationary BRAVO promoter is pBbravo ðxÞ ¼ PB ðWsbravo ðxÞ; xÞ, where Wsbravo ðxÞ is the stationary ‘Repression of a mobile activator’ mathematical model solution of Eqn 13. The same procedure is applied for the other models. This model reads: Parameter space exploration @Bðx; tÞ ¼ PB ðZ; xÞ  dB Bðx; tÞ; ð7Þ Parameter space exploration was done for the models above in one spatial @t (1D) dimension. For simplicity, in this 1D geometry, the VI region (e.g. @Zðx; tÞ @ 2 Zðx; tÞ light gray area in Fig. 2B-E, Fig. 3B-D,Fig. 4B-F,hFig. 5B,C)iwas  defined to ¼ PZ ðW ; BÞ  dZ Zðx; tÞ þ DZ ; ð8Þ LQC LQC @t @x2 be of the same size as the QC region LQC ; x [  2 ; 2 . Hence, the end of the VI is at position xVI=3LQC/2. To assess whether the simulation @W ðx; tÞ @ 2 W ðx; tÞ DEVELOPMENT ¼ gQC ðxÞ  dW W ðx; tÞ þ DW : ð9Þ results are compatible with experimental observations, we considered four @t @x2 different observables, ξ, RB, RBW and RW, defined as follows: For linear activations, the productions are PB(Z, x)=αZΘ(x+LQC/2) and 1. j ; LDxQC quantifies the expansion of the stationary BRAVO promoter PZ ðW ; BÞ ¼ b 1þcB W , whereas for non-linear saturating activations, we use expression in the bravo mutant compared with the WT, Δx, relative to  n  LQC (QC size), i.e. ξ=1 means that the stationary BRAVO promoter Zn PB ðZ; xÞ ¼ a Z n þk n Qðx þ LQC =2Þ and PZ ðW ; BÞ ¼ b W W n þk n 1 1þcB. These expression in the bravo mutant is expanded to a region as large as the B Z nonlinear PB(Z, x) and PZ(W, B) are the same functions as those in the QC compared with that in the WT. To define Δx (Fig. 2C) we compute immobilization model with intermediary Z, except that PZ(W, B) includes the value of the stationary BRAVO promoter expression in the WT at VI ; pB the end of the VI region and define this value as pBWT ðxVI Þ. WT the repression of BRAVO on Z. c sets the threshold of Z repression by BRAVO. Promoter activities in the stationary state are given by We then compute the spatial position at which the stationary BRAVO 10 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 promoter expression in the bravo mutant takes this value, and define the construction and implementation of the model in the realistic root layouts this position as x bravo (i.e. it is defined as pBbravo ðxbravo Þ ¼ pBWT VI ). Δx can be found in the supplementary Materials and Methods. is defined as Δx≡x bravo−xVI. Because of our simplified one- dimensional geometry, ξ values are not expected to be equal to Mixed model in a realistic root layout those in real roots. The equations for the rate of change of the concentrations of each type of ðx¼0Þ bravo protein across space ~ 2. RB ¼ pBpBWT ðx¼0Þ is the ratio of stationary BRAVO promoter expression rand time t are: at the center of the QC in the bravo mutant over that in the WT. @Bð~ r; tÞ Because the twofold change from experimental data (Fig. 1F) ¼ PB ðZ;~ rÞ  lBð~ r; tÞW ðx; tÞ  dB ð~ rÞBð~ r; tÞ; ð14Þ corresponds to the increase of the spatially integrated GFP expression @t across the midplane, and thus involves the increase at the QC as well @Zð~ r; tÞ as the increase (and thereby expansion) in other regions, we ¼ PZ ðW ; B;~ rÞ  dZ ð~ ~ Z ð~ r; tÞ þ r½D rÞZð~ ~ r; tÞ; rÞrZð~ ð15Þ @t considered that RB values consistent with experimental observations range between 1 and 2. @W ð~ r; tÞ pBWT ðxVI Þ ¼ PW ðW ;~ rÞ  lBð~ r; tÞW ð~r; tÞ  dW ð~ rÞW ð~ r; tÞ 3. RBW ¼ is the ratio of stationary BRAVO promoter @t pW WT ðx¼0Þ expression at the end of the VI over stationary WOX5 promoter ~ W ð~ þ r½D ~ ð~ rÞrW r; tÞ: ð16Þ expression at the center of the QC, both in the WT. ðx¼0Þ bravo The GFP proteins produced under the promoters of BRAVO (BGFP), 4. RW ¼ pW pW WT ðx¼0Þ is the ratio of WOX5 promoter expression at the QC of WOX5 (WGFP) and of Z (ZGFP) have the same production rate as in the bravo mutant over that in the WT. This observable is only used BRAVO, WOX5 and Z, respectively. All these GFP proteins have the same when the models include the (indirect) action of BRAVO on the diffusion coefficient and all degrade linearly with the same degradation rate WOX5 promoter, i.e. when there is WOX5 self-repression and (i.e. that of GFP, dGFP). Hence, the equations for the dynamics of these GFP BRAVO-WOX5 sequestration. proteins are: We compute these four observables in distinct parameter planes by solving, for each model for the WT and the bravo mutant, the corresponding @BGFP ð~ r; tÞ equations in time, until the stationary state is reached (see simulation details ¼ PB ðZ;~ rÞ  dGFP ð~ rÞBGFP ð~r; tÞ @t in the ‘Numerical implementation of the mathematical models’ subsection ~ GFP ð~ ~ GFP ð~ below). Each parameter space in the logarithmic scale is discretized into a þ r½D rÞrB r; tÞ; ð17Þ regular rectangular lattice of 15×15 pairs of values, for which ξ, RBW, RB and @ZGFP ð~ r; tÞ RW are evaluated. Heatmaps are generated using the Python contourf ¼ PZ ðW ; B;~ rÞ  dGFP ð~ rÞZGFP ð~r; tÞ function for each observable. The orange domains correspond to those @t regions where all observables meet the conditions compatible with ~ GFP ð~ þ r½D ~ GFP ð~ rÞrZ r; tÞ; ð18Þ experimental observations. @WGFP ð~ r; tÞ ¼ PW ðW ;~ rÞ  dGFP ð~ rÞWGFP ð~ r; tÞ @t Construction of realistic root layouts ~ GFP ð~ þ r½D ~ GFP ð~ rÞrW r; tÞ: ð19Þ In order to build two-dimensional realistic root layouts with which we could In all these equations, production and degradation only happens in pixels subsequently implement the corresponding reaction-diffusion equations labeled as cells. Within cells, the production functions PB ðZ;~ rÞ, PZ ðW ; B;~ rÞ (e.g. the mixed model in next subsection), we started by taking a confocal and PW ðW ;~ rÞ depend on Z, B and W as in the one-dimensional mixed model image of a middle plane of the root with propidium iodide-stained cell walls r;tÞn and we applied a segmentation routine that divides the root at the pixel scale with basal production of BRAVO, i.e. PB ðZ;~ rÞ ¼ að~rÞ Zð~Zð~ r;tÞn þkBn þ a0 ð~ rÞ,   and into its constituent cellular regions and cell walls (Fig. S8). To do this, W ð~r;tÞn n kW PZ ðW ; B;~ rÞ ¼ b W ð~r;tÞn þk n 1þcBð~r;tÞ and PW ðW ;~ 1 rÞ ¼ k n þW ð~r;tÞn gQC ð~ rÞ. In we made extensive use of the scikit-image collection of Python-based Z W algorithms (threshold_otsu, skeletonize and label) (van der Walt et al., these expressions, að~rÞ takes the constant value α only in the cells shown in 2014). In particular, we first defined the cell boundaries with the Fig. S9, being zero in all the other cells and regions; a0 ð~ rÞ takes the constant threshold_otsu method, which transforms the original image into a value α0 only in the pixels corresponding to the cells shown in Fig. S10, thresholded binary image in which only cell wall pixels remain. We then being zero in all other regions; gQC ð~rÞ takes the constant value γ only in QC used skeletonize to obtain a cell wall with a fixed width (2 pixels). We then cells (Fig. S9), being zero in all the other cells and regions. applied the label function on the modified image to label distinct regions We impose the restriction that inside cell walls only diffusion can happen, (each label constitutes the collection of pixels belonging to the particular whereas no reaction ( production, degradation or binding) can occur. Hence, region). We chose to define labels that enable us to distinguish between cells  cyt DX for pixels within cells and between the cell wall and the outside of the root as follows: all pixels DX ð~ rÞ ¼ ; within a cell have the same label, which is distinct from that of pixels in any Dwall X for pixels labeled as cell wall other cell. With this routine, we could access each cell as an individual  dX for pixels within cells entity, and we could modify its properties as a whole (e.g. change the dX ð~ rÞ ¼ ; 0 for pixels labeled as cell wall parameter values of the protein dynamics in all pixels of that cell). A single label is assigned to all the pixels in any cell wall. Thus, all cell walls  PX ðY; ~rÞ for pixels within cells constitute a single, the same, entity. Another single region is defined by all PX ðY ;~ rÞ ¼ ; 0 for pixels labeled as cell wall pixels outside of the root. DEVELOPMENT The pixel grid is the spatial grid on which the dynamical equations of protein where X stands for any of the modeled species and Y for those species that concentrations are set. For the images used, 1 pixel ~0:5 mm. We assigned the regulate the production of X. The diffusion coefficient in the cell wall same dynamical equations and parameter values in all pixels within each (denoted by superscript wall) is distinct from that inside cells (denoted by labeled region. These equations and parameter values can be distinct between superscript cyt). No differences between cells are set on degradation rates or regions. Specifically, the equations applied on the cell wall are distinct from diffusion coefficients. those within cells, as described in the next subsection. The diffusion coefficient All equations described up to this point correspond to the WT condition. of a protein within all pixels of the cell wall is the same. For simplicity, the To model the bravo mutant, the same equations with the same parameter diffusion coefficient of a protein is set to be the same in all the cell regions (i.e. values are used, but the BRAVO protein concentration is set to zero (i.e. the within any cell), but distinct from that in the cell wall. The only differences set mutant corresponds to Eqns 15-19 and Bð~ r; tÞ ¼ 0; 8~r; t.) Note that to between different cells are in the protein production terms. Further details on model this mutant, the GFP reporter for BRAVO promoter, BGFP ð~ r; tÞ, is 11 RESEARCH ARTICLE Development (2022) 149, dev200510. doi:10.1242/dev.200510 described with the same equations as in the WT (i.e. with Eqn 17) and hence original draft: J.M., M.I.; Writing - review & editing: I.B.-P., N.B., A.I.C.-D.; is not set to zero. Supervision: A.I.C.-D., M.I.; Project administration: A.I.C.-D., M.I.; Funding acquisition: In Fig. 6 and Figs S11-S14, pBRAVO:GFP corresponds to the variable A.I.C.-D., M.I. BGFP ð~r; tÞ computed at the stationary state (i.e. the activity of the BRAVO promoter as seen through its GFP reporter), whereas Bð~ r; tÞ represents the Funding stationary BRAVO protein concentration. Analogous definitions for WOX5 This work was supported by the Ministerio de Ciencia e Innovació n (MCIN), Agencia and Z are used. Fig. 6B-D use the same parameter values as in Fig. 6A Estatal de Investigació n (AEI) and European Regional Development Fund (Fondo (Table S3) except for c=0 and DcytZ ¼ DZ cellwall ¼ 0 (Fig. 6B); c=0 (Fig. 6C); Europeo de Desarrollo Regional, FEDER) program ‘Una manera de hacer Europa’ and λ=0 (Fig. 6D). MCIN/AEI/10.13039/501100011033/FEDER (PGC2018-101896-B-I00 to M.I.); Ministerio de Ciencia e Innovació n, Agencia Estatal de Investigació n and European Numerical implementation of the mathematical models Social Fund (Fondo Social Europeo, FSE) program ‘El FSE invierte en tu futuro’ (BES-2016-078218 to J.M.); Ministerio de Ciencia e Innovació n (MCIN), Agencia To find the stationary spatial profiles of the 1D models (‘immobilization by Estatal de Investigació n (AEI) and European Regional Development Fund (ERDF) sequestration’, ‘immobilization by sequestration with intermediary Z’, (BIO2016-78955 to A.I.C.-D.); the European Research Council (ERC) Consolidator ‘repression’ and mixed models), we first reduced the system of differential Grant (ERC-2015-CoG–683163 to A.I.C.-D.); Generalitat de Catalunya (M.I. and equations to second-order ordinary differential equations for the diffusible J.M. in Grup de Recerca Consolidat 2017 SGR 1061; FI-DGR 2016FI-B 00472, also variables. To do this, we first set to zero the equation for the non-diffusible funded by FSE, to N.B.); and the Ministerio de Educació n, Cultura y Deporte variables and substituted the result on the other equations. The remaining (FPU15/02822 to I.B.-P.). Open Access funding provided by Universitat de system could be cast as a boundary value problem, which we solved Barcelona. Deposited in PMC for immediate release. numerically by using the solve_bvp routine embedded in the Python-based SciPy library (Virtanen et al., 2020). In all these calculations, we used a Peer review history spatial step size δx=0.05 a.u. (arbitrary units), in a domain of x∈[−600, The peer review history is available online at https://journals.biologists.com/dev/ article-lookup/doi/10.1242/dev.200510. 600]a.u.. We use a QC size of LQC=15 a.u., and an equally long VI region. With this methodology, the explicit time dependence of the variables is not References computed, but only their stationary state. To double-check the validity of our Aida, M., Beis, D., Heidstra, R., Willemsen, V., Blilou, I., Galinha, C., solutions, we also simulated the whole temporal dynamics of the equations Nussaume, L., Noh, Y.-S., Amasino, R. and Scheres, B. (2004). The with a forward Euler method, obtaining the same stationary solutions as with PLETHORA genes mediate patterning of the Arabidopsis root stem cell niche. the solve_bvp method, thus confirming the results (Figs S1-S7). For all the Cell 119, 109-120. doi:10.1016/j.cell.2004.09.018 models, the values of B, W and Z at the boundaries (x=600 and x=− 600) are Bejarano, F., Pé rez, L., Apidianakis, Y., Delidakis, C. and Milá n, M. (2007). set to zero, except for the additional sequestrator variable S (Fig. S3), for Hedgehog restricts its expression domain in the Drosophila wing. EMBO Rep. 8, 778-783. doi:10.1038/sj.embor.7401003 which periodic boundary conditions are implemented. Berckmans, B., Kirschner, G., Gerlitz, N., Stadler, R. and Simon, R. (2020). The results displayed on the heatmaps were obtained by simulating the CLE40 signaling regulates root stem cell fate. Plant Physiol. 182, 1776-1792. dynamical equations corresponding to each model for the WT and the bravo doi:10.1104/pp.19.00914 mutant scenarios until the stationary state was reached. To compute the Betegó n-Putze, I., Mercadal, J., Bosch, N., Planas-Riverola, A., Marquès- temporal dynamics, we used a forward Euler method and discretized the 1D Bueno, M., Vilarrasa-Blasi, J., Frigola, D., Burkart, R. C., Martı́nez, C., Conesa, A. et al. (2021). Precise transcriptional control of cellular quiescence by Laplacian term as @ @xuðxÞ 2 2 ¼ d1x2 ðuiþ1 þ ui1  2ui Þ, where u(x) represents BRAVO/WOX5 complex in Arabidopsis roots. Mol. Syst. Biol. 17, e9864. either W and/or Z. The spatial step is δx=1 and uses a domain x∈[−400, doi:10.15252/msb.20209864 400]a.u.. The time step is dt=0.01 and the final time is tfinal=1000, where Cannataro, V. L., McKinley, S. 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The Development 119, 71-84. doi:10.1242/dev.119.1.71 Centre for Research in Agricultural Genomics (CRAG) is funded by the ‘Severo Eldar, A., Rosin, D., Shilo, B.-Z. and Barkai, N. (2003). Self-enhanced ligand degradation underlies robustness of morphogen gradients. Dev. Cell 5, 635-646. DEVELOPMENT Ochoa Programme’ from Centers of Excellence in R&D 2016-2019 (SEV-2015-485 0533). CRAG is also supported by grant CEX2019-000902-S funded by MCIN/AEI/ doi:10.1016/S1534-5807(03)00292-2 Forzani, C., Aichinger, E., Sornay, E., Willemsen, V., Laux, T., Dewitte, W. and 10.13039/501100011033, and by the CERCA Programme/Generalitat de Murray, J. A. H. (2014). WOX5 suppresses CYCLIN D activity to establish Catalunya. quiescence at the center of the root stem cell niche. Curr. Biol. 24, 1939-1944. doi:10.1016/j.cub.2014.07.019 Competing interests Grieneisen, V. A., Xu, J., Maré e, A. F. M., Hogeweg, P. and Scheres, B. (2007). The authors declare no competing or financial interests. 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