In animals and plants, stem-cell niches are local microenvironments that are tightly regulated to preserve their unique identity while communicating with adjacent cells that will give rise to specialized cell types. In the primary root of Arabidopsis thaliana, two transcription factors, BRAVO and WOX5, among others, are expressed in the stem-cell niche. Intriguingly, BRAVO, a repressor of quiescent center divisions, confines its own gene expression to the stem-cell niche, as evidenced in a bravo mutant background. Here, we propose through mathematical modeling that BRAVO confines its own expression domain to the stem-cell niche by attenuating a WOX5-dependent diffusible activator of BRAVO. This negative feedback drives WOX5 activity to be spatially restricted as well. The results show that WOX5 diffusion and sequestration by binding to BRAVO are sufficient to drive the experimentally observed confined BRAVO expression at the stem-cell niche. We propose that the attenuation of a diffusible activator can be a general mechanism acting at other stem-cell niches to spatially confine genetic activity to a small region while maintaining signaling within them and with the surrounding cells.
Both in animals and plants, stem cells are maintained in tightly regulated microenvironments called stem-cell niches (SCNs), in which they remain in an undifferentiated state and provide a continuous flux of precursors of more specialized cells that sustain growth and replace old or damaged tissues (Sablowski, 2007). SCNs usually consist of a few stem cells that are maintained by short-range signals produced by localized sources or ‘organizing centers’, groups of cells that sustain neighboring cells in a stem-cell state (Scadden, 2006; Spradling et al., 2001). As stem-cell daughters are placed outside the reach of these signals, they begin to differentiate, giving rise to more specialized cell types (Sablowski, 2004). In animals, diffusible ligands are common signals that preserve stem cells (Chen and McKearin, 2003; Han et al., 2008; Xie and Spradling, 1998; Zhang and Kalderon, 2001). Overproduction of these signals can drive an increase in the number of stem cells, resulting in enlarged niches and often leading to malfunctioning of the surrounding tissue or even whole organs (Spradling et al., 2001). Knowing the origin and function of these signals is therefore essential to understand the role of stem cells in the processes underlying organism development and sustenance.
In the model plant Arabidopsis thaliana, highly mobile hormones, such as auxin (Aida et al., 2004; Ding and Friml, 2010; Müller and Sheen, 2008), as well as short-range transcription factors like WUSCHEL and WUSCHEL-RELATED HOMEOBOX 5 (WOX5) (Sarkar et al., 2007; Yadav et al., 2011) are involved in specifying stem-cell niches. In the root apical meristem of Arabidopsis, the SCN lies at the tip of the root, a location known to be established by positional information conferred by the overlapping of the SCARECROW and SHORTROOT transcription factors, together with the activation of PLETHORA genes by the hormone auxin, the levels of which peak at the position where the SCN is established (Kidner et al., 2000; Shimotohno et al., 2018; van den Berg et al., 1995). This positional signaling allows for the necessary plasticity to establish a new niche when it has been destroyed or damaged, by a continual supply and renewal of stem cells at the same location (Sena et al., 2009).
The SCN is composed of a small group of rarely dividing pluripotent cells called the quiescent center (QC) and of the immediately surrounding stem cells, i.e. the vascular initials (VIs), columella stem cells (CSCs) and cortex-endodermis initials (CEIs) (Fig. 1A) (Dolan et al., 1993). Direct cell-cell contact from the QC to its surrounding stem cells is important for stem cell identity (Liu et al., 2017b; van den Berg et al., 1997) and can involve the transport of short-range signals from the QC. The homeodomain transcription factor WOX5 is specifically expressed at the QC (Sarkar et al., 2007) and can move towards adjacent cells (Berckmans et al., 2020; Clark et al., 2020; Pi et al., 2015). WOX5 itself has been proposed to act as the long-sought short-range signal to repress columella stem cell differentiation (Pi et al., 2015), albeit recent results challenge this view (Berckmans et al., 2020). Although short-range signaling is thought to ensure that stem-cell numbers are restrained and the SCN does not become displaced from the growing root tip, it is yet unclear how this is achieved (Matosevich and Efroni, 2021; Rahni et al., 2016; Scheres, 2007).
The R2R3-MYB transcription factor BRASSINOSTEROIDS AT VASCULAR AND ORGANIZING CENTER (BRAVO) has recently been linked to the maintenance of QC homeostasis (Betegón-Putze et al., 2021; Vilarrasa-Blasi et al., 2014). BRAVO is expressed at the QC and VI (Vilarrasa-Blasi et al., 2014) and colocalizes with WOX5 in the QC (Betegón-Putze et al., 2021). Both BRAVO and WOX5 have been shown to individually promote quiescence, as mutant roots with disrupted BRAVO or WOX5 exhibit increased QC divisions, supporting their role as essential factors for QC homeostasis (Forzani et al., 2014; Vilarrasa-Blasi et al., 2014). We have recently shown that BRAVO and WOX5 are co-dependent, promoting expression of each other and physically interacting, presumably through their binding into a protein complex (e.g. as a heterodimer) (Betegón-Putze et al., 2021). These results suggested that BRAVO promotes WOX5 expression indirectly through WOX5, via the formation of the complex and the subsequent disruption of a WOX5 negative feedback on itself. In contrast, since the expression of BRAVO increases upon WOX5 overexpression, WOX5 activates (directly or through targets) BRAVO expression (Betegón-Putze et al., 2021).
Our previous study revealed the effective regulations between BRAVO and WOX5, but did not address the mechanisms driving spatial changes in their expression. The expression of the BRAVO promoter, restricted to the QC and VI in wild-type (WT) roots, increases and expands towards the vasculature, cortex and endodermis in the bravo-2 loss-of-function mutant (Betegón-Putze et al., 2021) (Fig. 1B,C). This expansion suggests that BRAVO actively confines in space its own expression domain. Although our previous study considered the increase in total (spatially integrated) BRAVO promoter expression in the bravo-2 mutant through negative feedback (Betegón-Putze et al., 2021), it did not address how the expression of BRAVO is regulated in space to generate self-confinement. Here, we address this problem through mathematical and computational modeling.
To this end, we used the finding that WOX5 can activate BRAVO expression and can bind to BRAVO protein (Betegón-Putze et al., 2021) to evaluate how these interactions can be regulated in space to drive spatial self-confinement of BRAVO expression. We also used the finding that BRAVO and WOX5 can act antagonistically on common targets (Betegón-Putze et al., 2021) to propose an additional mechanism for self-confinement. To assess whether these different scenarios can drive a spatial confinement that is compatible with the reported experimental data, we first studied them through minimal models in one spatial dimension, separately and in combination. These simplified schemes enabled thorough parameter space exploration. Finally, to overcome the limitations set by the simplified geometry, we translated these scenarios to a realistic two-dimensional geometry. Our results support that WOX5 diffusion (Berckmans et al., 2020; Clark et al., 2020; Pi et al., 2015) is sufficient to drive the spatial self-confinement of BRAVO expression. BRAVO self-confinement also reduces the spatial domain of WOX5 activity, overall providing a natural way for stem-cell-specific factors to locally regulate SCN maintenance. Altogether, our results shed light on the regulatory principles balancing the confinement of transcription factors to a microenvironment and their communication with the surrounding cells.
BRAVO can confine its own expression domain by immobilizing WOX5
The expansion of BRAVO promoter expression towards the vasculature in the bravo-2 loss-of-function mutant compared with the WT (Betegón-Putze et al., 2021) (Fig. 1B-E) suggests that, in the WT, BRAVO confines its own expression to the SCN. Because there is no expansion in the loss-of-function wox5-1 mutant (Fig. 1D) nor in the bravo-2 wox5-1 mutant (Betegón-Putze et al., 2021), we considered that the mechanism for BRAVO self-confinement requires WOX5. To decipher how this self-confinement can be attained, we first considered that BRAVO transcription is ultimately activated by WOX5 (either directly or through WOX5 targets) (Betegón-Putze et al., 2021) and that WOX5 proteins are able to move from the QC to the VI (Berckmans et al., 2020; Clark et al., 2020). Thus, WOX5, by moving to the VI cells and shootwards, is expected to induce BRAVO expression in these cells. Additionally, we considered that BRAVO and WOX5 can physically interact at the protein level, presumably by binding together, as suggested by co-immunoprecipitation and fluorescence resonance energy transfer by fluorescence lifetime imaging (FRET-FLIM) analysis (Betegón-Putze et al., 2021).
Although the mobility of WOX5 (possibly through the plasmodesmata) has been experimentally tested in planta (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 and impeding the activation of BRAVO expression in the VI. Hence, the activation of the BRAVO promoter by WOX5 is spatially confined by BRAVO proteins, a restriction which can be released when BRAVO can no longer immobilize WOX5, e.g. in the bravo mutant.
To evaluate this mechanism, we constructed a minimal mathematical model, hereafter named the ‘immobilization by sequestration’ model (see Materials and Methods; Fig. 2A,B) in which the main assumptions are: (1) WOX5 induces BRAVO expression (Betegón-Putze et al., 2021); (2) the WOX5 protein diffuses (Berckmans et al., 2020; Clark et al., 2020), whereas BRAVO does not; (3) the BRAVO and WOX5 proteins reversibly bind together forming a complex that does not induce BRAVO expression (Betegón-Putze et al., 2021); (4) the BRAVO-WOX5 complex is immobile, i.e. it cannot diffuse; and (5) all molecules, the complex included, are degraded.
We studied the model in a one-dimensional space in which WOX5 is produced only at a localized region (dark grey shaded area in Fig. 2B,C) that we identify as the QC, and can diffuse in both directions, namely towards regions that could be identified as the CSCs and columella cells (negative values of the position in Fig. 2B,C), or towards the vasculature (positive values of the position in Fig. 2B,C, where the light gray shaded region is identified as VI cells). Conversely, BRAVO was set to be activated by WOX5 only in regions from the QC and towards the vasculature, but not towards the columella, thus mimicking the asymmetric activity of the BRAVO promoter in the Arabidopsis primary root.
In Fig. 2C-E, we show the stationary activity profiles of BRAVO and WOX5 promoters (pB and pW, respectively) and the stationary BRAVO and WOX5 protein concentrations (B and W, respectively), obtained by numerically solving the model equations with linear activation of BRAVO by WOX5 (see Materials and Methods). The concentration of the protein complex (which is proportional to the product BW) is shown in Fig. S1. The stationary profiles obtained when modeling the bravo mutant condition (see Materials and Methods) are also depicted. These results support that the ‘immobilization by sequestration’ mechanism can drive self-confinement and constitute a potential way for BRAVO to spatially confine its own expression in the WT root.
To address whether this mechanism drives BRAVO expression and self-confinement consistent with experimental observations, we defined three different observables (see Materials and Methods). The first observable, ξ, focuses on the spatial expansion of BRAVO promoter expression in the bravo mutant compared with that in the WT (Fig. 2C). We considered that expansions within 0.5 and five times the QC size were consistent with experimental observations (Fig. 1E). The second observable, RB, evaluates the increase of BRAVO expression levels in the bravo mutant (Fig. 2C). The model results are considered compatible with experimental data when BRAVO promoter activity at the QC increases up to twofold in the bravo mutant (Fig. 1F). The third observable, RBW, compares BRAVO and WOX5 expressions in WT, as BRAVO transcripts are less abundant (Fig. 1G) (Betegón-Putze et al., 2021; Clark et al., 2019). We considered that the simulation results also were compatible with observations from real roots if BRAVO promoter expression at the end of the VI over WOX5 promoter expression at the QC, both in the WT, was within the range of 0.05 and 0.5.
Exploration of different parameter spaces shows large domains in which the model is compatible with experimental data, for linear (Fig. 2F,G; Fig. S1A) and nonlinear regulatory functions (Fig. S2A,B). In this mechanism of self-confinement, the binding between WOX5 and BRAVO (mediated by the binding strength λ) as well as WOX5 diffusion are necessary (Fig. 2F). However, too much sequestration of WOX5 by BRAVO (large λ and high BRAVO synthesis rate αL) can lead to an overly exaggerated confinement, with BRAVO being highly expressed at the QC but not in VI cells (Fig. 2G; Fig. S1C), a situation which would disagree with the experimentally observed WT expression (Fig. 1B). The results also show that if WOX5 is set to have a large diffusion coefficient, BRAVO expression in the WT expands but gets dimmer (Fig. S1E). This suggests that WOX5 diffusion might be rather small, in agreement with observed protein distributions of WOX5 within the root SCN, which primarily reach cells adjacent to the QC (Berckmans et al., 2020; Clark et al., 2020). Overall, our results indicate that the ‘immobilization by sequestration’ mechanism is a plausible candidate model to explain the observed self-confinement of BRAVO expression.
As sequestration is a necessary factor for this mechanism to work, one could argue that the presence of other molecules also binding to BRAVO and/or WOX5 might impair or even destroy the confinement. Indeed, it is known that WOX5 and BRAVO proteins can bind to additional molecules, such as TOPLESS (TPL) and BES1 (Betegón-Putze et al., 2021; Vilarrasa-Blasi et al., 2014). These are relatively large molecules (∼39 kDa and ∼124 kDa, respectively; The Arabidopsis Information Resource, TAIR) and it would be possible for them to immobilize WOX5. We modeled this situation assuming the most limiting case, i.e. that these molecules (S) only act as competitors for the binding of BRAVO and WOX5 (see supplementary Materials and Methods; Table S2). The results show that BRAVO self-confinement takes place if S is much less produced than BRAVO (for the same degradation rate, Fig. S3). If instead S production is similar to that of BRAVO, then the BRAVO promoter expression domain is similarly confined in the WT and in the bravo mutant. In this mutant, S overcomes the absence of BRAVO by binding to WOX5 and immobilizing it (Fig. S3). As TPL and BES1 transcript abundance at the QC is not very high [ranging 0.57-3.08 times that of BRAVO and 0.13-0.39 times that of WOX5, based on read counts (Clark et al., 2019; Betegón-Putze et al., 2021)], the results suggest that the ‘immobilization by sequestration’ mechanism may take place.
To further assess the plausibility of the ‘immobilization by sequestration’ mechanism in the context of BRAVO and WOX5, we included the fact that BRAVO promotes WOX5 expression, as the expression of GFP under the WOX5 promoter decreases in the bravo-2 mutant (Betegón-Putze et al., 2021; Vilarrasa-Blasi et al., 2014). Specifically, we previously proposed that BRAVO promotes WOX5 expression through a mechanism that involves a WOX5 negative feedback (modeled as self-repression) and the binding of the WOX5 and BRAVO proteins (Betegón-Putze et al., 2021). WOX5 self-repression is consistent with increased WOX5 promoter expression in the wox5-1 mutant (Betegón-Putze et al., 2021). Therefore, as the ‘immobilization by sequestration’ model already included the binding between BRAVO and WOX5, we only added WOX5 self-repression to the model (Fig. 2H, Materials and Methods) and defined a fourth observable, RW, which measures the ratio between WOX5 promoter activity in the bravo mutant and that in the WT. We considered that simulation results within the range of 0.5 and 0.9 are compatible with experimental data (Betegón-Putze et al., 2021). The results show that the BRAVO expression profile remains similar when WOX5 self-repression is included (Fig. S2). RW sets additional constraints to the lower bound of BRAVO production rate (Fig. 2I) and limits the diffusion coefficient of WOX5 (Fig. 2J). This last restriction is not expected to be very limiting as the experimental data suggest a small amount of WOX5 diffusion (Berckmans et al., 2020; Clark et al., 2020).
Altogether, the ‘immobilization by sequestration’ mechanism constitutes a plausible candidate for explaining the self-confined BRAVO expression to both the QC and VI cells in the WT. Notably, the WT stationary profile of the WOX5 protein (W, not bound to BRAVO) obtained for this model is not symmetric, but decays differently above and below the QC (Fig. 2E). This behavior is caused by the presence and absence, respectively, of BRAVO proteins in the two distinct spatial regions. Above the QC, the gradient of WOX5 proteins is steeper than below the QC, where BRAVO cannot be activated. Hence, the presence of BRAVO makes the WOX5 gradient more abrupt, restricting the spatial domain in which WOX5 proteins are concentrated, and consequently confining the BRAVO expression domain. Therefore, this mechanism not only drives self-confined BRAVO expression but also results in a confined action of WOX5 proteins.
A mobile intermediary can facilitate BRAVO self-confinement
The self-confinement of BRAVO expression in the ‘immobilization by sequestration’ model requires WOX5 to diffuse. Experiments suggest that although it is mobile, WOX5 does not move very large distances (Berckmans et al., 2020; Clark et al., 2020). Hence, its low mobility may not be sufficient to explain the self-confinement of BRAVO expression observed in real roots. As activation of BRAVO by WOX5 could occur through intermediary molecules, we asked whether realistic BRAVO self-confinement can arise for smaller WOX5 diffusion coefficients if WOX5 activates BRAVO through a mobile intermediary molecule, Z. To evaluate this scenario, we included such an intermediary molecule in the ‘immobilization by sequestration’ model and studied its implications by numerically simulating the WT and bravo mutant in one dimension, as previously done. In this model (Fig. 3A,B) the assumptions are analogous to those mentioned earlier but include Z as follows: (1) WOX5 activates Z, which activates BRAVO; (2) Z and WOX5 diffuse, whereas the BRAVO protein does not; (3) BRAVO and WOX5 proteins reversibly bind together forming a complex that cannot induce Z; (4) the BRAVO-WOX5 complex is immobile, i.e. it cannot diffuse; (5) all molecules are degraded; and, (6) additionally, WOX5 self-represses as in Fig. 2H.
The results show that Z drives a wider BRAVO expression, as expected, which is still self-confined (Fig. 3C,D). The mobility of Z enables BRAVO self-confinement at lower diffusion coefficients of WOX5 (compare Fig. 3E with Fig. 2J, Fig. 3J) as BRAVO, when binding to WOX5, confines its expression by also attenuating the production of Z (Fig. S4).
BRAVO can confine its own expression domain by repressing its mobile activator
The previous model showed that a transcription factor can confine its own expression by reducing the production of its mobile activator Z. Based on this, we envisaged an additional scenario that drives a reduction of the mobile activator but is not based on the 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. 4H; Fig. S6).
Finally, we considered that BRAVO promotes WOX5 expression and assumed that this occurs through binding with WOX5 and WOX5 self-repression. We termed this the ‘mixed’ model, as it involves both the ‘repression’ and the ‘immobilization by sequestration’ models (Fig. 5A,B). Despite the sequestering of BRAVO by (large amounts of) WOX5 reduces the BRAVO-mediated repression, the presence of both the repression and immobilization mechanisms enhances self-confinement (Fig. 5C). In addition, BRAVO self-confinement that is compatible with experimental observations happens in larger parameter domains, i.e. when repression is not sufficient, sequestration takes its place (Fig. 5D-F). Note that sequestration is required to account for decreased WOX5 expression in the bravo mutant but not for BRAVO self-confinement. Similar results are found when basal production of BRAVO, i.e. independent of WOX5, is set at the VI region (Fig. S7) to mimic the situation in which these cells still show BRAVO promoter expression in the wox5-1 mutant (Betegón-Putze et al., 2021) (Fig. 1D).
The ‘immobilization by sequestration’ mechanism is sufficient and the ‘repression’ mechanism enhances self-confinement in Arabidopsis
Our previous results suggest that the mixed model is a conceivable candidate to explain the self-confined expression of BRAVO in the Arabidopsis root SCN. However, it remains unclear whether the degree of diffusion of WOX5 is sufficiently large to generate realistic expansions. To address this issue, we modeled the mixed model on realistic two-dimensional root layouts, for which the cellular geometry of the roots, as well as the presence of cell walls, was explicitly incorporated. We also modeled the dynamics of GFP produced by the BRAVO and the WOX5 promoters (pBRAVO:GFP and pWOX5:GFP, respectively; see Materials and Methods). 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 scale. Hence, the size of the cytoplasm and cell walls is determined by the number and localization of their corresponding pixels (Fig. S8). (2) The dynamics of the molecular components in the interior of the cells is distinct from the dynamics in the cell walls. Specifically, in all the pixels belonging to the interior of the cell, molecules can be produced, regulated or degraded, can sequester other molecules and are able to diffuse. Inside the cell walls, however, molecules are only able to diffuse. (3) The diffusion coefficient in cell walls is set to be smaller than in the interior of the cells. With this assumption, the physical boundaries between cells are naturally incorporated. These are to be expected for any molecule diffusing freely inside the cytoplasm but moving only occasionally from cell to cell, as it may happen, for example, in plasmodesmata-mediated transport. For simplicity (and lack of evidence), we assume that BRAVO and the BRAVO-WOX5 complex can only move inside cells, being unable to diffuse through cell walls. Conversely, WOX5 and Z proteins can move inside the cytoplasm with diffusion coefficients DWcyt and DZcyt, respectively, and between cells with diffusion coefficients DWwall and DZwall, respectively. Because GFP proteins are set to diffuse from cell to cell, they can reach cells in which there is no promoter activity. However, the degree of GFP diffusion is set sufficiently low so that this effect is small, such that GFP expression under the WOX5 promoter is mostly localized at the QC (Betegón-Putze et al., 2021; Vilarrasa-Blasi et al., 2014). (4) Cells are classified by cell type. We defined nine different cell types: quiescent center (QC), vascular initials (VI), vascular cells (V), cortex-endodermis initials (CEI), pericycle (PER), cortex (CX), endodermis (END), epidermis (EPI), columella stem cells (CSC), columella cells (CC) and lateral root cap cells (LRC) (Fig. 1A). This classification enables setting different dynamics for the proteins in distinct cell types, as well as different transport properties (through different diffusion coefficients). However, we set the diffusion coefficient of each molecular species to be the same in all cell types. In contrast, different cell types differ in their protein production dynamics. WOX5 only activates BRAVO in the QC, VI, vascular cells, cortex and endodermis, but not in the remaining cell types (Fig. S9). This assumption is based on the experimental observation that BRAVO is expressed only in inner tissues, i.e. from the SCN upwards. Basal production of BRAVO – independent of WOX5 – is set in a few of these cells (Fig. S10). For simplicity, WOX5 is set to be only produced at the QC [albeit low amounts of promoter expression are also present in the vascular initials (Berckmans et al., 2020)]. For each species, the degradation rate was assumed to be the same in all cell types.
We chose parameter values for WOX5 dynamics so that WOX5 proteins, in the absence of any regulatory factor, remain mostly localized at the QC, VI and CSCs (Fig. S11). This is consistent with reported data of GFP bound to WOX5 proteins (Berckmans et al., 2020; Clark et al., 2020). The other parameter values were chosen to drive pWOX5:GFP and pBRAVO:GFP expression patterns compatible with those observed in real WT roots (Table S3). The mixed model can explain the behavior of pBRAVO:GFP in the WT and the expansion of its domain in the bravo mutant (compare Fig. 6A with Fig. 1E). BRAVO self-confinement additionally involves the confinement of WOX5 proteins and of the diffusible target Z (Fig. S12). Although different layouts (Fig. S10) are used to model WT and bravo mutant roots (Fig. S10), this does not introduce significant differences (Fig. S13).
The small diffusion of WOX5 is sufficient to drive a realistic confinement of pBRAVO:GFP through the ‘immobilization by sequestration’ mechanism without the need of Z mobility (and thereby, of Z) (Fig. 6B). Yet, the presence of a mobile Z cannot be discarded. For such a mobile Z, sequestration and repression can provide an attenuation of Z that contributes to confine BRAVO expression (Fig. 6A,C,D; Fig. S14). Taken together, the results in the realistic root layout support the idea that the immobilization of WOX5 by BRAVO is sufficient for the self-confined expression of BRAVO in the root SCN.
We have previously shown that BRAVO and WOX5 regulate the expression of each other, and that their binding into a protein complex can be relevant for these regulations and for BRAVO and WOX5 action on QC divisions (Betegón-Putze et al., 2021). Here, we show that these interactions, together with the diffusion of WOX5, are sufficient to drive the spatially self-confined expression of BRAVO, as revealed by the ‘immobilization by sequestration’ mechanism. Furthermore, the opposite regulation of common targets by WOX5 and BRAVO, as inferred by transcriptional profiling (Betegón-Putze et al., 2021), proposes a complementary scenario in which this confinement is induced by negative feedback between BRAVO and a factor activated by WOX5 (the ‘repression’ mechanism). According to transcriptomics analyses at the QC, only the AT3G41762 gene has a significantly decreased expression in the wox5-1 mutant and increased expression in the bravo-2 mutant, making it a potential candidate to act as Z if it induces BRAVO. Notably, this gene is strongly expressed in the root SCN of WT Arabidopsis (Fig. S15) (Clark et al., 2019), but its function awaits to be characterized. The remaining genes that are antagonistically and significantly regulated by BRAVO and WOX5 show increased expression in the wox5-1 mutant and decreased expression in the bravo-2 mutant (Betegón-Putze et al., 2021). Hence, they are repressed by WOX5 and activated by BRAVO. If any of these genes inhibits a particular factor, this factor might be acting as Z (Fig. S15). Such a potential case is exemplified by peroxidase proteins, which are repressed by WOX5 but activated by BRAVO, and which remove the mobile hydrogen peroxide. Based on their expression patterns in the SCN (Fig. S15) (Clark et al., 2019), they may play a role in confining BRAVO expression through hydrogen peroxide, pointing to the relevance of preventing oxidative stress in the stem-cell microenvironment. Whether any of these candidates regulates BRAVO and can drive the ‘repression’ mechanism remains to be elucidated.
The ‘immobilization by sequestration’ and ‘repression’ mechanisms share the idea that the emergence of self-confinement involves negative feedback of BRAVO on itself with a mobile activator (WOX5 or Z) and an immobile inhibitor (BRAVO). As such, they mechanistically account, at least partially, for the effective negative self-regulation of BRAVO expression in the whole SCN that was previously proposed (Betegón-Putze et al., 2021). Each of these two mechanisms can be understood as a different regulatory way for this negative feedback to be accomplished: by immobilizing the activator or by reducing the production of the activator. Therefore, they can be encompassed within a general framework of self-confinement, i.e. an ‘attenuation of a mobile activator’ model. Both mechanisms drive the self-confinement of both the repressor and the activator. Yet, the mechanisms are not equivalent, each of them having distinct characteristics, as our analysis revealed. The most noticeable feature is perhaps the fact that the ‘immobilization by sequestration’ mechanism involves a change in the gradient profile of the activator (Fig. 2E; supplementary Materials and Methods), whereas the ‘repression’ mechanism does not.
A different mechanism to drive self-confined expression has been proposed for WUSCHEL in the shoot apical meristem of Arabidopsis. According to this mechanism, the WUSCHEL protein confines its own expression by activating its repressor, CLAVATA3, which is highly mobile (Yadav et al., 2011; Zhou et al., 2018). Both this mechanism and the ‘attenuation of a mobile activator’ mechanism studied here share the idea that negative feedback is responsible for the self-confinement. However, in the ‘attenuation of a mobile activator’ mechanism, the strongly mobile component is the activator and not the repressor. This mechanism is thus a distinct mechanism for self-confinement and, because of its minimal assumptions, we expect it, or a variant thereof, to take place in very distinct developmental contexts. For instance, Hedgehog signaling in the Drosophila wing confines its own expression by activating one of its repressors, the nuclear zinc finger master of thickveins (mtv, also known as sbb) (Bejarano et al., 2007). Thus, this Hedgehog self-confinement may be framed within the ‘attenuation of a mobile activator’ mechanism, in which Hedgehog acts as the mobile activator (Z in the repressor model) and master of thickveins as the immobile repressor.
The mechanism of ‘immobilization by sequestration’ can be related to mechanisms that generate robust morphogen gradient profiles (Eldar et al., 2003; Shilo and Barkai, 2016). Morphogens are ligand molecules that are produced at localized sources but can diffuse, generating an activity gradient that can then be interpreted by different genes, leading to their activation or repression in a concentration-dependent manner. It has been proposed that the sequestering of the morphogen by receptors might lead to an effective non-linear degradation of the ligand, resulting in concentration profiles that are robust to changes in the rate of production at the source (Eldar et al., 2003). We could make the correlation between such models and ours by identifying WOX5 as the morphogen and BRAVO as the receptor. In agreement with what has been described for morphogen gradients induced by non-linear degradation (Eldar et al., 2003), we find that the gradient of WOX5 decays more abruptly in the presence of BRAVO than without, thus suggesting that the specific regulations between BRAVO and WOX5 may be tuned to achieve robust activity profiles. Yet, we have not seen a robust response to changes in WOX5 expression levels for parameter values compatible with experimental observations (Fig. S16) and further analysis is required to address whether this model enhances BRAVO robustness to WOX5.
Modeling of root tissues has been most commonly done in terms of idealized realistic simplified geometries in which cells and cell walls are each subdivided in squares or rectangles with molecules diffusing between them (Grieneisen et al., 2007; Tian et al., 2014), or by considering real cellular layouts with diffusing molecules between but not within cells (Jönsson et al., 2005; Voß et al., 2014; Yadav et al., 2011). By using pixels as the basic unit for discretization, we are able to model the shapes of cells in a realistic manner and consider both the interactions within and between cells. A similar pixel-based approach has been used to model hormonal crosstalk in the Arabidopsis root (Liu et al., 2017a). Mathematically, our framework can be characterized as a reaction-diffusion model in heterogeneous media, in which the spatial inhomogeneities appear due to the presence of cell walls, which involve different diffusion coefficients. The realistic root layout used for the simulations can be extended to include internal structures within the cells (such as the nucleus) as well as structures in the cell walls (e.g. specific communication channels). Therefore, it has the potential to implement and evaluate much more complex scenarios in a manageable way.
The similarities between stem-cell niche organization in animals and plants may represent the outcome of convergent evolution (Dinneny and Benfey, 2008). Multicellularity – a necessary condition for stem-cell niches to emerge – is thought to have evolved independently in both kingdoms (Grosberg and Strathmann, 2007; Knoll, 2011), implying that the presence of stem-cell niches in widely disparate organisms may be a direct consequence of developmental constraints and not of historical contingencies. Similar mechanisms of niche regulation are therefore to be expected, not through common genes or molecules, but through more general regulatory principles. The fact that stem-cell niches consist of narrow regions of a few cells clustered together, in opposition to large numbers of cells distributed over the whole organism, possibly emerged as a way to ensure a proper balance between centralized renewal and genome integrity, by minimizing deleterious mutations that may be able to spread across whole cell lineages (Sena et al., 2009; Yoshiyamaa et al., 2009). Indeed, the smaller the population of stem cells and the lower their division rate, the less likely for deleterious mutations to accumulate in differentiated tissues (Cannataro et al., 2017). The mechanism proposed in this paper establishes a balance between the communication mediated by the activator (WOX5) with confining its action, ensuring communication remains local.
In the Arabidopsis SCN, these signals allow cells to communicate between them and with other cell types, at the same time as they create boundaries within which local information can be transmitted. Indeed, QC cells have been shown to influence neighboring cell types such as the CSCs, in which WOX5 can play a crucial role as a signaling agent (Pi et al., 2015). We propose that towards the vasculature, BRAVO can be a signaling molecule, which, by actively restraining its own expression from reaching cells far away from its source, ensures that the small microenvironment of the SCN remains confined within the root. The molecular processes underlying this spatial restriction and their implications for proper stem-cell renewal are just beginning to be uncovered. Mechanisms like the ones proposed here involve very general principles that contribute to the understanding of stem cell populations not only in plants, but in multicellular organisms on the whole.
MATERIALS AND METHODS
The models formulated set the rate of change of protein concentrations of BRAVO (B) and WOX5 (W), by using partial differential equations in which the transport of the mobile proteins is modeled through diffusion. In the models in which an intermediate factor (Z) is present, its dynamics are also considered. We only focus on the stationary solutions of the models, assuming these account for the experimentally reported expressions.
In all the models, the rate of synthesis of each protein is assumed to be proportional to its corresponding promoter activity and the quasi-stationary approximation for mRNAs (i.e. mRNAs dynamics are assumed to be very fast compared with the dynamics of proteins) is done (see supplementary Materials and Methods). For simplicity, proteins are assumed to degrade linearly. Mass action law is set for the rate at which two proteins form a complex (i.e. ∝ BW for the BRAVO-WOX5 complex). We only consider pairwise interactions, omitting higher order reactions. Complexes are taken to be unable to transcriptionally regulate any of the proteins considered (for simplicity, we name them inactive complexes, albeit they could regulate other factors not modeled herein). The supplementary Materials and Methods contain the derivation of the equations of the minimal models from the full set of equations that include mRNAs and complexes. The stationary solutions are the same for both (full and minimal) models.
The following subsections describe the one-dimensional minimal models in WT scenarios, the modeling of bravo mutants, the parameter space exploration with the definition of the observables and their comparison with experimental data, the construction of the realistic root layout, the equations used for the simulations in the realistic root layout and the numerical details of all models simulated.
‘Immobilization by sequestration’ mathematical model
The promoter activities of BRAVO and WOX5, which are computed at the stationary state (i.e. when all time derivatives are zero), are defined as pB(x)=PB(Ws(x), x) and pW(x)=PW(Ws(x), x), respectively, where Ws(x) denotes the spatial profile of WOX5 in the stationary state, as indicated by the subscript s. We also refer to these activities as promoter expressions.
Another version of this model where an additional sequestrator (S) affects the dynamics of BRAVO and WOX5 is described in the supplementary Materials and Methods (results in Fig. S3).
‘Immobilization by sequestration with intermediary Z’ mathematical model
‘Repression of a mobile activator’ mathematical model
For linear activations, the productions are PB(Z, x)=αZΘ(x+LQC/2) and , whereas for non-linear saturating activations, we use and . These nonlinear PB(Z, x) and PZ(W, B) are the same functions as those in the immobilization model with intermediary Z, except that PZ(W, B) includes the repression of BRAVO on Z. c sets the threshold of Z repression by BRAVO. Promoter activities in the stationary state are given by pB(x)=PB(Zs(x), x), pZ(x)=PZ(Ws(x)) and pW(x)=γQC(x), where again the subscript s indicates that the concentration profiles are those corresponding to the stationary state.
Modeling bravo mutants
Parameter space exploration
Parameter space exploration was done for the models above in one spatial (1D) dimension. For simplicity, in this 1D geometry, the VI region (e.g. light gray area in Fig. 2B-E, Fig. 3B-D, Fig. 4B-F, Fig. 5B,C) was defined to be of the same size as the QC region . Hence, the end of the VI is at position xVI=3LQC/2. To assess whether the simulation results are compatible with experimental observations, we considered four different observables, ξ, RB, RBW and RW, defined as follows:
quantifies the expansion of the stationary BRAVO promoter expression in the bravo mutant compared with the WT, Δx, relative to LQC (QC size), i.e. ξ=1 means that the stationary BRAVO promoter expression in the bravo mutant is expanded to a region as large as the QC compared with that in the WT. To define Δx (Fig. 2C) we compute the value of the stationary BRAVO promoter expression in the WT at the end of the VI region and define this value as . We then compute the spatial position at which the stationary BRAVO promoter expression in the bravo mutant takes this value, and define this position as xbravo (i.e. it is defined as ). Δx is defined as Δx≡xbravo−xVI. Because of our simplified one-dimensional geometry, ξ values are not expected to be equal to those in real roots.
is the ratio of stationary BRAVO promoter expression at the center of the QC in the bravo mutant over that in the WT. Because the twofold change from experimental data (Fig. 1F) corresponds to the increase of the spatially integrated GFP expression across the midplane, and thus involves the increase at the QC as well as the increase (and thereby expansion) in other regions, we considered that RB values consistent with experimental observations range between 1 and 2.
is the ratio of stationary BRAVO promoter expression at the end of the VI over stationary WOX5 promoter expression at the center of the QC, both in the WT.
is the ratio of WOX5 promoter expression at the QC in the bravo mutant over that in the WT. This observable is only used when the models include the (indirect) action of BRAVO on the WOX5 promoter, i.e. when there is WOX5 self-repression and BRAVO-WOX5 sequestration.
We compute these four observables in distinct parameter planes by solving, for each model for the WT and the bravo mutant, the corresponding equations in time, until the stationary state is reached (see simulation details in the ‘Numerical implementation of the mathematical models’ subsection below). Each parameter space in the logarithmic scale is discretized into a regular rectangular lattice of 15×15 pairs of values, for which ξ, RBW, RB and RW are evaluated. Heatmaps are generated using the Python contourf function for each observable. The orange domains correspond to those regions where all observables meet the conditions compatible with experimental observations.
Construction of realistic root layouts
In order to build two-dimensional realistic root layouts with which we could subsequently implement the corresponding reaction-diffusion equations (e.g. the mixed model in next subsection), we started by taking a confocal image of a middle plane of the root with propidium iodide-stained cell walls and we applied a segmentation routine that divides the root at the pixel scale and into its constituent cellular regions and cell walls (Fig. S8). To do this, we made extensive use of the scikit-image collection of Python-based algorithms (threshold_otsu, skeletonize and label) (van der Walt et al., 2014). In particular, we first defined the cell boundaries with the threshold_otsu method, which transforms the original image into a thresholded binary image in which only cell wall pixels remain. We then used skeletonize to obtain a cell wall with a fixed width (2 pixels). We then applied the label function on the modified image to label distinct regions (each label constitutes the collection of pixels belonging to the particular region). We chose to define labels that enable us to distinguish between cells and between the cell wall and the outside of the root as follows: all pixels within a cell have the same label, which is distinct from that of pixels in any other cell. With this routine, we could access each cell as an individual entity, and we could modify its properties as a whole (e.g. change the 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 constitute a single, the same, entity. Another single region is defined by all pixels outside of the root.
The pixel grid is the spatial grid on which the dynamical equations of protein concentrations are set. For the images used, 1 pixel . We assigned the same dynamical equations and parameter values in all pixels within each labeled region. These equations and parameter values can be distinct between regions. Specifically, the equations applied on the cell wall are distinct from those within cells, as described in the next subsection. The diffusion coefficient of a protein within all pixels of the cell wall is the same. For simplicity, the diffusion coefficient of a protein is set to be the same in all the cell regions (i.e. within any cell), but distinct from that in the cell wall. The only differences set between different cells are in the protein production terms. Further details on the construction and implementation of the model in the realistic root layouts can be found in the supplementary Materials and Methods.
Mixed model in a realistic root layout
In all these equations, production and degradation only happens in pixels labeled as cells. Within cells, the production functions , and depend on Z, B and W as in the one-dimensional mixed model with basal production of BRAVO, i.e. , and . In these expressions, takes the constant value α only in the cells shown in Fig. S9, being zero in all the other cells and regions; takes the constant value α0 only in the pixels corresponding to the cells shown in Fig. S10, being zero in all other regions; takes the constant value γ only in QC cells (Fig. S9), being zero in all the other cells and regions.
All equations described up to this point correspond to the WT condition. To model the bravo mutant, the same equations with the same parameter values are used, but the BRAVO protein concentration is set to zero (i.e. the mutant corresponds to Eqns 15-19 and .) Note that to model this mutant, the GFP reporter for BRAVO promoter, , is described with the same equations as in the WT (i.e. with Eqn 17) and hence is not set to zero.
In Fig. 6 and Figs S11-S14, pBRAVO:GFP corresponds to the variable computed at the stationary state (i.e. the activity of the BRAVO promoter as seen through its GFP reporter), whereas represents the stationary BRAVO protein concentration. Analogous definitions for WOX5 and Z are used. Fig. 6B-D use the same parameter values as in Fig. 6A (Table S3) except for c=0 and (Fig. 6B); c=0 (Fig. 6C); and λ=0 (Fig. 6D).
Numerical implementation of the mathematical models
To find the stationary spatial profiles of the 1D models (‘immobilization by sequestration’, ‘immobilization by sequestration with intermediary Z’, ‘repression’ and mixed models), we first reduced the system of differential equations to second-order ordinary differential equations for the diffusible variables. To do this, we first set to zero the equation for the non-diffusible variables and substituted the result on the other equations. The remaining system could be cast as a boundary value problem, which we solved 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 spatial step size δx=0.05 a.u. (arbitrary units), in a domain of x∈[−600, 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 computed, but only their stationary state. To double-check the validity of our solutions, we also simulated the whole temporal dynamics of the equations with a forward Euler method, obtaining the same stationary solutions as with the solve_bvp method, thus confirming the results (Figs S1-S7). For all the models, the values of B, W and Z at the boundaries (x=600 and x=− 600) are set to zero, except for the additional sequestrator variable S (Fig. S3), for which periodic boundary conditions are implemented.
The results displayed on the heatmaps were obtained by simulating the dynamical equations corresponding to each model for the WT and the bravo mutant scenarios until the stationary state was reached. To compute the temporal dynamics, we used a forward Euler method and discretized the 1D Laplacian term as , where u(x) represents either W and/or Z. The spatial step is δx=1 and uses a domain x∈[−400, 400]a.u.. The time step is dt=0.01 and the final time is tfinal=1000, where the steady state is reached. All variables are set to zero as the initial conditions and all variables are set to zero at the boundaries. The stationary profiles in Figs S1-S7 were computed with the Euler algorithm.
To simulate the dynamics of the mixed model (and its modifications) in the realistic root layout, we solved the corresponding reaction-diffusion equations with heterogeneous diffusion coefficients with a forward-time central-space scheme (FTCS), in which time is discretized in steps of size Δt=0.1 a.u. (so that after k steps, time is tk=kΔt) and space is discretized in steps of size (δx=1, δy=1) pixels, so that (xi, yj)=(iδx, jδy), with a lattice size of (Lx, Ly)=(228, 448) pixels for the WT root and (Lx, Ly)=(231, 448) pixels for the bravo mutant root. For the images used, 1 pixel≈0.5 μm.
We ran the simulations up to time t=3000 a.u., (Fig. 6; Figs S11-S14) this time point can be taken to correspond to the stationary state. All the diffusion coefficients outside the root layout are set to be zero, restricting the domain of the equations to the interior of the root. Additionally, this condition automatically implements reflecting boundary conditions at the root borders.
We acknowledge discussions with the members of A.I.C.-D.'s and M.I.'s labs. The Centre for Research in Agricultural Genomics (CRAG) is funded by the ‘Severo 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/10.13039/501100011033, and by the CERCA Programme/Generalitat de Catalunya.
Conceptualization: J.M., I.B.-P., N.B., A.I.C.-D., M.I.; Methodology: J.M., M.I.; Software: J.M.; Formal analysis: J.M.; Investigation: J.M., I.B.-P., N.B.; Writing - original draft: J.M., M.I.; Writing - review & editing: I.B.-P., N.B., A.I.C.-D.; Supervision: A.I.C.-D., M.I.; Project administration: A.I.C.-D., M.I.; Funding acquisition: A.I.C.-D., M.I.
This work was supported by the Ministerio de Ciencia e Innovación (MCIN), Agencia Estatal de Investigación (AEI) and European Regional Development Fund (Fondo Europeo de Desarrollo Regional, FEDER) program ‘Una manera de hacer Europa’ MCIN/AEI/10.13039/501100011033/FEDER (PGC2018-101896-B-I00 to M.I.); Ministerio de Ciencia e Innovación, Agencia Estatal de Investigación and European Social Fund (Fondo Social Europeo, FSE) program ‘El FSE invierte en tu futuro’ (BES-2016-078218 to J.M.); Ministerio de Ciencia e Innovación (MCIN), Agencia Estatal de Investigación (AEI) and European Regional Development Fund (ERDF) (BIO2016-78955 to A.I.C.-D.); the European Research Council (ERC) Consolidator Grant (ERC-2015-CoG–683163 to A.I.C.-D.); Generalitat de Catalunya (M.I. and J.M. in Grup de Recerca Consolidat 2017 SGR 1061; FI-DGR 2016FI-B 00472, also funded by FSE, to N.B.); and the Ministerio de Educación, Cultura y Deporte (FPU15/02822 to I.B.-P.). Open Access funding provided by Universitat de Barcelona. Deposited in PMC for immediate release.
Peer review history
The peer review history is available online at https://journals.biologists.com/dev/article-lookup/doi/10.1242/dev.200510.
The authors declare no competing or financial interests.