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Research Article

A Dynamic Model for Stem Cell Homeostasis and Patterning in Arabidopsis Meristems

  • Tim Hohm,

    Affiliations: Department of Medical Genetics, University of Lausanne, Lausanne, Switzerland, Swiss Institute of Bioinformatics, University of Lausanne, Lausanne, Switzerland, Computer Engineering and Networks Laboratory, Zürich, Switzerland

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  • Eckart Zitzler,

    Affiliation: Computer Engineering and Networks Laboratory, Zürich, Switzerland

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  • Rüdiger Simon mail

    ruediger.simon@uni-duesseldorf.de

    Affiliation: Institute of Genetics, Heinrich-Heine University, Düsseldorf, Germany

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  • Published: February 12, 2010
  • DOI: 10.1371/journal.pone.0009189

Abstract

Plants maintain stem cells in their meristems as a source for new undifferentiated cells throughout their life. Meristems are small groups of cells that provide the microenvironment that allows stem cells to prosper. Homeostasis of a stem cell domain within a growing meristem is achieved by signalling between stem cells and surrounding cells. We have here simulated the origin and maintenance of a defined stem cell domain at the tip of Arabidopsis shoot meristems, based on the assumption that meristems are self-organizing systems. The model comprises two coupled feedback regulated genetic systems that control stem cell behaviour. Using a minimal set of spatial parameters, the mathematical model allows to predict the generation, shape and size of the stem cell domain, and the underlying organizing centre. We use the model to explore the parameter space that allows stem cell maintenance, and to simulate the consequences of mutations, gene misexpression and cell ablations.

Introduction

Growth of the aerial parts of higher plants relies on a life-long supply with cells by the shoot apical meristem (SAM). The SAM contains a small population of non-differentiating stem cells in the central zone at the meristem tip [1]. After cell divisions in the stem cell domain (SCD), daughter cells are shifted towards the surrounding peripheral zone, where organ primordia are initiated and cells can enter a differentiation pathway. The architectural makeup of flower primordia, which gives rise to the plant's reproductive organs, resembles that of the SAM with the main difference that stem cell activity is switched off in flowers after generation of a species-specific number of organs. It is evident that land plants such as trees, which can grow in size and produce new organs for hundreds of years, must have developed robust regulatory systems that enable them to maintain active stem cell populations also under changing or adverse environmental conditions. Disturbing stem cell regulation can arrest the growth of a plant's shoot tip, or may result in gross tissue overproliferation and failure to reproduce. More subtle alterations in stem cell proliferation can affect overall size of a seed-producing inflorescence structures, such as a maize cob, the size of a fruit, or the number of petals in a horticultural flower. We are only just beginning to understand how the fate of the stem cell population is regulated in higher plants.

Maintenance of the undifferentiating stem cell population depends on signals from cells of the organizing centre or OC, which reside underneath the SCD in a deeper region of the meristem. Several gene products have been identified that enable these adjacent cell groups to communicate with each other. The stem cells of Arabidopsis thaliana secrete the CLAVATA3 (CLV3) peptide, consisting of 12 amino acids [2], [3], [4]. CLV3 was shown to interact with the LRR-receptor kinase CLAVATA1 (CLV1) that is expressed in and surrounding the OC [5], [6]. A second receptor system composed of the LRR-protein CLAVATA2 (CLV2) and the membrane associated kinase CORYNE (CRN) is more widely expressed in the meristem and vasculature, and also contributes to signal perception [7], [8]. CLV3 dependent activation of the two receptor systems represses the expression of WUSCHEL (WUS), a homeodomain transcription factor that is normally produced from OC cells, and which is required for the maintenance of stem cells [9], [10]. WUS itself acts non-cellautonomously to promote stem cell fate at the meristem tip. The WUS protein does not seem to move, and it could control the expression of other genes that generate a diffusible signal which ultimately promotes stem cell identity [11]. Searches for target genes showed that several ARABIDOPSIS RESPONSE REGULATOR (ARR) genes, which are negative regulators of cytokinin signalling, are repressed by WUS, thus involving cytokinin in meristem maintenance [12]. However, WUS induces stem cell fate only at the meristem tip, and not in the (WUS expressing) OC cells or other surrounding cells, indicating that a spatially restricted cofactor, or a competent cellular state is required to respond to WUS activity [13].

Because stem cells signal back to the OC via CLV3 and its receptors to restrict WUS expression, a feedback circuitry is established that maintains a stable stem cell population. Support for this model of stem cell homeostasis comes from a number of experimental observations: 1) loss-of-function mutants of WUS cannot maintain stem cells [10]; 2) loss-of-function mutants of CLV3 (or CLV1, CLV2 or CRN) allow for less restricted WUS expression and production of excessive stem cells [3], [4], [8], [14]; 3) constitutive high level expression of CLV3 represses WUS, causing stem cell loss [4]; 4) when WUS expression is uncoupled from repression by CLV3, e.g. when controlled from a heterologous promoter, the stem cell domain expands [15], [16]; 5) the CLV3/WUS circuitry is capable of self-organization. This was revealed by laser ablation experiments in tomato, showing that after elimination of both SCD and OC, new domains of WUS expression are generated at peripheral sites that then initiate new SCDs, which support further growth of the SAM [17].

However, all previous studies performed on various mutants or constitutive misexpression lines did not allow studying the immediate consequences of system perturbations. Cell ablation experiments are further complicated by wounding effects, and ectopic cell divisions in the SAM which are required for regeneration. Analyzing the dynamics of the CLV3/WUS circuitry at a shorter timescale required rapid and transient perturbations of gene expression. In a first study of this type, CLV3 expression was silenced by Dexamethason-induced expression of a foldback CLV3 RNA [18]. Live imaging of the SAM before and after CLV3 silencing showed that expression of a CLV3:GFP transgene, acting as a reporter for stem cell identity, extended into cells adjacent to the central zone within 24 hours after induction. Importantly, this re-specification of peripheral cells to stem cell identity was not preceded by cell divisions. In a similarly designed experiment, induction of high level CLV3 expression downregulated both WUS expression, and the stem cell marker CLV3, within 3 hours [14]. Together, these experiments showed that the CLV3/WUS circuitry is acting throughout development, and that the output, stem cell number, can be continuously readjusted in response to changing amounts of the signalling components. In line with this, fluctuations of central zone size were observed, indicating continuous activity of the circuitry [18].

However, the CLV3/WUS circuitry was also found to be surprisingly robust and to tolerate changes in CLV3 expression levels over a tenfold range [14], indicating that stem cells do not directly communicate their number via the amount of released CLV3 signal. Furthermore, while strong CLV3 signalling rapidly repressed WUS expression, a slowly acting compensation mechanism appeared to upregulate WUS with time. The components of this compensatory circuitry are unknown, but may be found among the gene set that controls WUS expression. SPLAYED (SYD) encodes a SNF2-type chromatin-remodelling ATPase that is required for WUS transcription [19]. BARD1, carrying BRCT and RING domains, interacts with and antagonizes SYD to restrict WUS expression to the OC [20]. HANABA TARANU (HAN), coding for a GATA-transcription factor, represses WUS postembryonically from the developing vasculature [21]. The interplay between these components is not understood, and they may act exclusively to establish a discrete WUS expression domain when meristems are generated. During development, a second feedback mechanism could operate via the cytokinin signalling pathway. WUS represses the expression of several ARRs in the meristem, which restrict cytokinin signalling [12]. In turn, continuous activation of ARRs arrests meristem activity and WUS expression, suggesting that WUS and ARRs mutually repress each other.

We have generated a computational model of stem cell fate regulation by the CLV3/WUS circuitry in the shoot apical meristem. Our model incorporates two feedback regulatory systems that merge upon WUS regulation. The driving force for modelling was to better understand the forces that shape the CLV3 and WUS expression domains, while making the minimal number of necessary assumptions about the factors to be involved. We used the model to study the effects of targeted system perturbations, and to explore the parameter space that allows for stem cell homeostasis under fluctuating conditions.

Results

Model Components and Basic Assumptions

We propose a partial differential equation (PDE) model to follow the dynamics of gene regulation across the SAM (Fig. 1A). Conceptually, at the centre of the model lies the regulation of WUS via two separable feedback operated reaction-diffusion systems, a commonly used type of differential equation models for developmental processes in biology [22]. Instead of representing the entire meristem structure, we here restrict the spatial component to two dimensions using an artificial longitudinal section through the SAM (Fig. 1B). Cells within this meristem section are modelled as discrete entities. We neglected growth and cell divisions for two reasons: firstly, we are concentrating on meristem homeostasis, i.e. meristem size remains unaltered, and secondly, the gene regulation that is considered in this study is faster than the cell cycle. The cellular or tissue framework thus remains static. The regulative processes within cells are mapped to a set of PDEs constituting a gene regulative program, which is executed in each cell. The components and underlying assumptions of our model are summarized as follows (Fig. 1A):

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Figure 1. Modelling components and cellular framework.

(A) Components and their interactions regulating stem cell homeostasis in Arabidopsis.(B) Two-dimensional meristem frame with assumed anchoring distribution indicated by colouring.

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Stemness.

Cells of the meristem can acquire stem cell identity, reflected in their level of stemness, which is controlled by a WUS-dependent signal (WUS-signal, see below). We avoided an artificial and static cut-off concentration for WUS-signal, above which cells switch to the stem cell status, and instead established a dynamic but sigmoidal response to WUS-signal, that results in variable levels of stemness to represent a cell's state.

Experimental evidence from WUS misexpression shows that only outer cell layers acquire stem cell identity. The underlying factors responsible for this are not known. We are now not postulating another signal, but take this observation into account by allowing only cells in outer cell layers to acquire stem cell identity. Therefore, only cells in the outer layers of the meristem are competent to react to WUS-signal and we restricted stemness to the outer layers. Stem cells express the signalling molecule CLV3, proportional to their stemness level. The stemness levels are expressed by the model variable [st].

CLV3.

CLV3 freely diffuses to neighbouring cells. To avoid flooding the entire model with CLV3, the CLV3 peptide is regarded to decay with time. We eliminated the need for receptor proteins or other signalling components because insufficient quantitative data are available to assess their contribution. Furthermore, CLV-signalling appears to be largely controlled by the amount of available CLV3 peptide [14]. Thus, the local CLV3 concentration is computed to directly restrict WUS expression. The CLV3 levels are expressed by the model variable [CLV3].

WUS.

We assume that WUS protein is not mobile and therefore remains mostly in the cells where the WUS gene is expressed [11], except for weak leakage diffusion. While all cells of the model meristem are in principal able to express WUS, we added a spatial parameter which makes cells that reside closer to the meristem tip more competent to activate WUS expression (Fig. 1B). This anchoring was found to be necessary in our model to ensure correct positioning of the two functional domains (SCD and OC) within the dome. Without the spatial component, immediately neighbouring SCD and OC are still formed, but at more random locations (Fig. 2). The requirement for a spatial component reflects the fact that our virtual meristem is not structured, i.e., all cells are intrinsically equal and carry no positional information. In plant meristems, such spatial information will be provided by signals within and between cell layers, or from the vasculature. We introduced a positive feedback loop for WUS via autoactivation. Although not experimentally proven, it is supported by the observation of a rapid upregulation of WUS expression in regenerating callus [23]. WUS promotes the expression of WUS-signal, which is mobile and can diffuse to neighbouring cells. The WUS levels are expressed by the model variable [WUS].

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Figure 2. System without anchoring distribution.

Equilibrium state of a simulation where the anchoring distribution guiding WUS expression is exchanged by a constant WUS reaction rate. As a result of the missing positional information, SCD and OC keep their positions relative to each other, but the SCD is now formed at a random location in the outer meristem layers.

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WUS-signal.

WUS-signal is generated by all WUS expressing cells, and the amount produced depends on the levels of WUS expression. Similar to CLV3, WUS-signal is mobile and degraded at a constant rate. Cells react to the amount of WUS-signal they receive with stemness. Only outer cell layers of the meristem are competent to respond to the WUS-signal. The WUS-signal levels are expressed by the model variable .

Factor X.

To account for CLV independent regulation of WUS expression, we incorporated a factor X (facX). At the start of the simulation, facX is expressed homogeneously in all cells and is freely diffusing. FacX induces WUS expression, but is itself under negative feedback regulation by WUS [14]. This is implemented through active degradation or consumption of facX by WUS. The interactions between WUS and facX are thus based on an activator-substrate-like mechanism that will generate a discrete WUS domain. The facX levels are expressed by the model variable [facX].

The described entities are compiled into a PDE representation of the intracellular gene regulative program given by Eqs. (0.1)–(0.5) (see Materials and Methods section). An overview of the interactions is given in Fig. 1A.

The resulting model depends on a set of parameters like kinetic constants; validation of our model therefore required first to identify a parameter setting that allow reproducing the two functional meristem domains, i.e. the CLV3-expressing SCD and the WUS-expressing OC, at approximately those locations which are experimentally observed in wildtype meristems (Fig. 3). The model parameters have been tuned by hand using a decomposition of the model (see Materials and Methods section).

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Figure 3. Time course simulation of a wild-type meristem.

Grading from blue (minimum) to red (maximum) illustrates the relative concentration of the indicated components. From left to right, the simulations were developed from close to zero concentrations for number of steps shown below, until the (stable) equilibrium state was reached.

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Simulation of Wildtype

Starting from almost zero concentrations of all considered components, the system was simulated until an equilibrium state was reached (Fig. 3); since we investigate system behaviour by means of numerical integration of the model equations, under ‘equilibrium state’ we understand a state where all derivates are zero or reasonably close to zero. In the wild-type scenario, a given meristem showed WUS expression first at the meristem tip, triggered by a sufficiently high level of facX. WUS then increased, thereby repressing facX at the same location. The centre of the OC shifted downwards. Distribution of WUS-signal overlaps with that of WUS, but since WUS-signal is diffusible, it is located in a wider domain and extended always to the meristem tip. Together with an increase of stemness at the meristem tip, CLV3 became expressed, pushing the OC downwards from its initial position. This time course reproduced the dynamic changes in gene expression patterns that are observed during embryonic development of the shoot meristem.

Simulations of mutants and system perturbations

I. Reducing CLV3 expression.

We next tested the consequences of reducing CLV3 expression. When CLV3 is downregulated during plant development, both the SCD and the OC expand laterally due to unrestricted WUS expression [4], [16], [18]. Furthermore, WUS expression is then no longer excluded from the meristem tip.

Our in silico analysis started from an equilibrated wild-type meristem, thus simulating a conditional knock-out of CLV3. When CLV3 expression was stopped, the SCD rapidly enlarged due to recruitment of lateral cells. At the same time, the OC expanded and shifted towards the meristem tip (Fig. 4). A similar, but less pronounced effect was seen when CLV3 was still expressed, but at reduced levels (see Fig. S1 and Text S1).

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Figure 4. Time course simulation for a clv3 loss-of-function mutant.

Only the concentrations of WUS, facX and CLV3 are shown, starting from an equilibrated wild-type (0). At calculation time point 1, CLV3 concentration is set to 0, and simulations proceed until equilibrium. Note that the WUS domain expands and shifts upwards.

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In addition, we simulated the effects of reducing CLV3 receptor activity, i.e. CLV1 or CLV2 and CRN constructs (see Fig. S2 and Text S2). These simulations showed behaviour comparable to reductions in CLV3 expression and thereby supports the assumption that from a modelling perspective it is sufficient to model CLV3 activity as a representative for CLV signalling.

II. Increasing CLV3 expression.

Plants that continuously express CLV3 fail to maintain a shoot meristem due to an early arrest of WUS expression and stem cell differentiation [4]. However, inducible overexpression during development was found to be compensated in some flower meristems, resulting in a recovery of WUS expression at later stages [14]. In our simulations, high level expression of CLV3 in all cells caused a rapid shrinkage of the OC, downregulation of WUS, and a reduction in stemness, concomitant with a reduction in CLV3 expression levels from the SCD (Fig. 5A). Overexpression of CLV3 set at an intermediate level resulted in WUS repression and a rapid loss of stemness, which recovered with time (see Fig. 5B). A similar behaviour was observed in plant floral meristems in response to induced overexpression of CLV3 [14]. Low level overexpression of CLV3 allowed the system to reach a new stable equilibrium state, with a smaller OC and SCD (see Fig. S3 and Text S3).

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Figure 5. Response to CLV3 overexpression in the entire meristem.

Time-evolution of WUS, FacX and CLV3 concentrations upon strong (A) or intermediate (B) level overexpression of CLV3 is shown, starting from equilibrated wild-type meristem (time-point 0) until the simulations reach a new equilibrium state. endoCLV3 = CLV3 expression from the endogenous promoter, taken as reporter for stem cell identity. Note that both a WUS expression domain and stem cells are reinitiated in (B), but not in (A).

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We further tested the robustness of the system against perturbations by analyzing the response to altered endogenous CLV3 expression in small, discrete steps: the effectiveness of stemness-dependent CLV3 expression is tested in a range from 10% to 620% in 10% steps. Varying CLV3 levels from 90% to 620% compared to wildtype affected the size of the SCD, while OC cell number remained constant (see Fig. S1B). This indicates that OC and SCD sizes are not strictly coupled, which has been also noted experimentally when analyzing the sizes of OC and SCD in plants grown under diverse environmental conditions [24]. In the simulations, we varied CLV3 expression in 10% steps in a range from 10% to 620%.

III. Altering WUS expression.

Lowering WUS expression levels reduces the sizes of both SCD and OC to a similar extent, and will cause a loss of both domains when WUS is fully repressed (see Fig. S4 and Text S4). Ectopic WUS expression was tested by changing the effect of CLV3 signalling on WUS activity from repressive to activating. In plants, this has been achieved by expressing WUS from the CLV3 promoter [15], which caused the coalescence of OC and SCD at the meristem tip together with lateral expansion of this joint domain. This cell behaviour is also observed in our simulations (Fig. 6).

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Figure 6. Simulation for WUS misexpression in the stem cell domain.

Starting from an equilibrated wild-type meristem, misexpression of WUS from the CLV3 promoter (CLV3WUS) is simulated until a new equilibrium state is reached. Cells in the meristem now aquire mixed identities and express both the OC and SCD marker genes.

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IV. Regeneration and de-novo generation of OC and SCD.

After ablating SCD and OC from the meristem by pointing a laser beam at the meristem tip, WUS becomes expressed at the periphery, and the OC and SCD are regenerated with time [17], highlighting that cells at the periphery are capable of, but normally inhibited from the acquisition of OC identity. We simulated the laser ablation experiment starting from an equilibrated wildtype meristem, and eliminated all WUS or CLV3 expressing cells from the meristem. We found that a new OC was generated which induced a SCD nearby (Fig. 7). This shows the self-generative capacity of our meristem model. During normal plant development, new meristems are generated during embryogenesis, flowering, and when axillary meristems are initiated. By simply altering facX expression levels in a given cellular framework, we could simulate the generation of a new OC and SCD, which coordinated increase in size when facX is further upregulated (see Fig. S5 and Text S5).

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Figure 7. Simulation for a cell ablation scenario.

The central region of the meristem was eliminated by virtual cell ablation. Note that even in the absence of an SCD and underlying OC, both domains can be partially restored.

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V. Role of facX and WUS feedback regulation.

In addition to the already described scenarios that are all inspired by previously conducted experiments, we analysed the role of facX and the interaction of WUS with facX. Without feedback of WUS to facX, facX could be exchanged for a constant expression of WUS. We tested this idea by eliminating the feedback term of WUS on facX (see Eq. (0.2) in Materials and Methods). However, using an evolutionary algorithm to search the parameter space of the resulting model, we could not identify any parameter settings resulting in the desired system behaviour, with separate but adjacent SCD and OC. We therefore conclude that facX and the feedback between WUS and facX are vital for the model to initiate stable Turing patterning. Candidates to realize the role of facX are genes and functions that control WUS expression. Because eliminating facX destabilizes the OC, we predict that mutations in genes contributing to facX function should result in either unrestricted WUS expression and meristem overproliferation (Fig. 8), or meristem arrest.

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Figure 8. System neglecting feedback of WUS on facX.

Archetype system behaviour for simulations without feedback of WUS on facX: the meristem overproliferates, WUS is expressed in all meristem cells and all cells in the outer layers acquire stem cell identity indicated by their expression of the stem cell marker CLV3.

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VI. Influence of anchoring factor distribution on system behaviour.

We have introduced artificial positional information in form of an anchoring distribution (Fig. 1B) to stimulate SCD and OC positioning at the meristem tip. To test the possible influence of this anchoring distribution on the pattern formation capabilities of the model, we exchanged the distribution for a constant value. Although positioning of SCD and OC became more variable now, we were still able to identify parameter sets that result in spatially confined and adjacent SCD and OC (Fig. 2), indicating that the anchoring distribution has no significant influence on the qualitative behaviour of our model.

Discussion

Mathematical modelling is a tool that allows asking the most stringent questions concerning the dynamic behaviour of predicted gene regulatory networks; it also quickly uncovers the restrictions and shortcomings of assumed interaction maps, and thus provides guidance to direct future experiments. We had initially attempted to build a model for the SCD and OC, based solely on the interaction between two activator-inhibitor based systems (CLV3/WUS, and WUS/facX) which were linked via WUS as the common node. Conceptually, the underlying assumption was that SCD and OC could originate independently of each other, but that their maintenance and relative position are controlled by mutual feedback regulation. However, such a model failed to reproduce the domain arrangement observed in actual meristems for the model's parameter space which we explored using a stochastic parameter estimation technique, namely an evolutionary algorithm [25]. For the evolutionary algorithm, we used an objective function described previously [26], extended by a simple domain recognition procedure capable of identifying circular domains. This indicates that an essential component was missing from this model. The most common outcome which we achieved was not juxtaposition, but an overlap of the SCD with the OC. To improve the spatial separation of the two domains, we considered that cells within an actual meristem differ from each other by their position. Several misexpression experiments using WUS had previously shown that the cellular response to WUS-derived signals depends on a cells' relative position within the meristem, corresponding to its developmental trajectory. Only by adding spatial components to our model we were able to achieve a realistic sizing and arrangement of the two domains within the meristem; removal of this spatial component causes extensive spatial overlapping of the SCD and OC.

Jönsson et al. had previously described a model for the WUS/CLV3 interaction that concentrated on the generation of the CLV3 domain (the SCD in our model) by a WUS derived signal [27]. This model did not yet include the negative feedback regulation of CLV3 signalling upon WUS expression, and the creation and maintenance of the WUS domain was not simulated. To confine the CLV3 domain to the meristem tip, the authors proposed that an (unidentified) factor diffusing from the outermost meristem layer, the L1, together with the WUS-dependent signal, induced CLV3 expression. They later used a reaction-diffusion model combined with two repressive signals, derived from the L1 and stem tissue, to activate WUS expression in a deeper meristem region [28]. Both models were successful at reproducing either the OC or the SCD, but did not incorporate the mutual interdependence between the factors that shape the two domains, and were less parsimonious with system components than the model we describe here.

A recently published model by Geier et al. [24] did not describe the spatial arrangement of domains, but addressed the observation that SCD and OC sizes vary strongly under changing environmental conditions. The model describes the SCD and OC as cell pools that are connected via differentiation rates and expand due to cell proliferation, which is regarded as an externally controlled parameter. Variation in the relative sizes of SCD and OC can be explained by assuming that a differentiation signal X is produced by OC or SCD, which can buffer the response of the cell pools against changes in proliferation rates. Although this model did not allow reproducing all mutant and overexpression experiments that we have simulated here, it combined modelling approaches with quantitative data, and highlighted the enormous developmental plasticity of the meristem.

We challenged our model by altering central system parameters to simulate mutant phenotypes and published transgenic experiments. In all experiments, our model proved to be robust against small-scale perturbations (see Text S6). This stability probably results from the combination of two feedback operated system, whereby one of them, the WUS/facX system, acts as a buffer that dampens fluctuations in WUS levels. Furthermore, the reaction rates that influence WUS expression in our simulations are one order of magnitude smaller than those controlling CLV3 levels. Increased stability against signalling noise was uncovered in the analysis of coupled positive feedback systems if the two linked regulatory loops operated at different speeds [29], [30]. Our modelling approach has revealed that combined feedback systems are sufficient to allow the generation and robust maintenance of two distinct cellular domains in a meristem, requiring only minimal assumptions about spatial restrictions of the system. The challenges ahead are now to extend the cell model into the third dimension, and incorporate cell divisions, but also to add other regulatory networks that control organ initiation and cell differentiation, approaching the goal of a virtual meristem.

Materials and Methods

Based on the gene interaction diagram shown in Fig. 1A, a system of coupled PDEs is set up that describes the temporal evolution of concentrations of the factors , , , and inside the cells of the SAM. For this model, the three-dimensional dome of cells constituting the SAM is restricted to a two-dimensional artificial longitudinal section (Fig. 1B). This section was generated by positioning cell centres and using a Voronoi decomposition in order to generate possible cell walls and thereby defining the cells. For the simulations we assume zero flux boundaries confining the simulated domain. The model equations are given in the following:(0.1)
(0.2)
(0.3)
(0.4)
(0.5)

The model depends on the following parameters: reaction rates , basal expression rates , degradation rates , and kinetic constants . While most parameters settings are similar for all cells of the considered simulated domain, there are two exceptions: (i) the reaction rate (Eqs. (0.1) and (0.2)) is given by a distribution with its maximum in the meristem tip (Fig. 1B). It is an artificial spatial component necessary for correct location of developing SCD and OC within the meristem. In addition is perturbed by a small uniformly random value which is a random influence necessary for this subsystem to produce patterns. (ii) The reaction term guiding depends on the indicator function . Since we assumed that only cells in the outer cell layers are competent to acquire stem cell identity, for competent cells the indicator function returns a value and otherwise. While the former parameters refer to processes taking place within the cells, the model includes interactions between neighbouring cells as well. This interaction is modelled by diffusion terms , where is the Laplace operator in two dimensions and is a diffusion rate. Note that although and are considered to be only locally active, their model equations (Eqs. (0.1) and (0.5)) contain diffusion terms since for these two factors we assume a weak leakage diffusion.

On top of the parameters we already described, the model contains a set of parameters that is used to accommodate the different modelled mutants and experiments. In this context, is used to simulate regulation of endogenous CLV3; modifies the reaction term guiding expression where simulates a wildtype situation while represents a knockout and values represent overexpression of endogenous . Choosing values , graded scenarios can be simulated. The parameter is used to simulate a CLAVATA background in all cells. In a wildtype scenario is set to 0 and with a range of graded background strengths can be simulated. In addition, for simulations testing conditions with respect to under which OC and SCD are generated the already introduced parameter is varied.

Parameter Calibration

The model parameter setting for the wildtype simulations as well as for the simulations of mutants are given in Table 1 and Table 2. Although the model parameters are dimensional, at this point we disregard their dimensionality: since only qualitative data is available to calibrate the model, the given parameter setting represents an equivalence class of parameter settings that can be generated starting from the given parameter setting by rescaling within the biologically feasible range. Therefore it is not possible to give exact dimensional values for corresponding biological parameters without at least some anchoring qualitative measurements.

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Table 1. Summary of constant model parameters.

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Table 2. Scenario dependent model parameters with their respective values.

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The model parameters have been tuned by hand using a type of hierarchical decomposition of the system: from a developmental perspective firstly the OC is formed which then induces the formation of a SCD. For the parameter tuning process we therefore divided the system into two parts. (i) Equations (0.1) and (0.2) which are responsible for the formation of the OC, and (ii) Eqs. (0.3) to (0.5) that constitute the SCD generating part of the system. During a three-stage process we started the tuning process of the model parameters by considering only the OC generating system, in order to identify parameters resulting in a single and spatially confined OC domain. For this subsystem, we started with parameter settings as documented previously (Koch and Meinhardt, 1994), and we were able to identify fitting parameters reasonably close to the initial settings. Using the resulting OC as input, we then tuned the parameters for the SCD generating part of the system. Here we ignored the feedback of on and aimed on the identification of parameters which resulted in an SCD of appropriate size and an area under influence that extends the SCD but remains spatially confined as well. In the last step we considered the full system adjusting the parameters responsible for the feedback process between the two subsystems.

Simulation

For the numerical simulation of the PDEs, we assume zero flux conditions on the boundaries of the cell plane. In order to simulate the time evolution of the model, the model equations are numerically integrated. The equations have to be discretized with respect to time and space, here using a constant time step and applying a finite difference scheme in cellular resolution to the space-dependent diffusion terms. Each cell is thereby represented by its centre and for the sake of simplicity, cell volumes are considered to show the gradients that would be assumed between the concentrations simulated for the cell centres. In addition, we assume free diffusion as the only means of communication between cells. Because the precise communication underlying WUS dependent signalling is still unknown, free diffusion represents a sort of ‘maximum entropy choice’. In terms of modelling complexity we benefit from this fact: with free diffusion, extracellular spaces, cell walls and membranes can be neglected during the simulation process. In addition, due to the Voronoi decomposition used to generate the considered section through the meristems, cell volumes and surfaces tend to even out. Since the model is not supposed to generate quantitative data, but rather to investigate qualitative behaviour, we neglect the influence of cell surfaces and volumes during simulations.

The diffusion terms in the considered system tend to be stiff and we therefore chose to use a variant of the second order implicit Crank-Nicolson integrator as presented previously by others [31] for these terms. To the only-time-dependent terms a faster explicit Adams-Bashford scheme [31] is applied instead. This implicit-explicit method is chosen in order to reach an appropriate trade-off between necessary computational effort and simulation accuracy.

The numerical simulations for the considered scenarios are done in a two-stage process. The first stage is used to equilibrate the system starting from the initial conditions. Here, under ‘equilibrium state’ we understand a system state in which all derivates are reasonably close to zero. During the second stage the parameters are adapted in order to accommodate the considered scenarios. As initial condition for the first stage, the level of all cells is homogeneously initialized with a starting concentration of 0.01. All other species are initialized with a value of 0. In the first stage, the system is simulated for 30000 time units. For the second stage we use the equilibrium concentrations obtained in the first stage as initial conditions, the parameters are adapted and in case of the laser ablation scenario the tissue topology is adapted. Afterwards the system is simulated for further 15000 time steps.

Modelling Background

To model biological systems there exists a range of different mathematical modelling approaches, e.g. stochastic molecular simulations, differential equation models, or discrete dynamic models. The available models vary with respect to possible level of detail where a gain in detail comes at the cost of additional computational effort. Since the focus of this study is to provide a model capable of reproducing the intricate dynamics underlying the maintenance of the SAM, we chose a differential equation model––an approach that provides the necessary level of detail and is commonly used to model the considered type of systems.

From a qualitative perspective, SAM maintenance is a question of developing and maintaining a patterning of a tissue with respect to distinct domains with specific gene expression profiles. A popular approach in developmental biology to capture pattern formation are reaction diffusion systems developed by Turing in the 1950s [22], [32]. Relying on diffusion, these systems are capable of producing spatially heterogeneous patterns of somewhat antagonistic reactions initiated by an initial small perturbation. Here we employ a similar mechanism: Eqs. (0.1)–(0.2) are a variant of the activator-substrate model [33] - a system that is known to produce circular domains that remain mobile and thereby allow an OC that is forming in the meristem tip to move down after initiation of the SCD. Eqs. (0.3)-(0.5) are derived guided by the law of mass action. Still, parameters like the Hill coefficients in Eq. (0.4) have been further tuned, e.g. large Hill coefficients are used in Eq. (0.4) in order to produce a sharper transition between cells showing a high level of stemness and neighbouring cells with low levels of stemness. In turn, such parameter choices reflect possible underlying biological reactions like the formation of homodimers or other forms of cooperativity.

Supporting Information

Figure S1.

Effects of reduced CLV3 levels on OC and SCD. (A) Time course simulation for a conditionally reduced CLV3 expression. (B) Impact of CLV3 expression levels on the sizes of OC and SCD. To assess the number of cells in the respective domains, the concentrations were discretized using thresholds relative to the wild type concentrations: For stemness: δst = 0.21, for WUS: δWUS = 0.31.

doi:10.1371/journal.pone.0009189.s001

(1.26 MB EPS)

Figure S2.

Simulating loss-of-function mutants in CLV1. (A) clv1 loss-of-function scenario (c1ko = 0), (B) a simulation where CLV1 retains some activity (c1ko = 0.2).

doi:10.1371/journal.pone.0009189.s002

(1.69 MB EPS)

Figure S3.

Gradual increase of exogenous CLV3 expression levels. (A) Time course simulation for a low level CLV3 overexpression ([CLV3ext] = 0.7). (B) Graded system response to different exogenous CLV3 expression levels. The [CLV3ext] level is varied in [0, 2]. To assess the number of cells in the respective domains, the concentrations of stemness and WUS were discretized using thresholds relative to the wild type concentrations. For stemness: δst = 0.21; for WUS: δWUS = 0.31 is used.

doi:10.1371/journal.pone.0009189.s003

(1.22 MB EPS)

Figure S4.

Altering WUS expression. (A) A simulation for slightly reduced WUS expression level (wko = 0.6), (B) a simulation for with intermediate WUS expression (wko = 0.4), (C) a simulation for a WUS loss-of-function scenario (wko = 0).

doi:10.1371/journal.pone.0009189.s004

(2.38 MB EPS)

Figure S5.

Regeneration and de-novo generation of OC and SCD. To assess the number of cells in the respective domains, concentrations were discretized using thresholds relative to the wild type concentrations. For stemness: δst = 0.21, for WUS: δWUS = 0.31 is used.

doi:10.1371/journal.pone.0009189.s005

(0.46 MB EPS)

Text S1.

Reducing CLV3 expression.

doi:10.1371/journal.pone.0009189.s006

(0.03 MB RTF)

Text S2.

Simulating loss-of-function mutants in CLV1.

doi:10.1371/journal.pone.0009189.s007

(0.86 MB RTF)

Text S3.

Gradual increase of exogenous CLV3 expression levels.

doi:10.1371/journal.pone.0009189.s008

(0.03 MB RTF)

Text S4.

Altering WUS expression.

doi:10.1371/journal.pone.0009189.s009

(0.18 MB RTF)

Text S5.

Regeneration and de-novo generation of OC and SCD.

doi:10.1371/journal.pone.0009189.s010

(0.08 MB RTF)

Text S6.

Sensitivity analysis for model parameters.

doi:10.1371/journal.pone.0009189.s011

(0.17 MB PDF)

Author Contributions

Conceived and designed the experiments: TH EZ RS. Performed the experiments: TH. Analyzed the data: TH RS. Wrote the paper: TH RS.

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