Complex interactions among diverse elements underlie the appearance of robust cell patterns during development. Understanding how the topology and dynamics of these interactions are related to phenotypical traits represents one of the most important challenges in systems biology, and is necessary to build a more general theory of development and evolution 1.

Cell-type determination and patterning are key developmental processes. Two overall modes of cell-type determination have been distinguished. The first is the lineage-based mode, which depends on factors transmitted from progenitor to daughter cells. The second one depends on positional information in the context of differentiating cells 2. The definition of these two types of cell fate determination is instrumental for studying and intervening in developmental systems, especially if one focuses on a particular spatio-temporal interval. The model we present here, however, suggests that the emergence of information needed for cell fate determination results from a dynamic process that depends upon molecular components that are both inside and outside cells. In other words, such information is dynamically generated by the interaction among molecular components within the undetermined cells and those in their context.

Since the generation and maintenance of most cellular patterns may depend on the interplay among gradients, cellular communication, environmental signals and lineage-related mechanisms, the intrinsic vs. extrinsic distinction that could be identified with lineage vs. positional modes of fate determination is sometimes blurred (see more examples of this claim in 3). Models that consider gene regulatory networks (GRNs), coupled by cell communication and subject to environmental signals, are thus helpful for an understanding of the emergence of the information needed for cell patterning. Such models will be useful in evaluating to what extent positional or lineage-related mechanisms are distinguishable, independent of undetermined cells, and necessary or sufficient for cell determination and patterning in a particular system.

Dynamical GRN models have been fruitfully used to study cell-fate determination 4567. In such models, the steady gene activation states (attractors) to which the dynamic system converges correspond to the multigene configurations that characterise different cell types 8. A discrete approach to modelling the dynamics of such genetic networks was first introduced by Kauffman 8 in order to describe qualitatively the concerted action of several genes during cell differentiation. This approach has been proven to capture the logic of regulatory interactions key to cell-type specification in several biological systems (e.g. 56789). Nevertheless, data availability has limited most GRN models to intracellular behaviour, and therefore cellular communication and other important features of pattern formation have been omitted from most models. On the other hand, mesoscopic models that have considered the latter features 101112 do not usually incorporate complex intracellular networks that regulate cell-type determination.

The epidermis of the model plant Arabidopsis thaliana is one of the most thoroughly studied cell-type determination and patterning systems. Such experimental efforts have enabled the postulation of GRNs for cell-fate determination in this system (see recent reviews in 131415). Moreover, relatively simple models may capture relevant aspects of this system as it may be accurately represented in a two-dimensional domain, and because Arabidopsis epidermal cells attain their fate in a fixed domain before they elongate. Actually, some theoretical studies on this system have been published 1617181920. Of these studies, only one 20 has considered the spatio-temporal dynamics of cell-fate determination by explicitly modelling GRNs, yet it focused on leaf patterning and was mainly aimed at presenting a useful modelling platform rather than studying a particular patterning mechanism.

We put forward a spatio-temporal model of coupled GRNs that integrates updated experimental evidence for cell-fate determination in Arabidopsis epidermis and incorporates cellular communication, domain characteristics (size and boundary conditions) and cell shape. We intend to provide a better understanding of how the dynamics of genetic interactions within a cell, in conjunction with cell-to-cell communication, give rise to robust spatial patterns of gene expression and, concomitantly, to cell type determination and arrangement.

Arabidopsis leaf epidermis bears hairs (trichomes) that tend to appear away from each other (Figure 1A; 21), while root epidermis exhibits trichoblast (root hair precursors) and atrichoblasts arranged in bands of a single cell type (Figure 1B; 2223). Despite the contrasting spatial patterns, epidermal cell type determination in root and leaf appears to be associated with similar GRNs (Figures 1E and 1F). Furthermore, although root and leaf GRNs exhibit some differences, they seem to be qualitatively equivalent in dynamical terms 1824.

The GRNs associated with epidermal cell type determination have been thought to behave as a so-called activator-inhibitor system 2526 and some theoretical work has been done in this direction 171819. In a previous paper, we simplified these GRNs to an activator-inhibitor system, so that we could explore the role of cell contextual or positional traits in cellular spatial patterns 18. It is often the case, however, that the structure of the GRN confers important dynamic properties on the system and that these properties are not recovered by simplified versions of the system 27. For example, coupled GRNs in cellularised spatial domains do not necessarily recover the attractors of single-cell GRNs. Moreover, some recent results indicate that the processes giving rise to the epidermal patterns may be richer than an activator-inhibitor system, as different nonlinear sub-networks of the GRN may generate or reinforce such patterns (e.g. 28). A meta-GRN model that encompasses these sub-networks is required to explore whether coupled GRNs in cellularised spatial domains recover the expected and robust gene activation configurations, and to study the role of sub-network redundancy in pattern formation.

Here we have modelled the meta-GRN by integrating the updated experimental data (Figures 1C–1F) into models that explicitly consider GRNs coupled via cell-to-cell communication in a cellularised domain. Interestingly, the attractors for the single cell GRN 18 were qualitatively preserved in the meta-GRN and the simulated spatial patterns of steady states corresponding to hair and non-hair cells matched those observed in actual Arabidopsis wild-type plants. The proposed model therefore accounts for the formation of cellular patterns from initially homogeneous domains containing cells with the same GRN. The model presented here is also useful for exploring the stability of such patterns in the face of environmental and developmental perturbations.

We validated the model by simulating mutant networks and comparing our results with reported gene expression profiles of mutant phenotypes. Then we used the model to postulate novel and testable predictions regarding additional regulatory interactions, the effect of cell shape on patterning and the link between the simulated GRN and the Gibberelic acid signalling pathway. Our results suggest that the proposed GRN constitutes the core of a complex regulatory module associated with Arabidopsis epidermal cell type-determination and patterning. Finally, this study contributes to uncovering both generic and specific aspects of GRN coupling mechanisms and patterning processes in biological systems.

The gene regulatory network model we postulate here is grounded on available experimental data up to June 2008. This evidence is summarised in the following paragraphs. In both root and leaf epidermis, TRANSPARENT TESTA GLABRA 1 (TTG1), bHLH proteins, GLABRA3 (GL3) and ENHANCER OF GLABRA 3 (EGL3), and Myb proteins, GLABRA1 (GL1) and WEREWOLF (WER) in leaf and root, respectively, form a complex that positively regulates the transcription of GLABRA2 (GL2). In turn, the expression of GL2 determines leaf trichome cell-fate and root atrichoblast identity. TRIPTYCHON (TRY), CAPRICE (CPC) and ENHANCER OF TRIPTYCHON AND CAPRICE 1, 2 and 3 (ETC1, 2 and 3) repress the leaf and root activators in partially redundant ways. All of these inhibitors are upregulated, directly or indirectly, by the root and leaf activators. Interestingly, GL2 positively regulates TRY in both root and leaf epidermis 2930.

WER and GL1 are assumed to be directly or indirectly upregulated by themselves. TTG1 is expressed in all the epidermal cells during hair and non-hair determination and the TTG1 protein moves through plasmodesmata among neighbouring cells in the leaf epidermis, accumulating in trichomes 2930. The fact that TTG1 moves among leaf epidermal cells and binds GL3 in the formation of a protein complex has recently led Bouyer and collaborators 28 to postulate a TTG1-trapping/depletion mechanism that could explain the accumulation of TTG1 in cells with GL3 (trichomes) and could suffice for trichome pattern formation. This mechanism is discussed later in the context of other experimental data and our results.

Despite being part of the activator complexes, GL3 and EGL3 are primarily expressed in root hair cells and their proteins move towards cells in which the other components of the complex are present. GL3 and EGL3 also seem to be activated by the inhibitor CPC and repressed by the root activator complex 3132. In leaves, GL3 expression occurs in trichome cells, where GL1 maximum expression is reported. Transcription of GL3 seems to be downregulated by itself and the localisation of the GL3 protein within the cell nucleus is regulated by TTG1 and GL1 31. Experimental evidence also shows that TRY acts non-cell autonomously in leaves and that the CPC protein moves through plasmodesmata among cells in both leaf and root epidermis 3334. In the root, SCRAMBLED (SCM) seems to be a crucial component of a positional signal downregulating WER in root-hair bands and therefore biasing the cellular pattern 3536.

In the proposed GRN model, elements or nodes of the network correspond to genes. The ETC node summarises the partially redundant activity of the three enhancers of TRY and CPC. Edges of the network stand for regulatory interactions between nodes (activation or repression; Figures 1E and 1F). Nodes representing genes for which there is enough information can attain three different states (0, no expression; 1, mild expression; 2, strong expression). In the leaf system, only ETC can either be expressed, 1, or not expressed, 0, and only ETC and TRY are binary in the root system. For simplicity, the state of all nodes was updated synchronically. In single-cell GRNs, the updating scheme does not seem to affect attractors when these are fixed-point type 37. Since the attractors of our GRN model are indeed fixed-point ones, it could be the case that asynchronicity does not affect them, yet this issue has not been explored in coupled GRN models.

In the GRN model, each node's state depends on that of other nodes, its regulators. The level of expression of a given gene is represented by a discrete variable g (0, 1 or 2) and it depends on the level of expression of other components of the network, g₁, g₂, ..., g_N. The state of every gene g therefore changes according to:

g_n(t+1) = F_n(g_n1(t), g_n2(t), ..., g_nk(t)).

In this equation g_n1, g_n2, ..., g_nk are the regulators of gene g_n, and F_n is a discrete function known as logical rule (logical rules were grounded on available experimental data and are graphically represented in Figures 1E, 1F, [see Additional file 1]). Given the logical rules, it is possible to follow the dynamics of the network for any initial configuration of the nodes expression states. Since the GRN is simulated as discrete, all possible combinations of gene activation states may be explored. One of the most important dynamical traits of a GRN is the existence of attractors. Starting out from an initial state, the reiterated application of equation (1) generates dynamics in which genes go through transient states until the whole network enters into a stationary or periodic profile of multi-gene expression. Such stationary or periodic expression profiles constitute the GRNs' attractors, which correspond to gene expression profiles characteristic of particular cell types.

In order to couple the GRNs in a compartmentalised (cellular) domain, we considered a discrete lattice of n × n elements, in which each element (i, j) represents a cell with a GRN. As in real organisms, all cells bear the same GRN. Modelled lattices are a simplified representation of leaf or root epidermal sections. Cells in the lattice have exactly four neighbours and there is no difference in permeability between them.

According to experimental data, some proteins codified by elements of the network move to neighbouring cells and affect gene expression in a non-cell-autonomous fashion (TRY, CPC and TTG1 in the leaf epidermis and CPC, GL3 and EGL3 in the root epidermis), giving rise to a network of coupled networks (herein meta-GRN). Although empirical evidence supporting cell-to-cell motion rather than aplopastic transport only exists for CPC and TTG1, all available data are congruent with the assumption that all mobile elements of the GRN move in a cell-to-cell manner. In the spatial model we therefore allowed for certain elements to move among neighbouring cells (Figures 1C, 1D and 2) following the equation:

g(t)_{T [i] [j]} = H(g(t)_{[i] [j]}+ D(g(t)_{[i+1] [j]}+g(t)_{[i-1] [j]} + g(t)_{[i] [j+1]}+g(t)_{[i] [j-1]}- 4(g(t)_{[i] [j]})),

where g(t)_{T [i] [j]} is the total amount of protein g in cell (i, j). D is a continuous variable that determines the proportion of g that can move from any cell to neighbouring ones and is correlated to the diffusion rate of g. H is a step function that converts the continuous values corresponding to the amount of g diffused into each cell into a discrete variable that may attain values of 0, 1 or 2. In order to simulate the effect on diffusion of protein attachment to a protein complex, our simulations consider that the mobile elements of the network diffuse with a lower rate from inside to outside when the other protein complex components are inside the same cell. So, for instance, when there is GL3 in a cell, the term -4(g(t)_{[i] [j-1]}) in the diffusion equation of TTG1 is removed or decreased (see also 28). More generally, we defined a term -A(g(t)_{[i] [j]}), in which de value of A depends on the presence of trapping proteins in a cell (i, j) at time t (A = 4 when no trapping proteins are present and A < 4 when these are present).

D was systematically varied from 0.0 to 0.25 with a step size of 0.01, enabling the generation of parameter spaces in which we plotted the average number of ectopic trichoblasts and atrichoblasts for every combination of parameter values. Four types of 'trapping' diffusion terms were also tested (-A(g(t)_{[i] [j]}), with A = 0,1,2,3; see Results). In the leaf system, boundary conditions were simulated as zero-flux, while in the root system, two borders were identified and two were kept at zero-flux, simulating a root-like cylinder. In the root model, the positional signal associated with SCM was simulated as a constant downregulation of WER every three cell files (two NH files between H files) 36.

Then, g(t)_{T [i] [j]} was considered in order to estimate the g(t)_{[i] [j]} value, which was then used to update the logical rules, according to the equation below:

g'(t)_{[i] [j]} = g(t)_{[i] [j]} +g(t)_{T [i] [j]} - k,

where k stands as a degradation constant. g'(t) was taken as an input to evaluate the logical rules and obtain g(t+1)_{[i] [j]}.

In brief, each cell's GRN is initialised with a random gene activation profile. Then, three steps are sequentially repeated until the whole lattice reaches a steady state (diagrammatic representation in Figure 2): (1) application of logical rules, (2) diffusion of mobile elements and (3) integration of diffused protein into the GRN inputs for the next updating step according to the logical rules.

In order to test the model, we also simulated networks that correspond to reported mutants. The loss and gain of function simulations were done by fixing the expression value of the altered gene to 0 or 2, respectively. The random seed was the same for both wild-type and mutant simulations so that results were comparable.

It should be noted that there are several dozens of genes involved in hair determination, patterning and differentiation in Arabidopsis 38. We have, however, incorporated in the GRN models only the elements that seem to be responsible for the decision: hair vs. non-hair. Changes in expression states of these elements do indeed yield changes on epidermal cell identity. In contrast, most of the genes that were not included seem to act downstream of the GRNs modelled here and may play important roles in morphogenesis once cells are committed to a particular fate. A few other genes that are not considered seem to act upstream of the GRN postulated here, probably linking the GRNs with signalling pathways or tissue-specific factors. Our simulations of both wild-type and mutant networks indeed suggest that the GRNs modelled here incorporate the necessary components to render the robust gene activation profiles characteristic of epidermal hair and non-hair cells in both roots and leaves, and thus uncover the core of a regulatory module that regulates epidermal cell patterning in Arabidopsis.

Graphics were elaborated in MATLAB and programs in C (codes available upon request).

We put forward an experimentally-grounded dynamical model that integrates experimental evidence and helps us to study the concerted action of multiple molecular components during Arabidopsis hair pattern formation. This constitutes one of the best experimental systems to address questions concerning cell differentiation and spatio-temporal arrangement of cell types. This study uncovered a regulatory module that is sufficient to recover the cell types and spatial configurations characteristic of Arabidopsis root and leaf epidermis. The model of coupled GRNs in a cellularised spatial domain was used to provide predictions regarding the effect of unknown interactions, signalling mechanisms acting on the GRNs and cell shape on such epidermal patterning.

Besides the insights of our meta-GRN model concerning the particular cell patterning system under consideration, it makes a general contribution to a fundamental issue in developmental biology: the origin of the information needed for the emergence of cell-type patterns from homogeneous domains. In contrast with single-cell models previously published, we achieved this by coupling single-cell GRNs by explicitly incorporating cell-to-cell communication in a cellularised spatial domain. This enabled us to evaluate whether the observed gene activation profiles are recovered in the meta-GRN.

In the case considered here, the information needed for cell-type arrangement consists of heterogeneous patterns of gene expression and is determined by complex interactions among multiple intra- and extra-cellular factors. Our model therefore recovers the de novo formation of positional information from such complex GRNs and suggests that the appearance of this information is very robust. It does not depend on particular initial conditions or domain size, and is resilient in the face of single perturbations in the logical rules, as well as changes in the protein diffusion rates. Robustness has been documented as well for single-cell GRN models 56.

As mentioned above, our model also enabled testable predictions on the behaviour of the particular system under study. The first prediction states that cell elongation, or other processes that could alter or bias diffusion rates in the x and y axes of the root (for example, differential transport rate), have a relevant role in stabilising the spatial pattern generated early in root development. This prediction points at a precise way in which cellular morphological traits may affect cellular patterning and could be tested by crossing a GL2:GUS or GFP marker line plants with altered cell shapes (e.g. 4546) and search for alterations in the spatial pattern of this genetic marker of epidermal cell type.

Another prediction concerns the action of GA signalling on the GRN, suggesting that this hormone signalling pathway acts on CPC or bHLH genes. This prediction is consistent with recent experiments with bHLH inducible overexpression lines 47 that suggest that an increase in bHLH expression suffices to produce more trichomes. Both bHLH and CPC actually have GA response motives (PLACE, http://www.dna.affrc.go.jp/PLACE/), but this does not imply that they respond directly to GA. According to our prediction, mutagenesis of the GA response motives should cause trichome density alterations in response to GA treatments. As data on other signal transduction pathways accumulate, the framework put forward here could be useful for an understanding of how diverse environmental cues are integrated by the regulatory module during epidermal cell-fate determination. This would contribute to a better comprehension of the plastic developmental responses of plant and animals to environmental cues.

The GRN modelled here is sufficient to recover the gene activation profiles and the hair spatial patterns typical of wt and mutants' root and leaf epidermis. Our results therefore support the idea that developmental processes, such as cell-type determination and patterning, depend on regulatory modules, which are semi-autonomous with respect to the rest of the genome 48. According to this and other studies (e.g. 6) the outputs of these modules (i.e. attractors that correspond to the possible alternative multigene expression profiles) are highly robust in the face of diverse perturbations, suggesting that these may act as patterning modules for a wide range of parameter values in kinetic functions of the network. Moreover, it seems that these modules may have been co-opted during evolution to regulate the decision underlying cell-fate determination of different structures in diverse tissues/organs (e.g. root, hypocotyl and leaf). Indeed, the latter could be the case in plant species with contrasting cellular patterns, in which local traits, such as cell size and shape, positional cues or domain geometry, may affect the type of pattern, even if the regulatory modules have been largely conserved during evolution.

The results presented here also support the idea that the coupled-GRNs system modelled here includes smaller coupled sub-networks that could themselves constitute mechanisms that contribute to the formation or maintenance of cellular patterns. In the leaf system, these mechanisms are: (i) the regulatory and protein-protein interactions between GL3 and TTG1 that give rise to the TTG1-trapping/depletion mechanism postulated by Bouyer and collaborators 28, (ii) the so-called activator-inhibitor system conformed by the activators of GL2 and its inhibitors 1718, and (iii) the feedback loop established between TRY and GL2 2930. Although these sub-networks or motifs could be important or even sufficient for the generation and maintenance of the epidermal cellular patterns, independently of the rest of the GRN, it is likely that they all act redundantly in the generation of such patterns in a robust manner. The latter seems to be the case as, for instance, our results suggest that TTG1 diffusion is not necessary for patterning when the whole GRN is present. Experimental data also show that wild-type cell configurations are observed in ttg1 mutants when GL3 is overexpressed 40. Since these mechanisms are not independent of each other, the meta-GRN model encompassing them all in conjunction with cell-cell communication is useful for studying the spatio-temporal patterning. The interlinkage of redundant sub-networks found in this system supports the hypothesis that interconnection among feed-back loops or small nonlinear systems confers robustness to biological networks and may constitute an important trait in network evolution 49.

The model and results presented enable us to conclude that cell-type determination and patterning in leaf and root epidermis of Arabidopsis are regulated by a complex GRN that encompasses redundant sub-networks. Since all these sub-networks involve cell-to-cell communication, the spatial meta-GRN model is central to studying how these interact with each other, giving rise to the information needed for cellular patterning. Unlike the importance of gene redundancy, the relevance and implications of sub-network redundancy remain largely unexplored. This work is one of the first contributions regarding this issue.

We could also conclude that the meta-GRN qualitatively reproduces pattern formation in the system under study. This is why this model could be used to state precise predictions, namely, that the GA pathway acts on CPC or the bHLH genes, that GL1 and WER are self-upregulated, that root cell shape has a stabilizing role on the banded pattern and that new cortex-related biasing signals may upregulate CPC.

Future developments of the meta-GRN model proposed here should allow for cell proliferation or continuous gradients in pattern formation. Nevertheless, the model presented here constitutes a starting-point from which to integrate future experimental evidence of the intracellular GRNs themselves (e.g. epigenetic regulation), as well as diverse cellular processes. As further meta-GRN models are postulated and validated for diverse biological systems, we will be able to evaluate which types of intracellular GRNs, regulatory motifs or sub-networks, as well as coupling mechanisms, underlie patterning and morphogenesis in living organisms.

As exemplified by the GRN studied here, development depends upon non-additive interactions occurring in a wide range of spatial and temporal scales. Accomplishing the paramount task of understanding how robust patterns arise during development may therefore be facilitated by computational or mathematical models that bridge the divide between dynamics of interactions at the microscopic level and the origin and evolution of morphological qualities. Indeed, models like the one put forward here constitute useful tools for integrating experimental work, hinting at new experiments and providing novel insights into the complex interactions that underlie patterning processes and, therefore, into developmental and evolutionary biology.

MB participated in the study design, specified the GRN model, wrote the computer programs and carried out the simulations, participated in the discussion of results and contributed to writing the manuscript. CES participated in the study and computer program design, as well as in the discussion of results. PPL participated in the study design, the mathematical formulation of the model, and in the discussion of results. EAB established the project and coordinated this study, participated in the study design, in the analysis of results and in writing the manuscript. All authors read and approved the final manuscript.

In the root meta-GRN model we assume that WER activates itself. This interaction is not essential for the formation of wild-type pattern, but it is required for the formation of the pattern observed in the absence of SCM. While this paper was under review, Savage and collaborators (2008) 50 put forward a mathematical model and a patterning mechanism that do not involve WER's self-activation. These authors also provide new data regarding WER regulation that are consistent with their proposal. However, the data does not reject the possibility that WER self-activates, as they are also consistent, for instance, with the combined activity of a self-activating loop and another input activating WER. According to available empirical data, both the model presented here and that proposed by Savage and collaborators (2008) 50 appear to be plausible. Further tests could help find if a mechanism involving WER's self-activation, another mechanism, or the coupled activity of more than one mechanism are important for cell-type arrangement in Arabidopsis root epidermis. Interestingly, the existence of more than one patterning mechanisms would emphasize the importance of sub-network redundancy in developmental networks.