Systems Biology Approaches to Development beyond Bioinformatics ...

BioScience Advance Access published April 6, 2016

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Systems Biology Approaches to Development beyond Bioinformatics: Nonlinear Mechanistic Models Using Plant Systems ELENA R. ÁLVAREZ-BUYLLA, JOSE DÁVILA-VELDERRAIN, AND JUAN CARLOS MARTÍNEZ-GARCÍA

Keywords: gene regulatory networks, genotype–phenotype map, epigenetic landscape, systems developmental biology, nonlinear stochastic modeling

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fundamental question in biology and biomedicine is how genes map to phenotypes. Such mapping depends on development, which includes the processes underlying cell differentiation and morphogenesis. In an attempt to understand the molecular basis of such mapping, the genomic era has produced an immense amount of data at the DNA, RNA, protein, and metabolic levels (Flintoft 2005, Kim and Przytycka 2013). The statistical analysis and integration of genome-scale data has produced important contributions to uncovering genotype–phenotype (G–P) associations and the regulatory modules involved in various normal and anomalous developmental processes (Brady et al. 2011, Liberman et al. 2012, Rogers et al. 2012, Li et al. 2013). Such practices are, mainly, of a descriptive nature, because they use statistical tools to recover patterns and to quantitatively describe them, uncovering associations with predictive power (Ellner and Guckenheimer 2011). The set of associative practices follows the so-called top-down approach of systems biology (Tomlin and Axelrod 2007). This approach has been very illuminating, but relying on a strictly associative and descriptive approach may hinder understanding of the underlying biological processes and

mechanisms, because these are highly nonlinear, stochastic, and dynamic. Ultimately, systems biology pursues an understanding of how such dynamic processes and systemslevel mechanisms underlie the emergence of the described patterns. We have sought an understanding of such underlying systems-level dynamical processes, which imply restrictions emanating from mutual interactions and feedback mechanisms at the genetic and nongenetic levels, as well as across temporal and spatial scales (as further explained below). Together, the multilevel or multiscale dynamic processes include the biological mechanisms that underlie cell differentiation and reprogramming. Therefore, we refer to biological systems-level mechanisms as those dynamic processes involved in the emergence of cell differentiation and morphogenesis. In contrast to addressing the molecular mechanism of the regulation of a particular gene or component, as is generally pursued in mechanistic molecular genetic approaches, we are interested in unraveling the integrated mechanisms and concerted action of the interacting molecular and physicochemical components that underlie development. Indeed, explaining how a particular gene is

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Biological systems are complex, stochastic, and nonlinear; therefore, understanding how genes map to phenotypes remains a challenge. A complexsystems mechanistic approach, emphasizing relations over associations, is required for understanding the emergence of cell differentiation and morphogenesis during development. An increasing number of contemporary studies that integrate biological data into dynamic, nonlinear, and stochastic models are providing novel explanations for development. Unfortunately, the adaptation of the biological research tradition to such quantitative and interdisciplinary approaches is not straightforward. In an attempt to contribute to this necessary transition, drawing mainly on our own studies as examples, we present here a nontechnical overview article of how such models are helping unravel the emergence of cell differentiation, pattern formation, and morphogenesis. The studies reviewed here suggest that we need to reevaluate how biological causal and functional roles are interpreted.

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previously by pioneer theoretical biologists (Kauffman 1993, Goodwin 1994, Solé and Goodwin 2008, Waddington 1957). Nonlinearity in genotype to phenotype maps Despite the fact that nonlinearity is intrinsic to biological and biochemical systems, reasoning in biology has traditionally followed a remarkably and instrumental linear cause–effect paradigm for particular components. This latter approach does not consider explicitly the conditional role of each part on others or the direct and indirect interactions among components that altogether constitute the system. In a cause–effect framework, which avoids the consideration of interactions and feedback loops, linear schemes of causation are frequently invoked (see, e.g., discussions in Huang 2011). The latter approach, which assumes that the behavior of a whole system can be understood on the basis of the behavior of its isolated components or their added behaviors, has been qualified as “reductionist.” In agreement with previous publications (Kaneko 2006, Alvarez-Buylla et al. 2010a, Huang 2011), we consider that a systems-level approach that explicitly considers interactions and the nonlinear stochastic nature of underlying processes is required to understanding the emergence of the observed biological patterns. Therefore, for the purpose of this article, by reductionist paradigm, we refer to an explanatory view in which the effect of interactions among components of interest (e.g., molecular species) is inadvertently ignored. Plant development: Emergent behavior from multilevel nonlinear dynamics Plants have been useful model systems for experimentally grounding and validating theoretical models. Here, we describe how dynamical models have contributed to understanding the emergence of cell phenotypes and morphogenetic patterns and processes as natural consequences of the restrictions imposed by nonlinear regulatory and dynamic systems. The cases summarized here imply dynamic emergent behaviors that scale from gene regulatory networks (GRNs) and epigenetic landscapes (ELs) to model cell-fate attainment at the cellular level and pattern formation at the tissue and organ levels (figures 1 and 2). Multistability and cell-fate dynamics: Intracellular regulatory networks. A theoretical understanding of the way in which the

concerted action of multiple genes, along with physicochemical constraints and environmental factors, regulate cell differentiation and morphogenesis is a foundational question in developmental biology (Turing 1952, Waddington 1957, Kauffman 1969, Thom 1983, Goodwin 1984, among others). Experimentally, biologists have access to the expression profiles, or gene configurations, characterizing the different cell types in a multicellular organism. These contrasting stable cellular phenotypes are manifested despite an underlying invariant genomic sequence. The process of cellular differentiation is therefore necessarily a consequence of epigenetic regulatory mechanisms. Given this knowledge, http://bioscience.oxfordjournals.org


regulated is very useful, but on its own, it does not explain how a cell type is specified, an organ is formed, or a disease is developed. Even if we study the so-called master genes, we are not able to understand the systems-level underlying mechanisms implied in its prominent role. The systemsbiology approach that implies the proposal of mechanistic, dynamic models to uncover G–P mapping algorithms is referred to as the bottom-up approach, and it has been mostly applied to well-curated regulatory modules with clear phenotypical consequences. The top-down and the bottom-up approaches complement each other and are increasing the explanatory and predictive power of systems biology (Bruggeman et al. 2007). A key aspect of the bottom-up mechanistic approach is that it considers nonlinear dynamical mapping models at different levels of organization (e.g., gene interactions and circuits, networks, cells, tissues, organs), incorporating both molecular–genetic (e.g., genes, proteins, miRNAs) and nongenetic (e.g., mechanical and elastic forces and fields, chemical concentrations and gradients) components to understand the G–P mapping and to uncover systems-level behaviors and traits. Such a dynamic models study cellular behavior as the inevitable manifestation of the intrinsic nonlinear and stochastic nature of underlying networks of interacting components. Such an endeavor implies uncovering the biological mechanisms of cell differentiation and morphogenesis. In summary, in our view, bottom-up systems biology is grounded in a complex systems and dynamic mechanistic approach. The importance of the nonlinear, stochastic dynamic nature of biological systems has been previously emphasized by many theoreticians (e.g., among others, Turing 1952, Kaufman 1969, Meinhardt 1982, Sawyer et al. 2010). More recently, dynamical models have been validated for particular biological systems by integrating mechanistic data for the system´s components and their interactions to study their concerted action and emergent structural and dynamical consequences (Mendoza and Alvarez-Buylla 1998, Çağatay et al 2009, Barrio et al. 2010). In this article, we provide an overview of contributions using such bottom-up approach of systems biology grounded on experimental data. We draw on almost two decades of research on complex network models applied to plant systems. We have focused on various plant organs (flower, root and leaf epidermis, root stem-cell niche, root cellular organization and growth, and shoot apical meristem transitions) and have taken advantage of the valuable experimental and molecular genetic data available for Arabidopsis thaliana (Mendoza and Alvarez-Buylla 1998, Espinosa-Soto et al. 2004, Alvarez-Buylla et al. 2007, 2008, 2010a, Benítez et al. 2007, 2008, Azpeitia et al. 2010, Azpeitia and Alvarez-Buylla 2012, Azpeitia et al. 2014). Our conceptual and theoretical explorations have illuminated the constructive role resulting from the inevitable interplay between feedback-based interactions among genetic and nongenetic components and stochasticity in real biological systems—a view that has been emphasized

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In GRN models, genes or proteins correspond to the nodes of a network, whereas the links among them stand for regulatory interactions (figure 3). A dynamical model describes mathematically the mechanisms by which the variables describing a system change in time (Ellner and Guckenheimer 2011). In the case of a GRN dynamical model, the activity (i.e., gene expression) of each component in the network is changing in time. The nonlinear character of the system resides in modeling the change of each gene in terms of the activity pattern of the genes regulating it along time. The mutual dependence of the genes in the network imposes constraints and restricts the future gene activity patterns of each gene. Whether a given gene will be expressed or not in future time-steps depends on the expression level of the genes regulating it but also on the way their interactions condition their individual effects, producing a nonadditive, joint regulatory effect on it. Because of such regulatory feedback among the genes, it is no longer possible to establish a direct causal effect for each individual regulatory gene (see figure 4). Therefore, the “functional role” of each gene can only be defined and understood in the context of the system in which it is immersed. Moreover, each gene in the network is simultaneously subject to this regulatory process; consequently, global, nonintuitive, and structured behavior emerges in Figure 1. Emergent models at different levels of organization: (a) single-cell a self-organized manner. multistable gene regulatory network models to study cell differentiation, A first striking behavior of such com(b) multicellular and epigenetic landscape models with no spatiotemporal plex regulatory networks is the existence restrictions to study the temporal pattern of cell-fate attainment, of a discrete number of genetic conand (c) mesoscopic morphogenetic models. figurations that are consistent with the restrictions imposed by the GRN and how can we explain, in system-level mechanistic terms, the that actually coincide with observed in vivo configurations. emergent stable gene expression profiles associated with difOnce the network presents such a configuration, the expresferent cell types? Why do the observed cellular phenotypes, sion of each gene in the network does not change anymore, at least partially related to and implemented by the molecubut it is stably maintained in a self-organized manner. lar profiles, constitute qualitative discrete entities? The Such stable configurations are referred to as attractors, and nonlinear dynamical analysis of GRNs has provided insights they have the property of being robust in the face of perinto these questions and phenomena through the concept turbations to the system’s state (see reviews and examples of multistability, as we further explain below. This approach in Alvarez-Buylla et al. 2007, 2010a). Qualitatively, cell was first proposed theoretically by Kauffman (1969) and types can be characterized or be considered as the in vivo later validated for several biological systems (Mendoza and manifestation of attractor configurations of the global GRN Álvarez-Buylla 1998, Von Dassow et al. 2000, Albert and coded in the genome of a multicellular organism. This has Othmer 2003; see reviews in Álvarez-Buylla et al. 2010a, been experimentally validated using human cells (Huang Dávila-Velderrain et al. 2014). et al. 2005).

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Multistability is the property of a given GRN of attaining more than one of these attractor states. In systems-biology literature, multistability is generally accepted as the formal dynamic mechanism explaining cellular differentiation, reprogramming, and dedifferentiation (Kauffman 1969, Mendoza and Alvarez-Buylla 1998, Laurent and Kellershohn 1999). Importantly, multistable systems are necessarily nonlinear (Ellner and Guckenheimer 2011). Floral organ primordial cell-fate specification. As a first example

of the nonlinear perspective to plant development, we have developed a cellular level model of cell-fate determination during floral-organ specification in Arabidopsis thaliana (see figures 1 and 2; Mendoza and Alvarez-Buylla 1998, EspinosaSoto et al. 2004, Alvarez-Buylla et al. 2010a, Azpeitia et al. 2014). Most flowers have four types of floral organs that are formed with a stereotypical pattern from the outside to the 4 BioScience • XXXX XXXX / Vol. XX No. X

inside: sepals, petals, stamens, and carpels (see, e.g., AlvarezBuylla et al. 2010a,b). Such a conserved pattern suggested an underlying robust mechanism that evolved before the origin of the flowering plant species. We aimed at discovering such a robust regulatory core. To this end, we proposed a GRN model grounded on experimental data to test, for the first time, that a multistable GRN could partition the space of possible configurations of gene expression states in a manner consistent with that seen in the different cell types of the floral meristem. This model demonstrated that in a real biological case, cellular phenotypes could indeed correspond to different attractors or stable configurations of gene expression states (Mendoza and Alvarez-Buylla 1998). This GRN was first published on the basis of limited available experimental data; however, since then, it has been updated to incorporate additional data as they become available (Espinosa-Soto et al. 2004, Alvarez-Buylla et al. 2010a). http://bioscience.oxfordjournals.org


Figure 2. Schematic representations of Arabidopsis thaliana plants before and after flowering. The cell-differentiation, tissue-patterning, or other developmental processes that have been studied using multistable GRN are indicated. In the text, the references from which each of the GRN models illustrated here were drawn are cited.

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Overall, these studies uncovered a set of components and interactions (i.e., a regulatory module) that are sufficient to recover observed patterns of gene expression in each one of the four types of primordial cells that later on form sepals, petals, stamens, and carpels (figure 2). The regulatory module incorporates the so-called ABC genes, which had been shown to be necessary for floral-organ specification in Arabidopsis using single mutants and their combinations (Coen and Meyerowitz 1991). These genes directly or indirectly interact with several other components and modules; we uncovered one possible set of such interactions that are sufficient to lead to the configurations observed during the early stages of flower development. Furthermore, theoretical analyses of this GRN have shown that it is robust to slight genetic perturbations, and it has also been validated for wild type, as well as for the gain and loss of function mutations (for details, see Alvarez-Buylla et al. 2007, 2010b). Finally, these GRN models have also proven to have predictive power, because some interactions that had not been uncovered or proposed before were predicted in the context of the GRN and later confirmed experimentally by other research groups (for details, see Alvarez-Buylla et al. 2007, 2010b). Cell–cell movement of GRN components: Restrictions underlying spatial pattern formation. To go from single-cell to tissue-level

coupled patterns of cell differentiation, we have explored higher-order systems in other plant organs. The molecular genetic basis of cell subdifferentiation (hair and nonhair) in the Arabidopsis thaliana epidermis of roots and leaves has been extensively studied experimentally (see reviews in Hui et al. 2013, Benítez et al. 2014). GRN models have integrated such molecular experimental data for this cell differentiation process and have been useful to uncover the multistable module that regulates the cell subdifferentiation process (figure 2; Mendoza and Álvarez-Buylla 1998, http://bioscience.oxfordjournals.org



Figure 3. A schematic representation of a gene regulatory network model.

Benítez et al. 2008). We proposed a model simulating the emergence of epidermal cell types in a static spatially explicit domain. In this model, the configurations at each location were restricted by the patterns of cell–cell movement of some of the GRN components according to experimental data, in addition to the interactions of the GRN itself. Interestingly, even though very similar GRNs are involved in both cell differentiation and patterning processes, cell-type spatial patterns are contrasting in roots (stripes) and leaves (between random and uniform; see figure 2). The model provided a clear example of the multistable nature of the dynamic systems underlying cell differentiation, as well as a case in which similar GRN modules are coopted for cell differentiation in different tissues and may yield contrasting phenotypes in different cellular contexts within the same organism (Benítez et al. 2008, Álvarez-Buylla and Benítez 2011, but see examples in other systems: Chang et al. 2006, Luo et al. 2013). Furthermore, similar regulatory motifs that involve local self-activation and lateral long-distance inhibition, such as those uncovered for the Arabidopsis models, have been found also for epidermal-cell patterning in animals (Sick et al. 2006). Our models for this case have also uncovered the role of nongenetic restrictions during cell differentiation, such as cell geometry and the relative position of one cell layer with respect to an underlying one, both being crucial for biasing the spatial pattern of hair and nonhair cell types in the Arabidopsis thaliana root with respect to the pattern found in leaves (see figure 2; Benítez et al. 2008). In addition, these models enabled predictions of the nodes that may connect the cell-differentiation regulatory module with gibberelic acid signaling pathways and eventually with environmental cues (Benítez et al. 2008). Therefore, this system offers an excellent opportunity to explore the constructive role of the environment in cell-pattern evolution and the role of epigenetic inheritance in plasticity and adaptive evolution. Different research groups independently uncovered and modeled particular subnetworks or modules involved in epidermal-cell subdifferentiation. Each one was shown to be sufficient to recover hair and nonhair cell types, as well as the contrasting spatial patterns observed in roots and leaves (Dolan 1996, Lee and Schiefelbein 2002, Schiefelbein 2003, Pesch and Hülskamp 2004). Interestingly, however, only when all these regulatory subnetworks or motifs were integrated in a single network were the spatial patterns robustly recovered (Benítez and Álvarez-Buylla 2010, Álvarez-Buylla and Benítez 2011). These results suggest that redundant regulatory motifs or subnetworks might be important for the emergence of systems-level characteristics and behaviors, such as the robustness and resilience of biological patterns (Álvarez-Buylla and Benítez 2011). Finally, this system also served to show that cell fate is not due to either internal or external factors; rather, it emerges from the feedback of both. It illustrates that positional information during cell differentiation emerges from the dynamic interaction of intracellular and extracellular components. This argument

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is further elaborated below, using other examples at higher levels of organization. Root stem-cell differentiation and spatial patterning. Stem-cell niches (SCN) are fundamental for multicellularity; they include proliferation and differentiation capabilities at the same time and in restricted spatial domains. These cellular structures have generic traits of cellular organization conserved in plants and animals, suggesting underlying biogeneric mechanisms (Sablowski 2004, Azpeitia and ÁlvarezBuylla 2012). Conserved patterns include the position of the organizer cells at the center of the SCN that have very low rates of proliferation. These cells are surrounded by stem cells with slightly higher proliferation rates that first transit to a domain of active proliferation, then to one of elongation, and later on to one of differentiation. Plant SCNs are more amenable to in vivo experimental studies than those on animals. Therefore, they may be useful for uncovering such biogeneric mechanisms. We have used the root SCN for integrating a dynamic GRN model based on available data. We used the model to test the sufficiency of uncovered pathways and then proposed a regulatory module able to recover the gene-expression configurations characteristic of the four precursor stem cell types or initial cells: vascular initials, cortex–endodermis initials, quiescent center, and columella—that is, epidermis-lateral root cap initials (figure 2; for details, see Azpeitia et al. 2010). This study showed that the two pathways that had been thought to be both necessary and sufficient to recover the cell types and their spatial patterns within the root SCN are not sufficient. 6 BioScience • XXXX XXXX / Vol. XX No. X

Further simulation studies were performed to postulate novel missing interactions (Azpeitia et al. 2013). Some of the interactions were later corroborated by independent experimental studies (Azpeitia et al. 2013), once again documenting the predictive power of GRN models. In this case, the spatial arrangement of each gene-expression configuration associated with each type of stem cell was also recovered by incorporating the patterns of intercellular movements for some of the SCN–GRN components (Azpeitia et al. 2011). Interestingly, the network of networks that followed the intracellular configuration for each cell at each spatial location (40 nodes) converged to a single robust attractor that mimicked the spatial arrangement of different geneexpression configurations observed in the actual Arabidopsis root SCN. Ongoing research is explicitly incorporating into this model hormone signaling, cell proliferation, and other components to understand how SCN size is established and spatiotemporally maintained during development. The results of all these models for single cells and static spatial domains have been useful to support the hypothesis that cell types emerge naturally as self-organized properties of the underlying molecular regulatory network, as well as additional restrictions that couple GRN dynamics, such as cell–cell movement of some of the networks components or cell geometry (figures 1 and 2). The detailed analysis of the proposed models through virtual mutation experiments provides insights into the impact on pattern formation that the different genes within the GRN can have. Importantly, it demonstrates that such effects depend on the whole regulatory system rather than http://bioscience.oxfordjournals.org


Figure 4. A schematic representation of gene feedback-based regulatory interactions: (a) Gene y is regulated by two different genes (w and x). A cause–effect view (here reductionist) would imply explaining the behavior (expression) of gene y exclusively as the additive behavior of the genes w and x, potentially ignoring feedback-based interactions. (b) A hypothetical feedback-based interaction: If the behavior of gene y regulates itself, its regulator (x), or both, a nonlinear view would be necessary in order to interpret and model it, given that in such scenario, the effect (behavior of y) becomes part of the cause.

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Populations of cells: Stochastic exploration of epigenetic landscapes and developmental dynamics In addition to nonlinearity, a property that is very relevant for the study of biomolecular processes and development is stochasticity (Tawfik 2010). What happens if we consider in our simple mechanistic models of development a more realistic approximation to the inner workings of biomolecular systems by including both nonlinearity and stochasticity? A quite interesting problem in developmental biology is the following: If molecular regulatory events at the cellular level are intrinsically stochastic, how is it possible that we observe a phenomenological, robust, and seemingly deterministic process during development? (Zernicka-Goetz and Huang 2010, Oates 2011). Models of plant development again have given interesting insights into this issue (figures 1 and 3; Alvarez-Buylla et al. 2008, Villarreal et al. 2012, Azpeitia et al. 2014). The GRN models briefly described in the previous section are deterministic. This means that for each gene in the network, we only need to specify which genes are regulating it and the functional form of their joint regulation in order to know exactly the effect on its expression rate of change given the current state of each gene in the network. Such models are useful tools, as was shown above. But in order to explain the patterns of transitions among cell states or attractors during actual developmental processes, further modeling considerations may be required. Stochastic GRN models are becoming useful as such means. In a stochastic GRN model, the joint regulatory effect discussed in the previous section also depends on a stochastic http://bioscience.oxfordjournals.org

influence, a noise effect that varies randomly each time. Introducing stochasticity into nonlinear dynamical models has provided interesting results, suggesting, for example, that noise may play a constructive role in biology and that present-day biological networks might have evolved in noisy conditions (Álvarez-Buylla et al. 2008, Tawfik 2010). In order to explore this possibility in a plant developmental process, we studied the role of stochastic perturbations on the GRN underlying floral-organ primordial cell fates. We addressed whether such stochastic GRN recovered, in addition to the observed cell states, the robust temporal morphogenetic pattern with which such states are attained during early flower development (Álvarez-Buylla et al. 2008). In flowering plants, a floral meristem is sequentially partitioned into four regions from which the floral-organ primordia are formed and sequentially give rise to sepals in the outermost whorl; petals in the second; stamens in the third; and, almost concomitantly, carpels in the fourth, in the central part of the flower. This spatiotemporal patterning is widely conserved among angiosperms. By introducing stochasticity (noise) into the GRN of organ identity genes, the most probable temporal order in which the uncovered attractors are attained in the simulated stochastic model is, in fact, consistent with the temporal sequence in which the specifications of corresponding primordial cell types are observed in vivo. The model provided, then, a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral-organ primordial cell specification. In addition, it also allowed predictions on the population dynamics of cells with different genetic configurations during development (for details, see Alvarez-Buylla et al. 2008, Azpeitia et al. 2014). This model enables the analysis of cell-state transition events in the establishment of multicellular hierarchies. Therefore, it constitutes a new approach to understanding a morphogenic process by uncovering restrictions that the GRN interactions impose also on the transitions among the attractors or cell fates and not only on the number and types of the stable states themselves. The stochastic GRN model of floral-organ specification is equivalent to a stochastic exploration of what Waddington called the “epigenetic landscape” (EL; figures 1 and 5). Such a landscape emerges from the GRN, and it can be characterized mathematically (Villareal et al. 2012). Several different approaches to formalize and to numerically analyze the EL have recently been proposed and are reviewed elsewhere (Wang et al. 2010, 2011, Zhou et al. 2012, Davila-Velderrain et al. 2015a). Nonetheless, given the multidimensional and nonlinear nature of such epigenetic landscapes, in most cases, a numerical exploration such as that proposed in Alvarez-Buylla and colleagues (2008) is required (see also figure 5). More recently, an approach to study the reconfiguration of the EL that emerges as a result of quantitative alterations of the decay rates of particular GRN components was proposed (Davila-Velderrain et al. 2015b). This study provides further explanations and predictions concerning the phenotypical impact of quantitative alterations of particular XXXX XXXX / Vol. XX No. X • BioScience 7


on individual genes or molecular components. Certain genes reach the master title when their strong effect on the phenotype is described experimentally in developmental functional genetic studies (see, e.g., Williams and Fletcher 2005). Dynamical GRN models enable a system-level mechanistic interpretation to their apparent “more relevant” role in development—relative to the other genes without the title— when their altered expression causes the GRN to attain altered stable configurations or attractors. An important contribution of nonlinearity in this respect is the realization, through dynamical modeling, of the fact that the quality that gives their higher rank to master genes is indeed a systemic property emerging from the joint effect of the gene in question and its interacting partners and not from itself: Out of justice to the collaborating genes, perhaps the master title should refer to a core regulatory module rather than to a gene alone. For example, in the floral-organ specification GRN model (Mendoza and Alvarez-Buylla 1998, Espinosa et al. 2004, Álvarez-Buylla et al. 2010a), a mechanistic explanation for the important role of the ABC master homeotic genes was provided. Their interactions with several other genes in the context of a dynamic GRN model guarantee that their loss of function mutations yield altered attractors that qualitatively correspond to those associated with the homeotic organs observed in the mutant flowers.

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Figure 5. A schematic representation of the epigenetic landscape associated with the GRN for flower development and uncovered by a stochastic exploration. Cell gene-expression configurations corresponding to the attractors are represented in the bottom of each basin (in this case, associated with the cell states characteristic of four inflorescence cell types and four floral organ primordial cell types: sepals, petals, stamens, and carpels). Noise (randomness represented by dice) or stochastic modeling is used in order to uncover the structure of the landscape, the latter being manifested by the differential likelihood of cell-state transitions (here represented with arrows). nodes. Such alterations can result from perturbations in signal transduction pathways or to the physicochemical fields connected to the GRN under analysis. Models to study how such signaling pathways process information and filter noise (Díaz and Álvarez-Buylla 2009) have been proposed for 8 BioScience • XXXX XXXX / Vol. XX No. X

ethylene in the root (figure 2) and can now be integrated with GRN models and EL reconfiguration analyses. An additional application of EL formalisms to understanding Arabidopsis developmental mechanisms concerns the GRN that underlies the transition from vegetative http://bioscience.oxfordjournals.org

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Spatiotemporal cellular patterns in plant tissues: Feedback of GRNs and extracellular sources of constraint The GRN models reviewed above only consider a singlecell developmental process, able to recover different gene expression profiles; a GRN coupled with cell–cell movement of some of its components in static spatial domains, able to attain arrangements of gene-expression configurations that mimic actual cellular patterns in plant tissues; or populations of cells, able to recover temporal patterns of transition among cell states by updating their states independently of each other (see figures 1 and 5). During multicellular developmental processes, in vivo groups or populations of cells attain distinct fates with certain spatial and temporal dynamics that occur concomitantly. Such spatiotemporal dynamics and patterns emerge from dynamic feedback interactions with physicochemical fields and cell proliferation dynamics. Such processes imply multilevel modeling http://bioscience.oxfordjournals.org

frameworks that consider various levels of organization and morphogenetic patterning. Key to these multilevel models is to postulate processes that generate the cells’ positional information at all times and which produce changes in the operation of the invariant underlying GRN accordingly. Such models can be used to address issues concerning the regulation of the size and dimension of tissues, as well as the relative position of organs (Alvarez-Buylla et al. 2007, Swat et al. 2015). Concordantly, we have started to put forward dynamic spatiotemporal models that consider GRNs in cellularized domains and that encompass the aforementioned sources of extrinsic constraint from fields of mechanic-elastic forces or hormones (Alvarez-Buylla et al. 2007, Barrio et al. 2010, 2013). In addition to understanding how mechanics, geometry, and growth contribute to the formation of functional and robust structures (Mirabet et al. 2011), we must consider that these sources of additional constraints not only influence each other but are also coupled with at least two other fundamental dynamics coming from regulatory networks and cell proliferation. In addition, these dynamical processes occur at different temporal and spatial scales. For example, chemical signals that are produced or excreted from cells to the extracellular matrix arrange themselves in space and time to form macroscopic patterns, which, in turn, affect GRNs in each cell, thereby biasing its dynamics toward different gene-expression configurations. In other words, in order to accomplish the extraordinary choreography implied by morphogenesis (without a choreographer!), the behavior of the chemicals or mechanic-elastic forces and communication mechanisms should be coupled with the dynamics of the GRN in such a way that the positional cues bias the attractor of the GRN, and, at the same time, the modified gene activity configuration of the GRN regulates the spatial pattern of chemical concentrations (Álvarez-Buylla et al. 2007). We refer to the models that capture such dynamics as models of cooperative nonlinear dynamics (Barrio et al. 2013). The general problem of cooperative nonlinear dynamics could be stated as follows: The GRN in each cell in the meristem is in a state of undifferentiated complacency but may be producing certain transcripts and signals that regulate other genes. The signals and molecules inside the cell, or in the intercellular space, may be altered via certain signal transduction pathways that respond to some physicochemical field that alters their regulation and molecular concentrations in different regions of space. Thus, a geometrical pattern is formed in space, which, in turn, provides each genetic network with a chemical environment that also depends on such spatial pattern. We developed a multilevel model of cooperative dynamics for the case of the GRN implied in floral-organ specification. The model suggested that a dynamically changing extracellular heterogeneous chemical concentration may be underlying the accommodation of the ABC genes in the observed spatial pattern (Álvarez-Buylla et al. 2007, Barrio et al. 2010). XXXX XXXX / Vol. XX No. X • BioScience 9


(VM) to inflorescence meristem (IM) and then into flower meristems (FM) at the shoot apical meristem (SAM; figure 2; Pérez -Ruiz et al. 2015). In this study, experiments on the role of a MADS-box gene, AGAMOUS LIKE14 (AGL12)/XANTAAL2(XAL2), yielded apparently contradictory data. The overexpression lines of this gene produced early flowering and at the same time caused a loss of determinacy in flowers, especially under short days. We put forward a multistable GRN module that incorporates a set of XAL2 interactions able to recover the observed gene expression profiles in different SAM stages for the main regulators discovered up to now to be involved in such developmental processes. We used this model to reconcile and to provide an explanation for the two apparently contradictory phenotypes of the 35S::XAL2 lines. An EL model for the proposed XAL2 GRN module showed that overexpression of this gene, in the context of the GRN module proposed, yielded a novel attractor that combined IM and FM characteristic genes (figure 2). Such a combined attractor is reminiscent of the mixed behavior (determinate and indeterminate) of the flower cells observed in overexpression lines. Furthermore, our EL analyses suggested that transitions from FM to this attractor could explain the loss of determinacy observed in the 35S::XAL2 lines and, at the same time, accelerate the transition from VM to IM and then to FM (figure 2). This model provides a clear hypothesis for the dynamic developmental mechanism that may underlie the phenotypes of this MADS overexpression and of several others. The model suggests that in this and likely other cases, we are not faced with a reprogramming of floral to inflorescence cells but rather with the emergence of a new cell type with mixed identities and behaviors, as well as an increased probability of transiting into it. This application illustrates that EL models and analyses can be used as well to further understand the dynamic mechanisms underlying cell reprogramming and dedifferentiation in plants and animals.

Overview Articles Mechanical forces provide cues for heterogeneous cellular behaviors by establishing sources of positional information, thereby contributing to the regulation of morphogenesis (Wolpert 1969, Meinhardt 1982, Alvarez-Buylla et al. 2007, Barrio et al. 2008, Hamant et al. 2010, Barrio et al. 2013). Recent work has started to uncover the molecular mechanisms by which mechanic-elastic forces are sensed by intracellular GRNs (Ning et al. 2009), as well as the role of myosin, actin, and tubulin fibers in cell structuring and in the transduction of changes in mechanical and elastic forces into GRN signals (Hamant et al. 2010, Mammoto et al. 2012, Romero et al. 2015).

Evolutionary implications of complex G–P feedback mapping: Eco–evo–devo and the constructive role of the environment Historically, population-level models in evolution have been developed under certain simplifying assumptions. Two salient assumptions are (1) the idea of genetic change as a direct indicator of phenotypic variation and (2) the additivity of genetic effects on the phenotype (Wilkins AS. 2008). A more faithful model of biological evolution should explicitly consider a G–P map and back, a developmental mechanism which specifies how phenotypic variation is generated (Alberch 1991) in an analogous way to that in which positional information emerges as a result of the feedback between internal and external restrictions (Wolpert 1969, Álvarez-Buylla et al. 2011). A nonlinear perspective is mandatory to understand how phenotypic variation is generated given a genetic background—or, in other words, to study the dynamic system-level mechanistic basis of the G–P map. 10 BioScience • XXXX XXXX / Vol. XX No. X



Organogenesis: The cooperative dynamics of physicochemical fields and cell proliferation in the root In line with our view on cooperative dynamics, we developed a two-dimensional simulation model that integrates the dynamics of physical and chemical fields coupled with cell proliferation dynamics, using Arabidopsis root as a study system of a SCN (figures 1 and 2; Barrio et al. 2013). In order to test our theoretical propositions, we quantified the characteristic cellular patterns in the root. Our model was able to reproduce cellular patterns (size and proliferation rates) observed in real roots. It provides a proof of principle that coupled physical fields and chemical processes, under active cell proliferation, give rise to patterns observed in multicellular organs and SCNs (Barrio et al. 2013, Romero et al. 2015). This mechanism of cooperation provides a general model for one possible way in which the dynamics of physicochemical fields and active cell proliferation give rise to positional information (Barrio et al. 2013). The proposed framework may be extended and modified to model other SCNs and also to propose GRNs coupled with the three types of dynamics incorporated in the root SCN model. Other multilevel modeling platforms and approaches have been recently put forward (Boudon et al. 2015).

Given the nongenetic character of developmental dynamics, phenotypic variation to a great extent has been neglected in the study of evolution. A deviation from a linear causation view of development would potentially affect the rate and direction of evolution, however (see, e.g., Alvarez-Buylla et al. 2007b, Jaeger et al. 2012). Empirical evidence of the evolutionary relevance of network structure and gene interaction—and therefore nonlinearity—on evolutionary dynamics at the molecular level is starting to emerge (Balleza et al. 2008). Evolutionary forces, functional constraints, and molecular interactions are conditionally dependent on the systems level and ensure that small changes within the GRN can yield large phenotypical alterations (Kauffman 1993, Purugganan 2004, Álvarez-Buylla et al. 2010a, DavilaVelderrain et al. 2014 and references therein). In addition, the origin of novel core GRNs may underly the emergence of evolutionary innovations (Wagner 2015). Following the above line of thought, in a recent study, we tested whether there is evidence supporting the idea that functional constraints associated with a GRN module underlying the dynamics of a developmental process could play a strong role in constraining evolutionary rates at the molecular level (Davila-Velderrain et al. 2014). We argue that the study of the molecular evolution of the genes involved in regulatory networks that have been uncovered with dynamical GRN models could help uncover general evolutionary principles, given that such models allow a rigorous distinction between structure and function (genotype and phenotype). We used as case of study the GRN of flower development described above. We reasoned that the functional constraint associated with a core regulatory module could also play a strong role in constraining evolutionary rates at the molecular level, and recently, we showed evidence suggesting that this is the case (for details, see DavilaVelderrain et al. 2014). Overall, the empirical evidence of the evolutionary relevance of network structure on gene evolutionary patterns is in agreement with our argument for the nonindividual role of master genes mentioned above and for the relevance of the nonlinear character of the regulatory systems in which they participate. Furthermore, biological networks of both uni- and multicellular organisms seem to have evolved to be able to cope with environmental impacts (i.e., to be robust) and, at the same time, to change and adapt to new conditions (i.e., to be plastic), a qualitative property consistent with what is referred to as criticality in nonlinear dynamics (Shmulevich et al. 2005, Balleza et al. 2008). Available studies suggest that critical dynamics is a generic characteristic of biological networks, which seems to imply evolutionary advantages for living organisms that are required to be robust and adaptable at the same time (Balleza et al. 2008). This trait of underlying epigenetic mechanisms of development implies important constraints to the evolutionary patterns of phenotypic variation (i.e., the resulting balance between phenotypic conservation and diversity) and, at the same time, postulates a dynamic mechanism for the large phenotypical

Overview Articles

Conclusions Even though genetic-based approaches have been favored, it is recently being accepted that the richness and robustness of biological forms are not encoded in the genes but rather emerge from their interactions and nongenetic components. Recent research is consistently showing that the physical forces and constraints imposed by biological structures are fundamental for understanding development, in agreement with previous studies (Selker et al. 1992). In summary, there are nontrivial interactions at all the levels of organization that involve genetic, cellular, physical, chemical, and geometrical interactions and constraints that need to be integrated to achieve a global, dynamic, system-level, and mechanistic understanding of development. Such an approach implies an understanding of how the cellular and morphogenetic patterns emerge at different levels of organization (see figure 1). A fundamental challenge in systems biology is developing the mathematical and computational approaches to explore how the different levels of complexity are coupled. As we reviewed in this article, for our own research in Arabidopsis thaliana, other recent studies in systems biology are extending GRN models to incorporate tissue-level patterning mechanisms such as cell–cell interactions, chemical signaling, cellular growth, proliferation, and senescence. These additional nonlinear sources of constraint inevitably impose physical limitations in terms of mechanical and elastic forces, which in turn affect cellular behavior and feedback to physical and chemical fields. Nonhomogenous GRN models that take into consideration heterogeneous http://bioscience.oxfordjournals.org

chemical and physical extrinsic conditions are the focus of ongoing research. The possibility of actually accounting experimentally for these physical processes—in an attempt to understand how cellular decisions occur during tissue patterning—will go beyond cell culture studies. Multilevel approaches, grounded in experimental data, will yield more accurate models of morphogenesis during normal development and also under altered conditions, such as those experienced in cancerous growths in humans. Plant systems are promising experimental models to propose and test such models and will continue to illuminate the biogeneric mechanisms at play during both plant and animal development. The merging of conceptually clear theories, computational-mathematical tools, and molecular–genomic data into coherent frameworks is at the basis of a much-needed nonlinear, dynamic, system-level explanatory and predictive approach to development and to evolution. Once again, plants are becoming critical systems for such an integrated and broad comparative approach. Acknowledgments ERAB’s work was supported by Consejo Nacional de Ciencia y Tecnología (CONACYT), Mexico (grant nos. 240180, 180380, and 152649), and by Universidad Nacional Autónoma de México–Dirección General Asuntos del Personal Académico– Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica (UNAM–DGAPA– PAPIIT grant nos. 203214, IN203814, and IN211516). We appreciate the help of Diana Romo in various tasks. We thank Estephania Sluhan for drawing figure 2. We thank the editor and three anonymous reviewers for their comments, which greatly helped improve the text. ERAB and JDV acknowledge the Centro de Ciencias de la Complejidad (C3) at UNAM. References cited

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