Wnt signaling is fundamentally important

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1 Toward a quantitative understanding of the Wnt/β-catenin pathway through simulation and experiment Bethan Lloyd-Lewis, 1,2, Alexander G. Fletcher 3,TrevorC.Dale 2, and Helen M. Byrne 3,4, Wnt signaling regulates cell survival, proliferation, and differentiation throughout development and is aberrantly regulated in cancer. The pathway is activated when Wnt ligands bind to specific receptors on the cell surface, resulting in the stabilization and nuclear accumulation of the transcriptional co-activator β- catenin. Mathematical and computational models have been used to study the spatial and temporal regulation of the Wnt/β-catenin pathway and to investigate the functional impact of mutations in key components. Such models range in complexity, from time-dependent, ordinary differential equations that describe the biochemical interactions between key pathway components within a single cell, to complex, multiscale models that incorporate the role of the Wnt/β-catenin pathway target genes in tissue homeostasis and carcinogenesis. This review aims to summarize recent progress in mathematical modeling of the Wnt pathway and to highlight new biological results that could form the basis for future theoretical investigations designed to increase the utility of theoretical models of Wnt signaling in the biomedical arena Wiley Periodicals, Inc. How to cite this article: WIREs Syst Biol Med 2013, 5: doi: /wsbm.1221 INTRODUCTION Wnt signaling is fundamentally important during the development and maintenance of multicellular organisms. It regulates diverse processes ranging from cell proliferation and stem cell maintenance to cell fate, migration, and polarity. Its aberrant regulation is implicated in a range of developmental disorders, degenerative diseases, and Correspondence to: DaleTC@cardiff.ac.uk; helen.byrne@maths. ox.ac.uk 1 Department of Pathology, University of Cambridge, Cambridge, UK 2 School of Biosciences, Cardiff University, Cardiff, UK 3 Mathematical Institute, University of Oxford, Oxford, UK 4 Department of Computer Science, University of Oxford, Oxford, UK These authors contributed equally to this work. Last authorship is shared between these authors. Conflict of interest: The authors have declared no conflicts of interest for this article. cancers, 1 and it is thus a key potential target for therapeutic intervention. To date, 19 Wnt proteins have been identified and shown to activate divergent intracellular Wnt signaling pathways, such as the planar cell polarity, the Wnt/calcium, and the more widely characterized canonical β-catenin/tcf (Tcell factor) pathway (Figure 1(a)), the focus of this review. For a comprehensive description of canonical and non-canonical pathways, see Refs 2 and 3. This article opens with a brief description of the Wnt signaling pathway and a summary of how theoretical models may be utilized to study biochemical transduction processes. Approaches that have been used to model the Wnt pathway at the biochemical, subcellular, cellular, and tissue scales are described, allowing a focus on state of the art multiscale models that couple cellular and tissue behavior with subcellular processes. We conclude by outlining promising directions for further investigation. Volume 5, July/August Wiley Periodicals, Inc. 391

2 wires.wiley.com/sysbio FIGURE 1 (a) Wnt signaling pathways. Different combinations of Wnt ligands and receptor at the cell surface can initiate specific Wnt pathways. The canonical β-catenin/tcf (T-cell factor) pathway is the best described (C), with the Ca2+ pathway (A) and planar cell polarity (PCP) pathway (B) also downstream of Frizzled receptors. Ryk and Ror are tyrosine kinase receptors also implicated in mediating Wnt signaling (pathways D and E). (b) Wnt/β-catenin signaling pathway. OFF state: ZNRF3 acts as a ubiquitin ligase to promote Frizzled and low-density lipoprotein receptor protein 6 (LRP6) turnover. β-catenin is held in the destruction complex where it is phosphorylated and targeted for degradation by the proteasome in the absence of Wnt ligand. TCF transcription factors are complexed with transcriptional inhibitors such as Transducer like Enhancer of split/groucho, and target genes are not transcribed. ON state: Rspo promotes ZNRF3 turnover by binding to ZNRF3 and LGR4, leading to stabilization of LRP6 and Frizzled and potentiation of Wnt signaling. Wnt binding causes disassociation of the complex in a Dvl-dependent manner, allowing the nuclear translocation of β-catenin and displacement of Groucho. Additional transcriptional co-activators and histone modifiers such as Brg1, CBP, Bcl9, and Pygopus are recruited to activate transcription of target genes. INTRODUCTION TO Wnt SIGNALING Binding of specific Wnt ligands to extracellular receptors, such as the Frizzled receptors, LRP5/6 (low-density lipoprotein receptor-related protein 5/6) co-receptors, 5 or the tyrosine kinases Ryk 6 and Ror2 7 stimulate distinct pathways, depending on the combinations of Wnt ligands and receptors. Initially each Wnt pathway was regarded as an isolated on/off switch, regulated by independent, linear cascades of proteins. However, extensive links between β-catenin/tcf and non-canonical pathways have since been identified (Figure 1(a)), including the ability of the Wnt5A/Ror2 pathway to regulate TCFdependent transcription. 8,9 Wnt components are also implicated in non-wnt pathways; for example, Axin regulates the function of the tumor suppressor p Canonical Wnt signaling is initiated downstream of the Frizzled-LRP receptor complex, leading to β-catenin stabilization and activation of target genes involved in differentiation and proliferation. 11,12 In the absence of a Wnt ligand, Wiley Periodicals, Inc. Volume 5, July/August 2013

3 WIREs Systems Biology and Medicine Wnt/β-catenin pathway through simulation and experiment FIGURE 1 Continued Volume 5, July/August Wiley Periodicals, Inc. 393

4 wires.wiley.com/sysbio β-catenin binds to a destruction complex of proteins, including Axin, adenomatous polyposis coli (APC), casein kinase 1α (CK1α), and glycogen synthase kinase 3β (GSK3β) Within this complex, β-catenin is sequentially phosphorylated by CK1α and GSK3β, 15 leading to its ubiquitination and subsequent degradation by the proteasome. 16 Meanwhile, in the nucleus, transcription factors of the TCF or lymphoid enhancing factors (LEF) family bind to transcriptional repressors, preventing the unnecessary activation of target genes 17 (Figure 1(b)). Binding of Wnt ligands to a Frizzled/LRP5 receptor complex 18 recruits the Dishevelled (Dvl) and Axin proteins to the cytoplasmic domains of the LRP receptors. 19 The resulting inhibition of the destruction complex causes β-catenin accumulation and translocation to the nucleus. There it binds to TCF/LEF complexes and displaces associatedrepressor proteins, leading to the transcription of Wnt/TCF-dependent target genes. 20 INTRODUCTION TO MATHEMATICAL AND COMPUTATIONAL MODELING Mathematical and computational models can be used to develop abstract representations of biological systems, test competing hypotheses, and generate new predictions that can then be validated experimentally. Modeling has yielded insight into subcellular processes such as cell cycle control, circadian rhythms, and cell death. 21 The approach used is typically determined by the question of interest and the quality of the data available. Most models of signaling pathways and gene regulatory networks involve either ordinary differential equations (ODEs) or Boolean networks. 22 ODE models yield dynamic and quantitative information about the concentrations of key network components (mrnas, proteins, and complexes). They are assembled by first establishing the effect of each network reaction on each network component: reactions that consume, transform, or degrade a species reduce its concentration, whereas those by which it is synthesized increase it. The time rate of change of each species is then determined by subtracting the sum of its rates of degradation from the sum of its rates of synthesis. In a Boolean model, each network component takes the value 0 (off) or 1 (on), which is updated over discrete iterations using Boolean logic rules, which are derived from the known network interactions. Both approaches have benefits and drawbacks. For example, Boolean models fail to provide quantitative information about the time evolution of a system; but because each variable exists in one of two states, and they contain no parameters, only rules about node node connectivity, it is easier to perform a comprehensive analysis of the possible qualitative behavior of such models and to establish which rules lead to qualitative agreement with time-series data. Several modifications to ODE approaches have been proposed. Time delays can be incorporated to account for the synthesis of gene products via transcript accumulation and translation. 23 The assumption that the cell is a well-mixed system can be relaxed by developing compartmental models. These models may be formulated using, for example, ODEs or Boolean rules, and used to study the effect of localizing reactions to specific cellular regions. For example, in the Wnt signaling pathway, phosphorylation of β-catenin to the destruction complex is localized in the cytoplasm, while transcription of target genes will occur in the nucleus. Partial differential equations (PDEs) can also be used to resolve spatial variation within a cell. 24 Finally, when molecules are present in small numbers, a deterministic, ODE approach may cease to be valid and a stochastic (or hybrid, ODE-stochastic) model should be used. 25 For a comprehensive review of the various formalisms used to model gene regulatory networks, see, for example, Ref 22. By embedding subcellular models into models that distinguish individual cells it is possible to link signaling pathways to cell stimuli (e.g., the binding of Wnt to Frizzled on the cell membrane) and cell decisions (e.g., the decision to divide). The construction of suites of interrelated models enables us systematically to increase our understanding of the behavior of complex biological systems. In the next section we illustrate how this approach has been applied to the Wnt signaling pathway. Mathematical Models of Wnt Signaling Mathematical models of Wnt signaling may be classified according to the scale at which they are built (Figure 2). At the biochemical level, there are detailed models of the core canonical Wnt signaling pathway, and its crosstalk with other pathways. Models at the subcellular level account for cellular compartmentalization. Finally, cellular and tissue level models describe how Wnt signaling regulates cellular decision-making and the development and maintenance of homeostasis within multicellular systems. Biochemical Models of Wnt Signaling The first quantitative model of the canonical pathway was proposed by Lee et al. 33 A system Wiley Periodicals, Inc. Volume 5, July/August 2013

5 WIREs Systems Biology and Medicine Wnt/β-catenin pathway through simulation and experiment FIGURE 2 Modeling state of the art. Mathematical models of Wnt signaling may be classified according to the scale at which they are built: biochemical, subcellular, cellular, and tissue. Different aspects of Wnt signaling biology is highlighted within each of the different scales, with circular boxes indicating biological complexity that has yet to be modeled in the mathematical studies indicated above. of coupled ODEs describes how the subcellular concentrations of key proteins such as β-catenin, Axin, and APC change over time in response to Wnt stimulation and other perturbations. The model accounts for protein synthesis and degradation, protein protein interactions, and phosphorylation and de-phosphorylation events. The Wnt pathway components that are included in this model are collated in Table 1. Where possible, parameter values were obtained from Xenopus oocyte extracts, experimental measurements being combined with modeling assumptions to generate the estimates. For example, total concentrations of Dvl, Axin, β-catenin, and other proteins were determined by comparing the signal intensities of Western blot analyses to signals of known amounts of recombinant protein. Binding constants were determined using biophysical studies of purified interacting proteins or protein domains. Many parameters were (and remain) unknown and values were chosen so that the rates of β-catenin degradation matched experimental data. Furthermore, most parameters were found to be insensitive to variation, suggesting that the model architecture produced robust dynamic responses. The validated model was used to generate new predictions, which were verified experimentally, regarding the differential regulation of β-catenin levels by the scaffold proteins, Axin, and APC. While both proteins are present in the destruction complex and, thereby, contribute to preventing levels of β-catenin from becoming too high, APC s additional role in promoting the degradation of Axin was found to ensure that the system was robust to variation in APC levels: when APC is scarce, Axin accumulates and plays a greater role in inhibiting β- catenin; by contrast, when APC is abundant it is the dominant regulator. Kruger and Heinrich 29 performed a systematic analysis of Lee et al. s model: differences in the timescales associated with the processes involved in the Wnt pathway and the existence of conservation laws were exploited to reduce the number of ODEs from 15 to 7. The authors also determined how the equilibrium concentrations of the signaling proteins changed when key system parameters were varied. Their sensitivity analysis revealed that, with the exception of the β-catenin/tcf complex, the Volume 5, July/August Wiley Periodicals, Inc. 395

6 wires.wiley.com/sysbio TABLE 1 List of Components of the Wnt Signaling Pathway that Have Been Incorporated into Mathematical Models to Date Wnt Pathway Component First Modeled Extracellular free Wnt Lee et al. 33 Free Axin Total free APC Axin/APC complex Free GSK3β Axin/APC/GSK3β destruction complex Axin 1 /APC 1 /GSK3β complex Total free β-catenin Axin 1 /APC 1 /GSK3β/β-catenin complex Axin 1 /APC 1 /GSK3β/β-catenin 1 complex Total free β-catenin 1 β-catenin/tcf complex Free TCF β-catenin/apc complex Inactive form of Dsh Active form of Dsh Axin RNA Wawra et al. 28 DKK1 RNA Free adhesion molecules van Leeuwen et al. 37 (α-catenin/e-cadherin) β-catenin/ adhesion molecule complex Wnt target protein β-catenin marked for ubiquitination Free cytoplasmic β-catenin Schmitz et al. 36 Free nuclear β-catenin Free cytoplasmic APC Free nuclear APC Cytoplasmic β-catenin/apc retention complex Nuclear β-catenin/apc retention complex Free Frizzled receptor Kogan et al. 31 Frizzled/Wnt complex Free LRP receptor Frizzled/Wnt/LRP ternary receptor complex Frizzled/Wnt/LRP bound to destruction complex Extracellular free secreted frizzle-related proteins Extracellular sfrp/wnt complex Extracellular free DKK DKK/LRP complex Unless stated otherwise, components indicate concentrations of proteins or protein complexes. 1 Phosphorylated forms. equilibrium values of all model variables were robust to variation in parameter values associated with Wnt stimulation. However, in the absence of Wnt, parameter fluctuations were found to be more likely to lead to inappropriate pathway activation. A more detailed asymptotic analysis of Lee et al. s model was performed by Mirams et al., 26 in order to identify which pathway components are dominant on the different timescales associated with Wnt signaling. These timescales range from that for the rapid phosphorylation of β-catenin to that for the slower degradation of β-catenin. Their analysis suggested that, on timescales associated with the half-life of β- catenin, Lee et al. s original model may be reduced to a single ODE for active β-catenin (ABC). However, their estimate of around 45 h for this timescale conflicts with more recent experimental measurements by Hernandez et al., 30 which indicate a half-life in the absence of a Wnt stimulus of around 16 minutes. A key aspect of several Wnt regulators is that they are positive or negative transcriptional targets of the canonical pathway [for example, Axin2, 46 Dickkopf, 47 zinc and ring finger 3 (ZNRF3), 48 and Arrow/LRP5 49 ] and, as such, could modulate absolute levels of Wnt pathway activity. 8 Extensions to Lee et al. s model have revealed that the inclusion of feedback due to Wnt-induced Axin2 and Dickkopf inhibitors can generate robust oscillations in certain pathway components. 28,35 More recently, a model by Kogan et al. 31 has predicted that the Wnt inhibitors secreted Frizzled-related protein and Dickkopf may have a synergistic effect when applied in combination as an anti cancer therapy. The additional Wnt pathway components that have been incorporated into these extended models are collated in Table 1. A common feature of the models described above is that absolute levels of β-catenin/tcf determine the pathway output (Figure 1(b)). Motivated by a reanalysis of Lee et al. s model, 33 Goentoro and Kirschner found that fold changes in β-catenin (pre/post Wnt ligand treatment) were more robust to perturbations in system parameters than changes in absolute levels of β-catenin. 27 These model predictions were validated by identifying functional responses of Xenopus embryos to pharmacological and genetic interventions that increased system-wide levels of β-catenin without triggering concentration-specific cellular responses unless they were coupled to fold changes following Wnt ligand treatment. The nature of the cellular sensor that carries out the time-dependent (before/after Wnt ligand) concentration measurement is currently unknown. A possible role for fold-change detection is to compensate for natural biological (both Wiley Periodicals, Inc. Volume 5, July/August 2013

7 WIREs Systems Biology and Medicine Wnt/β-catenin pathway through simulation and experiment environmental and genetic) noise so that, despite large variations in basal nuclear levels of β-catenin, the fold change is robust across all cells. 27 Extending this argument, we speculate that control of vital digital cellular decisions, such as entry to the cell cycle, should be affected by fold changes in β-catenin/tcf, while non-vital analog decisions, such as cell migration, may be regulated by absolute levels of β-catenin/tcf; these predictions remain to be verified. The data generated by Goentoro and Kirschner also suggest that Wnt-induced fold changes in β-catenin are buffered to perturbations in the components regulating degradation, but are sensitive to changes in the rate of β-catenin synthesis. 27 Interestingly, changes in the rates of β-catenin transcription and translation are emerging as routes for regulating Wnt/β-catenin signaling, but these have yet to be modeled. 50,51 A related area that merits further theoretical investigation is the impact of stochastic variation within, and between, individual cells. For example, Furusawa and Kaneko 52 suggest that noise in the signaling pathways of stem cells may enable them to attain the different stable states associated with differentiated cell types. Preliminary insight into the impact that such effects could have was described in Ref 53, where a stochastic model of Wnt signaling based on the work of Lee et al. 33 was shown to drive stochastic oscillations in β-catenin levels. The Wnt-dependent mechanism(s) that lead to β-catenin stabilization is a contentious subject. 54 Initially it was suggested that Wnt receptor complex formation allowed a Dvl/Frat protein complex to divert GSK3 from Axin within the β-catenin turnover complex. However, this mechanism is incompatible with results showing that mice lacking all three Frat genes are viable. 55 Subsequently, it has been suggested that the Wnt-induced phospho-pppspxs motifs in the intracellular tail of the LRP6 co-receptor directly inhibit GSK3β s activity against β-catenin by acting as a pseudo GSK3 substrate. 56,57 Additional proposed mechanisms include the restriction of β- catenin access to the degradation complex via loss of Axin or complex disaggregation and restriction of complex access to β-catenin by sequestration of the complex within receptor complexes or multivesicular bodies. 58 A recent study by Li et al. suggested that Wnt-dependent inhibition of phospho-β-catenin ubiquitination led to its saturation of the turnover complex. 59 However, a subsequent detailed study of phospho-β-catenin dynamics suggested that inhibition of β-catenin phosphorylation, rather than ubiquitination, was the key point of Wnt action in a number of cell lines. 30 High-quality time-dependent data in studies such as Hernandez et al. 30 and Tan et al. 32 allowed the construction and validation of mathematical models that are based on mammalian systems rather than Xenopus embryo extracts. 33 By suitably extending existing biochemical models of the Wnt pathway and validating them against relevant data, it should be possible to distinguish between the above hypotheses. Nonetheless, quantitative measurements of biochemical variables remain a bottleneck for parameter estimation in mathematical models: improved, high-throughput quantitative biochemical techniques will need to be developed fully to realize the potential of mathematical modeling. In addition to the core components (Figure 1(b)), hundreds of other proteins have been identified as Wnt regulators through studies that demonstrate their association with Wnt signaling components and their effects on TCF-dependent transcription/β-catenin stabilization Many non-core Wnt signaling components are expressed in tissue- or context-specific patterns and have been linked to physiological or pathological roles. These novel forms of regulation offer a wide range of modeling opportunities. One particular area that is still in its infancy is the modeling of interactions between Wnt and other signaling pathways. Wnt ligands and β-catenin/tcf signaling were observed to activate the extracellular signalregulated kinase (ERK) pathway components Ras and Raf-1, 63,64 and have been incorporated into an ODE model of Wnt-ERK crosstalk. 34 Both pathways have been shown to contribute to colorectal cancer progression. Kim et al. 34 performed a qualitative comparison of model predictions to experimental results and suggested that positive feedback between the two pathways stimulates ERK activity and, hence, upregulates β-catenin/tcf in a switch-like manner to increase the fidelity of the signal. While the model correctly predicts the consequences of Wnt/ERK crosstalk, it incorporates parameters that have yet to be experimentally validated, including assumptions regarding the significance of the ERK-induced phosphorylation of S9 on GSK3β, a site that can be removed in transgenic models with no detrimental developmental effects. 65 More generally, experimental studies of cross-pathway interactions have been undermined by the commercial supply of purified Wnt preparations that are frequently contaminated by non-wnt factors which activate parallel pathways, requiring the need for careful controls. 66 Subcellular Models of Wnt Signaling: Cellular Compartmentalization The models discussed above treat the cell as a wellmixed entity. In practice, cells comprise distinct Volume 5, July/August Wiley Periodicals, Inc. 397

8 wires.wiley.com/sysbio regions with proteins migrating between them and specific reactions localized to a particular domain. We now highlight examples that illustrate the spatial complexity of Wnt regulation and refer readers to recent reviews of Wnt signaling at the receptor, 67,68 β-catenin turnover complex, 2 and nuclear complex levels 69 for further information. The formation of a ligand-activated receptor complex is a key step in Wnt signaling, with many core Wnt receptor components at the plasma membrane (Figure 1(b)). For example, binding of Wnt ligands to cell surface LRP5/6 and Frizzled receptors results in signalosome formation, in which Dvl and Axin1 polymerize within receptor complexes that are visible as puncta by fluorescence microscopy. 70,71 The presence of macroscopically large complexes (>200 nm diameter) may conflict with the well-mixed assumptions made in most mathematical models: large complexes may only be able to disassemble from their surfaces and may not, as a result, be distributed uniformly throughout the cytoplasm. Such situations may be better described using a spatially resolved model in which the diffusion coefficients of the proteins and protein complexes depend on their molecular weights. The sequestering of active receptor complexes within endosomes and multivesicular bodies can also modulate receptor/ligand-dependent signals. 72,73 In addition, a key compartment-specific function for β- catenin takes place at the plasma membrane where it regulates cell adhesion as part of the cadherin/catenin complex. 74 This role is discussed below in the context of cellular interactions. Wnt-induced β-catenin stabilization results in its translocation to the nucleus, where it displaces transcriptional co-repressors and engages with DNAbound TCF transcription factors to activate the expression of Wnt target genes. 75 The mechanism of β-catenin nuclear translocation is poorly understood, β-catenin having been suggested to mediate its own import/export by its Armadillo (ARM) repeat domains independently of Ran GTPase importin/exportin pathways, 76 and of being retained in different cellular compartments via interactions with partners such as APC 77 and LEF Schmitz et al. 36 developed an ODE model to study the influence on Wnt signaling of nucleo-cytoplasmic shuttling of APC and β-catenin retention by APC. The additional Wnt pathway components present in their model are listed in Table 1. Model analysis revealed that APC s repressive role could be ameliorated by its shuttling of β-catenin from the nucleus, the latter enhancing the production of target genes in response to Wnt stimulation. Although this process has not yet been modeled, Axin has also been found in the nucleus and was suggested to regulate β-catenin localization. 79 Many components interact with β-catenin in the nucleus to regulate Wnt-dependent gene expression. β-catenin s N- and C-termini act as transcriptional activation domains that recruit chromatin remodeling complexes such as histone acetyltransferases and methyltransferases. 69 Lee et al. s model, and its extensions, considers a generic process of TCFdependent transcription initiation. However in vivo, tissue-specific transcription targets are specified by chromatin modification. Following signaling, and as a consequence of Wnt action, cell fate and responsiveness to Wnt ligands may be permanently altered owing to chromatin remodeling. When modeling cellular differentiation, the timescale over which chromatin modification occurs will need to be related to that of TCF-dependent transcription initiation. 80 Cellular Models of Wnt Signaling: Cell Cell Interactions Mathematical models that link Wnt signaling to changes in cell cell interactions frequently focus on the interaction between β-catenin and the adherens junction protein E-cadherin, an interaction that promotes cell cell adhesion. 81 Binding sites on β-catenin for TCF, E-cadherin and Axin are mutually exclusive, 82 with Wnt-dependent increases in β-catenin levels often predicted to enhance cell cell interactions by promoting the formation of E-cadherin:β-catenin complexes. While E-cadherin expression has been demonstrated to sequester β-catenin from transcriptionally active nuclear complexes, 83,84 it is clear that the loss/reduction of E-cadherin alone is insufficient to activate TCF-dependent transcription in the absence of Wnt ligands In addition to forming distinct protein complexes, β-catenin is differentially modified, in particular by phosphorylation. 88,89 The relationship between β-catenin levels, localization and activation of TCFdependent transcription is complex, with many reports demonstrating no correlation between absolute β-catenin levels and transcriptional activity. 90,91 Distinct molecular forms of β-catenin vary in stability and interact differentially with partners including TCF and cadherins, coupling β-catenin levels to different phenotypic outcomes. 82 Furthermore, the identity of the transcriptionally active form of β-catenin remains to be established. An N-terminal nuclear dephosphorylated isoform has been widely argued to be the primary transcriptionally active form of β-catenin (active β-catenin; ABC) as it is present in the nucleus Wiley Periodicals, Inc. Volume 5, July/August 2013

9 WIREs Systems Biology and Medicine Wnt/β-catenin pathway through simulation and experiment and has been shown to associate with TCF transcription factors. 90 Despite reports suggesting that this form is rare, 92 a recent study by Hernandez et al. argues that ABC represents as much as 80% of total β-catenin levels. 30 However, nuclear β-catenin levels are not a robust indicator of transcriptional activity and some studies are consistent with the spatially distinct phospho-serine 45 (PS45) being the key active isoform. 92 Phosphorylation by tyrosine kinases, such as Fer and Fyn, at specific residues can disrupt β-catenin s interaction with E-cadherin 93 and α-catenins, 94 and promote its interactions with transcriptional co-activators in the nucleus. 95 To model the distinct β-catenin complexes and cellular compartments, van Leeuwen et al. 37 generated an ODE model incorporating six different pools ofβ-catenin within three compartments, the nucleus, cytoplasm, and membrane. Table 1 shows a list of the Wnt pathway components introduced in this model. The model predicted that measurements of the abundance of β-catenin pools and transcriptional outputs would not be able to distinguish between mechanisms involving a unique transcriptionally active isoform of β-catenin and a simpler model in which all isoforms of β-catenin have similar transcriptional potential. However, the model predicted that dynamic measurements of E-cadherin levels following Wnt stimulation could be used to discriminate between the two hypotheses. This counter-intuitive prediction remains to be tested. Van Leeuwen et al. also used their model to predict that TCF-dependent transcription levels should be differentially sensitive to APC copy number in a manner that depends on the levels of endogenous Wnt ligand, a result supported by recent in vivo studies involving an allelic series of APC hypomorphic mutants. 96 The phenotypic consequences of Wnt-dependent alterations to cell cell adhesion may be highly contextdependent. During the development of epithelial tissues, changes in adhesion may promote or restrict the intercellular reorganization necessary for cell movement and mitotic cell division (discussed later). However, these changes are distinct from the complete loss of junctional complexes that characterize epithelial mesenchymal transition (EMT), a process that is associated with a subset of physiological cell migrations and cancer cell metastasis. 97 Activated Wnt signaling is strongly associated with EMT in a number of tumor types, including breast cancer. 97,98 Ramis- Conde et al. 39 developed a model linking β-catenin levels to EMT. They assumed that the activated β-catenin concentrations below a threshold promoted cadherin-mediated cell cell adhesion, while those above the threshold induced a cell fate change (via induction of TCF-target genes), reduced E-cadherinmediated adhesion, and increased cell migration. 99,100 The authors found that the intracellular dynamics of their model were sensitive to only two parameters, the constitutive rates of β-catenin production and proteasomal degradation. Importantly, the model recapitulated the heterogeneous patterns of nuclear β-catenin levels observed in tumors. 101 To model the repression of E-cadherin expression in EMT, Shin et al. 38 coupled Kim et al. s ODE model of Wnt and ERK pathway interactions to a model for the repression of E-cadherin via the induction of Snail and Slug transcriptional repressors. Their analysis suggested that positive feedback loops involving the Wnt and Erk pathways result in a switch-like behavior of E-cadherin expression that may be necessary for EMT. The role of β-catenin in contact inhibition has been analyzed in a spatially resolved compartmental model by Basan et al. 24 The authors used reactiondiffusion equations to describe the distributions of α-catenin and β-catenin (and various dimer combinations) within the cytoplasm, nucleus, and membrane. They suggested that de novo formation of E-cadherin complexes following cell contact should inhibit cell motility via an increase in α-catenin dimer levels. Their model qualitatively reproduced the impact of oncogenic transformations such as APC loss, β-catenin, and α-catenin mutations. While the above models recapitulate aspects of tumor development, they do not account for the behavior of many normal epithelia. For example, columns of actively migrating cells in the colonic epithelium were associated with the suppression of canonical β-catenin/tcf-dependent transcription via the expression of the non-canonical Wnt-5A ligands in nascent mesenchymal cells. 102 These Wnt-dependent cellular interactions and migration responses offer scope for future modeling opportunities. Tissue Models of Wnt Signaling: Multicellular Systems In this section, we focus on the effect of Wnt signaling outputs at the tissue level and the relationship between Wnt signaling and cell cycle control. Wnt Signaling in the Intestinal Crypt From a theoretical perspective, the best-studied tissue whose maintenance depends on Wnt signaling is the intestinal epithelium. It is formed from regularly spaced invaginations called crypts and (in the case of the small intestine) evaginations called villi. Functional stem cells toward the base of each crypt proliferate. 103 Volume 5, July/August Wiley Periodicals, Inc. 399

10 wires.wiley.com/sysbio As cells move up the crypt they cease proliferation and differentiate into specialized absorptive or secretory cells. Wnt signaling is known to play a central role in maintaining stem cells and regulating crypt dynamics. It has been proposed that the rates of cell proliferation, differentiation, and death are specified by a gradient of extracellular Wnt ligand up the crypt axis. 104 A variety of mathematical models have been used to study β-catenin function in intestinal crypts. The first tissue-level model to consider the role of Wnt signaling in the crypt was proposed by van Leeuwen et al. 44 and was based on an earlier model from the study by Meineke et al. 105 A fixed Wnt gradient imposed along the crypt axis was assumed to determine cell cycle progression, differentiation into the functioning cells that populate the crypt, and apoptosis, with high levels of Wnt at the crypt base driving cell proliferation and defining the stem cell niche. A threshold of Wnt signaling was assumed to be necessary for mitosis to occur. Van Leeuwen s model 44 represented the first attempt to link phenomena at the subcellular, cellular, and tissue levels of organization. It provided a framework for explicitly linking levels of β-catenin/tcf-dependent transcription to cell cell adhesion. In the van Leeuwen model, Wnt signaling controlled mechanical interactions between cells through the alteration of intercellular adhesion. However in the future, it may be possible to explore how mechanical forces in turn modulate cell adhesion, as stretching and compression alter β-catenin incorporation in junctional complexes. 106 In further work, Fletcher et al. 41 investigated the process of monoclonal conversion within the crypt, finding that the probability and timescale for this process to occur were sensitive to the precise geometry of the stem cell niche and to the Wnt stimulus threshold necessary for mitosis to occur, while Mirams et al. 40 used van Leeuwen et al. s crypt model theoretically to show that the probability of crypt domination by a mutant population was increased if Wnt signaling was raised near the base of the crypt. This emergent property incorporating wild type and mutant cells, different levels of Wnt signaling, proliferation, and adhesion offers a possible explanation for the experimental observation that basally located stem cells are preferential oncogenic targets. 107 An additional challenge for future models is to explain the observation that in the crypt, the highest levels of Wnt signaling are associated with slowerreplicating stem cells while lower levels of signaling drive rapid proliferation of transit amplifying cells located above the stem cells. 108 The stemness of intestinal stem cells is not exclusively Wnt ligand gradient-related however, and other signals, such as Notch, may act in combination with Wnt signaling in this context. Following an approach similar to that of van Leeuwen et al., 44 Buske et al. 45 developed a spatially resolved hybrid model to study the combined influence of Wnt and Notch signaling on cell proliferation and differentiation within a three-dimensional crypt model of the luminal epithelial monolayer. The authors computed cellular Notch activity by summing contributions from each cell s neighbors, allowing cell fate to be determined from a look-up table specifying distinct outcomes for particular Wnt and Notch activity levels. The authors also provided the first explicit representation of the basement membrane as a provider of mechanical support and as a promoter of an epithelial monolayer. This model recapitulated experimental observations on the spatial distributions of secretory Paneth and Goblet cells in wild-type crypts 109 ; conversion to monoclonality 110 ; and the effect of Wnt/Notch mutations on crypt organization. 111,112 In future, the presence of the basement membrane may allow the effects of cell matrix interactions and Wnt signaling to be accurately modeled. 113 A key feature for future modeling efforts will be the relationship between Wnt signaling and the mitotic cell cycle. Central to cell cycle regulation is that the temporal activation of several cyclin-dependent kinases (CDKs) is by cyclin regulatory subunits. Most cell cycle models are formulated as systems of coupled nonlinear ODEs that describe how levels of cyclins and CDKs change over time. 114 In these models, passage through cell-cycle checkpoints is assumed to occur when the levels of specific proteins rise above (or fall below) prescribed threshold values. Although recent studies have started to elucidate mechanisms linking Wnt signaling to events during the mitotic cell cycle 115 (Figure 3), no detailed mathematical model yet exists to describe this coupling. In many systems stimulation of the Wnt pathway drives cell proliferation. Surprisingly, observations of Wnt signaling suggest that β-catenin levels and TCF-dependent transcription oscillate during the cell cycle, being lowest during the G1/S-phase of the cell cycle and highest at G2/M. 116,117 Furthermore, several components of Wnt/β-catenin signaling including Dvl, Axin, β-catenin GSK3, and APC have been shown to associate with centrosomes and/or kinetochores during mitosis. Disruption of these components is associated with defects in chromosome segregation and has been suggested to promote chromosomal instability, a hallmark of cancer. 118 Wnt signaling also regulates the orientation of the mitotic spindle, thereby specifying the plane and symmetry of cell Wiley Periodicals, Inc. Volume 5, July/August 2013

11 WIREs Systems Biology and Medicine Wnt/β-catenin pathway through simulation and experiment FIGURE 3 Wnt signaling and the cell cycle. Recent studies have elucidated many mechanisms that link Wnt signaling to events during the mitotic cell cycle. 115 The Wnt target genes, CyclinD1 and Myc, are key regulators of G1, while Axin2 has been suggested to have a pro-proliferative role at G2/M. Dvl, Axin, β-catenin GSK3, and APC have been shown to associate with centrosomes and/or kinetochores during mitosis, with Wnt signaling also reported to regulate the orientation of the mitotic spindle. The G2/M-dependent induction of CyclinY activates the CDK14 kinase, leading to increased levels of LRP6 phosphorylation, 116 which enhances Wnt signaling and β-catenin accumulation. The increased levels of β-catenin may, via effects at G1/S and G2/M, allow progression through cell cycle checkpoints. Red arrows represent canonical Wnt/β-catenin pathway signaling, black arrows represent responses that result from cell cycle-dependent changes and dashed arrows represent further molecular binding/phosphorylation interactions. division. Importantly, the plane of cell division and the distribution of daughter cells may alter the function of stem cells in the intestine. By combining these observations, Hadjihannas et al. 118 have suggested that Wnt signaling may function at the G1/S checkpoint where it drives cells into S-phase, and again at G2/M to remove a spindle checkpoint. Interestingly, the Wnt target gene Axin2 has been suggested to have a pro-proliferative role at G2/M, 118 but has also been shown to be a negative regulator of Wnt/β-catenin transcriptional levels. Axin2 s dual role may therefore be key to the oscillations in β-catenin levels that are observed in many systems. 116,119 A further link between Wnt signaling and the cell cycle involves the Wnt receptor LRP6. Here the G2/M-dependent increase in plasma-membrane levels of CyclinY was shown to activate the CDK14 kinase, leading to increased levels of LRP6 phosphorylation, 116 which enhances Wnt signaling and β-catenin accumulation at G2/M. In turn, the increased levels of β-catenin may, via transcriptional regulation or through direct effects at centrosomes, allow progression through the G2/M checkpoint. Data now available on the timescales associated with transcriptional regulation, cell cycle progression, and β-catenin accumulation should allow the generation of theoretical models that link the dynamics of Wnt signaling with the cell cycle. These models should allow qualitative predictions to be made about the relationship between Wnt activity levels, stem cell identity, and both G1/S and G2/M checkpoints. DISCUSSION Over the last decade, mathematical and computational models have been used to increase our understanding of the spatiotemporal dynamics of the Wnt/β-catenin pathway and to investigate the functional impact of mutations in its components. These models have generated testable predictions at the biochemical, subcellular, cellular, and the tissue levels. Of particular interest to biologists are simple, caricature Volume 5, July/August Wiley Periodicals, Inc. 401

12 wires.wiley.com/sysbio models, based on a small set of high-quality experimental observations that generate counter-intuitive predictions. The ability of these idealized models to provide insight while modeling only a fraction of the processes associated with core Wnt signaling raises questions about the role of the large number (>200) of potential Wnt regulators Key roles for non-core components may be to allow the context-dependent linking of Wnt signaling with non-canonical and non- Wnt pathways such as Notch signaling. 120 While the incorporation of some of these additional components into existing models may extend their predictive range into new areas of biology, it may obscure less wellunderstood functions for core components that may be accessible within alternative model frameworks. A key challenge for biologists is to generate highquality quantitative information that can be used to validate and parameterize mathematical models and test any predictions that they generate. In order to improve our understanding of Wnt signaling, more detail about the interactions, localization, and kinetics of core pathway components is needed. In parallel, quantitative information on non-core components and their cellular and developmental contexts will be required to allow multiscale modeling solutions to be extended to other tissues and contexts. A practical limitation associated with the use of multiscale models for simulating large tissue regions remains their computational cost. This is of particular relevance when using such models in an exploratory manner, to compare alternative hypotheses or to conduct parameter sweeps. Motivated by this, several recent theoretical studies have focused on the derivation of coarse-grained tissue-scale descriptions from underlying cell-based models, and the relationships between these two representations. 42,43 This alternative approach may be particularly suitable in the absence of high-quality cell-level data or when considering the relationships between cell proliferation, signaling, and geometric changes associated, for example, with the buckling and fission of crypts in which Wnt signaling has become dysregulated. 121,122 Various biological tools, such as in vitro biochemical binding assays, RNAi screens, and in vivo genetic experiments in model organisms from Caenorhabditis elegans and Drosophila to mice have been utilized to further our understanding of the Wnt/β-catenin signaling pathway. A systems biology approach may be required to adequately describe the dynamics of biological changes and their responses to specific perturbations. 123 This relies on the availability of quantitative spatial and temporal data on key signaling proteins and complexes, information that is often unavailable or limited owing to the nonquantitative nature of many techniques and the tendency to extrapolate data between what may be dissimilar tissues/systems. High-resolution microscopy enables the compartmentalization of cells and, thus, the quantitative temporal and spatial assessment of key signaling proteins, such as β-catenin and APC. In addition, time-lapse microscopy is valuable for assessing the dynamics of the intracellular translocation of fluorescently tagged signaling proteins, and can provide useful insights into protein complex formation. Furthermore, quantitative microscopy enables multiparametric and single-cell analysis, which may reveal insights into the degree of stochasticity associated with Wnt/β-catenin signaling in different cell populations and how such effects influence processes such as cell death, proliferation, and migration. Such additional information should be considered during the development of future Wnt models, particularly hybrid ones that combine deterministic and stochastic equations, the latter being used when ODEs are no longer be valid. 124 For example, intracellular Axin concentrations may be so low in certain contexts that an ODE is unable to describe its evolution accurately. In summary, over the last decade the development of simple yet useful mathematical models has begun to provide scientific insight into the complex processes involved in Wnt signaling. However, as we have highlighted, much more qualitative information is known about the biology than has been incorporated into existing models, and many hypotheses remain properly to be tested (including those outlined in this review). Closer collaboration and increasing cross-fertilization of ideas between experimentalists and theoreticians remain key to future progress in this endeavor. ACKNOWLEDGMENTS AGF is supported by EPRSC (EP/I017909/1) and Microsoft Research, Cambridge. This publication was based on work supported in part by Award No. KUK , made by King Abdullah University of Science and Technology (KAUST). BLL was supported by Cancer Research UK. TCD was supported by Cancer Research UK, the Breast Cancer Campaign and Tenovus Wiley Periodicals, Inc. Volume 5, July/August 2013

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