Updating policy.

Two updating policies are mainly used. According to the synchronous policy, all components are updated simultaneously at each step; consequently, each state has at most one successor. In contrast, according to the asynchronous policy, only one variable can be updated at each step and all the possible successors of a state are computed. Mixed policies based on the notion of priority classes can also be defined using GINsim [61], where subsets of components are ranked. At each step, highest rank variables are then updated in a synchronous or asynchronous way.

In this work, we have used the fully asynchronous updating policy, which usually generates more realistic behaviours [58].

State transition graphs and attractors.

The dynamics of a logical model can be represented in terms of a state transition graph (STG), in which nodes denote different states of the system (represented by a Boolean vector encompassing the values of all the components), whereas arcs represent enabled transitions between pairs of states. Of particular interest are the states forming attractors, i.e. (groups of) states from which the system cannot escape, which represent potential asymptotical behaviours.

Attractors can be ranged into two main classes:

• stable states, corresponding to fixed points (i.e. states without successors);
• cyclic attractors, corresponding to terminal cycles or to more complex terminal strongly connected components, comprising several intertwined cycles.

Leaning on a representation of the logical rules in terms of Multi-valued Decision Diagrams (MDD), an algorithm enables the computation of all the stable states of a logical model (independently of the initial conditions) [62]. The efficiency of the algorithm (which does not require to compute the state transition graph) makes this tool particularly useful when dealing with large logical models. However, other means are needed to assess the reachability of the stable states from specific initial states, or yet to identify cyclic attractors.

Deeper dynamical analyses imply the computation of the state transition graph. GINsim user can define a set of initial states and an updating strategy; the software then computes the state transition graph, highlighting stable states and cyclic attractors.

GINsim further eases the definition of perturbations, which are simulated by forcing the level of a subset of components to fixed values (or value intervals). For instance, in the Boolean case, we can reproduce a loss-of-function by setting a component to 0, whatever the levels of its regulators, whereas a gain-of-function can be simulated by forcing the corresponding component to 1. More subtle perturbations can be simulated by rewriting relevant logical rules.

As the size of the model considered increases, we are facing the well known problem of the exponential growth of the state transition graph. To cope with this problem, we used two methods that amount to compress the model before simulation or to compress the resulting state transition graph on the fly. These two methods are briefly described hereafter, along with a complementary method enabling the identification of regulatory circuits playing crucial dynamical roles.

Scope of the MAPK logical model.

In order to study the response of the MAPK network to specific stimuli, and its influence on cell fate decision, we built a dynamical model covering the mechanisms reported in the MAPK map. We derived a regulatory graph using the MAPK reaction map as an information source, as detailed in supplementary Text S1. In particular, we considered the following subset of stimuli in our model: EGFR stimulus, FGFR3 stimulus, TGFβR stimulus, and DNA damage. With reference to the latter stimulus, please notice that we did not consider explicitly here the DNA repair process following DNA damage, but we only account for the triggering of growth arrest and apoptosis following sustained DNA damage [68]. In our dynamical model, DNA damage thus corresponds to sustained stress or to the effects of therapies involving DNA-damaging agents.

The regulatory graph shown in Figure 2 covers the activation of MAPK targets that influence the choice between proliferation, growth arrest, and apoptosis. In particular, we consider MYC and p70 (in the absence of p21) as markers of cell cycle enablement, p21 as a marker of growth arrest, FOXO3 and p53 as markers of apoptosis, whereas ERK and/or BCL2 indicate apoptosis disablement. To ease the interpretation of phenotypes, we defined three output nodes denoting proliferation, growth arrest, and apoptosis, respectively. These nodes represent enablement/disablement of the corresponding processes, depending on the MAPK network state, but not necessarily all requirements for this phenotype. For instance, when proliferation is enabled (by the interplay of MYC, p70 and p21), we assume that Cyclin/CDK complexes are activated. In this context, the node “Proliferation” denotes entry into the cell cycle and does not account for alterations of later stages of the cell cycle. Similar considerations are applicable for the other phenotypes modelled. This simplification is justified by our focus on the specific contributions of the MAPK network to cell fate decision.

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Figure 2. Regulatory graph of the MAPK logical model.

Each node denotes a model component. Model inputs, phenotypes and MAPK proteins (ERK, p38, JNK) are denoted in pink, blue and orange, respectively. Green arrows and red T-arrows denote positive and negative regulations, respectively. A comprehensive documentation is provided in the Table S4, which includes a summary of all modelling assumptions, references (PubMed links) and the specification of the logical rule associated with each component. The source file is further provided as supplementary Dataset S4, which can be opened, edited, analysed and simulated with the softare GINsim (www://www.ginsim.org/beta).

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Each of the 52 components of the regulatory graph is modelled by a Boolean variable along with a logical rule (Table S2) specifying how the component activity depends on its regulators.

Reduced model versions.

To cope with the relatively high dimension (52 components) of our model, we took advantage of the model reduction function implemented in GINsim (see Methods). Indeed, it is difficult to perform simulations with the original model version, as it entails 2⁵² states. The choice of components to hide depends on the simulation performed. For instance, if we plan to simulate a p53 loss-of-function and observe its effects on p21, we better conserve p53 (otherwise we cannot define the perturbed version) and p21 (otherwise, we cannot explicitly observe its value). However, as we wanted to test several situations, almost half of the MAPK model components were needed to be observable. Consequently, we designed several reduced versions of the original MAPK model, each of which dedicated to a subset of simulations. This strategy enabled us to drastically reduce the computational cost of our in silico experiments (the dimension of the reduced models ranged from 16 to 18 components).

Altogether, we defined three different reduced model versions, whose component lists are reported in supplementary Table S3. These definitions are enclosed in the comprehensive model file (supplementary Dataset S4) and enable the generation of reduced versions according to simulation needs. Figure 3 shows the regulatory graph corresponding to one reduced version, namely, “red1”. Supplementary Dataset S3 lists the simulations performed for each reduced version. The three model reductions were found equivalent in terms of attractors and main dynamical properties. Briefly, reduction “red1” was used to obtain the results discussed in the sections “Coherence with well established bladder cancer deregulations” and “Predictions generated with the MAPK logical model”, while reductions “red2” and “red3” were used to obtain the results discussed in the section “Coherence with additional cancer-related facts”.

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Figure 3. Regulatory graph of a reduced version of the MAPK model.

The regulatory graph corresponds to the “red1” reduced model version (cf. supplementary Table S3, column 1). To obtain this version, the preservation of pink and blue nodes was imposed, along with that of {EGFR, FGFR3, p53, p14, PI3K, AKT, PTEN, ERK}, in order to investigate the effects of perturbations affecting these components. The remaining nodes {FRS2, MSK} were maintained by the reduction algorithm because of the occurrence of auto-regulatory loops during the reduction process. Green arrows and red T-arrows denote positive and negative regulations, respectively, whereas blue arrows denote dual interactions.

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Coherence with well established bladder cancer deregulations.

As we build our model around the comparison between EGFR over-expression and FGFR3 activating mutation in bladder cancer, our first simulations were dedicated to the assessment of the model behaviour in these circumstances.

Figure 4a reports a simplified view of the model dynamics following EGFR over-expression, obtained by setting EGFR to 1 throughout the simulation, in the absence of additional stimuli (see supplementary Dataset S3 for precise simulation settings). In response to EGFR over-expression, the asynchronous state transition graph encompasses two sets of trajectories: one characterised by p53 activation and ERK silencing, leading to an apoptotic stable state (i.e. Apoptosis = Growth_Arrest = 1, Proliferation = 0); the other characterised by p53 silencing and ERK activation, leading to a PI3K/AKT-dependent proliferation stable state (i.e. Apoptosis = Growth_Arrest = 0, Proliferation = 1) (cf. simulation provided in the supplementary Text S2).

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Figure 4. Coherence of the logical model with well established bladder cancer deregulations.

a) Simplified representation of the model dynamics following EGFR over-expression (EGFR = 1 throughout the simulation and all inputs set to 0 throughout the simulation). If p53 is activated first (right branch), an apoptotic attractor is reached, characterised by inactivation of ERK and AKT. If ERK and PI3K are activated first (left branch), then p53 is inactivated and AKT is activated, leading to a proliferative attractor. b) Simplified representation of the model dynamics following FGFR3 activating mutation (FGFR3 = 1 and all inputs set to 0 throughout the simulation). When p53 is activated first (right branch), an apoptotic attractor is reached, characterised by inactivation of ERK and AKT. If ERK is activated first (left and central branches), then p53 is inactivated. When PI3K is also activated (central), a proliferative attractor is reached, characterised by activated AKT. In contrast, when PI3K is not activated (left), the cell fails to make a clear decision at the level of the MAPK network. c) Attractors reached by the model in presence of receptor alterations, coupled with additional common deregulations observed in bladder cancer. Coloured circles denote the phenotypes characterising the attractors reached in each situation (we used the same colour code as in panels a and b, while empty spaces denote the loss of the corresponding branch in the state transition graph). Identifiers in rectangles (e.g.. r3, r4, etc.) point to simulation results reported in more details in Dataset S3 and Text S2.

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Alterations in the p53 pathway have been associated with more aggressive and invasive bladder cancers [50]. The fundamental role of p53 in the model can be observed by simulating a p53 loss-of-function (second row of Figure 4c). In this case, the proliferative attractor is kept, while the apoptotic attractor is lost. In contrast, still in presence of EGFR over-expression, when the system is also subjected to persistent DNA damage (third row of Figure 4c), we obtain a single apoptotic attractor. This behaviour is in agreement with the fact that when damage is moderate (i.e. absence of DNA damage stimulus), the cell is still able to escape apoptotic cell death by down-regulating p53 signalling (i.e. possible switch between the two attractors of Figure 4a), but p53 eventually induces apoptosis in cells subjected to extensive DNA damage [69].

A similar response is also predicted in the case of TGFBR stimulus, in line with the typical anti-proliferative role played by this pathway [70]. In this respect, TGFBR has also been shown to induce proliferation in tumours, in some circumstances, but the conditions (especially in relation with MAPK network) under which this occurs are still poorly understood [70]. Strong activation of PI3K/AKT pathway has also been associated with enhanced bladder tumour proliferation [50]. Accordingly, in our model, PI3K/AKT gain-of-functions counteract p53 pathway effects (fifth row of Figure 4c), impairing the apoptotic phenotype.

Two other important loss-of-function known to be associated with poorer prognosis in bladder cancers have been simulated. Deletions of either PTEN or p14 in our model are associated with a more aggressive phenotype (sixth and seventh row of Figure 4c). The former is a tumour suppressor shown to inhibit AKT expression [71]. The latter is induced by MYC and is able to enhance p53 activity by inhibiting MDM2 [51]. According to our model, p14 loss-of-function abrogates apoptosis, which accounts for the observed dual role of MYC [72]: on the one hand, MYC contributes to proliferation (MYC is a read-out for proliferation in the model); on the other hand, it is involved in p53-dependent apoptosis.

The behaviour of our model in the case of FGFR3 activating mutation is depicted in Figure 4b (cf. Text S2 for a complete simulation). We find again the “p53 versus ERK” pattern accounting for the fundamental role of p53 in cell fate decision. Interestingly, the non-apoptotic branch of the asynchronous state transition graph is now characterised by two different behaviours. On the one hand, similarly to EGFR over-expression, when PI3K is active, the system will eventually reach a proliferative attractor. On the other hand, a p53-independent PI3K/AKT pathway leads to an attractor characterised by all phenotypes set to 0, that we interpret as “no cell fate decision taken” at the level of MAPK network. This is coherent with the contention that FGFR3 mutations, mainly characterising non-invasive bladder carcinomas, relatively mildly induce proliferation due to the action of ERK cascade [47], [51], [73]. Indeed, the coexistence of these outcomes (proliferation versus no cell fate decision) tentatively explain the less aggressive phenotype generally observed in FGFR3-mutated bladder carcinomas, in comparison with urothelial tumours over-expressing EGFR (associated only with a proliferative attractor). The underlying mechanisms are further analysed below. By and large, the simulations of FGFR3 mutation correctly recapitulate the effects of the major bladder cancer deregulations listed in Figure 4c (left column), producing results that are qualitatively coherent with those described for EGFR over-expression.

Coherence with additional cancer-related facts.

To further assess the consistency of the behaviour of our model with current knowledge, we selected a list of established facts regarding the effects of perturbations on (human/mouse) cancer cells, not necessarily bladder-related. Based on this list, we defined a series of in silico experiments, combining relevant initial states and virtual perturbations (loss-of-functions and/or gain-of-functions of selected model components, as defined in Methods).

This analysis mainly consisted in cross-checking the attractors obtained in our simulations with the qualitative information retrieved from the literature, without focusing on the full dynamics of the system.

A summary of the results obtained is provided in Table 1 (for more details, see the supplementary Dataset S3). Additionally, all the simulations performed can be easily reproduced using the model files available as supplementary Dataset S4. Briefly, the involvement of MAPKs in cell fate decision was assessed through perturbations of relevant components. The model accounts for the pro-apoptotic role of p38 and JNK, as well as for the promotion of growth arrest by p38, and for the proliferative role of ERK. The model also recapitulates the p21-mediated tumour suppressor function of p53, along with the impairment of this function due to epigenetic silencing of GADD45. Additionally, we were also able to reproduce (i) the positive effects of EGFR/FGFR3/RAS/RAF over-expressions on ERK activation; (ii) the negative effects of HSP90-inhibitors on cell proliferation; (iii) the role of ERK against anti-proliferative TGFβ signalling; and (iv) the role of JNK against RAS-induced proliferation. These simulations cover the most salient behaviours of the model, showing a remarkable coherence with published data.

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Table 1. Coherence of model simulations with published experimental evidence.

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ERK-related feedback mechanisms.

So far, we have described the behaviour of our model in presence of tumoural deregulations of growth factor receptors. Let us consider now what happens when the expression of such receptors is not altered (i.e. unperturbed logical rules for either EGFR or FGFR3 variables), in presence of sustained growth factor stimulation (i.e. either EGFR_stimulus or FGFR3_stimulus set to 1 throughout the simulations). The attractor reached from normal EGFR stimulation is characterised by oscillations (between 0 and 1) of the values of EGFR, ERK and p53, thus leading to oscillations of phenotype variables, in particular for “Proliferation” (Dataset S3 – r1). This behaviour contrasts with the well defined phenotypes obtained following EGFR over-expression (Figure 4a). It can be interpreted as the impossibility to obtain sustained activation (or inactivation) of the considered actors in presence of sustained growth factor stimuli. In other words, ERK oscillations in our state transition graph correspond to the transient ERK activation in the ODE-based model from Orton et al. [38]. Similar results are obtained for FGFR3 stimulation (Figure 4b and Dataset S3 – r2), in agreement with literature [74].

The main negative feedbacks underlying such responses are acting directly on the receptors (Figure 2): one accounting for GRB2-dependent ubiquitination and degradation of the receptors [75]; the other accounting for PKC-mediated negative effects [74], [76]. Concomitant disruptions of these feedbacks in our model lead to simulation results equivalent to those obtained with receptor gain-of-function (cf. Figure 2 and Table S2), whereas disruption of any other downstream negative feedback does not qualitatively influence the outcome (data not shown). This is also in agreement with the results obtained by Orton et al., who proposed that the degradation of receptors (e.g. rather than SOS inhibition by RSK) could be the main mechanism determining a transient activation of ERK pathway in response to growth factors.

Role of Sprouty-mediated feedbacks.

According to our model simulations (Figure 4a), when EGFR is over-expressed (e.g. in the presence of an autocrine signal), in the absence of p53 activation, the outcome is proliferation. In contrast, when FGFR3 stimulus is present, two possible outcomes are observed in the absence of p53: a proliferative stable state and a non-proliferative stable state, the later with all phenotype variables set to 0.

General hypotheses involving the interplay between the p53, RB and ERK pathways have been proposed to explain the different phenotypes experimentally observed in bladder carcinomas [50], [73], but the precise mechanisms have not been elucidated yet. A closer analysis of the regulatory graph shown in Figure 2 reveals several feedbacks. Interestingly, ERK exerts a positive feedback on EGFR but a negative feedback on FGFR3-activated FRS2, through Sprouty (SPRY in the model). Intuitively, this suggests that ERK strengthens EGFR stimulus but weakens FGFR3 stimulus, potentially explaining the different phenotypes observed. Additionally, GRB2 exerts a negative feedback on FRS2, which is in turn specifically activated by FGFR3.

Disruption of EGFR activation by SPRY does not play an important role in the case of EGFR over-expression (which indeed corresponds to setting EGFR to 1 independently of its regulators). FRS2 inhibition by SPRY, but not by GRB2, tentatively plays an important role in the response of the MAPK network to FGFR3 activating mutation. Indeed, disrupting the latter inhibition (Figure 5) does not affect significantly the model behaviour. On the contrary, comparison of Figures 4a–b and Figure 5 indicates that SPRY-dependent inhibition of FRS2 might be the key to explain the difference between the responses to EGFR and FGFR3 stimulations (i.e. in the absence of this inhibition, the model behaves equivalently in the two cases).

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Figure 5. FGFR3 activating mutation and role of SPRY.

Simulations were performed under FGFR3 gain-of-function (FGFR3 = 1 and all inputs set to 0, throughout the simulations). Simplified model dynamics are shown as in Figure 4a–b. Results are shown for the wild type model (red1 model reduction), as well as for perturbed model versions obtained by disrupting the inhibition of interest.

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To further clarify these mechanisms, we considered the role of functional positive circuits, which are known to promote multi-stable behaviours (cf. Methods). According to our model analysis, the GAB1-PI3K-GAB1 circuit underlies the coexistence of the two stable states found in the presence of FGFR3 activating mutation, but in the absence of p53 activation. We thus propose that this circuit plays a fundamental role in FGFR3 signalling, constituting a switch between proliferative and non-proliferative phenotypes. The underlying mechanisms can be further clarified by a careful analysis of Figure 2. On the one hand, following FGFR3 stimulus, when PI3K activation is faster/stronger than ERK activation, cell proliferation is enabled, because PI3K is definitively activated (due to the action of GAB1-PI3K-GAB1 positive circuit) and can then inhibit p21 through the PDK1/AKT pathway. ERK is then also activated, enhancing proliferation together with PI3K. On the other hand, upon ERK activation (coming from a rapid GRB2 and/or PKC mediated signalling), if the inhibition of GRB2 through the SPRY/FRS2 feedback is faster/stronger than PI3K activation, then PI3K cannot be activated anymore. In this scenario, ERK would rather contribute to disable cell proliferation.

Our model thus predicts that the strength/rapidity of PI3K activation versus SPRY-mediated ERK negative feedback could underly the less aggressive phenotypes observed in FGFR3-mutated bladder carcinomas.

MAPK cross-talk mechanisms.

Using our logical model, we further addressed the roles of cross-talks between the different MAPK cascades, in particular those involving phosphatases.

We first analysed the negative cross-talks from p38/JNK cascades towards ERK cascade, which involve MEK inhibition by AP1 and the phosphatase PPP2CA [2]. In the context of either EGFR over-expression or FGFR3 activating mutation (Figure 6a), the disruption of these inhibitions mainly lead to the loss of apoptotic attractors (compare Figure 6a with Figure 4a–b). The lost attractors are “replaced” by new attractors characterised by growth arrest alone. These two cross-talks are thus presumably important for the triggering of apoptotic responses in the presence of growth factor receptor alterations. Indeed, in the absence of such cross-talks, p53 pathway is only able to induce growth arrest, but not apoptosis, precluding a complete anti-proliferative response. This is also true in the concomitant presence of DNA damage stimulus (data not shown).

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Figure 6. Analysis of MAPK cross-talks by disruptions of specific interactions.

a) Effects of the disruptions of the inhibitions of MEK by AP1 and the phosphatase PPP2CA. b) Effects of the disruption of the inhibition of p38 or of JNK by the phosphatase DUSP1. These simulations were performed after removing the corresponding interactions and blocking the level of the perturbed receptor to level 1 (with all inputs set to 0, throughout the simulation).

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Finally, we examined the roles of p38 and JNK inhibitions by DUSP1 [77]. Following receptor (either EGFR or FGFR3) alteration, disruption of any of these two inhibitions results in a persistent silencing of ERK and a persistent activation of p53, as well as an activation of PI3K and an inactivation of AKT, which ultimately lead the system towards an apoptotic attractor (compare Figure 6b with Figure 4a–b). DUSP1-mediated cross-talks between the MAPK cascades thus tentatively underly proliferative responses in presence of growth factor receptor alterations, presumably via the inhibition of p53 pathway.