Parameter graph representation of the parameter space. a The EMT network from [11]. b The EMT network that we use for the analysis in this manuscript. c An example of the many possible Morse graphs for the network in (b). d The factor parameter graph for Ovol2. Each node represents one way in which the inputs of Ovol2 are integrated and affect the downstream nodes of Ovol2. Each node is characterized by the corresponding inequalities given in (1). Nodes colored in red are associated to essential parameters

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... modeled a network that includes Ovol2, Zeb1, Snail1, miR34a, miR200 and TGFβ depicted in Fig. 2a. They show in their model, and also find experimental evidence, that there exists not one, but two intermediate states I 1 and I 2 . Using ODE models they show that both states are sensitive to Ovol2 levels and overexpression of Ovol2 leads to a transition of the system to the epithelial state. Similarly, a high level of TGFβ induces ...

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... PG(i) is the parameter graph for node i of the network. For more detailed and mathematically rigorous description of PG and PG(i), see "Methods" section and [26,27]. An example of a factor graph is shown in Fig. 2d and is explained in more detail in "EMT model" ...

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... example Morse graph of the EMT system that we consider in this paper is given in Fig. 2c. Each node has an inscription of either FP, followed by a sequence of six numbers that represents a label in S, or XC. The annotation FP stands for a fixed point representing a steady state, and XC for a partial cycle; that is, a cycle where the state s i is constant for at least one i. We append to each fixed point the state label in ...

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... that represents a label in S, or XC. The annotation FP stands for a fixed point representing a steady state, and XC for a partial cycle; that is, a cycle where the state s i is constant for at least one i. We append to each fixed point the state label in S corresponding to the location of the fixed point in phase space. In the Morse graph in Fig. 2c there are six stable steady states denoted by FP and five unstable periodic states denoted by XC. The parameter graph together with the corresponding Morse graph at each node of the parameter graph forms a DSGRN ...

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... study the EMT network in Fig. 2a, taken from [11], subject to a few modifications. First, we remove the negative self-edge on Snail1, in order to define STGs unambiguously, see Remark 2 in "Methods" section. This may cause our model to miss some of the intermediate ...

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... output edges.Therefore there is no natural way to subdivide their expression levels into discrete classes. We chose to characterize the E and M phenotypes without the biomarkers Ecad and Vimentin in the following way. Instead of directly tracking Ecad and Vimentin, we track the expression levels of their regulators Zeb1, Snail1 and Ovol2 (see Fig. 2a). Since Vimentin, a biomarker for the mesenchymal state, is up-regulated by Zeb1 and Snail1 and downregulated by Ovol2, the highest expression of Vimentin will happen when Zeb1 and Snail1 are at their highest levels and Ovol2 is at its lowest level. This represents the mesenchymal state. The opposite pattern with Zeb1 and Snail1 low ...

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... that the epithelial state is present in the Morse graph in Fig. 2c in the lower left. The Morse graph shows multistability between E together with five intermediate E/M states. For example, FP(2,0,1,1,0,1) represents a FP steady state where Snail1 and TGFβ are at their lowest level, miR200, Ovol2, and Zeb1 are at intermediate levels, and miR34a is at its highest level. In "Results" section we will ...

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... the EMT network in Fig. 2b has 6 nodes and 12 edges, parameter space is 6 + 3 * 12 = 42 dimensional. The corresponding parameter graph has more than 21 billion parameter nodes, each associated to a region in 42-dimensional parameter space. If we want to query the parameter graph for changes in steady states induced by changing expression level of a particular ...

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... an illustration, we describe an example of PG(k) where node k has one input edge and two output edges, as is true for Ovol2 in the EMT network. This factor graph is shown in Fig. 2d. Ovol2 has a single in-edge from Zeb1 and two out-edges to Zeb1 and TGFβ. For simplicity, denote γ Ovol2 , the degradation rate of Ovol2, by γ , and denote L Ovol2, Zeb1 and U Ovol2,Zeb1 by L and U, respectively. Recall that parameter nodes are associated to regions in parameter space defined by inequalities (see "Parameter graph" ...

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... some of these inequalities represent parameter choices when the network does not work as depicted in Fig. 2a. For instance, node A1 implies that the output edges from node Ovol2 will never get actuated for any choice of inputs. On one hand this does represent a very low level of expression of gene Ovol2 which is a valid state of this gene. On the other hand, at this parameter node the output edges from node Ovol2 do not carry any information. ...

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... characterized and then perhaps pharmacologically controlled. Previous modeling work using differential equations models considered one or two parameters at a time and found up to two intermediate steady states. In the following analysis, we characterize the number and location of intermediate E/M states as found by DSGRN using the network in Fig. 2b. Our method is somewhat analogous to a oneparameter bifurcation analysis, but the difference is that the remaining parameters are allowed to vary across all of parameter space, rather than being ...

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... Ovol2 parameter factor graph is shown in Fig. 2d, and the extreme points A1 and B1 correspond to Ovol2 at its lowest level. In other words, A1 and B1 represent parameter regions in which Ovol2 is always below all thresholds at which it actuates its downstream genes. Likewise, the points A6 and B6 represent parameter regions where Ovol2 is at its highest level, and above all ...

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... for the actuation of downstream targets. The parameter nodes in between these extremes represent a gradual increase in Ovol2 expression levels as measured by the number of downstream genes it actuates. To facilitate graphing dynamical properties as functions of increasing abundance of Ovol2, we compress the structure of the factor graph (Fig. 2d) into five layers denoted by the numbers on the horizontal axis representing qualitative Ovol2 expression levels. As the layer number increases by one, the Ovol2 expression level is able to actuate more of its downstream genes. The layers of the factor parameter graphs for other genes are also compressed in this way. The complexity of ...

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... tabulate where different types of FPs occur in parameter space. For every parameter in the Ovol2-general parameter graph, the projection of that parameter onto the Ovol2 factor graph in Fig. 2d occurs in one of the five layers. For each layer in the Ovol2-general parameter, we count how many times a given type of FP occurs. That is a measure of the prevalence of that FP within the parameter graph as a function of increasing ...

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... of attractors that the EMT network can exhibit, the frequency with which we observe the E and M states, and how often the E and M states are monostable. An attractor is monostable if it is the only stable node in the Morse graph. Multistability of attracting states means that multiple stable Morse nodes are present in the Morse graph (see e.g. Fig. 2c). We observe that in all parameter nodes there are only fixed point attractors. As illustrated in Fig. 2c there are Morse nodes with signature XC, which correspond to closed state transition paths along which several gene product abundances oscillate. However, these are always unstable in the model and so likely not experimentally ...

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... and how often the E and M states are monostable. An attractor is monostable if it is the only stable node in the Morse graph. Multistability of attracting states means that multiple stable Morse nodes are present in the Morse graph (see e.g. Fig. 2c). We observe that in all parameter nodes there are only fixed point attractors. As illustrated in Fig. 2c there are Morse nodes with signature XC, which correspond to closed state transition paths along which several gene product abundances oscillate. However, these are always unstable in the model and so likely not experimentally observable, or observable only as transients. Therefore the EMT network structure robustly exhibits stable ...

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... factor graphs for TGFβ, Snail1 and Zeb1 are different than the one for Ovol2 in Fig. 2d. TGFβ has two in-edges and one out-edge, Snail1 has two in-edges and three out-edges, and Zeb1 has three in-edges and three out-edges, as shown in Fig. 2b, unlike the one in-edge, two out-edge topology of Ovol2. The factor graph of TGFβ is isomorphic to the A1-A6 half of the Ovol2 factor graph in Fig. 2d, and so has five layers like ...

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... factor graphs for TGFβ, Snail1 and Zeb1 are different than the one for Ovol2 in Fig. 2d. TGFβ has two in-edges and one out-edge, Snail1 has two in-edges and three out-edges, and Zeb1 has three in-edges and three out-edges, as shown in Fig. 2b, unlike the one in-edge, two out-edge topology of Ovol2. The factor graph of TGFβ is isomorphic to the A1-A6 half of the Ovol2 factor graph in Fig. 2d, and so has five layers like Ovol2. Snail1 has a far more complex factor graph with 300 nodes and 13 layers. Zeb1 factor graph has 4242 nodes in 25 ...

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... are different than the one for Ovol2 in Fig. 2d. TGFβ has two in-edges and one out-edge, Snail1 has two in-edges and three out-edges, and Zeb1 has three in-edges and three out-edges, as shown in Fig. 2b, unlike the one in-edge, two out-edge topology of Ovol2. The factor graph of TGFβ is isomorphic to the A1-A6 half of the Ovol2 factor graph in Fig. 2d, and so has five layers like Ovol2. Snail1 has a far more complex factor graph with 300 nodes and 13 layers. Zeb1 factor graph has 4242 nodes in 25 ...