Mathematical modeling of MAPK and PI3K-AKT pathway

We consider a simplified signaling network composed of interactions between key proteins in an oncogenic rat sarcoma (RAS)-driven mitogen-activated protein kinase (MAPK) and phosphoinositide 3-kinase (PI3K)–protein kinase B (AKT, PKB) pathway (Fig 2A). This pathway has been studied extensively, and the positive or negative feedback regulations between proteins in the pathway are known [58–62]. Of note, mammalian cells express three RAS gene family members (HRAS, NRAS, and KRAS), and our model is based on empirical data obtained from a lung cancer cell line (A549); KRAS is always activated by a point mutation (Gly12Ser), whereas the other two RAS proteins (HRAS and NRAS) are wild type. Recent studies reported the importance of crosstalk between wild-type and mutant RAS proteins in cancers driven by oncogenic mutant RAS [59, 63, 64]. Therefore, we consider two different types of RAS, wild type (RAS_w) and mutant type (RAS_m). The network node connectivity is based on prior pathway information between signaling proteins [58–62]. Most interactions are feed-forward and positive (Fig 2A, green lines) except the one negative feedback regulation of epidermal growth factor receptor (EGFR) by extracellular receptor kinase (ERK) (Fig 2A, red line). While network connectivity is assumed fixed in the model, the strength of interactions is variable and is modeled using a weight matrix (W). Each element in the network (node x_i) is updated by solving the following equation, (1) where x_1,2 correspond to two inputs (growth factor and hepatocyte growth factor [HGF]), and χ_3,4…,n correspond to the relative change of protein activities or cell viability due to an inhibitor with respect to untreated conditions (log₂ (treated/untreated)) to be consistent with experimental data. Of note, all of the experimental measures in our study are relative values, normalized to the unperturbed condition. The absolute concentration or activity of signaling proteins as well as cell viability are difficult to acquire from experiments performed in our study, namely western blot experiments and cell viability measurement assays. Therefore, the weights in the model represent relative abundance or protein activities in treated conditions compared to treatment-naïve conditions. The model assumes that the rate of change of a variable is determined by the linear combination of neighboring nodes with corresponding weights. This additive linear function has successfully described protein reaction networks [54, 55, 65] although other functions such as Michaelis-Menten kinetics are viable options [51]. In the experiments we carried out, the microenvironmental conditions are growth factor and HGF. The growth factor (model variable x₁) is always present, while HGF (model variable x₂) is present in only some of the experimental conditions. In particular, the HGF is not present in the control condition. All experimental results are normalized to this control (no-HGF) condition. To represent these experimental conditions in the model, the input value x₁ is set to a nonzero constant value (e.g., x₁ ≡ C), while the variable x₂ is set to be 0 or nonzero (e.g., x₂ ≡ C) if HGF is present. We chose to set the parameter constant C to be 10. N(i) represents the neighborhood of a protein node i (a set of nodes connected to the node i), and α indicates a tendency to return to the untreated state. The transfer function T accounts for saturation effects, and the constants ε and β modulate amplitude and slope. In the model, we set ε to be 4.5 and β to be 0.5 to model a smooth sigmoidal behavior.

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Fig 2. Signaling pathway model development and model calibration.

(A) Simplified Signaling Network Model. Two inputs (growth factor and HGF), signaling protein nodes (EGFR, MET, RAS_m, RAS_w, PI3K/AKT, RAF, MEK, ERK, RSK), and one output (cell viability). Of note, RAS_m indicates a mutant RAS, while RAS_w indicates a wild-type RAS. A green line represents a positive relation (stimulation) and red line represents a negative relation (inhibition). (B) pMET (Y1234/5), pMEK, pERK, pAKT (both T308 and S473), and pRSK (T359) expression after different inhibitors (1 μM), MET inhibitor (METi, PHA665752), EGFR inhibitor (EGFRi, Erlotinib), RAF inhibitor (RAFi, LY3009120), MEK inhibitor (MEKi, GDC0623), ERK inhibitor (ERKi, SCH772984), and AKT inhibitor (AKTi, MK2208) in both control medium (DMSO) and after 2-hour stimulation-by-HGF (50 ng/mL) condition (DMSO plus HGF). (C) Relative cell viabilities after treatments. Cells were treated inhibitors (1 μM) for 72 hours. Cell viabilities were assessed by CellTiter-Glo assay (Promega). Representative triplicates (± SD) are presented, which showed similar results at least three times. (D) The western blots were quantified using ImageJ and relative changes (log2 scale) are reported. Average values of relative cell viabilities are also reported in log2 scale. All the data are normalized to the treatment-naïve control condition. pAKT (T308) readouts are quantified and used in the model. Of note, we didn’t quantify total protein levels because our primary interest was protein activity (protein phosphorylation). The effects of all inhibitors are modeled by assuming very small activity of a target protein (i.e., 1/16 of control, Methods section), and therefore pMET under METi is set be a very small number (i.e., 1/16 of control) for consistency. (E) Comparison between model predictions (gray box plots) and experimental data (black dots). A log2 fold change of pMET, pMEK, pAKT, pERK, pRSK, and cell viability after treatments of different inhibitors (METi, EGFRi, MEKi, ERKi, AKTi) in both a control medium and HGF-stimulated conditions. RMSE of each protein is following. pMET: 0.03; pMEK: 0.49; pAKT:0.33; pERK: 0.96; pRSK: 0.57; and cell viability: 0.47. The numerical data used in Fig 2 are included in the first sheet S1 Data. AKT (PKB), protein kinase B; DMSO, Dimethyl sulfoxide (control); EGFR, epidermal growth factor receptor; ERK, extracellular receptor kinase; HGF, hepatocyte growth factor; MEK, mitogen-activated protein kinase kinase; MET (c-MET), tyrosine-protein kinase Met or hepatocyte growth factor receptor (HGFR); PI3K, phosphoinositide 3-kinase; RAF, rapidly accelerated fibrosarcoma; RAS, rat sarcoma; RAS_m, mutated RAS; RAS_w, wild-type RAS; RMSE, root-mean-squared-error; RSK, ribosomal S6 kinase.

https://doi.org/10.1371/journal.pbio.2002930.g002

Drug inhibition is modeled by knocking out a corresponding protein activity. For example, if a drug inhibits a protein x_i, then the variable x_i is set to be a very small number (). It is worth noting that we modeled the effect of drugs tyrosine-protein kinase Met inhibitor (METi), EGFRi, and AKTi by inhibiting pMET, pEGFR, and pAKT activities, respectively. Two other drugs, mitogen-activated protein kinase kinase (MEKi), and ERKi, are modeled to inhibit pERK and pRSK, respectively (Fig 2B).

Logistic tumor growth

We combine the pathway model to a logistic growth ODE model to simulate the growth of in silico cells as a well-mixed population in a resource-limited environment. The model is defined as follows, (2) where y_i represents the number of in silico cell i, r_i is the cell intrinsic growth rate of the cell i, N is the total number of cell types, and K is the carrying capacity (set to be 1 billion). To model influence of the signaling pathway on cell population growth, we formulate a cell population growth rate, r_i, as a function of cell viability solutions and the treatment-naïve growth rate (r_i ≡ f (θ_i, r₀); θ_i: cell viability solution in linear scale not in log2 scale; r₀: growth rate in an untreated control condition). Specifically, the cell viability for each cell type is obtained by fitting pathway model to our experimental data, as described in the Results section. The cell viability of cell type i (θ_i) represents the number of cell type i that survived after being given therapy relative to an untreated control condition (i.e., θ_i = P_i(t)/P₀(t), θ_i: cell viability, P_i,0(t): number of cell type i at time t in a treated and untreated condition, respectively). To obtain a functional form for the growth rate, we make the following assumptions. We assume that a cell population initially grows exponentially for some time T). We also assume that the number of initial cell population is the same in a treated condition and in an untreated control condition (P_i(0) = P₀(0)). Then, cell viability is presented as a function of growth rates and time (). Solving the function for growth rate r_i, we obtain a functional form for the growth rate. We use the doubling time of A549 cells (22 hours from [66]) to obtain the treatment-naïve growth rate (r₀ = 0.76 per day). We use our cell viability assay experimental time point (t = 3 days, described in Results section). Now, we have a constant growth rate of cell type i for each treatment condition (500 cells x 28 treatment conditions, total 14,000 growth rates, r_i). Then, the ODE Eq (2) is solved to simulate a given treatment response of cell i over time. All of the in silico cells are solved simultaneously competing for limited resource (carrying capacity K).

HCA tumor model

The pathway model is integrated into an HCA model [67, 68] to simulate treatment responses in a spatially heterogeneous microenvironment. The model has the following assumptions. In the HCA, the cells are defined as points on a two-dimensional lattice that also contains continuous concentration fields of microenvironmental factors, together representing a cross-section of tumor composed of cancer cells and stroma (50 cells x 38 cells). Here, we define the tumor and stroma region explicitly based on an image segmentation of lung adenocarcinoma tissue from a patient. The tumor region contains cancer cells. Each cancer cell contains the pathway model, as developed above (Fig 2A), that links the microenvironment to cell phenotypes. The model grid can contain any number of possible microenvironmental variables. For simplicity, however, we consider only growth factors and HGF. The growth factors are assumed to be constant in the domain. We explicitly model HGF dynamics in space and time using the following partial differential equation, (3) where H(x,t) represents the concentration of HGF at a lattice point x in tumor region and at time t. D represents the diffusion rate, and λ is a decay rate. The parameter values used in a simulation are given in the corresponding figure legend. The concentration of HGF is fixed to be a constant value (H(x,t) = γ) in the stromal region. A Neumann boundary condition (, normal derivative = 0) was imposed on the domain boundary. A Dirichlet condition (H(x,t) = γ) was imposed at the interface between tumor and stroma.

The steady state solution of Eq (3) is fed into one of the inputs (HGF) in the pathway model in each cell (Fig 2A). The pathway determines cell viability and controls three different phenotypes—proliferation, quiescence, and death—as defined by the rules summarized in the flowchart (S1 Fig). Each cell is allowed to execute only one phenotype per time step (day). Of note, the model considers only orthogonal neighbors (north, west, south, and east) for space to divide or move. Cells are not allowed to leave nor enter across the boundary and are thus confined within the domain. We assume that the distribution of HGF barely changes during HCA simulation. For example, if a cell divides into two daughter cells, this increased number of cells does not impact the HGF distribution because we assume the consumption of HGF by cells negligible and that cells do not produce HGF. Because HGF consumption is minimal and the HGF diffusion timescale (approximate seconds) is a lot shorter than cell division time scale (approximate day) and the model domain size is small (50 cells x 38 cells), any consumption would quickly be equilibrated to the steady state.

Cell line

A549 lung adenocarcinoma cell line was maintained in RPMI 1640 medium supplemented with 10% FBS. Cells were confirmed to be free of mycoplasma using PlasmoTest (Invivogen, San Diego, CA).

Western blots

Cells were washed with ice-cold PBS, and whole cell extracts were prepared using lysis buffer (0.5% NP-40, 50 mM Tris-Cl, pH 8.0, 150 mM NaCl, 1 mM EDTA) supplemented with protease inhibitor (Roche, Mannheim, Germany) and phosphatase inhibitor cocktail (Sigma-Aldrich, Carlsbad, CA). Whole-cell extracts were resolved on SDS-PAGE and transferred to nitrocellulose membrane. The membrane was blocked in 5% skim milk/PBST and then incubated in primary antibody at 4°C overnight. Bound antibodies were visualized by horseradish peroxidase-conjugated secondary antibodies and SuperSignal West Pico Chemiluminescent Substrate (Thermo Scientific, Waltham, MA). Primary antibodies used for our study were purchased from Cell Signaling Technology (Danvers, MA) (except for β-actin, which was from Sigma-Aldrich, St. Louis, MO).

Cell viability measurement

Cells were plated on 96-well plate at 2,000 cells per well and then exposed to drugs for 72 hours. Cell viability was analyzed by CellTiter-Glo (Promega, Madison, WI) according to the manufacturer’s recommendations.

Proximity ligation assays

Proximity ligation assays were performed as described using Duolink Far Red kit (Sigma-Aldrich, Carlsbad, CA) with antibodies to the following: EGFR (clone B38, Cell Signaling, Danvers, MA), GRB2 (clone 81, BD, San Jose, CA), and AF488-conjugated pan cytokeratin (clone Ae1/Ae3, eBioSciences, San Diego, CA). Tissue specimen was from a de-identified patient treated at Moffitt Cancer Center and was obtained via an institutionally approved protocol.

Experimentally measured cancer cell response to various kinase inhibitors

Kirsten rat sarcoma (KRAS)-driven cancer treatment is an important clinical need that remains largely unmet due to limited targeted drug efficacy of key downstream effectors, including MAPK and PI3K-AKT pathways. We therefore choose the KRAS mutant non–small-cell lung cancer (NSCLC) cell line (A549 cell line) as our experimental model system. Using our simplified oncogenic KRAS signaling pathway (Fig 2A, Mathematical modeling section for an explanation of how we obtained this simplified pathway) as a guide, we pharmacologically inhibited individual proteins in the MAPK pathway (MET, EGFR, MEK, ERK inhibitors) and PI3K-AKT (AKT inhibitor) pathway in A549 cells in the both absence and presence of HGF. Drug-induced changes in the phosphorylation of pathway proteins, surrogates for protein activity, were measured by western blotting (Fig 2B). Cell viability was also assessed after 72 hours of drug treatment (Fig 2C). These experimental data were quantified using ImageJ (Fig 2D), and all data were normalized to the control experimental condition (treatment-naïve condition).

Model calibration

The quantified changes (Fig 2D) and the Monte Carlo simulation were then employed as an optimization procedure to estimate model parameters (weights, W_ij) that minimize our cost function. A network with lower cost represents the experimental data more accurately. The cost function is defined as follows, (4) where is the steady state activity of protein or cell viability x_i in treatment condition d, y_d represents experimental data, and M is the total number of treatment conditions. The weight w_jr indicates the weight between RAS_m (r) and its neighbors (N(r)), and the constants (χ, η) modulate the magnitude of the penalty. The first term explains the difference between model prediction and experimental data for a network W. The second term is directed at the activating RAS mutation and incorporates a penalty for estimated weights from the RAS_m node (mutant RAS) that are too small. We included this penalty because our model is based on empirical data of a KRAS mutant cancer cell line (A549 cell), where the resulting KRAS protein is constitutively active. We aimed to capture this activating mutation by penalizing small weights from RAS_m to its neighbors.

We used the following method to implement Monte Carlo simulations:

1. Initialize a sparse weight matrix (W, W_ij = 0, for no connection in Fig 2A) with random numbers.
2. Enforce the weight elements to satisfy the prior pathway information (W_ij = |W_ij| for green line; W_ij = −|W_ij| for red line; Fig 2A).
3. Update protein node values using the Eq (1).
4. Evaluate cost C_old = C (W) using the Eq (4).
5. Randomly select an element of weight matrix and perturb it () for some small σ, σ > 0) and enforce weight constraints (step 2).
6. Update cost C_new = C (W^new) (Eq [4]) with a perturbed weight matrix (W^new).
7. If C_new < C_old, accept the perturbation; otherwise, mostly reject the perturbation. Only accept the perturbation with a small probability of p, where p = exp(−ν(|C_new − C_old|) for some large ν.
8. Go to the step 5 and repeat 5 through 7 until achieving error = |C_new − C_old| < δ, where δ is a predefined tolerance (for small non-negative number, δ > 0).

The model calibration resulted in more than 5,000 weight matrices that fit to the experimental data. We selected the best 500 (top 10%) weight matrices and used these to define our 500 in silico cells. The distributions of in silico cells are presented as box plots in Fig 2E along with the experimental measures. Errors (root-mean-squared-error [RMSE] formula given in S1 Text) are in the range of (0.03–0.56, except ERK: 0.96). The fit of ERK was poor because of unexpected inhibition of pERK by the drug (ERK inhibitor, SCH772984) [69]. The trained networks (weights) are quite heterogeneous (S1 Table). The distribution for each weight is different (S1 Table, skewed, normal, bimodal distributions, with a range of heterogeneity [Shannon] index values).

The weights here may represent relative protein abundance or protein-binding activity. There is ample evidence for differential abundance of protein species across cellular populations. An excellent example was recently published showing that variations in adaptor protein abundance are a major source of regulation of the EGFR-MAPK pathway [70]. There are several examples of differential binding activity of proteins in cell signal transduction. It is well established that adaptors such as GRB2, SHC1, and GAB1 can be recruited to receptor tyrosine kinases (RTKs) either directly or indirectly. Therefore, stochastic variation in multiprotein complex composition at individual receptors exists, and this will vary both within and between cells. It is also accepted that activation-induced receptor degradation and phosphatase activity will affect not only RTK adaptor interactions but also downstream signaling molecules such as rapidly accelerated fibrosarcoma (RAF), MEK, and ERK. Additionally, we have previously shown that, in EGFR mutant cell lines, only a fraction of the receptor is phosphorylated, and cell lines harboring the same oncogenic mutation have different levels of phosphorylated Tyrosine (Tyr) and Serine/threonine (Ser/Thr) residues [71]. Collectively, these examples indicate that protein–protein interactions in response to growth factors are not simply on–off states and that multiple factors independent of protein abundance control final signaling output.

Distributional responses to kinase inhibitors

We simulated responses of in silico cells to seven different inhibitors (EGFRi, METi, RAS_mi, AKTi, RAFi, MEKi, and ERKi). Of note, cell viability in untreated conditions is set to be a single constant value (cell viability_untreated ≡ 1). It is also worth noting that RAS_mi is assumed to inhibit only RAS_m (RAS mutant), not RAS_w. Distributions of relative cell viability (log2 scale, log₂ (treated/untreated)) of all in silico cells are presented in Fig 3A. Similar to the experimental results (Fig 2B–2D), MEKi and ERKi reduced average cell viability significantly, whereas mean effects of EGFRi, METi, and AKTi are marginal. These results reveal quite heterogeneous responses to drugs, which could be assessed by experimental approaches [72, 73]. For example, the distribution of EGFRi treatment is bimodal due to a bimodal distribution of trained MEK-ERK weights (S1 Table and S2 Fig), suggesting the presence of a subpopulation that responds significantly differently to drug from the rest of the population (S2 Fig, green versus pink).

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Fig 3. Model prediction and validation.

(A) Histograms of relative cell viabilities (log2 scale) of all in silico cells after treatments of EGFRi, METi, RAS_mi, AKTi, RAFi, MEKi, and ERKi. First, a bimodal distribution EGFRi-treated in silico cells (one mode: −1.25; second mode: −0.5). Second, a skewed distribution after METi (a skew toward 0; 0: no change). Third, a uniform distribution in response to RAS_mi (almost uniform distribution from −1.5 to 1.5). Fourth, a slight bimodal distribution after AKTi. Fifth, a distributional response after RAFi. Sixth, a normal distribution in response to MEKi. Seventh, a normal distribution after ERKi. (B) Histograms of relative cell viabilities of all in silico cells in log2 scale. First: EGFRi only (blue), EGFRi/ERKi (orange), and EGFRi/MEKi (yellow). Second: METi only (blue), METi/ERKi (orange), and METi/MEKi (yellow). Third: RAS_mi only (blue), RAS_mi/RAFi (green), RAS_mi/ERKi (orange), and METi/MEKi (yellow). Fourth: AKTi only (blue), AKTi/RAFi (green), AKTi/ERKi (orange), and AKTi/MEKi (yellow). Fifth: RAFi only (blue), RAFi/ERKi (orange), and RAFi/MEKi (yellow). (C) Validation. Model predicted relative cell viabilities (red bars) and experimental data (gray bars) after 10 different treatments. First: EGFRi, EGFRi/MEKi, and EGFRi/ERKi. Second: METi, METi/MEKi, METi/ERKi. Third: AKTi, AKTi/RAFi, AKTi/MEKi, AKTi/ERKi. The numerical data used in Fig 3 are included in the second sheet S1 Data. AKT (PKB), protein kinase B; DMSO, Dimethyl sulfoxide (control); EGFR, epidermal growth factor receptor; ERK, extracellular receptor kinase; MEK, mitogen-activated protein kinase kinase; MET (c-MET), tyrosine-protein kinase Met or hepatocyte growth factor receptor (HGFR); RAF, rapidly accelerated fibrosarcoma; RAS, rat sarcoma.

https://doi.org/10.1371/journal.pbio.2002930.g003

In silico–predicted rational drug combinations

With our calibrated in silico cell lines, we next examine which drug combinations significantly reduce cell viability and which show marginal effects. For example, what should be cotargeted with AKTi to decrease cell viability significantly? The model predicts that activation of alternative pathways under therapy may provide an escape route to therapy. For example, AKT inhibitor induced increased activity of ERK and ribosomal S6 kinase (RSK) (S3 Fig). Cells with high ERK and RSK activity display resistance (cell viability >0) under the inhibitor, which implies that simultaneous inhibition of this alternative pathway would overcome resistance. We reasoned that combination targeting of proteins that are highly correlated with relative cell viability under a given treatment would be beneficial. In order to identify such protein nodes, scatter plots between predicted protein activity (phosphorylation) and predicted relative cell viability were considered (S4 Fig). Then, the Pearson’s coefficient was calculated for all cases. We observed that, under multiple treatment conditions—including inhibition of EGFR, MET, RAS_m, AKT, and RAF—ERK and RSK showed the highest correlation with relative cell viability (S4 Fig, pink box). This suggests that co-inhibition of ERK (by MEKi) or RSK (by ERKi) activity with other therapies would decrease cell viability most significantly. Further simulation revealed an additional application of either MEKi or ERKi to each—EGFRi, METi, RAS_mi, RAFi, and AKTi—significantly decreased cell viability compared to monotherapy of EGFRi, METi, RAS_mi, RAFi, and AKTi (Fig 3B). We tested some of these model predictions experimentally and validated, to some degree, the model’s predictive ability (Fig 3C).

Hierarchical clustering of in silico cell responses

We next systematically assessed cell viability reduction to all mono and combination therapies using an unsupervised hierarchical clustering approach to classify cell populations on the basis of their treatment response (relative cell viability). The treatments were categorized into multiple groups (Fig 4A, a tree diagram on the right end of heat-map). The combination of AKTi with MEKi is uniformly effective to all the cells (see the red asterisk [*] row, dark blue across all in silico cells, with little variation between cells). This combination (MEKi/AKTi) has previously been shown to be effective in NSCLC both in vitro and in vivo [74]. A striking variation is observed in response to the treatments of AKTi, RAS_mi, and RAS_mi/AKTi (Fig 4A, red bars in the first two groups versus dark blue to yellow bars in the rest of the groups). The first two clusters (pink and black color on the top of the heat-map) are associated with poor responses (little to no reduction of cell viability after a given therapy), while others are correlated with good treatment outcomes (significant reduction of cell viability after a given therapy).

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Fig 4. Model predicted combination therapy effect.

(A) Hierarchical clustering and heat-map of relative cell viabilities after 28 different treatments. Simulated treatment responses (cell viability changes) were clustered using an unbiased hierarchical method with a Euclidian distance function, resulting in 7 different clusters (pink, black, blue, green, purple, cyan, and orange color) indicated by color bar on the top of heat-map. Each row indicates an individual therapy. Each column indicates an individual in silico cell. Blue to yellow bars: decrease to increase. The asterisk [*] indicates relative cell viability after a combination therapy of AKTi with MEKi. (B–C) Chord diagrams and weighted network diagrams of the representative in silico cells a and b. Chord diagram: each node in the circle represents each protein node in the network model, represented by different colors. The thickness of chord between two protein nodes represents a weight between two protein nodes (weight, w_ij). The chords are directed, colored by originating sector color. For example, the interaction between RAS_m and RAF is depicted as a light blue chord because the direction is from RAS_m to RAF (RAS_m → RAF; color of RAS_m sector: light blue). Weighted network diagram: the width of each edge represents the weight. A thicker edge represents a larger weight between two proteins. The numerical data used in Fig 4 are included in the third sheet S1 Data. AKT (PKB), protein kinase B; EGFR, epidermal growth factor receptor; ERK, extracellular receptor kinase; HGF, hepatocyte growth factor; MEK, mitogen-activated protein kinase kinase; MET (c-MET), tyrosine-protein kinase Met or hepatocyte growth factor receptor (HGFR); PI3K, phosphoinositide 3-kinase; RAF, rapidly accelerated fibrosarcoma; RAS, rat sarcoma; RAS_m, mutated RAS; RAS_w, wild-type RAS; RSK, ribosomal S6 kinase.

https://doi.org/10.1371/journal.pbio.2002930.g004

Possible mechanisms of response and resistance

Why are some cells sensitive to a given therapy while others are resistant to the same therapy? We hypothesize that this differential drug sensitivity, at least within the context of our model, must be attributed to differential protein activity as modulated by protein–protein interactions (i.e., the weights). To highlight possible mechanisms, we visualized the weights between protein nodes (W_ij in our model) using both circular chord diagrams [75] and network diagrams with weighted edges (Fig 4, bottom panels; left: chord diagram; right: weighted network with different edge widths). In the circular diagrams, protein nodes are arranged around a circle with the weight between protein nodes connected to each other through the use of arcs. The width of each arc is determined proportionally by the weight between two protein nodes. To illustrate differences in signaling, we selected two representative in silico cells (Fig 4A and 4B), where cell a is resistant to AKTi, RAS_mi, and RAS_mi/AKTi and cell b is more sensitive to these therapies. In addition, we compared ranges of all weights of all in silico cells between clusters defined by the hierarchical clustering (S5A Fig). We observe heterogeneity of weights within each cluster and between clusters. Differences between clusters are significant for some weights such as weights of growth factor EGFR, EGFR-RAS_w, MET-RAS_w, RAS_w-RAF, and MEK-ERK (S5A Fig).

Heterogeneous responses to HGF stimulation

We next asked how an additional microenvironmental stimulation would modulate the responses to targeted treatments. To address this question, all in silico cells were treated in the presence of HGF, a significant stromal factor that contributes to drug resistance [35, 36]. An unsupervised hierarchical clustering on the basis of cell viability changes, from the no-HGF condition, separated the treatments into several groups (Fig 5A, clustering of treatments on the right tree diagram). The analysis also classified the in silico cells into several groups based on cell viability changes due to HGF stimulation (Fig 5A, tree diagram on the top of heat-map).

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Fig 5. Effect of HGF stimulation on treatment responses.

(A) Heat-map of cell viability changes due to HGF stimulation (log₂ (treated with HGF/untreated)-log₂ (treated without HGF/untreated)). An unbiased hierarchical clustering separated the cell population into 6 different clusters (indicated by colors on the top of heat-map). Each row indicates a treatment, and each column indicates an in silico cell. Gray to red: no change to increase due to HGF. (B) Model validation. Comparisons between experimental data (gray bars) and model predictions (red bars) after three different combinations (AKTi/MEKi, EGFRi /MEKi, and EGFRi/AKTi). Relative change of cell viability in treated-with-HGF condition to one in treated-without-HGF condition (no HGF) is reported. (C) Chord diagrams and weighted network diagrams that visualize weights between two protein nodes in the representative in silico cell a and b. The numerical data used in Fig 5 are included in the fourth sheet S1 Data. AKT (PKB), protein kinase B; EGFR, epidermal growth factor receptor; ERK, extracellular receptor kinase; HGF, hepatocyte growth factor; MEK, mitogen-activated protein kinase kinase; MET (c-MET), tyrosine-protein kinase Met or hepatocyte growth factor receptor (HGFR); PI3K, phosphoinositide 3-kinase; RAF, rapidly accelerated fibrosarcoma; RAS, rat sarcoma; RAS_m, mutated RAS; RAS_w, wild-type RAS; RSK, ribosomal S6 kinase.

https://doi.org/10.1371/journal.pbio.2002930.g005

To test some of these model predictions, we considered three different combination therapies, AKTi/MEKi, EGFRi/MEKi, and EGFRi/AKTi. Of note, the model predicted that the treatments AKTi/MEKi and EGFRi/MEKi are not affected by HGF stimulation (Fig 5B, first two red bar graphs). The experimental data matched well with these predictions (Fig 5B, first two gray bar graphs). The model also predicted that the effect of combination therapy EGFRi/AKTi is significantly modulated by HGF stimulation (Fig 5B, the third red bar). This was also corroborated by experiment (Fig 5B, the third gray bar).

Possible mechanisms explaining HGF modulation

The suite of in silico cells is differentially affected by HGF stimulation. For example, some cells are not affected by the HGF stimulation (e.g., gray bars in EGFRi/RAS_mi in Fig 5A), while others are significantly affected by stimulation (e.g., red bars in EGFRi/RAS_mi in Fig 5A). Why are some cells affected by HGF stimulation, while others are not? To understand why this is the case, we selected two representative cells (a and b) and visualized the weights between protein nodes using both chord diagrams and weighted network diagrams (Fig 5C, bottom panels). In cell a, the influence of MET on RAS_w is relatively small (Fig 5C, thin blue chords from MET → RAS_w). In contrast to cell a, the influence of MET on RAS_w in cell b is much stronger (Fig 5C, cell b, thick blue chord from MET → RAS_w). This may explain why cell b increased its viability significantly upon HGF stimulation compared to no-HGF condition. In addition, distributions of all weights in each cluster show differential activity of proteins and its effects across all six different clusters (S5B Fig).

Competition between in silico cells drives different dominant populations

So far, treatment responses were assessed as if all in silico cells were treated in isolation. To address the effects of cell competition on treatment outcomes, we simulated all cells growing together under treatment with various inhibitors in a homogeneous, resource-limited microenvironment (Methods section). The entire in silico cell population responds in a similar way on some therapies (Fig 6, e.g., EGFRi, METi, RAFi, EGFRi/RAS_mi, etc.), but under other therapies (Fig 6, e.g., MEKi, ERKi, MEKi/ERKi, AKTi/RAFi, etc.)—due to differential viability—some cells became dominant. Interestingly, some treatments simultaneously selected for the same dominant in silico cell (Fig 6, color-shaded boxes). Addition of HGF to some treatments (e.g., EGFRi, RAS_mi, RAFi) significantly changes the fitness of in silico cells and thus drives a different dominant in silico cell (S6 Fig, dotted lines in no-HGF vs solid lines in HGF).

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Fig 6. Effect of homogeneous, resource-limited microenvironment on treatment outcomes and heterogeneity.

(A) The growth dynamics of all in silico cells treated with various therapies for three months in a homogeneous and resource-limited microenvironmental condition. A color line at each treatment indicates the most dominant cell after a given therapy. Color-shaded boxes indicate treatments selecting for the same dominant in silico cell. Orange box: dominant cell (cell ID: 342) after treatments of EGFRi/METi, EGFRi/AKTi, METi/AKTi, EGFRi/RAFi, and AKTi/RAFi. Red box: dominant cell (ID: 417) after RAS_mi/METi and AKTi. Yellow box: dominant cell (ID: 337) after EGFRi/MEKi, METi/MEKi, and AKTi/MEKi. Blue box: dominant cell (ID: 36) after RAFi/MEKi, MEKi, RAFi/ERKi, and MEKi/ERKi. Purple box: dominant cell (ID: 156) after EGFRi/ERKi, METi/ERKi, and AKTi/ERKi. Green box: dominant cell (ID: 113) after ERKi and ERKi/RAS_mi. (B) Linear relationship between average relative cell viability (log2 scale) and post-treatment Shannon indexes. Linear correlation constant: 0.65. Square: monotherapy; circle: combination therapy. Red: EGFRi alone or combination with METi, RAS_mi, AKTi, RAFi, MEKi, or ERKi. Blue: METi alone or combination with RAS_mi, AKTi, RAFi, MEKi, or ERKi. Pink: RAS_mi alone or combination with RAFi, AKTi, MEKi, or ERKi. Green: AKTi alone or combination with RAFi, MEKi, or ERKi. Gray: RAFi, RAFi/MEKi, RAFi/ERKi. Yellow: MEKi, MEKi/ERKi. Orange: ERKi. (C) Linear relationship between average relative cell viability (log2 scale) and post-treatment (with HGF stimulation) Shannon indexes. Linear correlation constant: 0.63. Square: monotherapy; circle: combination therapy. Color definitions are the same as panel B. (D) Therapies affected by HGF significantly. Each arrow starts from a point without HGF stimulation to a point with HGF stimulation. Gray: RAFi/ERKi and RAFi/MEKi. Blue: RAS_mi, RAS_mi/AKTi, RAS_mi/ERKi. Red: EGFRi, EGFRi/RAS_mi, EGFRi/AKTi, EGFRi/MEKi, and EGFRi/ERKi. The numerical data used in Fig 6 are included in the fifth sheet S1 Data. AKT (PKB), protein kinase B; corr, linear correlation; EGFR, epidermal growth factor receptor; ERK, extracellular receptor kinase; HGF, hepatocyte growth factor; MEK, mitogen-activated protein kinase kinase; MET (c-MET), tyrosine-protein kinase Met or hepatocyte growth factor receptor (HGFR); PI3K, phosphoinositide 3-kinase; RAF, rapidly accelerated fibrosarcoma; RAS, rat sarcoma; RAS_m, mutated RAS; RAS_w, wild-type RAS.

https://doi.org/10.1371/journal.pbio.2002930.g006

Assessment of post-treatment in silico cell population heterogeneity

We assessed post-treatment population heterogeneity by measuring the Shannon index (H(x) = −∑_ip_i(x)log(p_i(x)), where p_i(x) is the probability of finding an in silico cell i after a given therapy). We compared the index with average cell viability change (Fig 6B). The two are linearly correlated (ρ = 0.65), implying that the less effective a treatment is in controlling the average in silico population growth, the more heterogeneous the post-treatment population would be (Fig 6B). Combination therapies not only display a better average treatment response but also a less diverse post-treatment population (Fig 6B, boxes versus circles). Among all combination therapies, those that combined either with ERKi or MEKi display much better average therapeutic responses (Fig 6B, small cell viability). These treatments not only effectively decrease average cell viability but also lead to a less diverse post-treatment population (e.g., Fig 6B, EGFRi/[ERKi or MEKi] vs EGFRi/[METi,AKTi,RAFI] in red-color circles). HGF stimulation minimally affected the linear relationship between post-treatment heterogeneity and average cell viability reduction (Fig 6C, ρ = 0.63 vs Fig 6B, ρ = 0.65). However, a few treatments did elicit significant changes in both average response and heterogeneity due to HGF stimulation (Fig 6D).

Impact of microenvironmental heterogeneity on targeted therapy outcomes

Because activated receptors require multiple protein interactions to activate downstream signaling, we have utilized proximity ligation assays that measure the functional association of RTKs and adaptor proteins. Using NSCLC patient specimens and xenograft models, we have previously identified an association of EGFR:GRB2 complexes and response to EGFR inhibition [76] and more recently identified a correlation between MET:GRB2 complexes and response to MET kinase inhibitors [77]. Using this approach, we consistently observe spatial heterogeneity in abundance of RTKs binding to adaptor complexes. The abundance is often highest at tumor regions that are adjacent to stromal regions (Fig 7A).

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Fig 7. Effect of microenvironmental heterogeneity on a targeted therapy outcome and combination therapy.

(A) Heterogeneous EGFR activity in a lung squamous cell carcinoma. Representative image of red foci showing EGFR:GRB2 proximity. Tumor cells (green) are stained with a cytokeratin antibody demarcating epithelial origin. Nuclei (blue) are stained with DAPI. Image was acquired at 200x. (B) HGF distribution. The Eq (3) was solved assuming the following parameters: γ = 1.0, diffusion rate D = 0.04, and decay rate λ = 0.001. To be consistent with the choice of parameter in the pathway model (Eq [1]), we use a scaling factor ω (ω ≡ 10) (i.e., HGF input in the pathway model x₂ = ωH(x,t), H(x,t): solution obtained using the assumed parameters). (C) First, an initial randomized configuration of cells (domain size: 50 cells × 38 cells). Color represents different in silico cells. Second, a snapshot of RAS_m inhibitor simulation (day 180). (D) Simulated protein activity after 180 days of RAS_m inhibition. The activities of MET, EGFR, RAS_w, and RAF are heterogeneous (yellow to blue color), while those of AKT, MEK, ERK, and RSK are less heterogeneous (blue color). (E) Comparison of combination therapies for 400 days. First, a snapshot of simulation (day 400) after a sequential therapy of RAS_mi for the first 200 days and then RAFi for the rest 200 days (200 days of RAS_mi → 200 days of RAFi). Color represents different in silico cells. Second, a snapshot of simulation (day 400) after a sequential therapy of RAFi for the first 200 days, then RAS_mi for the remaining 200 days (200 days of RAFi → 200 days of RAS_mi). Third: the number of cells over time during the two sequential therapies of RAS_mi and RAFi (blue: RAS_mi → RAFi, red: RAFi → RAS_mi) and a concurrent therapy of the two (green: RAS_mi/RAFi). The numerical data used in Fig 7 are included in the sixth sheet S1 Data. AKT (PKB), protein kinase B; DAPI, 4',6-diamidino-2-phenylindole; EGFR, epidermal growth factor receptor; ERK, extracellular receptor kinase; GRB2, growth factor receptor bound protein 2; HGF, hepatocycte growth factor; MEK, mitogen-activated protein kinase kinase; MET (c-MET), tyrosine-protein kinase Met or hepatocyte growth factor receptor (HGFR); RAF, rapidly accelerated fibrosarcoma; RAS, rat sarcoma; RSK, ribosomal S6 kinase.

https://doi.org/10.1371/journal.pbio.2002930.g007

Motivated by this experimental observation (Fig 7A), we developed an HCA model to investigate the effects of microenvironmental heterogeneity on treatment outcomes (See Methods section and S1 Fig). A steady state configuration of HGF is considered throughout the whole simulation (See Methods section and Fig 7B). We randomly initialized in silico cells that contain calibrated signaling networks (Fig 2) in the domain to mimic a slice of tumor tissue (Fig 7C first, time step = 0; domain size: 50 cells × 38 cells). As proof of concept, we simulated RAS_m inhibition. After 180 days of the inhibition, distinct cells emerge near stroma (high HGF) in contrast to the nonstroma region (Fig 7C second, light- and dark-blue cells near stroma versus orange, red cells elsewhere). A clearer separation among the cell population emerges as therapy continues (S1 Movie). We also observed heterogeneous protein activity across the tissue (Fig 7D and S2 Movie). The signaling responses of some cells are affected by HGF (Fig 7D, e.g., high MET and RAF, black arrows near stroma), while signaling in other cells is not (Fig 7D, e.g., low activity of MET and RAF near stroma, red arrow).

The treatment selects for cells with high MET and RAF activity (phosphorylation), especially residing near the stroma (high HGF), which suggests that a therapy of METi or RAFi in addition to RAS_mi may be more effective. To test this suggestion, we simulated two sequential therapies of RAS_mi and RAFi and one concurrent therapy (Fig 7E and S3–S6 Movies). Depending on the order of the sequence (RAS_mi first versus RAFi first), different patterns of cells emerge after 400 days of treatment (Fig 7E, first versus second). Importantly, a concurrent combination of the two inhibitors was effective enough to eradicate all cells in this small region just after 30 days of treatment (S5 Movie and Fig 7E third, number of cells). The direct spatial competition of each cell within the tissue directly facilitated this result. To be more relevant to the clinical timeframe, we also simulated a shorter treatment schedule (e.g., 60 days) and observed similar cell behavior (S7 Fig).

Until now, we assumed that the initial states of the protein activities in our in silico cells were all zero. In order to examine the impact of changing this on the above treatment outcomes, we randomly seeded in silico cells in a slice of tissue (Fig 7C), and for each in silico cell, a random number was assigned to each initial protein activity. Then, we simulated RAS_m inhibitor for 30 days using our HCA model (S1 Fig) and repeated this process 100 times. The resulting configurations display some degree of heterogeneity (S8A Fig, shows representative results for three different initial configurations) due to cell–cell spatial competition as well as variable HGF modulation of the in silico cells. The HGF selects for cells whose cell viability is significantly modulated by HGF (violet to red cells near stroma). To illustrate this more accurately, we quantified the total number of surviving cells at time step 30 for each simulation (see S8B Fig for distributions) and classified them in terms of HGF modulation (S8C Fig). The treatment consistently selected for certain cells (S8C Fig), some of which are more influenced by HGF than others (S8C Fig).