SYSTEMS-LEVEL CONSEQUENCES OF LOW RAF ABUNDANCE FOR EGFR-ERK SIGNALING

Abstract

RAFs initiate the cascade leading to activation of the extracellular signal-regulated kinases (ERK). In a substantial fraction of cancer cells, RAFs are the least abundant pathway proteins between receptor tyrosine kinases (RTKs) and ERK. In some cases, active RAF kinases are present at the plasma membrane at just hundreds of copies per cell, but the consequences of such limited RAF abundance are unclear. By developing continuum and stochastic computational models of the epidermal growth factor receptor (EGFR)-ERK pathway, we showed that low RAF abundance creates signaling bottlenecks between RTKs and ERK with a potential for stochastic RAF dynamics that can propagate especially to low-abundance downstream pathway proteins. RAF bottlenecks were also predicted to impede ERK activation by oncogenic RAS mutants. Advanced parameter sensitivity and Sloppiness analyses identified RAS activation and RAS-RAF interactions as strong determinants of signaling in low-RAF settings and revealed an efficient model fitting approach. This work provides quantitative insight into a common, but unexplored, regime for EGFR-ERK signaling and a systematic approach to develop and characterize dynamic models of receptor-mediated signaling.

INTRODUCTION

The EGFR-ERK signaling pathway has been extensively studied due to its ubiquitous roles in normal physiology and disease (1). While the properties and functions of the constituent proteins are well characterized (2,3), it is only recently that mass spectrometry enabled their absolute quantification (4). We now understand that some proteins exist in surprisingly low abundance (e.g., ~1000 BRAF per HeLa cell (5)), suggesting potential rate-limiting steps in signal transduction. For example, in HeLa cells, the multifunctional adapter GRB2-associated binding protein 1 (GAB1) is expressed at ~1500 copies/cell (6), limiting its ability to bind and activate the abundant protein tyrosine phosphatase SHP2 (~300,000 copies/cell) (5,7). Receptors can also be limiting compared to their common adapters. For example, with less than half of EGFR proteins becoming tyrosine-phosphorylated in HeLa cells treated with near-saturating EGF concentrations (8), the peak ratio of phosphorated EGFR to GRB2 is estimated at 1:3.5 (9). While the concentrations and stoichiometries of network proteins clearly impact signaling dynamics (6), the quantitative effects of apparent stoichiometric limitations are poorly understood, in part because stoichiometries alone do not determine pathway fluxes.

A recent, unexpected discovery is that RAF1, a member of the RAF family kinases that link EGFR-RAS to the mitogen-activated protein kinases (MAPKs), can be expressed at fewer than 12,000 copies per cell (10). As a result, RAF1 abundance at the plasma membrane, which transiently reaches about 10–15% of total cellular RAF1, can be as low as 200 molecules per cell when ERK activity peaks 6–8 min after EGFR activation (10). Thus, cells with strikingly low RAF1 abundance at the membrane still activate ERK robustly and with dynamics that are much more protracted than those with which RAF1 peaks at the membrane. Although RAF1 contains structural domains that mediate interactions with multiple proteins, MEK is its only definitive substrate, suggesting that changes in RAF1 abundance would primarily impact ERK signaling (11). Low RAF1 abundance has been observed in multiple cell backgrounds. HeLa cells express an estimated ~7800 RAF1/cell based on quantitative immunoblotting using GST-tagged standards and an assumed cell diameter of 12.9 μm (12). Human mammary epithelial and triple-negative breast cancer cells express an estimated 6750 and 9720 RAF1/cell based on high-resolution chromatography and mass spectrometry (6). Whereas ARAF and BRAF preferentially bind to KRAS, RAF1 binds all GTP-loaded RAS proteins (13). Additionally, RAF1 has higher kinase activity towards MEK than ARAF (11,14), and heterodimerization with membrane-localized RAF1 may be required for BRAF’s activity towards MEK (11,15,16). Thus, RAF1 is the most important isoform for determining RAF-dependent signaling limitations.

RAF1 activation by RAS is complex and involves dimerization, membrane localization, interactions with phospholipids and scaffold proteins, and phosphorylation-dephosphorylation at multiple residues (17,18). RAF1 kinase activity is suppressed by its phosphorylation at Ser259 and Ser621 (19), and RAF1 is activated upon binding to GTP-loaded RAS and phosphorylation of the N-region and activation loop including Ser338 (17,19). RAF1 activation dynamics have mainly been studied experimentally using overexpressed RAF1 and RAF1 mutants (20). The consequences of the low membranous RAF1 abundance are therefore challenging to intuit, but their understanding will elucidate poorly understood aspects of pathway regulation. For example, low protein copy numbers generate noise and cell-to-cell heterogeneity in signaling (21,22). Limiting RAF1 expression could play an unappreciated role in these phenomena. Low RAF1 abundance may also create a signaling bottleneck (i.e., an overall rate-limiting step involving RAF1) in ERK activation via EGFR or other RTKs and explain the poor correlation between activating RAS mutations and ERK activity that have been observed in vitro and in vivo (23–25). Improved understanding of the role of RAF1 abundance on EGFR-ERK signaling could provide key insights into how this pathway is dysregulated in cancer (26).

Computational models of signaling based on coupled differential equations have become a mainstay for interpreting network dynamics and identifying operating regimes (3,27). Parameter sensitivity analyses have frequently been used to identify rate processes that strongly control model outputs (7,10,28–32), with some insights on protein abundance effects. For example, the low abundance of Par proteins was predicted to determine sensitivity of polarized C. elegans cells to chemical or mechanical perturbations (33), and low-abundance RSK was correctly predicted to dictate the magnitude and duration of the ribosomal protein S6 kinase activity in response to erythropoietin receptor stimulation in BaF3 murine cells (34). Parameter sensitivity analyses include univariate and multivariate approaches, which respectively quantify model output changes for perturbations to individual parameters or multiple parameters simultaneously (35,36). Sensitivity analysis can also aid model calibration by identifying parameters that strongly control fit (37). This can also be accomplished by Sloppiness analysis, a Bayesian inference method (38) that assesses parameter uncertainty of a model constrained by experimental data and prior knowledge of parameter values (39). A byproduct of Sloppiness analysis is the identification of directions in parameter space that control model fit. This is particularly useful for biochemical network models, where individual parameters are often poorly constrained and groups of parameters control dynamics (40). Sloppiness and sensitivity analyses thus yield different views of parameter importance, but the common goal of model calibration has obfuscated the difference (41).

Here, we developed an ordinary differential equation (ODE)-based mechanistic model of EGFR-ERK signaling for the setting of low RAF abundance, which is common in human cancer cells. To distill model output, we used a data-driven approach to compute model sensitivity to multivariate parameter perturbations and compared results with model Sloppiness analysis. The results identified RAF1:RAS-GTP binding kinetics as critical determinants of EGFR-ERK signaling, revealed the ability of low-abundance RAF1 to limit ERK activation by oncogenic RAS mutants, and demonstrated that stochastic RAF1 signaling can arise and propagate to low-abundance downstream nodes. Results are sequentially organized to present model fit and characteristics, discoverable inferences from parameter variation, implications for cancer cell settings, and predictions for settings where stochastic dynamics set in.

METHODS

Computational Methods

Clinical Proteomic Tumor Analysis Consortium (CPTAC) Data Analysis

We used CPTAC pancreatic ductal adenocarcinoma (PDAC), glioblastoma multiforme (GBM), colon adenocarcinoma (COAD), and lung squamous cell carcinoma (LSCC) proteomics to compare relative protein expression of EGFR, RAS (HRAS, NRAS, and KRAS), RAF (RAF1 and BRAF, where both were available), MEK1/2, and ERK1/2. CPTAC proteomics were accessed in Python via the cptac package. Proteins were labeled by corresponding gene symbols, as in CPTAC. Only samples with data for all proteins of interest were included in Figure 1.

Figure 1. In a substantial fraction of tumor and cancer cell settings, RAF is the least expressed EGFR-ERK pathway protein.

(A) Fractions of patient tumor samples from CPTAC pancreatic ductal adenocarcinoma (PDAC), glioblastoma multiforme (GBM), colon adenocarcinoma (COAD), and lung squamous cell carcinoma (LSCC) proteomics data with EGFR, RAS (HRAS, KRAS, NRAS), RAF (RAF1 and BRAF), MAP2K (MAP2K1/2), or MAPK (MAPK1/3) as the least abundantly expressed protein are shown. Proteins are labeled with the gene name that encodes for each protein, as reported in CPTAC proteomics data. (B) Fraction of cell lines from CCLE proteomics data that have the indicated proteins as the lowest-expressing protein is shown. Proteins are labeled with the gene name that encodes for each protein, as reported in CCLE proteomics data. (C) For CCLE cell lines where RAF was the least abundantly expressed protein, mean normalized expression of the indicated proteins was calculated, and standard errors of data are indicated by bars. (D) Mean-centered, variance-scaled EGFR, RAS, RAF, MAP2K, and MAPK expressions of cell lines from analysis in C were subjected to PCA. Percent variance explained by each principal component of a PCA model is shown. (E) A PC2 vs. PC1 biplot for the PCA described in D is shown. The scores of each observation (cell lines, dots) are colored by z-scored RAF1 expression and weights of each feature (arrows) are indicated. Cell lines where RAF was the least abundantly expressed pathway protein (blue bar in C) are labeled. (F) Average silhouette scores for PCA models built with different numbers of clusters are shown. Number of clusters used for k-means clustering followed by PCA in E is highlighted in red.

We also used CPTAC proteomics and phosphoproteomics to compare phosphorylated ERK in lung adenocarcinoma (LUAD) and COAD tumors that were classified as ‘low’ or ‘high’ RAF abundance based on median expression thresholding. Data were accessed in R via the reticulate package (42). R was used because the LUAD data did not become available on the cptac Python package until after we performed these calculations. For LUAD, RAF1 measurements were not available, and BRAF measurements data were used instead. RAF1 data were available for COAD. Proteomics data were merged with KRAS mutation data to classify tumors as ‘wild-type KRAS’ or ‘mutant KRAS.’ Only the available phosphorylation sites in CPTAC phosphoproteomics for each tumor – ERK1 pT202/pY204 and ERK2 pT185/pY187 for LUAD, and ERK1 pT202 and ERK2 pT185 for COAD – were used for comparisons. Python and R analyses are available at https://github.com/orgs/lazzaralab/repositories under the username “lazzaralab.” See Table S1 for details on R packages and Python libraries used and Table S2 for details on CPTAC data.

Cancer Cell Line Encyclopedia (CCLE) Data Analysis

To survey RAF expression in different cancer cells, we used the CCLE (43). Principal component analysis (PCA) and $k$-means clustering of cell lines based on CCLE mass spectrometry data were performed in R. 144 of 375 cell lines had complete protein expression measurements for EGFR, RAS, RAF, MEK, and ERK and were included in the analysis. Proteins were labeled by corresponding gene symbols, as reported in the CCLE. For PCA, the optimal number of principal components (PCs) was chosen as the number needed to explain at least 50% of data variance. $k$-means clustering was performed using the cluster package, and the optimal number of clusters to include in the PCA biplot of loadings and scores was chosen as the number required for the highest average silhouette score. FactoMineR, factoextra, and cluster R packages were used for figures. See Table S2 for details on the CCLE dataset.

Drug Sensitivity Analysis

Drug sensitivity data (raw cell viability measured by Promega CellTiterGlo assay) from the Sanger Genomics of Drug Sensitivity in Cancer version 2 (GDSC2) dataset (969 cell lines and 297 compounds) were downloaded from the GDSC project website. To assess cell sensitivity to MEK inhibitors based on RAF expression and RAS mutational status, drug sensitivity data were combined with GDSC2 genetic feature data and CCLE proteomics and then filtered to extract results for PD0325901, refametinib, selumetinib, and trametinib (all investigational or FDA-approved MEK inhibitors). 239 cell lines with complete viability, RAF expression (sum of RAF1, BRAF, ARAF), and RAS mutational status data were included in the analyses. Cell lines with RAF protein expression below the 25^th percentile or above the 75^th percentile were categorized as “low” or “high” RAF expression, respectively. Cell lines with at least one mutant allele encoding KRAS, NRAS, or HRAS were categorized as “RAS mutant.” We compared assay fluorescence intensities for cells in the presence of PD0325901, refametinib, selumetinib, or trametinib at the maximum drug concentrations tested (2.5, 10, 10, or 1 μM, respectively). See Table S1 for details on all R packages used and Table S2 for details on the GDSC dataset.

Model Description

The computational model of EGFR-ERK signaling developed here used the RAS-ERK module from Surve et al. (10) as a basis for that part of the network. While the Surve et al. model used a forcing function to describe RAS-GTP dynamics (10), the model developed here explicitly considered upstream processes leading to RAS activation, beginning with EGF binding to EGFR. This enabled prediction of how EGFR dynamics and other processes upstream of RAS affected RAF1 membrane localization. BRAF and RAF1 were the only RAF isoforms included as they are the primary binders of RAS with strong kinase activity towards MEK in response to EGF, and RAF1 achieves full kinase activity through dimerization, either as a homodimer or heterodimer with BRAF (11,16). Based on prior quantification (5), RAF1 abundance was assumed to be more than ten times greater than BRAF abundance, making RAF1 the rate-limiting RAF isoform.

The model consists of 46 coupled ODEs for 46 species (Table S3) and 40 parameters (Table S4). Initial conditions for each model species were based on protein abundances measured in HeLa cells (Table S4). The model describes EGF-EGFR binding, EGFR dimerization, EGFR dimer phosphorylation-dephosphorylation, GRB2 binding to phosphorylated EGFR (with or without SOS in complex), RAS activation, RAF1 binding to RAS-GTP, phosphorylation of MEK and ERK, and negative feedback regulation through phosphorylation of SOS by ERK that disrupts SOS association with GRB2 and through phosphorylation of RAF by ERK that leads to RAF1 dissociation from RAS-GTP (Fig. 2A). The model was simulated with or without the RAF kinase inhibitor sorafenib. SHC was not included because it plays a minor role in determining how much GRB2 is EGFR-bound (9) and has a minor role in determining phenotypes (9,44). Activated SHP2 complexes were not included because they regulate ERK signaling (45) via a cytosolic reaction-diffusion mechanism that occurs distal from EGFR (7,45) in a complex rate process not in scope for this model.

Figure 2. Fitted model recapitulates signaling dynamics measured in EGF-treated HeLa cells.

(A) Schematic of the EGFR-ERK signaling pathway model. The dynamic model describes signaling initiated by epidermal growth factor (EGF) binding to its receptor, leading to EGFR dimerization and phosphorylation of a representative C-terminal tyrosine on each receptor. Phosphorylated EGFR is bound by the adapter protein growth factor receptor-bound protein 2 (GRB2) and the guanine nucleotide exchange factor Son of Sevenless (SOS) to catalyze the exchange of guanosine diphosphate (GDP) to guanosine triphosphate (GTP) on membrane-bound RAS. GTP-loaded RAS (RAS-GTP) recruits RAF1 to the plasma membrane and induces an ‘open’ conformation for RAF1 to become phosphorylated, hetero- or homodimerize with BRAF. Catalytically active, phosphorylated RAF1 dimers recruit and phosphorylate mitogen-activated protein kinase (MEK), which catalyzes phosphorylation of extracellular signal-regulated protein kinase (ERK). Phosphorylated ERK participates in negative feedback loops, wherein ERK phosphorylates active and sorafenib-inhibited RAF1/BRAF (iRAF1, iBRAF), causing their release from RAS-GTP at the membrane, and SOS, disrupting its association with GRB2. Negative feedback regulations are indicated by dotted arrows. Reaction network was built in VCell, and model schematic was created with Biorender.com. (B) Model rate constants were fitted against immunoblot measurements for RAS-GTP in HeLa/HRAS-mVenus cells reported by Pinilla-Macua et al. (55) and live-cell imaging of membrane RAF1 with or without 10 μM sorafenib and immunoblot measurements for pMEK and pERK in HeLa/RAF1-mVenus reported by Surve et al. (10). Time points at which 5% of RAF1 remained active at the plasma membrane are indicated by vertical dotted lines. Throughout the panels, the means of model outputs generated from 1751 parameter sets are indicated by dotted lines, best-fit model outputs are indicated by bold lines, 68% confidence intervals (CI) about the means are represented by shaded areas, data are indicated by colored dots, and standard errors of data are indicated by bars. (C) Predicted maximum and time-averaged concentrations of the indicated proteins in response to 10 ng/mL EGF for 60 min with or without 10 μM sorafenib from the best-fit model are shown. pEGFR includes pEGFR, pEGFR bound to GRB2, and pEGFR bound to GRB2-SOS, RAS-GTP includes all RAS-GTP bound species, membrane RAF1 indicates all RAF1-bound species at the plasma membrane. Vertical cross bars indicate whether the dataset was included in model fitting. Model fitting was performed in SloppyCell.

The rationale for including specific kinetic processes involved in RAF binding to RAS-GTP, RAF dimerization, phosphorylation of RAF1, MEK, and ERK, and sorafenib binding can be found in Surve et al. (10). Upstream of RAS, these processes were modeled as first-order, reversible reactions: EGF-EGFR binding, EGFR phosphorylation, and binding of phosphorylated EGFR by GRB2 or GRB2-SOS. RAS activation was modeled using Michaelis-Menten kinetics with RAS-GDP as the substrate and phosphorylated EGFR bound to GRB2-SOS as the enzyme that converts inactive RAS-GDP to active RAS-GTP. This choice enables RAS-mediated ERK activation to be switch-like if the Michaelis constant for SOS-catalyzed GDP-to-GTP exchange on RAS, K_mgneslow, is large (46). Hydrolysis of RAS-bound GTP was modeled as a first-order, irreversible reaction. All other reactions were assumed to follow mass-action rates.

Model Implementation

The system of ODEs was solved and analyzed in MATLAB, Python, or Julia. For model equation generation, we used Virtual Cell (VCell), which provides a user-friendly graphical interface for drawing reaction network diagrams (47,48). Equations generated in VCell were exported to MATLAB as a MATLAB-compatible ODE function, which was solved using ode15s, and to the SloppyCell package in Python 2 as a Systems Biology Markup Language 2 (SBML2) file (49,50). The VCell BioModel is available in the public domain at http://vcell.org/vcell-models under the shared username “LazzaraLab” and deposited in BioModel (ID: MODEL2501150001). Model equations were also generated in the Julia programming language (51) by writing the model reactions and subsequently generating the model equations using the Catalyst.jl (52) and ModelingToolkit.jl (53) libraries, followed by solving the model equations using the DifferentialEquations.jl library (54). All MATLAB and Julia analyses are available on GitHub under the username “lazzaralab.” See Table S1 for details on versions and software packages used for model analyses.

Model Fitting in SloppyCell

The model was fit to prior measurements of total GTP-bound RAS with or without sorafenib, phosphorylated MEK without sorafenib, and phosphorylated ERK without sorafenib in EGF-treated HeLa cells expressing an HRAS-Venus fusion from the endogenous locus (HeLa/HRAS-mVenus), plus measurements of membrane-associated RAF1 with or without sorafenib in EGF-treated HeLa cells expressing a RAF1-mVenus fusion from the endogenous locus (HeLa/RAF1-mVenus) (10,55). At least five experimental replicates were available for any measurement (64 data points total). Total GTP-bound RAS and phosphorylated MEK and ERK concentrations were measured over the first 60 min following EGF treatment, and membrane RAF1 concentrations were measured during the first 30 min. In the prior data, measurements included RAS activity by RAS-GTP pull-down, membranous RAF1 by live-cell imaging, and phosphorylated MEK-ERK levels by immunoblot. Model fitting was performed to minimize the negative log likelihood, referred to as the cost function, $C\left(\mathbf{\theta }\right)$, given by

$$C\left(\mathbf{\theta }\right)=\frac{1}{2}\sum _i \frac{{\left[y_i-f(t,\mathbf{\theta })\right]}^2}{{\sigma }_i^2}$$

where $f(t,\mathbf{\theta })$ represents model-predicted protein concentration for $i$th data point using the vector of best-fit parameter values $\left(\mathbf{\theta }\right)$ at a time point $\left(t\right)$ and experimental time-course data point ($y_i$) normalized by the variance of data point $({\sigma }_i^2$) (56). To rule out the possibility of overfitting the model to datasets with a relatively large number of data points, we tested a standard least squares approach that weights the cost function by the number of datapoints in each dataset. The alternative approach did not improve the model fit, causing us to retain the use of Eq. [1]. Prior to fitting, baseline values were specified for as many rate constants as possible using previously reported values to ensure that well-known parameters did not substantially deviate from physiologically realistic values. Initial conditions were based on literature values and fixed.

Confidence intervals for SloppyCell fits were calculated in Python using a combination of Nelder-Mead derivative-free (57) and Levenberg-Marquardt derivative-based (58,59) optimization by generating a thermal ensemble of 1751 independent parameter sets. We computed an average and standard deviation for model-predicted concentrations of RAS-GTP, membrane RAF1, pMEK, and pERK. For RAS-GTP, pMEK, and pERK, standard errors of the mean (SEM) based on biological replicates were available and used for generating confidence intervals. SEM for membrane RAF1 concentrations could not be quantified because membrane RAF1 was measured by quantification of live-cell images of different cells at different time points (10). In this case, Poisson errors, which scale with the number of data points, of 17% of total RAF1 concentration (12,000 molecules/cell) were assumed for all data points. See Table S1 for software packages used.

Model Sloppiness Analysis in SloppyCell

Stiff and Sloppy parameters were identified by eigen-decomposition of the Fisher Information Matrix (FIM) as a function of changes in log-parameter values (40). Stiff parameters tend to project strongly on the FIM, meaning that changes to their values have relatively strong influence on the model’s ability to be constrained by the given data, whereas Sloppy parameters have the opposite effect. However, the hallmark of a Sloppy model is log-linear eigenvalue scaling in the FIM, making it impossible to make a clear cut between “important” and “unimportant” parameters (60).

Sensitivity Analyses in MATLAB

We used partial least squares regression (PLSR) as a method to conduct multivariate sensitivity analysis and refer to the approach as “PLSR SA.” PLSR has been widely used as a dimensionality reduction technique to predict cell phenotypes from multivariate signaling data (61,62). However, the application of PLSR for multivariate parameter sensitivity analysis remains limited; it efficiently reduces feature space by identifying latent variables or PLS components (linear combinations of the measured independent variables) that best predict variance in related dependent variables. PLSR was performed here in MATLAB using plsregress. The independent variable matrix was created as 3000 rows × 40 columns. Each row represented an independent set of the 40 parameters, with values randomly sampled from uniform distributions spanning upper and lower boundaries created by multiplying and dividing, respectively, best-fit parameters by 10. Where indicated, the independent variable matrix was alternatively created with Latin Hypercube sampling (LHS). We used the LHS_Call function (36) to divide each of the parameter ranges described above into 3000 non-overlapping bins with equal probability and randomly sampled parameter values from the stratified parameter space. Once the matrix of sampled parameters was generated, entries in each column were z-scored to eliminate potential effects of different absolute values for each parameter. The size of the dependent variable matrix varied for different analyses. For assessing sensitivity of the EGFR-ERK system to changes in model parameters, a 3000×6 matrix was generated by concatenating the predicted maximum values for GTP-loaded RAS with and without sorafenib, membrane RAF1 with and without sorafenib, phosphorylated MEK, and phosphorylated ERK concentrations in response to 10 ng/mL EGF for each parameter set. For evaluating sensitivity of ERK activity in mutant RAS relative to wild-type RAS to changes in model parameters, a 3000×1 matrix was generated by computing the difference in time-integrated phosphorylated ERK concentrations between wild-type and mutant RAS for each parameter set. Entries in each column of the dependent variable matrices were z-scored. A PLSR model based on six latent variables was chosen because metrics of fit (R²X, R²Y) and predictive power (Q²Y) plateaued after that point. This number also matched the number of modes identified in Sloppiness analysis. Details on other sensitivity analyses can be found in the Supporting Material.

Comparison of PLSR SA and Sloppiness Analysis

The number of PLS components and modes to compare was determined by finding the total number of PLS components that produced a cumulative Q²Y > 50% and the number of stiff modes at which eigenvalues of the FIM (λ) were > 0. For comparing aggregate importance of parameters across PLS components in PLSR SA and modes in Sloppiness analysis, we first calculated variable importance in projection (VIP) scores, which represent the sum of projection over all PLS components of each parameter $x$, weighted by the change in model outputs explained by the same parameter (63,64), for up to six components. The equivalent metric in Sloppiness analysis was calculated by taking the squares of individual parameter’s projection in modes up to 5, with projection in each mode weighted by the corresponding λ. Similarity between the magnitudes of projection of each parameter onto each mode in Sloppiness analysis and PLS component in PLSR SA was calculated using the corr function in MATLAB, which provides Pearson’s correlation coefficients and their associated P-values using two-tailed Student’s t-test. Similarity between the two methods was also calculated using the Jaccard index, defined as

$$J(A,B)=\frac{\left|A\bigcap B\right|}{\left|A\bigcup B\right|}$$

where $A$ and $B$ represent a set of parameters that have scaled sensitivity values above a threshold value in each PLS component in PLSR SA and mode in Sloppiness analysis.

Additional Computational Methods Information

Methods for PLSR SA-based model fitting, comparison of PLSR SA and Sloppiness analysis after filtering parameter sets, IC50 analyses, stochasticity sensitivity analyses, and identifiability analyses are provided in the Supporting Material.

Experimental Methods

Cell culture

Parental HeLa cells were maintained in Dulbecco’s Modified Eagle’s Medium (DMEM) supplemented with 10% FBS (VWR), 1% 200 mM L-Glutamine (Thermo Fisher), and 1% 10,000 μg/mL penicillin-streptomycin (Thermo Fisher). Cells were confirmed to be mycoplasma-negative using the Mycoplasma Detection Kit (CUL001B; R&D Systems). Cells were used within 15 passages of thawing frozen vials and maintained at 5% CO₂ and 37°C in a Thermo Fisher Scientific Forma Steri-Cycle i160 incubator.

RAF1 transient expression

Previously described pEYFP-N1 and RAF1-YFP DNA constructs (10,65) were propagated in chemically competent DH5α E.coli, and bacterial outgrowth in super optimal broth with catabolite repression medium was streaked across Luria-Bertani agar plates with 50 μg/mL kanamycin. Transformed E.coli colonies were selected the next day and grown in 5-mL LB media cultures with 50 μg/mL kanamycin. Plasmid DNA was purified the following day using QIAGEN miniprep kits. Resulting pEYFP-N1 and RAF1-YFP plasmids were confirmed by diagnostic digestion and whole-plasmid sequencing (Plasmidsaurus).

Immunofluorescence imaging and automated image analysis

HeLa cells were plated on 18-mm coverslips in 6-well culture plates at a density of 2×10⁵ cells per well. Media was changed to complete DMEM without antibiotics 3 hr before transfection. Cells were then transfected in serum-free DMEM without antibiotics using Fugene 6 (Promega). Parental, pEYFP-N1-trafected, and RAF1-YFP-transfected cells were treated with 10 ng/mL EGF for 5 min. Cells were then put on ice, washed twice with ice-cold PBS, and fixed in 4% paraformaldehyde for 20 min. Fixed cells were washed twice with PBS, permeabilized with 0.25% Triton X-100 in PBS, and stained overnight at 4°C with an antibody against ERK1/2 pT202/pY204 (CST, #4370, 1:400) diluted in Intercept Blocking Buffer (IBB) (LI-COR, 927–60001). Coverslips were incubated with Alexa Fluor 647 anti-rabbit secondary antibody (1:750) and Hoescht DNA stain (1:2000) diluted in IBB for 1 hr at 37°C. Coverslips were mounted on slides using Prolong Gold Antifade (Thermo Fisher, P36930) and imaged on a Zeiss Axiovert Observer.Z1 using a 20× objective. Fluorescence images were analyzed using CellProfiler (66). Images were corrected for uneven illumination using the CorrectIlluminationCalculate and CorrectIlluminationApply modules. Image segmentation was performed using DNA as the primary object and the IdentifyPrimaryObject module and pERK as a secondary cytoplasmic signal using IdentifySecondaryObjects. CellProfiler outputs were exported to R. Cells were classified as YFP-positive or -negative based on whether their mean per-cell YFP intensity was greater or lower than the median of the mean per-cell YFP intensity of parental cells. Integrated pERK intensities of YFP-positive or -negative cells were compared per cell.

Statistical analyses

For single-cell measurements based on immunofluorescence imaging, mixed effects models were constructed separately for each EGF treatment time point, with YFP expression set as the main effect and the number of biological replicates and fields of views as random effects using the lme4 package in R, followed by Tukey multiple comparisons test using the multcomp R package. For patient data analyses, wherein we compared ERK1/2 phosphorylation levels, cell viability in response to MEK inhibitors, and IC50 values of MEK inhibitors in wild-type and mutant RAS-expressing patients with low and high RAF1 expression, independent Student’s t-tests with Bonferroni correction were performed using the rstatix package in R.

RESULTS

Low RAF abundance is characteristic of a substantial fraction of tumors and cancer cells.

To assess the frequency of low RAF abundance in cancer settings, we first examined relative expression of the key EGFR-ERK pathway proteins using CPTAC mass spectrometry data. Rank-ordering the expression of EGFR, RAS (HRAS, KRAS, and NRAS), RAF (RAF1 and BRAF), MEK1/2, and ERK1/2 across tumors of different types revealed that 14% of PDAC, 13% of GBM, 9% of COAD, and 3% of LSCC patients expressed RAF as the least abundant protein (Fig. 1A). Performing the same analysis using CCLE mass spectrometry data revealed that 12% of cell lines with complete expression profiles of the queried proteins had RAF as the least abundant protein, 4.5-fold lower on average than the other proteins assessed (Fig. 1B,C). A two-dimensional PCA projection of the CCLE cell lines using EGFR, RAS, RAF, MEK, and ERK as features demonstrated that cells separated along an axis spanning two main groups – cells with relatively high expression of MAPK cascade proteins (RAF, MEK, ERK) and low expression of EGFR and RAS and cells with relatively high expression of EGFR and/or RAS but low MAPK cascade protein expression (Fig. 1D-E). Based on that, k-means clustering on the two-component PCA projections with $k=2$ identified a cluster of cells with relatively low RAF expression (Fig. 1E,F). Thus, low RAF abundance is a defining characteristic of a substantial fraction of tumors and cancer cell lines and may create a signaling bottleneck in those settings. It is important to remember though that abundances and stoichiometries alone do not dictate the locations of bottlenecks in signaling dynamics. When RAF abundance is low, it is particularly low relative to EGFR and RAS upstream (Fig. 1C,E), and MEK and ERK downstream (Fig. 1C,E). That fact and its position as the initiating kinase in the three-tiered MAPK cascade make RAF unique as a low-abundance protein.

The fitted network model captures RAF-limited EGFR-ERK pathway dynamics.

To understand the regulation of EGFR-ERK signaling dynamics in the context of low RAF1 abundance, we constructed a computational model using coupled ODEs. A schematic showing model species (Table S3) and their interactions is shown in Fig. 2A. We fit the model to previously reported data from HeLa cells expressing fluorescently tagged endogenous HRAS or RAF1, treated with 10 ng/mL EGF in the presence or absence of 10 μM sorafenib (Fig. 2B). Data for total GTP-bound RAS and whole-cell pMEK1/2 and pERK1/2 were collected up to 60 min after EGF treatment, whereas membrane-associated RAF1 data were collected up to 30 min after EGF treatment (10,55). Measurements with sorafenib strengthened model training by providing a condition with a RAF perturbation. When sorafenib was absent, GTP-bound RAS and membranous RAF1 peaked 2.5 to 5 min after EGF treatment, respectively, while pMEK and pERK displayed sustained activation through the first hour of response. Standard deviations of pMEK and pERK measurements were relatively large, and there were no membrane-associated RAF1 measurements beyond 30 min. As a result, model fits for membrane RAF1 showed relatively large confidence intervals. Unlike the activities measured by immunoblot at other pathway nodes, membrane RAF1 levels were measured by live-cell microscopy using smaller time intervals (0.25 – 3 min) for less overall time (up to 30 min after EGF addition) to prevent photobleaching. The tendency for RAF1 to remain membrane-associated for longer times in the presence of sorafenib (second panel of Fig. 2B) has been documented and explained (10,67,68). Based on that prior work and because sorafenib is a specific inhibitor of RAF, we assumed that RAS-GTP loading was unchanged by sorafenib and therefore used the same model training data for RAS-GTP with or without sorafenib. Local structural parameter identifiability analysis, which assesses whether model parameters can be determined from experimental data given the model structure (69), indicated that 26 of 32 parameters could be estimated from our data, not necessarily uniquely (Fig. S1A). Although global structural identifiability analysis, which examines whether parameters can be uniquely determined from an ideal dataset given the model structure (70), was not performed, it would likely indicate that fewer parameters are uniquely identifiable.

We used the best-fit parameter values from SloppyCell (Table S4), obtained by a combination of Nelder-Mead derivative-free (57) and Levenberg-Marquardt derivative-based (58,59) optimization, because it produced a smaller cost function value than a fit in MATLAB using only Nelder-Mead described in Supplemental Methods in the Supporting Material (Table S5). Quality of fit was reflected by order-of-magnitude equality between the SloppyCell-predicted confidence interval bounds and experimental errors (average difference of 54%; Fig. 2B). Note that the small model-predicted differences in RAS-GTP levels with or without sorafenib resulted from minor differences in fitted values of the four sorafenib-relevant parameters ($k_{iBf},k_{iBr},k_{iR1f},k_{\text{iR1r}}$) compared to their respective fitted values in the absence of sorafenib (Table S4). Comparison of model-predicted maximum and time-averaged concentrations of the six signaling species used in model training highlights the low abundance of RAF1 at the membrane compared to other species and a relatively small difference between maximum and time-averaged levels of pMEK and pERK species, which exhibited protracted activation (Fig. 2C).

Low membrane RAF1 abundance limits EGFR-ERK pathway flux.

We first tested the hypothesis that, in the fitted model, increased RAF1 abundance would lead to predicted increases in RAF1 membrane localization and ERK activity (Fig. 3A). After confirming this, we identified the parameters that control EGFR-ERK signaling dynamics, even outside the bounds used for model training, by performing multivariate sensitivity analyses using partial least squares regression (referred to as PLSR SA). This was done three different ways: 1) perturbing all kinetic rate constants and RAF1 abundance 2) perturbing all kinetic rate constants with fixed base-model RAF1 abundance, and 3) perturbing all rate constants but with a 100-fold increase in RAF1 abundance. In each case, the dependent variables were set to be the six signaling nodes used for model training. Using the PLSR model with six PLS components, we calculated VIP scores, which represent the aggregate importance of each parameter over all PLS components (i.e., latent variables) (63,64). RAF1 abundance was one of the strongest determinants of EGFR-ERK signaling dynamics when it was included among the varied parameters, and rate constants related to RAF1 membrane localization were strong determinants of signaling dynamics for calculations with low, invariant RAF1 abundance (Fig. 3B). Thus, in our calculations, low RAF1 abundance limited the magnitude and duration of ERK activity, and overall EGFR-ERK signaling dynamics were strongly controlled by RAF1 membrane translocation.

Figure 3. EGFR-ERK signaling system behavior is primarily controlled by RAS activation and RAF1:RAS-GTP unbinding rates.

(A) Membrane RAF1 and pERK dynamics are simulated using the best-fit parameter values from Fig. 2B with baseline RAF1 expression, 10-fold and 100-fold increase in RAF1 expression. (B) PLSR SA was performed using 3000 uniformly sampled, random parameter sets. The independent variable matrix (3000 rows × 41 columns) consisted of parameter perturbations, including variations in RAF1 abundance. The dependent variable matrix (3000 rows × 6 columns) contained the predicted maximum concentrations of RAS-GTP with or without sorafenib, membrane RAF1 with or without sorafenib, pMEK, and pERK in response to 10 ng/mL EGF for each parameter set. The analysis was repeated using only rate constants in the independent variable matrix (3000 rows × 40 columns), with fixed RAF1 expression at baseline and at a 100-fold increase. RAF1-related parameters are highlighted in light blue, and sorafenib-inhibited RAF1-related parameters are highlighted in dark pink. (C) In the top plot, the natural log of eigenvalues $\left(\mathrm{\lambda }\right)$ of the FIM matrix, which indicates sensitivity of cost $C\left(\mathbf{\theta }\right)$ to perturbations along orthogonal directions in parameter space, is indicated by asterisks, and percent model fit explained by each mode from Sloppiness analysis for increasing numbers of modes is indicated by circles. The bottom plot shows the cumulative predictive power (Q²Y) of PLSR SA model for increasing numbers of PLS components. In the plots, the selected numbers of modes or PLS components for comparison are highlighted by shaded areas and indicated by colored circles and asterisks. (D) Sloppiness analysis was conducted by eigen-decomposition of the FIM as a function of log-parameter values from the 1751 parameter sets described in Fig. 2B. PLSR SA was performed as described in B for fixed, baseline RAF1 expression. Projection of parameters onto the first six modes from model Sloppiness analysis (top) and parameter loadings in the first six PLS components from PLSR SA (bottom) were compared. PLSR SA parameter loadings were scaled to the range of projections of the FIM in Sloppiness analysis (−0.6 to 0.6) and plotted as scaled sensitivity. For each parameter, aggregate weights across modes in Sloppiness analysis λ(FIM_nn)²), calculated by adding the squares of projection onto each mode (FIM_nn) weighted by the corresponding eigenvalues (λ), and VIP scores from PLSR SA were also plotted. (E) The model was fitted by allowing all 40 parameters or a subset of parameters with VIP scores > 1 from PLSR SA in Fig. 3D to vary. Convergence of cost function value $C\left(\mathbf{\theta }\right)$ over iterations and corresponding computational wall time (total time required for the optimization to run) are shown. The vertical dashed lines show the position of iterations corresponding to the wall times required to satisfy the convergence criteria described in Methods.

To identify the rate processes that control EGFR-ERK signal transduction within the bounds of measurements used for model training, we conducted model Sloppiness analysis and compared the results against PLSR SA with fixed base-model RAF1 abundance. RAF1 expression was not varied for Sloppiness calculations because of the intended purpose of the method to stay within the bounds of known measurements. Sloppiness analysis determines the sensitivity of model fit to experimental data for changes in products of parameters (linear combinations of log parameters), represented as modes (71). By the sixth mode in Sloppiness analysis and the sixth PLS component in PLSR SA, plateaus were observed in the explained variance in model fit to data (>99%) and model predictive power as determined by cross validation (>50%), respectively (Fig. 3C). Parameter loadings in PLSR SA were scaled to the range of parameter projections on the FIM in Sloppiness analysis (−0.6 to 0.6), and that interval was defined as the scaled sensitivity. Both Sloppiness analysis and PLSR SA emphasized the importance of RAS (de)activation ($k_{\text{Rgneslow}},k_{\text{Rhydro}}$ in the first, most important mode or PLS component) (Fig. 3D, Fig. S1B). With the constraint of low membrane RAF1 abundance present in Sloppiness analysis, parameters for sorafenib-inhibited RAF1:RAS-GTP binding ($k_{\text{iR1f}}$ in Mode 0) and RAF1:RAS-GTP (un)binding rates ($k_{\text{R1f}},k_{\text{R1r}}$ in Mode 1) controlled the system most strongly as indicated by large absolute scaled sensitivity values (Fig. 3D). In PLSR SA with fixed RAF1 abundance but without the constraint of membrane RAF1 abundance being low, parameters for RAS activation ($K_{\text{mgneslow}}$ in PLS component 1) and MEK (de)phosphorylation rates ($K_{\text{pMEK}},K_{\text{dpMEK}}$ in PLS component 2) controlled the system strongly, as indicated by large absolute scaled sensitivity values (Fig. 3D, Fig. S1B). A qualitative summary of Fig. 3D revealed that the correspondence between PLS components and Sloppiness modes was strong upstream in the pathway (EGFR to RAF) but weakened farther down the cascade (Fig. S1C). Inferences from PLSR SA and Sloppiness analysis were more aligned when we repeated the comparison using only the 1000 random parameter sets for PLSR SA that yielded the lowest rank-ordered cost values (Fig. S1D-G). The importance of RAF1:RAS-GTP binding was accentuated in PLSR SA for the reduced number of parameter sets (Fig. S1F), and overlap between the two analyses improved, indicated by a larger Jaccard index of unique parameters with scaled sensitivity values > 0.3 (Fig. S1G). Overall, these analyses demonstrate that pathway dependence on RAF1 membrane localization rates is characteristic of low-RAF1 abundance.

Additional insights can be gleaned from Sloppiness analysis and PLSR SA that we now briefly review. PLSR SA can be used to prioritize parameters to increase model fitting efficiency. Indeed, varying only rate constants with VIP scores > 1 yielded a 75% reduction in computational time with an error that was only ~25% larger than that achieved using all parameters for the same convergence criterion (Fig. 3E, Fig. S2A-B, Table S6). Use of LHS (36) to select the values of the perturbed parameters did not produce a clear benefit (Fig. S2C), likely due to the large number of parameter sets required for a convergent PLSR cross-validation score (Table S6). Selecting parameters to vary based on PLSR SA also provided a way to avoid the issue of some fitted parameter values being incongruous with known values (Fig. S3). For example, varying only parameters with PLSR SA VIP scores > 1 allowed for the enforcement of an implied equilibrium dissociation constant for pEGFR binding to GRB2 $(K_d=k_{\text{G2r}}/k_{\text{G2f}}=0.48\mathrm{\mu }\mathrm{M}$) consistent with the reported range of 0.097–0.65 μM (72,73). Comparisons among the best-fit parameter estimates, obtained by varying all parameters or just those with PLSR SA VIP > 1, and the reference values (initial guesses) based on literature are shown in Table S7. Parameters with discrepancies between their fitted values and literature-based, physiological values greater than an order of magnitude (e.g. $k_{G2f},k_{G2r},k_{Bf},k_{nfbBR1}$) were also identified as Sloppy by Sloppiness analysis (Fig. 3D). Fixing Sloppy parameters (i.e., those with weak projection in modes 0 or 1), such as those for pTyr-SH2 interaction ($k_{\text{G2f}},k_{\text{G2r}}$), to biologically known values had negligible effect on model fit when the remaining parameters were varied (Fig. S2D,E). Conversely, moving the stiff RAF1:RAS-GTP (un)binding parameters ($k_{\text{R1f}},k_{\text{R1r}}$) to model-fitted values reported previously (20) led to an obviously erroneous fit result that produced virtually no RAS activation (Fig. S2D,E). Thus, although the best-fit parameter estimates obtained by varying all parameters did include some deviations from known parameter values or parameter ratios, those deviations could be corrected without a major impact on model fit. Both local sensitivity analysis, performed by calculating partial derivatives of model output with respect to the small changes in a parameter (74), and univariate sensitivity analysis, which quantifies the effects of a fixed range of perturbation in individual parameters on the combined model outputs, emphasized the nearby activation rates of the species (k_Rgneslow, k_R1f, k_pMEK, k_pMERK) at each tier of the cascade rather than membrane RAF1-related parameters (Fig. S4).

Low membrane RAF1 expression dampens ERK activity driven by oncogenic KRAS.

The emphasis of parameter sensitivity results on RAS- and RAF1-related parameters led us to question if RAF1 is a signaling bottleneck even when RAS is mutated. Constitutively active RAS mutants are generally thought to drive cancer in part by increasing RAF/MEK/ERK signaling, but elevated ERK activity is not always well correlated with RAS mutation (24,25). KRAS is the most abundant RAS isoform in many cell backgrounds and the second most abundant in HeLa cells (after HRAS) (5,6). Thus, inferences from a model that lumps all RAS isoforms together may be reasonably expected to reflect the scenario when KRAS is mutated. KRAS G12D, G12V, G13D, and G12C mutants accounting for 83% of all clinically observed mutants (75). We focused on KRAS^G12V (the second-most common KRAS mutation in colorectal, lung, and pancreas cancers) because it has the largest GTP hydrolysis rate constant among common KRAS mutants (76,77), which is advantageous in predicting upper-limit effects. To model KRAS^G12V, the rate constant for RAS-GTP hydrolysis was reduced 16-fold to match the reported GTP hydrolysis rate for the KRAS^G12V mutant (76). No retraining of the model was performed.

The model predicted only a ~1.3-fold difference in maximum EGFR-induced pERK levels between KRAS^G12V and wild-type KRAS, a smaller effect than for other pathway nodes (Fig. 4A). Parameters that most strongly controlled the predicted maximum pERK difference included rate constants for phosphorylation and dephosphorylation of ERK ($k_{\text{pERK}},k_{\text{dpERK}}$) or MEK ($k_{\text{pMEK}},k_{\text{dpMEK}}$) and RAF1:RAS-GTP unbinding ($k_{\text{R1r}}$) (Fig. 4B). All model reactions including the parameters $k_{\text{dpMEK}},k_{\text{pMEK}},k_{\text{dpERK}}$, and $k_{\text{pERK}}$ had pERK as a reactant or a product, making the strong effects of these parameters predictable. It was surprising, however, that the rate constants for ERK-mediated negative feedback phosphorylation of RAF1, SOS, and BRAF were not among the most important parameters because the small difference in ERK activity between wild-type and mutant KRAS has previously been partly attributed to stronger ERK-mediated feedback loops for wild-type versus mutant KRAS (23). RAF1 dissociation from RAS was a stronger determinant of the pERK difference between wild-type and mutant RAS than negative feedbacks, again highlighting RAF1 as the bottleneck for RAS mutants in the settings modeled here.

Figure 4. Oncogenic KRAS^G12V signaling is dampened by low RAF1 abundance.

(A) Model-predicted dynamics of RAS-GTP, membrane-bound RAF1 with or without 10 μM sorafenib, pMEK, and pERK signaling dynamics in response to 10 ng/mL EGF for WT RAS and KRAS^G12V are shown. KRAS^G12V was simulated by a 16-fold decrease in RAS-GTP hydrolysis rate constant from its baseline value. Throughout the panels, the means of model outputs generated from the 1751 parameter sets from Fig. 2B are indicated by dotted lines, best-fit model outputs are indicated by bold lines, and 68% confidence intervals (CI) about the means are represented by shaded areas. Fold differences between maximum concentrations of indicated model species in WT and mutant dynamics (Δ max.) are boxed. (B) PLSR SA was performed using the 3000 uniformly sampled, random parameter sets (3000 rows × 40 columns) as independent variable matrix and predicted absolute differences in time-integrated pERK concentrations between WT RAS and KRAS^G12V from each parameter set (3000 rows × 1 column) were used as the dependent variable matrix. VIP scores of parameters are shown, with VIP > 1 parameters highlighted in red. (C) pERK dynamics for WT RAS or KRAS^G12V with baseline RAF1 expression or a 10-fold increase in RAF1 expression were simulated using the best-fit parameter values from A. Differences in time-integrated pERK concentrations between WT and mutant RAS with baseline RAF1 expression or with 10-fold increase in RAF1 expression are labeled and indicated by shaded areas. Fold-differences (x) between time-integrated pERK concentrations of WT and mutant RAS with baseline RAF1 expression and with 10-fold increase in RAF1 expression during 60-min or 120-min response to 10 ng/mL EGF were compared. (D) pERK dynamics for WT RAS or KRAS^G12V in the absence of negative feedback on SOS, BRAF, and RAF1 by pERK with baseline RAF1 expression or with a 10-fold increase in RAF1 expression were simulated using the best-fit parameter values from A. Differences in time-integrated pERK concentrations between WT and mutant RAS in the absence of feedback with baseline RAF1 expression or with 10-fold increase in RAF1 expression are labeled and indicated by shaded areas. Fold-difference (x) between time-integrated pERK concentrations of WT and mutant RAS with baseline RAF1 expression and with 10-fold increase in RAF1 expression in the absence of feedback during 60-min response to 10 ng/mL EGF were calculated. (E) Absolute differences in time-integrated concentrations of all model species between WT RAS and KRAS^G12V with baseline RAF1 expression and 10-fold increase in RAF1 expression are plotted.

To demonstrate the extent of the RAF1 bottleneck for RAS mutation, we increased RAF1 abundance in simulations. Compared to baseline, a 10-fold bump in RAF1 expression increased the difference in time-integrated ERK activity between wild-type RAS and KRAS^G12V by 52% for a 60-min response to EGF; this difference more than doubled for a 120 min response to EGF (Fig. 4C). When negative feedback loops were turned off, the difference in time-integrated ERK activation between wild-type and mutant RAS for baseline RAF1 expression decreased, and a 10-fold increase in RAF1 expression led to nearly identical pERK levels for wild type and mutant RAS (Fig. 4D). A comparison of Figs. 4C and 4D reveals that the effect of turning off negative feedback was much stronger for wild-type RAS than mutant RAS, consistent with a report of weaker ERK-mediated negative feedback in RAS mutant cells than wild-type (23). Interestingly, weaker feedback for the RAS mutant arose naturally in our model because of saturated RAS activity even with negative feedbacks in place. With baseline RAF1 expression of 12,000 molecules/cell, the effect of constitutively active RAS on the duration of ERK signaling was most strongly dampened by RAS-BRAF-pRAF1-tetramer and RAS-BRAF-RAF1-tetramer, which are RAS-GTP-bound active RAF1 species (Fig. 4E). The remaining effect was distributed across RAS-GDP, RAS-GTP and other RAS-bound RAF1 and BRAF species, the most proximal species to RAS-GTP hydrolysis. On the other hand, when RAF1 expression was increased 10-fold, the effect was more heavily distributed across downstream species, such as phosphorylated MEK and ERK, suggesting that low RAF1 expression limits oncogenic RAS signaling.

Low RAF abundance correlates with low ERK activity in cancer cells and tumors and with cell sensitivity to MEK inhibition.

To experimentally test the model-predicted effect of RAF1 abundance on ERK activity, we ectopically expressed RAF1-YFP in HeLa cells. YFP expression in cells transfected with RAF1-YFP was lower than that in cells transfected with empty vector YFP, presumably because of the smaller size of the empty vector (Fig. 5A). Cells expressing relatively high levels of RAF1-YFP (YFP-positive) exhibited greater EGF-induced pERK than cells expressing relatively low RAF1-YFP (YFP-negative) or cells transfected with the empty expression vector (Fig. 5A). Combined with results in Surve et al. that showed a positive correlation between RAF1 abundance and RAF1 membrane localization (10), these data suggest that low RAF1 abundance limits ERK activity.

Figure 5. Low RAF abundance reduces ERK phosphorylation in lung adenocarcinoma tumors expressing wild-type KRAS and mutant KRAS and sensitizes cancer cells to MEK inhibitors.

(A) HeLa cells were transfected with an empty vector encoding YFP only (EV) or a vector encoding RAF1-YFP. Cells were treated with 10 ng/mL EGF for 5 min. Immunofluorescence was then performed with staining for pERK1/2, and image analysis was performed to quantify integrated pERK intensity of YFP− or YFP+ cells. Representative images are shown. High-magnification split-channel images are shown at right of micrographs for regions marked by white rectangles. Data are represented as mean ± SEM. Mixed effects analysis with the Tukey multiple comparisons test for integrated pERK intensity per cell were performed. n = 3. (B) BRAF expression in CPTAC lung adenocarcinoma tumor samples expressing WT or mutant KRAS was classified as ‘low’ or ‘high’ based on a standard median split and mapped to abundances of MAPK1 (ERK2) or MAPK3 (ERK1) phosphorylated at the indicated sites. Fold-differences in pERK1/2 for ‘low’ and ‘high’ BRAF expression are indicated, and P values were determined using a two-tailed Student’s t-test. Calculations were performed in R. (C) Normalized cell viability fluorescence intensity measurements (CellTiterGlo assay) indicating response to MEK inhibitors at the maximum concentrations tested for each: PD0325901 (2.5 μM), refametinib (10 μM), selumetinib (10 μM), and trametinib (1 μM) are plotted. The reported fluorescence intensities were normalized to fall within a range bounded by measurements for DMSO (control) and blank wells and were compared for low or high RAF expression. The plot at right shows the same results aggregated over all four inhibitors. *, P < 0.05; **, P < 0.01; ***, P < 0.001; ****, P < 0.0001; n.s., nonsignificant.

Model predictions about the effects of RAF expression were also tested by analyzing LUAD and COAD tumor proteomics. In those data, 15.6% of LUAD tumors and 16.9% of COAD tumors harbored KRAS mutation. RAF1 data were unavailable for LUAD, so BRAF data were used instead. In LUAD expressing wild-type or mutant KRAS, ERK1 phosphorylation on T202 and Y204 and ERK2 phosphorylation on T185 and Y187 were significantly higher in tumors with high BRAF abundance than in tumors with low BRAF abundance (Fig. 5B). In COAD, a similar effect was observed for ERK1 phosphorylation on T202 as a function of RAF1 abundance for wild-type RAS, but this relationship was not observed for ERK2 phosphorylation on T185 (Fig. S5A). No significant correlation was observed between BRAF abundance with ERK1/2 phosphorylation in wild-type or mutant KRAS in COAD (not shown). Thus, in two cancer contexts, low RAF abundance correlated with dampened ERK activity. It is worth noting that we expected to observe a larger difference due to variable RAF expression in tumors with mutant RAS than those with wild-type RAS based on our model predictions (Fig. 4C). However, the lack of dynamic data for acute responses to growth factor in patient tumors may obfuscate comparisons to our model results.

To understand how low overall RAF expression, defined as the sum of RAF1, BRAF, and ARAF expression, affects cell phenotypes, we compared the response of CCLE cell lines with low (below 25^th percentile) or high (above 75^th percentile) RAF expression (Fig. S5B) to four MEK inhibitors at their maximum tested concentrations. Raw viability data normalized by the DMSO treatment condition (for a fixed cell seeding density) revealed that cells with low RAF expression were generally more sensitive to MEK inhibitors than cells with high RAF expression (Fig. 5C). At the same time, the half-maximal inhibitory concentrations (IC50) of the MEK inhibitors for the proliferation phenotype were unaffected by RAF expression (Fig. S5C). Both observations can be true because the absolute change in viability effected by a drug is distinct from the rate at which the total possible effect manifests as drug concentration increases. Curiously, when we simulated MEK inhibition in the setting of response to EGF by simply reducing the initial MEK abundance by up to 100% (a choice that obviated the need to introduce additional model species and rate constants), we observed that the predicted time-integrated ERK activity was more sensitive to increases in MEK inhibition for low RAF abundance (Fig. S5D). Thus, while the model predicts that ERK is more responsive to MEK inhibition in a low-RAF context, that biochemical effect (if it indeed occurs in cells) may be insufficient to drive a viability phenotype dose-response effect.

RAF1 stochasticity results from low RAF1 abundance and processes upstream of RAF.

Protein copy numbers as low as hundreds per cell often engender stochastic dynamics (48). The low membrane RAF1 abundance measured in prior work (10) (~200 RAF1 molecules per cell at 6–8 min after EGFR activation) raises a question of whether certain levels of the EGFR-ERK pathway operate stochastically in RAF-bottleneck settings. There is precedent for this notion. For example, qualitative differences in cell motility have been shown among cells due to stochastic signaling dynamics for Rho GTPases expressed at 2000–3000 molecules per cell (78). To begin to address these issues, we simulated the responses of RAS-GTP, membranous RAF1, pMEK, and pERK to EGF using a stochastic solver for five different random seeds. Random seeds were used to generate reaction rates sampled from an exponential distribution (79), with the seed-generating function set so that the mean of the simulated distribution converges to the theoretical rates predicted by Michaelis-Menten or first-order mass action (80). As expected, dynamics were more stochastic at the level of membrane-localized RAF1 than at other pathway nodes, but the effect was modest with small deviations among seeds compared to the magnitude of the signal (Fig. 6A).

Figure 6. Stochasticity arising from low membrane RAF1 abundance is most strongly controlled by RAF1:RAS-GTP interactions.

(A) Stochastic model predictions of RAS-GTP, membrane RAF1, pMEK, and pERK concentrations in response to 10 ng/mL EGF with the best-fit parameter values from Fig. 2B (N=5 differentially seeded, stochastic runs) are shown. Membrane RAF1 stochasticity score is boxed. (B) Stochastic membrane RAF1 dynamics were simulated using the 1000 parameter sets described in Results. Membrane RAF1 dynamics simulated using the parameter sets that produced the top 10% of membrane RAF1 stochasticity scores, as defined in Results, are highlighted (top). Each box indicates the corresponding membrane RAF1 stochasticity score. Stochastic pERK dynamics predicted from the same parameter sets are shown below (bottom). (C) PLSR SA was performed using 1000 LHS parameter sets (1000 rows × 40 columns) as the independent variable matrix, and membrane RAF1 stochasticity scores for each parameter set (1000 rows × 1 column) were used as dependent variable matrix. The top 10 parameters ranked by PLSR SA VIP scores are shown. Confidence intervals for the VIP scores were obtained by running the PLSR SA model 10,000 times by random sampling of the X and Y matrices with resampling. Error bars indicate 95% confidence intervals. Parameters highlighted in pink have lower bounds of the confidence intervals of VIP scores > 1. (D) Results shown are similar to C, except the membrane RAF1 stochastic simulations with the top 10% of stochasticity scores were excluded prior to analysis. Error bars and highlighting are as in C. (E) PLSR coefficients from bootstrapped PLSR SA described in C are shown. Dots indicate the means of 10,000 bootstrapped PLSR coefficient samples, and bars indicate 95% confidence intervals. Parameters with mean PLSR coefficients highlighted in pink have nonzero lower and upper bounds of the confidence intervals of the coefficients. (F) Stochastic membrane RAF1 dynamics were simulated in the absence (left) or presence (right) of 10 μM sorafenib, using the parameter set that produced the most stochastic behavior under both conditions. (G) Stochastic pERK dynamics were simulated using the parameter set that produced the highest membrane RAF1 stochasticity score with baseline protein expression levels (left) or with 10-fold decreases in MEK and ERK expression (right). All calculations were performed in Julia.

To identify conditions that would promote even more RAF1 stochasticity at the membrane, we generated 1000 parameter sets using LHS wherein individual parameters were perturbed by factors ≤ 10 above or below baseline values. To quantify the effects across different parameter sets, we defined a stochasticity score as the variance in time-integrated membrane RAF1 dynamics among the five differentially seeded simulations. Certain parameter sets resulted in substantial membrane RAF1 stochasticity, while others did not, as was apparent in simulations among the top 10% of the most stochastic (Fig. 6B). The first two stochastic simulations (parameter sets 675, 571) in Fig. 6B represent extreme cases of low membrane localized RAF1 abundance in the tens of molecules, a possible scenario in the presence of sub-saturating concentrations of EGF (Fig. S6A). Other parameter sets that yielded RAF1 stochasticity (including parameter set 497) exhibited less extreme levels of membrane RAF1, in the hundreds of molecules, as observed in Surve et al (10). PLSR SA with the 1000 LHS parameter sets from Fig. 6B as independent variables and the corresponding stochasticity scores as the dependent variables revealed that the parameters that most strongly controlled stochastic membrane RAF1 levels were similar, but not identical, to those that controlled deterministic membrane RAF1 solutions shown in Fig. S1F, even when the top 10% of the most stochastic solutions were removed (Fig. 6C-E). Similar results were obtained when ~200 stochastic simulations were compared to deterministic simulations in VCell (Fig. S6B-D). Notably, parameters controlling phosphorylation of ERK and MEK $\left(k_{\text{pERK}},k_{\text{pMEK}}\right)$ decreased in importance for stochastic membrane RAF1 solutions relative to deterministic simulations, suggesting that MEK and ERK do not control membrane RAF1 stochasticity. The stochastic model also predicted that membrane RAF1 in the presence of sorafenib could have qualitatively different behaviors depending on the variable time steps used in stochastic stimulation (i.e., the different seeds) (Fig. 6F). This difference was manifested as a divergence in trajectories among random seeds, which is distinct from noisy, local oscillations in stochastic simulation results in the absence of sorafenib.

To determine if the effects of membrane RAF1 stochasticity could propagate down the pathway, we selected a parameter set with one of the top stochasticity scores and inspected the degree of stochasticity predicted for pERK. Stochasticity was modest for baseline protein expression levels, but increased substantially when the expression of MEK and ERK were decreased by just ten-fold (Fig. 6G), leaving them in the tens-of-thousands of copies per cell and well within the range of expression values observed in some cell lines, including HMEC breast epithelial cells and HS578T breast cancer cells (6). Decreased MEK and ERK expression did not influence membrane RAF1 levels (Fig. S6E). Thus, stochasticity at the level of membrane RAF1 could indeed plausibly propagate to ERK in certain cell settings.

DISCUSSION

Our results suggest that, in some cancer cell settings, RAF acts as a low-abundance EGFR-ERK signaling bottleneck and that low RAF expression impacts cancer cell phenotypes. RAF is not typically recognized as playing these roles, but prior work (6,10) and our analysis suggest that the low-RAF abundance setting is reasonably common. In principle, any pathway intermediate can limit signaling flux, especially when that protein participates in multiple signaling pathways. A protein becomes a signaling bottleneck when it creates an overall rate-limiting step in signaling flux. While abundance and stoichiometry do not uniquely determine the location of a bottleneck in a dynamic network, bottlenecks often arise at locations where a pathway intermediate is sandwiched between more abundant species. Along with SOS and GAB1, RAF1 is one of few proteins that satisfies this condition in the EGFR-ERK pathway in many cancer settings (6). In normal breast epithelial cells and breast cancer cells, SOS and GAB1 can be even less abundant than RAF1 and may be rate-limiting for ERK activation (6). Consistent with this, our model predicted that when RAF1 stoichiometric limitations were relieved, GRB2/SOS interactions and SOS abundance regulated peak ERK activation more strongly than did RAF-related parameters. For stoichiometries such as those in HeLa cells, sensitivity and Sloppiness analysis revealed that EGFR-ERK signaling (at six nodes spread throughout the pathway) was most sensitively regulated at the level of RAF1 even with an overall SOS abundance (7700 molecules/cell) that was lower than that of RAF1 (12,000 molecules/cell). This occurred in part because translocation to the membrane was much less efficient for RAF1 (10–15%) than for SOS (55%), and the binding affinity of RAF1 at the membrane was ~79 fold lower than that of SOS. Thus, the extremely low tendency for RAF1 to localize to the membrane makes it unique as a low-abundance pathway intermediate.

Comparison of the influential parameters nominated by our analysis with those identified by others should be made with caution, in part because we considered six pathway nodes simultaneously for a holistic assessment of system performance. The primary output of interest in this system is typically phosphorylated ERK (29,81–83). We considered model outputs at multiple pathway nodes because many more parameter combinations are likely to be able to explain pERK dynamics alone than can explain dynamics at all six nodes we considered simultaneously. Thus, our approach is more likely to yield a model that is predictive of scenarios not considered in model training. While some well-known regulatory modes, including induced expression of dual specificity phosphatases (8,84), receptor activation/deactivation cycling (8), and receptor dimerization (85), have previously been identified as strong regulators of system-wide control, our analysis points to RAF1:RAS-GTP binding as a stronger determinant of overall system behavior with low RAF1 abundance. In our model, the outputs of peak and time-integrated ERK activity are secondarily sensitive to changes in other receptor- and adapter-level rate constants and expression levels (e.g., GRB-SOS binding, receptor dimerization, and SOS concentration), consistent with other reports (6,8,83,85,86).

Stochasticity in signaling is typically considered, if at all, at the transcriptional level. Many transcription factors are expressed at low copy numbers (e.g., as low as 5 copies/cell for c-Fos (87)), and their intrinsic fluctuations produce noisy phenotypes (88). However, our results demonstrate how stochasticity can arise upstream of transcription factors, a conclusion also reached using a model of the tumor necrosis factor pathway (89). Stochasticity caused by low RAF1 abundance is predicted to be smoothed by more abundant downstream pathway nodes or to be increasingly propagated by similarly low abundance downstream nodes. The model prediction that low RAF1 abundance leads to intrinsic stochasticity (non-deterministic behavior) at specific nodes of the EGFR-ERK pathway may explain why some experimental measurements are persistently technically noisy. For example, the observed lower variance in membrane RAF1 concentrations with sorafenib (10) could have arisen from the increased abundance of RAF1 at the membrane. Our base model exhibits incipient stochasticity with ~1200 RAF1 molecules translocating to the plasma membrane, and cell lines expressing fewer RAF1 than HeLa cells may experience greater RAF1 stochasticity. 73% (1101) of the 1517 DepMap cancer cell lines have lower RAF1 expression than HeLa. Even for cell lines with a similar or higher RAF1 expression than HeLa, small amounts of RAF1 will translocate to the membrane when EGFR is activated by sub-saturating ligand concentrations (Fig. S6A). RAF1 could also become limited to just tens of molecules/cell when the RAF1:RAS-GTP dissociation rate constant ($k_{R1,r}$) is high, a phenomenon encountered in RAF1 mutant-expressing RASopathies and in cancer cells with limited calcium influx (90–92). Thus, the stochasticity we identified may be reasonably common.

In discussing stochasticity, care must be taken to specify the source and type. For example, a classification model predicted that fluctuations in RAF1 and BRAF expression engendered cell-to-cell variation in ERK activity and proliferation among MCF10A cells (93). While this may appear to contrast with our finding that stochasticity effects for low-abundance RAF1 do not propagate to ERK, the issue lies in the type of stochasticity described. Because RAF1 and BRAF are critical determinants of ERK signaling, cell-to-cell variation in their expression (sometimes referred to as stochasticity) leads to cell-to-cell variability in ERK activity. That can be true even without the effects of intrinsic stochasticity or non-deterministic system behavior, wherein cells with identical expression of the same pathway intermediates have different reaction trajectories simply because protein abundance is too low for deterministic behaviors.

Our understanding of how RAF1 abundance affects EGFR-ERK pathway dynamics may also be impacted by model assumptions and technical limitations. For example, broadly replacing our typical assumption of first-order enzyme-catalyzed kinetics by saturable Michaelis-Menten kinetics may more realistically constrain reaction rates, as in Futran et al. (94), but this would simultaneously increase the number of species and rate constants. As another example, in the stochasticity sensitivity analysis we performed, the degree of stochasticity would be more generalizable if it included extrinsic factors such as diffusion through use of the Gillespie algorithm with more simulations and incorporation of intrinsic (e.g., intermolecular collisions) and extrinsic (e.g., cell cycle effects) fluctuations (95). The model formulation could also be expanded to include other known EGFR activity-induced feedbacks, such as the ERK-dependent expression of Sprouty2, which sequesters the Cbl ubiquitin ligase and slows EGFR degradation (96). Including ERK-regulating cytosolic mechanisms distal from EGFR that involve spatial gradients, such as those involving GAB1-SHP2 complexes (7), could reveal other bottlenecks. However, the increased complexity of solving the system with spatial effects would require substantially increased computation time (97). As it becomes more computationally feasible to address these issues in a system of the size considered here, it will be important to assess the impact of the types of assumptions we made on qualitative and quantitative conclusions.

STATEMENT OF SIGNIFICANCE.

RAF kinases are critical intermediates between receptors and ERK, but RAFs are greatly outnumbered by other pathway proteins in a substantial fraction of cancer cells. To understand the potential for low RAF abundance to create unrecognized signaling bottlenecks, we trained a novel computational model of EGFR-ERK signaling and characterized it comprehensively using multivariate sensitivity and Sloppiness analyses. The model predicted that low RAF abundance suppresses EGFR-mediated ERK activation, limits the effects of oncogenic RAS mutants, and potentiates stochastic RAF dynamics that can propagate downstream. Thus, the canonical EGFR-ERK signaling pathway exhibits divergent behaviors in a region of parameter space that is considerably populated by cancer cells.

DATA AND CODE AVAILABILITY

All code generated is available on the Lazzara lab GitHub site (https://github.com/lazzaralab/Lee-et-al_EGFR-RAF-ERK-model). CPTAC proteomics and phosphoproteomics data were accessed by importing the cptac Python package. GDSC2 drug sensitivity data were downloaded from the GDSC website. Links to data repositories are provided in Table S2. All raw data used in the analyses of the effect of RAF1 expression on ERK activity in RAF1-YFP- and eYFP-transfected HeLa cells are deposited in Zenodo (https://doi.org/10.5281/zenodo.18025412).