Figures and data in Signaling pathways as linear transmitters

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7 figures, 4 tables and 1 additional file

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Figure 1

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The Wnt, ERK, and Tgfβ pathways transmit input using different core transmission architecture.

(A) Signaling pathways transmit inputs from ligand-receptor interaction to a change in output, the level of transcriptional regulator (white circle). (**B-D**) The core pathway for each metazoan signaling pathway is defined by distinct architectural features. In the Wnt pathway (B), the output is regulated by a futile cycle of continual synthesis and rapid degradation. In the ERK pathway (C), the output is regulated by a kinase cascade coupled to negative feedback. In the Tgfβ pathway (D), the output is regulated through continual nucleocytoplasmic shuttling.

https://doi.org/10.7554/eLife.33617.002

Figure 2 with 7 supplements

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The Wnt, ERK, and Tgfβ pathways are linear signal transmitters.

(**A-C**) Network diagrams of the signaling pathways. The Tgfβ diagram is modified from Schmierer et al. (2008). In the network diagram in A, DC refers to the β-catenin destruction complex. Below the network diagrams: the parameter groups and linearity equations we analytically derived in this study. Parameter groups and input functions are color-coded to the corresponding reactions in the network diagrams. Parameters that do not appear in the parameter groups either drop out due to irreversible reaction steps (such as k₁₀ and k₁₁ in the Wnt pathway) or negligible (as indicated by ellipses). (**D-F**) Our analysis reveals that in physiologically relevant parameter values, these pathways generate a linear input-output relationship. The outputs are β-catenin, dpERK, and nuclear Smad complex for the Wnt, ERK, and Tgfβ pathway, respectively. The input functions $u$ describe the effect of ligand-receptor interactions on the core pathway. Specifically: $u (W n t)$ is the rate by which Dishevelled/Dvl inhibits the destruction complex upon Wnt ligand activation, where $k_{3}$ and $k_{- 6}$ are defined in the figure and [Dvl]_a is the concentration Wnt-activated Dishevelled (see Equations A15); u(EGF) is concentration of EGF-activated Ras (Ras-GTP); and u(Tgfβ) is the fraction of Tgfβ -activated receptors. Red and blue lines, respectively: analytical and numerical solutions with measured parameters (plotted against the left y-axis). Grey line: examples of numerical solutions outside measured parameters (plotted against the right y-axis).

https://doi.org/10.7554/eLife.33617.003

Figure 2—source code 1: https://doi.org/10.7554/eLife.33617.011
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Figure 2—figure supplement 1

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Model simulations for the ERK pathway.

(A) Parameter groups in the ERK model are constant to within 10%, over the physiologically relevant range of u considered here, justifying the inclusion of variables into the parameter groups. (**B–C**) The dpERK output is an ultrasensitive function of both free and total phosphorylated Raf. The values $E_{s}$ and $R_{s}$ are illustrated in (B), and are defined in Appendix 1. (**D–F**) Numerical simulation of pulsatile response in the ERK pathway. (D) A pulse of input, RasGTP, is generated by EGF addition in an ERK model that includes details of receptor desensitization (Schoeberl et al., 2002). Basal activity of Ras is included to ensure constitutive negative feedback (Fritsche-Guenther et al., 2011). (E) dpERK output also exhibits a pulsatile response, peaking within 10 min. (F) We plot the peak dpERK output against peak input for a range of physiologically relevant u(EGF) doses, and find that it matches our steady-state predictions for linear input-output behavior. (G) Five-fold Raf overexpression does not break the linear input-output behavior.

https://doi.org/10.7554/eLife.33617.004

Figure 2—figure supplement 2

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The predicted linearity extends throughout the dynamic range of the ERK and Tgfβ pathways.

(**A–B**) Numerical simulation of the ERK and Tgfβ models. (A) The ERK model shows linear input-output relationship up to 93% of dpERK activation. (B) The Tgfβ pathway shows linear input-output relationship throughout the entire input range (from 0 to 1). Linearity was analyzed using the L1-norm or least absolute deviation (see Materials and methods). The blue range indicates where L1-norm was computed.

https://doi.org/10.7554/eLife.33617.005

Figure 2—figure supplement 3

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Model simulations for the Tgfβ pathway.

(A) Nuclear Smad4 concentration is constant to within 2%, over a physiologically relevant range of u(Tgfβ) considered here, justifying its inclusion into parameter group $α$ . (**B–D**) Numerical simulation of pulsatile response in the Tgfβ pathway. (B) A pulse of input, active Tgfβ receptor, is generated by Tgfβ addition in a model that includes details of receptor desensitization (Vizán et al., 2013). (C) S24n output also exhibits a pulsatile response. (D) We plot the peak S24n output against peak input, and find that it matches our steady-state predictions for linear input-output behavior.

https://doi.org/10.7554/eLife.33617.006

Figure 2—figure supplement 4

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Incorporating into the Wnt model the dual function of GSK3β in phosphorylating β-catenin and LRP5/6.

We include the role of GSK3β in phosphorylating LRP5/6 into the input function u(Wnt), such that u(Wnt) is a function of GSK3β and the phosphorylation rate k₉ (for simplicity, the same rate as GSK3β phosphorylation of β-catenin), and the reverse rate k_r. In both models, β-catenin increases in response to GSK3β inhibition (*e.g.*, by CHIR99021). However, only the model with the dual function of GSK3β shows a decrease in input range that we observed experimentally.

https://doi.org/10.7554/eLife.33617.007

Figure 2—figure supplement 5

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The requirements for linear signal transmission in the Wnt, Tgfβ, and ERK pathway.

In each plot, we varied S, defined in the equation shown on the x-axis, and simulated the input-output curve over the dynamic range of the model. The parameters in the equations are as defined in the main text. For the ERK and Tgfβ pathway, $α$ and γ are linked in such a way that they could not easily be varied independently. Linearity was assessed using the L-1 norm, which ranges from 0 to 0.5, with L-1 norm < 0.1 indicating linearity. L1-norm analysis was performed over the full dynamic range of the system, i.e., u(Wnt) = 1- 6, u(ERK) = 0 to 110,000 molecules of Ras-GTP, which gave 90% activation of [dpERK] in unperturbed cells, and u(Tgfβ) = 0 to 1.

https://doi.org/10.7554/eLife.33617.008

Figure 2—figure supplement 6

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Linear signal transmission occurs over a range of parameters in the model.

In this analysis, the parameter groups in each model were varied as indicated e.g., 3x is three-fold increase, 0.3x is three-fold decrease. 1x corresponds to the measured parameters. Plotted in each box is the input-output relationship, numerically simulated over the full dynamic range of the models, i.e., 1-6 for u(Wnt), 0-10⁵ for u(EGF), and 0-1 for u(Tgfβ). For simplicity, all outputs are normalized from 0 to 1. Grey shade: the unperturbed state. Purple shade: linear input-output response, as defined by L-1 norm < 0.1.

https://doi.org/10.7554/eLife.33617.009

Figure 2—figure supplement 7

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Numerical simulation of the input-output relationship of the NF-κB pathway.

We used the model first built by Hoffman *et al.* in 2002 (Hoffmann et al., 2002) and later revised by Ashall *et al.* in 2009 (Ashall et al., 2009). The parameters in the model have been measured or fitted to single-cell dynamics in multiple cell types (Hoffmann et al., 2002; Ashall et al., 2009). We simulated the model here over a physiologically observed dynamic range, i.e., Lee et al. (2014) observed in HeLa cells that at saturating ligand dose (10 ng/mL TNFα, set to one in the model),~25% of NF-κB pool is nuclear. Linearity is assessed using the L1-norm, where L1-norm < 0.1 indicates linear relationship (see Materials and methods).

https://doi.org/10.7554/eLife.33617.010

Figure 3 with 9 supplements

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Linearity was observed experimentally in the Wnt and ERK pathways.

(A) Measurements of the input-output relationship in the Wnt pathway. In these experiments, RKO cells were stimulated with 0–1280 ng/mL purified Wnt3A ligand, harvested at 6 hr after ligand stimulation, and lysed for Western blot analyses. Shown on top is a representative Western blot. The data plotted come from seven independent experiments (total N = 66). Each circle indicate the mean intensities of the phospho-LRP5/6 (x-axis) and β-catenin (y-axis) bands for all Western blot biological replicates, and error bars indicate the standard error of the mean. For each gel, we normalize the unstimulated sample (i.e. 0 ng/mL of Wnt3A) to one, and scale the magnitude of the dose response to the average of all gels (described in Materials and methods). The grey line is a least squares regression line, and ρ is the Pearson’s coefficient, where ρ = 1 is a perfect positive linear correlation. (B) Measurements of the input-output relationship in the ERK pathway. In these experiments, H1299 cells were stimulated with 0–50 ng/mL purified EGF ligand, harvested at 5 min after ligand stimulation, and lysed for Western blot analyses. Shown on top is a representative Western blot. The data plotted here come from five independent experiments (total N = 30). Each circle indicates the mean intensities of dpERK1/2 bands across Western blot biological replicates, and the error bars indicate standard error of the mean. Single replicates are plotted without error bars. All data is plotted relative to unstimulated sample. The grey line is a least squares regression line, and r² is the coefficient of correlation where r² = 1 is a perfect linear correlation. (C) As in (A), except that cells were treated with 1 μM CHIR99021 (detailed in Materials and methods). The data plotted here come from five independent experiments (total N = 59). The grey line is a least squares regression, and ρ is the Pearson’s coefficient, where ρ = 1 is a perfect positive linear correlation. Shown in the subplot are the same least squares regression line (solid line), overlaid with the model prediction (dashed line). (D) As in (B), but measurements were performed in H1299 cells expressing mutant Raf S289/296/301A. The data plotted here come from three independent experiments (total N = 15). The grey line is a fit using the ERK model. We first fitted the gain of the model to the data (i.e. the y-range), and afterward, varied the strength of dpERK feedback (k₂₅) to find the best fit. We used the weighted Akaike Information Criterion, w(AICc), to verify that the nonlinear fit from the ERK model outperforms a linear least squares fit (see Materials and methods). 0 < w(AICc) < 1, with higher w(AICc) indicates better performance by the non-linear fit. In all figures, linearity was additionally assessed using the least absolute deviations, L1-norm (see Methods). L1-norm can range from 0 to 0.5, with L1-norm < 0.1 indicate a linear relationship. Blue vs grey circles in each figure are explained in the main text. Source files of all Western blot gel images and numerical quantitation data are available in Figure 3—source data 1.

https://doi.org/10.7554/eLife.33617.012

Figure 3—source data 1: https://doi.org/10.7554/eLife.33617.022
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Figure 3—figure supplement 1

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LRP5/6 phosphorylation and β-catenin accumulation are already at steady state at 6 hr after Wnt stimulation.

RKO cells were treated with 160 ng/mL Wnt3A for the specified times, and then assayed for phospho-LRP5/6 and β-catenin level by Western blot. Error bars are standard error of the mean from 2 to 4 biological replicates. Data are plotted relative to the sample at time zero, and normalized to the average maximal activation across experiments. The grey lines connect the mean of each time point.

https://doi.org/10.7554/eLife.33617.013

Figure 3—figure supplement 2

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The dynamic range of Wnt signaling in RKO cells.

RKO cells were treated with the specified dose of Wnt3A for six hours, and then assayed for phospho-LRP5/6 and β-catenin by quantitative Western blot. Data are plotted relative to unstimulated samples. (**A–B**) In wt cells, phospho-LRP5/6 (A) shows > 90% of maximal response at 200 ng/mL Wnt3A, while β-catenin (B) shows 70% of maximal response at 200 ng/mL, and subsequently incremental response until 640 ng/mL Wnt3A. (**C–D**) In cells pre-treated with 1 µM CHIR99021, phospho-LRP5/6 (C) shows > 90% of maximal response at 80 ng/mL Wnt3A, while β-catenin (D) shows 70% of maximal response at 80 ng/mL and continues incremental activation at higher doses. (E). Cells were treated with 160 ng/mL Wnt3A and assayed for phospho-LRP5/6. Cells pre-treated with 1 µM CHIR99021 (N = 3) exhibited 50% the level of phospho-LRP5/6 as untreated cells (N = 3). The grey lines simply connect the means of data.

https://doi.org/10.7554/eLife.33617.014

Figure 3—figure supplement 3

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ERK activation peaks at 5 min after EGF stimulation.

H1299 cells were treated with 1 ng/mL EGF for the specified times, and then assayed for dpERK1/2 by Western blot. Data is plotted relative to the samples at time zero, with at least three biological replicates per time point.

https://doi.org/10.7554/eLife.33617.015

Figure 3—figure supplement 4

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Single-cell immunofluorescence measurements show graded ERK response to EGF.

In these experiments, H1299 cells were treated with varying doses of EGF for 5 min, and then fixed and analyzed for immunofluorescence against doubly phosphorylated ERK (dpERK).

(A) Representative images of cells treated with the indicated doses of EGF. (B) The intensity of nuclear level of dpERK staining across individual cells. Cell nuclei were delineated using DAPI staining (for EGF doses 0, 0.3, 1.3, and 2.0 ng/mL, N = 453, 381, 373, and 413 cells, respectively).

https://doi.org/10.7554/eLife.33617.016

Figure 3—figure supplement 5

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WT Raf-1 overexpression does not affect linear dose-response.

H1299 cells over-expressing Raf-1 were treated with the indicated dose of EGF for 5 min, and then assayed for dpERK1/2 by Western blot. The grey line is a fit from a linear model with r² = 0.99. Data is plotted relative to unstimulated samples, with total N = 9.

https://doi.org/10.7554/eLife.33617.017

Figure 3—figure supplement 6

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Expression of Raf S29/289/296/301/642A induces non-linear dose-response.

H1299 cells expressing the Raf mutant Raf S29/289/296/301/642 were treated with the indicated dose of EGF for 5 min, and then assayed for dpERK1/2 by Western blot. Data is plotted relative to unstimulated samples, with total N = 5.

https://doi.org/10.7554/eLife.33617.018

Figure 3—figure supplement 7

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Technical variability from Western blot.

(A) The level of β-catenin and phosphorylated LRP, measured across different lanes. (B) Ligand-stimulated change in β-catenin and phophorylated LRP level, measured in six independent Western blots. CV is coefficient of variation, defined as standard deviation/mean.

https://doi.org/10.7554/eLife.33617.019

Figure 3—figure supplement 8

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Linearity is not an artifact of loading control normalization.

In these two independent experiments, RKO cells were stimulated with a range of Wnt3A dose (0-160 ng/mL), the cells lysed after 6 hr, and analyzed for Western blot against β-catenin and phosphorylated LRP5/6 (pLRP5/6).

**Top row:** In each experiment, GAPDH intensity varies with <10% CV across samples. **Bottom row:** Raw β-catenin and LRP intensity data without normalization with GAPDH loading control. The measurements are plotted relative to unstimulated cells. Grey lines are least squares regression lines, and $ρ$ is the Pearson correlation coefficient.

https://doi.org/10.7554/eLife.33617.020

Figure 3—figure supplement 9

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Linearity was observed across independent experiments.

(A) In these two independent experiments, RKO cells were stimulated with a range of Wnt3A doses, lysed after 6 hr, and analyzed for Western blot against β-catenin and phospho-LRP5/6. (B) In these four independent experiments, H1299 cells were stimulated with a range of EGF doses, lysed after 5 min, and analyzed for Western blot against doubly-phosphorylated ERK. All measurements are plotted relative to unstimulated cells. Grey lines are least squares regression, ρ is Pearson correlation coefficient, and r² is correlation coefficient.

https://doi.org/10.7554/eLife.33617.021

Figure 4

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Benefits of linearity.

(A) Linearity enables multiplexing of inputs to a signaling pathway. Multiplexed signals can be independently decoded downstream, and therefore regulate distinct transcriptional events. (B) Illustration for how linearity between the receptor occupancy and downstream outputs gives rise to dose-response alignment (Andrews et al., 2016). (C) Linearity can produce fold-changes in output that are robust to variation in cellular parameters. To illustrate this, we added lognormal noise (0.1 CV) to all parameters of the Wnt model, and simulated the level of β-catenin before and after Wnt stimulation (blue circles). As long as the model operates in the regime of linear signal transmission (i.e. $Y = a \cdot u$ , where $Y$ is output, $u$ is input, and $a$ is a scalar that is a function of parameters), variation in parameters affects stimulated and basal level of β-catenin equally, and we get a constant fold change in β-catenin (i.e. red line, where $F C = Y_{s t i m u l a t e d} / Y_{b a s a l}$ is independent of parameter variations).

https://doi.org/10.7554/eLife.33617.023

Appendix 1—scheme 1

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Network of four proteins.
https://doi.org/10.7554/eLife.33617.026

Appendix 1—scheme 2

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Reaction set corresponding to protein network.
https://doi.org/10.7554/eLife.33617.027

Appendix 1—scheme 3

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Toy model of the ERK pathway.
https://doi.org/10.7554/eLife.33617.028

Tables

Appendix 1—table 1

Parameters, variables, and equations of the Wnt model.

https://doi.org/10.7554/eLife.33617.029

Parameter	Label	Value
Activation rate of Disheveled/Dvl by Wnt	$k_{1}$	$0.182$	$m i n^{- 1}$
Inactivation rate of Dvl	$k_{2}$	$1.82 \cdot 10^{- 2}$	$m i n^{- 1}$
Dissociation of destruction complex (DC) by active Dvl	$k_{3}$	$5.00 \cdot 10^{- 2}$	${n M}^{- 1} m i n^{- 1}$
Phosphorylation of DC	$k_{4}$	$0.267$	$m i n^{- 1}$
Dephosphorylation of DC	$k_{5}$	$0.133$	$m i n^{- 1}$
Forward rate for DC binding	$k_{6}$	$9.09 \cdot 10^{- 2}$	$n M^{- 1} m i n^{- 1}$
Reverse rate for DC binding	$k_{- 6}$	$0.909$	$m i n^{- 1}$
Dissociation constant for APC:axin binding	$K_{7}$	$50$	$n M$
Dissociation constant for β-catenin:DC binding	$K_{8}$	$120$	$n M$
Phosphorylation rate of β-catenin	$k_{9}$	$206$	$m i n^{- 1}$
Rate of phosphorylated β-catenin release from DC	$k_{10}$	$206$	$m i n^{- 1}$
Degradation rate of phosphorylated β-catenin	$k_{11}$	$0.417$	$m i n^{- 1}$
Synthesis rate of β-catenin	$v_{12}$	$0.423$	$n M m i n^{- 1}$
Degradation rate of β-catenin	$k_{13}$	$2.57 \cdot 10^{- 4}$	$m i n^{- 1}$
Synthesis rate of axin	$v_{14}$	$8.22 \cdot 10^{- 5}$	$n M m i n^{- 1}$
Degradation rate of axin	$k_{15}$	$0.167$	$m i n^{- 1}$
Dissociation constant for β-catenin:TCF binding	$K_{16}$	$30$	$n M$
Dissociation constant for β-catenin:APC binding	$K_{17}$	$1200$	$n M$
Total concentration of Disheveled	${D v l}_{t o t}$	$100$	$n M$
Total concentration of adenomatous polyposis coli	$A P C_{t o t}$	$100$	$n M$
Total concentration of T-cell factor	${T C F}_{t o t}$	$15$	$n M$
Total concentration of glycogen synthase kinase 3β	$G S K 3_{t o t}$	$50$	$n M$
Independent Variable		Label
Active Disheveled		$X_{2}$
APC/axin/GSK3 (* denotes phosphorylated)		$X_{3}$
APC/axin/GSK3		$X_{4}$
β-catenin/APC/axin*/GSK3		$X_{9}$
β-catenin*		$X_{10}$
β-catenin		$X_{11} (β c a t)$
axin		$X_{12}$
Dependent Variable		Label
Inactive Disheveled		$X_{1}$
GSK3		$X_{5}$
APC/axin		$X_{6}$
APC		$X_{7}$
β-catenin/APC/axin/GSK3		$X_{8}$
TCF		$X_{13}$
β-catenin/TCF		$X_{14}$
β-catenin/APC		$X_{15}$
Differential Equations
$\dot{[X_{2}]} = k_{1} \cdot W n t \cdot (D v l_{t o t} - [X_{2}]) - k_{2} \cdot [X_{2}]$
$(1 + \frac{[X_{11}]}{K_{8}}) \cdot \dot{[X_{3}]} + \frac{[X_{3}]}{K_{8}} \cdot \dot{[X_{11}]} = k_{4} \cdot [X_{4}] - k_{5} \cdot [X_{3}] - \frac{k_{9} \cdot [X_{3}] \cdot [X_{1} 1]}{K_{8}} + k_{10} \cdot [X_{9}]$
$\dot{[X_{4}]} = - (k_{3} \cdot [X_{2}] + k_{4} + k_{- 6}) \cdot [X_{4}] + k_{5} \cdot [X_{3}] + k_{6} \cdot G S K 3_{t o t} \cdot \frac{K_{17} \cdot [X_{12}] \cdot A P C_{t o t}}{K_{7} \cdot (K_{17} + [X_{11}])}$
$\dot{[X_{9}]} = \frac{k_{9} \cdot [X_{3}] \cdot [X_{11}]}{K_{8}} - k_{10} \cdot [X_{9}]$
$\dot{[X_{10}]} = k_{10} \cdot [X_{9}] - k_{11} \cdot [X_{10}]$
$(1 + \frac{[X_{3}]}{K_{8}} + \frac{K_{16} \cdot T C F_{t o t}}{(K_{16} + [X_{11}])^{2}} + \frac{K_{17} \cdot A P C_{t o t}}{(K_{17} + [X_{11}])^{2}}) \cdot \dot{[X_{11}]} + \frac{[X_{11}]}{K_{8}} \cdot \dot{[X_{3}]} = v_{12} - (\frac{k_{9} \cdot [X_{3}]}{K_{8}} + k_{13}) \cdot [X_{11}]$
$\begin{array}{ll} (1 + \frac{K_{17} \cdot A P C_{t o t}}{K_{7} \cdot (K_{17} + [X_{11}])}) \cdot \dot{[X_{12}]} - \frac{K_{17} \cdot [X_{12}] \cdot A P C_{t o t}}{K_{7} \cdot (K_{17} + [X_{11}])^{2}} \cdot \dot{[X_{11}]} \\ = k_{3} \cdot [X_{2}] \cdot [X_{4}] - k_{6} \cdot G S K_{t o t} \cdot \frac{K_{17} \cdot [X_{12} \cdot A P C_{t o t}]}{K_{7} \cdot (K_{17} + [X_{11}])} + k_{- 6} \cdot [X_{4}] + v_{14} \\ - k_{15} \cdot [X_{12}] \end{array}$
Equations for fast equilibrium reactions
$[X_{1}] = D v l_{t o t} - [X_{2}]$
$[X_{5}] = G S K 3_{t o t}$
$[X_{6}] = \frac{K_{17} \cdot [X_{12} \cdot A P C_{t o t}]}{K_{7} \cdot (K_{17} + [X_{11}])}$
$[X_{7}] = \frac{K_{17} \cdot A P C_{t o t}}{K_{17} \cdot (K_{17} + [X_{11}])}$
$[X_{8}] = \frac{[X_{3}] \cdot [X_{11}]}{K_{8}}$
$[X_{13}] = \frac{K_{16} \cdot T C F_{t o t}}{K_{16} + [X_{11}]}$
$[X_{14}] = \frac{[X_{11}] \cdot T C F_{t o t}}{K_{16} + [X_{11}]}$
$[X_{15}] = \frac{[X_{11}] \cdot A P C_{t o t}}{K_{17} + [X_{11}]}$

Appendix 1—table 2

Parameters, variables, and equations of the ERK model.

Values highlighted in yellow have been changed from the original model (explained in section ‘ERK Model’).

https://doi.org/10.7554/eLife.33617.030

Parameter	Label	Value
Forward rate for Raf:RasGTP binding	$k_{3}$	$1.67 \cdot 10^{- 6}$	$m o l e c u l e^{- 1} s^{- 1}$
Reverse rate for Raf:RasGTP binding	$k_{b 3}$	$5.3 \cdot 10^{- 3}$	$s^{- 1}$
Phosphorylation rate for Raf by RasGTP	$k_{4}$	$1$	$s^{- 1}$
Forward rate of pRaf:P1 binding	$k_{7}$	$1.18 \cdot 10^{- 4}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of pRaf:P1 binding	$k_{b 7}$	$0.2$	$s^{- 1}$
Dephosphorylation rate of pRaf by P1	$k_{8}$	$1$	$s^{- 1}$
Forward rate of MEK:pRaf binding	$k_{9}$	$1.95 \cdot 10^{- 5}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of MEK:pRaf binding	$k_{b 9}$	$3.3 \cdot 10^{- 2}$	$s^{- 1}$
Phosphorylation rate of MEK by pRaf	$k_{10}$	$3.5$	$s^{- 1}$
Forward rate of pMEK:pRaf binding	$k_{11}$	$1.95 \cdot 10^{- 5}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of pMEK:pRaf binding	$k_{b 11}$	$3.3 \cdot 10^{- 2}$	$s^{- 1}$
Phosphorylation rate of pMEK by pRaf	$k_{12}$	$2.9$	$s^{- 1}$
Forward rate of dpMEK:P2 binding	$k_{13}$	$2.38 \cdot 10^{- 5}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of dpMEK:P2 binding	$k_{b 13}$	$0.8$	$s^{- 1}$
Dephosphorylation rate of dpMEK by P2	$k_{14}$	$5.8 \cdot 10^{- 2}$	$s^{- 1}$
Forward rate of pMEK:P2 binding	$k_{15}$	$4.5 \cdot 10^{- 7}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of pMEK:P2 binding	$k_{b 15}$	$0.5$	$s^{- 1}$
Dephosphorylation rate of pMEK by P2	$k_{16}$	$5.8 \cdot 10^{- 2}$	$s^{- 1}$
Forward rate of ERK:dpMEK binding	$k_{17}$	$8.9 \cdot 10^{- 5}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of ERK:dpMEK binding	$k_{b 17}$	$1.83 \cdot 10^{- 2}$	$s^{- 1}$
Phosphorylation rate of ERK by dpMEK	$k_{18}$	$16$	$s^{- 1}$
Forward rate of pERK:dpMEK binding	$k_{19}$	$8.9 \cdot 10^{- 5}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of pERK:dpMEK binding	$k_{b 19}$	$1.83 \cdot 10^{- 2}$	$s^{- 1}$
Phosphorylation rate of pERK by dpMEK	$k_{20}$	$5.7$	$s^{- 1}$
Forward rate of pERK:P3 binding	$k_{21}$	$8.33 \cdot 10^{- 6}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of pERK:P3 binding	$k_{b 21}$	$0.5$	$s^{- 1}$
Dephosphorylation rate of pERK by P3	$k_{22}$	$0.246$	$s^{- 1}$
Forward rate of dpERK:P3 binding	$k_{23}$	$2.35 \cdot 10^{- 5}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of dpERK:P3 binding	$k_{b 23}$	$0.6$	$s^{- 1}$
Dephosphorylation rate of dpERK by P3	$k_{24}$	$0.246$	$s^{- 1}$
Forward rate of Raf:dpERK binding	$k_{25}$	$1 \cdot 10^{- 6}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of Raf:dpERK binding	$k_{b 25}$	$1$	$s^{- 1}$
Hyper-phosphorylation rate of Raf by ppERK	$k_{26}$	$10$	$s^{- 1}$
Forward rate of pRaf:dpERK binding	$k_{27}$	$0$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of pRaf:dpERK binding	$k_{b 27}$	$1$	$s^{- 1}$
Hyper-phosphorylation rate of phosphorylated Raf by dpERK	$k_{28}$	$10$	$s^{- 1}$
Forward rate of Rafi:P4 binding	$k_{29}$	$5 \cdot 10^{- 5}$	${m o l e c u l e}^{- 1} s^{- 1}$
Reverse rate of Rafi:P4 binding	$k_{b 29}$	$0.2$	$s^{- 1}$
Dephosphorylation rate of Rafi by P4	$k_{30}$	$0.5$	$s^{- 1}$
Total Raf	$R a f_{t o t}$	$4 \cdot 10^{4}$	$m o l e c u l e s$
Total MEK	$M E K_{t o t}$	$2.1 \cdot 10^{7}$	$m o l e c u l e s$
Total ERK	$E R K_{t o t}$	$2.21 \cdot 10^{7}$	$m o l e c u l e s$
Total phosphatase P1	$P 1_{t o t}$	$4 \cdot 10^{4}$	$m o l e c u l e s$
Total phosphatase P2	$P 2_{t o t}$	$4 \cdot 10^{5}$	$m o l e c u l e s$
Total phosphatase P3	$P 3_{t o t}$	$1 \cdot 10^{7}$	$m o l e c u l e s$
Total phosphatase P4	$P 4_{t o t}$	$4 \cdot 10^{4}$	$m o l e c u l e s$
Variable		Label
Unphosphorylated Raf		$R a f$
Raf bound to RasGTP		$R a f : R a s G T P$
Phosphorylated Raf		$p R a f$
Phosphatase for phosphorylated Raf		$P 1$
Phosphorylated Raf bound to its phosphatase		$p R a f : P 1$
Unphosphorylated MEK		$M E K$
MEK bound to its kinase		$M E K : p R a f$
Phosphorylated MEK		$p M E K$
Phosphorylated MEK bound to its kinase		$p M E K : p R a f$
Doubly-phosphorylated MEK		$d p M E K$
MEK phosphatase		$P 2$
Doubly-phosphorylated MEK bound to its phosphatase		$d p M E K : P 2$
Phosphorylated MEK bound to its phosphatase		$p M E K : P 2$
Unphosphorylated ERK		$E R K$
ERK bound to its kinase		$E R K : d p M E K$
Phosphorylated ERK		$p E R K$
Phosphorylated ERK bound to its kinase		$p E R K : d p M E K$
Doubly-phosphorylated ERK		$d p E R K$
ERK phosphatase		$P 3$
Phosphorylated ERK bound to its phosphatase		$p E R K : P 3$
Doubly-phosphorylated ERK bound to its phosphatase		$d p E R K : P 3$
Raf bound to doubly-phosphorylated ERK		$R a f : d p E R K$
Hyper-phosphorylated, ‘inactive’ Raf		$R a f i$
Phosphorylated Raf bound to doubly-phosphorylated ERK		$p R a f : d p E R K$
Phosphatase for hyper-phosphorylated Raf		$P 4$
Hyper-phosphorylated Raf bound to its phosphatase		$R a f i : P 4$
Differential Equations
$\begin{array}{ll} \dot{[R a f]} = - k_{3} \cdot [R a f] \cdot u (E G F) + k_{b 3} \cdot [R a f : R a s G T P] + k_{8} \cdot [p R a f : P 1] - k_{2} 5 \cdot [R a f] \\ \cdot [d p E R K] + k_{b 25} \cdot [R a f : d p E R K] + k_{30} \cdot [R a f i : P 4] \end{array}$
$\dot{[R a f : R a s G P T]} = k_{3} \cdot [R a f] \cdot u (E G F) - (k_{b 3} + k_{4}) \cdot [R a f : R a s G T P]$
$\begin{array}{ll} \dot{[p R a f]} = & k_{4} \cdot [R a f : R a s G T P] - k_{7} \cdot [p R a f] \cdot [P 1] + k_{b 7} \cdot [p R a f : P 1] - k_{9} \cdot [M E K] \cdot [p R a f] \\ + (k_{b 9} + k_{10}) \cdot [M E K : p R a f] - k_{11} \cdot [p M E K] \cdot [p R a f] + (k_{b 11} + k_{12}) \\ \cdot [p M E K : p R a f] - k_{27} \cdot [p R a f] \cdot [d p E R K] + k_{b 27} \cdot [p R a f : d p E R K] \end{array}$
$\dot{[P 1]} = - k_{7} \cdot [p R a f] \cdot [P 1] + (k_{b 7} + k_{8}) \cdot [p R a f : P 1]$
$\dot{[p R a f : P 1]} = k_{7} \cdot [p R a f] \cdot [P 1] - (k_{b 7} + k_{8}) \cdot [p R a f : P 1]$
$\dot{[M E K]} = - k_{9} \cdot [M E K] \cdot [p R a f] + k_{b 9} \cdot [M E K : p R a f] + k_{16} \cdot [p M E K : P 2]$
$\dot{[M E K : p R a f]} = k_{9} \cdot [M E K] \cdot [p R a f] - (k_{b 9} + k_{10}) \cdot [M E K : p R a f]$
$\begin{array}{ll} \dot{[p M E K]} = & k_{10} \cdot [M E K : p R a f] - k_{11} \cdot [p M E K] \cdot [p R a f] + k_{b 11} \cdot [p M E K : p R a f] + k_{14} \\ \cdot [d p M E K : P 2] - k_{15} \cdot [p M E K] \cdot [P 2] + k_{b 15} \cdot [p M E K : P 2] \end{array}$
$\dot{[p M E K : p R a f]} = k_{11} \cdot [p M E K] \cdot [p R a f] - (k_{b 11} + k_{12}) \cdot [p M E K : p R a f]$
$\begin{array}{ll} \dot{[d p M E K]} = & k_{12} \cdot [p M E K : p R a f] - k_{13} \cdot [d p M E K] \cdot [P 2] + k_{b 13} \cdot [d p M E K : P 2] - k_{17} \cdot [E R K] \\ \cdot [d p M E K] + (k_{b 17} + k_{18}) \cdot [E R K : d p M E K] - k_{19} \cdot [p E R K] \cdot [d p M E K] \\ + (k_{b 19} + k_{20}) \cdot [p E R K : d p M E K] \end{array}$
$\begin{array}{ll} \dot{[P 2]} = & - k_{13} \cdot [d p M E K] \cdot [P 2] + (k_{b 13} + k_{14}) \cdot [d p M E K : P 2] - k_{15} \cdot [p M E K] \cdot [P 2] \\ + (k_{b 15} + k_{16}) \cdot [p M E K : P 2] \end{array}$
$\dot{[d p M E K : P 2]} = k_{13} \cdot [d p M E K] \cdot [P 2] - (k_{b 13} + k_{14}) \cdot [d p M E K : P 2]$
$\dot{[p M E K : P 2]} = k_{15} \cdot [p M E K] \cdot [P 2] - (k_{b 15} + k_{16}) \cdot [p M E K : P 2]$
$\dot{[E R K]} = - k_{17} \cdot [E R K] \cdot [d p M E K] + k_{b 17} \cdot [E R K : d p M E K] + k_{22} \cdot [p E R K : P 3]$
$\dot{[E R K : d p M E K]} = k_{17} \cdot [E R K] \cdot [d p M E K] - (k_{b 17} + k_{18}) \cdot [E R K : d p M E K]$
$\begin{array}{ll} \dot{[p E R K]} = & k_{18} \cdot [E R K : d p M E K] - k_{19} \cdot [p E R K] \cdot [d p M E K] + k_{b 19} \cdot [p E R K : d p M E K] - k_{21} \\ \cdot [p E R K] \cdot [P 3] + k_{b 21} \cdot [p E R K : P 3] + k_{24} \cdot [d p E R K : P 3] \end{array}$
$\dot{[p E R K : d p M E K]} = k_{19} \cdot [p E R K] \cdot [d p M E K] - (k_{b 19} + k_{20}) \cdot [p E R K : d p M E K]$
$\begin{array}{ll} \dot{[d p E R K]} = & k_{20} \cdot [p E R K : d p M E K] - k_{23} \cdot [d p E R K] \cdot [P 3] + k_{b 23} \cdot [d p E R K : P 3] - k_{25} \cdot [R a f] \\ \cdot [d p E R K] + (k_{b 25} + k_{26}) \cdot [R a f : d p E R K] - k_{27} \cdot [p R a f] \cdot [d p E R K] \\ + (k_{b 27} + k_{28}) \cdot [p R a f : d p E R K] \end{array}$
$\begin{array}{ll} \dot{[P 3]} = & - k_{21} \cdot [p E R K] \cdot [P 3] + (k_{b 21} + k_{22}) \cdot [p E R K : P 3] - k_{23} \cdot [d p E R K] \cdot [P 3] \\ + (k_{b 23} + k_{24}) \cdot [d p E R K : P 3] \end{array}$
$\dot{[p E R K : P 3]} = k_{21} [p E R K] \cdot [P 3] - (k_{b 21} + k_{22}) \cdot [p E R K : P 3]$
$\dot{[d p E R K : P 3]} = k_{23} \cdot [d p E R K] \cdot [P 3] - (k_{b 23} + k_{24}) \cdot [d p E R K : P 3]$
$\dot{[R a f : d p E R K]} = k_{25} \cdot [R a f] \cdot [d p E R K] - (k_{b 25} + k_{26}) \cdot [R a f : d p E R K]$
$\dot{[R a f i]} = k_{26} \cdot [R a f : d p E R K] + k_{28} \cdot [p R a f : d p E R K] - k_{29} \cdot [R a f i] \cdot [P 4] + k_{b 29} \cdot [R a f i : P 4]$
$\dot{[p R a f : d p E R K]} = k_{27} \cdot [p R a f] \cdot [d p E R K] - (k_{b 27} + k_{28}) \cdot [p R a f : d p E R K]$
$\dot{[P 4]} = - k_{29} \cdot [R a f i] \cdot [P 4] + (k_{b 29} + k_{30}) \cdot [R a f i : P 4]$
$\dot{[R a f i : P 4]} = k_{29} \cdot [R a f i] \cdot [P 4] - (k_{b 29} + k_{30}) \cdot [R a f i : P 4]$
Algebraic Equations for conserved species
$R a f_{t o t} = [R a f] + [R a f : R a s G T P] + [p R a f] + [p R a f : P 1] + [M E K : p R a f] + [p M E K : p R a f] + [R a f : d p E R K] + [R a f i] + [p R a f : d p E R K] + [R a f i : P 4]$
$M E K_{t o t} = [M E K] + [M E K : p R a f] + [p M E K] + [p M E K : p R a f] + [d p M E K] + [d p M E K : P 2] + [p M E K : P 2] + [E R K : d p M E K] + [p E R K : d p M E K]$
$E R K_{t o t} = [E R K] + [E R K : d p M E K] + [p E R K] + [p E R K : d p M E K] + [d p E R K] + [p E R K : P 3] + [d p E R K : P 3] + [R a f : d p E R K] + [p R a f : d p E R K]$
$P 1_{t o t} = [P 1] + [p R a f : P 1]$
$P 2_{t o t} = [P 2] + [d p M E K : P 2] + [p M E K : P 2]$
$P 3_{t o t} = [P 3] + [p E R K : P 3] + [d p E R K : P 3]$
$P 4_{t o t} = [P 4] + [R a f i : P 4]$

Appendix 1—table 3

Parameters, variables and equations of the Tgfβ model.

https://doi.org/10.7554/eLife.33617.031

Parameter	Label	Value
Phosphorylation rate of Smad2	$k_{p h o s}$	$4.0 \cdot 10^{- 4}$	$n M^{- 1} s^{- 1}$
Dephosphorylation rate of Smad2	$k_{d e p h o s}$	$6.6 \cdot 10^{- 3}$	$n M^{- 1} s^{- 1}$
Nuclear import rate of Smad2	$k_{i n 2}$	$2.6 \cdot 10^{- 3}$	$s^{- 1}$
Nuclear export rate of Smad2	$k_{e x 2}$	$5.6 \cdot 10^{- 3}$	$s^{- 1}$
Nuclear import rate of Smad4	$k_{i n 4}$	$2.6 \cdot 10^{- 3}$	$s^{- 1}$
Nuclear export rate of Smad4	$k_{e x 4}$	$2.6 \cdot 10^{- 3}$	$s^{- 1}$
Smad complex import factor	$C I F$	$5.7$
Forward rate for Smad complex binding	$k_{o n}$	$1.8 \cdot 10^{- 3}$	$n M^{- 1} s^{- 1}$
Reverse rate for Smad complex binding	$k_{o f f}$	$1.6 \cdot 10^{- 2}$	$s^{- 1}$
Cytoplasmic to nuclear volume ratio	$a$	$2.3$
Total Smad2 (initialized to cytoplasm)	$S 2_{t o t}$	$73.0$	$n M$
Total Smad4 (initialized to cytoplasm)	$S 4_{t o t}$	$73.0$	$n M$
Total phosphatase in nucleus	$P P a s e$	$1$	$n M$
Total Receptors	$R_{t o t}$	$1$	$n M$
Variable		Label
Cytoplasmic Smad2		$S 2 c$
Cytoplasmic phosphorylated Smad2		$p S 2 c$
Cytoplasmic Smad4		$S 4 c$
Cytoplasmic Smad2:Smad4 complex		$S 24 c$
Cytoplasmic Smad2:Smad2 complex		$S 22 c$
Nuclear Smad2		$S 2 n$
Nuclear phosphorylated Smad2		$p S 2 n$
Nuclear Smad4		$S 4 n$
Nuclear Smad2:Smad4 complex		$S 24 n$
Nuclear Smad2:Smad2 complex		$S 22 n$
Differential Equations
$\dot{[S 2 c]} = - k_{p h o s} \cdot u (T g f β) \cdot [S 2 c] - k_{i n 2} \cdot [S 2 c] + k_{e x 2} \cdot [S 2 n]$
$\begin{array}{ll} \dot{[p S 2 c]} = & k_{p h o s} \cdot u (T f g β) \cdot [S 2 c] - k_{i n 2} \cdot [p S 2 c] - k_{o n} \cdot [p S 2 c] \cdot ([S 4 c] + 2 \cdot [p S 2 c]) + k_{o f f} \\ \cdot ([S 24 c] + 2 \cdot [S 22 c]) + k_{e x 2} \cdot [p S 2 n] \end{array}$
$\dot{[S 4 c]} = - k_{i n 4} \cdot [S 4 c] - k_{o n} \cdot [p S 2 c] \cdot [S 4 c] + k_{o f f} \cdot [S 24 c] + k_{e x 4} \cdot [S 4 n]$
$\dot{[S 24 c]} = k_{o n} [p S 2 c] \cdot [S 4 c] - k_{o f f} \cdot [S 24 c] - k_{i n 2} \cdot C I F \cdot [S 24 c]$
$\dot{[S 22 c]} = k_{o n} \cdot [p S 2 c] 2 - k_{o f f} \cdot [S 22 c] - k_{i n 2} \cdot C I F \cdot [S 22 c]$
$\dot{[S 2 n]} = a \cdot k_{i n 2} \cdot [S 2 c] - a \cdot k_{e x 2} \cdot [S 2 n] + k_{d e p h o s} \cdot P P a s e \cdot [p S 2 n]$
$\begin{array}{ll} \dot{[p S 2 n]} = & a \cdot k_{i n 2} \cdot [p S 2 c] - a \cdot k_{e x 2} \cdot [p S 2 n] - k_{d e p h o s} \cdot P P a s e \cdot [p S 2 n] - k_{o n} \cdot [p S 2 n] \\ \cdot ([S 4 n] + 2 \cdot [p S 2 n]) + k_{o f f} \cdot ([S 24 n] + 2 \cdot [S 22 n]) \end{array}$
$\dot{[S 4 n]} = a \cdot k_{i n 4} \cdot [S 4 c] - a \cdot k_{e x 4} \cdot [S 4 n] - k_{o n} \cdot [p S 2 n] \cdot [S 4 n] + k_{o f f} \cdot [S 24 n]$
$\dot{[S 24 n]} = a \cdot k_{i n 2} \cdot C I F \cdot [S 24 c] + k_{o n} \cdot [p S 2 n] \cdot [S 4 n] - k_{o f f} \cdot [S 24 n]$
$\dot{[S 22 n]} = a \cdot k_{i n 2} \cdot C I F \cdot [S 22 c] + k_{o n} \cdot [p S 2 n] 2 - k_{o f f} \cdot [S 22 n]$
Algebraic Equations for conserved species
$S 2_{t o t} = [S 2 c] + [p S 2 c] + [S 24 c] + 2 \cdot [S 22 c] + (2 \cdot [S 22 n] + [S 24 n] + [p S 2 n] + [S 2 n])$
$S 4_{t o t} = [S 4 c] + [S 24 c] + \frac{1}{a} ([S 24 n] + [S 4 n])$

Appendix 1—table 4

Examples of biological systems where the Wnt, ERK, and Tgfβ pathways have been shown to produce graded response in single-cell level.

https://doi.org/10.7554/eLife.33617.032

Pathway	Systems where graded response has been observed	References
Live imaging of single cells
Tgfβ pathway	Mouse myoblasts	Frick et al., 2017; Warmflash et al. (2012)
	Human epidermal keratinocytes	Nicolás et al. (2004); Warmflash et al. (2012) ; Schmierer et al. (2008); Vizán et al. (2013)
	Human cervical epithelial cells	Nicolás et al. (2004)
	Human breast epithelial cells	Strasen et al. (2018)
Canonical Wnt pathway	Human embryonic kidney cells	Kafri et al. (2016) (this is the only published live single-cell imaging study in the Wnt pathway so far)
ERK pathway	Mouse fibroblasts	Toettcher et al. (2013)
	Mouse embryonic fibroblasts	Mackeigan et al. (2005)
	Human non-small cell lung carcinoma	Cheong et al., 2011
	Human mammary gland cells	Selimkhanov et al. (2014); Perrett et al., 2013 Perrett et al., 2013
	Human cervical epithelial cells	Voliotis et al. (2014); Whitehurst et al. (2004); Perrett et al., 2013 Perrett et al., 2013
	Human foreskin fibroblasts	Whitehurst et al. (2004)
Immunofluorescence and FACS studies
Tgfβ pathway	Xenopus embryo	Schohl and Fagotto (2002)
	Mouse testes	Itman et al. (2009)
	Zebrafish embryo	Dubrulle et al. (2015)
Canonical Wnt pathway	Xenopus embryo	Schneider et al. (1996); Fagotto and Gumbiner (1994); Schohl and Fagotto (2002)
	Mouse embyo	Aulehla et al., 2008
	Planaria	Sureda-Gómez et al., 2016
	Sea anemone embryo	Wikramanayake et al., 2003
ERK pathway	Chick embryo	Delfini et al. (2005)
	Xenopus embryo	Schohl and Fagotto (2002)
	Human T lymphocyte cells	Lin et al. (2009)
	Rat adrenal gland cells	Santos et al., 2007