1. Cell Biology
  2. Computational and Systems Biology
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Signaling pathways as linear transmitters

  1. Harry Nunns  Is a corresponding author
  2. Lea Goentoro  Is a corresponding author
  1. California Institute of Technology, United States
Research Article
Cite this article as: eLife 2018;7:e33617 doi: 10.7554/eLife.33617
7 figures, 4 tables and 1 additional file

Figures

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
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(Wnt) is the rate by which Dishevelled/Dvl inhibits the destruction complex upon Wnt ligand activation, where k3 and k6 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—figure supplement 1
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 Es and Rs 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
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
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
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 kr. 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
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
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
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
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—figure supplement 1
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
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
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
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
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
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
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
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
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
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=au, 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 FC=Ystimulated/Ybasal is independent of parameter variations).

https://doi.org/10.7554/eLife.33617.023
Appendix 1—scheme 1
Network of four proteins.
https://doi.org/10.7554/eLife.33617.026
Appendix 1—scheme 2
Reaction set corresponding to protein network.
https://doi.org/10.7554/eLife.33617.027
Appendix 1—scheme 3
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
ParameterLabelValue
Activation rate of Disheveled/Dvl by Wnt

k1

0.182

min-1

Inactivation rate of Dvl

k2

1.82102

min-1

Dissociation of destruction complex (DC) by active Dvl

k3

5.00102

nM-1 min-1

Phosphorylation of DC

k4

0.267

min-1

Dephosphorylation of DC

k5

0.133

min-1

Forward rate for DC binding

k6

9.09102

nM-1 min-1

Reverse rate for DC binding

k-6

0.909

min-1

Dissociation constant for APC:axin binding

K7

50

nM

Dissociation constant for β-catenin:DC binding

K8

120

nM

Phosphorylation rate of β-catenin

k9

206

min-1

Rate of phosphorylated β-catenin release from DC

k10

206

min-1

Degradation rate of phosphorylated β-catenin

k11

0.417

min-1

Synthesis rate of β-catenin

v12

0.423

nM min-1

Degradation rate of β-catenin

k13

2.57104

min-1

Synthesis rate of axin

v14

8.22105

nM min-1

Degradation rate of axin

k15

0.167

min-1

Dissociation constant for β-catenin:TCF binding

K16

30

nM

Dissociation constant for β-catenin:APC binding

K17

1200

nM

Total concentration of Disheveled

Dvltot

100

nM

Total concentration of adenomatous polyposis coli

APCtot

100

nM

Total concentration of T-cell factor

TCFtot

15

nM

Total concentration of glycogen synthase kinase 3β

GSK3tot

50

nM

Independent VariableLabel
Active Disheveled

X2

APC*/axin*/GSK3 (* denotes phosphorylated)

X3

APC/axin/GSK3

X4

β-catenin*/APC*/axin*/GSK3

X9

β-catenin*

X10

β-catenin

X11 (βcat)

axin

X12

Dependent VariableLabel
Inactive Disheveled

X1

GSK3

X5

APC/axin

X6

APC

X7

β-catenin/APC*/axin*/GSK3

X8

TCF

X13

β-catenin/TCF

X14

β-catenin/APC

X15

Differential Equations

[X2]˙=k1Wnt(Dvltot[X2])k2[X2]

(1+[X11]K8)[X3]˙+[X3]K8[X11]˙=k4[X4]k5[X3]k9[X3][X11]K8+k10[X9]

[X4]˙=(k3[X2]+k4+k6)[X4]+k5[X3]+k6GSK3totK17[X12]APCtotK7(K17+[X11])

[X9]˙=k9[X3][X11]K8k10[X9]

[X10]˙=k10[X9]k11[X10]

(1+[X3]K8+K16TCFtot(K16+[X11])2+K17APCtot(K17+[X11])2)[X11]˙+[X11]K8[X3]˙=v12(k9[X3]K8+k13)[X11]



(1+K17APCtotK7(K17+[X11]))[X12]˙K17[X12]APCtotK7(K17+[X11])2[X11]˙=k3[X2][X4]k6GSKtotK17[X12APCtot]K7(K17+[X11])+k6[X4]+v14k15[X12]


Equations for fast equilibrium reactions

X1=Dvltot-X2

X5=GSK3tot


[X6]=K17[X12APCtot]K7(K17+[X11])

[X7]=K17APCtotK17(K17+[X11])

[X8]=[X3][X11]K8

[X13]=K16TCFtotK16+[X11]

[X14]=[X11]TCFtotK16+[X11]

X15=X11APCtotK17+X11
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
ParameterLabelValue
Forward rate for Raf:RasGTP binding

k3

1.67106

molecule1s1

Reverse rate for Raf:RasGTP binding

kb3

5.3103

s-1

Phosphorylation rate for Raf by RasGTP

k4

1

s-1

Forward rate of pRaf:P1 binding

k7

1.18104

molecule-1s-1

Reverse rate of pRaf:P1 binding

kb7

0.2

s-1

Dephosphorylation rate of pRaf by P1

k8

1

s-1

Forward rate of MEK:pRaf binding

k9

1.95105

molecule-1s-1

Reverse rate of MEK:pRaf binding

kb9

3.3102

s-1

Phosphorylation rate of MEK by pRaf

k10

3.5

s-1

Forward rate of pMEK:pRaf binding

k11

1.95105

molecule-1s-1

Reverse rate of pMEK:pRaf binding

kb11

3.3102

s-1

Phosphorylation rate of pMEK by pRaf

k12

2.9

s-1

Forward rate of dpMEK:P2 binding

k13

2.38105

molecule-1s-1

Reverse rate of dpMEK:P2 binding

kb13

0.8

s-1

Dephosphorylation rate of dpMEK by P2

k14

5.8102

s-1

Forward rate of pMEK:P2 binding

k15

4.5107

molecule-1s-1

Reverse rate of pMEK:P2 binding

kb15

0.5

s-1

Dephosphorylation rate of pMEK by P2

k16

5.8102

s-1

Forward rate of ERK:dpMEK binding

k17

8.9105

molecule-1s-1

Reverse rate of ERK:dpMEK binding

kb17

1.83102

s-1

Phosphorylation rate of ERK by dpMEK

k18

16

s-1

Forward rate of pERK:dpMEK binding

k19

8.9105

molecule-1s-1

Reverse rate of pERK:dpMEK binding

kb19

1.83102

s-1

Phosphorylation rate of pERK by dpMEK

k20

5.7

s-1

Forward rate of pERK:P3 binding

k21

8.33106

molecule-1s-1

Reverse rate of pERK:P3 binding

kb21

0.5

s-1

Dephosphorylation rate of pERK by P3

k22

0.246

s-1

Forward rate of dpERK:P3 binding

k23

2.35105

molecule-1s-1

Reverse rate of dpERK:P3 binding

kb23

0.6

s-1

Dephosphorylation rate of dpERK by P3

k24

0.246

s-1

Forward rate of Raf:dpERK binding

k25

1106

molecule-1s-1

Reverse rate of Raf:dpERK binding

kb25

1

s-1

Hyper-phosphorylation rate of Raf by ppERK

k26

10

s-1

Forward rate of pRaf:dpERK binding

k27

0

molecule-1s-1

Reverse rate of pRaf:dpERK binding

kb27

1

s-1

Hyper-phosphorylation rate of phosphorylated Raf by dpERK

k28

10

s-1

Forward rate of Rafi:P4 binding

k29

5105

molecule-1s-1

Reverse rate of Rafi:P4 binding

kb29

0.2

s-1

Dephosphorylation rate of Rafi by P4

k30

0.5

s-1

Total Raf

Raftot

4104

molecules

Total MEK

MEKtot

2.1107

molecules

Total ERK

ERKtot

2.21107

molecules

Total phosphatase P1

P1tot

4104

molecules

Total phosphatase P2

P2tot

4105

molecules

Total phosphatase P3

P3tot

1107

molecules

Total phosphatase P4

P4tot

4104

molecules

VariableLabel
Unphosphorylated Raf

Raf

Raf bound to RasGTP

Raf:RasGTP

Phosphorylated Raf

pRaf

Phosphatase for phosphorylated Raf

P1

Phosphorylated Raf bound to its phosphatase

pRaf:P1

Unphosphorylated MEK

MEK

MEK bound to its kinase

MEK:pRaf

Phosphorylated MEK

pMEK

Phosphorylated MEK bound to its kinase

pMEK:pRaf

Doubly-phosphorylated MEK

dpMEK

MEK phosphatase

P2

Doubly-phosphorylated MEK bound to its phosphatase

dpMEK:P2

Phosphorylated MEK bound to its phosphatase

pMEK:P2

Unphosphorylated ERK

ERK

ERK bound to its kinase

ERK:dpMEK

Phosphorylated ERK

pERK

Phosphorylated ERK bound to its kinase

pERK:dpMEK

Doubly-phosphorylated ERK

dpERK

ERK phosphatase

P3

Phosphorylated ERK bound to its phosphatase

pERK:P3

Doubly-phosphorylated ERK bound to its phosphatase

dpERK:P3

Raf bound to doubly-phosphorylated ERK

Raf:dpERK

Hyper-phosphorylated, ‘inactive’ Raf

Rafi

Phosphorylated Raf bound to doubly-phosphorylated ERK

pRaf:dpERK

Phosphatase for hyper-phosphorylated Raf

P4

Hyper-phosphorylated Raf bound to its phosphatase

Rafi:P4

Differential Equations

[Raf]˙=k3[Raf]u(EGF)+kb3[Raf:RasGTP]+k8[pRaf:P1]k25[Raf][dpERK]+kb25[Raf:dpERK]+k30[Rafi:P4]

[Raf:RasGPT]˙=k3[Raf]u(EGF)(kb3+k4)[Raf:RasGTP]

[pRaf]˙=k4[Raf:RasGTP]k7[pRaf][P1]+kb7[pRaf:P1]k9[MEK][pRaf]+(kb9+k10)[MEK:pRaf]k11[pMEK][pRaf]+(kb11+k12)[pMEK:pRaf]k27[pRaf][dpERK]+kb27[pRaf:dpERK]

[P1]˙=k7[pRaf][P1]+(kb7+k8)[pRaf:P1]

[pRaf:P1]˙=k7[pRaf][P1](kb7+k8)[pRaf:P1]

[MEK]˙=k9[MEK][pRaf]+kb9[MEK:pRaf]+k16[pMEK:P2]

[MEK:pRaf]˙=k9[MEK][pRaf](kb9+k10)[MEK:pRaf]

[pMEK]˙=k10[MEK:pRaf]k11[pMEK][pRaf]+kb11[pMEK:pRaf]+k14[dpMEK:P2]k15[pMEK][P2]+kb15[pMEK:P2]

[pMEK:pRaf]˙=k11[pMEK][pRaf](kb11+k12)[pMEK:pRaf]

[dpMEK]˙=k12[pMEK:pRaf]k13[dpMEK][P2]+kb13[dpMEK:P2]k17[ERK][dpMEK]+(kb17+k18)[ERK:dpMEK]k19[pERK][dpMEK]+(kb19+k20)[pERK:dpMEK]

[P2]˙=k13[dpMEK][P2]+(kb13+k14)[dpMEK:P2]k15[pMEK][P2]+(kb15+k16)[pMEK:P2]

[dpMEK:P2]˙=k13[dpMEK][P2](kb13+k14)[dpMEK:P2]

[pMEK:P2]˙=k15[pMEK][P2](kb15+k16)[pMEK:P2]

[ERK]˙=k17[ERK][dpMEK]+kb17[ERK:dpMEK]+k22[pERK:P3]

[ERK:dpMEK]˙=k17[ERK][dpMEK](kb17+k18)[ERK:dpMEK]

[pERK]˙=k18[ERK:dpMEK]k19[pERK][dpMEK]+kb19[pERK:dpMEK]k21[pERK][P3]+kb21[pERK:P3]+k24[dpERK:P3]

[pERK:dpMEK]˙=k19[pERK][dpMEK](kb19+k20)[pERK:dpMEK]

[dpERK]˙=k20[pERK:dpMEK]k23[dpERK][P3]+kb23[dpERK:P3]k25[Raf][dpERK]+(kb25+k26)[Raf:dpERK]k27[pRaf][dpERK]+(kb27+k28)[pRaf:dpERK]

[P3]˙=k21[pERK][P3]+(kb21+k22)[pERK:P3]k23[dpERK][P3]+(kb23+k24)[dpERK:P3]

[pERK:P3]˙=k21[pERK][P3](kb21+k22)[pERK:P3]

[dpERK:P3]˙=k23[dpERK][P3](kb23+k24)[dpERK:P3]

[Raf:dpERK]˙=k25[Raf][dpERK](kb25+k26)[Raf:dpERK]

[Rafi]˙=k26[Raf:dpERK]+k28[pRaf:dpERK]k29[Rafi][P4]+kb29[Rafi:P4]

[pRaf:dpERK]˙=k27[pRaf][dpERK](kb27+k28)[pRaf:dpERK]

[P4]˙=k29[Rafi][P4]+(kb29+k30)[Rafi:P4]

[Rafi:P4]˙=k29[Rafi][P4](kb29+k30)[Rafi:P4]
Algebraic Equations for conserved species

Raftot=[Raf]+[Raf:RasGTP]+[pRaf]+[pRaf:P1]+[MEK:pRaf]+[pMEK:pRaf]+[Raf:dpERK]+[Rafi]+[pRaf:dpERK]+[Rafi:P4]

MEKtot=[MEK]+[MEK:pRaf]+[pMEK]+[pMEK:pRaf]+[dpMEK]+[dpMEK:P2]+[pMEK:P2]+[ERK:dpMEK]+[pERK:dpMEK]

ERKtot=[ERK]+[ERK:dpMEK]+[pERK]+[pERK:dpMEK]+[dpERK]+[pERK:P3]+[dpERK:P3]+[Raf:dpERK]+[pRaf:dpERK]

P1tot=[P1]+[pRaf:P1]

P2tot=[P2]+[dpMEK:P2]+[pMEK:P2]

P3tot=[P3]+[pERK:P3]+[dpERK:P3]

P4tot=[P4]+[Rafi:P4]
Appendix 1—table 3
Parameters, variables and equations of the Tgfβ model.
https://doi.org/10.7554/eLife.33617.031
ParameterLabelValue
Phosphorylation rate of Smad2

kphos

4.0104

nM1s1

Dephosphorylation rate of Smad2

kdephos

6.6103

nM1s1

Nuclear import rate of Smad2

kin2

2.6103

s1

Nuclear export rate of Smad2

kex2

5.6103

s1

Nuclear import rate of Smad4

kin4

2.6103

s1

Nuclear export rate of Smad4

kex4

2.6103

s1

Smad complex import factor

CIF

5.7

Forward rate for Smad complex binding

kon

1.8103

nM1s1

Reverse rate for Smad complex binding

koff

1.6102

s1

Cytoplasmic to nuclear volume ratio

a

2.3

Total Smad2 (initialized to cytoplasm)

S2tot

73.0

nM

Total Smad4 (initialized to cytoplasm)

S4tot

73.0

nM

Total phosphatase in nucleus

PPase

1

nM

Total Receptors

Rtot

1

nM

VariableLabel
Cytoplasmic Smad2

S2c

Cytoplasmic phosphorylated Smad2

pS2c

Cytoplasmic Smad4

S4c

Cytoplasmic Smad2:Smad4 complex

S24c

Cytoplasmic Smad2:Smad2 complex

S22c

Nuclear Smad2

S2n

Nuclear phosphorylated Smad2

pS2n

Nuclear Smad4

S4n

Nuclear Smad2:Smad4 complex

S24n

Nuclear Smad2:Smad2 complex

S22n

Differential Equations

[S2c]˙=kphosu(Tgfβ)[S2c]kin2[S2c]+kex2[S2n]

[pS2c]˙=kphosu(Tfgβ)[S2c]kin2[pS2c]kon[pS2c]([S4c]+2[pS2c])+koff([S24c]+2[S22c])+kex2[pS2n]

[S4c]˙=kin4[S4c]kon[pS2c][S4c]+koff[S24c]+kex4[S4n]

[S24c]˙=kon[pS2c][S4c]koff[S24c]kin2CIF[S24c]

[S22c]˙=kon[pS2c]2koff[S22c]kin2CIF[S22c]

[S2n]˙=akin2[S2c]akex2[S2n]+kdephosPPase[pS2n]

[pS2n]˙=akin2[pS2c]akex2[pS2n]kdephosPPase[pS2n]kon[pS2n]([S4n]+2[pS2n])+koff([S24n]+2[S22n])

[S4n]˙=akin4[S4c]akex4[S4n]kon[pS2n][S4n]+koff[S24n]

[S24n]˙=akin2CIF[S24c]+kon[pS2n][S4n]koff[S24n]

[S22n]˙=akin2CIF[S22c]+kon[pS2n]2koff[S22n]
Algebraic Equations for conserved species

S2tot=[S2c]+[pS2c]+[S24c]+2[S22c]+(2[S22n]+[S24n]+[pS2n]+[S2n])

S4tot=[S4c]+[S24c]+1a([S24n]+[S4n])
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
PathwaySystems where graded response has been observedReferences
Live imaging of single cells
Tgfβ pathwayMouse myoblastsFrick et al., 2017; Warmflash et al. (2012)
Human epidermal keratinocytesNicolás et al. (2004); Warmflash et al. (2012) ; Schmierer et al. (2008); Vizán et al. (2013)
Human cervical epithelial cellsNicolás et al. (2004)
Human breast epithelial cellsStrasen et al. (2018)
Canonical Wnt pathwayHuman embryonic kidney cellsKafri et al. (2016) (this is the only published live single-cell imaging study in the Wnt pathway so far)
ERK pathwayMouse fibroblastsToettcher et al. (2013)
Mouse embryonic fibroblastsMackeigan et al. (2005)
Human non-small cell lung carcinomaCheong et al., 2011
Human mammary gland cellsSelimkhanov et al. (2014); Perrett et al., 2013Perrett et al., 2013
Human cervical epithelial cellsVoliotis et al. (2014); Whitehurst et al. (2004); Perrett et al., 2013Perrett et al., 2013
Human foreskin fibroblastsWhitehurst et al. (2004)
Immunofluorescence and FACS studies
Tgfβ pathwayXenopus embryoSchohl and Fagotto (2002)
Mouse testesItman et al. (2009)
Zebrafish embryoDubrulle et al. (2015)
Canonical Wnt pathwayXenopus embryoSchneider et al. (1996); Fagotto and Gumbiner (1994); Schohl and Fagotto (2002)
Mouse embyoAulehla et al., 2008
PlanariaSureda-Gómez et al., 2016
Sea anemone embryoWikramanayake et al., 2003
ERK pathwayChick embryoDelfini et al. (2005)
Xenopus embryoSchohl and Fagotto (2002)
Human T lymphocyte cellsLin et al. (2009)
Rat adrenal gland cellsSantos et al., 2007

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All data generated or analysed during this study are included in the manuscript and supporting files. Source data files have been provided for Figures 2 and 3.

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