15404684340920 1..37 - elife-33617-v4.pdf

*For correspondence:

hnunns@caltech.edu (HN);

goentoro@caltech.edu (LG)

Competing interests:

The

authors declare that no

competing interests exist.

Funding:

See page 15

Received:

16 November 2017

Accepted:

09 September 2018

Published:

19 September 2018

Reviewing editor:

Wenying

Shou, Fred Hutchinson Cancer

Research Center, United States

Goentoro. This article is

distributed under the terms of

the

Creative Commons

Attribution License,

which

permits unrestricted use and

redistribution provided that the

original author and source are

credited.

Signaling pathways as linear transmitters

Harry Nunns*, Lea Goentoro*

Division of Biology and Biological Engineering, California Institute of Technology,

Pasadena, United States

Abstract

One challenge in biology is to make sense of the complexity of biological networks. A

good system to approach this is signaling pathways, whose well-characterized molecular details

allow us to relate the internal processes of each pathway to their input-output behavior. In this

study, we analyzed mathematical models of three metazoan signaling pathways: the canonical Wnt,

MAPK/ERK, and Tgf

pathways. We find an unexpected convergence: the three pathways behave

in some physiological contexts as linear signal transmitters. Testing the results experimentally, we

present direct measurements of linear input-output behavior in the Wnt and ERK pathways.

Analytics from each model further reveal that linearity arises through different means in each

pathway, which we tested experimentally in the Wnt and ERK pathways. Linearity is a desired

property in engineering where it facilitates fidelity and superposition in signal transmission. Our

findings illustrate how cells tune different complex networks to converge on the same behavior.

DOI: https://doi.org/10.7554/eLife.33617.001

Introduction

Cells must continually sense, interpret, and respond to their environment. This is orchestrated by sig-

naling pathways: networks of multiple proteins that transmit signals and initiate cellular response.

Signaling pathways are critical to animal development and physiology, and yet there are fewer than

20 classes of metazoan signaling pathways (

Gerhart, 1999

). These signaling pathways evolved prior

to the Cambrian and remain highly conserved across animal phyla (

Gerhart, 1999

;

Pires-

daSilva and Sommer, 2003

). Each signaling pathway, therefore, governs a wide range of cellular

events, both within and across organisms.

Insights into the versatility of signaling pathways may be gleaned from pathway architectures.

Indeed, distinct architectural features define each pathway. Studies over the past several decades

have revealed distinct signaling capabilities that arise from pathway architecture, for example, all-or-

none response in the MAPK/ERK pathway (

Huang and Ferrell, 1996

;

Ferrell and Machleder, 1998

oscillations in the NF

B pathway (

Hoffmann et al., 2002

), or asymmetrical cell signaling in the

Notch/Delta pathway (

Sprinzak et al., 2010

). Alternatively, analysis of pathway architectures may

also reveal shared signaling capabilities that emerge from the distinct architectures, pointing to a

fundamental property that pathways have converged upon despite their separate evolutionary tra-

jectories. In this study, we sought to identify shared properties between conserved signaling

pathways.

To this end, we examined three signaling pathways, the canonical Wnt, ERK and Tgf

pathways.

These pathways are activated by an extracellular ligand binding to a membrane receptor

(

Figure 1A

). The ligand-receptor activation initiates a series of biochemical reactions within the cell,

culminating in a buildup of transcriptional regulator, which regulates transcription of broad gene tar-

gets. Since the ligand-receptor module is relatively plastic across organisms (e.g. flies have one EGF

receptor whereas humans have four [

Citri et al., 2003

]), we focused on the conserved core pathway

(

Figure 1A

). We define the input to the core pathway as the ligand-receptor activation, and the out-

put as the level of transcriptional regulator.

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RESEARCH ARTICLE

The Wnt, ERK, and Tgf

pathways transmit input using different core transmission architecture

(

Figure 1B–D

). In the Wnt pathway, signal transmission is characterized by a futile cycle of synthesis

and rapid degradation (

Kimelman and Xu, 2006

;

Saito-Diaz et al., 2013

;

Hoppler and Moon,

2014

). We use the term futile cycle to highlight that

-catenin is continually synthesized only to be

quickly targeted for degradation and kept at low concentration, as opposed to, for instance, being

synthesized only as needed. Ligand-receptor input diminishes the degradation arm of this cycle,

leading to accumulation of

-catenin output (

Kimelman and Xu, 2006

;

Stamos and Weis, 2013

;

Nusse and Clevers, 2017

). In the ERK pathway, signal transmission is characterized by a cascade of

phosphorylation events coupled to feedbacks, leading to an increase in phosphorylated ERK output

(

Kolch, 2005

;

Yoon and Seger, 2006

;

Avraham and Yarden, 2011

;

Lake et al., 2016

). Finally, sig-

nal transmission in the Tgf

pathway is characterized by continual nucleocytoplasmic protein shut-

tling (

Inman et al., 2002

;

Nicola

s et al., 2004

;

Xu and Massague

, 2004

;

Schmierer and Hill, 2005

;

Massague

et al., 2005

). Ligand-receptor input effectively increases the rate of nuclear import, lead-

ing to an increase in output, the nuclear Smad complex (

Schmierer et al., 2008

Importantly for our approach, the architectures of the three pathways are captured by mathemati-

cal models that have been refined by years of experiments. Although by no means complete, the

mathematical models have track records of success in predicting systems-level behaviors across mul-

tiple biological systems. For instance, the Wnt model (

Lee et al., 2003

) captures the dynamics of

destruction complex well enough as to enable prediction of robustness in fold-change response

(

Goentoro and Kirschner, 2009

) and the differential roles of the two scaffolds in the pathway

(

Lee et al., 2003

); the ERK model (

Huang and Ferrell, 1996

;

Ferrell and Bhatt, 1997

;

Schoeberl et al., 2002

;

Sturm et al., 2010

) captures the ultrasensitivity in the phosphorylation cas-

cade (

Huang and Ferrell, 1996

); and the Tgf

model (

Schmierer et al., 2008

) reveals the roles of

Figure 1.

The Wnt, ERK, and Tgf

pathways transmit input using different core transmission architecture. (

)

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 (

), the output is regulated by a futile cycle of continual

synthesis and rapid degradation. In the ERK pathway (

), the output is regulated by a kinase cascade coupled to

negative feedback. In the Tgf

pathway (

), the output is regulated through continual nucleocytoplasmic shuttling.

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

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nucleocytoplasmic shuttling in transducing the duration and intensity of ligand stimulation

(

Schmierer et al., 2008

We studied these mathematical models to identify what, if any, behaviors converge across path-

ways. The Wnt (

Lee et al., 2003

), ERK (

Sturm et al., 2010

), and Tgf

(

Schmierer et al., 2008

) mod-

els consist of 7, 26, and 10 coupled, nonlinear ODEs, respectively, with 22, 46, and 13 parameters.

Because of their large sizes, they are typically solved numerically to simulate experimental observa-

tions and generate new predictions. However, for the questions posed here, we found that numeri-

cal simulations are not sufficient. Rather, we needed analytics to uncover exactly how the pathway

behaviors depend on the underlying biochemical processes. While we previously derived an analyti-

cal solution to the Wnt pathway (

Goentoro and Kirschner, 2009

), analytical treatment of the Tgf

and ERK pathways has not been attempted due to the complex, nonlinear equations involved. To

address this problem, we employed various analytical techniques, including graph theory-based vari-

able elimination and dimensional analysis, to derive analytical or semi-analytical solutions to the

steady-state output of each pathway. Our analysis, along with subsequent experimental verification,

reveals a striking convergence across the Wnt, Tgf

, and ERK pathways: cells operate in the parame-

ter regime where the complex, nonlinear interactions in each pathway give rise to linear signal

transmission.

Results

Mathematical analysis identifies the Wnt, ERK, and Tgf

b

pathway as

linear transmitters

We began our analysis using established models of the Wnt (

Lee et al., 2003

), ERK (

Sturm et al.,

2010

), and Tgf

(

Schmierer et al., 2008

) pathways. These models capture the salient features of

each pathway, and include biochemical details such as synthesis, degradation, binding, dissociation

and post-translational modifications. In all the models, biochemical parameters have been directly

measured or fitted to kinetic measurements from cell, embryo or extract systems. Numerical simula-

tion of each model has predicted a wide range of pathway behaviors over the years (e.g. Wnt refs.

[

Lee et al., 2003

;

Goentoro and Kirschner, 2009

;

Herna

ndez et al., 2012

]; ERK refs. [

Huang and

Ferrell, 1996

;

Ferrell and Machleder, 1998

;

Schoeberl et al., 2002

;

Sturm et al., 2010

;

Fritsche-

Guenther et al., 2011

]; Tgf

refs. [

Schmierer et al., 2008

;

Gonza

lez-Pe

rez et al., 2011

;

Andrieux et al., 2012

;

Viza

n et al., 2013

;

Wang et al., 2014

]). Below, we describe our analysis of

each pathway and the unifying behavior that emerges from all three pathways.

Canonical Wnt pathway

In this pathway, cells sense ligand-receptor input by monitoring

-catenin protein (

Kimelman and

Xu, 2006

;

Stamos and Weis, 2013

;

Nusse and Clevers, 2017

;

MacDonald et al., 2009

;

Clevers and Nusse, 2012

-catenin is continually synthesized and rapidly degraded by a large

destruction complex, comprised of multiple proteins including APC, Axin, and GSK3

. The destruc-

tion complex binds and phosphorylates

-catenin, tagging it for degradation by the ubiquitin/pro-

teosome machinery (

Kimelman and Xu, 2006

;

Stamos and Weis, 2013

). Wnt ligands, through

binding to Frizzled and LRP receptors, inhibit the destruction complex, leading to accumulation of

catenin.

-catenin then regulates the expression of broad target genes (

Stamos and Weis, 2013

;

Nusse and Clevers, 2017

The model of the Wnt pathway (

Figure 2A

) was published in 2003 by a collaboration between

the Kirschner and Heinrich labs (

Lee et al., 2003

). The Wnt model consists of seven nonlinear differ-

ential equations and 22 parameters. Applying dimensional analysis, we previously derived the analyt-

ical solution to

-catenin concentration at steady-state (

Goentoro and Kirschner, 2009

cat

17

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

4

1

2

1

(1)

GSK3

tot

APC

tot

(2)

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Figure 2.

The Wnt, ERK, and Tgf

pathways are linear signal transmitters. (

A-C

) Network diagrams of the signaling pathways. The Tgf

diagram is

modified from

Schmierer et al. (2008)

. In the network diagram in A, DC refers to the

-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

Figure 2 continued on next page

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12

13

17

(3)

where the input function

u Wnt

ðÞ

is the rate of inhibition of the destruction complex (DC) via

Dishevelled/Dvl, a function of ligand-receptor activation. As illustrated in

Figure 2A

, K

’s are equilib-

rium dissociation constants, k

’s are rate constants, and v

’s are synthesis rates.

and

Equation 1

are dimensionless parameter groups defined in

Equations 2 and 3

characterizes

-catenin degra-

dation by the destruction complex, and

characterizes the extent to which

-catenin binds to APC

independently of the destruction complex.

Equation 1

demonstrates that, in general,

-catenin concentration is a nonlinear function of the

input

. Many parameters of the model were directly measured in

Xenopus

extracts, and the remain-

ing calculated from measurements in the same system (

Appendix 1—table 1

). In this study, we

examined how the analytical solution (

Equation 1

) behaves with these measured parameters. The

measured parameters (

Appendix 1—table 1

) indicate that

66

1

4

, and for maximal stimula-

tion,

6

0

. The large

reflects how

-catenin stability is primarily dictated by the destruction com-

plex, that is,

1

means that non-Axin-dependent degradation is minimal, and

means

that the positive feedback from sequestration by APC is minimal. Indeed, the rapid action of the

destruction complex in the Wnt pathway is a recurring observation across biological systems

(

Kimelman and Xu, 2006

;

Saito-Diaz et al., 2013

;

Hoppler and Moon, 2014

). With

1

Equation 1

simplifies to

cat

17

(4)

with detailed derivations presented in Appendix 1. Therefore, within physiologically relevant param-

eter values, the steady-state

-catenin concentration becomes a linear function of the input

(red

line,

Figure 2D

). The linear input-output relationship holds for the entire dynamic range of the

model, until the system saturates at maximal stimulation (

6

0

). We confirmed that the numerical

solution of the full model matches the analytical solution in

Equation 4

(blue line,

Figure 2D

), and

Figure 2 continued

steps (such as k

and k

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

-catenin, dpERK, and nuclear Smad complex for the

Wnt, ERK, and Tgf

pathway, respectively. The input functions

describe the effect of ligand-receptor interactions on the core pathway. Specifically:

Wnt

is the rate by which Dishevelled/Dvl inhibits the destruction complex upon Wnt ligand activation, where

and

are defined in the figure

and [Dvl]

is the concentration Wnt-activated Dishevelled (see

Equations A15

); u(EGF) is concentration of EGF-activated Ras (Ras-GTP); and u(Tgf

) is

the fraction of Tgf

-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).

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

The following source data and figure supplements are available for figure 2:

Source code 1.

DOI: https://doi.org/10.7554/eLife.33617.011

Figure supplement 1.

Model simulations for the ERK pathway.

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

Figure supplement 2.

The predicted linearity extends throughout the dynamic range of the ERK and Tgf

pathways.

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

Figure supplement 3.

Model simulations for the Tgf

pathway.

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

Figure supplement 4.

Incorporating into the Wnt model the dual function of GSK3

in phosphorylating

-catenin and LRP5/6.

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

Figure supplement 5.

The requirements for linear signal transmission in the Wnt, Tgf

, and ERK pathway.

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

Figure supplement 6.

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

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

Figure supplement 7.

Numerical simulation of the input-output relationship of the NF-

B pathway.

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

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that the response becomes nonlinear when

is decreased, breaking the requirement

1

(grey line,

Figure 2D

Source codes for the numerical simulations in

Figure 2D–F

(grey and black lines) are available in

Figure 2—source code 1

ERK pathway

The unexpected linearity that emerges from the model of the Wnt pathway prompted us to wonder

if such simplicity may be found in other pathways. Strikingly, we observed the same linearity in the

ERK and Tgf

pathways. In the ERK pathway (

Figure 2B

), ligand-receptor input is transmitted via a

cascade of protein phosphorylation (

Kolch, 2005

;

Yoon and Seger, 2006

). In particular, ligand-

receptor interactions activate Ras, which leads to membrane recruitment and phosphorylation of

Raf. Phosphorylated Raf subsequently doubly phosphorylates MEK, which in turn doubly phosphory-

lates ERK (

Kolch, 2005

). Doubly-phosphorylated ERK (dpERK) is a transcriptional regulator that

affects a broad array of genes (

Yoon and Seger, 2006

). The multi-step topology of the kinase cas-

cade, combined with distributive phosphorylation of each kinase, gives rise to ultrasensitivity – first

demonstrated in the seminal work by the Ferrell lab (

Huang and Ferrell, 1996

;

Ferrell and

Machleder, 1998

). In other contexts, the pathway also exhibits a graded response

(

Whitehurst et al., 2004

;

Mackeigan et al., 2005

;

Cohen-Saidon et al., 2009

;

Ahmed et al., 2014

)

that is thought to arise from the incorporation of negative feedbacks (

Lake et al., 2016

), one of

which is the inhibition of Raf by dpERK through hyper-phosphorylation of serine residues

(

Sturm et al., 2010

;

Dougherty et al., 2005

;

Hekman et al., 2005

The ERK model (

Sturm et al., 2010

) is the product of more than two decades of refinement

(

Huang and Ferrell, 1996

;

Ferrell and Machleder, 1998

;

Schoeberl et al., 2002

;

Sturm et al.,

2010

;

Fritsche-Guenther et al., 2011

). The model, which captures ultrasensitivity and Raf feedback,

consists of 26 differential equations and 46 parameters. To derive an analytical expression for the

ERK pathway, we used a variable elimination technique developed for networks of mass action kinet-

ics (

Feliu and Wiuf, 2012

). The technique utilizes an algebraic framework, linear elimination of varia-

bles, and mass conservation laws to parameterize steady-state in terms of core variables (described

in Appendix 1). We derived an analytical relationship between the steady-state output of the path-

way

dpERK

and the input to the phosphorylation cascade

dpERK

Raf

tot

pRaf

1

(5)

(6)

b29

(7)

MEK

(8)

26

25

26

...

(9)

Detailed derivations of

Equation 5

are presented in Appendix 1. The input

u EGF

ðÞ

Equa-

tion 5

is the concentration of active Ras, which is activated via GTP loading at the ligand-receptor

complex (

Kolch, 2005

). The parameter groups

, and

Equation 5

are defined in

Equa-

tions 6–9

, where the ellipses indicate additional small terms (expanded in Appendix 1). The relative

magnitudes of

, and

indicate how the Raf pool partitions during signaling (

Equations A21

A29

–

A31

). The dimensionless group

relates to the amount of free, phosphorylated Raf (

, blue-

shaded in

Figure 2B

dpERK

describes the amount of Raf inhibited through negative feedback

by dpERK (

, red-shaded in

Figure 2B

relates to the amount of unphosphorylated (

, blue-

shaded in

Figure 2B

), and

relates to the amount of phosphorylated Raf bound to other proteins

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(e.g. to MEK, brown-shaded in

Figure 2B

Equation 5

is not a closed solution, as it includes the

term

pRaf

, and there are variables included in parameter groups

. We confirmed that the

parameter groups remain constant over the course of signaling (within 10%,

Figure 2—figure sup-

plement 1

), justifying treating the latter variables as parameters.

Next, we considered how the analytical expression (

Equation 5

) behaves within a specific param-

eter regime observed in experiments. First, experiments in several mammalian cell systems have

shown that feedback is strong, such that a significant fraction of the Raf pool is inhibited (

Fritsche-

Guenther et al., 2011

;

Dougherty et al., 2005

). This means that

dpERK

ðÞ

. Sec-

ond, as has been observed in multiple contexts ([

Huang and Ferrell, 1996

;

Ferrell and Machleder,

1998

;

Schoeberl et al., 2002

;

Sturm et al., 2010

]

Appendix 1—table 2

), ERK phosphorylation is

ultrasensitive to the amount of pRaf (the ultrasensitive cascade is shaded green in

Figure 2B

Denoting

as the relative change of

dpERK

with respect to

pRaf

, ultrasensitivity entails that

1

. In this range, small changes in pRaf level have very large effects on dpERK level (e.g., in

model simulations, a 30% change in pRaf level results in a 900% change in dpERK level,

Figure 2—

figure supplement 1

). We find analytically that in the parameter regime where

dpERK

ðÞ

and

1

, the negative feedback holds the level of pRaf constant

(

pRaf

, details in Appendix 1). With these two features, strong negative feedback and ultrasen-

sitivity, dpERK becomes a linear function of the input

dpERK

Raf

tot

(10)

The full derivation is given in Appendix 1, and includes a toy model to illustrate the intuition for

how ultrasensitivity combines with negative feedback to produce linearity.

Equation 10

is plotted in

Figure 2E

(red line). We confirmed that the numerical solution of the full model matches the analyt-

ics in

Equation 10

, and becomes nonlinear when the negative feedback is weakened (grey line,

Figure 2E

). Although the analytical expression describes up until 50% of ERK activation, we verified

numerically that the predicted linearity extends to 93% of ERK activation (

Figure 2—figure supple-

ment 2

The linearity derived here applies across different dynamic ERK responses. The model we ana-

lyzed gives a sustained dpERK response. In some contexts, however, the ERK pathway shows a pul-

satile response, which has been attributed to receptor desensitization (

Schoeberl et al., 2002

Using a larger model that includes details of receptor desensitization (

Schoeberl et al., 2002

), we

numerically verified that the linearity holds for pulsatile responses - that is, the peak level of dpERK

increases linearly with the peak level of

(

Figure 2—figure supplement 1

Tgf

b

pathway

Finally, we examined signal transduction within the Tgf

pathway (

Figure 2C

). In the Tgf

pathway,

input from ligand-receptor interactions is transmitted by the Smad proteins. There are several clas-

ses of Smad proteins, including the receptor-regulated Smads (R-Smads) and the common Smad

(co-Smad or Smad4) (

Massague

et al., 2005

). Ligand-activated receptors phosphorylate R-Smads.

Phosphorylated R-Smads bind to the co-Smad, and shuttle into the nucleus and regulate broad tar-

get genes. In the nucleus, the Smad complex dissociates and R-Smads are constitutively de-phos-

phorylated and shuttled out to the cytoplasm, where the cycle of phosphorylation and complex

formation begins again (

Schmierer et al., 2008

). This dynamic translocation in and out of the

nucleus forms a continual nucleocytoplasmic shuttling of Smads, a known integral feature of the

Tgf

pathway (

Inman et al., 2002

;

Nicola

s et al., 2004

;

Xu and Massague

, 2004

;

Schmierer and

Hill, 2005

The Tgf

model (

Schmierer et al., 2008

) was published in 2008 by the Hill lab, and consists of 10

differential equations and 13 parameters. Even though the model was fitted to R-Smad2 data, the

general architecture of signal transmission is conserved across all five R-Smads (

Massague

et al.,

2005

;

Schmierer et al., 2008

). Using the variable elimination technique described before (

Feliu and

Wiuf, 2012

), we derived an analytical expression of the steady-state concentration of Smad complex

in the nucleus:

24

tot

(11)

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S4n

ex2

off

(12)

PPase

dephos

phos

tot

ex2

in2

(13)

ex2

PPase

dephos

ex2

CIF

in2

(14)

Equation 11

, the input function

u Tgf

ðÞ

is the active fraction of Tgf

receptors. The param-

eter

is the nucleocytoplasmic volume ratio. The dimensionless parameter groups

, and

Equation 11

are defined in

Equations 12–14

, where the ellipses indicate additional small terms

(expanded in Appendix 1).

, and

describe how the Smad2 pool partitions during signaling

(

Equations A44, A50, A51

relates to the amount of nuclear Smad complex (

, blue-shaded in

Figure 2C

, captures the parameters related to complex formation and translocation to the nucleus),

relates to the amount of free, unphosphorylated Smad2 (

, red-shaded in

Figure 2C

, captures the

parameters related to complex dissociation and translocation to the cytoplasm), and

loosely

relates to the remaining Smad2 pool (

is brown-shaded in

Figure 2C

). Phosphorylated Smad2

quickly forms complex (

Lagna et al., 1996

), so

essentially corresponds to total monomeric Smad2.

Finally,

Equation 11

is not a closed solution, since variable

4

appears in

. We numerically

tested that it is constant within 2% for non-saturating inputs (

Figure 2—figure supplement 3

), justi-

fying treating it as a parameter.

As in the Wnt and ERK pathway, the analytical expression for nuclear Smad complex (

Equa-

tion 11

) allows us to see that the behavior dramatically simplifies with parameters observed in

experiment. We consider the case for non-saturating inputs (

). Protein concentrations in the

Tgf

model were measured in human keratinocyte cells and the rate constants fitted to kinetic data

measured in the cells (

Schmierer et al., 2008

). With the measured parameters (

Appendix 1—table

), we find that

46

1

5

, and

0

7

. In this parameter regime, once Smad2 is imported to

the nucleus, it is rapidly dephosphorylated and exported. Dynamic Smad2 translocation maintains

monomeric Smad2 in excess to Smad complex (

ðÞ

). and forms the continual nucleocyto-

plasmic shuttling that is characteristic of the Tgf

pathway. Even under maximal Tgf

stimulation, it

has been estimated that phosphorylated Smad2 comprises only 36% of the Smad2 pool

(

Schmierer and Hill, 2005

;

Gao et al., 2009

). With

ðÞ

, the first term in the denominator

Equation 11

is small, and concentration of nuclear Smad complex becomes a linear function of

input:

24

tot

(15)

Equation 15

is plotted in

Figure 2F

(red line), and we confirmed that numerical simulations reca-

pitulates

Equation 15

(blue line,

Figure 2F

). Although the analytical solution is valid only for small

values of u, we numerically verified that the predicted linearity holds for the entire range of input u

(from 0 to 1,

Figure 2—figure supplement 2

). We confirmed that the pathway becomes nonlinear

when the R-Smad phosphatase is inhibited such that

ðÞ

(grey line,

Figure 2F

). While the

model analyzed here gives a sustained Smad response, we verified numerically that the linearity

holds for a larger model that includes receptor desensitization and gives a pulsatile Smad response

(

Figure 2—figure supplement 3

) (

Viza

n et al., 2013

Linearity in the Wnt and ERK pathways was observed experimentally

Analytical expressions for the Wnt, ERK, and Tgf

pathways reveal that the three pathways behave

as linear signal transmitters within parameter regimes measured in cells. To confirm the linearity, we

directly measured the input-output relationships in human cell lines. We focused our efforts on the

Wnt and ERK pathways, since we are limited by available antibodies in the Tgf

pathway.

To analyze the canonical Wnt pathway, we performed quantitative Western blot measurements in

RKO cells, a model system for Wnt signaling. To track the input, we measured the level of

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phosphorylated LRP5/6 receptors (on Ser1490), which increases within minutes of ligand-receptor

complex formation (

Tamai et al., 2004

). To track the output, we measured the level of

-catenin.

We confirmed that the level of phosphorylated LRP5/6 and

-catenin increase upon Wnt simulation

and reach steady-state within 6 hr (

Figure 3—figure supplement 1

). Accordingly, all subsequent

measurements were done at 6 hr after Wnt stimulation.

To measure the input-output relationship in the Wnt pathway, we treated RKO cells with varying

doses of purified Wnt3A and measured how

-catenin (output) correlates with phosphorylated LRP

(input). As shown in

Figure 3A

, the level of

-catenin increases linearly with the level of phosphory-

lated LRP. The linearity persists until saturation of the input, defined as 90% of maximal phosphory-

lated LRP response (blue circles,

Figure 3A

;

Figure 3—figure supplement 2

). Notably, at high

doses of Wnt3A,

-catenin continues to show incremental activation, despite saturation in phosphor-

ylation of LRP (grey circles,

Figure 3A

). This can be explained within some findings that, while Friz-

zled/LRP complex is the primary receptor input in

-catenin activation,

-catenin can be activated

independently of LRP (e.g.

Rotherham and El Haj, 2015

Consistent with the mathematical analysis, we observed in RKO cells that the Wnt pathway

behaves as a linear transmitter throughout the dynamic range of the input. As a control that is

expected from the Michaelis-Menten kinetics that describe ligand binding in the model, we con-

firmed that the linearity does not extend upstream to Wnt dose: both phospho-LRP5/6 and

-cate-

nin show nonlinear response to Wnt dose (

Figure 3—figure supplement 2

). Therefore, in the Wnt

pathway, a nonlinear ligand-receptor processing step is followed by linear signal transmission

through the core intracellular pathway.

Next, to measure the input-output relationship in the ERK pathway, we performed quantitative

Western blots in H1299 cells, one of the model systems used in the field. Linearity in the ERK path-

way has been suggested in different parts of the pathway, e.g.

Knauer et al. (1984)

used experimental and modeling analyses to infer linearity between receptor occupancy and the

downstream cellular proliferation;

Oyarzu ́n et al. (2014)

suggests linearity in ligand-receptor proc-

essing. Here, we specifically probe linearity in the core transmission step of the pathway. Detecting

the input level, EGF-activated Ras GTP, requires a pull down step that makes it less quantifiable.

Therefore, motivated by

Oyarzu ́n et al. (2014)

, we tested EGF ligand itself as the input. To track

the output, we measured the level of doubly-phosphorylated ERK1/2 (on Thr202/Tyr204), dpERK.

We first characterized the kinetics of response: dpERK peaks 5 min after EGF stimulation (

Figure 3—

figure supplement 3

), and saturates at 4 ng/ml EGF (grey circles,

Figure 3B

). Accordingly, all subse-

quent measurements were performed at 5 min after EGF stimulation, and linearity was assessed

over the input range of 0–4 ng/mL EGF (blue circles,

Figure 3B

We observed linearity in the input-output relationship of the ERK pathway, with the level of

dpERK increasing linearly with EGF dose (

Figure 3B

). The linearity holds throughout the dynamic

range of the system, over at least 12-fold activation of dpERK. As the ERK pathway is sometime

observed to show bimodal response that would be masked by bulk measurements, we confirmed

that the H1299 cells indeed show to graded dpERK response in single-cell level (

Figure 3—figure

supplement 4

), in agreement with a previous single-cell, live imaging study (

Cohen-Saidon et al.,

2009

). Therefore, as in the Wnt pathway, signals are transmitted linearly in the ERK pathway

throughout the dynamic range of the cell. Moreover, the linearity in the ERK pathway is more exten-

sive than in the Wnt pathway, as linearity extends all the way upstream, such that the level of dpERK

directly reflects the dose of extracellular EGF ligand.

Linearity in the Wnt and ERK pathways is modulated by perturbation to

parameters

Finally, the analytical expressions we derived in this study not only reveal linear signal transmission,

but also the mechanisms by which it arises. In the model of the Wnt pathway, linear transmission

occurs due to the futile cycle of

-catenin, in the parameter regime where

-catenin is continually

synthesized and rapidly degraded (i.e.

1

). This regime is not infinite: for instance, a ten-

fold decrease in

(e.g. by inhibiting the destruction complex) will break the futile cycle (grey line,

Figure 2D

To test if the futile cycle is indeed required for linear signal transmission, we inhibited the destruc-

tion complex using CHIR99021, an inhibitor of GSK3

kinase. As before, we measured the input-out-

put relationship,

-catenin vs. phospho-LRP5/6 level, up to 90% of maximal phospho-LRP5/6 input

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Figure 3.

Linearity was observed experimentally in the Wnt and ERK pathways. (

) 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

-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

is the Pearson’s coefficient, where

= 1 is a perfect positive linear correlation. (

) 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

is the coefficient of correlation where r

= 1 is a perfect linear correlation. (

) As in (

), except that cells were treated with 1

M 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

is the Pearson’s coefficient, where

= 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). (

) As in (

), 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 (k

) 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

Figure 3 continued on next page

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(blue circles,

Figure 3C

). As expected, we found that inhibiting the destruction complex (decreasing

in the model) reduced the range of linearity. The non-treated cells (blue circles,

Figure 3A

) exhibit

a linear input-output relationship over a 4.4-fold range of LRP input, whereas the CHIR-treated cells

show a linear input-output relationship over only a 2.8-fold range of LRP input (blue circles,

Figure 3C

Further, our measurements also reveal an unexpected feature of the Wnt pathway. In the model,

inhibiting GSK3

causes

-catenin response to become nonlinear for larger inputs (dashed line,

Figure 3C

subplot). In CHIR-treated RKO cells, however, this nonlinearity cannot be reached, as the

maximal amount of phosphorylated LRP (input) is reduced by 50% (grey circles,

Figure 3

;

Figure 3—

figure supplement 2

), consistent with the dual-function of GSK3

identified by

Zeng et al. (2005)

;

Zeng et al. (2008)

in phosphorylating

-catenin for degradation as well as phosphorylation LRP for

activation. Incorporating this dual-role of GSK3

into the model, we found that this expanded model

can indeed recapitulate the data (

Figure 2—figure supplement 4

). Therefore, our data indicate two

findings: first, that inhibiting GSK3

reduces the range of linear input-output behavior in the Wnt

pathway, as predicted by our analytics, and second, that GSK3

co-regulation of

-catenin and LRP

unexpectedly constrains the system within the linear regime.

Next, we examine the requirements for linearity in the ERK pathway.

Equation 10

reveals that lin-

earity in the ERK pathway depends upon the coupling of strong nonlinearities – ultrasensitivity and

negative feedback. As in the Wnt pathway, this regime is not infinite, for example, decreasing the

strength of feedback

enables the system to exit the ultrasensitive regime, and therefore reduces

linearity (grey line,

Figure 2E

To test this requirement, we examined the effects of weakening the negative feedback. We cre-

ated a stable H1299 cell line expressing Raf S289/296/301A, a Raf-1 mutant in which three serine

residues that are phosphorylated by dpERK are mutated to alanine (

Dougherty et al., 2005

;

Hekman et al., 2005

). Assessing the dynamic range of the input as before (0–4 ng/mL EGF), we

now found that dpERK responds nonlinearly to EGF dose (blue circles,

Figure 3D

), consistent with

model predictions (grey line,

Figure 3D

). As a control, we found that overexpressing WT Raf-1 to a

similar level does not perturb linearity (experiments,

Figure 3—figure supplement 5

; modeling,

Figure 3 continued

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

Figure 3—source data 1

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

The following source data and figure supplements are available for figure 3:

Source data 1.

DOI: https://doi.org/10.7554/eLife.33617.022

Figure supplement 1.

LRP5/6 phosphorylation and

-catenin accumulation are already at steady state at 6 hr after Wnt stimulation.

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

Figure supplement 2.

The dynamic range of Wnt signaling in RKO cells.

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

Figure supplement 3.

ERK activation peaks at 5 min after EGF stimulation.

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

Figure supplement 4.

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

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

Figure supplement 5.

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

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

Figure supplement 6.

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

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

Figure supplement 7.

Technical variability from Western blot.

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

Figure supplement 8.

Linearity is not an artifact of loading control normalization.

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

Figure supplement 9.

Linearity was observed across independent experiments.

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

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Figure 2—figure supplement 1

). Lastly, mutating all five direct ERK feedback sites on Raf-1 to ala-

nine had a similar effect to Raf S289/296/301A (

Figure 3—figure supplement 6

). Our results sup-

port the model requirement that strong negative feedback is critical to linear signal transmission in

the ERK pathway.

Discussion

Our study suggests that the canonical Wnt pathway, the ERK pathway, and the Tgf

pathway have

converged upon a shared strategy of linear signal transmission. Our mathematical analysis reveals

that, despite their distinct architectures, the three signaling pathways behave in some physiological

contexts as linear transmitters. Not only is linearity is predicted within measured parameter regimes,

the analysis shows that linearity is a property of the systems that occurs through a considerable

range of parameters (

Figure 2—figure supplements 5

and

). We then showed direct measure-

ments of the linear input-output relationship in the canonical Wnt and ERK pathway.

It would be interesting to further probe the generality of linear signal transmission. Linear behav-

ior requires that single cells responds to ligand in a graded manner. Although there are reports of

oscillatory or bimodality in signaling pathways, there are also multiple observations across biological

contexts of single cells responding to ligand in a graded manner (

Appendix 1—table 4

). Besides

the systems analyzed here, NF-

B is another signaling pathway that has been modeled rigorously

(

Hoffmann et al., 2002

;

Ashall et al., 2009

;

Lee et al., 2014

). Numerical simulations of a well-estab-

lished NF-

B model (

Ashall et al., 2009

) over the range of nuclear NF-

B translocation observed in

human epithelial cells (

Lee et al., 2014

) reveal that the peak of the nuclear NF-

B pulse correlates

linearly with ligand concentration (

Figure 2—figure supplement 7

). Finally, linearity extends beyond

metazoan signaling pathways. In the yeast pheromone sensing pathway, a homolog of the ERK cas-

cade, transcriptional output correlates linearly with receptor occupancy (

Yu et al., 2008

). The linear-

ity is mediated by negative feedback by Fus3 acting on Sst2, a feedback that is not conserved in the

mammalian ERK system. These further argue for linear signal transmission as a convergent property

across independently evolving signaling pathways, as well as between conserved pathways that

diverged 1.5 billion years ago.

What are potential advantages to linear signal transmission? Linearity is a feature of many engi-

neering systems, where it serves several practical purposes. In particular, linear signal transmission

enables the superposition of multiple signals, where the output of two simultaneous inputs is equal

to the sum of the outputs for each input separately. Superposition enables multiple, dynamic signals

to be faithfully transmitted and processed independently. Thus, for instance, linearity enables people

to listen to a phone call and interpret speech amongst background noise, and allows a car radio to

tune into one station out of multiple broadcasting on separate carrier frequencies. Notably, linearity

is also a desired goal in synthetic biology, where it is often implemented using negative feedback

(

Nevozhay et al., 2009

;

Del Vecchio et al., 2016

). Analogous to engineered circuits, linearity in bio-

logical signaling pathways may facilitate multiplexing inputs into a single pathway (

Figure 4A

A second benefit is that linearity might underlie two phenomena that are increasingly found

across signaling pathways. First, a linear transmitter naturally gives rise to dose-response alignment

(

Andrews et al., 2016

), where one or more downstream responses of a pathway closely follows the

fraction of occupied receptor (

Figure 4B

). Dose response alignment appears in many biological sys-

tems and is thought to improve the fidelity of information transfer through signaling pathways

(

Oyarzu ́n et al., 2014

;

Yu et al., 2008

;

Andrews et al., 2016

;

Becker et al., 2010

). Second, linearity

facilitates fold change detection, where cells sense fold changes in signal, rather than absolute level,

to buffer cellular noise (

Goentoro and Kirschner, 2009

;

Cohen-Saidon et al., 2009

;

Lee et al.,

2014

;

Thurley et al., 2014

;

Frick et al., 2017

). In linear input-output systems, the stimulated output

correlates linearly to the basal output; thus, the fold-change in output is robust to variations in cellu-

lar parameters (

Figure 4C

). Indeed, for the signaling pathways studied here, it has been shown

experimentally that the robust outcome of ligand stimulation is the fold-change in the level of tran-

scriptional regulator (

Goentoro and Kirschner, 2009

;

Cohen-Saidon et al., 2009

;

Lee et al., 2014

;

Frick et al., 2017

). Therefore, selecting for linearity may naturally confer the benefits of superposi-

tion, dose-response alignment, and a robust fold-change in output.

Interestingly, unlike synthetic circuits whose linearity is often designed to extend across multiple

orders of magnitude (

Nevozhay et al., 2009

;

Nevozhay et al., 2013

), the linearity we observed in

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the three natural pathways extends only one order of magnitude, which is also the dynamic range of

the pathways. However, we know that natural pathways can convey inputs varying across multiple

orders of magnitude, for example, vision. Thus, an advantage of linearity in natural pathways may be

that, in conjunction with fold-change detection at the receptor-level (

Olsman and Goentoro, 2016

the system as a whole can continually adapt to a given input, hence maintaining sensitivity to future

signals.

Why evolve complexity in signaling pathways only to produce seemingly simple behavior? We

offer two thoughts. First, complexity of each pathway might afford tunability, in the sense that

parameters can be tuned to produce different behaviors in different contexts. For instance, the ERK

pathway produces digital, all-or-none response in some contexts (

Huang and Ferrell, 1996

), and

analog response in others (

Whitehurst et al., 2004

;

Mackeigan et al., 2005

). Second - to take an

example from engineering - in order to utilize physical processes that are not naturally linear, engi-

neers must implement complex design features to approximate linearity. Similarly, many biochemical

processes are inherently nonlinear, meaning that linearity does not arise from a reduction in com-

plexity. Indeed, in each pathway we analyzed here, linearity emerges

from

complex interactions: a

futile cycle in the Wnt pathway, ultrasensitivity coupled to feedback in the ERK pathway, and contin-

ual nucleocytoplasmic shuttling in the Tgf

pathway. Therefore, analogous to engineered systems,

complexity in the biochemical pathways we analyzed here might have evolved in part to produce

linearity.

Materials and methods

Expression constructs

pBABEpuro-CRAF that contains the wt human Raf-1 clone was a gift from Matthew Meyerson

(Addgene plasmid # 51124). Mutant Raf (S289/296/301A) and (S29/289/296/301/642A) were gener-

ated using the Q5 site-directed mutagenesis kit (New England Biolabs, E0554S). The mutant and wt

Raf-1 were then placed downstream of a CMV promoter.

Figure 4.

Benefits of linearity. (

) Linearity enables multiplexing of inputs to a signaling pathway. Multiplexed signals can be independently decoded

downstream, and therefore regulate distinct transcriptional events. (

) Illustration for how linearity between the receptor occupancy and downstream

outputs gives rise to dose-response alignment (

Andrews et al., 2016

). (

) 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

-catenin

before and after Wnt stimulation (blue circles). As long as the model operates in the regime of linear signal transmission (i.e.

, where

output,

is input, and

is a scalar that is a function of parameters), variation in parameters affects stimulated and basal level of

-catenin equally, and

we get a constant fold change in

-catenin (i.e. red line, where

stimulated

basal

is independent of parameter variations).

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

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Cell lines and cell culture

RKO cells (ATCC, CRL-2577) and H1299 cells (ATCC, CRL-5803) were authenticated by STR profiling

and supplied by ATCC. RKO cells were cultured at 37

C and 5% (vol/vol) CO2 in DMEM (Thermo-

Fisher Scientific; 11995) supplemented with 10% (vol/vol) FBS (Invitrogen; A13622DJ), 100 U/mL

penicillin, 100

g/mL streptomycin, 0.25

g/mL amphotericin, and 2 mML-glutamine (Invitrogen).

H1299 cells were cultured at 37C and 5% (vol/vol) CO2 in RPMI (ThermoFisher Scientific; 11875) sup-

plemented with 10% (vol/vol) FBS (Invitrogen; A13622DJ), 100 U/mL penicillin, 100

g/mL strepto-

mycin, 0.25

g/mL amphotericin, and 2 mML-glutamine (Invitrogen). Both cell lines tested negative

for mycoplasma contamination.

Transfection of Raf-1 constructs

H1299 cells were transfected with the mutant and wt Raf-1 constructs using Lipofectamine 3000

(ThermoFisher Scientific, L3000). Stable expression was selected using puromycin at a concentration

of 1.5

g/mL for 2 weeks.

Reagents and antibodies

The following antibodies were purchased from Cell Signaling Technologies: anti-Phospho-p44/42

MAPK (Erk1/2) (Thr202/Tyr204) (E10) Mouse mAb #9106, anti-histone H3 (D1H2) XP Rabbit mAb

#4499, anti-c-Raf Antibody #9422, anti-phospho-LRP6 (Ser1490) Antibody #2568, anti-GAPDH

(D4C6R) Mouse mAb #97166. Anti-Beta-catenin mouse mAb was purchased from BD Transduction

Laboratories (#610153) and anti-GAPDH rabbit antibody was purchased from Abcam (ab9485). The

following fluorescent secondary antibodies were purchased from Fisher Scientific: IRDye 800CW

Goat anti-Mouse IgG (926–32210) and IRDye 680LT Goat anti-Rabbit IgG (926-68021).

Recombinant human Wnt3A was purchased from Fisher Scientific (5036WN), and recombinant

human EGF was purchased from Sigma (E9644). CHIR99021 was purchased from Sigma (SML1046).

Halt Protease and Phosphatase Inhibitor Cocktail (100X) was purchased from Fisher Scientific

(78440).

CHIR99021 treatment

RKO cells were pre-treated with 1

M CHIR99021 for 24 hr before adding replacement media con-

taining 1

M CHIR99021 and Wnt3A for 6 hr.

Cell lysis

RKO cells at 70% confluency were scraped in PBS, pelleted, and snap-frozen, and then thawed in

NP-40 lysis buffer containing Halt inhibitor cocktail. Samples were spun down, and the supernatants

were transferred to Laemmli sample buffer and boiled. The samples were then run onto a Bolt 4–

12% Bis-Tris Plus Gel (Thermofisher, NW04120BOX). H1299 cells at 70% confluence were scraped in

NP-40 lysis buffer containing Halt inhibitor cocktail, and further lysed in Laemmli sample buffer. Sam-

ples were spun down, and the supernatants were boiled. The samples were then run onto a Novex

4–20% Tris-Glycine Mini Gel (ThermoFisher, XP04200BOX).

Quantitative Western blots

Proteins were transferred onto nitrocellulose membranes, blocked for one hour at

room temperature (RT) with blocking buffer (Odyssey Blocking Buffer (TBS) (927–50000) or 5% milk

powder in TBS) and stained overnight at 4

C with primary antibody diluted in blocking buffer. The

membranes were then stained with fluorescent IR secondary antibodies diluted in blocking buffer for

one hour at RT. The fluorescent signal was then imaged using the LiCOR Odyssey Imager and quan-

tified using Odyssey Application software version 3.0. The background-subtracted intensity of the

protein bands were normalized to the loading control, GAPDH and/or Histone H3 (for RKO) or His-

tone H3 (for H1299). These values were then normalized to the reference lanes within each gel, to

allow comparison across gels. For

-catenin and phospho-LRP5/6, variation in the fold-activation

from experiment to experiment could artificially stretch the data along the x- and y-axis, and intro-

duce artifacts into the relationship between phospho-LRP5/6 and

-catenin. Therefore, for Wnt3A

dose responses, the data from each gel was scaled such that the mean of 80 ng/mL and 160 ng/mL

samples was equal to the mean across all gels. Finally, for each antibody used in the study, we did

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careful characterization of the linear range, and verified that our measurement conditions were

within the linear range of the antibody.

Technical variability of Western blot quantitation

. To con-

firm the effects reported, we verified that quantitation of the same sample loaded in multiple lanes

in a gel gives CV < 10%, and quantitation of the same sample across multiple independent gels

gives CV < 10% (

Figure 3—figure supplement 7

). As further control, we verified that normalization

with loading control did not produce artificial distortion of the input-output relationship: linearity

was observed without normalization in cases where loading was already uniform (

Figure 3—figure

supplement 8

L-1 and L2-norm analysis

L1-norm analysis was performed as described in

Nevozhay et al. (2013)

. Briefly, the data is fitted

with a cubic Hermite polynomial, and rescaled along the x and y axis to [0, 1]. The L1-norm is com-

puted as the area between the polynomial fit and the diagonal. Linearity is defined in this context as

L1-norm < 0.1. L2-norm analysis for Wnt pathway data was performed using a Pearson’s coefficient,

and L2-norm analysis for ERK pathway data was performed using the coefficient of correlation,

2

Akaike information criterion

To score the validity of nonlinear model fits for

Figure 3D

, we used the bias-corrected Akaike Infor-

mation Criterion as described in ref. (

Spiess and Neumeyer, 2010

), which assesses goodness-of-fit

and model parsimony. The weighted Aikaike

w AIC

ðÞ

provides a comparison of all considered mod-

els, which in our case is the nonlinear ERK pathway model fit and a linear fit, with the higher score

indicating a more valid model.

Acknowledgements

We would like to thank Rob Oania for providing advice on experiments, Michael Abrams, Christo-

pher Frick, Kibeom Kim, and Noah Olsman for comments on the manuscript, and Michael Elowitz

and Richard Murray for discussions on the study.

Additional information

Funding

Funder

Grant reference number Author

James S. McDonnell Founda-

tion

220020365

Lea Goentoro

National Science Foundation NSF.145863

Lea Goentoro

National Institutes of Health 5T32GM007616-37

Harry Nunns

The funders had no role in study design, data collection and interpretation, or the

decision to submit the work for publication.

Author contributions

Harry Nunns, Conceptualization, Data curation, Software, Investigation, Methodology, Writing—

original draft, Writing—review and editing; Lea Goentoro, Conceptualization, Funding acquisition,

Methodology, Writing—original draft, Writing—review and editing

Author ORCIDs

Harry Nunns

https://orcid.org/0000-0002-9669-0039

Lea Goentoro

https://orcid.org/0000-0002-3904-0195

Decision letter and Author response

Decision letter

https://doi.org/10.7554/eLife.33617.034

Author response

https://doi.org/10.7554/eLife.33617.035

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Additional files

Supplementary files

.

Transparent reporting form

DOI: https://doi.org/10.7554/eLife.33617.024

Data availability

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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