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Reconstructing and comparing signal transduction networks
from single cell protein quantification data
Tim Stohn,1,2 Roderick van Eijl,4Klaas W. Mulder,4Lodewyk F.A. Wessels2,3
and Evert Bosdriesz1,2,*
1Computer Science Department, Center for Integrative Bioinformatics (IBIVU), Vrije Universiteit Amsterdam, Amsterdam,
The Netherlands, 2Division of Molecular Carcinogenesis, The Oncode Institute, The Netherlands Cancer Institute, Amsterdam,
The Netherlands, 3Faculty of EEMCS, Delft University of Technology, Delft, The Netherlands and 4Department of Molecular
Developmental Biology, Radboud Institute for Molecular Life Sciences, Radboud University, Nijmegen, The Netherlands
⇤Corresponding author. e.bosdriesz@vu.nl
Abstract
Motivation: Signal transduction networks regulate a multitude of essential biological processes and are frequently aberrated in
diseases such as cancer. Developing a mechanistic understanding of such networks is essential to understand disease or cell population
specific signaling and to design e↵ective treatment strategies. Typically, such networks are computationally reconstructed based on
systematic perturbation experiments, followed by quantification of signaling protein activity. Recent technological advances now allow
for the quantification of the activity of many (signaling) proteins simultaneously in single cells. This makes it feasible to reconstruct
signaling networks from single cell data.
Results: Here we introduce single cell Comparative Network Reconstruction (scCNR) to derive signal transduction networks by
exploiting the heterogeneity of single cell (phospho)protein measurements. scCNR treats stochastic variation in total protein
abundances as natural perturbation experiments, whose e↵ects propagate through the network. scCNR reconstructs cell population
specific networks of the same underlying topology for cells from diverse populations. We extensively validated scCNR on simulated
single cell data, and we applied it to a dataset of EGFR-inhibitor treated keratinocytes to recover signaling di↵erences downstream
of EGFR and in protein interactions associated with proliferation. scCNR will help to unravel the mechanistic signaling di↵erences
between cell populations by making use of single-cell data, and will subsequently guide the development of well-informed treatment
strategies.
Availability and implementation: scCNR is available as a python module at https://github.com/ibivu/scmra. Additionally,
code to reproduce all figures is available at https://github.com/tstohn/scmra_analysis.
Supplementary information: Supplementary information and data are available at Bioinformatics online.
Key words: signaling, network reconstruction, single cell
1 Introduction
Signal transduction networks play a vital role in cell physiology,
where they regulate various biological processes like di↵erentiation,
proliferation, and apoptosis. Extracellular signals are commonly
transmitted through intracellular networks of interacting proteins
called kinases, which activate each other through post-
translational modifications such as phosphorylation. Diseases such
as cancer often are a consequence of aberrations in this signaling
machinery (Hanahan and Weinberg, 2011; Kolch et al., 2015), and
many cancer drugs specifically target proteins within the signaling
networks. However, adaptation and rewiring of the network
(Ahronian et al., 2015) in response to treatment often limits the
durability of a clinical response (Gerosa et al., 2020; Ho↵man
et al., 2023; Tognetti et al., 2021). Obtaining a mechanistic (Does
protein A activate protein B?) and quantitative (How strong is
the influence of protein A on protein B?) understanding of those
networks, and how they di↵er between cell populations (such as
resistant, mutant cells), is a key challenge in cellular biology and
has important implications for the design of treatment strategies
(Bosdriesz et al., 2022; Klinger et al., 2013).
Var i o us me t hods h a ve b e e n dev elo p e d t o sol ve th i s p rob l em.
Ordinary di↵erential equations based models are detailed and
quantitative (Fey et al., 2015; Raue et al., 2015), but rely on
numerous measurements and are restricted to small systems.
Boolean logical network models have simpler formulations, but
can only represent non-quantitative network interactions and
are often not able to explain cyclic structures and feedback-
loops (Oates et al., 2012; Grieco et al., 2013; Saez-Rodriguez et al.,
2009; Hill et al., 2012). Modular response analysis (MRA) finds
the middle ground by modelling networks quantitatively without
the complexity of fully dynamical models (Kholodenko et al.,
2002; Bruggeman et al., 2002). MRA determines the interaction
strengths between proteins based on systematic perturbation
experiments, in which the states of all nodes in the network
are recorded before and after the perturbations. MRA is able to
detect cross-talks and feedback loops, and has been successfully
employed to quantify novel network topologies (Dorel et al.,
2018; Klinger et al., 2013). Various alternative formulations have
been developed, such as optimisation based approaches (Bosdriesz
et al., 2018), maximum-likelihood approaches (Klinger et al., 2013;
Ahlmann-Eltze and Huber, 2023) or Bayesian methods (Halasz
et al., 2016; Santra et al., 2013; Rukhlenko et al., 2022). While
most methods model signaling networks in a specific context,
di↵erences between networks derived from di↵erent contexts are
1
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2Reconstructing and comparing signal transduction networks from single cell protein quantification data
often most informative. For example, comparing networks derived
from a cell line before and after acquiring resistance to a targeted
inhibitor can aid in elucidating the signaling changes that drive
the resistance mechanism (Bosdriesz et al., 2018), models of cell
lines with and without a particular oncogenic mutation can help
in prioritizing drug combinations that are specific to a particular
genetic background (Bosdriesz et al., 2022), and modeling the
dependence of a signaling network on cell-states can help predict
interventions that control cell fate decisions (Rukhlenko et al.,
2022). To compare networks across contexts, we recently developed
Comparative Network Reconstruction (CNR) (Bosdriesz et al.,
2018), which aims to identify quantitative di↵erences between
the signaling networks derived from cell populations in di↵erent
contexts.
Most methods that model signal transduction networks were
developed for bulk data. However, even in isogenic populations
in a homogeneous environment, cells in di↵erent cell states are
known to respond di↵erently to the same instructions (Kim
et al., 2018; Wang et al., 2022; Kramer et al., 2022). This has
important implications for drug resistance and cancer treatment
design (Korkut et al., 2015; Aissa et al., 2021) and e↵ortshave
been made to link signaling networks to their e↵ect on the cell
state transition in perturbed RAF inhibitor-resistant cancer cells
(Rukhlenko et al., 2022). Obtaining such insights has long been
elusive because we lacked the right data to study them, but
recent development of technologies for high dimensional single-
cell quantification of (phospho-)proteins and post-translational
modifications, based on mass cytometry (Tracey et al., 2021),
DNA-barcoded antibodies (Eijl et al., 2018; Stoeckius et al., 2017;
Sheng et al., 2022) and spatial methods such as iterative indirect
immunofluorescence imaging (Gut et al., 2018) now allow us for
the first time to study and model the mechanisms underlying the
heterogeneity of signal transduction in a data-driven manner.
Here, we describe single-cell Modular Response Analysis
(scMRA) and single-cell Comparative Network Reconstruction
(scCNR). scMRA is a method to infer signaling networks from
single-cell quantification of active and total protein abundance.
Importantly, it exploits stochastic variability of total protein
counts between cells as ’natural perturbation experiments’, similar
to perturbations at the bulk level in classical MRA (Kholodenko
et al., 2002), thereby eliminating the need for extensive
perturbation experiments. By considering the data captured
from each single cell as a data-point, scMRA vastly increases
the number of observations from which the reconstructions
are derived. Typically, the signaling diversity between cell
populations is of most interest, e.g. due to cell state e↵ects
such as the cell cycle and di↵erentiation, or due to treatment
e↵ects or disease progression or the emergence of resistant
sub-populations. Therefore, we extended scMRA to single-
cell Comparative Network Reconstruction (scCNR), in order
to identify which interactions di↵er quantitatively between cell
populations. Similar to CNR (Bosdriesz et al., 2018), scCNR
reconstructs a single shared network topology with cell population-
specific interaction strengths. We extensively validate scMRA and
scCNR on simulated single cell data of the epidermal growth
factor receptor (EGFR) signaling pathway and showed that both
methods perform well using as few as hundred cells as input, and in
the presence of considerable noise. Furthermore, we applied scCNR
to a dataset where we quantified 70 (phospho)proteins of key
signaling nodes using single-cell ID-seq (Eijl et al., 2018) in EGFR-
inhibitor treated keratinocytes, and showed that scCNR recovers
meaningful biological diverged protein interactions downstream of
EGFR and related to proliferation.
2 Materials and Methods
2.1. Formulation of single-cell MRA and CNR
Here, we briefly describe the formalism underlying single-cell
Modular Response Analysis (scMRA) and single-cell Comparative
Network Reconstruction (scCNR). More details and a full
derivation of the equations can be found in the Supplementary
Information 5.2.
scMRA and scCNR exploit stochastic variation in protein
abundances to identify and quantify the interaction strengths
between di↵erent nodes in a signal transduction network. As
input, they require single-cell measurements of the abundance and
activity of all nodes in the network, and potentially the applied
perturbations. These are related through the following set of linear
equations (one for each protein iin each cell a):
Ri,a ⇡X
j6=i
r⇠
ij ·Rj,a +s⇠
i·Rtot
i,a 8a, i, ⇠.(1)
where Ri,a ⌘(xi,a hx⇠
ai)/hx⇠
aiis the deviation of the measured
abundance of active protein xfrom the population average, Rtot
i,a ⌘
(xtot
i,a hxtot,⇠
ai)/hxtot,⇠
aiis the deviation of the measured total
protein abundance xtot from the population average, r⇠
ij ⌘
hx⇠
ji
hx⇠
ii
@x⇠
i
@x⇠
j
is the population-specific interaction strength between
node jand node i, with rii ⌘1, s the population-specific
sensitivity of node iactivity to a change in total protein, and
pim ⌘pm
hxii
@xi
@pm
pm
pmis the direct e↵ect that perturbation mhas
on the activity of node i.
scMRA and scCNR both solve an MIQP optimization problem
that aims find the values of r⇠
ij ,s⇠
iand pim that minimize the
squared error ✏between the measured active protein abundance
per cell and the prediction of the model as described by
equation (1). Additionally, two L0-regularization penalties are
added: one for the number of edges in the network, and one for the
number of population-specific interaction strengths, sensitivities to
change in total protein and perturbation e↵ects. The full MIQP
formulation then reads as follows:
minimize: X
a,i,j,m
h(1 ⌘)·✏2
i,a +⌘·Iedge
ij +✓(Ir
ij +Is
i+Ip
im)i
subject to:
Nn
X
j=1
r⇠
ij ·Rj,a +s⇠
i·Rtot
i,a +
Np
X
m=1
p⇠
im =✏i,a 8a, i
Iedge
ij =0)rij =0 8i, j
Ir
ij =0)r⇠
ij =hrij i8i, j
Is
i=0)s⇠
i=hsii8i
Ip
im =0)p⇠
im =hpimi8i, m
Iedge
ij ,Ir
ij ,Is
i,Ip
im 2{0,1}
where Iedge
ij is a binary indicator for the presence or absence of
an edge between nodes iand j.Ir
ij and Is
iare binary indicators for
a population-specific interaction strength and sensitivity to total
protein changes, respectively, and are only present for scCNR.
The hyperparameter ⌘and ✓determine the relative weight of the
regularization penalties. For the inclusion of perturbation e↵ects
see supplementary 5.4. The optimization problem was solved using
IBM ILOG CPLEX solver (version 20.1.0). CPLEX is available
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3
free of charge for academic purposes and guarantees an optimal
solution within small numerical tolerances for our problem.
2.2. Simulating single-cell protein data
Single-cell protein abundance data was simulated using an
existing ODE-based dynamic model of the EGFR signalling
pathway (Orton et al., 2009). Single-cell total protein abundances
were sampled from a log-normal distribution (µ=0,=0.1)
and used to calculate the resulting quasi-steady state. Drug
perturbations data was simulated as a 25% decrease in the
catalytic activity of all protein-activating reactions. BRAF and
RAS mutantions were modelled as a 100-fold reduction in the
deactivation rate of active BRAF and RAF, respectively. Noise
was added to the input data by multiplying the data with a
random number drawn from a normal distribution with zero mean
and a standard deviation equal to the noise. The ground-thruth
interaction strengths between proteins of the Orton model were
numerically calculated as the partial derivatives of active protein
with respect to its upstream active protein.
2.3. scID-seq analysis
For full detail, see Supplementary Methods sections 5.1.2- 5.1.6.
Briefly, Primary pooled human epidermal stem cells where treated
with the EGFR inhibitor AG1478 or vehicle (DMSO) for 48
hours. Subsequetly, the abundance of 70 phosphorylated and total
proteins in 282 cells was quantified using single-cell ID-seq (Eijl
et al., 2018). We ran scCNR to detect signaling di↵erences between
the untreated and EGFR-inhibitor treated cell population and
used a prior literature derived network topology.
3 Results
3.1. scMRA reconstructs networks from total protein variation
in single-cell protein data
Single cell Modular Response Analysis (scMRA) reconstructs
signal transduction networks from the quantification of phospho-
and total proteins in single cells. scMRA exploits the stochastic
variability of total protein level s betwee n si ng le c el ls a s na tu ra l
perturbations to the signaling network (Fig. 1A). Stochastic
di↵erences in the abundance of total proteins between cells directly
a↵ect the abundance of the corresponding phosphoproteins. These
changes in phosphoprotein levels then propagate through the
network leading to distinct steady states of the network in
each individual cell. Each cell is considered as an independent
measurement of the underlying signaling network.
scMRA reconstructs a unique set of interaction strengths for a
cell population (yellow and blue boxes representing populations
AandBinFig. 1). Tomodelthevariationbetweencell
populations (which may represent for instance di↵erent cell states,
cells with and without an oncogenic mutation, cells before and
after acquiring resistance to a drug, or cells that are cultured
for a long time in the presence or absence of an inhibitor),
we developed single cell Comparative Network Reconstruction
(scCNR) to identify population specific interaction strengths,
just as CNR models condition-dependent interaction strengths
in the bulk setting (Bosdriesz et al., 2018). This allows for the
identification of the most relevant di↵erences between the cell
populations.
Input to the methods are deviations of abundances of total
(Rtot) and phosphoprotein (R) of each cell from the cell population
mean, for each node in the network (Fig. 1B, ’Input’). The
output is the network topology described by interaction strengths
between phosphoproteins (r), and sensitivities of phosphoproteins
to deviations in its total protein (s) (Fig. 1B, ’Output’). scMRA
is formulated as a MIQP problem and fits a model that (i)
for each node in each cell aims to explain the deviations of
phosphoprotein abundance from the population mean and (ii)
penalizes the model complexity (number of interactions) to derive
a core signaling network. scCNR also derives a core signaling
network with a small number of interaction strength di↵erences
between cell populations, while still producing a good model fit
by (iii) penalizing the number of population-specific interaction
strengths (Fig. 1B, ’Algorithm’).
Often, not all total protein abundance measurements are
available for all nodes in the network. However, additional
perturbations, e.g. by small molecular inhibitors, can be included
in the experimental design and model to facilitate network
reconstruction. Furthermore, the formulation of the algorithm
allows for easy integration of prior network information for cases
where the topology might be established and the main interest lies
in quantifying the interactions (Fig. 1B, ’Input’).
3.2. scMRA faithfully reconstructs the topology of signal
transduction networks
To evaluate how well scMRA recovers network topology and node
interaction strengths, we first set out to test it on simulated data
for which the ground truth is known. To this end, we simulated
single-cell data for the epidermal growth factor receptor (EGFR)
signaling pathway using a previously published dynamic model
described by ordinary di↵erential equations (Orton model), and
reconstructed the network (Orton et al., 2009). Note that since
scMRA and scCNR aim to explain the steady-state deviation of
phosphoprotein abundance from the cell population mean only,
the scMRA and scCNR network reconstructions will be much
simpler than the original model. Nevertheless, the true interactions
between active proteins are unambiguously defined (Fig. 2A).
We si mula t ed si n g le ce l ls by s a mpl i ng to t a l pro t ein a b u nda n ces
from a log-normal distribution, and calculated the corresponding
phosphoprotein abundances at the steady state.
As a first test, we ran simulated 1000 cells, and added 20%
noise to the data. We repeatedly reconstructed the signaling
network using scMRA and progressively increased the complexity
penalty ⌘, resulting in increasingly sparse networks. Fig. 2B
shows the reconstructed interaction strengths of the network, with
true positive edges indicated in blue and false positives in black.
Importantly, when increasing the ⌘-penalty, false positive edges are
the first to be eliminated, while the majority of true positive edges
are retained. Furthermore, in reconstructions with low ⌘-penalties
and thus many edges, the false positive edges have typically weaker
interaction strengths than the true positive edges. However, in
reconstructions with a low ⌘-penalty the strengths of the true
positive edges are typically slightly di↵erent from their true values,
as indicated by the alpha value of the points in Fig. 2B. As the
penalty increases and false positive edges vanish, the reconstructed
true edges in the network converge towards their actual interaction
strength. This is further exemplified by the strong correlation
between reconstructed and true interaction strengths (Fig. 2C).
3.3. scMRA works well in the presence of noise and with few
input cells
As single-cell data is inherently noisy, we investigated how this
impacts the performance of scMRA by adding varying amounts of
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4Reconstructing and comparing signal transduction networks from single cell protein quantification data
Describe phosphoprotein counts in
every cell:
Cell - protein matrix
Optional:
Prior network:
Perturbations:
Single cells
Cell population ACell population B
Total
Phospho
Proteins
Input: Algorithm: Output:Single cell
heterogeneity
Cell population B
scMRA
One interaction strength matrix for
one population
scCNR
Combined reconstruction
with population specific interaction
strength matrices
Minimize objective function:
scMRA:
Fitting error +
Penalty on number of edges +
scCNR:
Fitting error A+ Fitting error B +
Penalty on number of edges +
Penalty on number of population specific
interaction strengths
• Interaction strength matrix (r )
• Sensitivity to total protein (s )
Influence of total protein deviation
Deviation of phospho-protein i from population mean =
Influence of upstream phospho-protein deviation +
ir
ij
!"
!#
$%$
!#
+si
Describe deviations in active protein:
for ain cells do
for iin proteins do
Ri;a
⇡P
j
6
=irij
·
Rj;a +si
·
Rtot
i;a
Describe deviations in active protein:
for ain cells do
for iin proteins do
Ri;a
⇡P
j
6
=irij
·
Rj;a +si
·
Rtot
i;a
Stochastic differences
in total protein as ‘nat-
ural perturbation ex-
periments’
Cell population A
A B
Stochastic variability
in total protein
Total protein
Phosphoprotein
Fig. 1: Schematic overview of scMRA and scCNR.(A) The methods exploit natural heterogeneity of phospho- and total protein
abundances between cells to infer the network topology and quantify the interaction strengths between network nodes. (B) The methods
take phospho- and total protein abundances from single cells as input, with additional cell population annotations for scCNR. Optionally
the methods can be enriched with perturbation data and a prior network topology. The algorithms exploit deviations of total protein
from the population mean (Rtot) as ’natural perturbations’. They fit the data to describe single-cell deviations of phosphoprotein
from its population mean (R) for a single population (scMRA) or several populations in parallel (scCNR) to derive (cell population-
specific) interaction strengths (r). The algorithms penalize the number of edges in the network. scCNR further penalizes the number of
population-specific interaction strengths.
noise on the input data. Similar to the analysis described above, we
repeatedly reconstructed the Orton network while decreasing the
network connectivity (by increasing the ⌘parameter). Comparing
the recovered edges to the ground truth network we generated
receiver-operating characteristic (ROC) curves. With 20% noise a
true positive rate of >75% can be achieved while keeping the false
positive rate below 2.5% (Fig. 2D, left panel). Even with as much
as 50% noise added to the data, a 70% true positive rate is attained
at a false positive rate <6%. To put the numbers into perspective:
a reconstructed network with 13 edges from a simulation with 1000
cells and 20% noise yields a network with two false positive edges.
In addition to successfully recovering the network topology, the
reconstructed interaction strengths are very similar to their true
values, a nd o nl y two e dges with we ak i nteraction strengt hs a re n ot
recovered (Fig. 2C).
Profiling large numbers of cells is not always feasible, as for
instance even in large datasets specific cell populations of interest
might be under-represented. Hence, we tested the influence of
the number of cells measured on the performance of scMRA by
simulating populations of various sizes, each with 20% noise added.
The resulting ROC-curves are shown in Fig. 2D, right panel. For
500 cells or more, a true positive rate of >75% with a 2.5% false
positive rate is attained. With as few as 50 cells a true positive
rate of 75% with a 5% false positive rate can be attained. Taken
together, this illustrates that scMRA can accurately reconstruct
the network from few cells in the presence of considerable noise.
3.4. Perturbation experiments can compensate for missing
total protein measurements
Ideally, scMRA uses measurements of phospho- and total protein
for all nodes in the network, but this might not always be
feasible due to experimental constraints. To assess the sensitivity
of scMRA to missing total protein measurements we removed
total protein information of randomly selected proteins from
the simulated data. To quantify the reconstruction accuracy we
calculated the root-mean-square error (RMSE) of the true and
reconstructed interaction strengths, from simulations of 1000
cells with 20% noise added. To allow a fair comparison between
reconstructed networks we compared networks with 13 edges.
As expected, increasing the number of proteins for which their
total abundance is missing increases the RMSE of interaction
strength reconstructions (Fig. 2E, left panel) from a mean RMSE
of 0.07 when all total proteins are measured to 0.22 when
they are all missing. To compensate for missing total protein
measurements, scMRA can take perturbation data as input to
improve network identification. To examine to which extent this
could help to mitigate the negative e↵ect of missing total protein,
we simulated perturbations to a node as a 25% reduction in the
catalytic activity of reactions activating that node. We randomly
selected nodes to be perturbed, performed the perturbations and
then reconstructed the network. When including the perturbations
in the reconstruction, we kept the total number of cells to 1000.
Adding perturbation data can decrease the RMSE from an average
of 0.22 to 0.12 in the scenario with complete absence of total
protein, but with perturbation data for all nodes in the network
(Fig. 2E, right panel). This confirms that additional perturbations
can partially compensate for missing total protein measurements
and could be an option to include in the experimental design when
total protein measurements might not be fully available.
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5
PI3K
SOS
RAF1
ERK
C3G
BRAF
AKT
RAS
MEK
RAP1
P90RSK
E
A B
D
Ground truth network
C
P90RSK - SOS
RAS - SOS
SOS - RAF1
Fig. 2: Evaluation of scMRA.(A) The EGFR signaling pathway according to the Orton Model. Arrow width is equivalent to the
interaction strength derived from the Orton Model. (B) Interaction strengths reconstructed using scMRA for reconstructions of decreasing
network complexity (increasing edge penalty ⌘). True positives are indicated in blue and false positives in black. (C) Correlation
between true and reconstructed interaction strengths for a network reconstruction with 1000 cells and 20% noise. (D) Receiver-operating
characteristic curves for the e↵ect of noise and number of cells on the reconstruction of the network topology. (E) RMSE between true and
reconstructed interaction strengths for simulations with randomly removed total protein measurements and simulations with completely
absent total protein measurements but additional perturbations for randomly selected nodes.
3.5. scCNR identifies signaling di↵erences between simulated
wild-type and mutant cell populations
Tissue samples and cell cultures are inherently heterogeneous
and can contain various cell types and states. To enable
the investigation of di↵erences in signaling between such
populations we developed single cell Comparative Network
Reconstruction (scCNR). scCNR is based on Comparative
Network Reconstruction(Bosdriesz et al., 2018) that aims to
identify what the most important quantitative di↵erences are
in signaling between cell populations. To this end, in scCNR
we reconstruct a single network topology for all populations
simultaneously, while allowing distinct interaction strengths for
each population. scCNR outputs one shared network topology with
a set of population-specific interaction strengths. In addition to
the complexity penalty on the number of edges (⌘), scCNR also
penalizes the number of population-specific interaction strengths
(✓). This greatly reduces the total number of model parameters,
thereby reducing overfitting and focusing on the most important
di↵erences. An additional advantage of this approach is that
the reconstruction of each population is based on the full
dataset, hence allowing information to be ”borrowed” between cell
populations.
We evalu a ted s c CNR on s i mu l ated s i ngl e c e ll pop u lat i o ns
of mutant RAS and BRAF cells, which we compared to
the simulated wild-type population. These mutations cause
constitutive activation of RAS and BRAF, respectively, which we
modeled similarly to Orton et al. by a 100-fold reduction of the
deactivation rate of the mutant protein (Orton et al., 2009). We
simulated 250 cells per population and added 20% noise. Next,
we reconstructed the signaling networks and compared them to
the ground truth network of the Orton model. We estimated an
appropriate value for ✓, which reduces the mean squared error
of the objective function while keeping the model complexity
low (Fig. S2). This resulted in approximately four edges that
di↵ered between the mutant and wild-type populations.
The simulations of the wild-type and RAS mutant cell
population result in major di↵erences upstream of BRAF, which
scCNR recovered truthfully (Fig. 3A and Fig. S3A). The BRAF
mutant network, on the other hand, mainly di↵ered downstream
of BRAF within the BRAF-MEK-ERK-P90RSK cascade, which
scCNR identified correctly as well (Fig. S4A and Fig. S3D). To
quantitatively compare the scCNR reconstruction to the ground
truth network, we reconstructed the signaling network for the
wild-type and mutant cell population 20 times with 250 cells
per population and 20% noise. Similar to previous simulations
we aimed to reconstruct networks with 13 edges. The average
reconstructed interaction strengths for both the RAS mutant and
BRAF mutant cell populations correlated well with the true values
for most edges (Fig. 3B and Fig. S4B). With the restriction
on detecting four population-specific interaction strengths, small
interaction strength di↵erences (such as BRAF-MEK), were not
recovered in every simulation, as expected. However, edges with
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6Reconstructing and comparing signal transduction networks from single cell protein quantification data
A B
Reconstructed differences between
RASmut & WT network
-0.65
+0.75
-0.24
+0.22
RAF1
ERK
C3G
BRAF
AKT
RAS
MEK
RAP1
P90RSK
PI3K
SOS
P90RSK - SOS
BRAF - MEK
RAF1 - MEK
RAP1 - BRAF
RAS - BRAF
ERK -
P90RSK
selected number of
non-equal edges
C
D
Fig. 3: scCNR identifies signaling di↵erences between cell populations. (A) Reconstructed di↵erences between the RAS mutant
and wild-type network. Gray background edges indicate the true network topology. Black and red edges highlight the positive and
negative di↵erences of the RAS mutant compared to the wild-type network. (B) Correlation between reconstructed and true interaction
strengths averaged over 20 simulations. Blue points correspond to interaction strengths in the RAS mutant population, yellow in the
wild-type population, and in gray points correspond to edges with the same interaction strength (di↵erence below 0.1). Error bars
visualize the range of reconstructed interaction strength values across simulations. Black lines connect corresponding wild-type and
mutant interaction strengths. (C) RMSE of interaction strength di↵erences between the two populations for increasing numbers of
population-specific interaction strengths. (D) RMSE for joined scCNR (blue) with 4 population-specific edges or separate scMRA (gray)
reconstructions of the network, with various levels of noise.
clear di↵erences between the wild-type and mutant population
were recovered well.
3.6. Joined network reconstruction with scCNR improves
performance over reconstruction with scMRA
In addition to highlighting the major di↵erences between
populations, we hypothesized that scCNR might improve network
reconstruction by pooling cells from all populations into a single
optimization problem, hence increasing the power. To test this
hypothesis we reconstructed the Orton network based on simulated
wild-type and RAS mutant single-cell data of 500 cells in total,
with 20% noise, and setting ⌘to obtain networks with 13 edges.
We qu a nti fi ed ho w wel l a c omb ine d r e con s truc t ion u s ing sc C NR
recovers di↵erences between mutant and wild-type networks, and
compared this with two scMRA reconstructions where the wild-
type and mutant populations are reconstructed independently. We
calculated the di↵erence of interaction strengths between the two
populations (wild-type and mutant) for both the ground truth
and the reconstructed networks and compared these. Fig. 3C
shows the RMSE of the di↵erence in interaction strengths as a
function of the number of edges that have a di↵erent interaction
strength between the two populations. The scCNR reconstructions
(blue points) are consistently more accurate that the scMRA
reconstructions (gray line), indicating the benefit of a joint
network reconstruction. Increasing the number of di↵erences above
four results in little further improvement. With increasingly noisy
input data, the increased performance of a joint reconstruction
with scCNR becomes even more pronounced as seen in Fig. 3D
and Fig. S3. Together, this shows that by pooling information
from cells of multiple populations scCNR can more accurately
reconstruct relevant di↵erences between populations as compared
to independent scMRA reconstructions.
3.7. scCNR detects signaling di↵erence in response to EGFR
inhibition
To further demonstrate the utility of scCNR we applied the
method to real experimental single-cell measurements of EGFR-
inhibitor or vehicle treated primary human keratinocytes, with
the aim to quantify how prolonged EGFR inhibition a↵ects signal
transduction. While the EGFR pathway is well studied, how
prolonged exposure to inhibitors alters signal transduction within
the network is still an ongoing field of research (Klinger et al.,
2013). In this experiment, we measured the abundance of 70
phosphorylated and total proteins of 282 cells using single-cell
ID-seq (Eijl et al., 2018), involving 31 key nodes of the EGFR
pathway, upon treatment with an EGFR inhibitor (AG1478) or
vehicle (DMSO) control for 48 hours.
As expected, EGFR inhibition induced a reduction in
phosphoprotein abundance of proteins downstream of EGFR,
including RPS6 and AKT (Fig. S5A) (Fan et al., 2009;
Phuchareon et al., 2015). We detect reduced levels of phospho-RB
after treatment, while CDK4 and Cycline-E stayed active, which
marks an arrest of cells in the early cell-cycle stage (Fig. S5A).
Furthermore, EGFR inhibition pushes cells into a di↵erentiated
cell state, which is marked by a decrease in ITGB1. We also detect
an increase of BMPR and it has been shown that BMP signaling
goes up during di↵erentiation (Eijl et al., 2018). EGFR inhibition
additionally raises the level of phosphorylated p38, which has been
linked to keratinocyte di↵erentiation and cell-cycle arrest (Saha
et al., 2014; Connelly et al., 2011; Adhikary et al., 2010; Efimova
et al., 1998). Next to the increased phosphorylation of p38, rising
levels of phosphorylated JNK have been reported for keratinocytes
after EGFR-inhibitor treatment (Lu et al., 2011). Together, this
demonstrates that the data contains biologically meaningful signal,
and so we continued to model the underlying signaling di↵erences
between the two cell populations.
We ap p l ied s c CNR t o i d ent ify d i ↵e ren c es of th e E GFR s i gnal i ng
pathway between the untreated and EGFR-inhibitor treated
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(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is
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7
population. We ran scCNR with a prior-network topology of
known protein interactions, and set the ✓-parameter to identify
twelve population-specific interaction strengths in the network
(Fig. 4A) to balance model complexity and model fit (Fig. S5B).
To assess the statistical significance of the di↵erences
in interaction strengths between the treated and untreated
populations identified by scCNR, we permuted the population
labels and reconstructed the interaction strengths 1000 times,
while fixing which of the interaction strengths can be population
specific from the unpermuted solution. In this null-model we
expect that the interaction strengths do not di↵er between
populations. We calculated the empirical p-value as the fraction
of permutations that led to a more extreme interaction strength
di↵erence than the one obtained from the unpermuted network
reconstruction. At a 20% false positive rate, 11 of the 12
population-specific interaction strengths had significant di↵erences
between populations (Fig. 4A). As expected from an inhibition
of EGFR, we predominantly recover di↵erences directly linked
to that node. Four of five outgoing edges from EGFR di↵er
in interaction-strength between the cell populations. EGFR
inhibition leads to cell-cycle arrest, and we recover signaling
di↵erences around the cell-cycle related node phospho-RB.
However, phospho-RB follows a bimodal distribution in the
untreated cell population and represents cells in di↵erent cell-
cycle stages, which makes this observation hard to interpret.
Nevertheless, the CDC2-RB edge is significant with a false
discovery rate below 10% (Fig. S5C).
Next, we conducted a bootstrapping analysis to verify that
the selection of edges is not primarily driven by outlier cells.
We su b - sam p led c e l ls wi t h repl a cem e nt f r o m b oth p o p u lat i ons,
repeated the network reconstruction 1000 times, and counted for
each edge how often it was identified to be population specific.
Edges around EGFR and phosho-RB consistently show up in
the reconstruction, supporting the evidence of those di↵erences
between cell populations. Moreover, all di↵erentially recovered
edges are far above the theoretical occurrence of a randomly
selected edge (Fig. 4B).
Finally, to investigate how reliable di↵erences in interaction
strength are quantitatively, we performed a bootstrapping analysis
where we fixed all edges that were identified to di↵er in strength
between populations, bootstrapped cells from both populations,
and re-ran scCNR to recover interaction strengths. These 1000
bootstraps provided us with distributions of interaction strengths
for both populations (Fig. 4C). Most interaction strengths
showed clear di↵erences between the cell populations. Notably,
we observed that interactions with phospho-RB move to zero
upon EGFR inhibition. Furthermore, among the four population-
specific edges downstream of EGFR, three decrease in interaction
strength following inhibition. Interactions with AKT123 and
SRC diminish to zero and also edges further downstream of
EGFR decrease in interaction strength upon EGFR inhibition,
such as ERK12-RSK, JNK-CJUN or MAPKAP2-AKT123. From
this, we conclude that despite the limitations imposed by the
relatively small sample size, scCNR is able to detect significant and
biologically meaningful di↵erences in signal transduction between
populations.
4 Discussion
How signal transduction di↵ers between cell states, in disease, or
upon treatment, plays a fundamental role in cell biology and has
potential clinical implications. Reconstructing signaling networks
has long been limited to bulk data, but recent advances in single-
cell (phospho-)protein measurement technologies allows us to take
advantage of many data points to study signaling networks on the
single cell level. To this end, we developed single cell Modular
Response Analysis (scMRA) and single cell Comparative Network
Reconstruction (scCNR), which exploit the stochastic variability
of protein abundances between cells as natural ’perturbation
experiments’. By penalizing network complexity and, in the case of
scCNR, the di↵erences in signaling between cell population, core
signaling networks and their most relevant di↵erences are found.
Additionally, prior network information can easily be integrated.
We ex t e nsi vel y val i dat e d s cMR A o n sim u lat e d dat a a n d sho we d
that the method works well even in presence of considerable noise
and with as few as 500 cells. Additionally, we applied scCNR
to a real-world single-cell ID-seq dataset of cells treated either
with an EGFR inhibitor or with a vehicle control, and recovered
biologically meaningful and expected signaling di↵erences between
the treated and untreated cell populations .
While our findings underscore the method’s capacity to identify
relevant signaling di↵erences, it is essential to also acknowledge
the limitations. While scCNR explains phosphoprotein counts for
every cell individually, it does rely on clustering of cells into
discrete groups to infer population specific network parameters.
In addition, some care needs to be taken in interpreting edges
that the model proposes, as these might be indirect and
mediated through unobserved nodes. Therefor, we expect the
scCNR results to typically serve as a valuable foundation for
generating hypotheses regarding mechanistic interactions, which
can subsequently be subjected to further in-depth exploration.
Future progress in single-cell protein measurement techniques will
enhance the detection of cell-to-cell variability and will improve
network inference with scCNR.
In contrast to the classical MRA and CNR formulation, the
scMRA and scCNR optimization problems are typically over-
determined since there are many more linear equations - one for
every cell and every protein - than possible edges. Nonetheless, the
computational complexity of the optimization problem increases
exponentially with the number of nodes in the network. However,
biological networks are often sparsely connected, and sparse
solutions can generally be found efficiently. For instance, the Orton
model can be solved in under 20 seconds for 500 cells with 20%
noise and 13 edges on a standard laptop. The number of nodes
in the network and the number of edges and population-specific
interaction strengths influence the search space and therefore the
run time. In cases where many interactions have to be inferred the
run time can be reduced drastically by incorporating prior network
information or by setting hyperparameter settings that limit the
number of model parameters that need to be inferred.
Several methods to reconstruct population specific networks
from single cell data have been proposed before(Rukhlenko et al.,
2022; Brandt et al., 2019; Kumar et al., 2020). However, these
methods use the single cell nature of the data only to cluster
cells in distinct groups, and then aggregate the cells within one
population. As such, they do not exploit the within population
variation in si gn al in g ac ti vi ty and the y re qu ir e ma ny p er tu rb at ions
for network inference. DREMI introduced the concept of using
cell-to-cell variation in protein activity to gain insight into the
strength of protein interactions (Krishnaswamy et al., 2014), but
this method only considers protein pairs, and does not reconstruct
networks. Compared to these methods, the advantages of scCNR
.CC-BY-NC-ND 4.0 International licensemade available under a
(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is
The copyright holder for this preprintthis version posted April 1, 2024. ; https://doi.org/10.1101/2024.03.29.587331doi: bioRxiv preprint
8Reconstructing and comparing signal transduction networks from single cell protein quantification data
significant not significant
Significant edges with FDR < 0.2 :
GSK3B
P65
MKK36
P38D
CREB
CDK4
H3
CMYC
CDC2
RB
ITGB1
RPS6
MTOR
EGFR
CJUN
CFOS
RSK
IKBA
RELA
SRC
JAK1
CYCLINE
FAK
LRP6
JNK
STAT5
AKT123
STAT3
AKT1
STAT1
ERK12
P38
MAPKAP2
A B
C
Prior network with significant edge differences Edge detection after bootstrapping cells
Interaction strength distribution after edge-fixation and bootstrapping cells
Fig. 4: Analysis of signaling di↵erences between untreated and EGFR-inhibitor treated keratinocytes.(A) Prior knowledge
network highlighting reconstructed di↵erences between EGFR-inhibitor treated and untreated cells. (B) Bootstrapping analysis to detect
edges that are consistently identified to have population-specific strengths. After every bootstrap scCNR identified 12 population-specific
interaction strengths and we counted their occurrence within all bootstrapped runs. Significant edges from A are coloured in red. The
gray line corresponds to the theoretical expectation of randomly selecting an edge to have a population-specific interaction strength. (C)
Distributions of interaction strengths from 1000 bootstrapping runs, for the untreated (black) and EGFR inhibitor treated cells (blue).
The red dot indicates the recovered interaction strength from the whole data, as depicted in A.
are that it uses variability within the single-cell population, with
the possibility to easily integrate perturbations, prior networks
and various cell populations, to identify cell population specific
signaling networks.
In conclusion, scMRA enables the reconstruction of signal
transduction networks from single-cell data, and scCNR
additionally recovers the most relevant di↵erences in signaling
between cell populations. We envision that scCNR will help in
the recovery of signaling networks in diverse biological settings,
especially by shedding light on clinically relevant signaling
di↵erences, e.g., between cell states and upon treatment.
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