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Predicting Essential Components of Signal
Transduction Networks: A Dynamic Model
of Guard Cell Abscisic Acid Signaling
Song Li1, Sarah M. Assmann1, Re´ka Albert2*
1 Biology Department, Pennsylvania State University, University Park, Pennsylvania, United States of America, 2 Physics Department, Pennsylvania State University, University
Park, Pennsylvania, United States of America
Plants both lose water and take in carbon dioxide through microscopic stomatal pores, each of which is regulated by a
surrounding pair of guard cells. During drought, the plant hormone abscisic acid (ABA) inhibits stomatal opening and
promotes stomatal closure, thereby promoting water conservation. Dozens of cellular components have been
identified to function in ABA regulation of guard cell volume and thus of stomatal aperture, but a dynamic description
is still not available for this complex process. Here we synthesize experimental results into a consistent guard cell
signal transduction network for ABA-induced stomatal closure, and develop a dynamic model of this process. Our
model captures the regulation of more than 40 identified network components, and accords well with previous
experimental results at both the pathway and whole-cell physiological level. By simulating gene disruptions and
pharmacological interventions we find that the network is robust against a significant fraction of possible
perturbations. Our analysis reveals the novel predictions that the disruption of membrane depolarizability, anion
efflux, actin cytoskeleton reorganization, cytosolic pH increase, the phosphatidic acid pathway, or Kþ efflux through
slowly activating Kþ channels at the plasma membrane lead to the strongest reduction in ABA responsiveness. Initial
experimental analysis assessing ABA-induced stomatal closure in the presence of cytosolic pH clamp imposed by the
weak acid butyrate is consistent with model prediction. Simulations of stomatal response as derived from our model
provide an efficient tool for the identification of candidate manipulations that have the best chance of conferring
increased drought stress tolerance and for the prioritization of future wet bench analyses. Our method can be readily
applied to other biological signaling networks to identify key regulatory components in systems where quantitative
information is limited.
Citation: Li S, Assmann SM, Albert R (2006) Predicting essential components of signal transduction networks: A dynamic model of guard cell abscisic acid signaling. PLoS Biol
4(10): e312. DOI: 10.1371/journal.pbio.0040312
Introduction
One central challenge of systems biology is the distillation
of systems level information into applications such as drug
discovery in biomedicine or genetic modification of crops. In
terms of applications it is important and practical that we
identify the subset of key components and regulatory
interactions whose perturbation or tuning leads to significant
functional changes (e.g., changes in a crop’s fitness under
environmental stress or changes in the state of malfunction-
ing cells, thereby combating disease). Mathematical modeling
can assist in this process by integrating the behavior of
multiple components into a comprehensive model that goes
beyond human intuition, and also by addressing questions
that are not yet accessible to experimental analysis.
In recent years, theoretical and computational analysis of
biochemical networks has been successfully applied to well-
defined metabolic pathways, signal transduction, and gene
regulatory networks [1–3]. In parallel, high-throughput
experimental methods have enabled the construction of
genome-scale maps of transcription factor–DNA and pro-
tein–protein interactions [4,5]. The former are quantitative,
dynamic descriptions of experimentally well-studied cellular
pathways with relatively few components, while the latter are
static maps of potential interactions with no information
about their timing or kinetics. Here we introduce a novel
approach that stands in the middle ground of the above-
mentioned methods by incorporating the synthesis and
dynamic modeling of complex cellular networks that contain
diverse, yet only qualitatively known regulatory interactions.
We develop a mathematical model of a highly complex
cellular signaling network and explore the extent to which
the network topology determines the dynamic behavior of
the system. We choose to examine signal transduction in
plant guard cells for two reasons. First, guard cells are central
components in control of plant water balance, and better
Academic Editor: Joanne Chory, The Salk Institute for Biological Studies, United
States of America
Received April 3, 2006; Accepted July 21, 2006; Published September 12, 2006
DOI: 10.1371/journal.pbio.0040312
Copyright: 2006 Li et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author
and source are credited.
Abbreviations: ABA, abscisic acid; PP2C, protein phosphatase 2C; Atrboh, NADPH
oxidase; Ca2þ
c , cytosolic Ca2þ increase; CaIM, Ca2þ influx across the plasma
membrane; CIS, Ca2þ influx to the cytosol from intracellular stores; CPC, cumulative
percentage of closure; GCR1, G protein–coupled receptor 1; GPA1, heterotrimeric G
protein a subunit 1; KAP, Kþ efflux through rapidly activating Kþ channels (AP
channels) at the plasma membrane; KOUT, Kþ efflux through slowly activating
outwardly-rectifying Kþ channels at the plasma membrane; NO, nitric oxide; NOS,
nitric oxide synthase; PA, phosphatidic acid; ROS, reactive oxygen species
* To whom correspondence should be addressed. E-mail: [email protected]
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PLoS BIOLOGY
understanding of their regulation is important for the goal of
engineering crops with improved drought tolerance. Second,
abscisic acid (ABA) signal transduction in guard cells is one of
the best characterized signaling systems in plants: more than
20 components, including signal transduction proteins,
secondary metabolites, and ion channels, have been shown
to participate in ABA-induced stomatal closure. ABA induces
guard cell shrinkage and stomatal closure via two major
secondary messengers, cytosolic Ca2þ (Ca2þ
c) and cytosolic pH
(pHc). A number of signaling proteins and secondary
messengers have been identified as regulators of Ca2þ influx
from outside the cell or Ca2þ release from internal stores; the
downstream components responding to Ca2þ are certain
vacuolar and plasma membrane Kþ permeable channels, and
anion channels in the plasma membrane [6,7]. Increases in
cytosolic pH promote the opening of anion efflux channels
and enhance the opening of voltage-activated outward Kþ
channels in the plasma membrane [8–10]. Stomatal closure is
caused by osmotically driven cell volume changes induced by
both Kþ and anion efflux through plasma membrane–
localized channels. Despite the wealth of information that
has been collected regarding ABA signal transduction, the
majority of the regulatory relationships are known only
qualitatively and are studied in relative isolation, without
considering their possible feedback or crosstalk with other
pathways. Therefore, in order to synthesize this rich knowl-
edge, one needs to assemble the information on regulatory
mechanisms involved in ABA-induced stomatal closure into a
system-level regulatory network that is consistent with
experimental observations. Clearly, it is difficult to assemble
the network and predict the dynamics of this system from
human intuition alone, and thus theoretical tools are needed.
We synthesize the experimental information available
about the components and processes involved in ABA-
induced stomatal closure into a comprehensive network,
and study the topology of paths between signal and response.
To capture the dynamics of information flow in this network
we express synergy between pathways as combinatorial rules
for the regulation of each node, and formulate a dynamic
model of ABA-induced closure. Both in silico and in initial
experimental analysis, we study the resilience of the signaling
network to disruptions. We systematically sample functional
and dynamic perturbations in network components and
uncover a rich dynamic repertoire ranging from ABA
hypersensitivity to complete insensitivity. Our model is
validated by its agreement with prior experimental results,
and yields a variety of novel predictions that provide targets
on which further experimental analysis should focus. To our
knowledge, this is one of the most complex biological
networks ever modeled in a dynamical fashion.
Results
Extraction and Organization of Data from the Literature
We focus on ABA induction of stomatal closure, rather
than ABA inhibition of stomatal opening, because these two
processes, although related, exhibit distinct mechanisms, and
there is substantially more information on the former process
than on the latter in the literature. Experimental information
about the involvement of a specific component in ABA-
induced stomatal closure can be partitioned into three
categories. First, biochemical evidence provides information
on enzymatic activity or protein–protein interactions. For
example, the putative G protein–coupled receptor 1 (GCR1)
can physically interact with the heterotrimeric G protein a
component 1 (GPA1) as supported by split-ubiquitin and
coimmunoprecipitation experiments [11]. Second, genetic
evidence of differential responses to a stimulus in wild-type
plants versus mutant plants implicates the product of the
mutated gene in the signal transduction process. For
example, the ethyl methanesulfonate–generated ost1 mutant
is less sensitive to ABA; thus, one can infer that the OST1
protein is a part of the ABA signaling cascade [12]. Third,
pharmacological experiments, in which a chemical is used
either to mimic the elimination of a particular component, or
to exogenously provide a certain component, can lead to
similar inferences. For example, a nitric oxide (NO) scavenger
inhibits ABA-induced closure, while a NO donor promotes
stomatal closure; thus, NO is a part of the ABA network [13].
The last two types of inference do not give direct interactions
but correspond to pathways and pathway regulation. The
existing theoretical literature on signaling is focused on
networks where the first category of information is known,
along with the kinetics of each interaction. However, the
availability of such detailed knowledge is very much the
exception rather than the norm in the experimental
literature. Here we propose a novel method of representing
qualitative and incomplete experimental information and
integrating it into a consistent signal transduction network.
First, we distill experimental conclusions into qualitative
regulatory relationships between cellular components (signal-
ing proteins, metabolites, ion channels) and processes. For
example, the evidence regarding OST1 and NO is summar-
ized as both OST1 and NO promoting ABA-induced stomatal
closure. We distinguish between positive and negative
regulation by using the verbs ‘‘promote’’ and ‘‘inhibit,’’
represented graphically as ‘‘!’’ and ‘‘—j,’’ respectively, and
quantify the severity of the effect by the qualifier ‘‘partial.’’ A
partial promoter’s (inhibitor’s) loss has less severe effects than
the loss of a promoter (inhibitor), most probably due to other
regulatory effects on the target node. Using these relations,
we construct a database that contains more than 140 entries
and is derived from more than 50 literature citations on ABA
regulation of stomatal closure (Table S1). A number of entries
in the database correspond to a component-to-component
relationship, such as ‘‘A promotes B,’’ which is mostly
obtained by pharmacological experiments (e.g., applying A
causes B response). However, the majority of the entries
belong to the two categories of indirect inference described
above, and are of the type ‘‘C promotes the process (A
promotes B).’’ This kind of information can be obtained from
both genetic and pharmacological experiments (e.g., disrupt-
ing C causes less A-induced B response, or applying C and A
simultaneously causes a stronger B response than applying A
only). There are a few instances of documented independence
of two cellular components, which we identify with the
qualifier ‘‘no relationship.’’ Most of the information is
derived from the model species Arabidopsis thaliana, but data
from other species, mostly Vicia faba, are also included where
comparable information from Arabidopsis thaliana is lacking.
Assembly of the ABA Signal Transduction Network
To synthesize all this information into a consistent
network, we need to determine how the different pathways
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Model of Guard Cell ABA Signaling
suggested by experiments fit together (i.e., we need to find the
pathways’ branching and crossing points). We develop a set of
rules compatible with intuitive inference, aiming to deter-
mine the sparsest graph consistent with all experimental
observations. We summarize the most important rules in
Figure 1; in the following we give examples for their
application.
If A ! B and C ! process (A ! B), where A ! B is not a
biochemical reaction such as an enzyme catalyzed reaction or
protein–protein/small molecule interaction, we assume that C
is acting on an intermediary node (IN) of the A–B pathway.
This IN could be an intermediate protein complex, protein–
small molecule complex, or multiple complexes (see Figure 1,
panel 1). For example, ABA ! closure, and NO synthase
(NOS) ! process (ABA ! closure); therefore, ABA ! IN !
closure, NOS ! IN. If A ! B is a direct process such as a
biochemical reaction or a protein–protein interaction, we
assume that C ! process (A ! B) corresponds to C ! A ! B.
A ! B and C ! process (A ! B) can be transformed to A
! C ! B if A ! C is also documented. This means that the
simplest explanation is to identify the putative intermediary
node with C. For example, ABA ! NOS, and NOS ! process
(ABA ! NO) are experimentally verified and NOS is an
enzyme producing NO, therefore, we infer ABA ! NOS !
NO (see Figure 1, panel 2).
A rule similar to rule 1 applies to inhibitory interactions
(denoted by —j); however, in the case of A —j B, and C —j
process (A —j B), the logically correct representation is: A !
IN —j B, C —j IN (see Figure 1, panel 3).
The above rules constitute a heuristic algorithm for first
expanding the network wherever the experimental relation-
ships are known to be indirect, and second, minimizing the
uncertainty of the network by filtering synonymous relation-
ships. Mathematically, this algorithm is related to the
problem of finding the minimum transitive reduction of a
graph (i.e., for finding the sparsest subgraph with the same
reachability relationships as the original) [14]; however, it
differs from previously used algorithms by the fact that the
edges can have one of two signs (activating and inhibitory),
and edges corresponding to direct interactions are main-
tained.
In the reconstructed network, given in Figure 2, the
network input is ABA and the output is the node ‘‘Closure.’’
The small black filled circles represent putative intermediary
nodes mediating indirect regulatory interactions. The edges
(lines) of the network represent interactions and processes
between two components (nodes); an arrowhead at the end of
an edge represents activation, and a short segment at the end
of an edge signifies inhibition. Edges that signify interactions
derived from species other than Arabidopsis are colored light
blue. We indicate two inferred negative feedback loops on
S1P and pHc (see below) by dashed light blue lines. Nodes
involved in the same metabolic reaction or protein complex
are bordered by a gray box; only those arrows that point into
or out of the box signify information flow (signal trans-
duction). Some of the edges on Figure 2 are not explicitly
incorporated in Table S1 because they represent general
biochemical or physical knowledge (e.g., reactions inside gray
boxes or depolarization caused by anion efflux).
A brief biological description of this reconstructed net-
work (Figure 2) is as follows. ABA induces guard cell
shrinkage and stomatal closure via two major secondary
messengers, Ca2þ
c and pHc. Two mechanisms of Ca2þ
c
increase have been identified: Ca2þ influx from outside the
cell and Ca2þ release from internal stores. Ca2þ can be
released from stores by InsP3 [15] and InsP6 [16], both of
which are synthesized in response to ABA, or by cADPR and
cGMP [17], whose upstream signaling molecule, NO [13,18], is
indirectly activated by ABA. Opening of channels mediating
Ca2þ influx is mainly stimulated by reactive oxygen species
(ROS) [19], and we reconstruct two ABA-ROS pathways
involving OST1 [12] and GPA1 (L. Perfus-Barbeoch and S. M.
Assmann, unpublished data), respectively. Based on current
experimental evidence these two pathways are distinct, but
not independent. The downstream components responding
to Ca2þ are certain vacuolar and plasma membrane Kþ
permeable channels, and anion channels in the plasma
membrane [6,7]. The mechanism of pH control by ABA is
less clear, but it is known that pHc increases shortly after ABA
treatment [20,21]. Increases in pHc levels promote the
opening of anion efflux channels and enhance the opening
of voltage-activated outward Kþ channels in the plasma
membrane [8–10]. Stomatal closure is caused by osmotically
driven cell volume changes induced by Kþ and anion efflux
through plasma membrane-localized channels, and there is a
complex interregulation between ion flux and membrane
depolarization.
In addition to the secondary-messenger–induced pathways,
there are two less-well-studied ABA signaling pathways
involving the reorganization of the actin cytoskeleton, and
the organic anion malate. ABA inactivates the small GTPase
protein RAC1, which in turn blocks actin cytoskeleton
disruption [22], contributing to an ABA-induced actin
cytoskeleton reorganization process that is potentially Ca2þ
c
dependent [23]. In our model system, Arabidopsis, ABA
regulation of malate levels has not been described. However,
in V. faba it has been shown that ABA inhibits PEP carboxylase
and malate synthesis [24], and that ABA induces malate
breakdown [25]. In some conditions sucrose is an osmoticum
that contributes to guard cell turgor [26,27] but no
mechanisms of ABA regulation of sucrose levels have been
described.
The recessive mutant of the protein phosphatase 2C (PP2C)
Figure 1. Illustration of the Inference Rules Used in Network
Reconstruction
(1) If A ! B and C ! process (A ! B), where A ! B is not a biochemical
reaction such as an enzyme catalyzed reaction or protein-protein/small
molecule interaction, we assume that C is acting on an intermediary
node (IN) of the A–B pathway.
(2) If A ! B, A ! C, and C ! process (A ! B), where A ! B is not a direct
interaction, the most parsimonious explanation is that C is a member of
the A–B pathway, i.e. A ! C ! B.
(3) If A —j B and C —j process (A —j B), where A —j B is not a direct
interaction, we assume that C is inhibiting an intermediary node (IN) of
the A–B pathway. Note that A! IN —j B is the only logically consistent
representation of the A–B pathway.
DOI: 10.1371/journal.pbio.0040312.g001
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Model of Guard Cell ABA Signaling
ABI1, abi1-1R, is hypersensitive to ABA [28,29]. ABI1 is
negatively regulated by phosphatidic acid (PA) and ROS, and
pHc can activate ABI1 [30–32]. ABI1 negatively regulates
RAC1 [22]. We hypothesize that ABI1 negatively regulates the
NADPH oxidase (Atrboh) because ABI1 negatively regulates
ROS production and Atrboh has been shown to be the
dominant producer of ROS in guard cells [33]. We also
assume that ABI1 inhibits anion efflux at the plasma
membrane, because the dominant abi1–1 mutant is known
to affect the ABA response of anion channels [34] and
because anion channels are documented key regulators of
ABA-induced stomatal closure [35]. Components functioning
downstream from ABI2 and its role in guard cell signaling are
not well established, so ABI2 is not included. The newly
isolated PP2C recessive mutants, AtP2C-HA [36] and AtPP2CA
[37], exhibit minor ABA hypersensitivity. However, their
Figure 2. Current Knowledge of Guard Cell ABA Signaling
The color of the nodes represents their function: enzymes are shown in red, signal transduction proteins are green, membrane transport–related nodes
are blue, and secondary messengers and small molecules are orange. Small black filled circles represent putative intermediary nodes mediating indirect
regulatory interactions. Arrowheads represent activation, and short perpendicular bars indicate inhibition. Light blue lines denote interactions derived
from species other than Arabidopsis; dashed light-blue lines denote inferred negative feedback loops on pHc and S1P. Nodes involved in the same
metabolic pathway or protein complex are bordered by a gray box; only those arrows that point into or out of the box signify information flow (signal
transduction).
The full names of network components corresponding to each abbreviated node label are: ABA, abscisic acid; ABI1/2, protein phosphatase 2C ABI1/2;
ABH1, mRNA cap binding protein; Actin, actin cytoskeleton reorganization; ADPRc, ADP ribose cyclase; AGB1, heterotrimeric G protein b component;
AnionEM, anion efflux at the plasma membrane; Arg, arginine; AtPP2C, protein phosphatase 2C; Atrboh, NADPH oxidase; CaIM, Ca2þ influx across the
plasma membrane; Ca2þ ATPase, Ca2þ ATPases and Ca2þ/Hþ antiporters responsible for Ca2þ efflux from the cytosol; Ca2þ
c , cytosolic Ca2þ increase;
cADPR, cyclic ADP-ribose; cGMP, cyclic GMP; CIS, Ca2þ influx to the cytosol from intracellular stores; DAG, diacylglycerol; Depolar, plasma membrane
depolarization; ERA1, farnesyl transferase ERA1; GC, guanyl cyclase; GCR1, putative G protein–coupled receptor; GPA1, heterotrimeric G protein a
subunit; GTP, guanosine 59-triphosphate; Hþ ATPase, Hþ ATPase at the plasma membrane; InsPK, inositol polyphosphate kinase; InsP3, inositol-1,4,5-
trisphosphate; InsP6, inositol hexakisphosphate; KAP, Kþ efflux through rapidly activating Kþ channels (AP channels) at the plasma membrane; KEV, Kþ
efflux from the vacuole to the cytosol; KOUT, Kþ efflux through slowly activating outwardly-rectifying Kþ channels at the plasma membrane; NADþ,
nicotinamide adenine dinucleotide; NADPH, nicotinamide adenine dinucleotide phosphate; NOS, Nitric oxide synthase; NIA12, Nitrate reductase; NO,
Nitric oxide; OST1, protein kinase open stomata 1; PA, phosphatidic acid; PC, phosphatidyl choline; PEPC, phosphoenolpyruvate carboxylase; PIP2,
phosphatidylinositol 4,5-bisphosphate; PLC, phospholipase C; PLD, phospholipase D; RAC1, small GTPase RAC1; RCN1, protein phosphatase 2A; ROP2,
small GTPase ROP2; ROP10, small GTPase ROP10; ROS, reactive oxygen species; SphK, sphingosine kinase; S1P, sphingosine-1-phosphate.
DOI: 10.1371/journal.pbio.0040312.g002
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Model of Guard Cell ABA Signaling
downstream targets remain elusive; thus, we incorporate
them as a general inhibitor of closure denoted AtPP2C.
Mutation of the gene encoding the mRNA cap-binding
protein, ABH1, results in hypersensitivity of ABA-induced
Ca2þ
c elevation/oscillation and of anion efflux in plants grown
under some environmental conditions [38,39]. We assume an
inhibitory effect of ABH1 on Ca2þ influx across the plasma
membrane (CaIM), which can explain both of these effects
due to the Ca2þ regulation of anion efflux. Since the abh1
mutation affects transcript levels of some genes involved in
ABA response, this mutation may also affect ABA sensitivity
by altering gene expression rather than by regulation of the
rapid signaling events on which our network focuses.
Mutations in the gene encoding the farnesyl transferase
ERA1 or the gene encoding GCR1 also lead to hypersensitive
ABA-induced closure; ERA1 has been shown to negatively
regulate CaIM and anion efflux [40,41], whereas GCR1 has
been shown to be interact with GPA1 [11]. We assume that
ERA1 negatively regulates CaIM and GCR1 negatively
regulates GPA1.
Another assumption in the network is that the protein
phosphatase RCN1/PP2A regulates nitrate reductase (NIA12)
activity as observed in spinach leaf tissue; this is expected to
be a well-conserved mechanism due to the high sequence
conservation of NIA-PP2A regulatory domains [42]. Figure 2
contains two putative autoregulatory negative feedback loops
acting on S1P and pHc, respectively. The existence of
feedback regulation can be inferred from the published
timecourse measurements of S1P [43] and pHc [21]—both
indicating a fast increase in response to ABA, then a
decrease—but the mediators are currently unknown. The
assembled network is consistent with our biological knowl-
edge with minimal additional assumptions, and it will serve as
the starting point for the graph analysis and dynamic
modeling described in the following sections.
Modeling ABA Signal Transduction
Signaling networks can be represented as directed graphs
where the orientation of the edges reflects the direction of
information propagation (signal transduction). In a signal
transduction network there exists a clear starting point, the
node representing the signal (here, ABA), and one can follow
the paths (successions of edges) from that starting point to
the node(s) representing the output(s) of the network (here,
stomatal closure). The signal–output paths correspond to the
propagation of reactions in chemical space, and can be
thought of as pseudodynamics [44]. When only static
information is available, pseudodynamics takes into account
the graph theoretical properties of the signal transduction
network. For example, one can measure the number of nodes
or distinct network motifs that appear one, two,. . .n edges
away from the signal node. Such motifs reflect different
cellular signaling processing capabilities and provide impor-
tant insights into the biological processes under investigation.
Graph theoretical measures can also provide information
about the importance (centrality) of signal mediators [45] and
can predict the changes in path structure when nodes or
edges in the network are disrupted. These disruptions,
explored experimentally by genetic mutations, voltage-
clamping, or pharmacological interventions, can be modeled
in silico by removing the perturbed node and all its edges
from the graph [46]. The absence of nodes and edges will
disrupt the paths in the network, causing a possible increase
in the length of the shortest path between signal (ABA) and
output (closure), suggesting decreased ABA sensitivity, or in
severe cases the loss of all paths connecting input and output
(i.e., ABA insensitivity).
We find that there are several partially or completely
independent (nonoverlapping) paths between ABA and
closure. The path of pH-induced anion efflux is independent
of the paths involving changes in Ca2þ
c. Based on the current
knowledge incorporated in Figure 2, the path mediated by
malate breakdown is independent of both Ca2þ and pH
signaling. This result could change if evidence of a suggested
link between pH and malate regulation [47] is found; note
that regulation of malate synthesis in guard cells appears to
have cell-specific aspects [48]. Increase in Ca2þ
c can be
induced by several independent paths involving ROS, NO,
or InsP6. Thanks to the existence of numerous redundant
signal (ABA)–output (closure) paths, a complete disconnec-
tion of signal from output (loss of all the paths) is possible
only if four nodes, corresponding to actin reorganization,
pHc increase, malate breakdown, and membrane depolariza-
tion, are simultaneously disrupted. This indicates a remark-
able topological resilience, and suggests that functionally
redundant mechanisms can compensate for single gene
disruptions and can maintain at least partial ABA sensitivity.
However, path analysis alone cannot capture bidirectional
signal propagation and synergy (cooperativity) in living
biological systems. For example, two nonoverlapping paths
that reach the node closure could be functionally synergistic.
Using only path analysis, disruption of either path would not
be predicted to lead to a disconnection of the signal (ABA)
from the output (closure), but due to the synergy between the
two paths, the closure response may be strongly impaired if
either of the two paths is disrupted experimentally. Because
of such limitations of path analysis, we turn from path
analysis to a dynamic description.
Dynamic models have as input information (1) the
interactions and regulatory relationships between compo-
nents (i.e., the interaction network); (2) how the strength of
the interactions depends on the state of the interacting
components (i.e., the transfer functions); and (3) the initial
state of each component in the system. Given these, the
model will output the time evolution of the state of the
system (e.g., the system’s response to the presence or absence
of a given signal). Given the incomplete characterization of
the processes involved in ABA-induced stomatal closure (as is
typical of the current state of knowledge of cell signaling
cascades), we employ a qualitative modeling approach. We
assume that the state of the network nodes can have two
qualitative values: 0 (inactive/off) and 1 (active/on) [49]. These
values can also describe two conformational states of a
protein, such as closed and open states of an ion channel, or
basal and high activity for enzymes. This assumption is
necessary due to the absence of quantitative concentration or
activity information for the vast majority of the network
components. It is additionally justified by the fact that in the
case of combinatorial regulation or cooperative binding, the
input–output relationships are sigmoidal and thus can be
distilled into two discrete output states [50].
Since ‘‘stomatal closure’’ does not usually entail the
complete closure of the stomatal pore but rather a clear
decrease in the stomatal aperture, and since there is a
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Model of Guard Cell ABA Signaling
significant variability in the response of individual stomata,
the threshold separating the off (0) and on (1) state of the
node ‘‘Closure’’ needs to invoke a population level descrip-
tion. We measured the stomatal aperture size distribution in
the absence of ABA or after treatment with 50 lM ABA (see
Materials and Methods). Our first observation was the
population-level heterogeneity of stomatal apertures even
in their resting condition (Figure 3A), a fact that may not be
widely appreciated when more standard presentations, such
as mean 6 standard error, are used (see Figure 3B). The
stomatal aperture distribution shifts towards smaller aper-
tures after ABA treatment, and also broadens considerably.
The latter result is inconsistent with the assumption of each
stomate changing its aperture according to a common
function that decreases with increasing ABA concentration,
and suggests considerable cell-to-cell variation in the degree
of response to ABA. Moreover, although there is a clear
difference between the most probable ‘‘open’’ (0 ABA) and
‘‘closed’’ (þ ABA) aperture sizes, there also exists an overlap
between the aperture size distribution of ‘‘open’’ and
‘‘closed’’ stomata. This result indicates the possibility of
differential and cell-autonomous stomatal responses to ABA.
In the absence of 6 ABA measurements on the same stomate,
we define the threshold of closure as a statistically significant
shift of the stomatal aperture distribution towards smaller
apertures in response to ABA signal transduction.
In our model the dynamics of state changes are governed
by logical (Boolean) rules giving the state transition of each
node given the state of its regulators (upstream nodes). We
determine the Boolean transfer function for each node based
on experimental evidence. The state of a node regulated by a
single upstream component will follow the state of its
regulator with a delay. If two or more pathways can
independently lead to a node’s activation, we combine them
with a logical ‘‘or’’ function. If two pathways cannot work
independently, we model their synergy as a logical ‘‘and’’
function. For nodes regulated by inhibitors we assume that
the necessary condition of their activation (state 1) is that the
inhibitor is inactive (state 0). As all putative intermediary
nodes of Figure 2 are regulated by a single activator, and
regulate a single downstream component, they only affect the
time delays between known nodes; for this reason we do not
explicitly incorporate intermediary nodes as components of
the dynamic model. Table 1 lists the regulatory rules of
known nodes of Figure 2; we give a detailed justification of
each rule in Text S1.
Frequently in Boolean models time is quantized into
regular intervals (timesteps), assuming that the duration of
all activation and decay processes is comparable [51]. For
generality we do not make this assumption, and in the
absence of timing or duration information we follow an
asynchronous method that allows for significant stochasticity
in process durations [52,53]. Choosing as a timestep the
longest duration required for a node to respond to a change
in the state of its regulator(s) (also called a round of update, as
each component’s state will be updated during this time
interval), the Boolean updating rules of an asynchronous
algorithm can be written as:
Sn
i ¼ BiðSmj
j ; Smk
k ; Sml
l ; ::Þ;
ð1Þ
where Si
n is the state of component i at timestep n, Bi is the
Boolean function associated with the node i and its regulators
j,k,l,.. and mj; mk; ml; :: 2 fn 1; ng, signifying that the time-
points corresponding to the last change in a input node’s
state can be in either the previous or current round of
updates.
Figure 3. Stomatal Aperture Distributions without ABA Treatment (gray
bars) and with 50 lM ABA (white bars)
(A) The x axis gives the stomatal aperture size and the y axis indicates the
fraction of stomata for which that aperture size was observed. The black
columns indicate the overlap between the 0 lM ABA and the 50 lM ABA
distributions.
(B) Classical bar plot representation of stomatal aperture for treatment
with 50 lM ABA (white bar, labeled 1) and without ABA treatment (gray
bar, labeled 2) using mean 6 standard error. This representation
provides minimal information on population structure.
DOI: 10.1371/journal.pbio.0040312.g003
Table 1. Boolean Rules Governing the States of the Known
(Named) Nodes in the Signal Transduction Network
Node
Boolean Regulatory Rule
NO
NO* ¼ NIA12 and NOS
PLC
PLC* ¼ ABA and Ca2þ
c
CaIM
CaIM* ¼ (ROS or not ERA1 or not ABH1) and not Depolar
GPA1
GPA1* ¼ (S1P or not GCR1) and AGB1
Atrboh
Atrboh* ¼ pHc and OST1 and ROP2 and not ABI1
Hþ ATPase Hþ ATPase* ¼ not ROS and not pHc and not Ca2þ
c
Malate
Malate* ¼ PEPC and not ABA and not AnionEM
RAC1
RAC1* ¼ not ABA and not ABI1
Actin
Actin* ¼ Ca2þ
c or not RAC1
ROS
ROS* ¼ ABA and PA and pHc
ABI1
ABI1* ¼ pHc and not PA and not ROS
KAP
KAP*¼ (not pHc or not Ca2þ
c) and Depolar
Ca2þ
c
Ca2þ
c*¼ (CaIM or CIS) and not Ca2þ ATPase
CIS
CIS* ¼ (cGMP and cADPR) or (InsP3 and InsP6)
AnionEM
AnionEM* ¼ ((Ca2þ
c or pHc) and not ABI1 ) or (Ca2þ
c and pHc)
KOUT
KOUT* ¼ (pHc or not ROS or not NO) and Depolar
Depolar
Depolar* ¼ KEV or AnionEM or not Hþ ATPase or not KOUT or Ca2þ
c
Closure
Closure* ¼ (KOUT or KAP ) and AnionEM and Actin and not Malate
The nomenclature of the nodes is given in the caption of Figure 2. The nodes that have
only one input are not listed to save space; a full description and justification can be
found in Text S1. The next state of the node on the left-hand side of the equation (marked
by *) is determined by the states of its effector nodes according to the function on the
right-hand side of the equation.
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Model of Guard Cell ABA Signaling
The relative timing of each process is chosen randomly and
is changed after each update round such that we are sampling
equally among all possibilities (see Materials and Methods).
This approach reflects the lack of experimental data on
relative reaction speeds. The internal states of signaling
proteins and the concentrations of small molecules are not
explicitly known for each stomate, and components such as
Ca2þ
c and cell membrane potential show various states even
in a homogenous experimental setup [54,55]. Accordingly, we
sample a large number (10,000) of randomly selected initial
states for the nodes other than ABA and closure (closure is
initially set to 0), and let the system evolve either with ABA
always on (1) or ABA always off (0). We quantify the
probability of closure (equivalent to the percentage of closed
stomata in the population) by the formula
PðclosureÞt ¼
X
N
j¼l
St
closureðjÞ=N
ð2Þ
where St
closure(j) is the state of the node ‘‘Closure’’ at time t in
the jth simulation and N is the total number of simulations, in
our case 10,000. We illustrate the main steps of our
simulation method in Figure 4.
As shown in Figure 5, in eight steps, the system shows
complete closure in response to ABA. In contrast, without
ABA, although some initial states lead to closure at the
beginning, within six steps the probability of closure
approaches 0. Initial theoretical analysis of the attractors
(stable behaviors) of this nonlinear dynamic system confirms
that when given a constant ABA ¼ 1 input, the majority of
nodes will approach a steady state value within three to eight
steps. This steady-state value does not depend on the initial
conditions. For example, OST1, PLC, and InsPK stabilize in
the on state, and PEPC settles into the off state within the first
timestep when ABA is consistently on. The exception is a set
of 12 nodes, including Ca2þ
c, Ca2þ ATPase, NO, Kþ efflux
from the vacuole to the cytosol, and Kþ efflux through rapidly
Figure 4. Schematic Illustration of Our Modeling Methodology and of the Probability of Closure
In this four-node network example, node A is the input (as ABA is the input of the ABA signal transduction network), and node D is the output
(corresponding to the node ‘‘Closure’’ in the ABA signal transduction network). The nodes’ states are indicated by the shading of their symbols: open
symbols represent the off (0) state and filled symbols signify the on (1) state. To indicate the connection between this example and ABA-induced
closure, we associate D ¼ off (0) with a picture of an open stomate, and D ¼ on (1) with a picture of a closed stomate. The Boolean transfer functions of
this network are A* ¼ 1, B* ¼ A, C* ¼ A, D* ¼ B and C (i.e., node A is on commencing immediately after the initial condition, the next states of nodes B
and C are determined by A, and D is on only when both B and C are on).
(A) The first column represents the networks’ initial states; the input and output are not on, but some of the components in the network are randomly
activated (e.g., middle row, node B). The input node A turns on right after initialization, signifying the initiation of the ABA signal. The next three
columns in (A) represent the network’s intermediary states during a sequential update of the nodes B, C, and D, where the updated node is given as a
gray label above the gray arrow corresponding to the state transition. This sequence of three transitions represents a round of updates from timestep 1
(second column) to timestep 2 (last column). Out of a total of 22 3 3! ¼ 24 possible different normal responses, two sketches of normal responses are
shown in the top two rows. The bottom row illustrates a case in which one node (shown as a square) is disrupted (knocked out) and cannot be
regulated or regulate downstream nodes (indicated as dashed edges).
(B) The probability of closure indicates the fraction of simulations where the output D ¼ 1 is reached in each timestep; thus, in this illustration the
probability of closure for the normal response (circles) increases from 0% at time step 1 to 100% at timestep 2. The knockout mutant’s probability of
closure (squares) is 0% at both time steps.
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Model of Guard Cell ABA Signaling
activating Kþ channels (AP channels) at the plasma membrane
(KAP), whose attractors are limit cycles (oscillations) accord-
ing to the model. Ca2þ
c oscillations have indeed been
observed experimentally [56,57]; no time course measure-
ments have been reported in the literature for the other
components, so it is unknown whether they oscillate or not.
We identified four subsets of behaviors for these nodes—
distinguished by different positions on the limit cycle—
depending on the initial conditions and relative process
durations. Due to the functional redundancy between Kþ
efflux mechanisms driving stomatal closure (see last entry of
Table 1), and the stabilization of the other regulators of the
node ‘‘Closure,’’ a closed steady state (Closure ¼ 1) is attained
within eight steps for any initial condition. The details of this
analysis will be published elsewhere.
Identification of Essential Components
After testing the wild-type (intact) system, we investigate
whether the disruption (loss) of a component changes the
system’s response to ABA. We systematically perturb the
system by setting the state of a node to 0 (off state), and
holding it at 0 for the duration of the simulation. This
perturbation mimics the effect of a knockout mutation for a
gene or pharmaceutical inhibition of secondary messenger
production or of kinase or phosphatase activity. We
characterize the effect of the node disruption by calculating
the percentage (probability) of closure response to a constant
ABA signal at each time step and comparing it with the
percentage of closure in the wild-type system.
The perturbed system’s responses can be classified into five
categories with respect to the system’s steady state and the
time it takes to reach the steady state. We designate responses
identical or very close to the wild-type response as having
normal sensitivity; in these cases the probability of closure
reaches 100% within eight timesteps. Disruptions that cause
the percentage of closed stomata to decrease to zero after the
first few steps are denoted as conferring ABA insensitivity (in
accord with experimental nomenclature). We observe re-
sponses where the probability of closure (the percentage of
stomata closed at any given timestep) settles at a nonzero
value that is less than 100%; we classify these responses as
having reduced sensitivity. Finally, in two classes of behavior
the probability of closure ultimately reaches 100%, but with a
different timing than the normal response. We refer to a
response with ABA-induced closure that is slower than wild-
type as hyposensitivity, while hypersensitivity corresponds to
ABA-induced closure that is faster than wild-type. Therefore,
the perturbed system’s responses can be classified into five
categories in the order of decreasing sensitivity defect:
insensitivity to ABA, reduced sensitivity, hyposensitivity,
normal sensitivity, and hypersensitivity.
We find that 25 single node disruptions (65%; compare
with Table 2) do not lead to qualitative effects: 100% of the
population responds to ABA with timecourses very close to
the wild-type response. In contrast, the loss of membrane
depolarizability, the disruption of anion efflux, and the loss of
actin cytoskeleton reorganization present clear vulnerabil-
ities: irrespective of initial conditions or of relative timing, all
simulated stomata become insensitive to ABA (Figure 5A).
Indeed, membrane depolarization is a necessary condition of
Kþ efflux, which is a necessary condition of closure, as is actin
cytoskeleton reorganization and anion efflux. The individual
disruption of seven other components—PLD, PA, SphK, S1P,
GPA1, Kþ efflux through slowly activating Kþ channels at the
plasma membrane (KOUT), and pHc increase —reduces ABA
sensitivity, as the percentage of closed stomata in the
population decreases to 20%—80% (see Figure 5B). At least
five components (S1P, SphK, PLD, PA, pHc) of these 7
predicted components have been shown to impair ABA-
Figure 5. The Probability of ABA-Induced Closure (i.e., the Percentage of
Simulations that Attain Closure) as a Function of Timesteps in the
Dynamic Model
In all panels, black triangles with dashed lines represent the normal (wild-
type) response to ABA stimulus. Open triangles with dashed lines show
that in wild-type, the probability of closure decays in the absence of ABA.
(A) Perturbations in depolarization (open diamonds) or anion efflux at
the plasma membrane (open squares) cause total loss of ABA-induced
closure. The effect of disrupting actin reorganization (not shown) is
identical to the effect of blocking anion efflux.
(B) Perturbations in S1P (dashed squares), PA (dashed circles), or pHc
(dashed diamonds) lead to reduced closure probability. The effect of
disrupting SphK is nearly identical to the effect of disrupting S1P (dashed
squares); perturbations in GPA1 and PLD, KOUT are very close to
perturbations in PA (dashed circles); for clarity, these curves are not
shown in the plot.
(C) abi1 recessive mutants (black squares) show faster than wild-type
ABA-induced closure (ABA hypersensitivity). The effect of blocking Ca2þ
ATPase(s) (not shown) is very similar to the effect of the abi1 mutation.
Blocking Ca2þ
c increase (black diamonds) causes slower than wild-type
ABA-induced closure (ABA hyposensitivity). The effect of disrupting
atrboh or ROS production (not shown) is very similar to the effect of
blocking Ca2þ
c increase.
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Model of Guard Cell ABA Signaling
induced closure when clamped or mutated experimentally
[8,31,43,58]. For these disruptions, both theoretical analysis
and numerical results indicate that all simulated stomata
converge to limit cycles (oscillations) driven by the Ca2þ
c
oscillations, yet the ratio of open and closed stomata in the
population is the same at any timepoint, leading to a constant
probability of closure. (The alternative possibility, of a subset
of stomata being stably closed and another subset stably open,
was not observed for any disruption.)
For all other single-node disruptions the probability of
closure ultimately reaches 100% (i.e., all simulated stomata
reach the closed steady state); however, the rate of con-
vergence diverges from the rate of the wild-type response (see
Figure 5C). Disruption of Ca2þ
c increase or of the production
of ROS leads to ABA hyposensitivity (slower than wild-type
response). In contrast, the disruption of ABI1 or of the Ca2þ
ATPase(s) leads to ABA hypersensitivity (faster than wild
type-response) (Figure 5C). The hyposensitive and hyper-
sensitive responses are statistically distinguishable (p , 0.05
for all intermediary time steps [i.e., for 0 , t , 8]) from the
normal responses. Our model predicts that perturbation of
OST1 leads to a slower than normal response that is
nevertheless not slow enough to be classified as hyposensitive.
Indeed, ost1 mutants are still responsive to ABA even though
not as strongly as wild-type plants [12].
After analyzing all single knockout simulations, we turned
to analysis of double and triple knockout simulations. First, to
effectively distinguish between normal, hypo- and hyper-
sensitive responses (all of which achieve 100% probability of
closure, but at different rates), we calculated the cumulative
percentage of closure (CPC) by adding the probability of
closure over 12 steps; the smaller the CPC value, the more
slowly the probability of closure reaches 100%, and vice
versa. Plotting the histogram of CPC values reveals a clear
separation into three distinct groups of response in the case
of single disruptions (Figure 6A). In contrast, the cumulative
effects of multiple perturbations lead to a continuous
distribution of sensitivities in a broad range around the
normal (Figure 6B and 6C). We use the single perturbation
results to identify three classes of response that achieve 100%
closure, but at varying rates. We define two CPC thresholds:
the midpoint between the most hyposensitive single mutant
and normal response, CPChypo ¼ 10.35; and the midpoint
between the normal and least hypersensitive single mutant
response, CPChyper ¼ 10.7. Disruptions with cumulative
closure probability , CPChypo are classified as hyposensitive,
disruptions with cumulative closure probability . CPChyper
are hypersensitive; and values between the two thresholds are
classified as normal responses. This hypo/hypersensitive
classification does not affect the determination of insensitive
or reduced sensitivity responses, which are identified by
observing a null or less than 100% probability of closure.
For double (triple) knockout simulations, some combina-
tions of perturbations exhibit sensitivities that are independ-
ent of the sensitivity of each of their components’
perturbation. Normal ABA-induced stomatal closure is
Table 2. Single to Triple Node Disruptions in the Dynamic Model
Number of
Nodes Disrupted
Percentage with
Normal Sensitivity
Percentage
Causing Insensitivity
Percentage Causing
Reduced Sensitivity
Percentage Causing
Hyposensitivity
Percentage Causing
Hypersensitivity
1
65%
7.5%
17.5%
5%
5%
2
38%
16%
27%
12%
6%
3
23%
25%
31%
13%
7%
In all the perturbations, there are five groups of responses. Normal sensitivity refers to a response close to the wild-type response (shown as black triangles and dashed line in Figure 5).
Insensitivity means that the probability of closure is zero after the first three steps (see Figure 5A). Reduced sensitivity means that the probability of closure is less than 100% (see dashed
symbols in Figure 5B). Hyposensitivity corresponds to ABA-induced closure that is slower than wild-type (black diamonds in Figure 5C). Hypersensitivity corresponds to ABA-induced
closure that is faster than wild-type (black squares in Figure 5C).
DOI: 10.1371/journal.pbio.0040312.t002
Figure 6. Classification of Close-to-Normal Responses
(A) For all the single mutants that ultimately reach 100% closure, we plot
the histogram of the cumulative probability of closure (CPC). We find
three distinct types of responses: hypersensitivity (CPC . 10.7, for abi1
and Ca2þ ATPase disruption); hyposensitivity (CPC , 10.35, for Ca2þ
c ,
atrboh, and ROS disruption); and normal responses ( 10.35 , CPC ,
10.7). For all the double (B) and triple (C) mutants that eventually reach
100% closure at steady state when ABA ¼ 1, we classify the responses
using the CPC thresholds defined by the single mutant responses. The
CPC threshold values are indicated by dashed vertical lines in the plot.
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Model of Guard Cell ABA Signaling
preserved in 38% (23%) of combinations (see Table 2). In
contrast, ABA signaling is completely blocked in 16% (25%)
of disruptions. In addition to perturbations involving the
three previously found insensitivity-causing single knockouts
(loss of membrane depolarizability, the disruption of anion
efflux, and the loss of actin cytoskeleton reorganization), a
large number of novel combinations are found. Interestingly,
perturbations of Ca2þ
c or Ca2þ release from stores, when
combined with disruptions in PLD, PA, GPA1, or pHc, lead to
insensitivity (see Figure 7 and Discussion). ABA-induced
closure is reduced (but not lost entirely) in 27% (31%) of the
cases. Hyposensitive responses are found for 12% (13%) of
double (triple) perturbations. All of the double perturbations
in this category involve a knockout mutation of Ca2þ
c,
Atrboh, or ROS. The triple perturbations involve a knockout
mutation of Ca2þ
c, Atrboh, or ROS, plus two other perturba-
tions, or combinations of three disruptions that alone are not
predicted to cause quantifiable effects (e.g., guanyl cyclase,
Ca2þ release from internal stores [CIS], and CaIM; see Figure
7). Around 6% (7%) of double (triple) perturbations, all
including a knockout mutation of ABI1 or Ca2þ ATPase, lead
to a hypersensitive response. In summary, accumulating
perturbations cause a dramatic decrease in the percentage
of normal response; the majority of triple knockouts are
either insensitive or have reduced sensitivity. The fraction of
hyposensitive and hypersensitive knockouts increases only
moderately.
Experimental Assessment of Model Predictions
As a first step toward experimental assessment of the
model’s predictions, we used a weak acid, Na-butyrate, to
clamp cytosolic pH, and then we treated the stomata with 50
lM ABA and observed the stomatal aperture responses. As
shown in Figure 8A, the stomatal aperture distributions
without butyrate treatments shift towards smaller apertures
after ABA treatment, forming a distribution that overlaps
with, but is clearly distinguishable from, the 0 ABA
distribution. However, when increasing concentrations of
butyrate are added in the solution, the ‘‘open’’ (0 ABA) and
‘‘closed’’ (þ ABA) distributions become increasingly over-
lapping (Figure 8B–8D). At the highest butyrate concentra-
tion (5 mM; Figure 8D), the 0 ABA and þABA populations of
stomatal apertures are statistically identical (the null hypoth-
esis that the two distributions are the same cannot be
Figure 7. Summary of the Dynamic Effects of Calcium Disruptions
All curves represent the probability of ABA-induced closure (i.e., the
percentage of simulations that attain closure) as a function of time steps.
Black triangles with dashed line represent the normal (wild-type)
response to ABA stimulus; open triangles with dashed lines show how
the probability of closure decays in the absence of ABA. CIS þ PA double
mutants (dashed circles) and Ca2þ
c þ pHc double mutants (dashed
diamonds) show insensitivity to ABA. Ca2þ ATPase þ RCN1 double
mutants (black circles) show hyposensitive (delayed) response to ABA.
Guanyl cyclase þ CIS þ CaIM triple mutants (black diamonds) also show
hyposensitivity; note that none of the guanyl cyclase or CIS or CaIM
single knockouts show changed sensitivity (data not shown). Ca2þ
ATPase mutants (black squares) show faster than wild-type ABA-induced
closure (ABA hypersensitivity).
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Figure 8. Effect of Cytosolic pH Clamp (Increasing Concentrations of Na-
butyrate from 0 to 5 mM) on ABA-Induced Stomatal Closure
The histograms show the distribution of stomatal apertures without ABA
treatment (gray bars) and with 50 lM ABA (white bars). Throughout, the
x-axis gives the stomatal aperture size and the y-axis indicates the
fraction of stomata for which that aperture size was observed. The black
columns indicate the overlap between the 0 lM ABA and the 50 lM ABA
distributions. Note that the data of (A) and those of Figure 3A are
identical; these data are reproduced here for ease of comparison with
panels (B–D).
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Model of Guard Cell ABA Signaling
rejected; two-tailed t test, p . 0.05). These results qualitatively
support our prediction of the importance of pHc signaling.
For a more quantitative comparison with the theoretically
predicted probability of closure corresponding to pH
clamping, one can define a threshold C between open and
closed stomatal states, such that stomata with apertures larger
than C can be classified as open and stomata with lower
apertures can be classified as closed. We identify the thresh-
old value C ¼ 4.3 lm by simultaneously minimizing the
fraction of stomata classified as closed in the control
condition and maximizing this fraction in the ABA treated
condition. Using this threshold we find that the fraction of
closed stomata in the 50 lM ABA þ 5 mM Na-butyrate
population is 26%, in agreement with the theoretically
predicted probability of closure (Figure 5B).
In plant systems, cytosolic pH changes in response to
multiple hormones such as ABA [20,59], jasmonates [21],
auxin [59], etc. The downstream effectors of pH changes
include ion channels [8], protein kinases [60], and protein
phosphatases [30]. Previous experiments with guard cells have
demonstrated the efficacy of butyrate in imposing a cytosolic
pH clamp [8,21]. While these prior experiments focused on a
single concentration of butyrate, here we used five different
concentrations (three shown), with 120 stomata sampled for
each treatment. As seen in Figure 8, we were able to monitor
the effect of butyrate in the þABA treatment in both
increasing the mean aperture size and reducing the spread
of the aperture sizes. There is a clear indication of saturation
between the two highest butyrate concentrations. While
detailed measurements of cytosolic pH constitute a full
separate study beyond the scope of the present article, the
results of Figure 8 support the suggestion from our model
that pHc should receive increased attention by experimen-
talists as a focal point for transduction of the ABA signal.
Discussion
Network Synthesis and Path Analysis
Logical organization of large-scale data sets is an important
challenge in systems biology; our model provides such
organization for one guard cell signaling system. As summar-
ized in Table S1, we have organized and formalized the large
amount of information that has been gathered on ABA
induction of stomatal closure from individual experiments.
This information has been used to reconstruct the ABA
signaling network (Figure 2). Figure 2 uses different types of
edges (lines) to depict activation and inhibition, and also uses
different edge colors to indicate whether the information was
derived from our model species, Arabidopsis, or from another
plant species. Different types of nodes (metabolic enzymes,
signaling proteins, transporters, and small molecules) are also
color coded. An advantage of our method of network
construction over other methods such as those used in
Science’s Signal Transduction Knowledge Environment
(STKE) connection maps [61] is the inclusion of intermediate
nodes when direct physical interactions between two compo-
nents have not been demonstrated.
As is evident from Figure 2, network synthesis organizes
complex information sets in a form such that the collective
components and their relationships are readily accessible.
From such analysis, new relationships are implied and new
predictions can be made that would be difficult to derive
from less formal analysis. For example, building the network
allows one to ‘‘see’’ inferred edges that are not evident from
the disparate literature reports. One example is the path
from S1P to ABI1 through PLD. Separate literature reports
indicate that PLDa null mutants show increased transpira-
tion, that PLDa1 physically interacts with GPA1, that S1P
promotion of stomatal closure is reduced in gpa1 mutants,
that PLD catalyses the production of PA, and that recessive
abi1 mutants are hypersensitive to ABA. Network inference
allows one to represent all this information as the S1P !
GPA1 ! PLD ! PA—j ABI1—j closure path, and make the
prediction that ABA inhibition of ABI1 phosphatase activity
will be impaired in sphingosine kinase mutants unable to
produce S1P.
Another prediction that can be derived from our network
analysis is a remarkable redundancy of ABA signaling, as
there are eight paths that emanate from ABA in Figure 2 and,
based on current knowledge (though see below) these paths
are initially independent. The prediction of redundancy is
consistent with previous, less formal analyses [62]. The
integrated guard cell signal transduction network (which
includes the ABA signal transduction network) has been
proposed as an example of a robust scale-free network [62].
To classify a network as scale-free, one needs to determine
the degree (the number of edges, representing interactions/
regulatory relationships) of each node, and to calculate the
distribution of node degrees (denoted degree distribution)
[45,46]. Scale-free networks, characterized by a degree
distribution described by a power law, retain their connec-
tivity in the face of random node disruptions, but break down
when the highest-degree nodes (the so-called hubs) are lost
[46]. While the guard cell network may ultimately prove to be
scale-free, the network is not sufficiently large at present to
verify the existence of a power-law degree distribution; thus,
the analogy with scale-free networks cannot be rigorously
satisfied.
Dynamic Modeling
Our model differs from previous models employed in the
life sciences in the following fundamental aspects. First, we
have reconstructed the signaling network from inferred
indirect relationships and pathways as opposed to direct
interactions; in graph theoretical terminology, we found the
minimal network consistent with a set of reachability
relationships. This network predicts the existence of numer-
ous additional signal mediators (intermediary nodes), all of
which could be targets of regulation. Second, the network
obtained is significantly more complex than those usually
modeled in a dynamic fashion. We bridge the incompleteness
of regulatory knowledge and the absence of quantitative
dose-response relationships for the vast majority of the
interactions in the network by employing qualitative and
stochastic dynamic modeling previously applied only in the
context of gene regulatory networks [53].
Mathematical models of stomatal behavior in response to
environmental change have been studied for decades [63,64].
However, no mathematical model has been formulated that
integrates the multitude of recent experimental findings
concerning the molecular signaling network of guard cells.
Boolean modeling has been used to describe aspects of plant
development such as specification of floral organs [65], and
there are a handful of reports describing Boolean models of
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Model of Guard Cell ABA Signaling
light and pathogen-, and light by carbon-regulated gene
expression [66–68]. Use of a qualitative modeling framework
for signaling networks is justified by the observation that
signaling networks maintain their function even when faced
with fluctuations in components and reaction rates [69]. Our
model uses experimental evidence concerning the effects of
gene knockouts and pharmacological interventions for
inferring the downstream targets of the corresponding gene
products and the sign of the regulatory effect on these
targets. However, use of this information does not guarantee
that the dynamic model will reproduce the dynamic outcome
of the knockout or intervention. Indeed, all model ingre-
dients (node states, transfer functions) refer to the node
(component) level, and there is no explicit control over
pathway-level effects. Moreover, the combinatorial transfer
functions we employed are, to varying extents, conjectures,
informed by the best available experimental information (see
Text S1). Finally, in the absence of detailed knowledge of the
timing of each process and of the baseline (resting) activity of
each component, we deliberately sample timescales and
initial conditions randomly. Thus, an agreement between
experimental and theoretical results of node disruptions is
not inherent, and would provide a validation of the model.
The accuracy of our model is indeed supported by its
congruency with experimental observation at multiple levels.
At the pathway level, our model captures, for example, the
inhibition of ABA-induced ROS production in both ost1
mutants and atrboh mutants [12,19,21] and the block of ABA-
induced stomatal closure in a dominant-positive atRAC1
mutant [22]. In our model, as in experiments, ABA-induced
NO production is abolished in either nos single or nia12
double mutants [13,18]. Moreover, the model reproduces the
outcome that ABA can induce cytosolic Kþ decrease by Kþ
efflux through the alternative potassium channel KAP, even
when ABA-induced NO production leads to the inhibition of
the outwardly-rectifying (KOUT) channel [70]. At the level of
whole stomatal physiology, our model captures the findings
that anion efflux [35,71] and actin cytoskeleton reorganiza-
tion [22] are essential to ABA-induced stomatal closure. The
importance of other components such as PA, PLD, S1P,
GPA1,
KOUT,
pH c
in
stomatal
closure
control
[8,20,31,43,58,72], and the ABA hypersensitivity conferred
by elimination of signaling through ABI1 [28,29], are also
reproduced. Our model is also consistent with the observa-
tion that transgenic plants with low PLC expression still
display ABA sensitivity [73].
The fact that our model accords well with experimental
results suggests that the inferences and assumptions made are
correct overall, and enables us to use the model to make
predictions about situations that have yet to be put to
experimental test. For example, the model predicts that
disruption of all Ca2þ ATPases will cause increased ABA
sensitivity, a phenomenon difficult to address experimentally
due to the large family of calcium ATPases expressed in
Arabidopsis guard cells (unpublished data). Most of the
multiple perturbation results presented in Figure 5 and
Table 2 also represent predictions, as very few of them have
been tested experimentally. Results from our model can now
be used by experimentalists to prioritize which of the
multitude of possible double and triple knockout combina-
tions should be studied first in wet bench experiments.
Most importantly, our model makes novel predictions
concerning the relative importance of certain regulatory
elements. We predict three essential components whose
elimination completely blocks ABA-induced stomatal closure:
membrane depolarization, anion efflux, and actin cytoskele-
ton reorganization. Seven components are predicted to
dramatically affect the extent and stability of ABA-induced
stomatal closure: pHc control, PLD, PA, SphK, S1P, G protein
signaling (GPA1), and Kþ efflux. Five additional components,
namely increase of cytosolic Ca2þ, Atrboh, ROS, the Ca2þ
ATPase(s), and ABI1, are predicted to affect the speed of
ABA-induced stomatal closure. Note that a change in
stomatal response rate may have significant repercussions,
as some stimuli to which guard cells respond fluctuate on the
order of seconds [74,75]. Thus our model predicts two
qualitatively different realizations of a partial response to
ABA: fluctuations in individual responses (leading to a
reduced steady-state sensitivity at the population level), and
delayed response. These predictions provide targets on which
further experimental analysis should focus.
Six of the 13 key positive regulators, namely increase of
cytosolic Ca2þ, depolarization, elevation of pHc, ROS, anion
efflux, and Kþ efflux through outwardly rectifying Kþ
channels, can be considered as network hubs [45], as they
are in the set of ten highest degree (most interactive) nodes.
Other nodes whose disruption leads to reduced ABA
sensitivity, namely SphK, S1P, GPA1, PLD, and PA, are part
of the ABA ! PA path. While they are not highly connected
themselves, their disruption leads to upregulation of the
inhibitor ABI1, thus decreasing the efficiency of ABA-
induced stomatal closure. Similarly, the node representing
actin reorganization has a low degree. Thus the intuitive
prediction, suggested by studies in yeast gene knockouts
[76,77], that there would be a consistent positive correlation
between a node’s degree and its dynamic importance, is not
supported here, providing another example of how dynamic
modeling can reveal insights difficult to achieve by less formal
methods. This lack of correlation has also been found in the
context of other complex networks [78].
Comparing Figure 3 and Figure 6C, one can notice a
similar heterogeneity in the measured stomatal aperture size
distributions and the theoretical distribution of the cumu-
lative probability of closure in the case of multiple node
disruptions. While apparently unconnected, there is a link
between the two types of heterogeneity. Due to stochastic
effects on gene and protein expression, it is possible that in a
real environment not all components of the ABA signal
transduction network are fully functional. Therefore, even
genetically identical populations of guard cells may be
heterogeneous at the regulatory and functional level, and
may respond to ABA in slightly different ways. In this case,
the heterogeneity in double and triple disruption simulations
provides an explanation for the observed heterogeneity in
the experimentally normal response: the latter is actually a
mixture of responses from genetically highly similar but
functionally nonidentical guard cells.
Importance of Ca2þ
c Oscillations to ABA-Induced
Stomatal Closure
Through the inclusion of the nodes CaIM, CIS, and the
Ca2þ ATPase node representing the Ca2þ ATPases and Ca2þ/
Hþ antiporters [79,80] that drive Ca2þ efflux from the
cytosolic compartment, our model incorporates the phenom-
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Model of Guard Cell ABA Signaling
enon of oscillations in cytosolic Ca2þ concentration, which
has been frequently observed in experimental studies
[56,81,82]. In experiments where Ca2þ
c is manipulated,
imposed Ca2þ
c oscillations with a long periodicity (e.g., 10
min of Ca2þ elevation with a periodicity of once every 20 min)
are effective in triggering and maintaining stomatal closure,
yet at 10 min (i.e., after just one Ca2þ
c transient elevation and
thus before the periodicity of the Ca2þ change can be
‘‘known’’ by the cell), significant stomatal closure has already
occurred [56]. This result suggests that the Ca2þ
c oscillation
signature may be more important for the maintenance of
closure than for the induction of closure [56,81], and that the
induction of closure might only be dependent on the first,
transient Ca2þ
c elevation.
According to our model, if Ca2þ
c elevation occurs, then
stomatal closure is triggered (consistent with numerous
experimental studies), but Ca2þ
c elevation is not required
for ABA-induced stomatal closure. Re-evaluation of the
experimental studies on ABA and Ca2þ
c reveals support for
this prediction. First, although Ca2þ elevation certainly can be
observed in guard cell responses to ABA, numerous exper-
imental results also show that Ca2þ
c elevation is only observed
in a fraction of the guard cells assayed [9,83]. Furthermore,
absence of Ca2þ
c elevation in response to ABA does not
prevent the occurrence of downstream events such as ion
channel regulation [84,85] and stomatal closure [86,87], a
phenomenon also predicted by our in silico analysis. Second,
it has been observed that some guard cells exhibit sponta-
neous oscillations in Ca2þ
c, and in such cells, ABA application
actually suppresses further Ca2þ
c elevation [88]; thus, ABA
and Ca2þ
c elevation are clearly decoupled.
Our model does predict that disruption of Ca2þ signaling
leads to ABA hyposensitivity, or a slower than normal
response to ABA. In the real-world environment, even a
slight delay or change in responsiveness may have significant
repercussions, as some stimuli to which guard cells respond
fluctuate on the order of seconds; and stomatal responses can
have comparable rapidity [74,75]. Moreover, our model
predicts that Ca2þ
c elevation (although not necessarily
oscillation) becomes required for engendering stomatal
closure when pHc changes, Kþ efflux or the S1P–PA pathway
are perturbed (see Figure 7). Thus, Ca2þ
c modulation confers
an essential redundancy to the network. Support for such a
redundant role can be found in a study by Webb et al. [89]
where Ca2þ concentration was reduced below normal resting
levels by intracellular application of BAPTA (such reduction
in baseline Ca2þ
c levels has been shown to reduce ABA
activation of anion channels [85]) and the epidermal tissue
was perfused with CO2-free air, a treatment that has been
shown to inhibit outwardly rectifying Kþ channels and slow
anion efflux channels [90]. The ABA insensitivity of stomatal
closure found by Webb et al. under these conditions [89]
therefore can be attributed to a combination of multiple
perturbations (of Ca2þ
c elevation, Kþ efflux, and anion efflux)
and is consistent with the predictions of our model.
Our model indicates that double perturbations of the Ca2þ
ATPase component and either of RCN1, OST1, NO, NOS,
NIA12, or Atrboh are hyposensitive (see Figure 7), consistent
with experimental results on disruptions in the latter
components [12,13,18,19,21,91]. Since the latter disruptions
alone, with unperturbed Ca2þ ATPase, are found to have a
close-to-normal response in our model, a Ca2þ ATPase–
disrupted and therefore Ca2þ
c oscillation–free model seems
to be closer to experimental observations on stomatal
aperture response recorded for these individual mutant
genotypes. This suggests that Ca2þ
c elevation (and not Ca2þ
c
oscillation) is the signal perceived by downstream factors that
control the induction of closure. Possibly, certain as-yet-
undiscovered interaction motifs, such as a synergistic feed-
forward loop [92] or dual positive feedback loops [93], could
transform the Ca2þ
c oscillation into a stable downstream
output.
Limitations of the Current Analysis
Network topology. Our graph reconstruction is incom-
plete, as new signaling molecules will certainly be discovered.
Novel nodes may give identity to the intermediary nodes that
our model currently incorporates. Discovery of a new
interaction among known nodes could simplify the graph
by reducing (apparent) redundancy. For example, if it is
found that GPA1 ! OST1, the simplest interpretation of the
ABA ! ROS pathway becomes ABA ! GPA1 ! OST1 !
ROS, and the graph loses one edge and an alternative
pathway. As an effect, the graph’s robustness will be
attenuated. Among likely candidates for network reduction
are the components currently situated immediately down-
stream of ABA because, in the absence of information about
guard cell ABA receptors [94], we assumed that ABA
independently regulates eight components. It is also possible
that a newly found interaction will not change the existing
edges, but only add a new edge. A newly added positive
regulation edge will further increase the redundancy of
signaling and correspondingly its robustness. Newly added
inhibitory edges could possibly damage the network’s robust-
ness if they affect the main positive regulators of the network,
especially anion channels and membrane depolarization. For
example, experimental evidence indicates that abi1 abi2
double recessive mutants are more sensitive to ABA-induced
stomatal closure than abi1 or abi2 single recessive mutants
[29], suggesting that ABI1 and ABI2 act synergistically. Due to
limited experimental evidence, we do not explicitly incorpo-
rate ABI2, but an independent inhibitory effect of ABI2
would diminish ABA signaling.
While it is difficult to estimate the changes in our
conclusions due to future knowledge gain, we can gauge the
robustness of our results by randomly deleting entries in
Table S1 or rewiring edges of Figure 2 (see Texts S2 and S3).
We find that most of the predicted important nodes are
documented in more than one entry, and more than one
entry needs to be removed from the database before the
topology of the network related to that node changes (Text
S2). Random rewiring of up to four edge pairs shows that the
dynamics of our current network is moderately resilient to
minor topology changes (Text S3 and Figure S1).
Dynamic model. In our dynamic model we do not place
restrictions on the relative timing of individual interactions
but sample all possible updates randomly. This approach
reflects our lack of knowledge concerning the relative
reaction speeds as well as possible environmental noise. The
significance of our current results is the prediction that
whatever the timing is, given the current topology of
regulatory relationships in the network, the most essential
regulators will not change. Our approach can be iteratively
refined when experimental results on the strength and timing
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Model of Guard Cell ABA Signaling
of individual interactions become available. For example, we
can combine Boolean regulation with continuous synthesis
and degradation of small molecules or signal transduction
proteins [95,96] as kinetic (rate) data emerge. Our model
considers the response of individual guard cell pairs to the
local ABA signal; however, there is recent evidence of a
synchronized oscillatory behavior of stomatal apertures over
spatially extended patches in response to a decrease in
humidity [97]. Our model can be extended to incorporate
cell-to-cell signaling and spatial aspects by including extrac-
ellular regulators when information about them becomes
available (see [51]).
Node disruptions. A knockout may either deprive the
system of an essential signaling element (the gene itself), or it
may ‘‘set’’ the entire system into a different state (e.g., by
affecting the baseline expression of other, seemingly unre-
lated signaling elements). Our analysis and current exper-
imental data only address the former. Because of this caveat,
in some ways rapid pharmacological inhibition may actually
have a more specific effect on the cell than gene knockouts.
Implications
Many of the signaling proteins present as nodes in our
model are represented by multigene families in Arabidopsis
[98], with likely functional redundancy among encoded
isoforms. Therefore, the amount of experimental work
required to completely disrupt a given node may be
considerable. It is also considerable work to make such
genetic modification in many of the important crop species
that are much less amenable than Arabidopsis to genetic
manipulation. It is also the case that, at present, there are no
reports of successful use of ratiometric pH indicators in the
small guard cells of Arabidopsis, suggesting that further
technical advances in this area are required. Facts such as
these indicate the importance of establishing a prioritization
of node disruption in experimental studies seeking to
manipulate stomatal responses for either an increase in basic
knowledge or an improvement in crop water use efficiency.
Our model provides information on which such prioritiza-
tion can be based. Future work on this model will focus on
predicting the changes in ABA-induced closure upon con-
stitutive activation of network components or in the face of
fluctuating ABA signals. Ultimately, the experimental infor-
mation obtained may or may not support the model
predictions; the latter instance provides new information
that can be used to improve the model. Through such
iteration of in silico and wet bench approaches, a more
complete understanding of complex signaling cascades can
be obtained.
Approaches to describe the dynamics of biological net-
works include differential equations based on mass-action
kinetics for the production and decay of all components
[99,100], and stochastic models that address the deviations
from population homogeneity by transforming reaction rates
into probabilities and concentrations into numbers of
molecules [101]. The great complexity of many cellular signal
transduction networks makes it a daunting task to recon-
struct all the reactions and regulatory interactions in such
explicit biochemical and kinetic detail. Our work offers a
roadmap for synthesizing incompletely described signal
transduction and regulatory networks utilizing network
theory and qualitative stochastic dynamic modeling. In
addition to being the practical choice, qualitative dynamic
descriptions are well suited for networks that need to
function robustly despite changes in external and internal
parameters. Indeed, several analyses found that the dynamics
of network motifs crucial for the stable dynamics and noise-
resistance of cellular networks, such as single input modules,
feed-forward loops [102,103] and dual positive feedback loops
[93], is correctly and completely captured by qualitative
modeling [104,105]. For example, at the regulatory module
level, several qualitative (Boolean and continuous/discrete
hybrid) models [51,53,96] reproduced the Drosophila segment
polarity gene network’s resilience when facing variations in
kinetic parameters [50], offering the most natural explan-
ation of which parameter sets will succeed in forming the
correct gene expression pattern [106]. We expect that our
methods will find extensive applications in systems where
modeling is currently not possible by traditional approaches
and that they will act as a scaffold on which more quantitative
analyses of guard cell signaling in particular and cell signaling
in general can later be built.
Our analyses have clear implications for the design of future
wet bench experiments investigating the signaling network of
guard cells and for the translation of experimental results on
model species such as Arabidopsis to the improvement of water
use efficiency and drought tolerance in crop species [107–
109]. Drought stress currently provides one of the greatest
limitations to crop productivity worldwide [110,111], and this
issue is of even more concern given current trends in global
climate change [112,113]. Our methods also have implications
in biomedical sciences. The use of systems modeling tools in
designing new drugs that overcome the limitation of tradi-
tional medicine has been suggested in the recent literature
[114]. Many human diseases, such as breast cancer [115] or
acute myeloid leukemia [116,117], cause complex alterations
to the underlying signal transduction networks. Pathway
information relevant to human disease etiologies has been
accumulated over decades and such information is stored in
several databases such as TRANSPATH [118], BioCarta (http://
www.biocarta.com), and STKE (http://www.stke.org). Our
strategy can serve as a tool that guides experiments by
integrating qualitative data, building systems models, and
identifying potential drug targets.
Materials and Methods
Plant material and growth conditions. Wild-type Arabidopsis (Col
genotype) seeds were germinated on 0.53MS media plates containing
1% sucrose. Seedlings were grown vertically under short-day
conditions (8 h light/16 h dark) 120 lmol m2 s1 for 10 d. Vigorous
seedlings were selected for transplantation into soil and were grown
to 5 wk of age (from germination) under short day conditions (8 h
light/16 h dark). Leaves were harvested 30 min after the lights were
turned on in the growth chamber.
Stomatal aperture measurements. Leaves were incubated in 20 mM
KCl, 5 mM Mes-KOH, and 1 mM CaCl2 (pH 6.15) (Tris), at room
temperature and kept in the light (250 lmol m2 s1) for 2 h to open
stomata. For pHc clamping, different amounts of Na-butyrate stock
solution (made up as 1M solution in water [pH 6.1]) were added into
the incubation solution, to achieve the concentrations given in Figure
8, 15 min before adding 50 lM ABA. Apertures were recorded after
2.5 h of further incubation in light. Epidermal peels were prepared at
the end of each treatment. The maximum width of each stomatal
pore was measured under a microscope fitted with an ocular
micrometer. Data were collected from 40 stomata for each treatment
and each experiment was repeated three times.
Model. The network in Figure 2 was drawn with the SmartDraw
software (http://www.smartdraw.com/exp/ste/home). The dynamic
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October 2006 | Volume 4 | Issue 10 | e312
1745
Model of Guard Cell ABA Signaling
modeling was implemented by custom Python code (http://www.
python.org). To equally sample the space of all possible timescales,
the random-order asynchronous updating method developed in [53]
was used. Briefly, every node is updated exactly once during each unit
time interval, according to a given order. This order is a permutation
of the N¼40 nodes in the network, chosen randomly out of a uniform
distribution over the set of all N! possible permutations. A new
update order is selected at each timestep. As demonstrated in [53],
this algorithm is equivalent to a random timing of each node’s state
transition.
Supporting Information
Figure S1. Probability of Closure in Randomized Networks where
Pairs of Positive or Negative Edges Are Rewired
Found at DOI: 10.1371/journal.pbio.0040312.sg001 (40 KB PDF).
Table S1. Synthesis of Experimental Information about Regulatory
Interactions between ABA Signal Transduction Pathway Components
Found at DOI: 10.1371/journal.pbio.0040312.st001 (407 KB DOC).
Text S1. Detailed Justification for Each Boolean Transfer Function
Found at DOI: 10.1371/journal.pbio.0040312.sd001 (149 KB DOC).
Text S2.
Verification of the Inference Process and the Resulting
Network
Found at DOI: 10.1371/journal.pbio.0040312.sd002 (45 KB DOC).
Text S3. Effect of Random Rewiring on the Network Dynamics
Found at DOI: 10.1371/journal.pbio.0040312.sd003 (36 KB DOC).
Accession Numbers
The Arabidopsis Information Resource (TAIR) (http://www.arabidopsis.
org) accession numbers for the genes discussed in this paper are
NIA12 (At1g77760/At1g37130), GPA1 (At2g26300), ERA1 (At5g40280),
AtrbohD/F (At5g47910/At4g11230), RCN1 (At1g25490), OST1
(At4g33950), ROP2 (At1g20090), RAC1 (At4g35020), ROP10
(At3g48040), AtP2C-HA/AtPP2CA (At1g72770/At3g11410), and GCR1
(At1g48270).
Acknowledgments
The authors thank Drs. Jayanth Banavar, Vincent Crespi, and Eric
Harvill for critically reading a previous version of the manuscript;
and Dr. Istva´n Albert for assistance with figure preparation.
Author contributions. SL, SMA, and RA conceived and designed
the experiments. SL performed the experiments. SL and RA analyzed
the data. SL, SMA, and RA wrote the paper.
Funding. RA gratefully acknowledges a Sloan Research Fellowship.
Research on guard cell signaling in SMA’s laboratory is supported by
NSF-MCB02–09694 and NSF-MCB03–45251.
Competing interests. The authors have declared that no competing
interests exist.
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Model of Guard Cell ABA Signaling
|
16968132
|
ROS = ( Atrboh )
PEPC = NOT ( ( ABA ) )
PLD = ( GPA1 )
HTPase = NOT ( ( pH ) OR ( Ca2_c ) OR ( ROS ) )
Ca2_c = ( ( CIS ) AND NOT ( Ca2_ATPase ) ) OR ( ( CaIM ) AND NOT ( Ca2_ATPase ) )
ROP10 = ( ERA1 )
RAC1 = NOT ( ( ABA ) OR ( ABI1 ) )
OST1 = ( ABA )
ROP2 = ( PA )
InsP6 = ( InsPK )
SphK = ( ABA )
Depolar = ( ( KOUT AND ( ( ( NOT AnionEM AND NOT Ca2_c AND NOT HTPase AND NOT KEV ) ) ) ) OR ( Ca2_c ) OR ( KEV ) OR ( HTPase AND ( ( ( NOT AnionEM AND NOT Ca2_c AND NOT KOUT AND NOT KEV ) ) ) ) OR ( AnionEM ) ) OR NOT ( AnionEM OR Ca2_c OR HTPase OR KOUT OR KEV )
RCN1 = ( ABA )
Ca2_ATPase = ( Ca2_c )
NOS = ( Ca2_c )
GPA1 = ( ( AGB1 ) AND NOT ( GCR1 ) ) OR ( S1P AND ( ( ( AGB1 ) ) ) )
Atrboh = ( ( OST1 AND ( ( ( pH AND ROP2 ) ) ) ) AND NOT ( ABI1 ) )
Malate = ( ( ( PEPC ) AND NOT ( AnionEM ) ) AND NOT ( ABA ) )
AnionEM = ( pH AND ( ( ( Ca2_c ) ) OR ( ( NOT ABI1 ) ) ) ) OR ( Ca2_c AND ( ( ( pH ) ) OR ( ( NOT ABI1 ) ) ) )
KAP = ( ( Depolar ) AND NOT ( Ca2_c AND ( ( ( pH ) ) ) ) )
pH = ( ABA )
CIS = ( InsP3 AND ( ( ( InsP6 ) ) ) ) OR ( cGMP AND ( ( ( cADPR ) ) ) )
InsP3 = ( PLC )
PA = ( PLD )
ABI1 = ( ( ( pH ) AND NOT ( PA ) ) AND NOT ( ROS ) )
CaIM = ( ( ( ABH1 AND ( ( ( NOT ERA1 ) ) ) ) AND NOT ( Depolar ) ) OR ( ( ERA1 AND ( ( ( NOT ABH1 ) ) ) ) AND NOT ( Depolar ) ) OR ( ( ROS ) AND NOT ( Depolar ) ) ) OR NOT ( ROS OR ERA1 OR ABH1 OR Depolar )
S1P = ( SphK )
NIA12 = ( RCN1 )
cGMP = ( GC )
PLC = ( ABA AND ( ( ( Ca2_c ) ) ) )
cADPR = ( ADPRc )
ADPRc = ( NO )
Actin = ( ( Ca2_c ) ) OR NOT ( RAC1 OR Ca2_c )
AGB1 = ( GPA1 )
Closure = ( ( KOUT AND ( ( ( AnionEM ) ) AND ( ( Actin ) ) ) ) AND NOT ( Malate ) ) OR ( ( KAP AND ( ( ( AnionEM ) ) AND ( ( Actin ) ) ) ) AND NOT ( Malate ) )
InsPK = ( ABA )
KEV = ( Ca2_c )
KOUT = ( pH AND ( ( ( Depolar ) ) ) ) OR ( ( Depolar ) AND NOT ( ROS AND ( ( ( NO ) ) ) ) )
GC = ( NO )
NO = ( NOS AND ( ( ( NIA12 ) ) ) )
|
A Logical Model Provides Insights into T Cell
Receptor Signaling
Julio Saez-Rodriguez1, Luca Simeoni2, Jonathan A. Lindquist2, Rebecca Hemenway1, Ursula Bommhardt2,
Boerge Arndt2, Utz-Uwe Haus3, Robert Weismantel3, Ernst D. Gilles1, Steffen Klamt1*, Burkhart Schraven2*
1 Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany, 2 Institute of Immunology, Otto-von-Guericke University, Magdeburg, Germany,
3 Institute for Mathematical Optimization, Otto-von-Guericke University, Magdeburg, Germany
Cellular decisions are determined by complex molecular interaction networks. Large-scale signaling networks are
currently being reconstructed, but the kinetic parameters and quantitative data that would allow for dynamic
modeling are still scarce. Therefore, computational studies based upon the structure of these networks are of great
interest. Here, a methodology relying on a logical formalism is applied to the functional analysis of the complex
signaling network governing the activation of T cells via the T cell receptor, the CD4/CD8 co-receptors, and the
accessory signaling receptor CD28. Our large-scale Boolean model, which comprises 94 nodes and 123 interactions and
is based upon well-established qualitative knowledge from primary T cells, reveals important structural features (e.g.,
feedback loops and network-wide dependencies) and recapitulates the global behavior of this network for an array of
published data on T cell activation in wild-type and knock-out conditions. More importantly, the model predicted
unexpected signaling events after antibody-mediated perturbation of CD28 and after genetic knockout of the kinase
Fyn that were subsequently experimentally validated. Finally, we show that the logical model reveals key elements
and potential failure modes in network functioning and provides candidates for missing links. In summary, our large-
scale logical model for T cell activation proved to be a promising in silico tool, and it inspires immunologists to ask new
questions. We think that it holds valuable potential in foreseeing the effects of drugs and network modifications.
Citation: Saez-Rodriguez J, Simeoni L, Lindquist JA, Hemenway R, Bommhardt U, et al. (2007) A logical model provides insights into T cell receptor signaling. PLoS Comput
Biol 3(8): e163. doi:10.1371/journal.pcbi.0030163
Introduction
Understanding how cellular networks function in a holistic
perspective is the main purpose of systems biology [1].
Dynamic models provide an optimal basis for a detailed study
of cellular networks and have been applied successfully to
cellular networks of moderate size [2–5]. However, for their
construction and analysis they require an enormous amount
of mechanistic details and quantitative data which, until now,
has been often lacking in large-scale networks. Therefore,
there has been considerable effort to develop methods based
exclusively on the often well-known network topology [6,7].
One may distinguish between studies on the statistical
properties of graphs [8–10] and approaches aiming at
predicting functional or dysfunctional states and modes.
For the latter, a large corpus of methods has been developed
for metabolic networks mainly relying on the constraints-
based approach [11,12]. However, for signaling networks,
methods facilitating a similar functional analysis—including
predictions on the outcome of interventions— have been
applied to a much lesser extent [6].
Here we demonstrate that capturing the structure of
signaling networks by a recently introduced logical approach
[13] allows the analysis of important functional aspects, often
leading to predictions that can be verified in knock-out/
perturbation experiments. Logical networks have until now
been used for studying artificial (random) networks [14] or
relatively small gene regulatory networks [15–18]. In contrast,
herein we study a large-scale signaling network, structured in
input (e.g., receptors), intermediate, and output (e.g., tran-
scription factors) layers. Compared with gene regulatory
networks, the behavior of signaling networks is mainly
governed by their input layer, shifting the interest to input–
output relationships. Addressing these issues requires parti-
ally different techniques, as compared with gene regulatory
networks. We use a special and intuitive representation of
logical networks (called logical interaction hypergraph (LIH); see
Methods), which is well-suited for this kind of input–output
analysis. By applying logical steady state analysis, one may
predict how a combination of signals arriving at the input
layer leads to a certain response in the intermediate and the
output layers. Additionally, this approach facilitates predic-
tions of the effect of interventions and, moreover, allows one
to search for interventions that repress or provoke a certain
logical response [13]. Furthermore, each logical network has a
unique underlying interaction graph from which other
important network properties such as feedback loops,
signaling paths, and network-wide interdependencies can be
evaluated.
Editor: Rob J. De Boer, Utrecht University, The Netherlands
Received February 6, 2007; Accepted July 5, 2007; Published August 24, 2007
A previous version of this article appeared as an Early Online Release on July 5,
2007 (doi:10.1371/journal.pcbi.0030163.eor).
Copyright: 2007 Saez-Rodriguez et al. This is an open-access article distributed
under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.
Abbreviations: LIH, logical interaction hypergraph; MHC, Major Histocompatibility
Complex; MIS, Minimal intervention set; TCR, T cell receptor
* To whom correspondence should be addressed. E-mail: inquiries regarding the
mathematical methodology should be addressed to Steffen Klamt, klamt@
mpi-magdeburg.mpg.de, and regarding the biological and experimental data to
Burkhart Schraven, [email protected]
PLoS Computational Biology | www.ploscompbiol.org
August 2007 | Volume 3 | Issue 8 | e163
1580
Importantly, we consider here a logical model to be
constructed by collecting and integrating well-known local
interactions (e.g., a kinase phosphorylates an adaptor
molecule). The logical model is then employed to derive
global information (e.g., stimulation of a receptor leads to the
activation of a certain transcription factor via several logical
connections). Thus, the available data on the global network
behavior is not used to construct the model; instead, it is used
to verify the model. The model may then be employed to
predict global responses that have not yet been studied
experimentally.
Here, we apply the logical framework to a carefully
constructed model of T cell receptor (TCR) signaling. T-
lymphocytes play a key role within the immune system:
cytotoxic, CD8þ, T cells destroy cells infected by viruses or
malignant cells, and CD4þ T helper cells coordinate the
functions of other cells of the immune system [19]. The
importance of T cells for immune homeostasis is due to their
ability to specifically recognize foreign, potentially danger-
ous, agents and, subsequently, to initiate a specific immune
response. T cell reactivity must be exquisitely regulated as
either a decrease (which weakens the defense against
pathogens with the consequence of immunodeficiency) or
an increase (which can lead to autoimmune disorders and
leukemia) can have severe consequences for the organism.
T cells detect foreign antigens by means of the TCR, which
recognizes peptides only when presented upon MHC (Major
Histocompatibility Complex) molecules. The peptides that
are recognized by the TCR are typically derived from foreign
(e.g., bacterial, viral) proteins and are generated by proteo-
lytic cleavage within so-called antigen presenting cells (APCs).
Binding of the TCR to peptide/MHC complexes and the
additional binding of a different region of the MHC
molecules by the co-receptors (CD4 in the case of T helper
cells and CD8 in the case of cytotoxic T cells), together with
costimulatory molecules such as CD28, initiates a plethora of
signaling cascades within the T cell. These cascades give rise
to a complex signaling network, which controls the activation
of several transcription factors. These transcription factors,
in turn, control the cell’s fate, particularly whether the T cell
becomes activated and proliferates or not [20]. Therefore, we
chose to focus on a limited number of receptors that are
known to be central to the decision making process. The high
number of kinases, phosphatases, adaptor molecules, and
their interactions give rise to a complex interaction network
which cannot be interpreted via pure intuition and requires
the aid of mathematical tools. Since no sufficient basis of
kinetic data is available for setting up a dynamic model of this
network, we opted to use logical modeling as a qualitative and
discrete modeling framework. Note that there are kinetic
models dealing with a smaller part of the network (e.g.,
[5,21,22]), as well as models of the gene regulatory network
governing T cell activation [23].
We recently introduced our approach for the logical
modeling of signaling networks [13], and, to exemplify it, we
presented a small logical model for T cell activation (40
nodes). However, this model only served to demonstrate
applicability and was too incomplete to address realistic
complex input–output patterns. In contrast, the model
presented herein has been significantly expanded to 94 nodes
and refined by a careful reconstruction process (see below). It
is thus realistic enough to be verified with diverse exper-
imental data and to test its predictive power.
In this report, the large-scale logical model describing T
cell activation and the analysis performed therewith will be
presented. First we will show that a number of important
structural features can be identified with this model. Then we
will show that the model not only reproduces published data
on wet lab experiments, but it also predicts non-intuitive and
previously unknown responses.
Results
Setup of a Curated, Comprehensive Logical Model of T
Cell Receptor Signaling
We have constructed a logical model describing T cell
signaling (see Methods and Figure 1), which comprises the
main events and elements connecting the TCR, its corecep-
tors CD4/CD8, and the costimulatory molecule CD28, to the
activation of key transcription factors in T cells such as AP-1,
NFAT, and NFjB, all of which determine T cell activation and
T cell function. In general, the model includes the following
signaling steps emerging from the above receptors: the
activation of the Src kinases Lck and Fyn, followed by the
activation of the Syk-related protein tyrosine kinase ZAP70,
and the subsequent assembly of the LAT signalosome, which
in turn triggers activation of PLCc1, calcium cascades,
activation of RasGRP, and Grb2/SOS, leading to the activa-
tion of MAPKs [20]. Additionally, it includes the activation of
the PI3K/PKB pathway that regulates many aspects of cellular
activation and differentiation, particularly survival. For the
activation of elements that play an important role, but whose
regulation is not well-known yet (e.g., Card11, Gadd45), an
external input was added. These elements can be considered
as points of future extension of the model.
As mentioned above, our model, which is documented in a
detailed manner in Tables S1 and S2, is based upon local
interactions (e.g., kinase ZAP70 phosphorylates the adaptor
molecule LAT) that are well-established for primary T cells in
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Author Summary
T-lymphocytes are central regulators of the adaptive immune
response, and their inappropriate activation can cause autoimmune
diseases or cancer. The understanding of the signaling mechanisms
underlying T cell activation is a prerequisite to develop new
strategies for pharmacological intervention and disease treatments.
However, much of the existing literature on T cell signaling is related
to T cell development or to activation processes in transformed T
cell lines (e.g., Jurkat), whereas information on non-transformed
primary T cells is limited. Here, immunologists and theoreticians
have compiled data from the existing literature that stem from
analysis of primary T cells. They used this information to establish a
qualitative Boolean network that describes T cell activation
mechanisms after engagement of the TCR, the CD4/CD8 co-
receptors, and CD28. The network comprises 94 nodes and can be
extended to facilitate interpretation of new data that emerge from
experimental analysis of T cell activation. Newly developed tools and
methods allow in silico analysis, and manipulation of the network
and can uncover hidden/unforeseen signaling pathways. Indeed, by
assessing signaling events controlled by CD28 and the protein
tyrosine kinase Fyn, we show that computational analysis of even a
qualitative network can provide new and non-obvious signaling
pathways which can be validated experimentally.
A Logical Model of T Cell Receptor Signaling
the literature. We did not use the known global information
(e.g., stimulation of a receptor leads to the activation of a
certain transcription factor) for the model construction.
Instead, in simulations, the local interactions give rise to a
global behavior which can be compared with available
experimental observations (and was thus used to verify the
model).
Each component in the logical model can be either ON
(‘‘1’’) or OFF (‘‘0’’). We consider a compound to be ON only if
it is fully activated and able to trigger downstream events
properly; otherwise, it is OFF. Furthermore, we consider two
timescales [13]: early (s ¼ 1) and late (s ¼ 2), involving
processes occurring during or after the first minutes of
activation, respectively (the time-scale for each interaction is
given in Table S2). Some key regulatory processes such as the
degradation of signaling proteins mediated by the E3
ubiquitin ligase c-Cbl [24–26] occur after a certain time,
and are thus assigned s ¼ 2. Therefore, as will be shown later,
analysis of signal propagation during the early events reveals
which elements become activated, and the consideration of
the late events allows a rough approximation to the dynamic
behavior (sustained versus transient) of the network.
The model comprises 94 different compounds and 123
interactions that give rise to a complex map of interactions
(Figure 1). It is, to the best of our knowledge, the largest
Boolean model of a cellular network to date.
Interaction-Graph-Based Analyses
The first step in our analysis was to examine the interaction
graph underlying the logical model. The former can be easily
derived from the latter when a special representation of
Boolean networks is used (see Methods). The interaction
graph is less constrained than the Boolean network since it
only captures direct (positive or negative) effects of one
Figure 1. Logical Model of T Cell Activation (Screenshot of CellNetAnalyzer)
Each arrow pointing at a species box is a so-called hyperarc representing one possibility to activate that species (see Methods). All the hyperarcs pointing
at a particular species box are OR connected. Yellow species boxes denote output elements, while green ones represent (co)receptors. In the shown
‘‘early-event’’ scenario, the feedback loops were switched off, and only the input for the costimulatory molecule CD28 is active (scenario in column 2 of
Table 1). The resulting logical steady state was then computed. Small text boxes display the signal flows along the hyperarcs (blue boxes: fixed values
prior to computation; green boxes: hyperarcs activating a species (signal flow is 1); red boxes: hyperarcs which are not active (signal flow is 0)).
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A Logical Model of T Cell Receptor Signaling
molecule upon another. Thus, unlike the logical model, the
interaction graph cannot describe how different causal
effects converging at a certain species are combined. For
example, in an interaction graph we may say that A and B
have a positive influence on another node C; the logical
network is more precise because it expresses that A AND B
(or A OR B) are required to activate C. Accordingly,
compared with the logical model, an interaction graph
requires less a priori knowledge about the network under
study which comes at the price that functional predictions
are limited. Nevertheless, as demonstrated in this section, a
number of important functional features can be revealed
from the graph model.
First we studied global properties of the graph. As
expected, the graph is connected (i.e., neglecting the arc
directions, there is always a path from one node to all others).
However, the directed graph contains as a core one strongly
connected component with 33 nodes (i.e., for each pair (a,b)
of nodes taken from this component there is a path from a to
b and from b to a). This structural organization is related with
the bow-tie structure found in other cellular networks (e.g.,
[7,27]) and implies that the rest of the network (not contained
in the strongly connected component) mainly consists in
relatively simple input and output layers (including branch-
ing cascades) feeding to and from this component.
We continued the interaction-graph-based analysis by
computing the feedback loops. Feedback loops are of major
importance for the dynamic behavior and functioning of
biological networks. Negative feedback loops control homeo-
static response and can give rise to oscillations, while positive
feedbacks govern multistable behavior (connected to irrever-
sible decision-making and differentiation processes) [15,28–
30]. The interaction graph underlying the logical T cell model
has 172 feedback loops, 89 thereof being negative. Remark-
ably, all feedback loops are only active in the second timescale
because each loop contains at least one process of the second
timescale. The elements of the MAPK cascade are involved in
92% of the feedback loops. This is due to the fact that there is
a connection from ERK to the phosphatase SHP1 from the
bottom to the top of the network [5]. Due to this connection,
the resulting feedback can return to ERK via many different
paths, thereby leading to a high number of loops. Indeed, if
the ERK ! SHP1 connection is not considered, the number
of loops is reduced dramatically from 172 to 13 (with only 11
being negative), all located in the upper part of the network.
c-Cbl is involved in ;85% of them, thus underscoring the
importance of c-Cbl in the regulation of signaling processes
[25,26].
There are 4,538 paths, each connecting one of the three
compounds from the input layer (TCR, CD4/CD8, CD28) with
one compound in the output layer (transcription factors and
other elements controlling T cell activation). The high
number of negative paths (2,058) can be traced back to the
presence of two negative connections (via DGK and Gab2). In
fact, considering the early signaling events within the
network, where DGK and Gab2 are not active yet, the
number of paths is reduced to 1,530, with only six of them
being negative. These paths are from the TCR and CD28 to
negative regulators of the cell cycle (p21, p27, and FKHR),
having thus a positive effect on T cell proliferation. These
and other global effects can be graphically inspected via the
dependency matrix [13,31], depicted in Figure 2. Importantly,
when considering the timescale s ¼ 1, there is no ambivalent
effect (i.e., via positive and negative paths) between any
ordered pair (A,B) of species, i.e., A is either a pure activator
of B (only positive paths from A to B), or a pure inhibitor of B
(only negative paths from A to B), or has no direct or indirect
influence on B at all. For example, during early activation, the
TCR can only have a positive effect upon AP1 (the array
element (TCRb, AP1) in Figure 2 is green). Note that this
changes for timescale s ¼ 2 where, in several cases, a
compound influences another species in an ambivalent
manner.
Analysis of the Logical Model
An important aspect that can be studied with a logical
model is signal processing and signal propagation and the
corresponding response (activation/inactivation) of the nodes
upon external stimuli and perturbations (see Methods). One
starts the analysis of a scenario by defining a pattern of input
stimuli, possibly in combination with a set of nodes that are
knocked-out or knocked-in. Then, by an iterative evaluation
of the Boolean rules in each node, the signal is propagated
through the network, switching each node ON or OFF,
respectively (see [13] and Methods). For example, since CD28
(an input) is (permanently) ON in the scenario shown in
Figure 1, it will (permanently) activate node X, which will in
turn (permanently) activate Vav1, and so forth. In the same
scenario, since the input CD4 is OFF, Lckp1 and therefore
Abl, ZAP70, and other components cannot become activated
and therefore are in the OFF state. In the ideal case, each
node can be assigned a uniquely determined state that follows
from a given input pattern. In terms of Boolean networks, the
set of determined node values then represents a logical steady
state. In some cases, in particular when negative feedback
loops are active, only a fraction of the elements can be
assigned a unique steady state value, whereas other (or even
all) nodes might oscillate [15]. However, since in the T cell
model all negative feedback loops become active only during
timescale s ¼ 2 (as described above), a complete logical steady
state follows for arbitrary input patterns when considering s
¼ 1.
Using this kind of logical steady state analysis, we first
analyzed the activation pattern of key elements upon differ-
ent stimuli (activation of the TCR and/or CD4 and/or CD28;
Table 1) for timescale s ¼ 1. The model was able to reproduce
data from both the literature and our own experiments,
providing a holistic and integrated interpretation for a large
body of data. The model also predicted a non-obvious
signaling event, namely that the activation of the costimula-
tory molecule CD28 alone leads not only to the activation of
PI3K—which is to be expected from a large body of literature
dealing with CD28 signaling showing that PI3K binds to the
motif YxxM of CD28 [32,33]—but also to the selective
activation of JNK, but not ERK. The model predicts a
pathway from CD28 to JNK which gives a holistic explanation
for this result: the pathway does NOT involve the LAT
signalosome, activation of PLCc1, and Calcium flux, but
clearly depends on the activation of the nucleotide exchange
factor Vav1 which activates MEKK1 via the small G-protein
Rac1 (Figure 1). Clearly, the activating pathway shown in
Figure 1 could be identified by a visual inspection of the map
(note that we have intentionally drawn the network in such a
way that this route can be easily seen). However, in large-scale
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A Logical Model of T Cell Receptor Signaling
networks the identification of long-distance pathways by
simply following all possible routes becomes infeasible and is
particularly complicated if AND connections are involved.
Furthermore, since the CD28-induced JNK activation path-
way was not expected, one would probably not have searched
specifically for this pathway, while the algorithm reveals the
whole response of the network.
The prediction made by the model is particularly surpris-
ing in light of published data which either suggest that CD28
stimulation alone does NOT activate JNK [34,35] or induces
only a weak activation [36]. Driven by this surprising
prediction, we performed the corresponding experiments in
vitro. As shown in Figure 3A, stimulation of mouse primary T
cells with a non-superagonistic CD28 antibody induced an
evident and sustained JNK phosphorylation, thus confirming
almost perfectly the predicted binary response. Note, the
model also predicted that JNK activation does not depend on
the activation of PI3K. Again, this prediction was verified by
applying a pharmacological inhibitor of PI3K (Figure 3D).
The discrepancies with the literature could be due either to
the different cellular systems (primary T cells versus T cell
lines) or to the different stimulation conditions.
The nature of the kinase involved in CD28-mediated
signaling remains unclear. Indeed, application of the Src-
kinase inhibitor PP2 that inactivates both Lck þ Fyn [37],
showed that Src-kinases, which were proposed to mediate
CD28 signaling [38], are dispensable for the CD28-mediated
activation of JNK (Figure 4). To fit the Src-kinase inhibitor
data with the model, it would have been possible to simply
bypass the Src-kinase and to draw a causal connection from
CD28 to Vav. Such a connection would indeed be justified
since it is well established that triggering of CD28 leads to the
activation of Vav ([39]; for more details, see Table S2,
reactions 35 and 48). However, we preferred to include a
to-be-identified kinase X that gets activated by CD28 (Figure
1), in order to keep within the model the information that
there is a component to be identified. Potential candidates
for kinase X would be members of the Tec-family of PTKs.
However, it is difficult to study the signaling properties of
these kinases in primary non-transformed cells since specific
inhibitors for Tec kinases are not yet available and the
corresponding knock-out mice show defects in thymic
development. Therefore, as we focused during model
generation on well-established data from primary T cells
and excluded data obtained from knockout mice showing
alterations of thymic development, we did not include it.
Figure 2. Dependency Matrix of the Logical T Cell Signaling Model (Figure 1) for the Early Events Scenario (s ¼ 1)
The color of a matrix element Mxy has the following meaning [13]: (i) dark green: x is a total activator of y; (ii) dark red: x is a total inhibitor of y; (iii) white:
no (direct or indirect) influence from x on y.
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A Logical Model of T Cell Receptor Signaling
The ability of the model to recapitulate the T cell
phenotype of a variety of previously described knock-out
mice was also tested (Table 1). Indeed, the model could
reproduce the phenotype of several knock-outs and again
reported a rather unexpected result: activation of the TCR in
Fyn-deficient cells selectively triggers the PI3K/PKB pathway.
This prediction was subsequently tested using peripheral
primary T cells prepared from spleen of Fyn-deficient mice.
As shown in Figure 3B, the wet-lab experiments corroborated
the model result again.
However, there was an experimental result which the
model could not reproduce: TCR-mediated JNK activation is
blocked by an inhibitor of PI3K (Figure 3C). In fact, this result
is not in accordance with the network because PI3K has no
influence upon JNK (see dependency matrix, Figure 2).
To identify potential connections that would explain the
experimental data, we applied the concept of Minimal
Intervention Sets (MISs; see Methods). A MIS is a irreducible
collection of actions (e.g., activation or inactivation of certain
compounds), that, if applied, guarantees that a certain goal (a
desired behavior) is fulfilled [13]. Here, we computed the MISs
by which JNK becomes activated under the experimentally
obtained constraint (see Figure 3C) that PI3K is OFF
(describing the effect of the PI3K inhibitor), ZAP70 is ON,
and that the TCR has been activated. These MISs (Table 2)
thus provide a list of minimal combinations of elements that
should be directly or indirectly affected by PI3K and thus
allow us to explain the observed response of JNK upon
inhibiting PI3K. Some of them are obvious, e.g., the first MIS
in Table 2 suggests that JNK activation could be directly
interacting with PI3K or elements that are located down-
stream of PI3K (e.g., PIP3). There is currently no convincing
experimental evidence for an effect of PI3K on JNK, though.
Other MISs in Table 2 suggest that a PI3K-mediated activation
of Vav (both 1 and 3 isoforms) is involved, which would be an
attractive possibility to explain the experimental data.
Indeed, Vav possesses a PH domain which can bind to PIP3,
and this mechanism could be important for Vav activation
[40], thus making it a reasonable extension of the model.
Another molecule that could be involved in PI3K-mediated
Table 1. Summary of Predicted Activation Pattern upon Different Stimuli and Knock-Out Conditions
Input/
Output
WT
WT
WT
PI3K
PI3K
PI3K
SLP76
Fyn
Fyn
Fyn
Rlk and
Itk
Lck and
Fyn
Lck and
Fyn
Lck and
Fyn
Input
TCR
1
0
1
1
0
1
1
1
1
1
1
1
0
1
CD4
0
0
0
0
0
0
0
0
1
0
0
0
0
0
CD28
0
1
1
0
1
1
0
0
0
1
0
0
1
1
Output
ZAP
1
0
1
1
0
1
1
0
1
0
1
0
0
0
LAT
1
0
1
1
0
1
1
0
1
0
1
0
0
0
PLCga
1
0
1
0
0
0
0
0
1
0
0
0
0
0
ERK
1
0
1
0
0
0
0
0
1
0
0
0
0
0
JNK
1
1
1
1
1
1
1
0
1
1
1
0
1
1
PKB
1
1
1
0
0
0
1
1
1
1
1
0
1
1
AP1
1
0
1
0
0
0
0
0
1
0
0
0
0
0
NFKB
1
0
1
0
0
0
0
0
1
0
0
0
0
0
NFAT
1
0
1
0
0
0
0
0
1
0
0
0
0
0
Reference
Figure
3A, 3C
Figure
3A, 3D
Figures
3A, 4
Figure 3C
Figure 3D
Figure 4
[49]
Figure
3B, [50]
Figure
3B, [50]
Figure
3B, [50]
[51]
Figure 4
Figure 4
Figure 4
The headings denote the perturbed (switched-off) element. In the case of PI3K and Lck and Fyn, the perturbation was done via a chemical inhibitor, and for the rest it was through a
genetic knock-out. The ‘‘Input’’ rows show the stimuli, and ‘‘Output’’ the predictions of the model for key elements of the network. Here, blue numbers denote results corroborated by
published data, while green ones were confirmed by our own data. The red number shows a discrepancy between model and experiment (see discussion in the main text). Finally, the row
labeled Reference indicates the Figure where the experimental results are shown or points to the literature reference.
doi:10.1371/journal.pcbi.0030163.t001
Figure 3. In Vitro Analysis of Model Predictions
(A) Activation of ERK and JNK upon CD28, TCR (CD3), or TCR þ CD28 stimulation in mouse splenic T cells.
(B) Activation of PKB upon TCR, TCRþ CD4, and TCR þ CD28 stimulation in Fyn-deficient and heterozygous splenic mouse T cells.
(C) Inhibition of PI3K with both Ly294002 and Wortmannin blocks the phosphorylation of PKB, ERK, and JNK, but not ZAP-70 in human T cells.
(D) Inhibition of PI3K with both Ly294002 and Wortmannin blocks the phosphorylation of PKB, but not of JNK in human T cells upon CD28 stimulation.
As a control, the total amount of ZAP70 (A) or b-actin (B–D) was determined. One representative experiment (of three) is shown.
doi:10.1371/journal.pcbi.0030163.g003
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A Logical Model of T Cell Receptor Signaling
activation of JNK is the serine/threonine kinase HPK1 (see
Figure 1 and Tables S1 and S2). Interestingly, HPK1 is
phosphorylated by Protein Kinase D1 (PKD1) [41], a kinase
whose activation depends on PKC (which in turn is depend-
ent on DAG, downstream of PI3K) for activation. Since the
regulation and functional roles of both PKD1 and PKC (with
the exception of the h isoform) are not yet well-established in
T cells, we did not include them in the model, but a
connection PI3K ! PIP3 ! Itk ! PLCc ! DAG ! PKC !
PKD1 ! HPK1 would be plausible (in which the path from
PKC to HPK1 via PKD1 would be new). An alternative could
be a Rac-dependent activation of HPK1 [42]; however, this is
again a not-well-established connection and thus was not
considered.
Definitely, the model requires a direct or indirect
connection from PI3K to JNK, and additional experiments
are required to assess which of the candidate links predicted
by the MISs are relevant in peripheral T cells. This particular
example illustrates another useful and important application
of our approach: the model not only reveals that a link is
missing, but also suggests candidates that can be verified
experimentally. Thus, MIS analysis is capable of guiding the
experimentalist and helps to plan the corresponding experi-
ments.
As an additional application of MISs, we computed
combinations of failures (constitutive activation or inactiva-
tion of elements caused for example by mutations) which lead
to sustained T cell activation without external stimuli. These
failure modes would cause uncontrolled proliferation and
thus may be connected to diseases such as leukemia or
autoimmunity. Interestingly, components occurring in the
MISs with few elements (Table 3) are in fact known
oncogenes: ZAP70 [43], PI3K [44], Gab2 [45], and PLCc1
[46] (and SLP76 is directly involved in PLCc1 activation).
Robustness and Sensitivity Analysis of the Logical Model
Strongly related to the idea of MISs is a systematic
evaluation of the network response if the model is confronted
with failures. By considering a failure as something that
happens to the cell by an internal or external event (e.g., a
mutation), we may assess the robustness—one of the most
important properties of living systems [47]—of the network.
In contrast, if we consider the failure as an error that has
been introduced during the modeling process (due to
incomplete knowledge), then we are assessing the sensitivity
of the model with respect to the predictions it makes.
Accordingly, to study robustness and sensitivity issues, we (i)
removed systematically each single interaction from the
network, (ii) recomputed the scenarios given in Table 1, and
(iii) compared the new predictions with the 126 original
predictions (Table 1), ranking the interactions according to
the number of introduced changes produced (Table 4). As an
average value, 4.76 errors were introduced per simulated
failure, which corresponds to 3.78% of the total numbers of
predictions. The most sensitive interactions are mainly
located in the upper part of the network and activate
components such as the T cell receptor (TCRb), ZAP70,
LAT, Fyn, or Abl. It is intuitively clear that the network is very
Figure 4. In Vitro Analysis of Src-Kinase Inhibition
Inhibition of Src-Kinases (Lck and Fyn) with PP2 blocks TCR-induced but affects only moderately CD28-induced PKB and JNK activation in human T cells;
therefore, we concluded that CD28 signaling is not strictly Src-kinase– dependent. The effect was compared with PI3K inhibition via Wortmannin (ccf.
Figure 3C and 3D), which blocks the phosphorylation of PKB but not of JNK. b-actin was included as the loading control. One representative experiment
(of three) is shown.
doi:10.1371/journal.pcbi.0030163.g004
Table 2. Application of the Minimal Intervention Sets To Identify
Candidates To Fill the Gap between PI3K and JNK
MIS
jnk
hpk1
rac1r
hpk1
sh3bp2
mekk1
mkk4
mekk1
mlk3
hpk1
mekk1
rac1p1
hpk1
mekk1
vav1
hpk1
mkk4
rac1p2
hpk1
mlk3
vav3
hpk1
rac1p1
rac1p2
hpk1
rac1p1
vav3
hpk1
rac1p2
vav1
hpk1
vav1
vav3
hpk1
mlk3
rac1p2
The MISs of maximal size 3 to obtain JNK off under the conditions (i) TCR on, (ii) PI3K off,
and (iii) ZAP70 on (as shown in the experiment, see Figure 3D and Table 1) were
computed, setting the rest of conditions to the standard values for the early events. Here,
each MIS represents one set of molecules that should be influenced by PI3K in order to be
consistent with the fact that PI3K inhibition blocks JNK activation. For species
abbreviations, see Tables S1 and S2.
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A Logical Model of T Cell Receptor Signaling
sensitive to failures (again, caused either by internal/external
events or modeling errors) in these upper nodes because all
pathways branching downstream are governed by them.
Accordingly, the validation of our model (with the data from
Table 1) is most sensitive to modeling errors in the upper part
of the network. We also note that species that can be
activated by more than one interaction (e.g., PI3K) are
significantly less sensitive to single interaction failures since
alternative pathways exist. Regarding robustness, it is worth
emphasizing that in the worst case about 30% of the original
predictions are affected after removal of an interaction,
indicating that there is no ‘‘all-or-nothing’’ interaction in the
network.
We have also performed the same analysis for the removal
of a species (instead of an interaction) which basically led to
the same results (unpublished data). However, the removal of
a node can be seen as a stronger intervention in the network
than deleting an interaction, as the former simulates the
simultaneous removal of all interactions pointing at that
species. Accordingly, deleting nodes implies some stronger
deviations from the original predictions.
Qualitative Description of the Dynamics
So far we have analyzed which elements within the
signaling network get activated upon signal triggering (i.e.,
for the first timescale s ¼ 1). This is due to the fact that a large
corpus of data for these conditions is available (see Table 1).
However, it is important to note that the model is also able to
roughly predict the dynamics upon different stimuli and
conditions.
The modus operandi goes as follows: first, one computes
the steady state values with no external input (s ¼ 0).
Subsequently, the steady state for s ¼ 1 is computed as
described above. Finally, one computes the state of the ‘‘slow’’
interactions (those only active at s ¼ 2) as a function of the
values at s¼1, and subsequently recomputes the steady states.
This provides the response at late events, s ¼ 2. The results
obtained can be plotted in a time-dependent manner (Figure
5). Here, one can also investigate the effect of different
knock-outs. For example, the absence of PAG has no effect on
key downstream elements of the cascade, due to the
redundant role of other negative regulatory mechanisms
(specifically, the degradation via c-Cbl and Cbl-b, and Gab-2–
mediated inhibition of PLCc1). Only a multiple knock-out of
these regulatory molecules leads to sustained activation of
key elements. Thus, these results point to a certain degree of
redundancy in negative feedbacks for switching off signaling.
This sort of qualitative analysis of the dynamics shows the
ability of the Boolean approach to reproduce the key
dynamic properties (transient versus sustained) of a signaling
process.
Discussion
In this contribution, a logical model describing a large
signaling network was established and analyzed. We set up a
Table 4. Robustness Analysis: Ranked List of the Most Sensitive
Interactions
Interaction
Caused Errors
if Removed
!ccblp1 þ tcrlig ! tcrb
39
!ccblp1 þ tcrp þ abl ! ¼ zap70
34
zap70 ! lat
27
tcrb þ lckr ! fyn
26
tcrbþfyn ! tcrp
26
fyn ! abl
26
pi3k þ !ship1 þ !pten ! pip3
21
lat ! plcgb
15
zap70 þ !gab2 þ gads ! slp76
15
lat ! gads
15
pip3 ! pdk1
13
lckp2 þ !cblb ! pi3k
11
lckr þ tcrb ! lckp2
11
!ikkab ! ikb
11
zap70 þ lat ! sh3bp2
10
plcgb þ !ccblp2 þ slp76 þ zap70 þ vav1 þ itk ! plcga
10
pdk1 ! pkb
10
pip3 þ zap70 þ slp76 ! itk
10
zap70 þ sh3bp2 ! vav1
10
!dgk þ plcga ! dag
9
!shp1 þ cd45 þ cd4 þ !csk þ lckr ! lckp1
8
cd28 ! x
8
tcrb þ lckp1 ! tcrp
8
lckp1 ! abl
8
mek ! erk
6
ras ! raf
6
ca ! cam
6
dag ! rasgrp
6
ip3 ! ca
6
lat ! grb2
6
grb2 ! sos
6
plcga ! ip3
6
raf ! mek
6
sos þ !gap þ rasgrp ! ras
6
x ! vav1
5
mkk4 ! jnk
5
mlk3 ! mkk4
5
rac1p1 ! mlk3
5
rac1r þ vav1 ! rac1p1
5
Each single non-input interaction was removed from the network followed by a
recomputation of the scenarios given in Table 1. The number of deviations from the 126
predictions made in Table 1 is shown. For abbreviations and comments on the
interactions, see Tables S1 and S2.
doi:10.1371/journal.pcbi.0030163.t004
Table 3. Minimal Intervention Sets To Produce the Full
Activation Pattern in T Cells
MIS
!gab2
pi3k
zap70
!gab2
pip3
zap70
pi3k
plcga
zap70
pi3k
slp76
zap70
pip3
slp76
zap70
pip3
plcga
zap70
pdk1
plcga
zap70
The MISs of maximal size 3 that induce sustained full activation (namely: ap1, bcat, bclxl,
cre, cyc1, nfkb, p70s, sre, and nfat are on, whereas fkhr, p21c, and p27k are off) of T cells
without external stimuli. The MISs were computed using CellNetAnalyzer. Note that the
exclamation mark ‘‘!’’ denotes ‘‘deactivation’’; species without this symbol have to be
activated (constitutively). Interestingly, the compounds involved in these MISs are
involved in oncogenesis (ZAP70, PI3K, Gab2, and PLCc1 are oncogenes, and SLP76 is
directly involved in PLCc1 activation, see Figure 1 and main text). Note that since PIP3 is a
second messenger and not ‘‘mutable’’, for the purpose of this analysis the MISs involving
its activation can be considered equivalent to those involving its activator PI3K (i.e., these
MISs are equivalent).
doi:10.1371/journal.pcbi.0030163.t003
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A Logical Model of T Cell Receptor Signaling
comprehensive Boolean model describing T cell signaling and
performed logical steady state analyses unraveling the
processing of signals and the global input–output behavior.
Moreover, by converting the logical model into an interaction
graph, we extracted further important features, such as
feedback loops, signaling paths, and network-wide interde-
pendencies. The latter can be captured in a dependency
matrix (as in Figure 2) which provides thousands of
qualitative predictions that can be falsified in perturbation
experiments. The logical model reproduces the global
behavior of this complex network for both natural and
perturbed conditions (knock-outs, inhibitors, mutations, etc.).
Its validity has been proven by reproducing published data
and by predicting unexpected results that were then verified
experimentally. Table 1 summarizes the results of 14 different
scenarios, in which the logical model predicted 126 states. For
44 of them, experimental data was available (15 from
literature and 29 from our own experiments) confirming
the predictions, except in the case discussed above.
Furthermore, we clearly show that the concept of inter-
vention sets allows one (a) to identify missing links in the
network, (b) to reveal failure modes that can explain the
effects of a physiological dysfunction or disease, and (c) to
search for suitable intervention strategies, while keeping
track of potential side effects, which is valuable for drug
target identification.
Compared with a kinetic model based on differential
equations, a Boolean approach is certainly limited regarding
the analysis of quantitative and dynamical aspects, and it
certainly cannot answer the same questions. However, to
establish such a model requires mainly the topology and only
a relatively small amount of quantitative data; hence, a
combination of information which is currently available in
large-scale networks. Although the model itself is qualitative
(i.e., discrete), it enables us not only to study qualitative
aspects of signaling networks, but it can also be validated by
semi-quantitative measurements such as those in Figures 3
and 4. In summary, with the network involved in T cell
activation as a case study, our approach proved to be a
Figure 5. Considering Different Time Scales, a Rough Description of the Dynamics Can Be Obtained
The activation of key elements upon activation of the TCR, the coreceptor CD4, and the costimulatory molecule CD28 is represented at the resting state,
s ¼ 0 (no inputs); early events s ¼ 1 (input(s), no feedback loops); and later-time events, s ¼ 2 (input(s), feedback loops). The black lines correspond to a
wild type while the green ones to a PAG KO. Note that the absence of PAG has no effect on key downstream elements of the cascade, due to the
redundant role of other negative regulatory mechanisms (degradation via c-Cbl and Cbl-b, Gab-2 mediated inhibition of PLCc1). Multiple knock-out of
these regulatory molecules leads to sustained activation of key elements (red lines).
doi:10.1371/journal.pcbi.0030163.g005
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A Logical Model of T Cell Receptor Signaling
promising in silico tool for the analysis of a large signaling
network, and we think that it holds valuable potential in
foreseeing the effects of drugs and network modifications.
Although sometimes the results of a logical model may
(afterward) appear to be obvious (as in the case of the CD28-
mediated JNK connection), it enables an exhaustive and
rigorous analysis of the information processing taking place
within a signaling network. Such a systematic analysis
becomes infeasible for a human being in large-scale systems.
In addition, the LIH can represent the situation of varying
cofactor functions; for example, that two substances A AND B
are required to activate a third substance C, but activation of
C in the presence of A and a fourth substance D requires B
not to be present.
Certainly, the logical model for T cell activation is far from
complete. We are just at the beginning of the reconstruction
process and other receptors and their pathways need to be
included. However, we feel that already in its current state,
the model may prove useful to inspire immunologists to ask
new questions which may first be answered in silico.
Furthermore, the model may also provide a framework for
those who may endeavor to quantitatively model TCR
signaling.
Methods
Logical network representation and analysis. We began construc-
tion of the signaling network for primary T cells by collecting data
from the literature and from our own experiments providing well-
established connections (Tables S1 and S2). As a first (intermediate)
result, we obtain an interaction graph. Interaction graphs are signed
directed graphs with the molecules (such as receptor, phosphatase, or
transcription factor) as nodes and signed arcs denoting the direct
influence of one species upon another, which can either be activating
(þ) or inhibiting (). For example, a positive arc leads from MEK to
ERK because the first phosphorylates and thereby activates the
second (Figure 1). From the incidence matrix of an interaction graph
we can identify important features such as feedback loops as well as
signaling paths and network-wide interdependencies between pairs
of species (e.g., perturbing A may have no effect on B as there is no
path connecting A to B). Algorithms related to these analyses are
well-known [48] and were recently presented in the context of
signaling networks [13]. However, from interaction graphs we cannot
conclude which combinations of signals reaching a species along the
arcs are required to activate that species. For example, in Figure 1,
Jun AND Fos are required to form active AP1.
For a refined representation of such relationships, we use a logical
(or Boolean) model in which we introduce discrete states for the
species (here the simplest (binary) case: 0 ¼ inactive or not present; 1
¼ active or present) and assign to each species a Boolean function.
Here we use a special representation of Boolean functions known as
disjunctive normal form (DNF, also called ‘‘sum of product’’
representation) which uses exclusively AND, OR, and NOT oper-
ators. A Boolean network with Boolean functions in disjunctive
normal form can be intuitively drawn and stored as a hypergraph
(LIH) [13], which is well-suited for studying the information flows
and input–output relationships in signal transduction networks
(Figure 1). In this hypergraph, each hyperarc connects its start nodes
with an AND operation (indicated by a blue circle in Figure 1) and
each hyperarc represents one possibility for how its end node can be
activated or produced (note that hyperacs may also have only one
start node, i.e., they are then ‘‘graph-like’’ arcs). Red branches
indicate species that enter the hyperarc with their negated value. For
example, PLCc-1 (PLCga in Figure 1) AND NOT DGK activates DAG
(see Figure 1). Note that each LIH has a unique underlying
interaction graph (which can be easily derived from the LIH
representation by splitting the AND connections), whereas the
opposite is, in general, not true.
Within this logical framework we may study the effect of a set of
input stimuli (typically ligands) on downstream signaling by comput-
ing the logical steady state [13] that results by propagating the signals
through the network from the input to the output layer. It seems
worthwhile to remark that the updating assumption (synchronous
versus asynchronous [14,15])—which must usually be made when
dealing with dynamic Boolean networks—is not relevant here as we
focus on the logical steady states, which are equivalent in both cases.
Sometimes a logical steady state is not unique or does not exist due to
the presence of feedbacks loops. However, many feedback loops
become active only in a longer timescale justifying setting them OFF
in the first wave of signal propagation (allowing them to be switched
ON for the second timescale). This has been used here for several
feedback loops (see main text and Table S2). The effect of knocking-
out a species can be tested by re-computing the (new) logical steady
state for the respective stimuli. MISs satisfying a given intervention
goal can be computed by systematically testing sets of permanently
activated or/and deactivated nodes [13,31].
All mathematical analyses and computations have been performed
with our software tool CellNetAnalyzer [31], a comprehensive user
interface for structural analysis of cellular networks. CellNetAnalyzer
and the T cell model can be downloaded for free (for academic use)
from http://www.mpi-magdeburg.mpg.de/projects/cna/cna.html.
Immunoblotting. Human or mouse T cells were purified using an
AutoMACS magnetic isolation system according to the manufac-
turer’s instructions (Miltenyi, http://www.miltenyibiotec.com). Mouse
T cells were stimulated with 10 lg/ml of biotinylated CD3e (a subunit
of the TCR) antibody (145–2C11, BD Biosciences, http://www.
bdbiosciences.com/), 10 lg/ml of biotinylated CD28 antibody (37.51,
BD Biosciences), CD3 plus CD28 mAbs, or with CD3 plus 10 lg/ml of
biotinylated CD4 (GK1.5, BD Biosciences) followed by crosslinking
with 25 lg/ml of streptavidin (Dianova, http://www.dianova.de) at 37
8C for the indicated periods of time. Human T cells were stimulated
with CD3e mAb MEM92 (IgM, kindly provided by Dr. V. Horejsi,
Prague, Czech Republic) or with CD3 plus CD28 mAbs (248.23.2).
Cells were lysed in buffer containing 1% NP-40, 1% laurylmaltoside
(N-dodecyl b-D-maltoside), 50 mM Tris pH 7.5, 140 mM NaCl, 10mM
EDTA, 10 mM NaF, 1 mM PMSF, 1 mM Na3VO4. Proteins were
separated by SDS/PAGE, transferred onto membranes, and blotted
with the following antibodies: anti-phosphotyrosine (4G10), anti-
ERK1/2 (pT202/pT204), anti-JNK (pT183/pY185), anti-phospho-Akt
(S473) (all from Cell Signaling, http://www.cellsignal.com/), anti-
ZAP70 (pTyr 319, Cell Signaling), anti-ZAP70 (cloneZ24820, Trans-
duction Laboratories, http://www.bdbiosciences.com/), or against b-
Actin (Sigma, http://www.sigmaaldrich.com/). Where PI3K and src-
kinase inhibitors were used, T cells were treated with 100 nM
Wortmannin (Calbiochem, http://www.emdbiosciences.com) or 10 lM
PP2 (Calbiochem) for 30 min at 37 8C prior to stimulation. All
experiments have been repeated three times and reproduced the
shown results.
Supporting Information
Table S1. List of Compounds in the Logical T Cell Model
Model name corresponds to the name in Figure 1 and Table S2.
Common abbreviations are those usually used in the literature, while
name is the whole name. Type classifies the molecules, if applies, as
follows: K ¼ Kinase, T ¼ Transcription Factor, P ¼ Phosphatase, A ¼
Adaptor Protein, R ¼ Receptor, G ¼ GTP-ase. In the case where two
pools of a molecule were considered, a ‘‘reservoir’’ was included
which was required for both pools. This allows us to perform a
simultaneous knock-out of both pools.
Found at doi:10.1371/journal.pcbi.0030163.st001 (56 KB PDF).
Table S2. Hyperarcs of the Logical T Cell Signaling Model (see Figure
1 and Methods)
Exclamation mark (‘‘!’’) denotes a logical NOT, and dots within the
equations indicate AND operations. The names of the substances in
the explanations are those used in the model and Figure 1; the
biological names are displayed in Table S1. In the case where two
pools of a molecule were considered (e.g., lckp1 and lckp2), a
‘‘reservoir’’ (lckr) was included which was required for both pools.
This allows us to perform a simultaneous knock-out of both pools
acting on the reservoir.
Found at doi:10.1371/journal.pcbi.0030163.st002 (183 KB PDF).
Acknowledgments
The authors would like to thank the members of the signaling group at
the Institute of Immunology (M. Smida, X. Wang, S. Kliche, R. Pusch, M.
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A Logical Model of T Cell Receptor Signaling
Togni, A. Posevitz, V. Posevitz, T. Drewes, U. Ko¨ lsch, S. Engelmann) and
I. Merida and J. Huard for essential biological input into the model.
Author contributions. JSR set up the model and performed the
analysis. LS, JAL, UB, BA, and BS, with the help of the signaling group
at the Institute of Immunology, gathered the biological details of the
model and analyzed the correctness of the results. LS and BA
performed the wet-lab experiments. RH supported the model setup,
analysis, and documentation. SK developed the theoretical methods
and tools (CellNetAnalyzer) and supported the analysis. UUH and RW
contributed to the theoretical methods with useful insights. EDG, BS,
and RW coordinated the project.
Funding. The authors are thankful for the support of the German
Ministry of Research and Education to EDG (Hepatosys), the German
Research Society to BS and EDG (FOR521), and the Research Focus
Dynamical Systems funded by the Saxony-Anhalt Ministry of
Education.
Competing interests. The authors have declared that no competing
interests exist.
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17722974
|
tcrlig = ( tcrlig )
vav1 = ( zap70 AND ( ( ( sh3bp2 ) ) ) ) OR ( xx )
lat = ( zap70 )
pi3k = ( ( xx ) AND NOT ( cblb ) ) OR ( ( lckp2 ) AND NOT ( cblb ) )
gap = ( unknown_input3 )
rac1r = ( unknown_input )
Dummy = ( ( plcgb AND ( ( ( vav1 AND zap70 AND itk AND slp76 ) ) ) ) AND NOT ( ccblp2 ) )
ca = ( ip3 )
pkb = ( pdk1 )
nfkb = NOT ( ( ikb ) )
lckp1 = ( ( cd45 AND ( ( ( lckr AND cd4 ) AND ( ( ( NOT csk ) ) ) ) ) ) AND NOT ( shp1 ) )
nfat = ( calcin )
bclxl = NOT ( ( bad ) )
gadd45 = ( unknown_input )
bad = NOT ( ( pkb ) )
pag = ( ( fyn ) AND NOT ( tcrb ) )
ccblp2 = ( ccblr AND ( ( ( fyn ) ) ) )
cyc1 = NOT ( ( gsk3 ) )
vav3 = ( sh3bp2 )
sre = ( rac1p2 ) OR ( cdc42 )
ikkg = ( card11a AND ( ( ( pkcth ) ) ) )
calcin = ( ( ( ( cam ) AND NOT ( calpr1 ) ) AND NOT ( akap79 ) ) AND NOT ( cabin1 ) )
cabin1 = NOT ( ( camk4 ) )
grb2 = ( lat )
ship1 = ( unknown_input2 )
plcga = ( ( plcgb AND ( ( ( vav1 AND zap70 AND itk AND slp76 ) ) ) ) AND NOT ( ccblp2 ) )
sh3bp2 = ( zap70 AND ( ( ( lat ) ) ) )
shp2 = ( gab2 )
pten = ( unknown_input2 )
mlk3 = ( hpk1 ) OR ( rac1p1 )
gsk3 = NOT ( ( pkb ) )
cd45 = ( unknown_input )
dag = ( ( plcga ) AND NOT ( dgk ) )
bcl10 = ( unknown_input2 )
gab2 = ( grb2 AND ( ( ( lat AND zap70 ) ) ) ) OR ( gads AND ( ( ( lat AND zap70 ) ) ) )
erk = ( mek )
ap1 = ( fos AND ( ( ( jun ) ) ) )
fos = ( erk )
cblb = NOT ( ( cd28 ) )
gads = ( lat )
pdk1 = ( pip3 )
itk = ( pip3 AND ( ( ( zap70 AND slp76 ) ) ) )
ras = ( ( sos AND ( ( ( rasgrp ) ) ) ) AND NOT ( gap ) )
ikb = NOT ( ( ikkab ) )
ccblp1 = ( ccblr AND ( ( ( zap70 ) ) ) )
p21c = NOT ( ( pkb ) )
tcrp = ( tcrb AND ( ( ( fyn OR lckp1 ) ) ) )
rac1p2 = ( rac1r AND ( ( ( vav3 ) ) ) )
rsk = ( erk )
fyn = ( lckp1 AND ( ( ( cd45 ) ) ) ) OR ( tcrb AND ( ( ( lckr ) ) ) )
hpk1 = ( lat )
cam = ( ca )
cre = ( creb )
akap79 = ( unknown_input2 )
p70s = ( pdk1 )
p38 = ( ( zap70 ) AND NOT ( gadd45 ) )
card11a = ( malt1 AND ( ( ( card11 AND bcl10 ) ) ) )
pip3 = ( ( ( pi3k ) AND NOT ( ship1 ) ) AND NOT ( pten ) )
card11 = ( unknown_input )
ccblr = ( unknown_input )
creb = ( rsk )
zap70 = ( ( tcrp AND ( ( ( abl ) ) ) ) AND NOT ( ccblp1 ) )
jun = ( jnk )
camk2 = ( cam )
jnk = ( mkk4 ) OR ( mekk1 )
sos = ( grb2 )
slp76 = ( ( zap70 AND ( ( ( gads ) ) ) ) AND NOT ( gab2 ) )
mkk4 = ( mekk1 ) OR ( mlk3 )
cdc42 = ( unknown_input2 )
lckp2 = ( lckr AND ( ( ( tcrb ) ) ) )
mekk1 = ( hpk1 ) OR ( cdc42 ) OR ( rac1p2 )
lckr = ( lckr_input )
plcgb = ( lat )
ip3 = ( plcga )
raf = ( ras )
ikkab = ( ikkg AND ( ( ( camk2 ) ) ) )
rlk = ( lckp1 )
fkhr = NOT ( ( pkb ) )
cd28 = ( cd28 )
malt1 = ( unknown_input )
pkcth = ( dag AND ( ( ( pdk1 AND vav1 ) ) ) )
p27k = NOT ( ( pkb ) )
rasgrp = ( dag )
mek = ( raf )
rac1p1 = ( rac1r AND ( ( ( vav1 ) ) ) )
tcrb = ( ( tcrlig ) AND NOT ( ccblp1 ) )
abl = ( fyn ) OR ( lckp1 )
csk = ( pag )
xx = ( cd28 )
shp1 = ( ( lckp1 ) AND NOT ( erk ) )
dgk = ( tcrb )
bcat = NOT ( ( gsk3 ) )
calpr1 = ( unknown_input2 )
camk4 = ( cam )
|
"BioMed Central\nPage 1 of 26\n(page number not for citation purposes)\nBMC Bioinformatics\nOpen Acc(...TRUNCATED) |
16464248
| "Lck = ( ( CD45 AND ( ( ( CD8 ) ) ) ) AND NOT ( PAGCsk ) ) \nZAP70 = ( ( TCRphos AND ( ( ( Lck (...TRUNCATED) |
"BioMed Central\nPage 1 of 18\n(page number not for citation purposes)\nTheoretical Biology and Medi(...TRUNCATED) |
16542429
| "IL4 = ( ( GATA3 ) AND NOT ( STAT1 ) ) \nSOCS1 = ( Tbet ) OR ( STAT1 ) \nIFNg = ( ( STAT4 ) AND (...TRUNCATED) |
"Emergent decision-making in biological signal\ntransduction networks\nToma´sˇ Helikar*, John Konv(...TRUNCATED) |
18250321
| "PTP1b = ( NOT ( ( EGFR AND ( ( ( EGF ) ) ) ) OR ( Stress ) ) ) OR NOT ( EGFR OR EGF OR Stress(...TRUNCATED) |
"Global control of cell cycle transcription by coupled CDK and\nnetwork oscillators\nDavid A. Orland(...TRUNCATED) |
18463633
| "MBF = ( CLN3 ) \nHCM1 = ( MBF AND ( ( ( SBF ) ) ) ) \nSWI5 = ( SFF ) \nYOX1 = ( MBF AND ( ( ( SB(...TRUNCATED) |
"BioMed Central\nPage 1 of 15\n(page number not for citation purposes)\nBMC Systems Biology\nOpen Ac(...TRUNCATED) |
18433497
| "TBK1 = ( External_Activator ) \nIL1R1 = ( External_Activator ) \nAPAF1gene = ( TP53nucleus ) \nILIB(...TRUNCATED) |
"Network model of survival signaling in large granular\nlymphocyte leukemia\nRanran Zhang†, Mithun(...TRUNCATED) |
18852469
| "FasT = ( ( NFKB ) AND NOT ( Apoptosis ) ) \nsFas = ( ( FasT ) AND NOT ( Apoptosis ) ) \nIL2 = ((...TRUNCATED) |
"BioMed Central\nPage 1 of 14\n(page number not for citation purposes)\nBMC Systems Biology\nOpen Ac(...TRUNCATED) |
19025648
| "SREBP_SCAP = ( ( Insig_SREBP_SCAP ) AND NOT ( Statins ) ) \nMevalonyl_pyrophosphate = ( Mevalonic(...TRUNCATED) |
"BioMed Central\nPage 1 of 20\n(page number not for citation purposes)\nBMC Systems Biology\nOpen Ac(...TRUNCATED) |
19118495
| "ERa = ( MEK1 ) OR ( Akt1 ) \np21 = ( ( ( ( ERa ) AND NOT ( Akt1 ) ) AND NOT ( CDK4 ) ) AND NO(...TRUNCATED) |
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