plosPLoS Comput BiolploscompPLoS Computational Biology1553-734X1553-7358Public Library of ScienceSan Francisco, USA08-PLCB-RA-0712R310.1371/journal.pcbi.1000390Research ArticleCell Biology/Cell SignalingComputational Biology/Signaling NetworksDevelopmental Biology/Cell DifferentiationComputational Models of the Notch Network Elucidate Mechanisms of
Context-dependent SignalingModels of Notch SignalingAgrawalSmita123¤aArcherColin123¤bSchafferDavid V.123*Department of Chemical Engineering, University of California Berkeley,
Berkeley, California, United States of AmericaDepartment of Bioengineering, University of California Berkeley,
Berkeley, California, United States of AmericaHelen Wills Neuroscience Institute, University of California Berkeley,
Berkeley, California, United States of AmericaMiyanoSatoruEditorUniversity of Tokyo, Japan* E-mail: schaffer@berkeley.edu
Current address: Institute of Human Genetics, University of Minnesota Twin
Cities, Minneapolis, Minnesota, United States of America
Current address: Australian Institute of Bioengineering and Nanotechnology,
The University of Queensland, Brisbane, Australia
Conceived and designed the experiments: SA DVS. Performed the experiments:
SA. Analyzed the data: SA. Contributed reagents/materials/analysis tools: SA
CA. Wrote the paper: SA.
The authors have declared that no competing interests exist.
52009225200955e1000390218200817420092009Agrawal et alThis 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.
The Notch signaling pathway controls numerous cell fate decisions during
development and adulthood through diverse mechanisms. Thus, whereas it functions
as an oscillator during somitogenesis, it can mediate an all-or-none cell fate
switch to influence pattern formation in various tissues during development.
Furthermore, while in some contexts continuous Notch signaling is required, in
others a transient Notch signal is sufficient to influence cell fate decisions.
However, the signaling mechanisms that underlie these diverse behaviors in
different cellular contexts have not been understood. Notch1
along with two downstream transcription factors hes1 and
RBP-Jk forms an intricate network of positive and negative
feedback loops, and we have implemented a systems biology approach to
computationally study this gene regulation network. Our results indicate that
the system exhibits bistability and is capable of switching states at a critical
level of Notch signaling initiated by its ligand Delta in a particular range of
parameter values. In this mode, transient activation of Delta is also capable of
inducing prolonged high expression of Hes1, mimicking the
“ON” state depending on the intensity and duration of the
signal. Furthermore, this system is highly sensitive to certain model parameters
and can transition from functioning as a bistable switch to an oscillator by
tuning a single parameter value. This parameter, the transcriptional repression
constant of hes1, can thus qualitatively govern the behavior of
the signaling network. In addition, we find that the system is able to dampen
and reduce the effects of biological noise that arise from stochastic effects in
gene expression for systems that respond quickly to Notch signaling.
This work thus helps our understanding of an important cell fate control system
and begins to elucidate how this context dependent signaling system can be
modulated in different cellular settings to exhibit entirely different
behaviors.
Author Summary
The Notch signaling pathway is an evolutionarily conserved signaling system that
is involved in various cell fate decisions, both during development of an
organism and during adulthood. While the same core circuit functions in various
different cellular contexts, it has experimentally been shown to elicit varied
behaviors and responses. On the one hand, it functions as a cellular oscillator
critical for somitogenesis, whereas in other situations, it can function as a
cell fate switch to pattern developing tissue, for example in the
Drosophila eye. Furthermore, malfunctioning of Notch
signaling is implicated in various cancers. To better understand the underlying
mechanisms that allow the network to function distinctly in different contexts,
we have mathematically modeled the behavior of the Notch network, encompassing
the Notch gene along with two of its downstream effector transcription factors,
which together form a network of positive and negative feedback loops. Our
results indicate that the qualitative and quantitative behavior of the system
can readily be tuned based on key parameters to reflect its multiple roles.
Furthermore, our results provide insights into alterations in the signaling
system that lead to malfunction and hence disease, which could be used to
identify potential drug targets for therapy.
This work was supported by the Helen Wills Neuroscience Institute and LDRD
3668DS. The funders had no role in study design, data collection and analysis,
decision to publish, or preparation of the manuscript.Introduction
Cells continuously receive signals from their microenvironments – including
factors present in the extracellular matrix, soluble media, and surrounding cells
– which collectively influence cell function and behavior via activating
intracellular signal transduction and gene regulation networks. These networks
generally involve complex, nonlinear interactions of proteins, such as
phosphorylation cascades (reviewed in [1]) and second messenger
signaling systems [2], whose structures feature positive and negative
feedback loops, feed-forward interactions, signal amplification, and cross-talk with
other pathways [3]. Mathematical models of these interactions are
therefore very insightful or even necessary avenues to analyze and understand the
regulation of cell behavior, as the properties of these networks can exceed an
intuitive understanding [4]–[6].
Notch is a signaling system required for numerous critical cell fate specification
events during the development of the nervous system, hematopoietic system, eye, and
skin [7]–[11]. The receptor for
this pathway is the single pass transmembrane protein Notch that, when bound by its
ligands Delta or Jagged, undergoes a series of cleavage events to release its
intracellular domain (NICD) [9],[12].
This NICD then translocates into the nucleus and acts as a transcriptional
upregulator of target genes, including members of the hes family,
through its interaction with the transcription factor RBP-Jκ [13]. In mammals
there are four different Notch proteins (Notch1-4) and 5 ligands (Delta 1, 3, and 4
and Jagged 1 and 2). For this study, we have focused primarily on the Notch1
signaling pathway.
In its role as a critical regulator of cell fate [7]–[11],
Notch has been known to function via lateral inhibition and induction mechanisms to
create fine-grained patterns in undifferentiated cells, a process required for
proper boundary formation and differentiation of various tissues [14],[15]. It
can also function as a binary cell fate switch, for example during differentiation
of the epidermis [16] and endodermal epithelium of the gut [17], to
promote differentiation of one cell type from precursor cells at the expense of
another. Furthermore, in some cases continuous Notch activity is not required for
cell fate specification. For example, transient Delta-Notch signaling has been shown
to be sufficient to induce T-cell [18] and NK cell differentiation [19] from their respective
precursor cells, and can induce an irreversible switch to gliogenesis in neural
crest stem cells [20]. Notch signaling also occurs only transiently in
many instances during the development of Drosophila[21],
zebrafish [22],[23], and mice [24]. It was also
recently shown that human embryonic stem cells (hESCs) require activation of Notch
signaling to form the progeny of all three embryonic germ layers, and subsequent
transient Notch signaling enhanced generation of hematopoietic cells from committed
hESCs [25].
The mechanisms by which a short Notch signaling pulse can permanently switch cell
fate are not elucidated.
The Notch system has also been shown to function as an oscillator. Specifically, the
expression levels of members of the hes family, a group of
downstream Notch target genes [26], have been shown to oscillate with a 2 hour
periodicity in some systems during development, which for example aids in
somitogenesis (i.e. the patterning of somites) [27]–[29]. Hes1
protein and mRNA concentrations have also been observed to oscillate with an
approximate 2 hr time period upon serum starvation in various cultured cell lines
including myoblasts, fibroblasts, and neuroblastoma cells [30]. Furthermore,
oscillations in the Notch network have been proposed to be important in maintaining
neural progenitor cells in an undifferentiated state [31]. Finally, there is
evidence that such oscillations may also afford cells the opportunity to repeatedly
test for the continued existence of a signal [32], thereby increasing
cellular response sensitivity and flexibility by allowing the cell to integrate the
results of many periodical evaluations of the signal before making an ultimate cell
fate decision.
The Delta-Notch signaling system has been previously modeled to elucidate its role in
fine-grained pattern formation through the action of lateral inhibition and
induction [33]–[35]. Collier et
al. developed a simple 2-parameter model that focuses on pattern formation
due to feedback inhibition between adjacent cells via Delta-Notch signaling [33]. Other
models build upon this simple model by adding more molecular detail at the
intercellular level [34],[35]. In addition, several studies have focused on
trying to understand the underlying mechanism of Notch system oscillations [32],[36], where a
Hes1 negative feedback loop composed of Hes1 protein repressing
hes1 transcription, likely plays a central role [37].
Delays related to transcription and translation were also proposed to be important
for the observed oscillations [38]. However, while several models have thus been
proposed and have yielded important insights into this system [30], [36], [38]–[40], they
have focused exclusively on Hes1 and not analyzed its interactions with other
signaling proteins in the Notch system. Additionally, all these models focus on a
particular aspect or mode of Notch signaling (e.g. lateral inhibition or
oscillation) but do not yet address how complex, alternative behaviors could arise
from the same network.
Here we mathematically model the Notch signaling system to analyze how the same
network is capable of functioning as a cell fate switch or an oscillator in
different biological contexts. This model, which includes the regulation of the
notch1-RBP-Jk-hes1 gene circuit, predicts that the Notch1-Hes1
system acts as a bistable switch in certain regions of parameter space, where Hes1
levels can change by 1–2 orders of magnitude as a function of the input
Delta signal. In addition, it predicts that a transient pulse of a high level of
Delta is capable of inducing high Hes1 expression levels for a duration that would
be sufficient to induce a cell fate switch. Moreover, the model elucidates how the
network can be ‘tuned’ to function in different regimes, either
as an oscillator or a cell fate switch, by changing a key parameter. Finally, low
numbers of reactants can lead to significant statistical fluctuations in molecule
numbers and reaction rates, making cells intrinsically noisy biochemical reactors
[41],[42]. Stochastic
simulations of the Notch system, which enable the analysis of the effect of
biological noise in the system arising due to stochastic variations in gene
expression, reveal that for systems that respond quickly to Notch signaling, the
network is able to dampen the effects of this biological noise and function in a
manner similar to what is predicted by the deterministic model. In summary, the
model enables analysis of the different behavioral responses of the Notch signaling
network observed over a broad spectrum of signaling inputs and parameter values and
can be further expanded to study Notch signaling in numerous contexts.
Methods
We developed a model of Notch signaling to investigate how this system can function
as either an oscillator or as a simple binary switch capable of responding to steady
state or transient inputs. Brief experimental work revealed that the
notch1 promoter is positively upregulated by its gene product and
is downregulated by Hes1 (Text S1, Fig. S1). We thus examined the behavior of the
notch1, RBP-Jκ, and
hes1 genes, which form a complex set of regulatory feedback loops
(Fig. 1). A deterministic
model composed of a system of differential equations was developed to analyze
dynamic changes in the levels of the network constituents. However, since the
concentrations of some of the species were low, stochastic simulations were also
conducted to examine whether noise in the levels of the network components could
significantly impact system behavior, as noise has the potential to undermine the
fidelity of cell fate choices [41],[43].
10.1371/journal.pcbi.1000390.g001
Schematic of the Notch1-RBP-Jκ-Hes1 signaling network.
(A) Each arrow represents a term or event in the differential equation model
including transcription, translation, mRNA and protein degradation, nuclear
import, TF binding, receptor-ligand binding and receptor processing. (B)
Schematic of the positive and negative feedback loops of the
Notch1-RBP-Jk-Hes1 network. (-|) represents repression and (->)
represents activation of target genes.
Deterministic Model Development
A set of differential equations was developed to track changes in the
concentrations of various species in the nucleus and cytoplasm of a cell as a
function of time following activation of Notch by its ligand. The cell is
modeled as a 10 µm diameter sphere with a 5 µm diameter
nucleus. Numerous processes were modeled as terms in the differential equation
system, including transcription, translation, transport, degradation or - in the
case of Notch - receptor cleavage (Fig. 1A, Text S1). As examples, the three equations
tracking the Hes1 cytoplasmic mRNA, Hes1 cytoplasmic protein and Hes1 nuclear
protein concentrations are given by:The rate of change (in units of ) of the cytoplasmic mRNA concentration of Hes1 is given by the
difference in the rates of it transcription and degradation. RfHcm is
the transcription rate of hes1 mRNA in the nucleus. We assume
instantaneous export of mRNA to the cytoplasm. A factor of 7 is included to take
into account the dilution due to export to the cytoplasm (Text S1).
kdHcm, kdHcp, and kdHnp denote the
degradation constants for the hes1 mRNA (Hcm), cytoplasmic
protein (Hcp), and nuclear protein (Hnp), respectively, which are assumed to
undergo first order degradation kinetics. ktrHc denotes the
translation constant (min−1) for conversion of cytoplasmic
hes1 mRNA into cytoplasmic protein. Transcriptional and
translational delay times are incorporated into the model, as these are
processes that inherently involve delays between initiation and the production
of a molecule of mRNA or protein, as previously described [38],[44].
Thus, the translation of Hes1 protein is based on the delayed
hes1 mRNA concentration HcmD (delayed by time TpHc,
the average time for translation of Hes1), which is the concentration of mRNA
present when the process of translation was initiated instead of the
concentration at the present time. kniHcp denotes the nuclear import
rate in units of min−1. A dilution factor of 7 is again
used to incorporate differences in nuclear and cytoplasmic volumes.
The hes1 Promoter
The transcription rates for notch1, hes1, and
RBP-Jκ are based on the states of their respective
promoters. Previous promoter analysis has been complemented with Genomatix Suite
Gene2Promoter transcription factor (TF) binding site prediction software to
identify potential TF binding sites in the promoters of the three genes in the
model.
Takebayashi et al. [37] observed that hes1
transcription is repressed by its own gene product through Hes1 protein binding
to sites in the hes1 promoter termed N-boxes. Through a series
of binding and transcriptional activity assays, the study determined that Hes1
bound strongly to three N-boxes found upstream of the transcriptional start site
and repressed transcription of the hes1 mRNA up to 40-fold.
Also, while the work concluded that there was a synergistic rather than an
additive effect of the N-box binding dependent repression of gene expression,
further mathematical analysis has indicated that there is no or very weak
synergy among the different binding sites [45]. Several positive
regulatory regions were also found in the hes1 promoter, and it
was also shown to have two adjacent RBP-Jκ binding sites [46],[47]. Thus, we have
modeled the hes1 promoter to have three equivalent N-boxes
where the Hes1 protein can bind and repress transcription, as well as two
equivalent RBP-Jκ sites. The presence of all other positive regulators
of transcription is lumped into a constant basal rate of transcription.
The notch1 Promoter
As it has not been extensively investigated, the notch1 promoter
sequence was analyzed in the Gene2Promoter software. One putative Hes1 site
(N-box) and two putative RBP-Jκ sites were found in the ∼1 kb
notch1 promoter analyzed. This may imply that
notch1 is both positively and negatively regulated by its
own gene product. To test this, a transcriptional activity experiment using the
dual luciferase assay system was conducted. The promoter of murine
notch1[48]
was used to drive expression of hRluc cDNA (Renilla luciferase). Co-transfection
studies with plasmids expressing RBP-Jκ, NICD, Hes1, and dNHes1 (a
dominant negative form of Hes1) indeed demonstrated that the
notch1 promoter is regulated negatively by Hes1 and
RBP-Jκ in the absence of NICD but is positively regulated by NICD in the
presence of RBP-Jκ (Text S1, Fig. S1).
The notch1 promoter was modeled with two RBP-Jκ sites
and one N-box.
The RBP-Jκ Promoter
A 418 bp sequence upstream of the RBP-Jκ gene as
characterized by Amakawa et al. [49] was analyzed in
the Gene2Promoter software for TF binding sites of interest. Three potential
Hes1 binding sites and three potential RBP-Jκ sites were found. Thus,
the RBP-Jκ gene also potentially undergoes
autoregulation under Notch signaling, and a three N-box, three RBP-Jκ
site model was utilized.
Modeling the Transcription Term
As discussed above, all three promoters have one or more binding sites for both
Hes1 (the N-box) and RBP-Jκ. It is assumed that Hes1 can bind and
repress transcription of the corresponding promoter only in its homodimer form,
and the dimerization reaction is assumed to be at steady state over timescales
of protein transcription, translation and import, driven by mass action kinetics
such that the concentration of the dimer is given by:Where, KaHp is the association equilibrium constant
for the dimerization reaction. Similarly, the time scales of transcription
factor binding to and dissociation from the promoter elements are also assumed
to be much faster than those of gene transcription and protein synthesis, such
that binding to the promoter is at pseudo steady state. In addition, it is
assumed that NICD can bind only when an RBP-Jk protein is bound to its site on
the promoter, and that this NICD binding converts RBP-Jκ from a
transcriptional repressor to an activator [13].
The level of promoter activation (i.e. rate of mRNA synthesis) is modeled by an
approach termed BEWARE [50],[51], in which the probabilities of a promoter
being in any one of its many possible states are calculated based on the
relative concentrations of the three transcription factors (Hes1, RBP-Jκ
and NICD), and their respective DNA binding affinities, using equilibrium
binding equations. The level of activation of the promoter is then given by: , where, P[Pi] is the probability
of the promoter being in state i, and vi is the activation rate of
gene transcription associated to the promoter being in that state i. When the
promoter is empty, the gene activation rate is assumed to be the basal
transcription rate (Vb) for that promoter. When a Hes1 dimer is bound
to an N-box, the rate is reduced by a factor rN that takes into
account the repressive effect of the Hes1 transcription factor, and when
RBP-Jκ is bound, the rate is reduced by a factor rR.
Furthermore, when the promoter is in its maximally activated state with the NICD
bound to the RBP-Jκ and no Hes1 dimers bound, the activation rate is
assumed to be at its maximum and is given by
(Vmax+Vb). In the case of multiple RBP-Jκ
binding sites, an additional factor tc (<1) is used to account
for states where not all RBP-Jκ sites bind NICD to represent the
decrease from the maximum possible activation rate. For a detailed expression of
transcription rates please refer to Supplemental Materials (Text S1).
Although explicit parameters have been included to account for cooperative
binding for Hes1 dimers to multiple N-boxes and for RBP-Jκ binding
(cooperativity factors Cn, Cr and Cnr - please
refer to Table 1 for model
parameters), they have been set to 1 for these simulations, as recent work
suggests there is very little if any cooperative effect in Hes1 binding to
N-boxes [45]. Finally, it is assumed that each mRNA produces a
fixed number of proteins, i.e. mRNA dynamics have been neglected [50].
10.1371/journal.pcbi.1000390.t001
Parameter values used for the models.
Description
Symbol
Value
Source
Degradation constant of Hes1 protein
(min−1)
kdHcp, kdHnp
0.0315
[30]
Degradation constant of Hes1 mRNA
(min−1)
kdHcm
0.029
[30]
Degradation constant of RBP-Jk protein
(min−1)
kdRcp
0.00231
[82]
Degradation constant of RBP-JK mRNA
(min−1)
kdRcm
0.0075
[82]
Degradation constant of full-length Notch1 protein
(min−1)
kdNp
0.017
[54], Text
Degradation constant of NICD protein
(min−1)
kdNcp,kdNnp
0.0014 or 0.00385
[55], Text
Degradation of Notch mRNA
(min−1)
kdNm
0.0058
[83]
Cooperativity factor for Hes1-DNA binding
Cn
1
Text
Cooperativity factor for RBP-Jk DNA binding
Cr
1
Text
Cooperativity factor for RBP-Jk Hes1 DNA binding
Cnr
1
Text
Rate of protein translation from Hes1 mRNA
(min−1)
KtrHc
4.5
[45], Text
Rate of protein translation from RBP-Jk mRNA
(min−1)
KtrRc
2.5
Text
Rate of protein translation from Notch1 mRNA
(min−1)
KtrN
1
Text
RBP-Jk DNA association constant
(M−1)
Kr
3.23×108
[84]
Hes1 DNA association constant
(M−1)
Kn
2×108
Text
RBP-Jk NICD association constant
(M−1)
Ka
1×108
Text
Hes1 dimer association constant
(M−1)
KaHp
1×109
Text
Transcriptional time delay for Hes1 (min)
TmHc
10
[52], Text
Translational time delay for Hes1 (min)
TpHc
2.35
[52], Text
Transcriptional time delay for RBP-Jk (min)
TmRc
20
[52], Text
Translational time delay for RBP-Jk (min)
TpRc
4.3
[52], Text
Transcriptional time delay for Notch1 (min)
TmNc
70
[52], Text
Translational time delay for Notch1 (min)
TpNc
21
[52], Text
Basal transcriptional rate for Hes1 (M/min)
Vbh
1.14×10−10
[45], Text
Basal transcriptional rate for RBP-Jk (M/min)
Vbr
4.3×10−11
Text
Basal transcriptional rate for Notch1 (M/min)
Vbn
1.23×10−11
Text
Maximal transcriptional rate for Hes1 (M/min)
Vmaxh
5×10−10
[53], Text
Maximal transcriptional rate for RBP-Jk (M/min)
Vmaxr
2×10−10
Text
Maximal transcriptional rate for Notch1 (M/min)
Vmaxn
5.5×10−11
Text
Nuclear import rate of Hes1 protein
(min−1)
kniHcp
0.1
[85], Text
Nuclear import rate of RBP-Jk protein
(min−1)
kniRcp
0.1
[85], Text
Nuclear import rate of NICD protein
(min−1)
kniNcp
0.1
[85], Text
NICD generation constant upon Delta binding
(M−1 min−1)
KfNcp
7.6×107
Text
Repression constant of Hes1 bound to N-box
rNbox
0.3
[30], Text
Repression constant of RBP-Jk alone bound to
promoter
rR
0.2
Text
Parameter Determination
Experimentally determined values for half-lives of proteins and mRNA, association
and dissociation constants of proteins to their respective DNA binding sites,
dimerization constants, and protein translation and transcription rates have
been used when possible (Table
1). These values are often not available for the exact species of
interest; however, the best available estimates based on similar protein classes
are used wherever applicable as the starting point. The time delays for
transcription and translation for each of the three genes are calculated as
previously described [52] and are detailed in the Supplemental
Materials (Text
S1). 4.5 transcripts per minute [45] and 20 transcripts
per minute [53] were used as initial estimates for
hes1 basal and maximum transcription rates respectively.
The transcription rates for RBP-Jκ and
notch1 were then determined from these estimates and the
estimates of their minimum transcription times (Text S1).
The degradation rates for the Hes1 protein and mRNA were determined
experimentally by Hirata et al. in fibroblasts [30]. They observed
similar values in other cultured cell types including myoblasts, neuroblastomas,
and teratocarcinomas. Pulse chase experiments of Logeat et al. [54]
were used to assess the degradation rates for the full-length Notch1 protein,
and an estimate of Notch1 protein half-life of ∼40 minutes was derived.
GSK3β has been shown to affect the stability of NICD [55].
Although there are conflicting results as to whether GSK3β helps to
stabilize [55] or destabilize the cleaved NICD [56],
our experimental results show that GSK3β is essential for the NICD
regulation of neural stem cell differentiation into astrocytes (Agrawal, Ngai,
and Schaffer, manuscript in preparation). Furthermore, we show that Notch1
signaling upregulates the expression of GSK3β in these cells. Thus, the
effect of GSK3β is incorporated into the model by increasing the
half-life of NICD from 3 to 8 hrs [55] above a threshold
concentration of Hes1 (which is assumed to directly or indirectly regulate the
expression of GSK3β). This increased NICD half-life does not however
change the qualitative behavior of the Hes1 switch (Fig. S3A).
The repression constant of Hes1 dimer bound to an N-box (rNbox) is
estimated from the results of Takebayashi et al. [37] that show that
in the presence of three N-boxes, transcription is repressed by ∼40
fold. This yields a repression value of ∼0.3 per N-box (Please refer to
Supplemental Materials (Text S1) for details). Since there are no
reliable estimates of the NICD generation constant upon Delta binding
(kfNcp), a lumped parameter of this constant with the Delta
concentration is used to report the strength of the Delta signal
(kfNcp*Delp). The initial parameters for which the
experimentally determined values are not accurately available were later
subjected to sensitivity analysis (See results).
Computational Methods and Initial Conditions
The differential equations described in the model were solved (with parameter
values given in Table 1)
using Berkeley Madonna 8.3.11 software (www.berkeleymadonna.com) with the Runge Kutta 4 module at a step
size of 1 min. To arrive at realistic initial conditions for the model, the
initial concentrations of all species were set to 0 with zero Delta signal, and
the simulations were run until the various species attained steady state
concentration levels. These steady state values (listed in Table 2) were then used as
the initial conditions for subsequent simulations. For the various experiments,
the system was run for 750 minutes without stimulation with the Delta ligand to
attain a basal steady state, and the Delta concentration was then increased to
different levels to initiate Notch1 signaling. Simulations were run either with
a constant Delta signal throughout or with varying duration pulses of the Delta
signal. The system was simulated for a duration of 5,000–10,000
minutes (∼3.5–7 days), as neural progenitor stem cells have
been previously shown to undergo differentiation upon Notch activation in
3–5 days ([57]). Longer simulations up to 50,000 minutes
were conducted when required to confirm Hes1 had reached steady state levels.
10.1371/journal.pcbi.1000390.t002
Initial conditions for deterministic and stochastic models.
Species
Deterministic Model (mol/l)
Stochastic Model (# of molecules/cell)
Hcm (Hes1 mRNA)
4.34 * 10−12
1
Hcp (cytoplasmic Hes1 protein)
1.48 * 10−10
41
Hnp (nuclear Hes1 protein)
3.30 * 10−9
130
Rcm (RBP-Jκ mRNA)
1.44 * 10−12
1
Rcp (cytoplasmic RBP-Jκ protein)
3.53 * 10−11
10
Rnp (nuclear RBP-Jκ protein)
1.07* 10−8
422
Nm (Notch1 mRNA)
1.66 * 10−11
5
Np (Notch1 protein)
9.74 * 10−10
269
Ncp (cytoplasmic NICD)
0
0
Nnp (nuclear NICD)
0
0
Stochastic Model Development
Since the levels of several protein species in the deterministic model
simulations were very low (Table
2), at the level of tens of molecules per cell, assumptions of mass
action kinetics and pseudo steady state may not hold true, and stochastic
effects may play an important role in the dynamics of the signaling network
[41],[58]. To analyze whether
noise in protein and mRNA concentrations would impact the dynamics of the
system, a stochastic simulation of the model using the Gillespie algorithm [43]
was implemented in C++ (code available upon request). To relax
the assumptions of mass action kinetics and pseudo steady state, we explicitly
simulated every reaction step, making a total of 299 reactions. For example,
every interaction between a transcription factor and a promoter was modeled as a
discrete reaction in the simulation. The τ-leap method [59]
was also incorporated into the algorithm to accelerate the stochastic
simulations and increase their efficiency.
ResultsThe Notch1-Hes1 Network as a Bistable Switch
The response of the Notch1-RBP-Jκ-Hes1 system to a step change in an
input Delta signal was analyzed. Simulations were initiated using the steady
state levels of the different species in the absence of any external Delta (also
listed in Table 2), and at
t = 750 minutes a Delta signal was applied.
Fig. 2A demonstrates
that when a low input Delta stimulus is applied, the Hes1 concentration settles
to a correspondingly low steady state value. However, when the input Delta
signal was increased (10-fold), Hes1 shows a rapid increase to a new, 20-fold
higher steady state value. Further steady state analysis at a range of input
Delta levels and initial conditions reveals that the system exhibits
bistability. At low levels of Delta signal, basal levels of Hes1 are maintained
in the cell (“OFF” state), but as the Delta signal strength
is increased beyond a threshold level, it stimulates the production of Hes1,
which is then maintained at high levels (“ON” state) through
the concerted regulation of the Notch1-RBP-Jκ-Hes1 network (Fig. 2B). Bistability
– which has previously been proposed as an advantageous mechanism to
mediate an unambiguous cell fate switch, including in stem cells [51],[60]
– is evident within an intermediate range of Delta signal values
(Fig. 2B).
10.1371/journal.pcbi.1000390.g002
Bistability in Notch signaling.
(A) Deterministic Hes1 trajectories as a function of time for two
different strengths of Delta signals given as a product of the Delta
concentration and the rate constant of formation of NICD upon
Delta-Notch binding
(kfNcp*Delp = kDelp). These
deterministic simulations were initiated using steady state values of
the system under no Delta signal and at
t = 750 minutes (indicated by vertical
arrow) the input Delta signal was applied. (B) Hysteresis in the
Notch1-Hes1 network, where Hes1 concentration can attain two possible
steady states for an intermediate range of Delta inputs. The point of
switching depends on whether the Delta signal is increasing or
decreasing.
Network Sensitivity to Biological Noise
The initial numbers of some protein and mRNA species in the system were in the
range of tens of molecules per cell (Table 2), such that stochastic fluctuations
in individual species may impact the dynamics of the network. In particular,
intracellular noise inherent in systems with small numbers of molecules and/or
slow biochemical reactions can randomize or undermine the
“accuracy” of cell fate choices [41],[58]. To analyze such
behavior, stochastic simulations based on the Gillespie algorithm [43],
distinct from the deterministic model, were developed. Steady state analysis
shows that at low, constant Delta signals, the Hes1 levels fluctuate about a low
mean value corresponding to the “OFF” state, as expected
(data not shown). However, if the Delta signal is increased to a level just
below the concentration at which the deterministic model would predict a switch
in state (Fig. 2B),
stochastic simulations reveal that noise in the network can induce some
trajectories to spontaneously switch states (Fig. 3A). Analogous to results previously
observed in other systems [51],[61],[62],
noise thus undermines the bistable switch and induces spontaneous flipping
between states. Analysis of the time it takes the system to initially pass from
the lower to the upper state reveals that as the strength of the input signal is
increased, this average first passage time (FPT) decreases, and the percentage
of trajectories that change state increases (Fig. 3B). However, this
“uncertainty” occurs within a narrow range of intermediate
Delta signal levels, and if this intermediate window is avoided, the system
effectively behaves deterministically.
10.1371/journal.pcbi.1000390.g003
Stochastic simulations demonstrate spontaneous
“OFF” to “ON” transitions.
(A) Representative stochastic Hes1 trajectories as a function of time
after application of a constant Delta stimulus at 750 min at levels just
below “ON” levels predicted by the deterministic
model. Some Hes1 trajectories remain at low levels
(“OFF” state) while others randomly switch state to
higher levels (“ON” state). (B) First passage time
(FPT) of stochastic trajectories for passage from
“OFF” to “ON” state as a
function of Delta signal strength in the bistable region. The mean and
standard deviation of 40–60 runs in each case are plotted. The
percentage of trajectories that switched to “ON”
state under the given Delta signal is indicated below each data point.
All points except those connected by the same letter (A) are
statistically distinct (p<0.01, 2-tail t-test).
In addition, “ON” to “OFF” transitions
were simulated by first stimulating with a high Delta signal for 4000 minutes to
induce high Hes1 expression levels. When Delta was then reduced to levels that
were in the predicted bistable region based on the deterministic model, the
system maintained high expression levels of Hes1 (Fig. 4A), as anticipated from the
deterministic results (Fig.
2B). Contrary to what was expected based on the deterministic model,
however, when the Delta signal was instead reduced to zero, some trajectories
remained in the high Hes1 expression (“ON”) state (Fig. 4B). This indicates the
role of stochastics in potentiating high Hes1 expression levels even in the
absence of continued signal.
10.1371/journal.pcbi.1000390.g004
The Notch system exhibits bistability under stochastic simulations.
(A) Hes1 stochastic trajectories are shown during high Delta levels for
4000 minutes, following which the Delta signal is brought down to levels
that failed to switch the state to “ON” when
provided for a prolonged duration
(kDelp = 4×10−4)
in the deterministic model. All the trajectories remain in the
“ON” state – corresponding to the region
of bistability seen in the deterministic simulations. (B) Hes1
stochastic trajectories are shown after application of a high Delta
signal for 4000 minutes, after which the Delta signal is brought down to
0. Some trajectories persist in the “ON” state.
Response of the System to Transient Delta Activation
It has been shown for neural crest stem cells [20] that a transient
Notch signal is sufficient to induce cell differentiation. Also, there are
numerous situations where transient Notch-Delta signaling determines the fates
of immature cells, both in tissue culture [18],[19]
and during organismal development [21]–[24]. Under continuous Delta stimulation, the system
can attain high steady-state Hes1 expression levels, thus acting as a switch,
but we next wanted to examine whether transient Delta activation was also
capable of eliciting high Hes1 expression. We thus examined the dynamic response
of the system to transient activation of the Notch1 pathway upon variation in
the strength and duration of an applied Delta signal.
When the system is stimulated for a short duration (10 minutes) with a moderate
strength Delta signal, the deterministic model predicts a transient peak in the
Hes1 expression that eventually decays to its low steady state value (Fig. 5A). However, the peak
expression of Hes1 continually increases with increasing input signal duration
up to ∼800 minutes, beyond which the maximum expression levels of Hes1
attained remain the same but the duration of prolonged high expression levels
progressively increases (Fig.
5A). Similarly, as the input Delta signal strength is increased for a
constant pulse duration, the peak Hes1 concentrations attained also increase up
to a maximum value, after which a further increase in the signal strength only
increases the duration of high Hes1 levels (Fig. 5B). The cell is thus able to attain
high Hes1 expression either under prolonged low intensity Delta signaling or a
short burst of high intensity Delta signaling.
10.1371/journal.pcbi.1000390.g005
Both input Delta signal strength and duration affect the output Hes1
expression levels.
(A) Effect of Delta signal duration on the Hes1 expression levels: a
transient Delta signal of
kfNcp*Delp = 5×10−3
was provided in the deterministic model for varying amounts of time
ranging from 10 minutes to 3000 minutes, and the resulting Hes1
trajectories were simulated up to 15000 minutes. (B) The effect of Delta
signal strength on Hes1 expression: a transient Delta signal of varying
strengths (expressed as
kDelp = kfNcp*Delp
(min−1)) was provided in the deterministic
model for 100 minutes, and the resulting Hes1 were simulated up to 20000
minutes.
Stochastic Effects on Transient Delta Signaling
We also examined the effect of stochastics on transient activation of the
network. Simulations were run using the parameter values as in the deterministic
model for various Delta pulse durations ranging from 10 minutes to 3000 minutes
and >40 trajectories per input duration value were analyzed. For Delta
pulse durations of less than 500 minutes, the stochastic simulations followed
the prediction of the deterministic model (data not shown). However, for a
500-minute Delta pulse, even though the deterministic model predicts a transient
Hes1 peak that does not attain the maximum possible expression level, a small
percentage of the stochastic trajectories in fact did switch to the
“ON” state (corresponding to high Hes1 expression levels)
(data not shown). Also, as the duration of the Delta pulse is increased, the
percentage of trajectories that remain in the “ON” state for
the simulated 15,000 minutes progressively increases even though the
deterministic model predicts that the system would revert back to the
“OFF” state within that time. Furthermore, the average first
passage time (FPT) of the trajectories that do switch state increases as the
Delta pulse duration increases (Fig. 6). It is likely that for shorter Delta pulse durations, if the
system is to undergo the spontaneous “OFF” to
“ON” transition, it does so early, soon after the
application of the Delta signal. However, in the case of longer duration input
signals, the continued presence of the signal allows trajectories to switch
state even much later in the simulation, resulting in an apparently longer first
passage time. Collectively, these results imply that even for very short signal
pulse, a small fraction of a population of cells receiving a pulse of Delta
signal could switch their state due to stochastic effects.
10.1371/journal.pcbi.1000390.g006
First passage time (FPT) for passage from “OFF”
to “ON” state as a function of Delta signal duration
in stochastic simulations.
The mean and standard deviation of >20 runs in each case are
plotted. The percentage of trajectories that switched to the
“ON” state under the given Delta signal is indicated
below each data point. All points except those connected by the same
letter (A,B,C) are statistically distinct (p<0.01, 2-tail
t-test).
Bifurcation Analysis
A number of parameters in the model have not been directly experimentally
measured and were estimated from data available for similar protein classes in
different contexts, and we thus performed sensitivity analysis for all such
parameters by varying them individually through a broad range of values in the
deterministic model (Table
3, Fig.
S2). Although in most cases the qualitative behavior of the system
remained unchanged, the system did exhibit considerable sensitivity to specific
parameters, which were then subjected to further analysis. These include: the
half-life of NICD, the equilibrium binding constant of NICD with RBP-Jκ
(Ka), the maximal transcription rates (Vmax), and the
repression constant of Hes1 (rNbox). NICD has a long half-life of a
few hours under normal physiological conditions [55]. However, our model
indicates that if the NICD half-life is drastically reduced, the system fails to
function as a switch and cannot express high levels of Hes1 (Fig. S3).
In addition, the equilibrium binding constant (Ka) of NICD to
RBP-Jκ in the model is 108 M−1, but as
Ka increases – denoting stronger interactions of NICD
with the promoter – bifurcation analysis demonstrates that the OFF-ON
transition occurs at accordingly lower values of the Delta signal
(kDelp) (Fig.
7A). Similarly, increasing the maximal transcription rate of Hes1
(Vmaxh) to indicate a stronger promoter shifts the OFF-ON
transitions to lower Delta signal strengths (Fig. 7B).
10.1371/journal.pcbi.1000390.g007
Bifurcation Analysis.
(A) Bifurcation analysis of how the switching points vary with the
equilibrium binding constant (Ka) of NICD to RBP-Jk. Stronger
interaction between NICD and RBP-Jk lowers the threshold of Delta signal
required to turn the system ON. (B) Bifurcation analysis of how the
switching points vary with the maximal transcription rate of Hes1
(Vmaxh). A higher maximal transcription rate, indicating
a stronger Hes1 promoter, also slightly shifts the region of bistability
towards lower Delta signal strengths.
10.1371/journal.pcbi.1000390.t003
Summary of results of sensitivity analysis documenting the effect of
increasing parameter values on the threshold of Delta signal strength
required to switch the system state from OFF to ON.
Parameter
Range of variation
Effect on threshold Kdelp (Delta signal strength)
with increasing parameter value
Results
Ka
107–109
(M−1)
Decrease over 2 orders of magnitude; no qualitative
effect
Fig. 7A
Vmaxh
0.3–0.9 nM/min
Slight decrease; no qualitative effect
Fig. 7B
rNbox
0–1
Drastic qualitative change in behavior of the
system
Fig. 8
Kn
107–109
(M−1)
Slight increase; no qualitative effect
Fig. S2A
KaHp
108–1010
(M−1)
Slight increase; no qualitative effect
Fig. S2B
rR
0–0.5
Slight decrease; no qualitative effect
Fig. S2C
kdNcp, kdNnp
0.001–0.04
Increase over 2 orders of magnitude
Fig. S3
The system exhibits a shift in the region of bistability, thus
changing the sensitivity of the system to the Delta signal, but the
qualitative nature of the gene network in most cases remains the
same for a broad range of the parameter values.
The Degree of Repression by Hes1 Determines the Qualitative Nature of the
Cellular Response to Delta Stimuli
Interestingly, the response of the deterministic model was most sensitive to the
extent to which Hes1 binding reduced or repressed expression of target genes
(rNbox). As the Hes1 repression constant (rNbox) is
progressively decreased (or the repressive strength of Hes1 progressively
increased) from 0.3 to 0.1, the final steady state concentrations of Hes1
progressively decrease for a given level of Delta signaling (Fig. S4),
but the system continues to exhibit bistability. Intriguingly, as the value of
rNbox is further decreased below 0.1, there is a dramatic
qualitative change in the response of the system. Specifically, the system
undergoes a bifurcation or transition from bistable to monostable behavior and
at such high repressive strengths is unable to attain high steady state Hes1
expression levels. Finally at very low values of rNbox
(<0.03), it once again undergoes a transition to a stable oscillatory
response where the Hes1 levels in the cell oscillate about a low mean steady
state value (Fig. 8A). A
phase plot of the response of the system with variable rNbox (Fig. 8B) demonstrates how the
same gene network can transition from behaving as a bistable switch to being an
oscillator. The model thus elucidates the versatility of the system, where
tuning of a single key parameter can convert its behavior from a switch to a
clock. Previous hes1 models showing sustained oscillations have
focused exclusively on the low rNbox region (i.e.
rNbox = 0) of such a phase plot
[30],[36],[38],[39].
10.1371/journal.pcbi.1000390.g008
Effect of the repression constant of Hes1 (rNbox) on the
Notch signaling network.
(A) At lower values or rNbox (higher repression constants for
Hes1), the network predicts oscillations in Hes1 levels. As the value of
rNbox is decreased to 0.03 and lower, the system exhibits
stable oscillations. (B) At a fixed Delta signal strength of
kDelp = 2×10−4,
as the rNbox is progressively decreased, the response of Hes1
transitions from behaving as a bistable switch to a brief region of
monostability to an oscillator.
Discussion
The Notch signaling system is an evolutionarily conserved network that functions in
multiple organs to orchestrate cell fate specification [63]–[65] in a
context dependent manner. In some cases, it can function as a binary cell fate
switch at the individual cell level [16],[17], whereas in other
situations cell-cell contact dependent Notch signaling can result in pattern
formation in an array of cells [14],[15], and in yet other contexts it can function as a
biological clock to govern pattern formation and differentiation during
somitogenesis [27]–[29]. Although several
additional components such as Fringe, Numb, and Presenilin can feed into and
modulate the Notch signaling cascade, the core of the signaling pathway is
relatively simple, where Notch acts as a membrane bound transcription factor that is
activated by ligand binding and induces transcription of target hes
genes via its interaction with the RBP-Jk transcription factor
[10].
However, the system can exhibit complex inter-regulation of its components. A better
understanding of the functioning and regulation of this signaling system –
and in particular how it exhibits diverse behaviors in different contexts
– is valuable from a basic biology standpoint, in understanding how
misregulation of the Notch signaling pathway can underlie disease, and from
regenerative medicine viewpoint in therapeutic applications of stem cells.
Mathematical modeling can provide valuable insights into the behavior of this gene
regulatory circuit. Previous models have focused either on the level of cell-cell
interactions to simulate the levels of Notch and Delta within adjacent cells and
thereby analyze pattern formation based on levels of Delta and Notch levels in an
array of cells [33]–[35], or on the autoregulation
of the hes genes in isolation to examine the oscillatory behavior
of the gene circuit [30],[36],[39],[40],[44],[45],[66],[67]. Here we have developed an integrative model that
takes into account the intracellular signaling network downstream of Notch
activation through its ligand Delta, leading to the activation of the
hes1 gene via interaction with RBP-Jκ. These three genes
potentially regulate the transcription of one another (Text S1, Fig S1) [37],[46],[47], forming
a network of positive and negative feedback loops (Fig. 1B). Our model begins to elucidate how a
cell can potentially tune key system parameters in the resulting Notch1-Hes1 gene
circuit to elicit diverse responses.
The behavior of the system was most sensitive to the repression constant of Hes1,
rNbox. The degree of Hes1 repression of a transcriptional target can
be modulated by the presence of co-factors. For example, whereas Groucho can act as
a transcriptional co-repressor for Hes1, Runx2 can act as a negative regulator of
the repressive activity of Hes1 by interfering with the interaction of Hes1 with the
TLE corepressors [68]. The repressive activity of Hes1 can also be
further potentiated by its interaction with the winged-helix protein brain factor 1
[69].
Therefore, because different cells can express these factors to different extents,
which can thereby modulate the value of rNbox, the same gene circuit can
be tuned to transduce an input Delta signal into qualitatively different responses
– oscillation vs. switching.
The model predicts that for low repressive strengths of Hes1
(0.1<rNbox<0.3), the Hes1 expression level functions as a
bistable switch in response to varying the strength of the Delta signal, thereby
providing an unambiguous fate switch that is insensitive to the presence of small
fluctuations in input signal (Fig.
2). Hysteresis has been previously observed experimentally in other
biological systems including the JNK signaling cascade [70],[71] and the Cdc2 cell
cycle regulation [72]. Parameters such as Ka (the
association binding constant of NICD to RBP-Jκ) and Vmax (the
maximal transcription rates) can shift the region of bistability, thus changing the
sensitivity of the system to the Delta signal, but the qualitative nature of the
gene network remains the same for a broad range of these parameter values. Positive
feedback loops with nonlinearity can yield bistability [51], and both Notch1
autoregulation and NICD-mediated conversion of RBP-Jκ into a transcriptional
activator that in turn upregulates Notch1 expression constitute positive feedback
loops that can drive this behavior.
Since the numbers of some protein and mRNA species in the model were low (Table 2), we developed a
stochastic model to examine the effect of biological noise and cell-to-cell
variability on the bistable response of the system to Delta signaling. Spontaneous
OFF to ON switching of states was observed even in regions not predicted by the
deterministic model. For example, as the Delta values are increased through the
bistable range, the percentage of trajectories switching to the ON state increases,
and the average FPT for these trajectories decreases (Fig. 3B). These results are consistent with
observations in other bistable systems [73], and computationally in
other signaling systems [51], where noise has been shown to cause spontaneous
switching of states. However, since the timescale of a system's downstream
response to the Notch network's state varies from a few hours (for example
during somitogenesis) [74] to a few days (for example during stem cell
differentiation) ([57]), the impact of stochastic noise on the fate
switch will also be different in different contexts. Thus, for very low Delta
signals, the average FPT is sufficiently high (>110 hrs) such that the cell
remains in the OFF state for prolonged periods of time and would be non-responsive
to Delta signaling over timescales of a few hours, whereas in the case of a
population of cells experiencing Notch signaling over a period of 4–6
days, spontaneous switching could undermine the genetic switch and cause some cells
to change fate at these low Delta input signals.
While the system can behave as a switch in a particular range of parameters at steady
state, there are also many situations in which Notch signaling is transient, yet is
sufficient to induce a switch in cell fate [18]–[24].
To simulate this, the model behavior was analyzed under transient Delta activation.
The network response to a transient Delta stimulus was a strong function of both the
signal intensity and duration, and either a high intensity signal for a short
duration or a low intensity signal for a prolonged duration was capable of inducing
transient increase in Hes1 expression levels for up to 2.5 days after withdrawal of
the signal (Fig. 5), a time
sufficient to initiate a biological response [57].
This prolonged expression of Hes1 upon transient Delta activation is due to the long
half-life of NICD [55]. The bistable switch is thus sensitive to the
degradation constant of NICD. If the NICD half-life were for example drastically
reduced, the model would predict that the system would fail to express high levels
of Hes1 regardless of Delta levels (Fig. S3). Hes1 is a repressive transcription
factor that in some systems plays a crucial role in suppressing the activation of
oncogenes. For example, in breast cancer cells, Hes1 can inhibit both estrogen- and
heregulin-beta1-stimulated growth via downregulation of E2F-1 expression [75]. Thus,
a malfunction in the Notch system, such as a reduction in NICD half-life, could
contribute to cell transformation. Indeed, aberrant Notch signaling is implicated in
many cancers (reviewed in [76]). For example, integrin-linked kinase (ILK),
which is either activated or overexpressed in many types of cancers including breast
cancer [77], can remarkably reduce the protein stability of
Notch1 and thus decrease its half-life drastically [78]. Interestingly, high ILK
and low NICD levels are detected in basal cell carcinoma and melanoma patients [78].
By increasing the repressive strength of the Hes1 dimer by 10-fold
(rNbox<0.03), the cell can transition from being a bistable
system, to a brief region of monostability, and finally to an oscillator (Fig. 8B). Oscillations occur with
a time period of approximately 2 hrs, similar to what Hirata et al. observed in cell
culture [30]. This value also compares well with the various
models that have been developed (for the Hes system in isolation) to explain
oscillations in the hes family of genes and their homologues. These
models assume complete repression in the presence of even a single Hes homodimer
bound to the promoter region [36],[39],[45],[66]. This corresponds to an rNbox value of
0, in which case there would be no difference between the repressive strength of
promoters with 1, 2 or 3 N-boxes. From the experimental observations of Takebayashi
et al. [37], where the repressive strength of the promoter did in
fact increase with the number of N-boxes, the estimated value of rNbox is
0.3. However, during somitogenesis, the factors expressed in the presomitic mesoderm
(PSM) may enhance the repression due to Hes1 such that the value of rNbox
is very low.
This current model represents the Notch signaling network core in a single cell, and
it can readily be extended to a field of cells to analyze the role of Notch in
patterning tissue formation [60]. In addition, there are numerous cell-specific
mechanisms and factors that feed into this important signaling core [79]–[81]. Additional molecular
species can be added to this model framework, or the parameter values of the current
model can readily be modulated for example to simulate changes in DNA binding
affinities, repressive constants, or the protein and mRNA stabilities as a function
of cell-specific factors. This simple but versatile model can therefore be expanded
by incorporation of additional molecular mechanism, specific to particular cell
types, to make predictions on the role of Notch signaling in diverse cells and
tissues.
In summary, we have theoretically and computationally analyzed the
Notch1-RBP-Jκ-Hes1 signaling network, which is responsible for cell fate
specification in numerous contexts. Our results indicate that the network,
consisting of both positive and negative feedback mechanisms, can be tuned to
function either as a bistable cell fate switch or an oscillator based on relatively
small changes in a key parameter value. Furthermore, the duration and strength of
the Delta signal regulate either the peak or the final steady state levels of Hes1
attained. Therefore, cells can readily tune the Notch system to regulate a variety
downstream cell fates and functions.
Supporting Information
Supplemental Materials
(0.09 MB DOC)
Transcriptional analysis of the Notch1 promoter. Relative fold changes in the
activity of the Notch1 promoter in the presence of exogenous Hes1 (H),
dNHes1 (dN), RBP-Jκ(R) and NICD (N) are shown. Relative amounts of
plasmids used in each case encoding the respective cDNA are indicated. For
example H0.1dN1R0.5N0.5 indicates
Hes1 = 0.1 µg,
dNHes1 = 1 µg,
RBP-Jk = 0.5 µg and
NICD = 0.5 µg in a total of 4
µg transfection.
(2.52 MB TIF)
Bifurcation Analysis. (A) Bifurcation analysis of how the switching points
vary with the Hes1 DNA association constant (Kn). Varying the association
constant over two orders of magnitude causes a slight shift in the strength
of the Delta signal required to switch the system state. Thus, stronger DNA
association of Hes1 (higher values of Kn) increases the threshold values of
Delta signal strength (Kdelp) required to turn the system ON. (B)
Bifurcation analysis of how the switching points vary with the Hes1
dimerization constant (KaHp). Varying the dimerization constant over two
orders of magnitude causes a slight increase in the strength of the Delta
signal required to switch the system state to ON. (C) Bifurcation analysis
of how the switching points vary with the repression constant of
RBP-Jκ (rR). Increasing the repression constant from 0 to 0.5
(corresponding to a decrease in the RBP-Jκ repressive strength), has
very little effect on the threshold of Delta signaling strength required to
turn the system ON.
(0.82 MB TIF)
Effect of half-life of NICD on the Hes1 switch. (A) Changing the half life
from 3 hrs (kdNcp = 0.00385) to 8 hrs
(0.0014) due to the effect of GSK3β causes a slight increase in the
steady state Hes1 concentration in response to a Delta signal of
kDelp = 5×10−4,
but no qualitative change in the switch. Increasing the degradation constant
by 10-fold (kdNcp = 0.0014 to 0.014)
however, causes complete suppression of the switch. (B) Analysis of how the
threshold value of Delta signal required to switch the system from OFF to ON
increases with the increasing degradation constant of NICD (decreasing NICD
half life).
(0.52 MB TIF)
Effect of repression through Hes1 (rNbox) on the high steady state
values of Hes1 expression.Decreasing rNbox progressively
decreases the steady state concentrations of Hes1 in the rNbox
range of 0.3 to 0.1.
(0.42 MB TIF)
We would like to thank Dr. Wei-Qiang Gao, Dr. Diane Hayward, Dr. R. Kopan, Dr. R.
Kageyama and Dr. C. Cepko for providing valuable plasmid constructs.
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