The Notch signaling system is an evolutionarily conserved network that functions in multiple organs to orchestrate cell fate specification 
in a context dependent manner. In some cases, it can function as a binary cell fate switch at the individual cell level 
, whereas in other situations cell-cell contact dependent Notch signaling can result in pattern formation in an array of cells 
, and in yet other contexts it can function as a biological clock to govern pattern formation and differentiation during somitogenesis 
. 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 
. 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 
, or on the autoregulation of the hes
genes in isolation to examine the oscillatory behavior of the gene circuit 
. 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
, forming a network of positive and negative feedback loops (). 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 
. The repressive activity of Hes1 can also be further potentiated by its interaction with the winged-helix protein brain factor 1 
. 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 (). Hysteresis has been previously observed experimentally in other biological systems including the JNK signaling cascade 
and the Cdc2 cell cycle regulation 
. 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 
, 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 (), 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 (). These results are consistent with observations in other bistable systems 
, and computationally in other signaling systems 
, 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) 
to a few days (for example during stem cell differentiation) (
), 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 
. 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 (), a time sufficient to initiate a biological response 
This prolonged expression of Hes1 upon transient Delta activation is due to the long half-life of NICD 
. 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 
. 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 
). For example, integrin-linked kinase (ILK), which is either activated or overexpressed in many types of cancers including breast cancer 
, can remarkably reduce the protein stability of Notch1 and thus decrease its half-life drastically 
. Interestingly, high ILK and low NICD levels are detected in basal cell carcinoma and melanoma patients 
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 (). Oscillations occur with a time period of approximately 2 hrs, similar to what Hirata et al. observed in cell culture 
. 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 
. 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. 
, 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 
. In addition, there are numerous cell-specific mechanisms and factors that feed into this important signaling core 
. 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.