Operating principles of Notch–Delta–Jagged module of cell–cell communication

Our framework has been introduced in [25], that extends previous work [26] by incorporating Jagged in addition to Delta and the asymmetric regulation of the proteins by the signal NICD that activates Notch and Jagged but represses Delta. Trans-activation (interaction between receptor of one cell and ligand of the neighboring cell) activates the Notch signaling pathway by releasing the signal (NICD), but cis-inhibition (interaction between the receptor and ligand belonging to the same cell) leads to the degradation of both proteins. We initially assume no effect of the glycosyltransferase Fringe, i.e., the affinity of Notch to Delta or Jagged is the same. In this case, the equations for the Notch receptor (N), the ligands Delta (D) and Jagged (J), and the signal NICD (I), are given by:

$$\begin{eqnarray}&&\frac{{\rm d}N}{{\rm d}t}={{N}_{0}}{{H}^{S+}}(I,{{\lambda }_{I,N}})-{{k}_{C}}N\left[ D+J \right]-{{k}_{T}}N\left[ {{D}_{{\rm ext}}}+{{J}_{{\rm ext}}} \right]-\gamma N,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\frac{{\rm d}D}{{\rm d}t}={{D}_{0}}{{H}^{S-}}(I,{{\lambda }_{I,D}})-{{k}_{C}}DN-{{k}_{T}}D{{N}_{{\rm ext}}}-\gamma D,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\frac{{\rm d}J}{{\rm d}t}={{J}_{0}}{{H}^{S+}}(I,{{\lambda }_{I,J}})-{{k}_{C}}JN-{{k}_{T}}J{{N}_{{\rm ext}}}-\gamma J,\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\frac{{\rm d}I}{{\rm d}t}={{k}_{T}}N[{{D}_{{\rm ext}}}+{{J}_{{\rm ext}}}]-{{\gamma }_{I}}I,\end{eqnarray} \tag{ 4 }$$

where N₀, D₀, and J₀ represent innate production rates of Notch, Delta and Jagged respectively. Their degradation rates have been considered to be the same, represented by γ. N_ext, D_ext and J_ext are the amounts of receptor Notch, and ligands Delta and Jagged available from neighboring cells. k_c and k_T represent the cis-inhibition rates and trans-activation rates of Notch with its ligands. γ_I denotes the degradation rate of NICD. H^S+(I,λ_I,N) and H^S+(I,λ_I,J) represent the transcriptional activation of Notch (N) and Jagged (J) by the signal NICD (I), and H^S−(I,λ_I,D) denotes the repression of Delta (D) by I. H^S+(I,λ_I,N), H^S+(I,λ_I,J) and H^S−(I,λ_I,D) are shifted Hill functions defined as H⁻(X) + λ_X,YH⁺(X) where H⁻(X) is inhibitory Hill function, and H⁺(X) is excitatory Hill function, and λ_X,Y denotes the fold-change in production of Y due to X. For activation, shifted Hill functions are depicted by H^S+ and λ_X,Y > 1; for inhibition, they are depicted by H^S− and λ_X,Y < 1. λ_X,Y = 1 denotes no effect in production rate [27].

NICD (Notch signal) degrades rapidly as compared to N, D and J [28]. Therefore, we can assume a quasi-steady approximation for it. In this case, equations (1)–(4) can be reduced to two equations by defining L = D + J and L_ext = D_ext + J_ext, where L represents the total amount of both ligands in the cell, and L_ext denotes the total amount of external ligands available to bind. Then, the reduced system is:

$$\begin{eqnarray}&&\frac{{\rm d}N}{{\rm d}t}={{N}_{0}}{{H}^{S+}}(I,{{\lambda }_{I,N}})-{{k}_{C}}NL-{{k}_{T}}N{{L}_{{\rm ext}}}-\gamma N,\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\frac{{\rm d}L}{{\rm d}t}={{D}_{0}}{{H}^{S-}}\left( I,{{\lambda }_{I,D}} \right)+{{J}_{0}}{{H}^{S+}}\left( I,{{\lambda }_{I,J}} \right)-{{k}_{C}}LN-{{k}_{T}}L{{N}_{{\rm ext}}}-\gamma L,\end{eqnarray} \tag{ 6 }$$

where, by quasi steady-state approximation, I = k_TNL_ext/γ_I.

Until now, we considered equal cis-inhibition rates (k_C) as well as equal trans-activation rates (k_T) for both ligands. However, when we include the effect of Fringe, a downstream target of Notch signaling [13], we consider two populations of Notch—one that is modified (glycosylated) by Fringe and the other that is not (see derivation in SI section S2). The modified Notch has a higher binding affinity for Delta but lower for Jagged as compared to the unmodified one, both in cis and in trans [14–17]. Now, cis-inhibition rates ${{k}_{{{C}_{D}}}}$ and ${{k}_{{{C}_{J}}}}$ and trans-activation rates ${{k}_{{{T}_{D}}}}$ and ${{k}_{{{T}_{J}}}}$ depend on the levels of NICD (I) that activates Fringe. Thus, when Fringe's effect is incorporated, ${{k}_{{{C}_{D}}}}$ = k_CH^S+(I,λ_F,D) and ${{k}_{{{T}_{D}}}}$ = k_TH^S+(I,λ_F,D) where λ_F,D > 1 (Fringe increases the binding of N to D and D_ext). Similarly, since Fringe decreases the binding of N to J and J_ext, λ_F,J < 1 in ${{k}_{{{C}_{J}}}}$ = k_CH^S−(I,λ_F,J) and ${{k}_{{{T}_{J}}}}$ = k_TH^S−(I,λ_F,J)., and the dynamics of N, D, J and I are given by:

$$\begin{eqnarray}&&\frac{{\rm d}N}{{\rm d}t}={{N}_{0}}{{H}^{S+}}\left( I,{{\lambda }_{I,N}} \right)-N\left[ {{k}_{{{C}_{D}}}}D+{{k}_{{{C}_{J}}}}J \right]-N\left[ {{k}_{{{T}_{D}}}}{{D}_{{\rm ext}}}+{{k}_{{{T}_{J}}}}{{J}_{{\rm ext}}} \right]-\gamma N,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\frac{{\rm d}D}{{\rm d}t}={{D}_{0}}{{H}^{S-}}(I,{{\lambda }_{I,D}})-{{k}_{{{C}_{D}}}}DN-{{k}_{{{T}_{D}}}}D{{N}_{{\rm ext}}}-\gamma D,\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\frac{{\rm d}J}{{\rm d}t}={{J}_{0}}{{H}^{S+}}(I,{{\lambda }_{I,J}})-{{k}_{{{C}_{J}}}}JN-{{k}_{{{T}_{J}}}}J{{N}_{{\rm ext}}}-\gamma J,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\frac{{\rm d}I}{{\rm d}t}=N[{{k}_{{{T}_{D}}}}{{D}_{{\rm ext}}}+{{k}_{{{T}_{J}}}}{{J}_{{\rm ext}}}]-{{\gamma }_{I}}I.\end{eqnarray} \tag{ 10 }$$

The effect of Fringe here is given by two independent parameters—λ_F,D and λ_F,J. Later, to elucidate the net effect, we collapse these two parameters into a single parameter (f) that represents the Fringe effect such that f = λ_F,D = 1/λ_F,J. f = 1 represents no effect of Fringe and the higher the value of f, the stronger is this effect.

Recent studies show that some cells can release soluble form of the ligands Delta and Jagged that activate Notch signaling in the neighbors in a paracrine manner [23, 29]. However, soluble ligands do not have enough mechanical pulling force to cleave Notch and generate NICD [30] and they require alternative mechanisms such as ligand multi-merization to generate sufficient mechanical force [31]. For this reason, soluble ligands are expected to have lower trans-activation rates than those of membrane-bound ligands [32]. Therefore, a model that considers both soluble and the membrane-bound ligands should have two terms in equation (7) (Notch equation) for interaction with both forms of these ligands, and equations (7)–(10) would be given by equations (11)–(14), where $D_{{\rm ext}}^{s}$ and $J_{{\rm ext}}^{s}$ represent soluble forms of ligands Delta and Jagged respectively, and $k_{{{T}_{D}}}^{s}$ and $k_{{{T}_{J}}}^{s}$ denote the trans-activation rates of Notch signaling for soluble ligands Delta and Jagged, respectively, such that $k_{{{T}_{D}}}^{s}=k_{T}^{s}{{H}^{S+}}$ (I, λ_F,D) and $k_{{{T}_{J}}}^{s}{{H}^{S-}}$ (I, λ_F,J).

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \frac{{\rm d}N}{{\rm d}t}={{N}_{0}}{{H}^{S+}}\left( I,{{\lambda }_{I,N}} \right)-N\left[ {{k}_{{{C}_{D}}}}D+{{k}_{{{C}_{J}}}}J \right]-N\left[ {{k}_{{{T}_{D}}}}{{D}_{{\rm ext}}}+{{k}_{{{T}_{J}}}}{{J}_{{\rm ext}}} \right] \\ \quad \quad \;\;\;-N\left[ k_{{{T}_{D}}}^{s}D_{{\rm ext}}^{s}+k_{{{T}_{D}}}^{s}\;J_{{\rm ext}}^{s} \right]-\gamma N, \\ \end{array}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\frac{{\rm d}D}{{\rm d}t}={{D}_{0}}{{H}^{S-}}(I,{{\lambda }_{I,D}})-{{k}_{{{C}_{D}}}}DN-{{k}_{{{T}_{D}}}}D{{N}_{{\rm ext}}}-\gamma D,\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\frac{{\rm d}J}{{\rm d}t}={{J}_{0}}{{H}^{S+}}(I,{{\lambda }_{I,J}})-{{k}_{{{C}_{J}}}}JN-{{k}_{{{T}_{J}}}}J{{N}_{{\rm ext}}}-\gamma J,\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\frac{{\rm d}I}{{\rm d}t}=N[{{k}_{{{T}_{D}}}}{{D}_{{\rm ext}}}+{{k}_{{{T}_{J}}}}{{J}_{{\rm ext}}}]+k_{T}^{s}N[D_{{\rm ext}}^{s}+J_{{\rm ext}}^{s}]-{{\gamma }_{I}}I.\end{eqnarray} \tag{ 14 }$$

For the case of two-cell system interacting solely with each other via NDJ signaling (i.e. the two cells do not respond to any external signal apart from the ligands of the neighboring cell), N_ext, D_ext, J_ext represent the Notch, Delta and Jagged respectively of the neighboring cell. The dynamics of cell #1 interacting with cell #2 can be described by equations (15)–(18) (similar equations represent the dynamics of cell #2 interacting with cell #1):

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} \frac{{\rm d}{{N}_{1}}}{{\rm d}t}={{N}_{0}}{{H}^{S+}}\left( {{I}_{1}},{{\lambda }_{{{I}_{1}},{{N}_{1}}}} \right)-{{N}_{1}}\left[ \left( {{k}_{C}}{{D}_{1}}+{{k}_{T}}{{D}_{2}} \right) \right.{{H}^{S+}}\left( {{I}_{1}},{{\lambda }_{F,D}} \right) \\ \quad \quad \;\;\;\;+\;\left( {{k}_{C}}{{J}_{1}}+{{k}_{T}}{{J}_{2}} \right){{H}^{S-}}\left. \left( {{I}_{1}},{{\lambda }_{F,J}} \right) \right]-\gamma {{N}_{1}}, \\ \end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\frac{{\rm d}{{D}_{1}}}{{\rm d}t}={{D}_{0}}{{H}^{S-}}\left( {{I}_{1}},{{\lambda }_{{{I}_{1}},{{D}_{1}}}} \right)-{{k}_{C}}{{H}^{S+}}\left( {{I}_{1}},{{\lambda }_{F,D}} \right){{D}_{1}}{{N}_{1}}-{{k}_{T}}{{H}^{S+}}\left( {{I}_{2}},{{\lambda }_{F,D}} \right){{N}_{2}}{{D}_{1}}-\gamma {{D}_{1}},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\frac{{\rm d}{{J}_{1}}}{{\rm d}t}={{J}_{0}}{{H}^{S+}}\left( {{I}_{1}},{{\lambda }_{{{I}_{1}},{{J}_{1}}}} \right)-{{k}_{C}}{{H}^{S-}}\left( {{I}_{1}},{{\lambda }_{F,J}} \right){{J}_{1}}{{N}_{1}}-{{k}_{T}}{{H}^{S-}}\left( {{I}_{2}},{{\lambda }_{F,J}} \right){{N}_{2}}{{J}_{1}}-\gamma {{J}_{1}},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\frac{{\rm d}{{I}_{1}}}{{\rm d}t}={{k}_{T}}{{N}_{1}}\left[ {{J}_{2}}{{H}^{S-}}\left( {{I}_{1}},{{\lambda }_{F,J}} \right)+{{D}_{2}}{{H}^{S+}}\left( {{I}_{1}},{{\lambda }_{F,D}} \right) \right]-{{\gamma }_{I}}{{I}_{1}},\end{eqnarray} \tag{ 18 }$$

where N₁, D₁, J₁ and I₁ represent the Notch receptor, Delta, Jagged and NICD respectively for cell #1, and similarly N₂, D₂, J₂ and I₂ represent the Notch receptor, Delta, Jagged and NICD respectively for cell # 2.

Similarly, the system can be extended to the case where each cell has n-neighbors, and the generalized equations are given by equations (19)–(22) given as:

$$\begin{eqnarray}&&\frac{{\rm d}N}{{\rm d}t}={{N}_{0}}{{H}^{S+}}(I,{{\lambda }_{I,N}})-N\left[ \left( {{k}_{C}}D+{{k}_{T}}\sum \limits_{i=1}^{n} {{D}_{i}} \right){{H}^{S+}}\left( I,{{\lambda }_{F,D}} \right)+\left( {{k}_{C}}J+{{k}_{T}}\sum \limits_{i=1}^{n} {{J}_{i}} \right){{H}^{S-}}\left( I,{{\lambda }_{F,J}} \right) \right]-\gamma N,\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&\frac{{\rm d}D}{{\rm d}t}={{D}_{0}}{{H}^{S-}}\left( I,{{\lambda }_{I,D}} \right)-{{k}_{C}}{{H}^{S+}}\left( I,{{\lambda }_{F,D}} \right)DN-{{k}_{T}}D\sum \limits_{i=1}^{n} {{N}_{i}}{{H}^{S+}}\left( {{I}_{i}},{{\lambda }_{F,D}} \right)-\gamma D,\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\frac{{\rm d}J}{{\rm d}t}={{J}_{0}}{{H}^{S+}}\left( I,{{\lambda }_{I,J}} \right)-{{k}_{C}}{{H}^{S-}}\left( I,{{\lambda }_{F,J}} \right)JN-{{k}_{T}}J\sum \limits_{i=1}^{n} {{N}_{i}}{{H}^{S-}}\left( {{I}_{i}},{{\lambda }_{F,J}} \right)-\gamma J,\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&\frac{{\rm d}I}{{\rm d}t}={{k}_{T}}N\sum \limits_{i=1}^{n} \left[ {{J}_{i}} \right.\left. {{H}^{S-}}\left( I,{{\lambda }_{F,J}} \right)+{{D}_{i}}{{H}^{S-}}\left( I,{{\lambda }_{F,D}} \right) \right]-{{\gamma }_{I}}I.\end{eqnarray} \tag{ 22 }$$

A detailed discussion of the values of the parameters used is presented in the SI section S1 (available at stacks.iop.org/NJP/17/055021/mmedia). The model is robust with respect to changes in the values of the most parameters as discussed in the section S3 in the SI, and is most sensitive to the production and degradation rates of Notch (N₀ and γ respectively), production rate of Delta D₀, and the trans-activation rate k_T. The codes were implemented in python using the PyDSTool [33].

ND signaling circuit

We first evaluate the dynamics of ND signaling for one-cell system by analyzing the reduced model given by equations (5)–(6) when J₀ = 0, i.e., no Jagged is produced. In this case, ND circuit is bistable, and the two stable steady states are:—(i) Sender (S)—in which the cell has (high Delta (ligand), low Notch (receptor)), and (ii) Receiver (R)—in which the cell has (low Delta (ligand), high Notch (receptor)) (figures 2(A) and (D)). Low levels of Delta in the neighboring cell (D_ext) cause the cell to behave as a Receiver (low Delta, high Notch), while high levels of D_ext cause it to behave as a Sender (high Delta, low Notch), i.e. the two adjacent cells adopt alternate fates (figure 2(A)). However, at intermediate levels of D_ext, we see a coexistence of the Sender and Receiver states, such that random fluctuations in the cell can lead to a transition between the states or a switching of the cell-fate (figure 2(D)).

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This switch-like behavior of ND signaling has been noted in previous studies as well [26, 34, 35]. A canonical example of this phenomenon, also known as lateral inhibition, is the AC/VU differentiation in C. elegans, where initially the two cells are identical (i.e. both neighbors have the same levels of ligand and receptor), but due to a random fluctuation, one cell has increased levels of ligand and/or the other has increased levels of receptor; and this stochastic difference is amplified by the mutually inhibitory feedback loop—thereby driving the cell fate diversification [34]. However, lateral inhibition need not be static, it can be dynamic too—i.e. cells can compete against their neighbors continuously, as seen during sprouting angiogenesis, where the migrating 'stalk' cells are continuously competing to become 'tip' cells [36]. Also, during neural development, lateral inhibition occurs between the dynamic clusters of neuronal precursors, in order to strike a balance between the number of cells differentiating to become neurons, and those staying as precursors for further rounds of neurogenesis [37].

In addition to alternate fate or checkerboard-like patterns, lateral inhibition can also give rise to sparse or non-checkerboard patterns, i.e. patterns with a spacing of more than one cell, when the cooperativity (Hill function coefficient) in interaction between Notch and Delta of adjacent cells is weak, or when the ND system is expanded to consider other molecules such as miR-124 [35, 38].

NJ signaling circuit

NDJ signaling circuit

The two ligands—Delta and Jagged—can be produced at different rates in the same cell due to stochastic events and/or the influence of external factors such as the inflammatory cytokine TNF-α that can promote the production of Jagged and inhibit that of Delta [48]. Therefore, we next explore the effect of varying the production rates of the ligands on cell-fate decisions mediated by NDJ signaling, for both a one-cell and a two-cell system. For the one-cell system, our results are represented as phenotype diagrams when the circuit is driven by two control parameters—external Delta (D_ext) and external Jagged (J_ext) (figure 3). Until now, we have considered the total amount of external ligands (L_ext = D_ext + J_ext) driving the circuit, but now we consider the two ligands D_ext and J_ext separately so as to incorporate the effect of Fringe that creates an asymmetry between the likelihood of Notch to interact with these two ligands. The circuit now is represented by equations (7)–(10). These diagrams consist of multiple phases (sets of co-existing phenotypes for the same parameter set): three monostable phases—{S}, {S/R} and {R}, three bistable phases—{S, R}, {S, S/R} and {S/R, R}, and a tristable phase—{S, S/R, R}.

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Different phenotype diagrams denote different combinations of the production rates of Jagged and Delta. At low production rates of Jagged (J₀), the increase in production rate of Delta (D₀) decreases the parameter range for the existence of phases containing the hybrid S/R phenotype (compare the area bounded by dashed lines in figure 3(C) versus that in figure 3(A)). Similar effects of high production rates of Delta (D₀) are observed at high production rates of Jagged (J₀) (compare the area bounded by dashed lines in figure 3(D) versus that in figure 3(B). Therefore, high levels of Delta inhibit NJ signaling and promote lateral inhibition or cell fate diversification. Conversely, high production rates of Jagged can significantly increase the parameter range for the existence of phases containing the hybrid S/R phenotype, and this effect is more pronounced at low production rates of Delta (compare the area bounded by dashed lines in figure 3(B) vs. that in figure 3(A). Hence, the two ligands act antagonistically with respect to their effect on cell-fate decisions in a cell population.

Next, we investigate the effect of different production rates of Delta (D_0,) and Jagged (J₀) in a two-cell system. Both these cells are identical and communicate with each other via the NDJ system. At low production rates of both ligands (Jagged and Delta) in both cells, they act as Receiver (R₁, R₂—low ligand, high receptor) because Notch (receptor) is still being produced at a fixed rate. However, when the production rate of Delta is higher than that of Jagged, one cell becomes the Sender (S) and the other the Receiver (R)—(S₁, R₂) or (R₁, S₂). Conversely, when the production rate of Jagged is higher than that of Delta, both cells attain the hybrid S/R state that enables them to communicate in a bidirectional manner via the NDJ system (S/R₁, S/R₂) (figures 4 and S4). Consistent with these results for a two-cell system, the different production rates of these ligands can also have opposite effects on the emergent tissue-level patterning. High production rates of Delta, but not that of Jagged, can facilitate checkerboard-like 'salt-and-pepper' pattern [25].

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In Drosophila as well as mammals, Fringe can promote ND signaling by increasing the binding affinity of Notch for Delta and decreasing that for Jagged, both for cis-inhibition and trans-activation interactions [14, 15]. Here, for both one-cell and two-cell systems, we explore how Fringe levels affect the parameter range for attaining the different steady states in NDJ circuit and their relative stability. We consider the effect of Fringe on NDJ signaling by a single variable f, such that f = λ_F,D = 1/λ_F,J, where λ_F,D and λ_F,J denote the fold-change in binding affinity (or equivalently the cis-inhibition and trans-activation rates) between (Notch and Delta) and (Notch and Jagged) respectively.

Fringe promotes lateral inhibition patterning

For one-cell system, we investigate the effect of Fringe when it is driven exclusively by different levels of D_ext and J_ext. We plot a bifurcation diagram for the Fringe effect and the external ligand driving the cell, and find that an increase in Fringe effect decreases the range of existence of the phases that contain the hybrid S/R state—{S/R}, {S, S/R}, {S/R, R} and {S, S/R, R} (figure S5), thereby being consistent with the NJ signaling attenuating role of Fringe. We next address the effect of Fringe on tissue patterning by simulating a one-dimensional layer of cells interacting via NDJ pathway, for different values of Fringe effect. Our results show that at high Fringe effect, the tissue level pattern of similar fates is disrupted and the 'salt-and-pepper' pattern starts to emerge (figure 5). Our results are consistent with experiments suggesting that Fringe promotes lateral inhibition during neurogenesis [51].

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Fringe alters the relative stability of the steady states of NDJ signaling

In order to evaluate the effect of noise in the NDJ circuit for two-cell system, we represent the phase space by an effective potential for different values of Fringe effect. The effective potential depends on the levels of Notch of each cell, and is defined as U = –log(P), where P = P(N₁, N₂) is the probability density at the two dimensional phase space (N₁ × N₂) which is calculated by using the Euler–Maruyama method to approximate the ordinary differential equation to a stochastic differential equation [52]. We observed three states of the system: (i) first cell behaves as a Sender (low Notch) and the second one as a Receiver (high Notch)—(S₁,R₂), (ii) first cell behaves as a Receiver (high Notch) and the second one as a Sender (low Notch)—(R₁,S₂), and (iii) both cells behave as both Sender and Receiver (both with intermediate level of Notch)—(S/R₁, S/R₂). For low or no effect of Fringe, both the cells tend to stay predominantly in the hybrid S/R state—(S/R₁, S/R₂) and alternate their fates only sporadically (figure 6(A)). However, when the Fringe effect is strong, the basin of attraction of the (S/R₁, S/R₂) state becomes shallower and therefore small fluctuations can move the cells to one of the two states that now have a deep basin of attraction; thus one cell starts behaving as a Receiver (R; high Notch), and the other as a Sender (S; low Notch)—(S₁, R₂) or (R₁, S₂) (figure 6(B)).

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These results indicate that Fringe promotes lateral inhibition by stabilizing alternate Sender and Receiver fates between neighboring cells. Thus, loss of Fringe is expected to stabilize the hybrid S/R state, i.e. the outcome of NJ signaling. Because NJ signaling is more often implicated than ND signaling in many aspects of tumor progression [19], our results on the role of Fringe in the inhibition of NJ signaling rationalize the fact that Fringe has been reported as a tumor suppressor in multiple cancers such as prostate, lung and basal-like breast cancer (BLBC) [53–55]. Further, loss of Fringe facilitates NJ signaling that is critical in the tumorigenesis of BLBCs [53].

In mammals, Fringe can have three homologues—Lunatic Fringe (Lfng), Manic Fringe (Mfng) and Radical Fringe (Rfng). It should be noted that the results about Fringe promoting ND signaling and lateral inhibition hold true for those homologues that have similar circuit architecture with Notch signaling as considered here, i.e. they promote ND signaling both in cis and in trans, and they are activated by NICD. Among the existing homologues, only Lfng has been shown to be a direct downstream target of NICD [13]; and both Lfng and Mfng, but not Rfng, promotes ND signaling [14]. Consistently, both Lfng and Mfng, but not Rfng, has been reported to be a tumor suppressor [53–55]. Future models should consider the effect of these different Fringe homologues distinctly.

Cis-inhibition, i.e. cell-autonomous binding and consequent degradation of the receptor and ligand, has been reported to be critical for lateral inhibition and pattern formation in multiple developmental contexts such as sensory organ precursors and Drosophila wing primordium [56–58], but it is not present in other contexts such as hair cell formation in the inner ear [59]. Furthermore, the role of cis-inhibition in angiogenesis remains enigmatic [48].

To decipher the role of cis-inhibition in lateral inhibition patterning, we evaluate its effect on the stability of cell-fate decisions, and represent the phase space by an effective potential for the case of both lower and higher cis-inhibition rate (k_C). At higher k_C, the basin of attraction for the (S/R₁, S/R₂) becomes shallow and those for the (S₁, R₂) and (R₁, S₂) become deeper, i.e. at higher levels of cis-inhibition, adjacent cells are much more likely to adopt alternate fates of one being the sender and the other receiver than similar fates of being hybrid S/R ones (figures 7(A) and (B)). These results for the role of cis-inhibition in NDJ signaling are consistent with previous observations about cis-inhibition in ND signaling that it can facilitate pattern formation even without any co-operativity in ND interaction, and usually confers faster dynamics and greater robustness to noise during patterning [26, 41, 56].

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Next, we evaluate the effect of cis-inhibition in tissue patterning. Increased cis-inhibition facilitates 'salt-and-pepper' or alternate Sender and Receiver patterning, whereas decreased cis-inhibition leads the cells to attain the same fate of hybrid S/R (figure 7(C)). Interestingly, the hybrid S/R state observed at lower cis-inhibition has very high levels of NICD (as compared to levels of NICD in other simulations in our analysis), Notch and Jagged, but low levels of Delta (figure 7(C), figure S6), thereby suggesting that cis-inhibition can maintain low levels of NICD and Notch in a cell-autonomous manner. Our results are corroborated by experiments showing that cis-inhibition can act as a buffer for cells against accidental Notch activity [60]. This proposed novel role of cis-inhibition is also indicative of its ability to attenuate NJ signaling similar to Fringe, and therefore cis-inhibition might as well act as a tumor suppressor.

Such insights into pattern formation are a direct outcome of coordinated cell-fate decisions happening at sub-cellular level, and can be gained only after analyzing a multi-cell model of NDJ signaling, as presented here. Results about relative stability of signaling states for one cell have implications about the spatiotemporal stability of tissue-level patterns. By considering multiple types of cells with different signaling modalities among themselves in a theoretical model, the pattern formation dynamics at a tissue level can give way to organ level dynamics, hence linking 'network biology' to 'network physiology'[1, 2]. Further, as shown recently, ND signaling can happen without cell–cell contact as well—through exosomes (cell-derived vesicles that can play a key role in long-range cell–cell communication) [61]. These exosomes, that can circulate in blood to reach other organs as well as be released by both tumor and stroma cells for targeted interaction[62], might constitute a key component of vertical integration from sub-cellular level to organ level both during development and disease.

In addition to transmembrane ligands, Notch signaling can also be activated by binding of Notch receptor to soluble ligands that can be secreted by other cells [23, 29]. Transmembrane ligands activate Notch signaling and hence affect cell-fate decisions in a juxtacrine (cell–cell contact) manner, but soluble ligands function through paracrine (one cell secretes a ligand that can bind to nearby cells and induce a response) manner, thereby sometimes competing for the available Notch receptors on the surface [63]. However, the soluble ligands usually activate Notch weakly, i.e. have a lower trans-activation rate for Notch [32].

To evaluate the role of soluble ligand-mediated paracrine signaling in cell-fate decision making, we incorporated the soluble forms of ligands Delta and Jagged in our two-cell system (two identical cells communicating via NDJ pathway), as described in equations (11)–(14) where D_ext and J_ext represents Delta and Jagged of the neighboring cell and D^s_ext and J^s_ext represent soluble Delta and Jagged respectively. Here, the soluble ligands have been incorporated in the model so as to mimic the biological situation where tumor cells are exposed to soluble Jagged from stromal cells, i.e. soluble ligands are only produced by stroma, and not tumor cells themselves that still interact via transmembrane ligands only [23]. Thus, levels of soluble ligands have been considered as a parameter and not a dynamic variable of the model.

For the case of paracrine communication via soluble Delta, both the cells tend to behave as Receiver (R₁, R₂) for low production rate of Jagged (J₀) or both of them behave as hybrid S/R (S/R₁, S/R₂) for high J₀. These cells can adopt opposite fates to each other only for intermediate levels of J₀ and high production rate of Delta (D₀) (figures 8(A) and (C), compare with figure 4(A)). Conversely, when the cells interact with soluble Jagged, the cells tend to attain similar fates—hybrid S/R (S/R₁, S/R₂)—unless the production rate of Delta (D₀) is high. For high D₀, these cells attain opposite fates- one behave as Sender, and the other Receiver (S₁, R₂) (figures 8(B) and (D)). Collectively, our results suggest that the soluble ligands—both Delta and Jagged—increase the range of parameters for which both cells attain same fate.

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Our results also offer an explanation into why stromal cells secrete Jagged1 more often than Delta [23]. Higher levels of soluble Jagged can enable many cells to adopt a hybrid S/R state (figures 8(B) and (D)), thereby enabling a both-way communication between cancer cells and/or between cancer and stromal cells. Not surprisingly, soluble Delta has been implicated in tumor suppression[64], but soluble Jagged1 facilitates cancer cell survival [65] as well as mediates communication between cancer cells and endothelial cells that leads to increased population of therapy-resistant tumor-initiating CSCs [23].

Being the key cell–cell communication pathway, Notch signaling circuit also couples to other aspects of cell-fate decision. Two of those modules play a crucial role in cancer metastasis—EMT module and stemness module (figure 9(A)). EMT allows cancer cells to initiate metastasis and travel to different organs in the body, and gaining stemness allows them to initiate a new tumor at those different organs. Here, we discuss the crucial role of hybrid S/R state (that allows bidirectional communication between cancer cells and/or between cancer–stroma) in tumor progression.

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Hybrid S/R state allows cancer cells to move and circulate collectively

Hybrid S/R state increases CSC population