Redundancy and cooperation in Notch intercellular signaling

The Notch signaling pathway, which is conserved across metazoans, is responsible for coordinating cell fate decisions among directly interacting cells (see Artavanis-Tsakonas et al., 1999; Ehebauer et al., 2006; Greenwald and Rubin, 1992 for reviews). This pathway orchestrates the development of different organs, tissues and cells, such as the sensory organs in vertebrates (Chitnis, 1995), the eye imaginal disk in Drosophila (Roignant and Treisman, 2009), the pancreas (Apelqvist et al., 1999) and intestine (VanDussen et al., 2012) in mice, and immune cells (Radtke et al., 2004) in humans. Furthermore, the Notch signaling pathway also plays a relevant role in different human diseases and cancer (Mašek and Andersson, 2017; Nowell and Radtke, 2017).

The Notch signaling pathway is involved in the mediation of two distinct coordinated cell fate decisions: lateral inhibition and lateral induction. In lateral inhibition, a cell inhibits adjacent cells from adopting the same cell fate (Cabrera, 1990; Goriely et al., 1991; Heitzler and Simpson, 1991; Simpson, 1990; Sternberg, 1988). This is important during development, for example, so that each specified hair cell in sensory organs is surrounded by supporting cells (Daudet and Lewis, 2005; Neves et al., 2013; Petrovic et al., 2014; Pickles and van Heumen, 2000; Zine et al., 2000), and so that only a few select cells become neurons in vertebrate retinae (Formosa-Jordan et al., 2013; Harris, 1997; Livesey and Cepko, 2001; Marquardt and Gruss, 2002). On the other hand, lateral induction occurs when a cell induces neighboring cells to adopt the same cell fate (Adam et al., 1998; Daudet and Lewis, 2005; Eddison et al., 2000; Hartman et al., 2010; Neves et al., 2011). This type of decision underlies the specification of prosensory patches during development of the inner ear in vertebrates (Daudet and Lewis, 2005; Hartman et al., 2010; Neves et al., 2011), the differentiation of the lens fibers in rats (Saravanamuthu et al., 2009), muscle differentiation in mice (Manderfield et al., 2012), as well as embryonic endocrine epithelial-mesenchymal transitions and oncogenic transformation (Timmerman et al., 2004).

Notch is a transmembrane receptor that binds to membrane-bound DSL ligands (Delta and Serrate/Jagged/Lag-2 ligands) of neighboring cells. This type of interaction is known as a trans-interaction (see Fortini, 2009; Kopan and Ilagan, 2009 for reviews). After binding, cleavage of the intracellular domain of Notch (hereafter referred to as the signal) translocates to the nucleus where it regulates the transcription of a battery of genes, including proneural repressor genes, which in turn regulate the expression of the DSL molecules. As such, Notch signaling can ultimately regulate the transcription of its own ligands in either a repressive or an inductive manner. For example, by activating the proneural repressor genes, Notch signaling can repress the transcription of Delta in the ommatidial crystal of Drosophila (Lubensky et al., 2011), and Delta-like 1 (Dll1) and Jagged2 (Jag2) in the developing inner ear of vertebrates (Kiernan et al., 2001; Neves et al., 2011). In contrast, Jagged1 (Jag1) and Delta-like 4 are Notch ligands that are induced by Notch signaling in the inner ear of vertebrates (Kiernan et al., 2001; Neves et al., 2011) and in angiogenesis (Benedito et al., 2009), respectively.

Our understanding of Notch-mediated lateral communication has typically been conceived through the action of a single type of ligand activating a single type of receptor (Fig. 1). In this case, lateral inhibition arises naturally when Notch signaling represses the ligand (Collier et al., 1996) (Fig. 1C). A cell expressing the ligand trans-activates Notch signaling in an adjacent cell, which, in turn, inhibits the production of this same ligand (Fig. 1C). Equivalent cells that mutually communicate through such lateral inhibition can become distinct upon amplifying small differences early on, and drive a periodic pattern of two cell types within a tissue, as previously shown by mathematical modeling (Collier et al., 1996) (Fig. 1A). These two cell types correspond to two distinct signaling states (Sprinzak et al., 2010): a sending cell (S), which strongly produces the ligand; and an adjacent receiving cell (R), which has high Notch signaling activity (Fig. 1A). In contrast, lateral induction arises when the ligand activates the signal in adjacent cells causing increased ligand production therein (Fig. 1D). Thus, cells communicating through lateral induction reach the same cell fate and signaling state, acting as both sending and receiving (S/R) cells (Boareto et al., 2015; Daudet and Lewis, 2005; Hartman et al., 2010; Jiménez et al., 2017; Matsuda et al., 2012; Neves et al., 2011; Petrovic et al., 2014) (Fig. 1B).

However, in vivo, the scenario is much more complicated, with communication through the Notch signaling pathway depending on numerous ligands and receptors. In this way, the various ligands can be antagonistically or equivalently regulated with respect to one another (Benedito et al., 2009; Gama-Norton et al., 2015; Kiernan et al., 2005; Okigawa et al., 2014; Petrovic et al., 2014; Preuße et al., 2015; Ye et al., 2017; Zine, 2003; Zine et al., 2000). Different Notch ligands typically activate the same canonical pathway such that the final process of lateral communication is the result of the combination of all signals (Fig. S1). Therefore, it is not always easy to predict the overall lateral communication that arises when multiple ligands activate Notch signaling. Along these lines, mathematical modeling has been useful in guiding our understanding of specific situations in which ligands are antagonistically regulated by Notch signaling (Boareto et al., 2015; Petrovic et al., 2014).

A factor to take into account when considering multiple Notch ligands is that each type can drive signaling with a characteristic efficiency (Benedito et al., 2009; Petrovic et al., 2014; Yang et al., 2005). Some ligands are so inefficient (hereafter named weak ligands) that even in high quantities they only weakly activate Notch signaling. For example, Jag1 activates Notch signaling less efficiently than Delta-like 4 (Dll4) in both angiogenesis (Benedito et al., 2009) and mouse haematopoietic stem cell development (Gama-Norton et al., 2015), and less efficiently than Delta-like 1 (Dll1) in chick inner ear development (Petrovic et al., 2014). When Notch signaling is activated by several ligands, differences in signaling efficiencies enable the cell to fine-tune the overall level of Notch signaling. In other words, multiple ligands do not just result in saturated activation. For example, whereas overexpression of a strong ligand can result in an increase in Notch signaling, overexpression of a weak ligand – when a strong ligand is also present – can drive a reduction in Notch signaling if the two ligands bind to the same type of Notch receptor (Petrovic et al., 2014). This type of modulation has been reported in both angiogenesis (Benedito et al., 2009) and inner ear development (Petrovic et al., 2014).

Differences in efficiency between Notch ligands can be attributed to various factors that drive distinct consequences. For example, binding between a ligand and a receptor of the same cell (cis-interaction) can prevent signaling through mutual inactivation (cis-inhibition) (Boareto et al., 2015; Fiuza et al., 2010; Formosa-Jordan and Ibañes, 2014; LeBon et al., 2014; Sakamoto et al., 2002; Sprinzak et al., 2011, 2010). Other much less studied factors that are expected to drive differences in ligand efficiencies are trans-interactions (Yang et al., 2005). For example, the various types of receptor-ligand complexes that are created through trans-interactions are all processed differently and likely lead to distinct levels of Notch signaling (Formosa-Jordan and Ibañes, 2014; Yang et al., 2005). The effect of overexpressing a weak ligand in a cell that also expresses a strong ligand depends on which factor causes the ligand to be weak. If the ligand is weak due to cis-inhibition, then its overexpression makes the cell better at sending but poorer at receiving signals (Sprinzak et al., 2010). In contrast, if the ligand is weak due to poor processing of trans-interactions, then its overexpression can cause the cell to become even worse at sending signals (Petrovic et al., 2014).

Previously, we have studied the effects of Jag1, a Notch-activated ligand that elicits weak signaling through trans-interactions, during avian inner ear development (Petrovic et al., 2014). Mathematical modeling predicted that this type of ligand, which drives lateral induction when acting alone, can facilitate patterning by lateral inhibition when acting together with a strong ligand that is repressed by Notch signaling (Petrovic et al., 2014). In this way, Jag1 changes its role: it drives Notch signaling in adjacent cells when acting in isolation, but through competitive inhibition, reduces signaling to adjacent cells when a strong ligand is present. Modeling also predicted that the inefficiency of the weak ligand enables the transition from the prosensory state to the hair cell specification stage through activation of the strong ligand (Petrovic et al., 2014).

As exemplified above, theoretical modeling has been useful for understanding the complexity of Notch signaling. Here, we pinpoint common principles behind overall Notch signaling when several ligands with different efficiencies due to the distinct processing dynamics of trans-interactions (i.e. not those affected by cis-inhibition) are at play. To unveil such common principles, we evaluate a wide variety of scenarios where ligands are either equivalently or antagonistically regulated by Notch signaling, and where ligands share or do not share limiting resources like the Notch receptor. Our results suggest that efficiency differences arising from alternative processing of trans-interactions can make Notch signaling more flexible: not only can signaling responses be enhanced, but through the combination of existing elements, new emergent responses can be created to promote evolvability.

We selected a modeling approach that strongly facilitates the theoretical study of spatially coordinated responses of multiple interacting cells across the parameter space. The tissue is modeled as a two-dimensional array of hexagonal cells (Fig. 1) in which we search for two types of coordinated responses – lateral inhibition and lateral induction. With respect to lateral inhibition, we searched for periodic patterning of distinct cell fates arising from equivalent cells (see linear stability analysis in the Materials and Methods). For outcomes of lateral induction, we chose the most restrictive definition (Matsuda et al., 2012; Petrovic et al., 2014), which states a ligand that is transiently and exogenously activated locally within a tissue propagates and drives the transition of the whole tissue from one stable state to another stable state, both being composed of entirely equivalent cells (i.e. both states are homogeneous and linearly stable under small perturbations) (see linear stability analysis in the Materials and Methods).

Our model is based on the activity dynamics of two ligands (d for ligand1 and z for ligand2), each of which activates Notch signaling (s) in adjacent cells through trans-interactions, and whose production is in turn regulated by this same signaling (see Materials and Methods). For the purpose of studying solely the effect of distinct ligand efficiencies in trans-activation, our model does not include any cis-interactions. Our model, which extends the model of Collier et al. (1996) by including a second ligand, is a simplified version of the model of Petrovic et al. (2014). Thus, in our model, the signaling activity of a cell changes in function of the average activity of each ligand over all adjacent cells (trans-interactions), and the ligand activity changes according to the signaling activity within the same cell (see Materials and Methods, Eqns 1-6).

Without loss of generality, the maximal activities of ligand1 and ligand2, as well as the maximal stationary signaling activity driven by ligand1 in isolation, were set to 1 (see Materials and Methods). To account for differences in trans-interaction ligand efficiencies, we set ligand2 to drive a maximal stationary signaling activity of value ɛ when it is the only ligand acting. Therefore, when ɛ=1, the two ligands are equally efficient, whereas when ɛ<1, ligand2 is a weak ligand and ligand1 is a strong ligand.

We defined the affinity of a ligand for the Notch receptor together with its maximal production rate in one parameter, r_l, hereafter, named trans-interaction strength (see Materials and Methods). Thus, r_d and r_z are the trans-interaction strengths for the two ligands d and z, respectively. The parameter ɛ_r≡r_z/r_d defines the relative strength of the trans-interactions mediated by ligand2 (z) compared with those mediated by ligand1 (d). When we refer to overexpression of ligand1 or ligand2, it means we have set high values for r_d or r_z, respectively.

Trans-activation of Notch signaling depends on both the trans-interaction strengths (r_d, r_z) and the trans-interaction efficiencies of the ligands (1 and ɛ, respectively, for ligand1 and ligand2) (see Materials and Methods). In addition, we considered two different scenarios of signal trans-activation (Fig. S1): ligands that use independent resources to signal (i.e. a different receptor type each), and ligands that share common resources (i.e. the same type of receptor) (see Materials and Methods, Eqns 5 and 6). These distinct scenarios were modeled such that the presence of a weak ligand makes the cell a better sending cell only if the weak ligand signals through its own specific receptor (see Materials and Methods). In contrast, as in the model of Petrovic et al. (2014), if the weak ligand shares resources with the strong ligand, then it can cause a reduction in signaling to adjacent cells. In such a case, the ligand is known as a partial agonist (see Materials and Methods).

Previously, we created a model based on a scenario in which a weak Notch-activated ligand shares resources with a strong Notch-repressed ligand (Petrovic et al., 2014) (Fig. 2A). In such a model, we found that the presence of the weak ligand facilitates pattern formation through competitive inhibition with the strong ligand (Petrovic et al., 2014). As expected, the model presented in the current study yields similar results when considering these two types of ligands (Fig. 2B). When only the strong ligand is present, it induces patterning unless overexpressed (Fig. 2B) (Collier et al., 1996). If overexpressed, patterning arises only when the strong ligand acts together with a weak ligand (Fig. 2B). In this way, both ligands cooperate to drive patterning, albeit only when the ligands share resources like the Notch receptor (Fig. S2).

We then analyzed the role of ligand-mediated feedback on pattern formation. As expected, we found that the strong ligand is able to mediate sufficient feedback to drive patterning (red area in Fig. 2B, defined in the Materials and Methods), a phenomenon that also occurs when the weak ligand is also present. Strikingly, the Notch-activated weak ligand (i.e. a ligand that, owing to its inefficiency, would not be able to drive any response by itself) can also mediate sufficient feedback to drive patterning, albeit concurrently with the strong ligand (gray area overlaps the red area, marked as striped in Fig. 2B and Fig. S2, see Materials and Methods).

Analysis of the stationary signal and ligand activities showed that patterning through strong and weak ligands that share resources can drive several distinct signaling states (Fig. 2C). When the weak ligand has a low trans-interaction strength (ɛ_r<< 1, Fig. 2C bottom), cells with a high activity of the strong ligand become surrounded by cells with strong activity of the weak ligand and strong Notch signaling. This is in accordance with ligand expression patterns observed in the inner ear of chick embryos (Fig. 2D), and as previously modeled (Petrovic et al., 2014). While the cells activating the strong ligand are called sending (S) cells, herein we term the surrounding ones blocking and receiving (B/R) cells. According to our definition, the blocking signaling state arises when there is activity of the weak ligand because it diminishes signaling to adjacent cells by the strong ligand. However, we find that a weak ligand with a strong trans-interaction strength drives distinct patterns of signaling, including signaling with three different states (Fig. 2C). Specifically, S cells can be in contact with both B/R and R cells (Fig. 2C, middle). While R cells exhibit intermediate signaling activities and do not significantly activate the weak ligand, B/R cells have high Notch and weak ligand activities.

In addition, we found that cooperation between weak and strong ligands has dynamic consequences for patterning. For example, patterning arises faster when there is cooperation between antagonistic strong and weak ligands than when there are two equal ligands (e.g. equally strong, both equally inhibited by Notch signaling) that do not share resources (Fig. S3 and Table S1, see Materials and Methods, and supplementary Materials and Methods). Therefore, co-expression of a weak ligand that is activated by Notch signaling can accelerate patterning (Fig. S3A) when compared with co-expression of an equally strong ligand that is repressed by Notch signaling and does not share common resources (Fig. S3B).

Subsequently, we analyzed the opposite scenario in which Notch signaling activates the strong ligand but represses the weak ligand (Fig. 3A). We set the strong ligand so that it could drive lateral induction on its own (Figs 1B, 3B). Exogenous application of this ligand on a few cells within an array of cells with almost no signaling or ligand results in ligand propagation and the sequential transition of all cells to a S/R state (Fig. 1B). In contrast, we set the weak ligand to be very inefficient and unable to drive any response on its own (Fig. 3B). Our analysis shows that when acting together, both ligands cooperate in such a way that the weak ligand facilitates lateral induction and enables transition under any trans-interaction strength or abundance of the strong ligand (Fig. 3B). This requires that the ligands share resources and signal with different efficiencies (Fig. S4). In such a cooperative case, cells can make the transition from a B state, in which high activity of the weak ligand strongly reduces signaling and therefore the strong ligand activity, to an S/R state that exhibits high levels of strong ligand and signaling (Fig. 3C). The reverse transition can also occur and each involves a distinct ligand that is propagated (Fig. S5). For example, when the trans-interaction strength of the weak ligand is low compared with that of the strong ligand (ɛ_r<< 1), only the strong ligand can propagate and induce the transition from B to S/R (Fig. 3C). This also corresponds to the analogous case in which the weak ligand is absent (Fig. 1B). In contrast, when the weak ligand has a stronger trans-interaction strength (ɛ_r>> 1), the opposite transition becomes possible through the propagation of the Notch-repressed weak ligand (Fig. S5). This highlights the relevant and active role of the weak ligand in driving an unexpected outcome.

In light of these findings, we decided to evaluate whether the weak ligand could still promote the same responses even when the strong ligand was unable to drive them in isolation. For example, both patterning and ligand propagation between bistable states require strong nonlinearities in ligand dynamics. If the strong ligand does not present such nonlinearities, then it is unable to elicit a coordinated response on its own (i.e. a pattern, if repressed by Notch signaling; or ligand propagation, if activated by Notch signaling). We found that the presence of a very weak antagonistically regulated ligand that is not capable of driving a response on its own, can help promote a coordinated response (Fig. 4). This response corresponds to the type of response that would arise if the strong ligand exhibited more nonlinear dynamics. In addition, the feedback mediated by the weak ligand, in presence of the strong ligand, becomes the only feedback sufficient to drive the patterning response (gray area in Fig. 4B). Thus, two antagonistic ligands, that, when in isolation, cannot drive a response, can drive a response when cooperating together. Importantly, this only occurs if the ligands have distinct efficiencies and share resources.

Altogether, these results show that antagonistic strong and weak ligands that share resources can cooperate to drive responses that are characteristic of the regulation of the strong ligand. This is because by reducing signaling to adjacent cells, the weak ligand effectively mediates the same type of lateral regulation as the antagonistically regulated strong ligand (see Materials and Methods) (Petrovic et al., 2014).

We next studied the case in which the two ligands are analogously regulated (i.e. both are either repressed or activated by Notch signaling) and drive Notch signaling with the same efficiency (i.e. ɛ=1). When both ligands are present and activate Notch signaling, they become redundant (i.e. the response can also be induced by just one of the two ligands) and drive the same response, independently of whether they share the same resources (Fig. 5A-C and Fig. S6A,B). When both ligands are repressed by Notch signaling they elicit a pattern formation response (Fig. 5B,C), whereas when both are activated by Notch signaling, ligands are propagated and cause cells to transition from a low ligand activity state to a higher ligand activity state (Fig. S6A,B). However, no response arises when either of the two ligands is overexpressed because Notch signaling becomes saturated (Figs S7A,B and S8A,B).

As expected, redundancy does not occur when one ligand is very weak (Fig. 5D,E, Fig. S6C,D). In addition, in such a case, overexpression of the weak ligand has very little effect compared with overexpression of the strong ligand when they do not share resources (Fig. 5D, Fig. S6C). However, when there is a single receptor type (i.e. shared resources), the strong ligand cannot drive a coordinated response when the weak ligand is overexpressed. This is true when both ligands are either repressed (Fig. 5E) or activated (Fig. S6D) by Notch signaling.

The responses observed in the scenarios in which ligands share common resources can provide explanations for the puzzling phenotypes that arise from mutations of Jag1, Jag2 and Dll1 during the specification and determination of hair cells in the mammalian inner ear (Kiernan et al., 2005; Liu et al., 2013; Zhang et al., 2000; Zine, 2003; Zine et al., 2000) (see Materials and Methods). These three ligands signal through only the Notch1 receptor despite Notch2 also being expressed (Kiernan et al., 2005; Zhang et al., 2000; Zine et al., 2000), and are distinctly regulated by Notch signaling. Whereas Jag1 is induced by Notch signaling, Jag2 and Dll1 are repressed (Daudet and Lewis, 2005; Hartman et al., 2010; Kiernan et al., 2005). It has been shown in vivo that Jag1 helps to mediate proper hair cell patterning (Zine et al., 2000). According to both Petrovic et al. (2014) and our current data (Fig. 2B), Jag1 should be a weak ligand. This would allow Jag1 to cooperate with a stronger ligand that is repressed by Notch to drive patterning. Analysis of a three-ligand scenario shows that, for this cooperation to emerge, Jag1 needs to be weaker than just one of the other two ligands (Fig. S9A). It has also been reported that Jag2 mutation disrupts inner hair cell (iHC) patterning, but not as much outer hair cell (oHC) patterning (Kiernan et al., 2005; Zhang et al., 2000). Our results suggest that Jag2 and Dll1 are similarly strong ligands in oHCs (albeit not necessarily equally strong), acting in a redundant manner (Fig. S9A), whereas Jag2 is stronger than Dll1 in iHCs, being required for patterning (Fig. S9B). Mutation of the glycosyltransferase lunatic fringe (Lfng) rescues the Jag2 mutant phenotype, suggesting that Lfng suppresses signaling by Dll1 (Zhang et al., 2000). In agreement, our results support that this rescue is to be expected when Lfng partially decreases Dll1 efficiency during iHC patterning (Fig. S9C) (see also supplementary Materials and Methods for further details). This partial change in efficiency would require that Notch signaling by Dll1 is altered after ligand-receptor binding (with no need for ligand-receptor affinity changes).

The disruption that overexpression of a weak ligand causes when sharing resources with a strong equally regulated ligand is not due to signal saturation, but rather because the weak ligand opposes the actions of the strong ligand (Fig. 5E, Fig. S6D, see Materials and Methods). For example, an overexpressed weak ligand that (just like the strong ligand) is repressed by Notch signaling mediates its own induction in adjacent cells and thereby impedes patterning (Fig. 5E, see Materials and Methods). Conversely, when both ligands are activated by Notch signaling, the weak ligand disrupts ligand propagation via a type of lateral inhibition (gray region in Fig. S6D). We reasoned that there must be a regime in which these kinds of antagonistic lateral regulations are functional.

Although a very weak ligand that is repressed by Notch signaling cannot drive any relevant response on its own, we found that it can mediate fully functional lateral induction when acting with a stronger ligand that is also repressed by Notch (Fig. 6A,B). This functional lateral induction corresponds to a propagation of the weak ligand and the transition of all cells from one stable state to another stable state (Fig. 6C). This response requires that the weak ligand becomes repressed under lower signaling activity than the stronger ligand (i.e. 1/b_z<1/b_d), thereby enabling mutual lateral induction between adjacent cells (Fig. 6A, Fig. S10). Such ligand propagation is an emergent response that cannot be driven by either ligand expressed in isolation, and is antagonistic to the expected response based on the regulation of each single ligand by Notch signaling.

In this kind of response, cells are simultaneously S/R cells and can transition to a B state through propagation of the weak ligand. In the S/R state, signal activity is sufficiently high to completely repress the weak ligand, but still low enough to only partially repress the strong ligand (Fig. 6C). After ligand propagation, signaling drops and repression of the two ligands is relieved. This corresponds to a B state because the weak ligand causes the neighboring cells to have a lower signaling activity compared with if only the strong ligand was expressed. During this transition, the signal decreases as the ligand propagates, and the transition does not occur if the strong ligand is locally increased. Thus, this response is mainly mediated by the weak ligand, because despite the presence of the strong ligand being a requirement, only the induction of the weak ligand (ligand2) drives this transition.

A robust emergent response also occurs when both the strong and weak ligands are activated by Notch signaling and share resources (Fig. 7A). In this case, weak ligand-mediated salt-and-pepper patterning arises (Fig. 7B,C). The weak ligand mediates an effective circuit of lateral inhibition that in turn drives patterning when the strong ligand is activated at a lower signal activity than the weak ligand (Fig. 7A, Fig. S10B). This emergent response is characterized by two distinct signaling states in which B/R cells are surrounded by S/R cells with low (but not null) signaling and ligands levels (Fig. 7C). The B/R cells are receiving as they have a high signaling activity that activates both ligands, and are blocking because they express the weak ligand and hence drive lower signaling in neighboring cells. In comparison, the S/R cells are sending because they drive signaling in adjacent cells (i.e. in the B/R cells) through low but sufficient activity of the strong ligand, and are receiving because their signal activity is sufficiently high to activate the strong ligand but not the weak one.

These results demonstrate that robust emergent responses can arise when two ligands are equivalently regulated by Notch signaling. For this to occur, however, both ligands – one being strong and the other weak – need to be present as well as share the receptor. Furthermore, each ligand must also be regulated by Notch signaling at different signaling thresholds, and the dynamics of the weak ligand be characteristically nonlinear (Fig. S10C-F). In such a scenario, the emergent response is opposite to the response driven by the strong ligand when acting alone. Thus, this response is unexpected when considering the type of regulation that Notch signaling mediates on the two ligands.

Single ligand-mediated responses have been thoroughly computationally addressed, and it is well known that these responses can be highly dependent on how Notch signaling regulates the production of the ligand (Collier et al., 1996; Matsuda et al., 2012, 2015; Petrovic et al., 2014; Wearing et al., 2000; Webb and Owen, 2004). However, when additional regulations are taken into account, such as receptor activation by Notch signaling as in C. elegans development (Wilkinson et al., 1994), elaborate responses can arise from a sole ligand. For example, theoretical modeling predicts that periodic patterns can emerge when Notch signaling activates both the ligand and the receptor (Owen et al., 2000). Commonly, various different ligands are expressed within tissues and activate Notch signaling at the same time (Benedito et al., 2009; Gama-Norton et al., 2015; Kiernan et al., 2005; Petrovic et al., 2014; Preuße et al., 2015). Owing to nonlinear coupling between the regulatory circuits mediated by each ligand, mathematical modeling becomes a powerful tool for addressing what functional response can be expected when two ligands are co-expressed.

We chose a minimal phenomenological description that, albeit simple, can provide insights into experimental data as previously shown for similar approaches (Cohen et al., 2010; Petrovic et al., 2014). We focused on two main features. First, ligands can activate Notch signaling with different trans-activation efficiencies. Second, ligands share resources to signal (i.e. there is only one Notch receptor type) or not (i.e. there are as many receptor types as ligands). Additional complexities, such as receptor activation by Notch signaling or cis-inhibition, were not included. Our results show redundancy and cooperation in the multi-ligand scenario (Table 1, Fig. 8). Ligands that exhibit different efficiencies and share resources can cooperate. Cooperation requires the dynamics of the weak ligand to be nonlinear. Cooperation between ligands antagonistically regulated by Notch enables the weak ligand to enhance and even drive the type of coordinated response that is expected from Notch regulation of the strong ligand. In contrast, similarly regulated ligands can cooperate to drive new emergent tissue responses. For this to happen, the ligands must be similarly regulated by Notch signaling yet at different activity thresholds. Otherwise, equivalently regulated ligands can be redundant and even detrimental to one another.

We exemplified that redundancy, together with enhancement, might be able to explain the distinct differentiation phenotypes observed in the cochlea of the mammalian inner ear (Fig. S9). Moreover, our results on emergent responses (Fig. 7A,B) may apply to angiogenesis. In this process, two cell types (tips and stalks) arise and two Notch ligands, one strong (Dll4) and one weak (Jag1), have opposing roles (Benedito et al., 2009). Through ligand Dll4-mediated signaling, tip cells strongly activate Notch signaling, which suppresses the tip phenotype, in adjacent (stalk) cells (Benedito et al., 2009). Jag1 is expressed in stalk cells and competes with Dll4, reducing Notch signaling (Benedito et al., 2009). Dll4 is activated by Notch signaling, raising the issue of how tip/stalk selection arises. Our analysis suggests that the selection can be mediated by Notch signaling if Jag1 is activated by Notch (Pedrosa et al., 2015) at a higher Notch signaling activity than Dll4. In this scenario, the selection is an emergent response that requires both Jag1 and a stronger ligand (Dll4) (Fig. 7A,B). This could be consistent with the altered selection phenotype found when either ligand is mutated or when Lfng, which can enhance differences in efficiency between the ligands, is mutated (Benedito et al., 2009).

To account for signaling states that arise when several ligands are present and share resources but signal with different efficiency, we have introduced a blocking state (B) in which the weak ligand diminishes signaling to adjacent cells. In this way, this state can describe the trans-inhibitory effect that can arise when different Notch ligands are at play (Benedito et al., 2009; Gama-Norton et al., 2015; Petrovic et al., 2014).

It is well known that the same regulatory interactions can drive distinct spatially coordinated outcomes. This can happen at different strengths of the interactions (i.e. different parameter values) or even at the same strengths (Corson et al., 2017; Lubensky et al., 2011; Palau-Ortin et al., 2015). A recent study focused on regulatory interactions that drive different coordinated responses depending on the basal level of activation of one element (Jiménez et al., 2017). It was found that this system does not necessarily decouple into two parts with each part performing one of the two coordinated responses, but, rather, one of the coordinated responses requires all the components (Jiménez et al., 2017). Our emergent states and enhanced responses fall into this category because they require all components and cannot be decoupled into submodules.

During the development of vertebrate neural-sensory organs, there are transitions from lateral induction to lateral inhibition (Daudet and Lewis, 2005; Hartman et al., 2010; Millimaki et al., 2007; Petrovic et al., 2014; Zine, 2003). To begin with, a group of cells specifies a prosensory domain by lateral induction. Then, through lateral inhibition, some cells differentiate into sensory cells while the others differentiate into supporting tissue. From the classical paradigm of one ligand-mediated lateral regulation, it is expected that these transitions can only occur via the sequential non-overlapping action of single antagonistic ligands. In other words, first a ligand activated by Notch drives lateral induction, then after it disappears, a ligand repressed by Notch emerges and drives lateral inhibition. However, it has been reported that these transitions can also involve the co-expression of multiple ligands (Daudet and Lewis, 2005; Petrovic et al., 2014; Zine, 2003). For example, we have previously shown that this type of transition can occur through the activation of a stronger, Notch-repressed ligand on tissues that also express a weak, Notch-activated ligand (Petrovic et al., 2014). Here, our current results suggest two additional ways in which transitions between lateral induction and lateral inhibition can occur: (1) by modulating the strength of trans-interactions of two co-expressed ligands that are equally regulated by Notch, share common signaling resources and signal with different efficiencies (arrowheads in Figs. 6B, 7B); and (2) by modulating the relative thresholds of ligand regulation (arrowheads in Fig. S10A,B).

In general, animals have more types of ligands than receptors, and as such when the different types of ligands and receptors interact, they can signal with different strengths (Bray, 2006; D'Souza et al., 2010; Kopan and Ilagan, 2009; Lendahl, 1998). In addition, this fact increases the likelihood that multiple ligands compete for common receptors. Here, our theoretical results suggest that when ligands compete for signaling, existing responses can be enhanced and new functional emergent responses can even arise. Furthermore, these emergent responses are inconsistent with the responses that arise in the absence of competition. Thus, as competition between ligands can drive new responses by merely combining already existing elements, we propose that competition confers both a robustness and evolvability to Notch signaling. Ultimately, we expect that our theoretical analyses will help envisage future experiments and quantifications for unraveling the effects of distinct ligands of Notch signaling.

In both cases, is the signal activity threshold for regulation of the ligand activity and h_l stands for the effective cooperativity of the process, which in this model is simplified to a single regulatory step. δ_l stands for the basal constant production of the ligand with respect to the maximal production (of value 1).

For most of the presented results, unless otherwise stated, we used v=v_l=1 for both ligands, h_z=h_d=4 and δ_l=0.0001. We set b_l=1 (b_l=10⁴) when the ligand is activated (repressed) by the signal to obtain ligand propagation (patterning) for each ligand acting in isolation.

To study the outcomes across the parameter space, we chose to perform linear stability analysis (LSA) of the homogeneous solutions, as described in the Materials and Methods subsections on LSA. This enables a vast analysis of the parameter space. Through LSA, we searched for those parameter space regions where there is a homogeneous stationary solution that is unstable with respect to inhomogeneous perturbations and a pattern can be expected to emerge (denoted as patt region and enclosed by blue dashed lines). We also evaluated the parameter space regions where there are two homogeneous states that are linearly stable (denoted as bis region and enclosed by black solid lines). These regions can sustain ligand propagation (i.e. the lateral induction phenotype). Numerical integration of the dynamics (see Materials and Methods in subsection ‘Programs and simulation details’) at specific parameter values was carried out to compute the ligand and signal activities over time of cells (depicted in figures with grayscale in hexagonal arrays) and to corroborate the LSA results (see supplementary Materials and Methods for further details).

We used custom-made programs (C language, available upon request) to determine the homogeneous steady states of Eqns 1 and 2 [(ds/dt)=(dl/dt)=0, taking l=d for ligand1 and l=z for ligand2]. We drop the subindexes i given the homogeneity of the steady states. Implementing the bisection method for root finding, roots were computed for each case described above, i.e. ligands activated and/or repressed by the signal and ligands sharing or not common resources. Hereafter, the stationary solution for any variable x is named as x₀. On these homogeneous solutions, we performed linear stability analysis as described below.

We defined R_x,y as the partial derivative of the dynamics of species x with respect to species y, evaluated at the homogeneous stationary state. Hence, the sign of R_l,sR_s,〈l〉 is used to determine the effective lateral regulation that the average ligand levels present in the neighborhood of a cell (〈l〉) mediates on the ligand activity in such a cell, evaluated in the homogeneous state. In particular, it evaluates whether ligand activity effectively inhibits (R_l,sR_s,〈l〉<0) or induces (R_l,sR_s,〈l〉>0) the ligand activity in adjacent cells. According to dynamics in Eqns 1 and 2, these terms are R_l,sR_s,〈l〉=v_l(∂f_l(s)/∂s)(∂f_s(〈l〉)/∂〈l〉) evaluated in the homogeneous state. When there is only one ligand, R_s,〈l〉=(∂f_s(〈l〉)/∂〈l〉)>0 is always fulfilled, and hence the type of lateral regulation a ligand mediates is dependent only on how the signal regulates the ligand activity (i.e. on the sign of R_l,s=v_l(∂f_l(s)/∂s)). The same happens if there are two ligands that do not share resources, as both ligands always effectively activate the signaling (R_s,〈d〉=(∂f_s,NN(〈d〉,〈z〉)/∂〈d〉)>0, R_s,〈z〉=(∂f_s,NN(〈d〉,〈z〉)/∂〈z〉)>0). In contrast, when there are two ligands signaling differently and sharing limited resources, the weak ligand can exhibit a partial agonist behavior, as described in the previous subsection, so that the condition R_s,〈l〉<0 would be fulfilled. Therefore, the sign of R_l,sR_s,〈l〉 for a partial agonist ligand is opposed to the one it mediates when acting in isolation (i.e. when R_s,〈l〉>0). In other words, the partial agonist mediates a lateral regulation of its ligand activity that is in opposition to the one it mediates when acting in isolation.

A theoretical prediction of how fast patterning is expected to be achieved was computed through the exponential growing rate of the perturbation that grows the fastest over the homogeneous state. The calculation of this rate in the case of two ligands results in ⁠. In the calculations depicted in Fig. S3, we assume v=1.

We used custom-made programs in C to evaluate the conditions set by LSA (conditions 9-11 defining patt and bis regions, and conditions defining red and gray regions), the theoretical growth rate obtained from LSA and the partial agonist condition across the parameter space, by defining a grid of parameter set values in logarithmic space. The results in the parameter space were plotted using custom-made Python scripts (version 2.7) with the matplotlib package (Hunter, 2007). These programs are available upon request.

Numerical integration of the dynamics was performed by using NDSolve function of Mathematica 8.0 for a perfect hexagonal grid of 12×12 cells with periodic boundary conditions (custom-made program available upon request). Time step of integration was variable between 0.001 and 0.05 chosen by default by the Mathematica function. Integration was carried out to the final non-dimensional time 500. Initial conditions for each dynamic variable x in each i cell was x_i(t=0)=x₀(1+0.2 × (0.5−η)), where x₀ is the stationary homogeneous solution and η is a random uniform number between 0 and 1. The signal and ligand activities are depicted in a linear grayscale with white being 0 and black being 1. Numerical integration of the dynamics was carried out to corroborate the patt and bis regions obtained through LSA results at some points of the parameter space and to evaluate the time for pattern formation in a point of the parameter space (see supplementary Materials and Methods for further details).

Transitions from a homogeneous state with low ligand activities to a homogeneous state with higher activities were allowed by imposing the low state as initial condition. Then, we selected a small cluster of four cells in the central region of the array and set their initial condition for the ligand as l(t=0)=1.

We acknowledge fruitful discussions with J. Petrovic, J. Neves and F. Giráldez. We thank F. Giráldez for careful reading of a draft of the manuscript.