Cell fate determination by lateral inhibition via Notch/Delta signalling has been extensively studied. Most formalised models consider Notch/Delta interactions in fields of cells, with parameters that typically lead to symmetry breaking of signalling states between neighbouring cells, commonly resulting in salt-and-pepper fate patterns. Here, we consider the case of signalling between isolated cell pairs, and find that the bifurcation properties of a standard mathematical model of lateral inhibition can lead to stable symmetric signalling states. We apply this model to the adult intestinal stem cell (ISC) of Drosophila, the fate of which is stochastic but dependent on the Notch/Delta pathway. We observe a correlation between signalling state in cell pairs and their contact area. We interpret this behaviour in terms of the properties of our model in the presence of population variability in contact areas, which affects the effective signalling threshold of individual cells. Our results suggest that the dynamics of Notch/Delta signalling can contribute to explain stochasticity in stem cell fate decisions, and that the standard model for lateral inhibition can account for a wider range of developmental outcomes than previously considered.

INTRODUCTION

The Notch/Delta signalling pathway is one of the main regulators of cellular differentiation during development and adult tissue maintenance (reviewed by Artavanis-Tsakonas et al., 1999; Ehebauer et al., 2006; Koch et al., 2013). It often drives mutually inhibitory interactions between cells, acting as a gate for differentiation. This mode of action has been termed lateral inhibition, and has been the object of experimental study as well as mathematical formalisation for decades (see, for example, Othmer and Scriven, 1971; Collier et al., 1996; Sprinzak et al., 2011; Petrovic et al., 2014). Quantitative models of lateral inhibition usually involve a field of cells expressing initially similar amounts of the receptor Notch and its membrane-bound ligand Delta. Delta trans-activates Notch in neighbouring cells and Notch, once activated, reduces in turn the ability of the cell to signal through Delta, leading to a state of mutual repression. This symmetry (and cell fate equivalence) is eventually broken by enforced biases and/or stochastic variation in Notch/Delta levels (Collier et al., 1996; Plahte, 2001; reviewed by Simpson, 1997) resulting in extended fine-grained spacing patterns (Othmer and Scriven, 1971; Collier et al., 1996; see also Shaya and Sprinzak, 2011) that have been experimentally characterized in depth in real developmental systems (reviewed by Greenwald, 1998; Arias and Stewart, 2002). By contrast, little attention has been paid so far to the effect of lateral inhibition in isolated cell pairs, beyond the trivial expectation that symmetry breaking will eventually take place, leading to cells taking opposing fates (see, for example, Collier et al., 1996; Rouault and Hakim, 2012). However, there has been no formal investigation of whether alternative steady states are possible, perhaps due to the lack of an experimental model to relate it to.

The cellular homeostasis of the adult Drosophila midgut (Fig. 1A-D) can provide this experimental scenario, as in this tissue Notch/Delta signalling occurs mostly in isolated pairs of cells (Ohlstein and Spradling, 2006; de Navascués et al., 2012; Goulas et al., 2012). The intestinal lining of the fly is maintained by intestinal stem cells (ISCs), which divide to both self-renew and provide committed progenitors (Fig. 1B). Progenitors specialise in producing either nutrient-absorbing enterocytes or secretory enteroendocrine cells (Guo and Ohlstein, 2015; Micchelli and Perrimon, 2006; Ohlstein and Spradling, 2006; Zeng and Hou, 2015). The precursors of enterocytes, called enteroblasts (EBs), are frequently found forming pairs with ISCs (Ohlstein and Spradling, 2006; Micchelli and Perrimon, 2006; Bardin et al., 2010; de Navascués et al., 2012) (Fig. 1C). These pairs are thought to result from an earlier division of an ISC and subsequent fate allocation by Notch signalling, before a new division or terminal differentiation event takes place (Goulas et al., 2012; de Navascués et al., 2012). Importantly, ISC divisions in the enterocyte lineage result in asymmetric fate (one ISC and an EB), symmetric self-renewal (two ISCs) or differentiation (two EBs), which globally result in balanced, homeostatic proportions (de Navascués et al., 2012) (Fig. 1D). This mode of tissue maintenance, whereby the balance between stem cell self-renewal and differentiation is achieved at the population level rather than within every stem cell lineage, is termed neutral competition (Klein and Simons, 2011) and is found in a growing number of self-renewing adult tissues (Simons and Clevers, 2011). Although no molecular mechanism has been fully elucidated so far for any case of neutral competition, in the fly gut it has been proposed to arise from lateral inhibition mediated by Notch/Delta (de Navascués et al., 2012), a pathway known to define the fate of the ISC offspring (Ohlstein and Spradling, 2006; Micchelli and Perrimon, 2006; Bardin et al., 2010).

Here, we explore the capacity of a standard model of lateral inhibition acting in pairs of interacting cells to result in steady states with different signalling states (either symmetric or asymmetric) co-existing in the tissue. We find that this is indeed possible, provided signalling thresholds vary across cell pairs. Next, we turn to the Drosophila midgut and find that the tissue displays high variability of contact area between pairs of ISC/EB cells, which can be associated with an effective heterogeneity in signalling thresholds between pairs of cells. When contrasting this variability with the distribution of fate combinations in pairs of ISC/EB cells, we find a correlation between the contact area of specific cell pairs and their fate profile. Moreover, our model can reproduce the distribution of fate outcomes from the contact area distribution determined experimentally.

Our results expand the repertoire of possible outputs of a system governed by lateral inhibition, and connect this mode of signalling with a mode of stem cell-based tissue maintenance (neutral competition) that is highly relevant in adult tissue homeostasis and tumourigenesis (Simons and Clevers, 2011; Vermeulen et al., 2013; Baker et al., 2014), and whose molecular regulation is poorly understood.

RESULTS

Lateral inhibition can result in stable, opposing symmetric signalling states

We study the steady-state behaviour of a standard model of lateral inhibition for the case of two cells (see Materials and Methods). For fixed r and h, the steady states of this system depend on two parameters, a and b (the dimensionless activation and inhibition thresholds, respectively; see Materials and Methods), which we allow to vary across the population of cell pairs. We then calculate the equilibrium state of the system in this two-dimensional parameter space, according to the resulting signalling profile: asymmetric (one cell positive for Notch activation and the other one negative), symmetric positive or symmetric negative for Notch activation (Fig. 1C). Thus, for a population of cell pairs with variable activation or inhibition thresholds (a and b), the three possible signalling state profiles occur (Fig. 2A, see Fig. 2B-E for a comparison of the dynamic evolution of examples of the three profiles, with Fig. 2B corresponding to the parameter values from Collier et al., 1996). The three signalling state profiles can be found within a relatively short range of parameter values (Fig. 2A). Importantly, this scenario does not change qualitatively when considering a wide range of threshold values for Notch activity classification, as defined in the Materials and Methods section (0.001≤Nthr≤0.8; Fig. S1B-E), or higher cooperativity values in Notch signalling or Delta inhibition [r, h=(2, 5) or (5, 2)] (Fig. S1F,G). However, we found that cooperativity was necessary: for r=1, h=1, the heterogeneous (asymmetric) steady states are lost (Fig. S1H). This requirement of cooperativity is in agreement with other theoretical works on lateral inhibition (Sprinzak et al., 2011) as well as with in vitro estimates for r of 1.7 (Sprinzak et al., 2010).

Fig. 1.

Tissue maintenance in the Drosophila adult midgut and effect of mild genetic perturbations of Notch signalling. (A) Confocal micrograph showing the cell types present in the midgut epithelium. ISCs are esg-GFP+ (blue), and EBs are esg-GFP+ and GBE-Su(H)-lacZ+ (green). The two differentiated cells, enteroendocrine cells and enterocytes, are recognisable by nuclear Prospero (Pros) expression (red) and their large polyploid nuclei (Hoechst, grey), respectively. (B) ISCs self-renew and produce EBs (which will terminally differentiate without further division). (C) Confocal micrographs showing examples of cell pair type profiles: asymmetric (i), symmetric Notch negative (two ISCs, ii) and symmetric Notch positive (two EBs, iii). (D) ISCs balance self-renewal and differentiation by dividing either asymmetrically (one ISC and an EB) or symmetrically into two ISCs or two EBs. (E) Observed frequencies of the pair types depicted in C and D for wild-type (n=235), hypermorphic (l(1)NB/+, n=209) and hypomorphic (N55e11/+, n=213) conditions for Notch. (F) Observed frequencies of the pair types depicted in C and D for da-GS, UAS-NRNAi flies after 3 days of feeding on 0 (‘mock’, n=234), 20 (n=192) and 50 µg (n=99) per vial of RU486. (G) Total numbers of ISCs and EBs in the pairs in the conditions described in E. Overall numbers of ISCs/EBs in these genotypes are 765/737 (wild type), 790/752 (N55e11/+) and 804/1148 [l(1)NB] (de Navascués et al., 2012). (H) Overall numbers of ISC/EB in the conditions described in F. In E-H, triangles with colour gradient indicate the relative levels of Notch signalling: wild type (white), excessive (red) or defective (blue). Scale bars: 20 µm.

Fig. 1.

Tissue maintenance in the Drosophila adult midgut and effect of mild genetic perturbations of Notch signalling. (A) Confocal micrograph showing the cell types present in the midgut epithelium. ISCs are esg-GFP+ (blue), and EBs are esg-GFP+ and GBE-Su(H)-lacZ+ (green). The two differentiated cells, enteroendocrine cells and enterocytes, are recognisable by nuclear Prospero (Pros) expression (red) and their large polyploid nuclei (Hoechst, grey), respectively. (B) ISCs self-renew and produce EBs (which will terminally differentiate without further division). (C) Confocal micrographs showing examples of cell pair type profiles: asymmetric (i), symmetric Notch negative (two ISCs, ii) and symmetric Notch positive (two EBs, iii). (D) ISCs balance self-renewal and differentiation by dividing either asymmetrically (one ISC and an EB) or symmetrically into two ISCs or two EBs. (E) Observed frequencies of the pair types depicted in C and D for wild-type (n=235), hypermorphic (l(1)NB/+, n=209) and hypomorphic (N55e11/+, n=213) conditions for Notch. (F) Observed frequencies of the pair types depicted in C and D for da-GS, UAS-NRNAi flies after 3 days of feeding on 0 (‘mock’, n=234), 20 (n=192) and 50 µg (n=99) per vial of RU486. (G) Total numbers of ISCs and EBs in the pairs in the conditions described in E. Overall numbers of ISCs/EBs in these genotypes are 765/737 (wild type), 790/752 (N55e11/+) and 804/1148 [l(1)NB] (de Navascués et al., 2012). (H) Overall numbers of ISC/EB in the conditions described in F. In E-H, triangles with colour gradient indicate the relative levels of Notch signalling: wild type (white), excessive (red) or defective (blue). Scale bars: 20 µm.

Fig. 2.

Parameter space and dynamic behaviour of the model. (A) Stable solutions of the system classified according to their resulting signalling state. Green stands for symmetric positive fates (EB-EB pairs), blue represents symmetric negative fates (ISC-ISC pairs) and orange denotes asymmetric fates (ISC-EB pairs). The threshold in Notch level for EB identification is taken to be equal to 0.1 (see the text for more details). Dotted line, boundary of stability for steady states with identical cells; these ‘homogeneous’ solutions are stable above the line. (B-E) Time evolution (in arbitrary units) of Notch and Delta activity in pairs of cells interacting with parameters from the points indicated as 1 to 4 in A. Parameter values in point 1 correspond to those used by Collier et al. (1996) (B), while parameter values in points 2-4 (C-E) correspond to examples of other asymmetric pairs, and symmetric-positive and symmetric-negative pairs.

Fig. 2.

Parameter space and dynamic behaviour of the model. (A) Stable solutions of the system classified according to their resulting signalling state. Green stands for symmetric positive fates (EB-EB pairs), blue represents symmetric negative fates (ISC-ISC pairs) and orange denotes asymmetric fates (ISC-EB pairs). The threshold in Notch level for EB identification is taken to be equal to 0.1 (see the text for more details). Dotted line, boundary of stability for steady states with identical cells; these ‘homogeneous’ solutions are stable above the line. (B-E) Time evolution (in arbitrary units) of Notch and Delta activity in pairs of cells interacting with parameters from the points indicated as 1 to 4 in A. Parameter values in point 1 correspond to those used by Collier et al. (1996) (B), while parameter values in points 2-4 (C-E) correspond to examples of other asymmetric pairs, and symmetric-positive and symmetric-negative pairs.

In a biological system, the existence of three possible signalling state profiles would be equivalent to having three different cell fate combinations across a population of initially uncommitted cell pairs interacting through Notch/Delta, with the specific fate combination of a given cell pair depending on the sensitivity to Delta activation and Notch inhibition of the pair. To investigate the potential of this lateral inhibition model, incorporating variable activation and inhibition thresholds, to describe a real biological system, we turned to the Drosophila midgut.

Cell pair type frequencies correlate with Notch activity

In the Drosophila midgut, Notch-negative cells correspond to ISCs, and Notch-positive cells to EBs. The Notch activity reporter GBE-Su(H) is hardly expressed above background levels in ISCs (Ohlstein and Spradling, 2007; J.d.N., unpublished), and hence our choice of a low-threshold value, Nthr=0.1. Symmetric positive pairs in the model will equate to an event of symmetric differentiation (EB/EB), symmetric negative pairs to symmetric self-renewal (ISC/ISC), and asymmetric pairs to asymmetric ISC fate (ISC/EB) (Fig. 2A). If this was the case, we would expect the relative frequencies of these pairs to correlate with overall Notch activity levels, as has been shown with the balance of the entire populations of ISCs and EBs (Biteau et al., 2008; de Navascués et al., 2012). Indeed, heterozygous conditions for the null allele N55e11 lead to more ISC-ISC pairs and fewer EB-EB pairs relative to the wild type, whereas heterozygosis for the hyperactive, Dl-dependent l(1)NB mutation favours EB-EB pairs at the expense of ISC-ISC pairs, the presence of which is negligible (Fig. 1E). To ensure that these differences are due to variations in Notch activity, rather than to the genetic background of the different mutant chromosomes, we carried out a serial knock-down of Notch in ISCs and EBs. To achieve this, we used the GeneSwitch system (Osterwalder et al., 2001) with the driver da-GS, which is largely specific to ISCs and EBs in the intestine (Fig. S2A; Nicolas Buchon, personal communication), and increasing concentrations of the inducer RU486. In these experiments, the flies share the same genetic background, and differ only in the amount of RU486 present in the food. Indeed, we find that mild knock-down of Notch (to a level that does not yet lead to ISC-like tumour growth; Fig. S2B-F) increases the number of ISC-ISC pairs, at the cost of ISC-EB and EB-EB pairs, in a dose-dependent manner (Fig. 1F). The baseline fractions of the control for the induction experiment (‘mock’ bars in Fig. 1F) are different from the control for the mutants (‘N+/+’ bars in Fig. 1E), even if at the overall level the total numbers of ISCs and EBs are balanced (Fig. 1G,H; de Navascués et al., 2012), highlighting the importance of controlling for genetic background in these experiments. From these data, we conclude that the frequency of undifferentiated pair types are a good readout of the strength of Notch signalling.

Cell contact area as modulator of activation threshold

To relate the model to real tissue, we need first to consider how the dimensionless parameters a and b are related to biological features displaying variability across undifferentiated (esg+) cell pairs. We assume that biochemical processes intrinsic to the cell, such as protein degradation rates (δD and δN), the maximal biosynthesis rates (α and β), and the threshold of Delta inhibition by Notch in the same cell (KD), will not be highly variable among cells with a common developmental identity. However, the threshold of Notch activation by neighbouring Delta (KN) depends directly on the interaction between the two cells, which could be variable for different pairs of cells. For example, owing to spatial heterogeneity of cell packing, the contact area between cell pairs could be substantially different from pair to pair. Indeed, tissue images reveal that undifferentiated cells in nests show irregular shapes and variable contact area (Fig. 3A). In the Notch/Delta system, the amounts of Notch and Delta are usually limiting, and this seems to hold true for the adult Drosophila gut, where haploinsufficiency has been described (Biteau et al., 2008; de Navascués et al., 2012; see also Fig. 1E-H). Therefore, it is expected that variations of ∼2-fold or more in contact area would lead to significant changes in the levels of Notch activation. This is captured in the model by expressing the dimensionless activation threshold a in terms of the contact area, following Eqn 7 in the Materials and Methods. Specifically, we assume that a is inversely related with the contact area (the larger the contact area, the easier it is for Delta to activate Notch, and thus the smaller the activation threshold). As shown in Fig. 2A (see also Fig. S3), variation in a best allows for heterogeneity in steady-state levels of Notch activity and therefore in fate choice. From this, we hypothesize that variation of contact area (or any other biological feature correlating with the threshold of Notch activation) is likely to allow the diversity in fate outcome predicted by the model.

Fig. 3.

Variability in contact area, and distribution of Notch and Delta at the membrane. (A) Confocal stacks projected in z, showing variability in contact length (as a proxy for area) in four different sample pairs (i-iv). Scale bar: 20 µm. (B) (i, ii) Side views of the intestinal epithelium, showing the apical-basal distribution of Notch and Arm. Lumen is at the top and basal at the bottom. (iii-v) Top view of the intestinal epithelium (iii) with zy and xz side views (iv, v) corresponding to the marked lines in iii. Lumen is at the top and right of the xz and zy views, respectively. (C) Top view of the intestinal epithelium, with zy and xz side views corresponding to the marked lines in the top view panel. The top view panel is a z-projection illustrating the membrane localisation of Delta, with side views showing its apical-basal distribution. Lumen is at the top and right of the xz and zy views, respectively. Scale bars: 10 µm in B,C. (D-F) Arm (D), Delta (E) and Notch (F) levels along the perimeter of the cell planes (colour lines) and mean (white). For each cell plane, position 0 corresponds to the centre of the contacting membranes (defined as the position that intersects the line connecting the cell centroids in that plane). Data in D and F are from 20 paired cells; data in E are from 43 cells.

Fig. 3.

Variability in contact area, and distribution of Notch and Delta at the membrane. (A) Confocal stacks projected in z, showing variability in contact length (as a proxy for area) in four different sample pairs (i-iv). Scale bar: 20 µm. (B) (i, ii) Side views of the intestinal epithelium, showing the apical-basal distribution of Notch and Arm. Lumen is at the top and basal at the bottom. (iii-v) Top view of the intestinal epithelium (iii) with zy and xz side views (iv, v) corresponding to the marked lines in iii. Lumen is at the top and right of the xz and zy views, respectively. (C) Top view of the intestinal epithelium, with zy and xz side views corresponding to the marked lines in the top view panel. The top view panel is a z-projection illustrating the membrane localisation of Delta, with side views showing its apical-basal distribution. Lumen is at the top and right of the xz and zy views, respectively. Scale bars: 10 µm in B,C. (D-F) Arm (D), Delta (E) and Notch (F) levels along the perimeter of the cell planes (colour lines) and mean (white). For each cell plane, position 0 corresponds to the centre of the contacting membranes (defined as the position that intersects the line connecting the cell centroids in that plane). Data in D and F are from 20 paired cells; data in E are from 43 cells.

The above-mentioned assumption by which the activation threshold depends inversely on the contact area requires Notch and Delta proteins to be homogeneously distributed across the cell surface (such that their probability of binding increases with the area of contact). To evaluate this, we used confocal microscopy to examine the localisation of both Notch and Delta proteins with respect to the membrane marker Armadillo/β-catenin (Arm), in both single and paired ISCs and EBs in the midgut epithelium (Fig. 3B,C). In Drosophila epithelia, Arm participates in the formation of adherens junctions, which localise mostly in the apical domain, with lower levels at the lateral membrane (Tepass and Hartenstein, 1994). Therefore, we used Arm staining to define membrane boundaries in 3D, and measured the intensity of Arm, Notch and Delta:GFP proteins at the cell membrane. The latter is an endogenously tagged protein that correctly represents the localisation of the wild-type Delta both visually and by co-localisation analysis (Fig. S4; see also Nagarkar-Jaiswal et al., 2015).

We could not find any strong pattern in the variations of Notch or Delta immunodetection intensity within confocal planes, and it would seem that both are only slightly enriched at the boundary between two undifferentiated cells (Fig. 3E,F). This indicates that Notch and Delta concentrations are largely independent of the position at the membrane along the cell perimeter, and in particular along the contact between esg+ cells. Moreover, the localisation of both Notch and Delta along the apical-basal axis of the cells is also largely homogeneous (Fig S5E,F). This is manifest in the small variation in the average amounts of Notch and Delta between different optical planes (Fig. S5B,C, left panels), and in the narrow distribution of mean values per plane, with low values of coefficients of variation per plane, of Notch intensity values (Fig. S5B,C, right panels). Therefore, the contact area between cells is a good approximation to the total amount of Notch and Delta proteins available for signalling.

We note that Arm largely parallels Notch and Delta localisation at the membrane (Fig 3D; Fig. S5D) but shows a stronger enrichment at the boundary (Fig. 3D), in agreement with previous reports (Maeda et al., 2008). Incidentally, these results also reveal that neither Arm nor Notch nor Delta are restricted to the apical domain in the midgut epithelium, and instead can be found in similar amounts along the apical-basal axis of the membrane in ISCs and EBs (Fig. S5D-F). This situation contrasts with Arm and Notch distribution in other Drosophila epithelia (Tepass and Hartenstein, 1994; Tepass et al., 2001; Sanders et al., 2009).

Taken together, our results suggest that both the Notch receptor and its ligand Delta are randomly and homogeneously distributed in the cell membrane, which suggests that measurements of membrane contact area may be relevant to the dynamics of Delta-Notch signalling as a proxy for the activation threshold a in our model.

Correlation of contact area values and cell fate profiles

We have shown that contact area can be used as a measure of the amount of Notch and Delta available for interaction. Therefore, our model predicts that contact area should correlate with the patterns of symmetric and asymmetric fates. Indeed, we observe increasing average contact area in ISC-ISC (7.67 µm2), ISC-EB (13.28 µm2) and EB-EB (14.79 µm2) cells (Fig. 4A). The contact area in ISC-ISC pairs was significantly lower than in the other two cell-pair types (with P-values of 0.002 and 0.005 when compared with ISC-EB and EB-EB pairs, respectively), whereas the latter could not be statistically distinguished from each other (P=0.5). We must also remark that EB cells increase in size after they progress from their original ISC state (Fig. 4B and Fig. S6B), which complicates our interpretation of the correlation between fate and contact area, especially in EB-EB pairs. This fact, together with the statistical similarity between ISC-EB and EB-EB pairs mentioned above, led us to focus on the distinction between ISC-ISC and ISC-EB, filtering the results coming from the latter cell-pair type based on the size of the EB cells (see below).

Fig. 4.

The model can reproduce the observed cell fate profiles. Statistical comparisons between data distributions in A,B,D-F were performed with the Kolmogorov–Smirnov test. (A) Swarm/box plots showing contact area values segregated by fate profiles (11 ISC-ISC, 218 ISC-EB and 42 EB-EB pairs). See main text. (B) Swarm/box plots showing cell size values (measured as projected areas) of ISCs and EBs. Projected areas correlate well with cell volumes (Fig. S6A), which also shows a similar difference between ISCs and EBs (Fig. S6B). (C) Frequency of contact area values for nests of two undifferentiated cells, irrespective of pair type. The red line marks the smooth kernel distribution (SKD) used to generate areas for the simulation. (D) Cumulative frequency of the contact area data for ISC-ISC and ISC-EB pairs. (E) Cumulative frequency of cell size (projected area) of ISCs and EBs. Blue and light green lines correspond to the data depicted in B. Increasingly darker green lines correspond to the cell size distributions of EBs, which are smaller than 1.0, 0.75 and 0.5 times the maximal ISC size, respectively. (F) Cumulative frequency of contact area of ISC-ISC (blue line) and ISC-EB (orange) pairs. Increasingly darker orange lines correspond to the contact area distributions of ISC-EB pairs with size limits of the EB of the pair is as in E. Data in B-F are from a separate, larger dataset than that in A. (G) Inset of the phase space (corresponding to the white square in Fig. 2A), with shaded boxes indicating the ranges of a and b values where the pair type frequencies from Fig. 1E (upper panel) and Fig. 1F (lower panel) best fit the model. For each experimental condition, the height of the box is determined by the neighbourhood of b values where H≤0.02; the values of a determining the width of the box are obtained from Eqn 7 using the best fitting value for c (when b equals the value at the mid-height of the box) and the SKD depicted in C to obtain experimentally supported values for A. Notice the difference in scale between the a and b axes, evident even with the log scale. (H) Fate outputs for the lateral inhibition model for different amounts of contact area. The order of fate outcomes with increasing contact area follows the experimental data in A.

Fig. 4.

The model can reproduce the observed cell fate profiles. Statistical comparisons between data distributions in A,B,D-F were performed with the Kolmogorov–Smirnov test. (A) Swarm/box plots showing contact area values segregated by fate profiles (11 ISC-ISC, 218 ISC-EB and 42 EB-EB pairs). See main text. (B) Swarm/box plots showing cell size values (measured as projected areas) of ISCs and EBs. Projected areas correlate well with cell volumes (Fig. S6A), which also shows a similar difference between ISCs and EBs (Fig. S6B). (C) Frequency of contact area values for nests of two undifferentiated cells, irrespective of pair type. The red line marks the smooth kernel distribution (SKD) used to generate areas for the simulation. (D) Cumulative frequency of the contact area data for ISC-ISC and ISC-EB pairs. (E) Cumulative frequency of cell size (projected area) of ISCs and EBs. Blue and light green lines correspond to the data depicted in B. Increasingly darker green lines correspond to the cell size distributions of EBs, which are smaller than 1.0, 0.75 and 0.5 times the maximal ISC size, respectively. (F) Cumulative frequency of contact area of ISC-ISC (blue line) and ISC-EB (orange) pairs. Increasingly darker orange lines correspond to the contact area distributions of ISC-EB pairs with size limits of the EB of the pair is as in E. Data in B-F are from a separate, larger dataset than that in A. (G) Inset of the phase space (corresponding to the white square in Fig. 2A), with shaded boxes indicating the ranges of a and b values where the pair type frequencies from Fig. 1E (upper panel) and Fig. 1F (lower panel) best fit the model. For each experimental condition, the height of the box is determined by the neighbourhood of b values where H≤0.02; the values of a determining the width of the box are obtained from Eqn 7 using the best fitting value for c (when b equals the value at the mid-height of the box) and the SKD depicted in C to obtain experimentally supported values for A. Notice the difference in scale between the a and b axes, evident even with the log scale. (H) Fate outputs for the lateral inhibition model for different amounts of contact area. The order of fate outcomes with increasing contact area follows the experimental data in A.

We measured the contact areas in 480 pairs of esg+ cells with both symmetric (ISC-ISC, n=74) and asymmetric (ISC-EB, n=406) fates. The data show that contact area in these pairs is highly variable, ranging from just around 1 µm2 to over 40 µm2 (Fig. 4C). This degree of variability (of more than one order of magnitude) indicates that contact area has the potential to be a regulatory mechanism of the system (through its influence on the dimensionless activation threshold a).

Next, we classified measurements of contact area according to the fate profile of their corresponding cell pair and compared their values (Fig. 4D). We found that the cumulative distributions of contact areas in ISC-ISC and ISC-EB pairs are clearly separated. On average, the contact area between two ISCs (11.59±0.73 µm2; mean±s.e.m.) is clearly smaller than that between an ISC and an EB (17.68±0.42 µm2) (P=8×10−7). To verify that this correlation is not simply a consequence of EBs being larger, we compared the distributions of contact areas between ISC-ISC pairs and different subsets of ISC-EB pairs, which included only the pairs where the EB member was smaller than a given size limit (1×, 0.75× and 0.5× the maximum ISC size). The effect of the thresholding on the distribution of cell sizes (measured in terms of the projected areas) is shown in Fig. 4E. As the threshold decreases, the average size of the filtered EBs becomes progressively smaller, until it can no longer be statistically discriminated from the distribution of ISC sizes (for the subset with size limit equal to half the maximum ISC size, P=0.07). However, the corresponding contact area distributions of these subsets of ISC-EB pairs are all strikingly similar, with values systematically larger than those of ISC-ISC pairs at a high level of significance (maximum P-value=2×10−4) (Fig. 4F). This is in good agreement with the fact that the cell size of the pair members and their contact area are not correlated (Fig. S6C,D).

We also wanted to check that the differences in cell-pair types discussed above are not associated with differences in adhesion properties between ISCs and EBs, as could be suggested by the fact that Arm levels are enriched at the contact area between these cell pairs (Fig. 3D). For this, we measured the values of Arm at the boundary of ISC-ISC pairs and ISC-EB pairs. As we cannot distinguish which cell of the pair originates the Arm signal at the boundary (see Materials and Methods), we measured the signal assigned to ISCs in either ISC-ISC pairs or ISC-EB pairs, as an approximation to the strength of adhesion at the boundary of these cell pairs. We found a similar, moderate increment in Arm levels at the cell pair boundary in both ISC-ISC and ISC-EB pairs (Fig. S6E-F), indicating that adhesion properties are not fundamentally different between cell pair types (at least as far as adherens junctions are concerned). This fits well with the observation that the transcriptional profiles of ISCs and EBs are not particularly enriched with genes involved in cell adhesion (Dutta et al., 2015).

Taken together, these results indicate that a smaller contact area correlates with the ISC-ISC pair profile, and this is likely linked to a higher activation threshold of Notch in ISC-ISC pairs, rather than to changes in cell shape, size or adhesion properties in the ISC-EB pairs posterior to EB fate acquisition.

Updating the model with area variation reproduces fate profile distributions

This finding gives us biological justification to consider the activation threshold a to be variable in the model (inversely proportional to the contact area), and to test the capacity of the model to produce the observed proportions of fate pairs in our various experimental conditions (Fig. 1E,F). To achieve this, we first generated a large sample of contact area values A, from a smooth kernel distribution based on the experimental data (Fig. 4C). To input values from A into the model, we simply used Eqn 7 (ac/A). We then analysed the stable steady states of the model, obtaining the proportions of the three possible fate pairs resulting from A, for different values of b and c.

In order to compare the fate distribution obtained from the model and the experimental datasets, we use the Kullback-Leibler relative entropy (H), which is a dissimilarity measure between two probability distributions (giving the value 0 if the distributions are equal; Kullback and Leibler, 1951). We found an excellent agreement between the proportions of EB-EB, EB-ISC and ISC-ISC pairs observed experimentally, and the distributions from the model for an extended range of values of b and c, as indicated by the low values of H between theoretical and experimental distributions (see Fig. S6G,H). Ranges of values for parameters a and b with good fit for each experimental condition are mapped to the model phase diagram in Fig. 4G (upper panel for Notch mutant and wild-type alleles; lower panel for RU486-induced Notch knock-down and the mock treatment), with corresponding pair frequencies (data and model) and parameter values (c, average b) given in Table 1 (with H values ranging from 1.8×10−4 to 1.6×10−2). Focusing on the control datasets (Notch wild type and Notch mock knock-down, Fig. 4G), one finds that ISC-ISC pairs occur at the lowest values of contact area (largest values of a), in good agreement with our experimental observations (Fig. 4A,D,F). In this region of parameter space, EB-EB and ISC-EB pairs are found at higher values of contact area (lower a), although we cannot distinguish statistically between the experimentally measured contact areas of these two cell-pair types (Fig. 4A). Importantly, we find that the regions of the parameter space where the model fits each of the mutant and knock-down experimental distributions correspond to parameter changes in line with the nature of the genetic perturbations. First, for all loss-of-function conditions of Notch (Notch knock-down and N55e11/+ heterozygotes), the model can account for the observed pair frequencies with small shifts in the b parameter and more substantial changes in the a parameter (note the very different scale ranges for a and b in Fig. 4G). The shifts towards higher a values can be explained as an increase in the KN parameter (i.e. lower activation of Notch for a given Delta stimulus; see Eqns 5 and 7), which fits the nature of a loss-of-function condition. In the case of Notch gain of function, represented by the Nl1N-B/+ hypermorphs, the model can reproduce the resulting pair frequencies by substantially reducing both a (therefore decreasing KN, i.e. more Notch activation for the same Delta stimulus) and b (which can be accounted for as an increase of α, the maximal rate of Notch activation; see Eqn 7); again, both notions sit well with the nature of the l(1)NB allele (a Delta-dependent hypermorphic allele).

Table 1.

Cell-fate profiles as obtained experimentally (‘data’ column) and theoretically (‘model’ column)

Cell-fate profiles as obtained experimentally (‘data’ column) and theoretically (‘model’ column)
Cell-fate profiles as obtained experimentally (‘data’ column) and theoretically (‘model’ column)

Our experimental observations thus agree with the model predictions, including both gain- and loss-of-function perturbations of Notch signalling. Hence, our results confirm that the contact area between pairs of cells can influence the fate outcome of Notch/Delta signalling in the Drosophila midgut (Fig. 4H), with a small contact area clearly favouring symmetric self-renewal.

DISCUSSION

We have considered a standard model of Notch/Delta-mediated lateral inhibition (Collier et al., 1996) and have investigated the effect of the trans-activation of Notch by Delta and the inhibition threshold of Delta by Notch signalling (here considered phenomenologically as the dimensionless thresholds a and b, respectively) on the dynamics of lateral inhibition for a system of two cells. We find that, provided there is a degree of variability in contact areas between cell pairs, three different signalling states (and therefore fate combinations) can occur under the same conditions. This is a considerable expansion of the model, the use of which has so far mostly been centred on solutions that provide fine-grained (checkerboard) patterns. The model reproduces the signalling outcomes observed in the Drosophila intestine, which translate into differentiation versus self-renewal fates. Thus, our results provide a mechanism whereby ISCs may undergo neutral competition, which is a widespread pattern of adult tissue maintenance in metazoans from Drosophila to humans.

The work by Collier et al. (1996) established a minimal model of lateral inhibition as a system leading to checkerboard patterns of stable, all-or-none signalling states. Their formalisation, and choice of parameters, has become a reference in the field (Sprinzak et al., 2011; Formosa-Jordan et al., 2012; Petrovic et al., 2014). However, to allow accommodating phenomenology that departs from the classical fine-grained all-or-none patterns, expansions of this model have required the introduction of additional genetic components (e.g. an extra ligand; Boareto et al., 2015) or noise components (de Back et al., 2013). By contrast, here we have left intact the general dynamics of the minimal model and simply introduced a degree of variability in the sensitivity of each cell pair to signal transduction.

Our work considers the contact area between cells engaged in signalling as the source of variation in signalling threshold. Contact area can be an effective tuning parameter of a biological system (Khait et al., 2016), as it can integrate mechanical constraints into signalling, as has been shown for cell density and proliferative control by the Hippo pathway (Schlegelmilch et al., 2011; Kim et al., 2011; Silvis et al., 2011). In a system such as the posterior midgut, where some differentiated cells are much larger than their progenitors (see Fig. 1A), differentiated and mature cell loss certainly would have a local impact in the packing geometry of cells interacting via Notch/Delta, and connect naturally with the fate outcome of stem cell divisions. This could be particularly useful in conditions of regeneration. Importantly, our theoretical framework could in principle accommodate any source of variation; for example, variation arising from the unequal (either random or regulated) inheritance of signalling components could result in variation in the capability of signal transduction in the population. It is interesting to consider that although shortly after division most of the ISC daughter cells display similar levels of Notch and Delta proteins (Ohlstein and Spradling, 2006), endosomes bearing the signalling molecule Sara display an inhomogeneous inheritance pattern (Montagne and González-Gaitán, 2014). It has recently been found that ISC divisions producing enteroendocrine cell precursors do seem to segregate Delta asymmetrically towards the precursor cell (Guo and Ohlstein, 2015), which suggests that ISCs switch between different types of cell division.

Understanding how Notch/Delta signalling results in stochastic cell fate patterns is of particular relevance in adult homeostatic tissues, as Notch signalling controls fate in many types of tissue stem cells (Koch et al., 2013). Moreover, many adult stem cells balance their fate via neutral competition (Krieger and Simons, 2015). Our model proposes a mechanism whereby Notch/Delta signalling could result in neutral competition of stem cells by lateral inhibition between sibling cells. This provides an alternative explanation to the neutral competition of Drosophila adult ISCs, which has been proposed to arise from Notch/Delta-mediated lateral inhibition involving the offspring of non-related ISCs, coinciding in space (de Navascués et al., 2012) and resolving 20% of the time in symmetric fate. Although the two proposals are compatible with each other, the latter faces the difficulty that ISC/EB nests rarely contain more than two cells (de Navascués et al., 2012). Moreover, we and others have found isolated pairs of ISCs or EBs frequently in the tissue (de Navascués et al., 2012; Goulas et al., 2012). Our model provides a potential explanation of how the offspring of a single ISC (pairs of Notch/Delta signalling cells) may reach a symmetric steady state, leading to symmetric self-renewal or differentiation.

It would be interesting to see how our model translates to a larger group of interacting cells, in particular in light of recent findings in the oesophageal epithelium. There, tissue is maintained by the neutral competition of basal progenitor cells (Doupé et al., 2012), and this competition is heavily influenced by Notch signalling, to the point that alterations in the pathway can lead to the fixation of mutant clones and poise the tissue for tumour initiation (Alcolea et al., 2014).

MATERIALS AND METHODS

The model: lateral inhibition mediated by Notch-Delta interaction

We consider that the rate of Notch activation in a cell is an increasing function of Delta concentration on its neighbour (signalling), and that the rate of Delta expression is a decreasing function of the level of activated Notch in the same cell (inhibition). We represent these interactions by means of a standard mathematical model of Notch/Delta signalling (Collier et al., 1996) between pairs of cells, which is given by:
formula
(Eqn 1)
formula
(Eqn 2)
formula
(Eqn 3)
formula
(Eqn 4)
Here, represent the levels of Notch activity in cells 1 and 2, and are the concentrations of Delta in each cell. α and β are the maximal production rates of Notch and Delta, respectively, whereas δN and δD are their corresponding degradation rates. The production terms for Notch (f) and Delta (g) are given by the Hill functions
formula
(Eqn 5)
where the first function represents the signalling effect of Delta on the neighbouring cell, and the second corresponds to the inhibition of Delta expression by activated Notch in the same cell. KN is the threshold of Notch activation by neighbouring Delta, KD is the threshold of Delta inhibition by Notch in the same cell, and the coefficients r and h represent the cooperative character of the two aforementioned processes. Similarly to (Collier et al., 1996) we rewrite Eqns 1-4 in dimensionless form as:
formula
(Eqn 6)
The parameter ν is the ratio between the degradation rates of Delta and Notch, δD/δN, a and b are the dimensionless thresholds for Notch activation by Delta in the neighbouring cell, and Delta inhibition by Notch in the same cell, respectively:
formula
(Eqn 7)
a and b are referred to as the activation and inhibition thresholds, and their values set the location of the half-maximal points of the Hill functions in Eqn 6. Importantly, the signalling threshold a is considered to depend explicitly on the contact area A between the two interacting cells, given that a larger contact area will effectively reduce the threshold of Notch activation, by increasing the number of receptors available to bind ligands from the sending cell. The constant c is treated as a parameter of the model.

We studied the behaviour of the system on a region of the a-b parameter space that spans a biologically plausible range, according to data from the literature for KN (Sprinzak et al., 2010; Pei and Baker, 2008), KD (Friedmann and Kovall, 2010), α and β (Agrawal et al., 2009), δN (Hsu et al., 2006; Agrawal et al., 2009), and δD (Hsu et al., 2006). For the fixed Hill coefficients r and h, the parameters a and b determine the steady state of the system.

Steady states and cell fate identification

The system of Eqn 6 has a homogeneous steady state in which Notch and Delta have the same values in the two cells:
formula
(Eqn 8)
This state corresponds to a situation in which both cells in the pair have the same fate. The stability boundary of this homogeneous steady state can be calculated using standard methods (Collier et al., 1996), and is represented by a dotted line in Fig. 2A. Above this line the homogeneous state is stable. Below it, a heterogeneous stable steady state appears in which the values of Notch and Delta are different between the two cells:
formula
(Eqn 9)
formula
(Eqn 10)

In parallel with this classification of steady states, a cell is considered to be Notch positive when the level of Notch surpasses a certain threshold Nthr (considered here to be 0.1), and Notch negative in the opposite case. In the case of the Drosophila midgut, a Notch-positive cell would correspond to an EB, and a Notch-negative cell to an ISC. In that way, a homogeneous steady state can represent either an ISC/ISC pair (symmetric Notch negative, blue region in Fig. 2A) or an EB/EB pair (symmetric Notch positive, green region in Fig. 2A). Most heterogeneous steady states, in turn, correspond to an ISC/EB pair (orange region in Fig. 2A), although heterogeneous states in which both values of Notch lie below (or above) the threshold Nthr still represent symmetric ISC/ISC (or EB/EB) pairs. This is reflected in Fig. 2A through the difference between the stability boundary (dotted grey line) and the boundaries between the fate-pair domains (shown in colour code).

Dynamic behaviour

We investigate the temporal evolution of the model by solving numerically the system of Eqn 6. For this purpose, we use a finite difference approximation (two-stage Runge-Kutta; LeVeque, 2007). In our calculations, we are considering r=h=2 and δN=δD, as in previous works (Collier et al., 1996; Sprinzak et al., 2010). Cells are considered initially negative for Notch activation (Ohlstein and Spradling, 2007; Guo and Ohlstein, 2015).

Drosophila culture and strains

Adult flies were raised in standard cornmeal medium, collected daily and maintained in fresh vials with added yeast (food replaced every 24-48 h). Untreated flies were dissected at 4-6 days of age. Mifepristone (RU486)-treated flies were treated as described above and transferred to RU486-containing medium at 3-8 days of age for 3 more days before dissection.

EBs were identified by co-expression of one of two enhancer trap reporters of the undifferentiated cell marker escargot (FlyBase: P{PTT-GB}esgYB0232; Quiñones-Coello et al., 2007; or P{lacW}esgk00606; Spradling et al., 1999) and one of two GBE-Su(H) synthetic reporters of Notch transcriptional activity (Bray and Furriols, 2001; de Navascués et al., 2012). ISCs were identified by expression of the esg reporter alone (Micchelli and Perrimon, 2006; Ohlstein and Spradling, 2006).

Delta levels were measured with the DeltaMI04868-GFSTF.1 transgenic insertion (FlyBase Mi{PT-GFSTF.1}DlMI04868-GFSTF.1), which tags all three annotated Delta isoforms with GFP at the endogenous locus (Nagarkar-Jaiswal et al., 2015).

Genetic perturbation of Notch activity was performed: (1) by knock-down with da-GS (FlyBase: P{da-GSGAL4.T}; Tricoire et al., 2009) and UAS-NotchRNAi (FlyBase: P{w[+mC]=UAS-N.dsRNA.P}14E; Presente et al., 2002), and activation with mifepristone (RU486, Sigma), with mock treatment as control; or (2) by using the molecular null allele N55e11 (Kidd et al., 1983) or the Delta-dependent, hypermorphic l(1)NB (FlyBase: Nl1N-B; Lyman and Young, 1993; Brennan et al., 1997), with Oregon R as wild-type control.

Immunohistofluorescence and imaging

Immunofluorescence was performed essentially as described previously (Bardin et al., 2010) but with a heat-fixation step (Miller et al., 1989). This fixation method was essential for the robust immunodetection of Armadillo (Fang et al., 2016).

Primary antibodies used were: chicken anti-β-galactosidase (Abcam, ab9361, 1:200), rabbit anti-GFP (Abcam, ab6556, 1:200), sheep anti-Notch (Muñoz-Descalzo et al., 2011; 1:1000), anti-Arm (mAb, N2-7A1, 1:50) and anti-Dl (mAb, C594.9B, 1:50). The N2-7A1 and C594.9B antibodies were obtained from the Developmental Studies Hybridoma Bank (NICHD and The University of Iowa). Secondary antibodies conjugated with Alexa fluorophores were from Invitrogen (A21203, A21071, A11039 and A11034, 1:500). Of note, immunodetection of GBE-Su(H)-lacZ expression in heat-fixed material was less sensitive than that of GBE-Su(H)-GFP:nls; this might have led to an underestimation of EB numbers, which in any case will have played against our hypotheses.

Confocal stacks were obtained in a Zeiss LSM 710 with an EC Plan-Neofluar 40× oil immersion objective (numerical aperture 1.3). For the quantification of membrane features, voxel sizes were 0.22×0.22×0.6, 0.21×0.21×1 or 0.18×0.18×0.6 (contact area), 0.16×0.16×0.7 µm (distribution of Notch and Armadillo), or 0.14×0.14×1 µm (distribution of Delta), with airy units adjusted so that there was negligible oversampling in Z, or not at all.

Image analysis

To measure contact area, stacks were analysed with a combination of ImageJ macros and python scripts, as follows. (1) All esg-GFP+ cells in the z-projection of the stack were manually identified. (2) Expression of the GBE-Su(H)-lacZ reporter was automatically thresholded in 3D to determine its expression status (positive or negative) in every esg+ cell, followed by a step of manual correction. (3) Nests of esg+ cells were manually outlined; outlines determined the automatic cropping of 3D stacks containing one esg+ cell pair each. (4) The contact membrane of each esg+ cell pair was semi-automatically determined using FIJI for each optical plane, by binarising the immunofluorescence of Armadillo/β-catenin (Arm). Arm labels the membrane throughout the apical-basal axis (see Results), which allows measuring the amount of contacting membrane in each cell pair as the number of Arm+ voxels shared between the two cells (expressed in µm2).

To measure cell size, the cellular perimeter in 3D was used to measure the volume enclosed. Alternatively, we projected the maximum intensity of Arm in the Z dimension and calculated the area enclosed in the projected perimeter of the cell as the ‘projected area’. This measurement correlated well with the cell volume (Fig. S6A).

To evaluate the colocalisation between Delta:GFP and anti-Delta, Delta+ cells appropriately immunostained were manually identified and individually outlined in 2D in several fields of view (comprising a few hundred enteric cells each). For each outline, the enclosing 3D stack was automatically extracted, and the Pearson correlation and the Manders colocalisation (Manders et al., 1993) coefficients were calculated between different pairs of channels.

For measuring Notch, Delta and Arm distribution at the membrane, the membrane contours (three or four pixels wide) of cells in pairs were manually determined in each plane. At the spatial resolution of our micrographs, it is not possible to distinguish, for a pixel at the boundary, to which cell of the pair it belongs, so we took the approximation of splitting the thickness of the contact between the two cells. Intensity data from those positions were used as described in what follows.

Intensity normalisation

For each plane in the confocal stack, an empty 50×50 pixel square nearby each of the cell pairs in that stack was manually selected. The signal therein (for all the planes in the stack where membrane was detectable) was averaged for all the squares of the same stack, and this value was taken as background. Notch, Delta and Arm intensity values for each stack were normalised by dividing by the background value.

Distribution along cell perimeter

In each confocal plane, each membrane pixel position was assigned an angular value respect to the centroid by calculating its tangent arc (±π depending on the quadrant). Thirty overlapping sliding windows (of 2π/15 rad with half window overlap) were delimited in each plane, and their pixel intensities were normalised and averaged.

Distribution along the apical-basal axis

Each cell was sliced in 10 overlapping angular windows (2π/5 rad with half overlap). For each window, a normalised, average intensity measurement was taken per confocal plane (i.e. along the apical-basal axis). Apical-basal positions were normalised from 0 to 1. Intensity data points along the apical-basal axis were obtained by interpolation from average normalized intensity values.

Acknowledgements

We thank Alfonso Martínez Arias for his encouragement and support, and Nicole Gorfinkiel and David Sprinzak for critical comments on the manuscript.

Author contributions

Conceptualization: N.G., J.G.O., J.d.N.; Methodology: N.G., J.G.O., J.d.N.; Investigation: N.G., R.M.C., J.d.N.; Software: R.M.C., N.G., J.d.N.; Visualization: R.M.C., N.G., J.d.N.; Writing – Original Draft: N.G., J.G.O., J.d.N.; Writing – Review & Editing: N.G., R.M.C., J.G.O., J.d.N.; Funding Acquisition: J.G.O., J.d.N.; Resources: N.G., J.G.O., J.d.N.; Project Administration: J.G.O. and J.d.N.; Supervision: J.G.O., J.d.N.

Funding

We acknowledge support from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) to N.G. and from Cardiff University to J.d.N. J.G.O. and R.M.-C. are supported by the Ministerio de Economía, Industria y Competitividad and Fonds Européen de Développement Économique et Régional (FEDER) (through project FIS2015-66503-C3-1-P), and by the Institució Catalana de Recerca i Estudis Avançats (ICREA) Academia Programme. R.M.-C. also acknowledges financial support from La Caixa Banking Foundation.

References

Agrawal
,
S.
,
Archer
,
C.
and
Schaffer
,
D. V.
(
2009
).
Computational models of the notch network elucidate mechanisms of context-dependent signaling
.
PLoS Comput. Biol.
5
,
e1000390
.
Alcolea
,
M. P.
,
Greulich
,
P.
,
Wabik
,
A.
,
Frede
,
J.
,
Simons
,
B. D.
and
Jones
,
P. H.
(
2014
).
Differentiation imbalance in single oesophageal progenitor cells causes clonal immortalization and field change
.
Nat. Cell Biol.
16
,
615
-
622
.
Arias
,
A. M.
and
Stewart
,
A.
(
2002
).
Molecular Principles of Animal Development
.
USA
:
Oxford University Press
.
Artavanis-Tsakonas
,
S.
,
Rand
,
M. D.
and
Lake
,
R. J.
(
1999
).
Notch signaling: cell fate control and signal integration in development
.
Science
284
,
770
-
776
.
Baker
,
A.-M.
,
Cereser
,
B.
,
Melton
,
S.
,
Fletcher
,
A. G.
,
Rodriguez-Justo
,
M.
,
Tadrous
,
P. J.
,
Humphries
,
A.
,
Elia
,
G.
,
McDonald
,
S. A. C.
,
Wright
,
N. A.
, et al. 
(
2014
).
Quantification of crypt and stem cell evolution in the normal and neoplastic human colon
.
Cell Rep.
8
,
940
-
947
.
Bardin
,
A. J.
,
Perdigoto
,
C. N.
,
Southall
,
T. D.
,
Brand
,
A. H.
and
Schweisguth
,
F.
(
2010
).
Transcriptional control of stem cell maintenance in the Drosophila intestine
.
Development
137
,
705
-
714
.
Biteau
,
B.
,
Hochmuth
,
C. E.
and
Jasper
,
H.
(
2008
).
JNK activity in somatic stem cells causes loss of tissue homeostasis in the aging Drosophila gut
.
Cell Stem Cell
3
,
442
-
455
.
Boareto
,
M.
,
Jolly
,
M. K.
,
Lu
,
M.
,
Onuchic
,
J. N.
,
Clementi
,
C.
and
Ben-Jacob
,
E.
(
2015
).
Jagged–Delta asymmetry in Notch signaling can give rise to a sender/receiver hybrid phenotype
.
Proc. Natl Acad. Sci. USA
112
,
E402
-
E409
.
Bray
,
S.
and
Furriols
,
M.
(
2001
).
Notch pathway: making sense of suppressor of hairless
.
Curr. Biol.
11
,
R217
-
R221
.
Brennan
,
K.
,
Tateson
,
R.
,
Lewis
,
K.
and
Arias
,
A. M.
(
1997
).
A functional analysis of Notch mutations in Drosophila
.
Genetics
147
,
177
-
188
.
Collier
,
J. R.
,
Monk
,
N. A. M.
,
Maini
,
P. K.
and
Lewis
,
J. H.
(
1996
).
Pattern formation by lateral inhibition with feedback: a mathematical model of delta-notch intercellular signalling
.
J. Theor. Biol.
183
,
429
-
446
.
de Back
,
W.
,
Zhou
,
J. X.
and
Brusch
,
L.
(
2013
).
On the role of lateral stabilization during early patterning in the pancreas
.
J. R. Soc. Interface
10
,
20120766
.
de Navascués
,
J.
,
Perdigoto
,
C. N.
,
Bian
,
Y.
,
Schneider
,
M. H.
,
Bardin
,
A. J.
,
Martinez-Arias
,
A.
and
Simons
,
B. D.
(
2012
).
Drosophila midgut homeostasis involves neutral competition between symmetrically dividing intestinal stem cells
.
EMBO J.
31
,
2473
-
2485
.
Doupé
,
D. P.
,
Alcolea
,
M. P.
,
Roshan
,
A.
,
Zhang
,
G.
,
Klein
,
A. M.
,
Simons
,
B. D.
and
Jones
,
P. H.
(
2012
).
A single progenitor population switches behavior to maintain and repair esophageal epithelium
.
Science
337
,
1091
-
1093
.
Dutta
,
D.
,
Dobson
,
A. J.
,
Houtz
,
P. L.
,
Gläßer
,
C.
,
Revah
,
J.
,
Korzelius
,
J.
,
Patel
,
P. H.
,
Edgar
,
B. A.
and
Buchon
,
N.
(
2015
).
Regional cell-specific transcriptome mapping reveals regulatory complexity in the adult drosophila midgut
.
Cell Rep.
12
,
346-258
.
Ehebauer
,
M.
,
Hayward
,
P.
and
Arias
,
A. M.
(
2006
).
Notch, a universal arbiter of cell fate decisions
.
Science
314
,
1414
-
1415
.
Fang
,
H. Y.
,
Martinez-Arias
,
A.
and
de Navascués
,
J.
(
2016
).
Autocrine and paracrine Wingless signalling in the Drosophila midgut by both continuous gradient and asynchronous bursts of wingless expression
.
F1000Res.
5
,
317-314
.
Formosa-Jordan
,
P.
,
Ibanes
,
M.
,
Ares
,
S.
and
Frade
,
J. M.
(
2012
).
Regulation of neuronal differentiation at the neurogenic wavefront
.
Development
139
,
2321
-
2329
.
Friedmann
,
D. R.
and
Kovall
,
R. A.
(
2010
).
Thermodynamic and structural insights into CSL-DNA complexes
.
Protein Sci.
19
,
34
-
46
.
Goulas
,
S.
,
Conder
,
R.
and
Knoblich
,
J. A.
(
2012
).
The par complex and integrins direct asymmetric cell division in adult intestinal stem cells
.
Cell Stem Cell
11
,
529
-
540
.
Greenwald
,
I.
(
1998
).
LIN-12/Notch signaling: lessons from worms andflies
.
Genes Dev.
12
,
1751
-
1762
.
Guo
,
Z.
and
Ohlstein
,
B.
(
2015
).
Stem cell regulation. Bidirectional Notch signaling regulates Drosophila intestinal stem cell multipotency
.
Science
350
,
aab0988
.
Hsu
,
C.-P.
,
Lee
,
P.-H.
,
Chang
,
C.-W.
and
Lee
,
C.-T.
(
2006
).
Constructing quantitative models from qualitative mutant phenotypes: preferences in selecting sensory organ precursors
.
Bioinformatics
22
,
1375
-
1382
.
Khait
,
I.
,
Orsher
,
Y.
,
Golan
,
O.
,
Binshtok
,
U.
,
Gordon-Bar
,
N.
,
Amir- Zilberstein
,
L.
and
Sprinzak
,
D.
(
2016
).
Quantitative analysis of Delta-like-1 membrane dynamics elucidates the role of contact geometry on Notch signaling
.
Cell Rep.
14
,
225
-
233
.
Kidd
,
S.
,
Lockett
,
T. J.
and
Young
,
M. W.
(
1983
).
The Notch locus of Drosophila melanogaster
.
Cell
34
,
421
-
433
.
Kim
,
N.-G.
,
Koh
,
E.
,
Chen
,
X.
and
Gumbiner
,
B. M.
(
2011
).
E-cadherin mediates contact inhibition of proliferation through Hippo signaling-pathway components
.
Proc. Natl Acad. Sci. USA
108
,
11930
-
11935
.
Klein
,
A. M.
and
Simons
,
B. D.
(
2011
).
Universal patterns of stem cell fate in cycling adult tissues
.
Development
138
,
3103
-
3111
.
Koch
,
U.
,
Lehal
,
R.
and
Radtke
,
F.
(
2013
).
Stem cells living with a Notch
.
Development
140
,
689
-
704
.
Krieger
,
T.
and
Simons
,
B. D.
(
2015
).
Dynamic stem cell heterogeneity
.
Development
142
,
1396
-
1406
.
Kullback
,
S.
and
Leibler
,
R. A.
(
1951
).
On information and sufficiency
.
Ann. Math. Statist.
22
,
179
-
86
.
LeVeque
,
R. J.
(
2007
).
Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems
. Vol.
98
.
Philadelphia, PA
:
SIAM
.
Lyman
,
D.
and
Young
,
M. W.
(
1993
).
Further evidence for function of the Drosophila Notch protein as a transmembrane receptor
.
Proc. Natl. Acad. Sci. USA
90
,
10395
-
10399
.
Maeda
,
K.
,
Takemura
,
M.
,
Umemori
,
M.
and
Adachi-Yamada
,
T.
(
2008
).
E-cadherin prolongs the moment for interaction between intestinal stem cell and its progenitor cell to ensure Notch signaling in adult Drosophila midgut
.
Genes Cells
13
,
1219
-
1227
.
Manders
,
E. M. M.
,
Verbeek
,
F. J.
and
Aten
,
J. A.
(
1993
).
Measurement of co-localization of objects in dual-colour confocal images
.
J. Microsc.
169
,
375
-
382
.
Micchelli
,
C. A.
and
Perrimon
,
N.
(
2006
).
Evidence that stem cells reside in the adult Drosophila midgut epithelium
.
Nature
439
,
475
-
479
.
Miller
,
K. G.
,
Field
,
C. M.
and
Alberts
,
B. M.
(
1989
).
Actin-binding proteins from Drosophila embryos: a complex network of interacting proteins detected by F-actin affinity chromatography
.
J. Cell Biol.
109
,
2963
-
2975
.
Montagne
,
C.
and
Gonzalez-Gaitán
,
M.
(
2014
).
Sara endosomes and the asymmetric division of intestinal stem cells
.
Development
141
,
2014
-
2023
.
Muñoz-Descalzo
,
S.
,
Tkocz
,
K.
,
Balayo
,
T.
and
Arias
,
A. M.
(
2011
).
Modulation of the ligand-independent traffic of Notch by Axin and Apc contributes to the activation of Armadillo in Drosophila
.
Development
138
,
1501
-
1506
.
Nagarkar-Jaiswal
,
S.
,
Lee
,
P. T.
,
Campbell
,
M. E.
and
Chen
,
K.
(
2015
).
A library of MiMICs allows tagging of genes and reversible, spatial and temporal knockdown of proteins in Drosophila
.
Elife
4
,
e05338
.
Ohlstein
,
B.
and
Spradling
,
A.
(
2006
).
The adult Drosophila posterior midgut is maintained by pluripotent stem cells
.
Nature
439
,
470
-
474
.
Ohlstein
,
B.
and
Spradling
,
A.
(
2007
).
Multipotent Drosophila intestinal stem cells specify daughter cell fates by differential notch signaling
.
Science
315
,
988
-
992
.
Osterwalder
,
T.
,
Yoon
,
K. S.
,
White
,
B. H.
and
Keshishian
,
H.
(
2001
).
A conditional tissue-specific transgene expression system using inducible GAL4
.
Proc. Natl. Acad. Sci. USA
98
,
12596
-
12601
.
Othmer
,
H. G.
and
Scriven
,
L. E.
(
1971
).
Instability and dynamic pattern in cellular networks
.
J. Theor. Biol.
32
,
507
-
537
.
Pei
,
Z.
and
Baker
,
N. E.
(
2008
).
Competition between Delta and the Abruptex domain of Notch
.
BMC Dev. Biol.
8
,
4
.
Petrovic
,
J.
,
Formosa-Jordan
,
P.
,
Luna-Escalante
,
J. C.
,
Abello
,
G.
,
Ibanes
,
M.
,
Neves
,
J.
and
Giraldez
,
F.
(
2014
).
Ligand-dependent Notch signaling strength orchestrates lateral induction and lateral inhibition in the developing inner ear
.
Development
141
,
2313
-
2324
.
Plahte
,
E.
(
2001
).
Pattern formation in discrete cell lattices
.
J. Math. Biol.
43
,
411
-
445
.
Presente
,
A.
,
Shaw
,
S.
,
Nye
,
J. S.
and
Andres
,
A. J.
(
2002
).
Transgene-mediated RNA interference defines a novel role for notch in chemosensory startle behavior
.
Genesis
34
,
165
-
169
.
Quiñones-Coello
,
A. T.
,
Petrella
,
L. N.
,
Ayers
,
K.
,
Melillo
,
A.
,
Mazzalupo
,
S.
,
Hudson
,
A. M.
,
Wang
,
S.
,
Castiblanco
,
C.
,
Buszczak
,
M.
,
Hoskins
,
R. A.
, et al. 
(
2007
).
Exploring strategies for protein trapping in Drosophila
.
Genetics
175
,
1089
-
1104
.
Rouault
,
H.
and
Hakim
,
V.
(
2012
).
Different Cell Fates from Cell-Cell Interactions: core architectures of two-cell bistable networks
.
Biophysj
102
,
417
-
426
.
Sanders
,
P. G. T.
,
Muñoz-Descalzo
,
S.
,
Balayo
,
T.
,
Wirtz-Peitz
,
F.
,
Hayward
,
P.
and
Arias
,
A. M.
(
2009
).
Ligand-independent traffic of Notch buffers activated Armadillo in Drosophila
.
PLoS Biol.
7
,
e1000169
.
Schlegelmilch
,
K.
,
Mohseni
,
M.
,
Kirak
,
O.
,
Pruszak
,
J.
,
Rodriguez
,
J. R.
,
Zhou
,
D.
,
Kreger
,
B. T.
,
Vasioukhin
,
V.
,
Avruch
,
J.
,
Brummelkamp
,
T. R.
, et al. 
(
2011
).
Yap1 acts downstream of α-catenin to control epidermal proliferation
.
Cell
144
,
782
-
795
.
Shaya
,
O.
and
Sprinzak
,
D.
(
2011
).
From Notch signaling to fine-grained patterning: Modeling meets experiments
.
Curr. Opin. Genet. Dev.
21
,
732
-
739
.
Silvis
,
M. R.
,
Kreger
,
B. T.
,
Lien
,
W.-H.
,
Klezovitch
,
O.
,
Rudakova
,
G. M.
,
Camargo
,
F. D.
,
Lantz
,
D. M.
,
Seykora
,
J. T.
and
Vasioukhin
,
V.
(
2011
).
α-catenin is a tumor suppressor that controls cell accumulation by regulating the localization and activity of the transcriptional coactivator Yap1
.
Sci. Signal.
4
,
ra33-ra33
.
Simons
,
B. D.
and
Clevers
,
H.
(
2011
).
Strategies for homeostatic stem cell self-renewal in adult tissues
.
Cell
145
,
851
-
862
.
Simpson
,
P.
(
1997
).
Notch signalling in development: on equivalence groups and asymmetric developmental potential
.
Curr. Opin. Genet. Dev.
7
,
537
-
542
.
Spradling
,
A. C.
,
Stern
,
D.
,
Beaton
,
A.
and
Rhem
,
E. J.
(
1999
).
The Berkeley Drosophila Genome Project gene disruption project: single P-element insertions mutating 25% of vital Drosophila genes
.
Genetics
153
,
135
-
177
.
Sprinzak
,
D.
,
Lakhanpal
,
A.
,
LeBon
,
L.
,
Santat
,
L. A.
,
Fontes
,
M. E.
,
Anderson
,
G. A.
,
Garcia-Ojalvo
,
J.
and
Elowitz
,
M. B.
(
2010
).
Cis-interactions between Notch and Delta generate mutually exclusive signalling states
.
Nature
465
,
86
-
90
.
Sprinzak
,
D.
,
Lakhanpal
,
A.
,
LeBon
,
L.
,
Garcia-Ojalvo
,
J.
and
Elowitz
,
M. B.
(
2011
).
Mutual inactivation of notch receptors and ligands facilitates developmental patterning
.
PLoS Comput. Biol.
7
,
e1002069
.
Tepass
,
U.
and
Hartenstein
,
V.
(
1994
).
The development of cellular junctions in the Drosophila embryo
.
Dev. Biol.
161
,
563
-
596
.
Tepass
,
U.
,
Tanentzapf
,
G.
,
Ward
,
R.
and
Fehon
,
R.
(
2001
).
Epithelial cell polarity and cell junctions in Drosophila
.
Annu. Rev. Genet.
35
,
747
-
784
.
Tricoire
,
H.
,
Battisti
,
V.
,
Trannoy
,
S.
,
Lasbleiz
,
C.
,
Pret
,
A.-M.
and
Monnier
,
V.
(
2009
).
The steroid hormone receptor EcR finely modulates Drosophila lifespan during adulthood in a sex-specific manner
.
Mech. Ageing Dev.
130
,
547
-
552
.
Vermeulen
,
L.
,
Morrissey
,
E.
,
van der Heijden
,
M.
,
Nicholson
,
A. M.
,
Sottoriva
,
A.
,
Buczacki
,
S.
,
Kemp
,
R.
,
Tavaré
,
S.
and
Winton
,
D. J.
(
2013
).
Defining stem cell dynamics in models of intestinal tumor initiation
.
Science
342
,
995
-
998
.
Zeng
,
X.
and
Hou
,
S. X.
(
2015
).
Enteroendocrine cells are generated from stem cells through a distinct progenitor in the adult Drosophila posterior midgut
.
Development
142
,
644
-
653
.

Competing interests

The authors declare no competing or financial interests.

Supplementary information