Spatial modelling of NOTCH signalling predicts tuneable dynamic heterogeneity in endothelia

Abstract

Endothelial cells (ECs) are phenotypically heterogeneous to perform specialised functions across different organs and tissues. Although environmental inputs are considered key drivers, we previously observed persistent diversity in NOTCH pathway activation within quiescent endothelia, but the underlying mechanism remains unclear. Here, we developed a computational–experimental framework linking core NOTCH dynamics to emergent EC heterogeneity in space and time. We built an agent-based model of NOTCH signalling incorporating lateral inhibition, lateral induction, cis interactions, and optional HES1 autoregulation, and constrained timescales using HUVEC cultures under γ-secretase inhibition. Model exploration identified regimes with emergent asynchronous HES1 oscillations driven by productive cis interactions or by autoregulation. Contact-modulated cis/trans interactions reproduced the experimentally observed spatial distributions, whereas autoregulation produced globally synchronised dynamics inconsistent with spatial heterogeneity. Additionally, oscillation amplitude and local synchrony were tuneable by varying biologically relevant parameters. Consistent with these predictions, sparse-versus-confluent experiments showed that intercellular contact is required for spatial heterogeneity. Overall, our results support the hypothesis that intrinsic, contact-dependent NOTCH dynamics can generate heterogeneous EC phenotypes, providing a mechanistic basis for functional plasticity in quiescent monolayers. We provide the calibrated model as an open-source, extensible resource for reproducible testing of alternative NOTCH interaction hypotheses against future quantitative datasets.

Introduction

Endothelial cells (ECs) form the inner lining of all blood and lymphatic vessels and preserve vascular homoeostasis, tissue perfusion, and organ development. Despite their common lineage, ECs display remarkable phenotypic diversity across different tissues and vascular beds. This heterogeneity is observed both in vivo and in vitro and is increasingly understood to be a functional adaptation to the complex signalling environments ECs encounter throughout development and adult life^1,2. Single-cell transcriptomic studies have highlighted diverse EC phenotypes across different tissues and within the same tissue, pointing toward local signalling dynamics as a source of variability².

In the context of human development, the cardiovascular system, including the heart, large, and small blood vessels, is the first to become functional, driving flow-dependent specialisation. Additionally, dynamic EC specialisation is crucial for organ-specific vascularisation, like the development of a tight blood–brain barrier and to instruct organ development via angiocrine signalling³.

The NOTCH pathway is a highly conserved cell–cell communication mechanism involved in binary fate decisions during embryogenesis, including somitogenesis, neurogenesis, and vascular patterning^4–6. For instance, NOTCH signalling plays a central role in arterial-venous specification, tip-stalk cell competition during angiogenic sprouting, and endothelial-to-hematopoietic transition^5,7,8. These processes require precise temporal and spatial control of NOTCH activity coordinated through lateral inhibition, lateral induction, and cis-receptor–ligand interactions^9–12.

Our previous work demonstrated that quiescent EC monolayers exhibit marked heterogeneity in expression of the NOTCH target gene HES1 in vitro¹³. We highlighted that the distributions of intensity signal of HES1 were skewed and weakly multimodal, demonstrating the existence of multiple cell “states” within an otherwise homogeneous cell population. Additionally, per-image spatial autocorrelation analysis statistically demonstrated the existence of cell clusters with either homogeneous or “chessboard”—like signal patterns interspersed within cells with varying signal levels without significant autocorrelation indices¹³. However, the clusters did not have an ordered global pattern and were instead randomly dispersed throughout the EC monolayer. In contexts like somitogenesis, arteriogenesis, angiogenesis, neurogenesis, myogenesis, hair follicles, and intestinal crypts, heterogeneous and oscillatory expression of NOTCH pathway targets has been linked to spatiotemporal progenitor maintenance and differentiation control^11,14,15. Importantly, in all these contexts, emergent heterogeneity is limited in time, eventually inducing a stable phenotypic switch in the cells involved (except for angiogenesis, where the NOTCH-mediated tip cell phenotype is transient). Our previous results in phenotypically stable EC monolayers (in terms of cell lineage) demonstrate persistent heterogeneity of NOTCH signalling (i.e., we detected similar signal distributions in EC cultured from one to several days and after several passages in culture)¹³.

Overall, ours and others' raise questions on the mechanistic basis of endothelial NOTCH heterogeneity, including whether it could emerge from various sources of noise (cell geometry, stochasticity in timing/extent of gene transcription, etc.)^11,16,17 or from a structured signal, and if the signal at the single-cell level is stable or varying over time.

In this study, we set out to investigate whether dynamic and spatial interactions within the NOTCH signalling network can alone generate the phenotypic heterogeneity observed in quiescent ECs. In addition, we explore whether a data-informed NOTCH signalling network yields stable or dynamically changing EC phenotypes.

We focus on spatially explicit models that incorporate realistic cell geometries and intercellular contact dynamics to explore whether asymmetric local cell–cell interactions could suffice to drive emergent heterogeneity or whether more complex models like “intrinsic” gene oscillators should be evoked as previously shown, albeit in different contexts.

By integrating single-cell in vitro measurements and maps, en face immunostaining of murine aortic endothelium, and spatially resolved agent-based models, we assess the capacity of NOTCH dynamics to explain plasticity in EC monolayers.

Results

HES1 expression is spatially heterogeneous in the murine aortic endothelium

To investigate spatial heterogeneity of NOTCH pathway activity in vivo, we analysed HES1 expression in the murine aortic endothelium. Since mural cell-derived signals could modulate NOTCH signalling¹⁸, it was important to assess whether heterogeneity persists in this native context.

We performed whole-mount en face immunostaining of murine thoracic aortas¹⁹, visualising endothelial nuclei and HES1 protein expression. Representative confocal images (Fig. 1A) revealed clear heterogeneity in nuclear HES1 intensity across endothelial cells (ECs), including subsets of cells with either low or high signal levels. This pattern is reminiscent of what we observe in cultured EC monolayers (Fig. 2C¹³).

Fig. 1. Heterogeneity of HES 1 intensities in Endothelial monolayers.

A Confocal image of en face immunostaining of murine aortas. Hoechst highlights cell nuclei. CDH5 (VE-Cadherin) highlights EC-EC junctions. HES1 is used as a readout of NOTCH signalling. The rightmost panel is an intensity map where segmented nuclei were painted according to their measured nuclear HES1 intensity. B Density plots corresponding to single cell measurements of HES1 intensities in murine aortas (red, n = 329) and cultured HUVEC (black, dashed, n = 6431).

Fig. 2. Creation and calibration of a spatially explicit NOTCH signalling model in EC.

A Top panels: regular square and experiment-derived cell maps extracted from CC3D simulations at MCS = 0. The maps are coloured according to single-cell intensities of tmDN (randomly initialised). Bottom panels: Schematic representations of cell–cell relationships with regular or experiment-derived (irregular) cell shapes. Irregular cell shapes cause asymmetric signal transmission in neighbour cells. B Schematics of the intracellular P/GRN driving NOTCH signalling in our model. a, b Dll4-NOTCH1 in trans and cis (productive and non-productive via cr parameter). c, d Jag1-NOTCH1 in trans and cis (non-productive, NOTCH1 receptor competition). e HES1 autoregulation. Each species production/degradation is regulated by its respective kinetic law. Black arrows in the diagram indicate modulatory interactions (arrows promote reaction, caps inhibit reaction), red arrows indicate that interactions promote involved species degradation. Key parameters examined throughout the paper are indicated in green (see Supplementary Note 1 for full details). C Representative images relative to DAPT inhibition experiments. Top panels: Nuclei intensity maps relative to HES1 measurements in untreated (NT) or DAPT-treated HUVEC at 6 or 24 h upon treatment. Bottom panels: Representative CDH5 immunofluorescence images of HUVEC treated as in top panels demonstrating formation of serrated junctions and cell elongation (mean % increase of elongated cells/well, el vs. NT ± SD) upon DAPT treatment. D Time course quantification of low NICD and HES1 expressors (% of cells in the 5th percentile vs. total cells per image) in HUVEC upon DAPT treatment. *Indicate pairwise statistical significance vs. NT, # indicates pairwise significance between indicated timepoints, by Kruskal–Wallis followed by Dunn’s Post hoc test.

To further characterise this heterogeneity, we analysed HES1 intensity distributions across three independent aortic preparations and compared them to cultured human umbilical vein (HUVEC) (red and black dashed traces, respectively, Fig. 1B). Overlaying density distributions from cultured and ex vivo EC monolayers demonstrated a qualitatively similar pattern. Therefore, similarly to what we previously demonstrated in cultured EC from different organs¹³ and further corroborated by single-cell transcriptomics data², ex vivo murine aortic EC have spatially heterogeneous nuclear HES1 levels which follow a weakly multimodal distribution. Together, these data demonstrate that spatial heterogeneity in HES1 expression is a physiological feature of the murine aortic endothelium.

Importantly, the overlapping expression patterns observed ex vivo and in vitro, where mural cell signalling is absent, suggest that the underlying mechanisms are intrinsic to ECs, validating cultured ECs as a tool to investigate these aspects in vitro.

Creation and calibration of a spatialised multiscale model of NOTCH signalling in endothelial cells

To investigate whether NOTCH signalling is sufficient to explain the spatial HES1 heterogeneity observed in native and cultured EC monolayers, we developed a spatially explicit multiscale model. We encoded a range of hypothesised signalling rules, including lateral inhibition (classic NOTCH model where sender cells inhibit receiver cells)¹², lateral induction (opposite action, often associated with Jag1)^20,21, and cis interactions (ligand–receptor interaction in the same cell)^10,22, to assess which combinations could reproduce the observed heterogeneity in silico. We also explored whether EC heterogeneity arises from multiple stable phenotypic states or from dynamic, oscillatory behaviours, as previously proposed in developmental systems including neural and mesodermal lineages^{11,14,17,23,24}. To link in silico behaviours to experimental observations we used experimentally derived cell maps and single-cell fluorescence intensity data for model calibration.

Conceptually, the model consists of two interlinked layers: a spatial cell-based layer that captures the topological and contact-dependent interactions among ECs (Fig. 2A) and a deterministic ordinary differential equations (ODE) based subcellular model describing the dynamics of core NOTCH signalling components and transcriptional regulation (schematics in Fig. 2B and Supplementary Note 1).

To define cell–cell spatial relationships, we used two types of cell arrangements, regular square lattices and experimental cell maps derived from in vitro imaging data. The first arrangement parallels previous work in the field^9,10, the latter embeds the system in a realistic context where irregular cell shapes can drive asymmetric signalling among neighbours (Fig. 2A top panels). In both cases, the model simulates how differential membrane contact areas modulate trans- and cis-signalling intensities between cells, linking geometry to signal propagation (Fig. 2A bottom panels).

The subcellular layer is formulated as a system of ODEs representing a hybrid protein/gene regulatory network (P/GRN), adapted from existing NOTCH signalling models^9,16. This layer includes canonical lateral inhibition and lateral induction via Delta-like 4 (Dll4) and Jagged1 (Jag1), respectively. Additionally, we encoded productive (NICD production) and non-productive (ligand–receptor pair degradation) cis interactions, and transcriptional autoregulation of HES1^11,16,17 (Fig. 2B). In the model, cis interactions are local same-cell reactions and can occur in isolated cells. However, in confluent monolayers, their effective impact is driven by the multicellular context, because trans interactions with neighbouring cells alter the pool of ligand–receptor pairs available for cis binding. We parameterised cis behaviour using two quantities: (cr), the fraction of available same-cell ligand–receptor pairs that engage in cis interactions, and (kconv), the fraction of cis-engaged pairs that productively contribute to NICD generation. This formulation allows the model to represent purely inhibitory cis interactions, in which cis binding removes ligand–receptor pairs without signalling, as well as partially productive cis interactions, both of which have been previously reported^22,25.

We encoded the ODE system using Antimony/Tellurium, a human-readable SBML-compatible language and execution environment. For the base model, we translated previous models’ architecture (systems of interconnected Hill and Mass Action functions), paralleling previously established parameters and boundary conditions (Supplementary Note 1).

Since the pace of NOTCH signal transduction can vary depending on cell lineage and context, we set out to experimentally estimate key timescales of the pathway in EC. Ligand/receptors dynamics is challenging to study quantitatively via single-cell image analysis due to their diffuse and uneven distribution in the cell; therefore, we retained previously established parametrisation for their production/degradation rates. However, active fractions of second messengers like NICD and transcription factors like HES1 can be measured via analysis of their nuclear localisation in time-series experiments using established methods¹³.

To calibrate key temporal parameters, including degradation and production of NICD and HES1 in EC, we treated confluent HUVEC monolayers with the γ-secretase inhibitor DAPT, which inhibits cleavage of NOTCH receptors and the production of the second messenger NICD. We performed time-series experiments where we added or removed (after 24 h treatment) DAPT at different timepoints and used our EC profiling tool (ECPT, see the “Methods” section)¹³, we quantified nuclear NOTCH1 (used as a proxy of NICD) and HES1 levels.

By inspecting the density estimate plots relative to treated and untreated cells (Supplementary Fig. 1A, aggregate single cells measurements across six replicates ~7000 cells/condition), it is evident that DAPT treatment alters the distributions of cell signal intensities, especially at the tails of the distributions (e.g., 5th percentile, Supplementary Fig. 1A). Notably, DAPT treatment increase single-cell heterogeneity per image quantified by comparing standard deviations of intensities (Supplementary Fig. 1B), but it does not abolish signal in all cells as highlighted by long-term treatment maps and signal intensity distributions (Fig. 2C and Supplementary Fig. 1A).

To estimate the net temporal effect of DAPT treatment, we quantified, for each image, the percentage of nuclei with nuclear NOTCH1 or HES1 intensities falling within the low-intensity range defined from the lower tail of the control distribution. Figure 2D shows that DAPT treatment increased the fraction of low nuclear NOTCH1 cells, with a corresponding decrease after DAPT removal. Changes in nuclear NOTCH1 were already detectable at the earliest sampled time point after treatment or washout (20 min), indicating that NICD levels respond within the temporal resolution of the assay. However, because measurements were obtained from fixed-cell images and the sampling interval did not resolve times below 20 min, these data do not allow precise fitting of a characteristic NICD turnover rate. We therefore interpret the 20 min response as an experimentally resolved upper bound for rapid NICD dynamics rather than as a fitted kinetic constant.

By comparison, nuclear HES1 showed slower population-level dynamics. DAPT treatment increased the fraction of low-HES1 nuclei within the first hour, but recovery after DAPT removal was substantially delayed, with partial recovery over 2–4 h and full recovery only by 24 h. These slower dynamics are consistent with the expected delay between perturbation of NOTCH second messenger signalling and downstream transcriptional/protein-level changes in HES1.

Within these temporal-resolution constraints, the DAPT perturbation experiments were used to guide the relative timescales implemented in the model. We assigned NICD dynamics to a fast timescale, compatible with changes detectable within the first 20 min sampling interval, and HES1 dynamics to a slower timescale on the order of 1–2 h. The CC3D simulation time resolution was set to 10 min per Monte Carlo step (MCS; 6 MCS = 1 h), while remaining parameters were chosen to remain consistent with previous modelling frameworks.

Exploration of EC-calibrated model identifies distinct NOTCH signalling regimes

To examine how intercellular interactions and feedback mechanisms shape signalling in our model, we simulated a series of configurations in which three key processes were independently switched ON or OFF: cis interactions, Jagged1 induction, and HES1 autoregulation. Each parameter combination was simulated using both idealised square lattices (Supplementary Fig. 2) and experimentally derived endothelial cell (EC) geometries (Figs. 2A and 3). Comparing the two series allowed us to directly assess how cell geometry-bound asymmetries in signal relay influence the emergence of signalling patterns.

Fig. 3. Configurations of the spatial NOTCH model with experimental cell maps.

Left panels are representative intensity maps and right panels are corresponding time series tracking 3-cell neighbours over 100 MCS. In the absence of cis interactions and HES1 autoregulation (A, B), the system generates stable phenotypes, i.e., constant traces and checkerboard pattern with (+ Jag1) or without intermediate expressors. When cis interactions are active beyond a critical cr threshold and without HES1 autoregulation (E, F), the system generates asynchronous oscillations in the HES1 levels. HES1 autoregulation induces synchronous oscillations under all tested conditions (C, D, G, H). All simulations were initialised with random HES1 intensities and run for 4500 MCS before sampling HES1 for 100 MCS.

In the absence of cis interactions and autoregulation (Fig. 3 and Supplementary Fig. 2 panel A), simulations recapitulated the expected behaviour of classical lateral inhibition: a stable checkerboard pattern of alternating high and low-HES1 expression (bistability). However, experimental cell maps where cells have irregular geometries displayed a higher degree of heterogeneity (multi-stability) in comparison to regular square cells. For each configuration shown in Fig. 3, the cell map represents the full simulated monolayer, while the accompanying time series tracks one randomly selected representative local neighbourhood consisting of a centre cell and two directly neighbouring cells, allowing one to appreciate individual cell trajectories and local coordination.

Enabling Jagged1 induction (Fig. 3 and Supplementary Fig. 2, panel B) in this context partially disrupted the regularity of the pattern and promoted the emergence of a multistable pattern, especially in irregular cell maps as previously reported⁹.

Introducing HES1 autoregulation alone (Fig. 3 and Supplementary Fig. 2, panel C) generated synchronous, high-amplitude oscillations in both geometries. These oscillations were stable over time and insensitive to the presence or absence of Jagged1, which contributed only minor modulation of phenotype distribution under these conditions (Fig. 3 and Supplementary Fig. 2, panel D).

Activating productive cis interactions (Fig. 3 and Supplementary Fig. 2, panels E–H) markedly altered the system’s behaviour. In the absence of autoregulation (Fig. 3 and Supplementary Fig. 2, panel E), cis interactions disrupted the checkerboard pattern and generated stable oscillations. Baseline Jagged1 induction (Fig. 3 and Supplementary Fig. 2, panel F) in this context affected the amplitude, period, and phase of the oscillations. To investigate the effects of cis interactions in greater detail, we have performed a higher-resolution parameter scan, as discussed later in the manuscript and illustrated in Fig. 6.

Fig. 6. Phase diagrams correlating cis interactions strength ligand/receptor production rates to qualitative model behaviour.

Phase diagrams of 900 simulations across A the cr and kconv parameters, and B bN1 versus vD4, and bJ1 versus J1ind (cr = 0.1, kconv = 0.08). Individual simulations are coloured according to the calculated peak-to-peak measure (Ptp, left panels) and spatial correlation of time series (Corr, right panels) resulting from the indicated parameters settings. Defaults were: cr = 0.116, kconv = 0.084, bN1 = 1.0, bD4 = 0.69, bJ1 = 0.96, J1ind = 0.06, corresponding to the top-performing simulation in Fig. 4.

When cis interactions were combined with autoregulation (Fig. 3 and Supplementary Fig. 2, panels G, H), the system entered a high-amplitude, synchronous oscillatory regime displaying HES1 fluctuations regardless of cell geometries and Jagged1 inducibility, suggesting that in our model, autoregulation outcompetes cis-induced oscillations.

Overall, we show that autoregulation is sufficient to drive oscillatory behaviour, but also that, under the constraints of our model, the oscillations are synchronous across the population (regardless of different, randomly defined initial values of ligand/receptors), yielding patterns that are not observed in experimental data (Fig. 2C).

Cis interactions can also break stability, producing oscillatory patterns which, unlike autoregulation-induced ones, are not synchronous and qualitatively more reminiscent of experimental data.

HES1 spatial heterogeneity constrains our model to oscillatory NOTCH dynamics

Next, we asked whether any of the different patterns produced by our model fit experimental data for HES1 heterogeneity. To identify model configurations that best matched our experimental observations, we performed a fine-grained parameter search using an evolutionary algorithm (Nevergrad, Differential Evolution) to efficiently explore the parameter space. We modulated basal expression rates of ligand/receptor (Dll4, NOTCH1, Jagged1), cis-associated parameters (cr and kconv controlling the rate of cis-complexes formation, and rate of conversion to active species, respectively), and whether the simulation used the standard or autoregulatory HES1 sub model.

Simulated and experimental HES1 distributions were compared using the Kolmogorov–Smirnov (KS) distance between ECDFs. We used KS distance as a continuous optimisation and ranking metric, rather than as a formal goodness-of-fit test. Experimental data were sampled from 18 untreated HUVEC images, while simulated data were collected from 150 consecutive post-relaxation time points. This sampling strategy was intended to identify simulations that maintain persistent heterogeneity over time, rather than regimes that transiently resemble the experimental distribution. This latter constraint reflects the consistent presence of HES1 heterogeneity across our experimental dataset. A noise-only or transient explanation would predict occasional homogeneous samples, which are not observed experimentally, despite extensive sampling across hundreds of images¹³. As shown in Fig. 1, the preservation of this pattern in vivo further supports heterogeneity as a robust endothelial feature.

For visualisation, simulations with KS < 0.1, which had ECDF discrepancy within the scale of experimental subsampling variability (Supplementary Fig. 5), were highlighted in parameter-space plots as low-distance candidate regimes (Fig. 4A, yellow dots). Candidate simulations were then inspected using corresponding spatial maps and time series to distinguish persistent spatial heterogeneity from transient or globally synchronous behaviour.

Fig. 4. Parameters optimisation against experimental data.

A Scatterplot representing individual simulation outputs versus simulation parameters from an evolutionary search across the parameter space. D < 0.1 are considered positive hits (yellow hues), dashed circle in bottom panels highlights optimal results D < 0.06. B Representative results of the best-fitting simulation (D = 0.04, cr = 0.116, kconv = 0.084, bN1 = 1.0, bD4 = 0.69, bJ1 = 0.96, J1ind = 0.06). Left panel: Representative snapshot of H1 (HES1) cell map. Middle panel: ECDF corresponding to simulated and experimental (NT HUVEC) data series. Right panel: Time series plot tracking 3 neighbouring cells across 100 MCS (~20 h). C Scatterplots representing individual simulations’ KS D versus indicated simulation parameters scanned systematically across the indicated ranges (30 points per axis). Top and middle panels, scan across bN1 and bD4, and bJ1 and J1ind values respectively (cr and kconv as indicated). Bottom panel: scan across cr and bJ1. In all panels: D = KS Distance between simulated and experimental data (NT HUVEC).

Results from the parameter search showed that low-distance candidate regimes required active and productive cis interactions, with both cr and kconv above ~0.07 (Fig. 4A). The best-performing simulations accumulated in a relatively narrow region around cr ~ 0.11 and kconv ~ 0.08. Within this search, model performance showed weaker dependence on basal NOTCH1, Dll4 or Jagged1 expression, although lower bD4 and bJ1 values generally gave better results. Simulations using the HES1 autoregulatory submodel alone consistently performed less well and failed to reproduce the observed spatial heterogeneity.

Visual inspection of the best-performing simulations without HES1 autoregulation demonstrated persistent spatiotemporal heterogeneity (Fig. 4B). Figure 4B shows one representative low-distance simulation whose ECDF closely follows the experimental HES1 distribution. The simulated cell map reproduced the qualitative structure of the experimental maps, with high- and low-HES1 cells scattered among mid-range expressors. Corresponding time series revealed asynchronous periodic HES1 dynamics with a characteristic period of approximately 4 h, consistent with prior endothelial NOTCH dynamics studies²⁶. Because KS distance is relatively insensitive to differences in distribution tails, we additionally repeated optimisation using the Anderson–Darling statistic as a tail-sensitive loss metric. This recovered the same qualitative model regime and yielded a representative simulation comparable to the KS-ranked solution.

By contrast, autoregulation-only simulations could generate time-dependent variability when pooled across time, but cell maps did not show persistent spatial heterogeneity because autoregulation-induced spatial synchronisation under the tested conditions, consistent with Fig. 3C, D, G, H. Accordingly, simulations including the autoregulatory submodel consistently yielded KS statistics > 0.1.

Thus, top-ranked simulations were dynamic rather than static, but only contact-modulated cis/trans interaction regimes reproduced both the experimental distribution and the spatial organisation of HES1 heterogeneity.

Having identified a contact-modulated cis/trans interaction regime that reproduced the experimental HES1 distribution and spatial patterning, we next assessed the sensitivity of this behaviour to ligand and receptor expression levels. We performed fine-grained scans across basal Dll4 and NOTCH1 expression, basal and inducible Jagged1 expression, and cis-interaction strength while retaining the remaining optimised parameters.

Modulating basal NOTCH1 and Dll4 expression under otherwise optimised parameters identified a restricted region of parameter space (bN1 ~ 1.0 ± 0.2; bD4 ~ 0.9 ± 0.3) with low distributional distance from the experimental data (Fig. 4C, top panel). This indicates that sustained ligand and receptor expression is required to maintain the signalling regime that best reproduces HES1 heterogeneity. Low-distance simulations were concentrated within this region, while the KS metric varied smoothly across neighbouring parameter combinations, indicating that model performance is tuneable rather than dependent on an abrupt threshold.

Modulating basal and inducible Jagged1 expression under otherwise optimised parameters showed that model performance was primarily sensitive to basal Jagged1 levels, with weaker dependence on inducible Jagged1 expression (J1ind) (Fig. 4C, middle panel). Low-distance simulations were obtained across a broad range of J1ind values, whereas the best-performing simulations clustered at intermediate-to-high basal Jagged1 expression.

In our model, Jagged1 competes with Dll4 in both cis and trans through non-productive interactions. We therefore examined whether model performance depended on the balance between basal Jagged1 expression and cis-interaction strength cr. Low-distance simulations occupied a discrete region in the bJ1-cr plane (Fig. 4C, bottom panel), showing a mild inverse relationship in which stronger cis engagement allowed comparable performance at lower basal Jagged1 levels. Within this region, KS distance varied smoothly, again indicating that the system is continuously tuneable.

These observations are consistent with previous computational and experimental studies showing that both cis-interaction strength and Jagged1 levels can be modulated in EC, for example, through Fringe-mediated glycosylation^9,22,27. In our model, kconv is used to represent the continuum between purely inhibitory cis interactions and partially productive cis interactions. However, in the absence of time-resolved data directly constraining these parameters in endothelial monolayers, we treat them as explored model parameters rather than fixed measured quantities.

Taken together, these results support the hypothesis that persistent HES1 heterogeneity in quiescent endothelial monolayers is maintained by contact-dependent NOTCH dynamics. Within the explored model space, this behaviour is reproduced by regimes in which cis interactions are sufficiently productive to destabilise fixed spatial patterns, while neighbour-derived trans signalling and contact geometry modulate the effective cis/trans balance. Thus, cis interactions act as a local regulatory mechanism embedded within a multicellular contact-dependent system, rather than as an isolated cell-autonomous source of heterogeneity.

Endothelial HES1 heterogeneity is driven by cell-contact-dependent signalling

We showed above that cell-autonomous HES1 autoregulation, as implemented in our model, did not reproduce the experimental HES1 distributions as well as tuneable contact-dependent interactions. To test whether HES1 heterogeneity in endothelial monolayers depends on cell–cell contact, we compared sparse and confluent HUVEC cultures. If endothelial HES1 heterogeneity were primarily generated by cell-autonomous oscillations, sparse cells would be expected to retain heterogeneous snapshot distributions because individual cells would be captured at different oscillatory phases. Conversely, if heterogeneity depends on contact-mediated interactions, sparse cells should show reduced distributional heterogeneity.

Consistent with our previous observations and shown in Supplementary Fig. 1A, confluent HUVECs displayed broad, weakly multimodal single-cell HES1 distributions, including low- and high-intensity subpopulations. Conversely, sparse HUVECs showed a narrower HES1 distribution, concentrated around a higher median level, and lacking the outlier populations observed in confluent cultures. Representative immunofluorescence images supported this interpretation: sparse endothelial cells showed relatively uniform nuclear HES1, whereas confluent monolayers contained spatially interspersed high- and low-HES1 cells (Fig. 5A).

Fig. 5. NOTCH signal activity in sparse vs. confluent EC.

A Box plots comparing single-cell nuclear NOTCH1 and HES1 intensities between confluent (n = 6431) and sparse (n = 1081) HUVEC. P values calculated by the Shapiro–Wilk test to assess per-data series normality. Values < 0.05 reject the normal distribution hypothesis. B Representative NOTCH1 and HES1 immunofluorescence images relative to sparse and confluent HUVEC.

Importantly, sparse cells retained nuclear NOTCH1 signal (Fig. 5B), used here as a proxy for NICD. Thus, sparse cells did not simply lack NOTCH pathway activity. Instead, the data suggest that basal NOTCH/NICD activity can support relatively uniform HES1 expression in isolated cells, whereas the structured HES1 heterogeneity observed in confluent monolayers requires cell–cell contact.

We next simulated sparse-cell conditions in the competing model architectures (Supplementary Fig. 3). In the contact/cis-interaction model, isolated cells converged toward uniform HES1 levels, with the final steady state depending on (Cr, fraction of same-cells ligand–receptor pairs engaged in cis interactions) and (kconv, fraction of productive cis interactions). This matched our experimental observation of reduced heterogeneity under sparse conditions. In contrast, inclusion of an autonomous HES1 autoregulatory oscillator produced persistent oscillations in isolated cells, generating phase-dependent snapshot heterogeneity that was not observed experimentally. Moreover, autoregulation in the absence of a positive NOTCH/cis input yielded zero HES1 expression, indicating that negative autoregulation alone is insufficient to maintain the experimentally observed HES1 baseline. This is consistent with developmental HES/HER oscillators, where negative-feedback dynamics require sustained upstream signalling inputs rather than isolated autoregulation alone²⁸.

Together, these results support a contact-dependent origin of HES1 heterogeneity in confluent endothelial monolayers. Sparse cells retain basal NOTCH/NICD-HES1 activity, but they do not display the heterogeneous HES1 distributions predicted by a dominant autonomous oscillator, which also requires external inputs to operate.

Modulation of cis interactions strength yields locally synchronised oscillations

We have shown that HES1 autoregulation-driven globally synchronous oscillations, in which all cells oscillate in phase, do not reproduce the experimentally observed HES1 distributions. However, coordinated endothelial NOTCH dynamics have been reported experimentally in angiogenic vessels, although Dll4 rather than NICD/HES1 oscillations were the main readout²⁶. Considering that in our model, Dll4 and HES1 dynamics are tightly linked, we asked whether local rather than global synchronisation might occur in our model and whether it was tuneable. Specifically, we asked whether neighbouring cells in simulated endothelia exhibit coordinated HES1 dynamics (e.g., time-correlated or phase-locked signalling trajectories) despite differences in signal levels across the wider cell population. Qualitative inspection of simulated time series (Figs. 3A and 4B) suggested such local coordination.

To measure local synchronisation in our simulations, we tracked HES1 signal in a neighbourhood of three cells under varying cr and Kconv values and calculated the peak-to-peak distance of oscillations (Ptp, span across the dynamic range) (0 in stable simulations), and a pairwise time series correlation index (see methods) for each series. Corresponding phase portraits (Fig. 6A) show how the two parameters drive different regimes in our model and whether the signals in the cell neighbourhood are synchronised. The Ptp portrait demonstrates a wide region of stable behaviour (amplitude = 0), a transition zone where low-amplitude oscillations emerge, and a region demonstrating wide stable oscillations (Ptp > ~0.5), paralleling results in Fig. 3E, F. The correlation portrait demonstrates that simulations across the transition zone had a mid-range correlation index (0.5–0.7, green dots), indicating partial synchronisation among cell neighbours. Solutions in the top-right quadrant of the plot had correlation indices > 0.9, indicating phase-locked neighbours, demonstrating how the system can generate locally synchronous oscillations in the HES1 signal.

To further illustrate these results, we have analysed selected time series by plotting Poincare sections across tracked cell traces (Supplementary Fig. 4). This analysis graphically demonstrates how increasing cr values generate oscillations with increasing amplitude (Supplementary Fig. 4 upper panels) and synchronisation (Supplementary Fig. 4 lower panels). The rightmost panels in Supplementary Figure 4 demonstrate the behaviour of the top result of our parameter search (Fig. 4B), which demonstrates partial synchronisation, paralleling previously described results²⁶ under the experimental conditions of our reference data (NT HUVEC).

As before, to evaluate whether ligand/receptor expression could tune oscillations’ synchronicity in our optimised model we systematically scanned across relative basal and inducible production rates (bN1, bD4, bJ1, J1ind, Fig. 6B).

Inspection of phase plots relative to bN1 versus bD4 highlighted a region of parameter space where HES1 levels were oscillating with increasing amplitudes proportional to bN1 and bD4 values. This space was subdivided into two sub-regions with higher or milder synchronicity indices dependently on relative proportion of ligand/receptor productions (bN1/bD4 ratio) with relatively higher bD4 promoting desynchronisation, and relatively higher bN1 promoting synchronisation.

Inspection of phase plots relative to bJ1 versus J1ind highlighted that oscillation amplitude is inversely proportional to the basal expression rate of Jagged1, with values > ~1 completely dampening the oscillations. Inducibility did not dramatically affect the amplitude of oscillations, except for a trend in the high range showing cumulative effects with basal expression (HES1 oscillations dampening). The corresponding correlation phase plot does not show a clear correlation of bJ1 versus J1ind with the synchronicity index. However, several combinations with mid-range bN1 (~0.5) yielded highly asynchronous oscillations scattered among more synchronous, the transitions around these points appeared less gradual than in other comparisons.

Overall, these results show that our model can reproduce both locally synchronised and non-synchronised dynamics depending on the extent of cis interactions involved. Additionally, under optimised parameters, amplitude and synchronicity of HES1 oscillations are tuneable by varying basal expression rates of NOTCH1, Dll4, and Jagged1.

Discussion

Vascular endothelial cells (EC) are highly diverse and adaptable^1–3, and this plasticity allows the endothelium to operate as a distributed sensing system, rather than as a uniform layer of equivalent cells. In this framework, individual ECs contribute partial and locally distinct information, which is integrated via cell–cell connections to enable collective responses to complex or noisy stimuli, as proposed by McCarron and colleagues²⁹.

In our previous work, we observed spatially heterogeneous NOTCH–HES1 activity in cultured endothelial monolayers¹³, resembling the patchwork-like organisation described in this distributed sensing paradigm. Building on these observations, we asked whether such spatial heterogeneity could, in principle, arise intrinsically from endothelial NOTCH signalling dynamics, providing a mechanistic basis for phenotypic diversity in otherwise quiescent monolayers.

We demonstrate that HES1, a key transcriptional target of NOTCH, exhibits spatially heterogeneous expression in the murine aortic endothelium, closely resembling patterns seen in vitro. This observation supports the use of cultured EC monolayers to dissect the mechanistic origins of such heterogeneity under controlled conditions.

Directly tracking NOTCH pathway dynamics in primary EC is challenging; we attempted to better understand EC heterogeneity theoretically via spatially resolved, multiscale modelling, which integrated experimentally grounded kinetics and realistic cellular geometries.

Building on prior theoretical results^{9,10,12,16,17} our model incorporated lateral inhibition and induction, productive and non-productive cis interactions, and HES1 autoregulation. To ground these interactions in realistic dynamics, we calibrated key temporal parameters, including NICD degradation and the delayed suppression of HES1, using time-course experiments following γ-secretase inhibition in HUVEC. These experiments revealed a rapid loss of NICD followed by a slower, temporally offset HES1 response, establishing a separation of timescales that proved essential for reproducing the range of signalling behaviours observed in confluent monolayers. While these timescales are consistent with previous estimates from other systems^9,17, constraining them experimentally in endothelial cells allowed us to anchor the model dynamics to an EC-specific regulatory context, even in the absence of precise knowledge of all molecular interactions within the NOTCH pathway.

In our prior¹³ and current experiments, NOTCH–HES1 heterogeneity was consistently observed across many endothelial monolayers cultured under a range of conditions, making it unlikely to reflect experimental noise or transient fluctuations. Instead, it points to an underlying regulatory mechanism capable of sustaining heterogeneity in a tissue composed of terminally differentiated yet highly plastic and self-renewing cells like EC.

One potential explanation is that heterogeneity arises from cell-autonomous oscillations in NOTCH signalling, which have been measured directly in other contexts, including recent reports of oscillatory NOTCH dynamics at the single-cell level^16,17. In parallel, theoretical work has shown that specific NOTCH interaction motifs, particularly productive and non-productive cis interactions, can generate stable and multistable signalling states in spatially coupled systems^9,22,25. However, whether these cell-autonomous or interaction-driven mechanisms are sufficient to account for the persistent, spatially structured heterogeneity observed in endothelial monolayers remains unclear, motivating us to explicitly examine alternative hypotheses to better understand endothelial NOTCH dynamics.

To address these possibilities, we systematically explored the model’s parameter space to identify signalling regimes compatible with persistent endothelial heterogeneity. This analysis revealed that oscillatory HES1 dynamics, characterised by asynchronous fluctuations with a period of ~4–6 h, emerge robustly in regimes where cis engagement and productive cis conversion are sufficiently strong. Consistent with previous reports, transcriptional autoregulation of HES1 alone was also sufficient to generate sustained oscillations under physiologically plausible conditions. However, when embedded in a spatially connected tissue context, HES1 autoregulation by itself failed to reproduce the locally heterogeneous signalling patterns observed experimentally, generating instead highly synchronised oscillations, indicating that cell-autonomous oscillations are insufficient within our model formulation.

These modelling predictions were supported experimentally by comparing sparse and confluent endothelial cultures. Sparse cells, which lack contact with neighbours, exhibited uniformly high HES1 levels and retained active NOTCH signalling, consistent with the ability of cell-autonomous mechanisms, such as cis activation, to maintain expression. In contrast, spatial heterogeneity was lost in the absence of intercellular communication, reinforcing the conclusion that cell–cell interactions are required not simply to modulate, but to sustain heterogeneous NOTCH dynamics in endothelial monolayers.

Finally, we asked whether our model could reproduce different degrees of coordination in NOTCH dynamics, ranging from locally asynchronous to locally synchronous behaviour, both of which have been reported experimentally²⁶. A high-resolution exploration of cis-interaction parameters identified different signalling regimes, including stable states, low-amplitude oscillations, and robust oscillatory behaviour. Within these explorations, oscillations remained globally asynchronous across the whole parameter space, while neighbouring cells could exhibit varying levels of local coordination.

Extending this analysis, we found that the degree of local coordination could be further modulated by relative ligand and receptor expression levels, as well as by cell geometry. Together, these results indicate that the extent of local coordination in NOTCH–HES1 dynamics is a tuneable property of our model, shaped by interaction strength, signalling balance, and spatial context.

Notably, several of these parameters are known to be modulated in vivo, for example, cis interactions can be influenced by Fringe proteins²⁵, while Jagged1 competes with Dll4 during angiogenesis^26,30 and is downregulated by VEGF²⁷, suggesting that the range of behaviours captured by the model is compatible with experimentally observed modes of NOTCH regulation.

In the absence of additional quantitative data to further constrain these parameters, we do not attempt to assign specific regulatory mechanisms to individual regimes. Instead, our model provides an open-access, flexible, experimentally grounded framework that can be further explored and extended as new measurements of NOTCH dynamics and interactions become available.

Although our model captures a range of endothelial NOTCH–HES1 behaviours consistent with the experimental heterogeneity we observe, several limitations should be noted.

First, our framework is intentionally minimal, only focussing on core NOTCH interactions, not including additional regulators and pathway crosstalk known to influence HES-family dynamics in other systems (e.g., inputs from WNT/FGF³¹). Additionally, our model does not represent other potentially important layers of regulation, such as chromatin-mediated delays or coupled feedback architectures. Especially, we note that all cells within a given simulation shared the same kinetic parameter values. This assumption favours comparison of minimal mechanistic architectures but may underestimate heterogeneity that could arise from extrinsic cell-to-cell variability in protein abundance, cell state, or kinetic rates. However, introducing such variability would add an additional unconstrained mechanism, whereas the contact/cis-interaction model reproduced the key confluent and sparse-cell observations without requiring imposed parameter heterogeneity.

Second, while the DAPT time-course data constrain key timescales and restrict the space of plausible dynamics, they do not uniquely identify parameter values or molecular implementations. Accordingly, we interpret the modelling results as sufficient tests of candidate interaction rules, rather than as proof that a single mechanism is exclusive in vivo. Similarly, the KS and Anderson–Darling metrics constrain model behaviour at the level of distributional similarity, but they do not establish a unique statistical fit. Their main role is to compare candidate mechanisms under a common quantitative criterion, which we combine with spatial maps, time-series behaviour and sparse-cell experiments to interpret model plausibility.

Third, the spatial simulations do not capture cell rearrangements, growth, or longer-term remodelling, which may further shape coupling in dynamic endothelial contexts. These features are already implemented in our model, but they were switched off as we didn’t expect them to be highly relevant within the relatively short observation time windows evaluated in the present work.

Finally, our experimental readouts provide indirect access to pathway dynamics and NICD localisation can be difficult to interpret unambiguously, underscoring the need for future work using live reporters²⁰ and targeted perturbations to more directly resolve endothelial NOTCH dynamics and discriminate between alternative regulatory contributions. Additionally, our findings highlight the challenges in predicting EC behaviour from static transcriptomic or imaging data alone. The non-linear, spatially contingent nature of NOTCH dynamics implies that subtle shifts in geometry, contact, or feedback strength can qualitatively alter signalling outcomes.

In this context, we further demonstrate the utility of combining high-throughput, high-content imaging with mechanistic modelling to study heterogeneous endothelial signalling, building on our previous work and providing a framework that can be readily extended as new experimental capabilities emerge. In future efforts, these convenient and sustainable in vitro systems³² will allow appreciating heterogeneity and dynamics across cell populations, multiple conditions, and spatial contexts, providing the statistical power required to resolve subtle but persistent signalling behaviours.

Functionally, our results reinforce the idea that EC heterogeneity is not fixed but dynamic, a property that could enhance tissue robustness by enabling phenotypic switching without reliance on progenitor cells. Such dynamic allocation of roles across ECs would allow parallel sensing, barrier maintenance, and immune interaction functions to be redistributed in response to perturbation²⁹. This flexibility may be particularly relevant in adult homoeostasis and pathological contexts where spatial relationships between ECs and tissues are altered, e.g., during wound healing, inflammation, or tumour angiogenesis^33,34.

Although our theoretical analysis does not allow us to assert with certainty that the NOTCH-dependent EC phenotype is dynamically changing, it highlights the importance of in silico analyses to better understand potentially complex cell–cell dynamics with relevance in research and clinical contexts. Recent, more advanced in vitro systems are opening the way to directly observe the dynamics of intracellular signalling³⁵, and it will be important to apply such technologies to EC to fully appreciate the complex regulation of endothelial phenotype.

In conclusion, we present a computational and experimental framework that connects NOTCH signalling dynamics to emergent EC heterogeneity in space and time. Our model demonstrates how local rules can generate global diversity in endothelial monolayers, offering a framework to better understand endothelial heterogeneity. Additionally, we provide our open-source model as an extensible platform to inform future experiments and to develop and test alternative or more refined hypotheses regarding endothelial NOTCH signalling.

Methods

Cell culture

All in vitro data shown in the present work have been generated as described previously (Chesnais et al., 2022)¹³. Briefly, HUVECs (PromoCell) were plated on 10 μg/ml fibronectin (Promocell)-coated flasks, grown in EGMV2 medium (Promocell), detached with Accutase (Thermo Fisher Scientific, Waltham, MA), and used by passage 5. For experiments, 4 × 10⁴ ECs per well were seeded for confluent experiments and 1 × 10³ ECs per well for sparse experiments in fibronectin-coated 96-well plates (μClear, Greiner). Cells were cultured for 24 h for sparse and 96 h for confluent under basal conditions (EGMV2, Promocell) or treated with DAPT (5 μM, Tocris bioscience, UK).

Immunostaining and image acquisition

Cells were fixed with 2% paraformaldehyde in phosphate-buffered saline (PBS) for 10 min at room temperature. Cells were blocked 1 h with PBS supplemented with 1% foetal bovine serum (FBS) and permeabilised with 0.1% Triton X-100. Cells were then incubated for 1 h at room temperature with primary antibodies against CDH5 (VE-cadherin; Novusbio NB600-1409, 1 μg/ml final), NOTCH1 (Abcam, ab194122, Alexa Fluor 647-conjugated, 1 μg/ml final), and Hes1 (Abcam, ab119776, 1 μg/ml final). Plates were washed and incubated 1 h with 1 μg/ml secondary Alexa Fluor 488-conjugated and Alexa Fluor 555-conjugated antibody (Thermo Fisher Scientific), Hoechst 33342 (1 μg/ml, Sigma). We obtained the images with an Operetta CLS system (PerkinElmer, Waltham, MA) equipped with a ×40 water immersion lens (NA 1.1). We acquired 10 images per well, in triplicate wells, in two independent experiments for each experimental condition (60 images in total/exp condition). We used HUVEC from two different donors in the independent experiments.

En face preparation and whole mount staining of mouse aorta

For en face preparation of aortas, 2–8-month-old mice (C57BL/6J, Jax Strain 000664) euthanised for standard colony maintenance under Schedule 1 methods were used. For preparation and immunostaining, we used an adaptation of a previously established protocol¹⁹. In brief, upon euthanasia, animals were immediately perfused with PBS by intracardial injection and subsequently perfusion fixed by injection of 2% PFA–PBS. Dissected aortas were fixed for an additional 1 h by immersion in 2% PFA–PBS, washed extensively with PBS and blocked with FBS (5% FBS + 0.01% Triton-X) overnight at 4 °C. Immunostaining was performed after permeabilization (0.1% Triton-X for 10 minutes at RT) using primary antibodies diluted in blocking solution (VE-Cadherin, #14-1441-82, 5 μg/ml final and HES1, #PA5-28802 5 μg/ml final, Thermo Fisher Scientific) for 48 h at 4 °C, followed by extensive washing using blocking solution at RT. Aortas were then incubated with Alexa Fluor 488-conjugated and Alexa Fluor 555-conjugated antibody (Thermo Fisher Scientific) and Hoechst 33342, for 24 h at 4 °C, washed, and mounted on a coverslip with Mowiol (Sigma-Aldrich). The aortas were imaged, and z-stacks were obtained using a Leica Sp8 confocal microscope with a ×40 air objective. We acquired one representative image from three independent preparations. Images were post-processed and quantified with ImageJ, and relative cell maps were generated in Python.

Image and data analysis

Analysis of in vitro images obtained in this study has been done with ECPT (endothelial cells profiling tool)¹¹. In brief, serial images for each experimental condition were exported in TIFF format, normalised using standard algorithms (ImageJ), and quantified using our pipeline. The ECPT was developed to assess phenotypic heterogeneity in medium-high-throughput screenings of endothelial monolayers concurrently assessing cell proliferation and junctional states, along with intensity of selected markers in subcellular compartments (e.g., nuclei) like NICD and HES1 following a standardised workflow. ECPT allows measuring features of interest in many images per experimental condition, and cross-experiment comparison, thus, to identify and control key sources of experimental variability. This allows for appraising heterogeneity at local (individual image, ~100 cells) and global (multiple images from the same monolayer, thousands of cells), or across conditions (different treatments, or different EC donor or body site) levels. The ECPT, original data used in the publication¹³, and associated documentation, including a step-by-step walkthrough, are all freely accessible at https://github.com/LVeschini78/ECPT. Data were exported as MySQL databases and analysed in Python using standard libraries like pandas, scipy.stats, numpy, and matplotlib. To perform parameter scans using the spatial CC3D model, we developed custom scripts leveraging the CC3D Python API³⁶.

Ex vivo images were segmented by cell painting to precisely annotate relevant nuclei and avoid artifacts. HES1 intensity in the region of interest (ROIs) was measured with the ImageJ measure feature.

All the code used in this work is available at https://github.com/LVeschini78/EC_NDJ_Dynamics.

Spatial model of cell monolayers in CompuCell3D

The multicellular spatialised model of EC monolayers was developed using a cellular Potts model formalism (CPM, or Glazier-Graner-Hogeweg model (Swat et al., 2012) using the software CompuCell3D (CC3D version 4.2.5, www.compucell3d.org,). Cell maps were generated via a custom R script taking ECPT-segmented images as input and outputting a lattice description file (.piff), which can be directly read by CC3D. The spatialised model can simulate moving or frozen cells, for simplicity, and assuming that cell movements within the observation window are limited, all results presented in the work refer to simulations where EC were “frozen” (i.e., immobile).

The model is composed of five separate modules. However, the first four steppables are standard CC3D modules and extensive documentation on how to set these up is available on the CC3D website and manuals. The fifth step, NDJ_Interactions, is dedicated to evaluating neighbours’ interactions as described in the present work.

In brief, the module calculates contact areas (CA) between each cell and its neighbour’s CA, which is then factored when calculating the extent of trans interactions between individual pairs of cells at each MCS, thus linking cell shape and disposition with the extent of NOTCH signalling. The Antimony code encoding the ODE model and calls to all steppables are included as additional modules. Full explanation of the model scripts' functioning is provided in Supplementary Note 1.

For each parameter set tested during simulations, kinetic and interaction parameters were kept identical across all cells so that parameter scans represent alternative global model wirings rather than imposed cell-to-cell parameter variability. Initial HES1 values for each cell were randomised and no additional stochastic variation in kinetic parameters was introduced between cells unless explicitly stated.

Ordinary differential equations model of NOTCH signalling

We developed the mathematical framework of NOTCH signalling as an ordinary differential equations (ODE) system, as proposed previously and encoding a hybrid protein/gene regulatory network (P/GRN) using the Tellurium package and the Antimony language (SBML-compatible). We assumed a well-stirred bioreactor cell model where we explicitly encoded kinetic laws for key genes/proteins.

To build our GRN, we selected NOTCH pathway components expressed in EC; the choices were mainly based on existing literature, especially the single-cell transcriptomic atlas of murine endothelial cells (https://endotheliomics.shinyapps.io/ec_atlas/), details in Supplementary Note 1.

In brief, we built the model as a system of interconnected ODEs using standard mass action and Hill functions to model production/degradation and recovery response curves for each species. Dll4, Notch1, Jag1 participate in cis and trans interactions as schematised in Fig. 1, relaying downstream NICD, which in turn regulates HES1 production. HES dynamics was modelled either as a standard Hill function without autoregulation, or a more complex network including delays that oscillates autonomously. Parameters of the HES1 subsystem were calibrated to reproduce previously reported oscillations’ amplitude and period. The autoregulatory HES1 submodule produces an ~2 h oscillation when considered in isolation, consistent with canonical HES/HES1 negative-feedback models. In the full multicellular endothelial model, this intrinsic timescale is modified by contact-dependent NOTCH activation, cis/trans interactions, neighbour coupling, and cell geometry, yielding slower best-fitting HES1 dynamics of ~4–6 h.

Cis interactions were controlled by two parameters. (cr) represents the fraction of available same-cell ligand–receptor pairs assigned to cis-interaction. (kconv) represents the fraction of cis-engaged pairs that productively generate NICD, while the remaining cis interactions are treated as non-productive/inhibitory interactions that remove ligand–receptor pairs without contributing to signalling. Thus, (kconv = 0) corresponds to purely inhibitory cis interactions, whereas (kconv > 0) allows partially productive cis signalling.

To perform parameter scans, simulations were run automatically via the CC3D Python API. To visually inspect simulated maps and to generate snapshots, the simulations were run via the CC3D player application. Snapshots were generated at runtime via export facilities available in the CC3D player.

Statistical analysis and data visualisation

All data wrangling and statistical analyses were performed in Python. To perform the evolutionary parameter optimisation, we used the Python Nevergrad library (https://facebookresearch.github.io/nevergrad/).

As distributional fit metrics, we compared simulated and experimental HES1 intensity data using empirical cumulative distribution functions (ECDFs). Experimental data were generated by randomly sampling 18 untreated HUVEC images from our database, while simulated data were collected at runtime from 150 consecutive post-relaxation time points (corresponding to 25 h). We chose this consecutive sampling strategy to identify parameter regimes that maintain persistent HES1 heterogeneity over time, as it penalises regimes in which heterogeneity appears only transiently or after pooling temporally disconnected phases. The Kolmogorov–Smirnov (KS) distance was used as the primary continuous loss/ranking metric during Nevergrad optimisation and was calculated within a CC3D steppable, allowing the CC3D model to act as a loss function that receives input parameters and returns a distributional distance to experimental data. Because the KS distance is relatively insensitive to tail-region differences, candidate simulations were also evaluated using the Anderson–Darling (AD) statistic, and an additional optimisation was performed using AD as the loss metric. To contextualise these values, we performed an experimental resampling analysis in which per-image untreated HUVEC subsamples were compared with the aggregate experimental HES1 distribution; KS, AD statistic and AD p-values were calculated for each comparison (Supplementary Fig. 5). The KS < 0.1 value highlighted in parameter-space plots was therefore used as an indicative visual guide for low-distance simulations, not as a statistical acceptance criterion.

Ptp and Corr indices: We quantified local temporal coordination of simulated time series using a lag-aware cross-correlation score, denoted Corr, between the centre-cell trajectory and each neighbouring-cell trajectory. For each pair, traces were standardised and the strongest absolute correlation was extracted within a fixed lag window (plus/minus 20 time steps), allowing phase-shifted coordination to be captured. Pairs with very low variance were treated as non-dynamic and assigned a score of zero. The Corr-based synchronicity index was defined as the mean of these centre-to-neighbour pairwise scores, so higher values indicate stronger local temporal coordination. In parallel, dynamic magnitude was summarised by Ptp (peak-to-peak), which captures oscillation amplitude.

For representative time-series visualisations in Fig. 3 and Supplementary Figs. 2 and 5, the time series tracks one randomly selected local neighbourhood consisting of a centre cell and two directly neighbouring cells. This three-cell neighbourhood was used for visual clarity: plotting all cells in the monolayer would obscure individual trajectories, whereas tracking neighbouring cells allows local temporal coordination or desynchronisation to be visually assessed. The same visualisation strategy was applied to one representative simulation for each model configuration.

All scripts available at https://github.com/LVeschini78/EC_NDJ_Dynamics.

Author contributions

F.C. Conducted all *in vitro* and *ex vivo* investigations. J.H. Contributed to *in vitro* investigations and image analysis. E.E. Drafted intracellular NOTCH model and its molecular interactions network. D.D. Contributed to project design. M.B. Conducted *in vivo* experiments. T.J.S. contributed to the development of the CC3D model. J.A.G. and L.V. designed, supervised and finalised the project, and gathered fundings. L.V. designed, calibrated and finalised the integrated model, wrote the analytic code and conducted all statistical analyses, maintains GitHub Repo. All authors contributed to manuscript writing, reviewing, and figure preparation.

Data availability

All data relative to the present manuscript are available on Zenodo: https://doi.org/10.5281/zenodo.18865760.

Code availability

All code to reproduce all aspects and figures of the present work is available at https://github.com/LVeschini78/EC_NDJ_Dynamics.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41540-026-00786-0.